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% David R. Wilkins
% School of Mathematics, Trinity College, Dublin 2, Ireland
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% Trinity College, 1st June 1999.

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\centerline{\Largebf ON SYMBOLICAL GEOMETRY}

\vskip24pt

\centerline{\Largebf By}

\vskip24pt

\centerline{\Largebf William Rowan Hamilton}

\vskip24pt

\centerline{\largerm (The Cambridge and Dublin Mathematical Journal:}
\centerline{\vbox{\halign{\largerm #\hfil\cr
   vol.~i (1846), pp.\ 45--57, 137--154, 256--263,\cr
   vol.~ii (1847), pp.\ 47--52, 130--133, 204--209,\cr
   vol.~iii (1848), pp.\ 68--84, 220--225,\cr
   vol.~iv (1849), pp.\ 84--89, 105--118.)\cr}}}

\vskip36pt

\vfill

\centerline{\largerm Edited by David R. Wilkins}

\vskip 12pt

\centerline{\largerm 1999}

\vskip36pt\eject

\pageno=-1

\null\vskip36pt

\centerline{\Largebf NOTE ON THE TEXT}

\bigskip

The paper {\it On Symbolical Geometry}, by Sir William Rowan Hamilton,
appeared in installments in volumes i--iv of {\it The Cambridge and
Dublin Mathematical Journal}, for the years 1846--1849.

The articles of the paper appeared as follows:

\bigskip

\hskip\parindent\hbox{\vbox{\halign{#\hfil &\quad #\hfil\cr
   introduction and articles 1--7 &vol.~i (1846), pp.\ 45--57,\cr
   articles 8--13 &vol.~i (1846), pp.\ 137--154,\cr
   articles 14--17 &vol.~i (1846), pp.\ 256--263,\cr
   articles 18, 19 &vol.~ii (1847), pp.\ 47--52,\cr
   articles 20, 21 &vol.~ii (1847), pp.\ 130--133,\cr
   articles 22--24 &vol.~ii (1847), pp.\ 204--209,\cr
   articles 25--32 &vol.~iii (1848), pp.\ 68--84,\cr
   articles 33--39 &vol.~iii (1848), pp.\ 220--225,\cr
   articles 40, 41 &vol.~iv (1849), pp.\ 84--89,\cr
   articles 42--53 &vol.~iv (1849), pp.\ 105--118.\cr}}}

\bigbreak

Various errata noted by Hamilton have been corrected.  In addition,
the following obvious corrections have been made:---

\smallbreak

\item{}
in article~13, the sentence before equation (103), the word
`considered' has been changed to `considering';

\smallskip

\item{}
a missing comma has been inserted in equation (172);

\smallskip

\item{}
in equation (209), `$= 0$' has been added;

\smallskip

\item{}
in article~31, the sentence after equation (227), ${\rm d}'$
has been corrected to ${\rm d}''$;

\smallskip
 
\item{}
in equation (233), ${\rm a}^\backprime$ has been replaced in
the second identity by ${\rm a}^{\backprime\backprime}$;

\smallskip

\item{}
in equation (294), a superscript ${}^2$ has been applied to
the the scalar term in the identity.

\bigskip

In the original publication, points in space are usually denoted
with normal size capital roman letters (${\rm A}$, ${\rm B}$,
${\rm C}$ etc.), but with `small capitals' (${\sc a}$, ${\sc b}$,
${\sc c}$ etc.) towards the end of the paper.  In this edition,
the latter typeface has been used throughout to denote points of
space.

In the original publication, the operations of taking the vector
and scalar part are usually denoted with capital roman letters
${\rm V}$ and ${\sc S}$, but are on occasion printed in italic
type.  In this edition, these operations have been denoted
by roman letters throughout.

\bigbreak\bigskip

\line{\hfil David R. Wilkins}

\vskip3pt

\line{\hfil Dublin, June 1999}

\vfill\eject

\pageno=1

\null\vskip36pt

\centerline{ON SYMBOLICAL GEOMETRY.}

\vskip12pt

By Sir {\sc William Rowan Hamilton}, LL.D., Dub.\ and Camb.,
P.R.I.A., Corresponding Member of the Institute of France and of
several Scientific Societies, Andrews' Professor of Astronomy in
the University of Dublin, and Royal Astronomer of Ireland.

\bigbreak

\centerline{\sc INTRODUCTORY REMARKS.}

\nobreak\bigskip

The present paper is an attempt towards constructing a symbolical
geometry, analogous in several important respects to what is
known as symbolical algebra, but not identical therewith; since
it starts from other suggestions, and employs, in many cases,
other rules of combination of symbols.  One object aimed at by
the writer has been (he confesses) to illustrate, and to exhibit
under a new point of view, his own theory, which has in part been
elsewhere published, of algebraic quaternions.  Another object,
which interests even him much more, and will probably be regarded
by the readers of this Journal as being much less unimportant,
has been to furnish some new materials towards judging of the
general applicability and usefulness of some of those principles
respecting symbolical language which have been put forward in
modern times.  In connexion with this latter object he would
gladly receive from his readers some indulgence, while offering
the few following remarks.

An opinion has been formerly published\footnote*{{\it Trans.\
Royal Irish Acad.}, vol.~{\sc xvii}.  Dublin, 1835.}
by the writer of the present paper, that it is possible to regard
Algebra as a {\it science}, (or more precisely speaking) as a
{\it contemplation}, in some degree {\it analogous to Geometry},
although not to be confounded therewith; and to separate it, as
such, in our conception, from its own {\it rules of art\/} and
{\it systems of expression\/}: and that when so regarded, and so
separated, its ultimate subject-matter is found in what a great
metaphysician has called the inner intuition of {\it time}.  On
which account, the writer ventured to characterise Algebra as
being the {\it Science of Pure Time\/}; a phrase which he also
expanded into this other: that it is (ultimately) the Science of
{\it Order and Progression}.  Without having as yet seen cause to
abandon that former view, however obscurely expressed and
imperfectly developed it may have been, he hopes that he has
since profited by a study, frequently resumed, of some of the
works of Professor Ohm, Dr.~Peacock, Mr.~Gregory, and some
other authors; and imagines that he has come to seize their
meaning, and appreciate their value, more fully than he was
prepared to do, at the date of that former publication of his own
to which he has referred.  The whole theory of the laws and logic
of symbols is indeed one of no small subtlety; insomuch that (as
is well known to the readers of the {\it Cambridge Mathematical
Journal}, in which periodical many papers of great interest and
importance on this very subject have appeared) it requires a
close and long-continued attention, in order to be able to form a
judgment of any value respecting it: nor does the present writer
venture to regard his own opinions on this head as being by any
means sufficiently matured; much less does he desire to provoke a
controversy with any of those who may perceive that he has not
yet been able to adopt, in all respects, their views.  That he
has adopted {\it some\/} of the views of the authors above
referred to, though in a way which does not seem to himself to be
contradictory to the results of his former reflexions; and
especially that he feels himself to be under important
obligations to the works of Dr.~Peacock upon Symbolical Algebra,
are things which he desires to record, or mark, in some degree,
by the very {\it title\/} of the present communication; in the
course of which there will occur opportunities for acknowledging
part of what he owes to other works, particularly to
Mr.~Warren's Treatise on the Geometrical Representation of the
Square Roots of Negative Quantities.

\nobreak\bigskip
{\it Observatory of Trinity College, Dublin, Oct}. 16, 1845.

\bigbreak

\centerline{\it Uniliteral and Biliteral Symbols.}

\nobreak\bigskip

1.
In the following pages of an attempt towards constructing a
symbolical geometry, it is proposed to employ (as usual) the
roman capital letters ${\sc a}$, ${\sc b}$, \&c., with or without
accents, as symbols of {\it points\/} in space; and to make use
(at first) of binary combinations of those letters, as symbols of
straight {\it lines\/}: the symbol of the beginning of the line
being written (for the sake of some analogies\footnote*{The
writer regards the line to ${\sc b}$ from ${\sc a}$ as being in
some sense an interpretation or construction of the symbol
${\sc b} - {\sc a}$; and the evident possibility of reaching the
point~${\sc b}$, by going along that line from the
point~${\sc a}$, may, as he thinks, be symbolized by the formula
${\sc b} - {\sc a} + {\sc a} = {\sc b}$.})
towards the right hand, and the symbol of the end towards the
left.  Thus ${\sc b} {\sc a}$ will denote the line {\it to\/}
${\sc b}$ {\it from\/} ${\sc a}$; and is not to be confounded
with the symbol ${\sc a} {\sc b}$, which denotes a line having
indeed the same extremities, but drawn in the opposite direction.
A biliteral symbol, of which the two component letters denote
determined and different points, will thus denote a finite
straight line, having a determined length, direction, and
situation in space.  But a biliteral symbol of the particular
form ${\sc a} {\sc a}$ may be said to be a {\it null\/} line,
regarded as the limit to which a line tends, when its extremities
tend to coincide: the conception or at least the name and symbol
of such a line being required for symbolic generality.  All lines
${\sc b} {\sc a}$ which are not null, may be called by contrast
{\it actual\/}; and the two lines ${\sc a} {\sc b}$ and
${\sc b} {\sc a}$ may be said to be the {\it opposites\/} of each
other.  It will then follow that a null line is its own opposite,
but that the opposites of two actual lines are always to be
distinguished from each other.

\bigbreak

\centerline{\it On the mark $=$.}

\nobreak\bigskip

2.
An {\it equation\/} such as
$${\sc b} = {\sc a}
   \eqno (1),$$
between two uniliteral symbols, may be interpreted as denoting
that ${\sc a}$ and ${\sc b}$ are {\it two names for one common
point\/}; or that a point~${\sc b}$, determined by one
geometrical process, coincides with a point~${\sc a}$ determined
by another process.  When a formula of the kind (1) holds good,
in any calculation, it is allowed to {\it substitute\/}, in any
other part of that calculation, either of the two equated symbols
for the other; and every other equation between two symbols of
one common class must be interpreted as to allow a similar
substitution.  We shall not violate this principle of symbolical
language by interpreting as we shall interpret, an equation such
as
$${\sc d} {\sc c} = {\sc b} {\sc a}
   \eqno (2),$$
between two biliteral symbols, as denoting that the two
lines,\footnote*{The writer regards the relation between the two
lines, mentioned in the text, as a sort of interpretation of the
following symbolic equation,
${\sc d} - {\sc c} = {\sc b} - {\sc a}$;
which may also denote that the point~${\sc d}$ is ordinally
related (in space) to the point~${\sc c}$ as ${\sc b}$ is to
${\sc a}$, and may in that view be also expressed by writing the
{\it ordinal analogy},
${\sc d} \mathbin{..} {\sc c} :: {\sc b} \mathbin{..} {\sc a}$;
which admits of {\it inversion\/} and {\it alternation}.  The
same relation between four points may, as he thinks, be thus
symbolically expressed
${\sc d} = {\sc b} - {\sc a} + {\sc c}$.
But by writing it as an equation between lines, he deviates less
from received notation.}
of which the symbols are equated, have {\it equal lengths and
similar directions}, though they may have different situations in
space: for if we call such lines {\it symbolically equal}, it
will be allowed, in {\it this\/} sense of equality, which has
indeed been already proposed by Mr.~Warren, Dr.~Peacock, and
probably by some of the foreign writers referred to in
Dr.~Peacock's Report, as well as in that narrower sense which
relates to magnitudes only, and for lines in space as well as
those which are in one plane, to assert that lines {\it equal\/}
to the same line are equal to each other.  (Compare {\it
Euclid}, {\sc xi}.~9.)  It will also be true, that
$${\sc d} = {\sc b},
   \quad\hbox{if } {\sc d} {\sc a} = {\sc b} {\sc a}
   \eqno (3),$$
or in words, that the ends of two symbolically equal lines
coincide if the beginnings do so; a consequence which it is very
desirable and almost necessary that we should be able to draw,
for the purposes of symbolical geometry, but which would not have
followed, if an equation of the form (2) had not been interpreted
so as to denote {\it only\/} equality of lengths, or {\it only\/}
similarity of directions.  The opposites of equal lines are equal
in the sense above explained; therefore the equation (2) gives
also this {\it inverse\/} equation,
$${\sc c} {\sc d} = {\sc a} {\sc b}
   \eqno (4).$$

Lines joining the similar extremities of symbolically equal lines
are themselves symbolically equal ({\it Euc}.\ {\sc i}.~33);
therefore the equation (2) gives also this {\it alternate\/}
equation,
$${\sc d} {\sc b} = {\sc c} {\sc a}
   \eqno (5).$$

The {\it identity\/} ${\sc b} {\sc a} = {\sc b} {\sc a}$ gives,
as its alternate equation,
$${\sc b} {\sc b} = {\sc a} {\sc a}
   \eqno (6),$$
which symbolic result may be expressed in words by saying that
any two null lines are to be regarded as equal to each other.
Lines equal to opposite lines may be said to be themselves
opposite lines.

\bigbreak

\centerline{\it On the mark $+$.}

\nobreak\bigskip

3.
The equation\footnote*{On the plan mentioned in former notes,
this equation would be written as follows:
$$({\sc c} - {\sc b}) + ({\sc b} - {\sc a})
   =  {\sc c} - {\sc a}.$$

It might be thus expressed: the ordinal relation of the
point~${\sc c}$ to the point~${\sc a}$ is compounded of the
relations of ${\sc c}$ to ${\sc b}$ and of ${\sc b}$ to ${\sc
a}$.}
$${\sc c} {\sc b} + {\sc b} {\sc a} = {\sc c} {\sc a}
   \eqno (7) $$
is true in the most elementary sense of the notation, when
${\sc b}$ is any point upon the finite straight line
${\sc c} {\sc a}$; but we propose now to {\it remove this
restriction for the purposes of symbolical geometry}, and to
regard the formula (7) as being universally {\it valid, by
definition, whatever three points of space may be denoted\/} by
the three letters ${\sc a} {\sc b} {\sc c}$.  The equation (7)
will then {\it express nothing about those points}, but will
serve to {\it fix the interpretation of the mark~$+$ when
inserted between any two symbols of lines\/}; for if we meet any
symbol formed by such insertion, suppose the symbol
${\sc h} {\sc g} + {\sc f} {\sc e}$, we have only to draw, or
conceive drawn, from any assumed point~${\sc a}$, a line
${\sc b} {\sc a} = {\sc f} {\sc e}$,
and from the end~${\sc b}$ of the line so drawn, a new line
${\sc c} {\sc b} = {\sc h} {\sc g}$;
and then the proposed symbol
${\sc h} {\sc g} + {\sc f} {\sc e}$
will be interpreted by (7) as denoting the line
${\sc c} {\sc a}$, or at least a line equal thereto.  In like
manner, by defining that
$${\sc d} {\sc c} + {\sc c} {\sc b} + {\sc b} {\sc a}
   = {\sc d} {\sc a}
   \eqno (8),$$
we shall be able to interpret any symbol of the form
$${\sc k} {\sc i} + {\sc h} {\sc g} + {\sc f} {\sc e},$$
as denoting a determined (actual or null) line; at least if we
now regard a line as {\it determined\/} when it is {\it equal\/}
to a determined line: and similarly for any number of biliteral
symbols, connected by marks~$+$ interposed.  Calling {\it this\/}
act of connection of symbols, the operation of {\it addition\/};
the added symbols, {\it summands\/}; and the resultant symbol, a
{\it sum\/}; we may therefore now say, that the sum of any number
of symbols of given lines is itself a symbol of a determined
line; and that this symbolic sum of lines represents the
{\it total\/} (or final) {\it effect\/} of all those successive
rectilinear {\it motions}, or translations of a point in space,
which are represented by the several summands.  This {\it
interpretation of a symbolic sum of lines\/} agrees with the
conclusions already published by the authors above alluded to;
though the methods of symbolically obtaining and expressing it,
here given, may possibly be found to be new.  The same
interpretation satisfies, as it ought to do, the condition that
the sums of equals shall be equal (compare the demonstration of
{\it Euclid}, {\sc xi}.~10); and also this other condition,
almost as much required for the advantageous employment of
symbolical language, that those lines which, when added to equal
lines, give equal sums, shall be themselves equal lines: or that
$${\sc f} {\sc e} = {\sc d} {\sc c},
   \quad\hbox{if, }
  {\sc f} {\sc e} + {\sc b} {\sc a}
   =  {\sc d} {\sc c} + {\sc b} {\sc a}
   \eqno (9).$$

It shews too that the sum of two opposite lines, and generally
that the sum of all the successive sides of any closed polygon,
or of lines respectively equal to those sides, is a null line:
thus
$${\sc a} {\sc a}
   =  {\sc a} {\sc b} + {\sc b} {\sc a}
   =  {\sc a} {\sc c} + {\sc c} {\sc b} + {\sc b} {\sc a}
   =  \hbox{\&c.}
   \eqno (10).$$

The symbolic sum of any two lines is found to be {\it independent
of their order}, in virtue of the same interpretation; so that
the equation
$${\sc f} {\sc e} + {\sc h} {\sc g}
   = {\sc h} {\sc g} + {\sc f} {\sc e}
   \eqno (11),$$
is true, in the present system, {\it not as an independent
definition}, but rather as one of the modes of {\it symbolically
expressing that elementary theorem of geometry\/}, ({\it Euclid},
{\sc i}.~33), on which was founded the rule for deducing, from
any equation (2) between lines, the {\it alternate\/} equation
(5).  For if we assume, as we may, that three points
${\sc a}$,~${\sc b}$,~${\sc c}$,
have been so chosen as to satisfy the equations
${\sc f} {\sc e} = {\sc b} {\sc a}$,
${\sc h} {\sc g} = {\sc c} {\sc a}$;
and that a fourth point~${\sc d}$ is chosen so as to satisfy the
equation
${\sc d} {\sc c} = {\sc b} {\sc a}$;
the same points will then, by the theorem just referred to,
satisfy also the equation
${\sc d} {\sc b} = {\sc c} {\sc a}$;
and the truth of the formula (11) will be proved, by observing
that each of the two symbols which are equated in that formula is
equal to the symbol ${\sc d} {\sc a}$, in virtue of the definition
(7) of $+$, without any new definition: since
$${\sc f} {\sc e} + {\sc h} {\sc g}
   =  {\sc d} {\sc c} + {\sc c} {\sc a}
   =  {\sc d} {\sc a}
   =  {\sc d} {\sc b} + {\sc b} {\sc a}
   =  {\sc h} {\sc g} + {\sc f} {\sc e}.$$
A like result is easily shown to hold good, for any number of
summands; thus
$${\sc f} {\sc e} + {\sc h} {\sc g} + {\sc k} {\sc i}
   = {\sc k} {\sc i} + {\sc h} {\sc g} + {\sc f} {\sc e}
   \eqno (12);$$
since the first member of this last equation may be put
successively under the forms
$$({\sc f} {\sc e} + {\sc h} {\sc g}) + {\sc k} {\sc i},\quad
  {\sc k} {\sc i} + ({\sc f} {\sc e} + {\sc h} {\sc g}),\quad
  {\sc k} {\sc i} + ({\sc h} {\sc g} + {\sc f} {\sc e}),$$
and finally under the form of the second member; the stages of
this successive transformations of symbols admitting easily of
geometrical interpretations: and similarly in other cases.
{\it Addition of  lines in space\/} is therefore generally (as
Mr.~Warren has shewn it to be for lines in a single plane) a
{\it commutative operation\/}; in the sense that the summands may
interchange their places, without the sum being changed.  It is
also an {\it associative\/} operation, in the sense that any
number of successive summands may be associated into one group,
and collected into one partial sum (denoted by enclosing these
summands in parentheses); and that then this partial sum may be
added, as a single summand, to the rest: thus
$$({\sc k} {\sc i} + {\sc h} {\sc g}) + {\sc f} {\sc e}
   =  {\sc k} {\sc i} + ({\sc h} {\sc g} + {\sc f} {\sc e})
   =  {\sc k} {\sc i} + {\sc h} {\sc g} + {\sc f} {\sc e}
   \eqno (13).$$

\bigbreak

\centerline{\it On the mark $-$.}

\nobreak\bigskip

4.
The equation\footnote*{On the plan mentioned in some former
notes, this equation would take the form
$$({\sc c} - {\sc a}) - ({\sc b} - {\sc a})
   =  {\sc c} - {\sc b}.$$}
$${\sc c} {\sc a} - {\sc b} {\sc a} = {\sc c} {\sc b}
   \eqno (14) $$
is true, in the most elementary sense of the notation, when
${\sc b}$ is on ${\sc c} {\sc a}$; but we may remove this
restriction by a {\it definitional extension\/} of the formula
(14), for the purposes of symbolical geometry, as has been done
in the foregoing article with respect to the formula (7); and
then the equation (14), so extended, will express {\it nothing
about the points\/} ${\sc a}$,~${\sc b}$,~${\sc c}$, but will
serve to fix the {\it interpretation of any symbol}, such as
${\sc k} {\sc i} - {\sc f} {\sc e}$, formed by {\it inserting the
mark~$-$ between the symbols of any two lines}.  This general
meaning of the effect of the mark~$-$, so inserted, is consistent
with the particular interpretation which suggested the formula
(14); it is also consistent with the usual symbolical opposition
between the effects of $+$ and $-$; since the comparison of (14)
with (7) gives the equations
$$({\sc c} {\sc a} - {\sc b} {\sc a}) + {\sc b} {\sc a}
  =  {\sc c} {\sc a}
   \eqno (15),$$
and
$$({\sc c} {\sc b} + {\sc b} {\sc a}) - {\sc b} {\sc a}
   =  {\sc c} {\sc b}
   \eqno (16),$$
either of which two equations, if regarded as a general formula,
and combined with the formula (7), would include, reciprocally,
the definition (14) of $-$, and might be substituted for it.

Symbolical {\it subtraction\/} of one line from another is thus
equivalent to the {\it decomposition\/} of a given rectilinear
{\it motion\/} (${\sc c} {\sc a}$) into two others, of which one
(${\sc b} {\sc a}$) is given; or to the {\it addition of the
opposite\/} (${\sc a} {\sc b}$) of the line which was to be
subtracted: so that we may write the symbolical equation
$$- {\sc b} {\sc a} = + {\sc a} {\sc b}
   \eqno (17),$$
because the second member of (14) may be changed by (7) to
${\sc c} {\sc a} + {\sc a} {\sc b}$.  These conclusions
respecting symbolical subtraction of lines, differ only in their
notation, and in the manner of arriving at them, from the results
of the authors already referred to, so far as the present writer
is acquainted with them.  In the present notation, when an
isolated biliteral symbol is preceded with $+$ or $-$, we may
still interpret it as denoting a line, if we agree to prefix to
it, for the purpose of such interpretation, the symbol of a null
line; thus we may write
$$+ {\sc a} {\sc b} = {\sc a} {\sc a} + {\sc a} {\sc b}
   =  {\sc a} {\sc b},\quad
  - {\sc a} {\sc b} = {\sc a} {\sc a} - {\sc a} {\sc b}
   =  {\sc b} {\sc a}
   \eqno (18);$$
$+ {\sc a} {\sc b}$ will, therefore, on this plan, be another
symbol for the line ${\sc a} {\sc b}$ itself, and
$- {\sc a} {\sc b}$ will be a symbol for the opposite line
${\sc b} {\sc a}$.

\bigbreak

\centerline{\it Abridged Symbols for Lines.}

\nobreak\bigskip

5.
Some of the foregoing formul{\ae} may be presented more
concisely, and also in a way more resembling ordinary Algebra, by
using now some new {\it uniliteral\/} symbols, such as the small
roman letters ${\rm a}$,~${\rm b}$,~\&c., with or without
accents, as symbols of lines, instead of binary combinations of
the roman capitals, in cases where the lines which are compared
are not supposed to have necessarily any common point, and
generally when the {\it situations\/} of lines are disregarded,
but not their lengths nor their directions.  Thus we shall have,
instead of (11) and (12), (13), (15) and (16), these other
formul{\ae} of the present Symbolical Geometry, which agree in
all respect with those used in Symbolical Algebra:
$${\rm a} + {\rm b} = {\rm b} + {\rm a},\quad
  {\rm a} + {\rm b} + {\rm c} = {\rm c} + {\rm b} + {\rm a}
   \eqno (19);$$
$$({\rm c} + {\rm b}) + {\rm a}
   =  {\rm c} + ({\rm b} + {\rm a})
   =  {\rm c} + {\rm b} + {\rm a}
   \eqno (20);$$
$$({\rm b} - {\rm a}) + {\rm a} = {\rm b},\quad
  ({\rm b} + {\rm a}) - {\rm a} = {\rm b}
   \eqno (21);$$
and because the isolated but {\it affected\/} symbols
$+ {\rm a}$, $- {\rm a}$, may denote, by (18), the line~${\rm a}$
itself, and the opposite of that line, we have also here the
usual {\it rule of the signs},
$$+ ( + {\rm a} ) = - ( - {\rm a} ) = + {\rm a},\quad
  + ( - {\rm a} ) = - ( + {\rm a} ) = - {\rm a}
   \eqno (22).$$

\bigbreak

\centerline{\it Introduction of the marks $\times$ and $\div$.}

\nobreak\bigskip

6.
Continuing to denote lines by letters, the formula
$$({\rm b} \div {\rm a}) \times {\rm a} = {\rm b}
   \eqno (23),$$
which is, for the relation between multiplication and division,
what the first of the two formul{\ae} (21) is for the relation
between addition and subtraction, will be true, in the most
elementary sense of the multiplication of a length by a number,
for the case when the line~${\rm b}$ is the sum of several
summands, each equal to the line~${\rm a}$, and when the number
of those summands is denoted by the quotient
${\rm b} \div {\rm a}$.
And we shall now, for the purposes of symbolical generality,
{\it extend\/} this formula (23), so as to make it be valid,
{\it by definition}, {\it whatever two lines\/} may be denoted by
${\rm a}$ and ${\rm b}$.  The formula will then {\it express
nothing respecting those lines\/} themselves, which can serve to
distinguish them from any other lines in space; but will furnish
a {\it symbolic condition}, which we must satisfy by the
{\it general interpretation\/} of a {\it geometrical quotient},
and of the {\it operation of multiplying a line\/} by such a
quotient.

To make such a general interpretation consistent with the
particular case where the quotient becomes a {\it quotity}, we
are led to write
$${\rm a} \div {\rm a} = 1,\quad
  ({\rm a} + {\rm a}) \div {\rm a} = 2,
   \enspace\hbox{\&c.}
   \eqno (24),$$
and conversely
$$1 \times {\rm a} = {\rm a},\quad
  2 \times {\rm a} = {\rm a} + {\rm a}
   \enspace\hbox{\&c.}
   \eqno (25);$$
and because, when quotients can be thus interpreted as quotities,
the four equations
$$({\rm c} \div {\rm a}) + ({\rm b} \div {\rm a})
   = ({\rm c} + {\rm b}) \div {\rm a}
   \eqno (26),$$
$$({\rm c} \div {\rm a}) - ({\rm b} \div {\rm a})
   = ({\rm c} - {\rm b}) \div {\rm a}
   \eqno (27),$$
$$({\rm c} \div {\rm a}) \times ({\rm a} \div {\rm b})
   =  {\rm c} \div {\rm b}
   \eqno (28),$$
$$({\rm c} \div {\rm a}) \div ({\rm b} \div {\rm a})
   =  {\rm c} \div {\rm b}
   \eqno (29),$$
are true in the most elementary sense of arithmetical operations
on whole numbers, we shall now {\it define\/} that these four
equations are valid, {\it whatever three lines\/} may be denoted
by ${\rm a}$,~${\rm b}$,~${\rm c}$; and thus shall have
conditions for the general {\it interpretations of the four
operations $+$~$-$~$\times$~$\div$ performed on geometrical
quotients}.

We shall in this way be led to interpret a quotient of which the
divisor is an actual line, but the dividend a null one, as being
equivalent to the symbol $1 - 1$ or {\it zero\/}; so that
$$({\rm a} - {\rm a}) \div {\rm a} = 0,\quad
  0 \times {\rm a} = {\rm a} - {\rm a}
   \eqno (30).$$

{\it Negative\/} numbers will present themselves in the
consideration of such quotients and products as
$$(- {\rm a}) \div {\rm a} = 0 - 1 = -1,\quad
  (-1) \times {\rm a} = - {\rm a},
   \enspace\hbox{\&c.}
   \eqno (31);$$
{\it fractional\/} numbers in such formul{\ae} as
$${\rm a} \div ({\rm a} + {\rm a})
   =  1 \div 2 = {\textstyle {1 \over 2}},\quad
  {\textstyle {1 \over 2}} \times ({\rm a} + {\rm a}) = {\rm a},
   \enspace\hbox{\&c.}
   \eqno (32);$$
and {\it incommensurable\/} numbers, by the conception of the
connected {\it limits\/} of quotients and products, and by the
formula, which symbolical language leads us to assume,
$$\left(  \lim {n \over m} \right) \times {\rm a}
   = \lim \left( {n \over m} \times {\rm a} \right)
   \eqno (33).$$

If then we give the name of {\sc scalars} to all numbers of the
kind called usually {\it real}, because they are all contained on
the one {\it scale\/} of progression of number from negative to
positive infinity; and if we agree, for the present, to denote
such numbers generally by small italic letters
$a$,~$b$,~$c$,~\&c.; and to insert the mark $\parallel$ between
the symbols of two lines when we wish to express that the
directions of those lines are either exactly similar or exactly
opposite to each other, in each of which two cases the lines may
be said to be {\it symbolically parallel\/}; we shall have
generally two equations of the forms
$${\rm b} \div {\rm a} = a,\quad
  a \times {\rm a} = {\rm b},
  \quad\hbox{when } {\rm b} \parallel {\rm a}
   \eqno (34).$$
That is to say, the {\it quotient of two parallel lines\/} is
generally a {\it scalar number\/}; and, conversely, to multiply a
given line (${\rm a}$) by a given scalar (or real) number~$a$, is
to determine a new line (${\rm b}$) parallel to the given line
(${\rm a}$), the direction of the one being similar or opposite
to that of the other, according as the number is positive or
negative, while the length of the new line bears to the length of
the given line a ratio which is marked by the same given number.
So that if ${\sc a}_0$~${\sc a}_1$~${\sc a}_a$ denote any three
points on one common axis of rectilinear progression, which are
related to each other, upon that axis, as to their order and
their intervals, in the same manner as the three scalar numbers
$0$,~$1$,~$a$, regarded as ordinals, are related to each other on
the scale of numerical progression from $-\infty$ to $+\infty$,
then the equations
$${\sc a}_a {\sc a}_0 \div {\sc a}_1 {\sc a}_0 = a,\quad
  a \times {\sc a}_1 {\sc a}_0 = {\sc a}_a {\sc a}_0
   \eqno (35) $$
will be true by the foregoing interpretations.

It is easy to see that this mode of interpreting a quotient of
parallel lines renders the formul{\ae} (26) (27) (28) (29)
consistent with the received rules for performing the operations
$+$~$-$~$\times$~$\div$ on what are called the real numbers,
whether they be positive or negative, and whether commensurable
or incommensurable; or rather reproduces those rules as
consequences of those formul{\ae}.

\bigbreak

\centerline{\it On Vectors, and Geometrical Quotients in
general.}

\nobreak\bigskip

7.
The other chief relation of directions of lines in space, besides
parallelism, is perpendicularity; which it is not unusual to
denote by writing the mark~$\perp$ between the symbols of two
perpendicular lines.  And the other chief class of geometrical
quotients which it is important to study, as preparatory to a
general theory of such quotients, is the class in which the
dividend is a line perpendicular to the divisor.  A quotient of
this latter class we shall call a {\sc vector}, to mark its
connection (which is closer than that of a {\it scalar\/}) with
the conception of {\it space}, and for other reasons which will
afterwards appear: and if we agree to denote, for the present,
such vector quotients (of perpendicular lines) by small Greek
letters, in contrast to the scalar class of quotients (of
parallel lines) which we have proposed to denote by small italic
letters, we shall then have generally two equations of the forms
$${\rm c} \div {\rm a} = \alpha,\quad
  {\rm c} = \alpha \times {\rm a},
   \quad\hbox{if } {\rm c} \perp {\rm a}
   \eqno (36).$$

{\it Any\/} line~${\rm e}$ may be put under the form
${\rm c} + {\rm b}$, in which ${\rm b} \parallel {\rm a}$, and
${\rm c} \perp {\rm a}$; a {\it general geometrical quotient\/}
may therefore, by (26) (34) (36), be considered as the
{\it symbolic sum of a scalar and a vector}, zero being regarded
as a common limit of quotients of these two classes; and
consequently, if we adopt the notation just now mentioned, we
have generally an equation of the form
$${\rm e} \div {\rm a} = \alpha + a
   \eqno (37).$$

This {\it separation of the scalar and vector parts\/} of a
general geometrical quotient corresponds (as we see) to the
decomposition, by {\it two separate projections}, of the dividend
line into two other lines of which it is the symbolic sum, and of
which one is parallel to the divisor line, while the other is
perpendicular thereto.  To be able to mark on some occasions more
distinctly, in writing, than by the use of two different
alphabets, the conception of such separation, we shall here
introduce two new symbols of operation, namely the abridged words
${\rm Scal.}$ and ${\rm Vect.}$, which, where no confusion seems
likely to arise from such farther abridgment, we shall also
denote more shortly still by the letters ${\rm S}$ and ${\rm V}$,
prefixing them to the symbol of a general geometrical quotient
in order to form separate symbols of its scalar and vector parts;
so that we shall now write more generally, for any two lines
${\rm a}$ and ${\rm e}$,
$${\rm e} \div {\rm a}
   =  {\rm Vect.} ({\rm e} \div {\rm a})
    + {\rm Scal.} ({\rm e} \div {\rm a})
   \eqno (38);$$
or more concisely,
$${\rm e} \div {\rm a}
   =  {\rm V} ({\rm e} \div {\rm a})
    + {\rm S} ({\rm e} \div {\rm a})
   \eqno (39);$$
in which expression the order of the two summands may be changed,
in virtue of the definition (26) of addition of geometrical
quotients, because the order of the two partial dividends may be
changed without preventing the dividend line ${\rm e}$ from being
still their symbolic sum.  A scalar cannot become equal to a
vector, except by each becoming zero; for if the divisor of the
vector quotient be multiplied separately by the scalar and the
vector, the products of these two multiplications will be (by
what has been already shown) respectively lines parallel and
perpendicular to that divisor, and therefore not symbolically
equal to each other, except it be at the limit where both become
null lines, and are on that account regarded as equal.  A scalar
quotient ${\rm b} \div {\rm a} = a$,
(${\rm b} \parallel {\rm a}$), has been seen to denote the
relative length and relative direction (as similar or opposite)
of two parallel lines ${\rm a}$,~${\rm b}$: and in like manner a
vector quotient ${\rm e} \div {\rm a} = \alpha$,
(${\rm c} \perp {\rm a}$), may be regarded as denoting the
{\it relative length and relative direction\/} (depending on
{\it plane\/} and {\it hand\/}) {\it of two perpendicular
lines\/} ${\rm a}$,~${\rm c}$; or as indicating at once {\it in
what ratio\/} the length of one line~${\rm a}$ must be altered
(if at all) in order to become equal to the length of another
line~${\rm c}$, and also {\it round what axis}, perpendicular to
both these two rectangular lines, the direction of the divisor
line~${\rm a}$ must be caused or conceived to turn,
right-handedly, through a right angle, in order to attain the
original direction of the dividend line~${\rm c}$.  A line drawn
in the direction of this {\it axis of\/} (what is here regarded
as) {\it positive rotation}, and having its length in the same
ratio to some assumed {\it unit\/} of length as the length of the
dividend to that of the divisor, may be called the {\sc index} of
the vector.  We shall thus be led to substitute, for any equation
between two vector quotients, an equation between two lines,
namely between their indices; for if we define that two vector
quotients, such as ${\rm c} \div {\rm a}$ and
${\rm c}' \div {\rm a}'$ if ${\rm c} \perp {\rm a}$ and
${\rm c}' \perp {\rm a}'$, are {\it equal\/} when they have
{\it equal indices}, we shall satisfy all conditions of
symbolical equality, of the kinds already considered in
connection with other definitions; we shall also be able to say
that in every case of two such equal quotients, the two dividend
lines (${\rm c}$ and ${\rm c}'$) bear to their own divisor lines
(${\rm a}$ and ${\rm a}'$), respectively, one common ratio of
lengths, and one common relation of directions.  We shall thus
also, by (23), be able to {\it interpret the multiplication\/} of
any given line ${\rm a}'$ by any given vector
${\rm c} \div {\rm a}$, {\it provided that the one is
perpendicular to the index of the other}, as the operation of
deducing from ${\rm a}'$ another line ${\rm c}'$, by altering
(generally) its length in a given ratio, and by turning (always)
its direction round a given axis of rotation, namely round the
index of the vector, right-handedly, through a right angle.  And
we can now {\it interpret an equation between two general
geometrical quotients}, such as
$${\rm e}' \div {\rm a}' = {\rm e} \div {\rm a}
   \eqno (40),$$
as being equivalent to a {\it system of two separate equations},
one between the scalar and another between the vector parts,
namely the two following:
$${\rm S} ({\rm e}' \div {\rm a}')
   =  {\rm S} ({\rm e} \div {\rm a});\quad
  {\rm V} ({\rm e}' \div {\rm a}')
   =  {\rm V} ({\rm e} \div {\rm a})
   \eqno (41);$$
of which each separately is to be interpreted on the principles
already laid down; and which are easily seen (by considerations
of similar triangles) to imply, when taken jointly, that the
length of ${\rm e}'$ is to that of ${\rm a}'$ in the same ratio
as the length of ${\rm e}$ to that of ${\rm a}$; and also that
the same rotation, round the index of either of the two equal
vectors, which would cause the direction of ${\rm a}$ to attain
the original direction of ${\rm e}$, would also bring the
direction of ${\rm a}'$ into that originally occupied by
${\rm e}'$.  At the same time we see how to interpret the
operation of multiplying any given line~${\rm a}'$ by any given
geometrical quotient ${\rm e} \div {\rm a}$ of two other lines,
{\it whenever the three given lines
${\rm a}$,~${\rm e}$,~${\rm a}'$,
are parallel to one common plane\/}; namely as being the complex
operation of altering (generally) a given length in a given
ratio, and of turning a given line round a given axis, through a
given amount of right-handed rotation, in order to obtain a
certain new line ${\rm e}'$, which may be thus denoted, in
conformity with the definition (23),
$${\rm e}' = ({\rm e} \div {\rm a}) \times {\rm a}'
   \eqno (42).$$
The relation between the four lines
${\rm a}$,~${\rm e}$,~${\rm a}'$,~${\rm e}'$,
may also be called a {\it symbolic analogy}, and may be thus
denoted:
$${\rm e}' : {\rm a}' :: {\rm e} : {\rm a}
   \eqno (43);$$
${\rm a}'$ and ${\rm e}$ being the {\it means}, and ${\rm e}'$
and ${\rm a}$ the {\it extremes\/} of the analogy.  An analogy or
equation of this sort admits (as it is easy to prove) of
{\it inversion\/} and {\it alternation\/}; thus (43) or (42)
gives, {\it inversely},
$${\rm a}' : {\rm e}' :: {\rm a} : {\rm e},\quad
  {\rm a}' \div {\rm e}' = {\rm a} \div {\rm e}
   \eqno (44),$$
and {\it alternately},
$${\rm e}' : {\rm e} :: {\rm a}' : {\rm a},\quad
  {\rm e}' \div {\rm e} = {\rm a}' \div {\rm a}
   \eqno (45).$$
These results respecting analogies between {\it co-planar lines},
that is, between lines which are in or parallel to one common
plane, agree with, and were suggested by, the results of
Mr.~Warren.  But it will be necessary to introduce other
principles, or at least to pursue farther the track already
entered on, before we can arrive at an interpretation of a {\it
fourth proportional to three lines which are not parallel to any
common plane\/}: or can interpret the multiplication of a line by
a quotient of two others, when it is not perpendicular to what
has been lately called the index of the vector part of that
quotient.

\bigbreak

\centerline{\it Determinateness of the first Four Operations on
Geometrical Fractions (or Quotients).}

\nobreak\bigskip

8.
Meanwhile the principles and definitions which have been already
laid down, are sufficient to conduct to clear and determinate
interpretations of all operations of combining geometrical
quotients among themselves, by any number of additions,
subtractions, multiplications, and divisions: each
{\it quotient\/} of the kind here mentioned being regarded, by
what has been already shown, as the {\it mark of a certain
complex relation between two straight lines in space}, depending
not only on their {\it relative lengths}, but also on their
{\it relative directions}.  If we denote now by a symbol of
fractional form, such as
$\displaystyle {{\rm b} \over {\rm a}}$,
the quotient thus obtained by dividing one line~${\rm b}$ by
another line ${\rm a}$, when directions as well as lengths are
attended to, the definitional equations (26), (27), (28), (29),
will take these somewhat shorter forms:\footnote*{On the
principles alluded to in former notes, the formul{\ae} for the
addition, subtraction, multiplication, and division, of any two
geometrical fractions, might be thus written:
$$\eqalign{
      {{\sc d} - {\sc c} \over {\sc b} - {\sc a}}
    + {{\sc c} - {\sc a} \over {\sc b} - {\sc a}}
   &= {{\sc d} - {\sc a} \over {\sc b} - {\sc a}},\cr
   \noalign{\vskip3pt}
      {{\sc d} - {\sc a} \over {\sc b} - {\sc a}}
    - {{\sc c} - {\sc a} \over {\sc b} - {\sc a}}
   &= {{\sc d} - {\sc c} \over {\sc b} - {\sc a}},\cr
   \noalign{\vskip3pt}
      {{\sc d} - {\sc a} \over {\sc c} - {\sc a}}
      \times
      {{\sc c} - {\sc a} \over {\sc b} - {\sc a}}
   &= {{\sc d} - {\sc a} \over {\sc b} - {\sc a}},\cr
   \noalign{\vskip3pt}
      {{\sc d} - {\sc a} \over {\sc b} - {\sc a}}
      \div
      {{\sc c} - {\sc a} \over {\sc b} - {\sc a}}
   &= {{\sc d} - {\sc a} \over {\sc c} - {\sc a}};\cr}$$
${\sc a}$, ${\sc b}$, ${\sc c}$, ${\sc d}$ being symbols of any
four points of space, and ${\sc b} - {\sc a}$ being a symbol of
the straight line drawn to ${\sc b}$ from ${\sc a}$.  If we
denote this line by the biliteral symbol ${\sc b} {\sc a}$, we
obtain the following somewhat shorter forms, which do not however
all agree so closely with the forms of ordinary algebra:
$$\eqalign{
      {{\sc d} {\sc c} \over {\sc b} {\sc a}}
    + {{\sc c} {\sc a} \over {\sc b} {\sc a}}
   &= {{\sc d} {\sc a} \over {\sc b} {\sc a}},\cr
   \noalign{\vskip3pt}
      {{\sc d} {\sc a} \over {\sc b} {\sc a}}
    - {{\sc c} {\sc a} \over {\sc b} {\sc a}}
   &= {{\sc d} {\sc c} \over {\sc b} {\sc a}},\cr
   \noalign{\vskip3pt}
      {{\sc d} {\sc a} \over {\sc c} {\sc a}}
      \times
      {{\sc c} {\sc a} \over {\sc b} {\sc a}}
   &= {{\sc d} {\sc a} \over {\sc b} {\sc a}},\cr
   \noalign{\vskip3pt}
      {{\sc d} {\sc a} \over {\sc b} {\sc a}}
      \div
      {{\sc c} {\sc a} \over {\sc b} {\sc a}}
   &= {{\sc d} {\sc a} \over {\sc c} {\sc a}}.\cr}$$}
$${{\rm c} \over {\rm a}} + {{\rm b} \over {\rm a}}
   = {{\rm c} + {\rm b} \over {\rm a}};\quad
  {{\rm c} \over {\rm a}} - {{\rm b} \over {\rm a}}
   = {{\rm c} - {\rm b} \over {\rm a}};
   \eqno (46),$$
$${{\rm c} \over {\rm a}} \times {{\rm a} \over {\rm b}}
   = {{\rm c} \over {\rm b}};\quad
  {{\rm c} \over {\rm a}} \div {{\rm b} \over {\rm a}}
   = {{\rm c} \over {\rm b}};
   \eqno (47),$$
which agree in all respects with the corresponding formul{\ae} of
ordinary algebra, and serve to fix, in the present system, the
meanings of the operations $+$, $-$, $\times$, $\div$, on what
may be called {\it geometrical fractions}.  These {\sc fractions}
being only other forms for what we have called {\it geometrical
quotients\/} in earlier articles of ths paper, we may now write
the identity,
$${{\rm b} \over {\rm a}} = {\rm b} \div {\rm a}
   \eqno (48).$$

For the same reason, an {\it equation between any two such
fractions}, for example the following,
$${{\rm f} \over {\rm e}} = {{\rm b} \over {\rm a}}
   \eqno (49),$$
is to be understood as signifying, 1st, that the {\it length\/}
of the one {\it numerator\/} line~${\rm f}$ is to the length of
its own {\it denominator\/} line~${\rm e}$ {\it in the same
ratio\/} as the length of the other numerator line~${\rm b}$ to
the length of the other denominator line~${\rm a}$; 2nd, that
these four lines are {\it co-planar}, that is to say, in or
parallel to one common plane; and 3rd, that the {\it same amount
and direction of rotation}, round an axis perpendicular to this
common plane, which would bring the line~${\rm a}$ into the
direction originally occupied by ${\rm b}$, would also bring the
line~${\rm e}$ into the original direction of ${\rm f}$.  The
same complex relation between the same four lines may also (by
what has been already seen) be expressed by the {\it inverse\/}
equation
$${{\rm e} \over {\rm f}} = {{\rm a} \over {\rm b}}
   \eqno (50),$$
or by the {\it alternate\/} form
$${{\rm f} \over {\rm b}} = {{\rm e} \over {\rm a}}
   \eqno (51).$$
Two fractions which are, in this sense, {\it equal\/} to the same
third fraction, are also equal to each other; and the
{\it value\/} of such a fraction is not altered by altering the
lengths of its numerator and denominator in any common ratio; nor
by causing both to turn together through any common amount of
rotation, in a common direction, round an axis perpendicular to
both; nor by transporting either or both, without rotation, to
any other positions in space.  When the lengths and directions of
any three co-planar lines, ${\rm a}$, ${\rm b}$, ${\rm e}$, are
given, it is always possible to determine the length and
direction of a fourth line~${\rm f}$, which shall be co-planar
with them, and shall satisfy an equation between fractions, of
the form (49).  It is therefore possible to {\it reduce any two
geometrical fractions to a common denominator\/}; or to satisfy
not only the equation (49), but also this other equation,
$${{\rm h} \over {\rm g}} = {{\rm c} \over {\rm a}}
   \eqno (52),$$
by a suitable choice of the three lines
${\rm a}$,~${\rm b}$,~${\rm c}$, when the four lines
${\rm e}$,~${\rm f}$,~${\rm g}$,~${\rm h}$, are given; since,
whatever may be the given directions of these four lines, it is
always possible to find (or to conceive as found) a fifth
line~${\rm a}$, which shall be at once co-planar with the pair
${\rm e}$,~${\rm f}$ and also with the pair ${\rm g}$,~${\rm h}$.
For a similar reason it is always possible to transform two given
geometrical fractions into two others equivalent to them, in such
a manner, that the new denominator of one shall be equal to the
new numerator of the other; or to satisfy the two equations
$${{\rm h} \over {\rm g}} = {{\rm c}' \over {\rm a}'},\quad
  {{\rm f} \over {\rm e}} = {{\rm a}' \over {\rm b}'}
   \eqno (53),$$
by a suitable choice of the three lines
${\rm a}'$,~${\rm b}'$,~${\rm c}'$,
whatever the four given lines
${\rm e}$,~${\rm f}$,~${\rm g}$,~${\rm h}$ may be.  Making then
for abridgment
$${\rm c} + {\rm b} = {\rm d},\quad
  {\rm c} - {\rm b} = {\rm d}'
   \eqno (54),$$
and interpreting a sum or difference of lines as has been done in
former articles, we see that it is always possible to choose
eight lines
${\rm a}$,~${\rm b}$,~${\rm c}$,~${\rm d}$,
${\rm a}'$,~${\rm b}'$,~${\rm c}'$,~${\rm d}'$,
so as to satisfy the conditions (49), (52), (53), (54); and thus,
by (46) and (47), to interpret the sum, the difference, the
product, and the quotient of {\it any two\/} given geometrical
fractions,
$\displaystyle {{\rm f} \over {\rm e}}$ and
$\displaystyle {{\rm h} \over {\rm g}}$,
as being each equal to {\it another given fraction\/} of the same
sort, as follows:
$${{\rm h} \over {\rm g}} + {{\rm f} \over {\rm e}}
   = {{\rm d} \over {\rm a}},\quad
  {{\rm h} \over {\rm g}} - {{\rm f} \over {\rm e}}
   = {{\rm d}' \over {\rm a}}
   \eqno (55),$$
$${{\rm h} \over {\rm g}} \times {{\rm f} \over {\rm e}}
   = {{\rm c}' \over {\rm b}'},\quad
  {{\rm h} \over {\rm g}} \div {{\rm f} \over {\rm e}}
   = {{\rm c} \over {\rm b}}
   \eqno (56),$$
any variations in the new numerators and denominators, which are
consistent with the foregoing conditions, being easily seen
to make no changes in the values of the fractions which result.
The {\it interpretations\/} of these four symbolic combinations,
which are the first members of the four equations (55) and (56),
are thus entirely {\it fixed\/}: and we are {\it no longer at
liberty, in the present system}, to introduce arbitrarily any
{\it new meanings\/} for those symbolical forms, or to subject
them to any {\it new laws\/} of combination among themselves,
without examining whether such meanings or such laws are
consistent with the principles and definitions which it has been
thought right to establish already, as appearing to be more
simple and primitive, and more intimately connected with the
application of symbolical language to geometry, or at least with
the plan on which it is here attempted to make that application,
than any of those other laws or meanings.  If, for example, it
shall be found that, in virtue of the foregoing principles, the
{\it successive addition\/} of any number of geometrical
fractions gives a result which is independent of their order,
this consequence will be, for us, a {\it theorem}, and not a
definition.  And if, on the contrary, the same principles shall
lead us to regard the {\it multiplication\/} of geometrical
fractions as being in general a {\it non-commutative\/}
operation, or as giving a result which is {\it not\/} independent
of the order of the factors, we shall be obliged to accept this
conclusion also, that we may preserve consistency of system.

\bigbreak

\centerline{\it Separation of the Scalar and Vector parts of Sums
and Differences of Geometrical Fractions.}

\nobreak\bigskip

9.
To develope the geometrical meaning of the first equation (46),
we may conceive each of the two numerator lines
${\rm b}$,~${\rm c}$, and also their sum~${\rm d}$, to be
orthogonally projected, first on the common denominator
line~${\rm a}$ itself, and secondly on a plane perpendicular to
that denominator.  The former projections may be called
${\rm b}_1$,~${\rm c}_1$,~${\rm d}_1$; the latter
${\rm b}_2$,~${\rm c}_2$,~${\rm d}_2$; and thus we shall have the
nine relations,
$$\left. \matrix{
\displaystyle
{\rm b}_2 + {\rm b}_1 = {\rm b},\quad
{\rm b}_1 \parallel {\rm a},\quad
{\rm b}_2 \perp     {\rm a},\cr
   \noalign{\vskip 3pt}
{\rm c}_2 + {\rm c}_1 = {\rm c},\quad
{\rm c}_1 \parallel {\rm a},\quad
{\rm c}_2 \perp     {\rm a},\cr
   \noalign{\vskip 3pt}
{\rm d}_2 + {\rm d}_1 = {\rm d},\quad
{\rm d}_1 \parallel {\rm a},\quad
{\rm d}_2 \perp     {\rm a},\cr}
   \right\}
   \eqno (57),$$
together with the three equations
$${\rm c} + {\rm b} = {\rm d},\quad
  {\rm c}_1 + {\rm b}_1 = {\rm d}_1,\quad
  {\rm c}_2 + {\rm b}_2 = {\rm d}_2
   \eqno (58);$$
of which the last two are deducible from the first, by the
geometrical properties of projections.  We have, therefore, by
(46),
$${{\rm c} \over {\rm a}} + {{\rm b} \over {\rm a}}
   = {{\rm d} \over {\rm a}}
   = {{\rm d}_2 \over {\rm a}} + {{\rm d}_1 \over {\rm a}}
   \eqno (59),$$
$${{\rm d}_1 \over {\rm a}}
   = {{\rm c}_1 \over {\rm a}} + {{\rm b}_1 \over {\rm a}},\quad
  {{\rm d}_2 \over {\rm a}}
   = {{\rm c}_2 \over {\rm a}} + {{\rm b}_2 \over {\rm a}}
   \eqno (60).$$

Since the three projections
${\rm b}_1$,~${\rm c}_1$,~${\rm d}_1$, are parallel to ${\rm a}$
(in that sense of the word {\it parallel\/} which does not
exclude coincidence), the three quotients in the first equation
(60) are what we have already named {\it scalars}; that is, they
are what are commonly called real numbers, positive, negative, or
zero: they are also the scalar parts of the three quotients in
the first equation (59), so that we may write
$${{\rm b}_1 \over {\rm a}} = {\rm S} {{\rm b} \over {\rm a}},\quad
  {{\rm c}_1 \over {\rm a}} = {\rm S} {{\rm c} \over {\rm a}},\quad
  {{\rm d}_1 \over {\rm a}} = {\rm S} {{\rm d} \over {\rm a}}
   \eqno (61),$$
using the letter~${\rm S}$ here, as in a former article, for the
characteristic of the operation of {\it taking the scalar part\/}
of any geometrical quotient, or fraction.  (If any confusion
should be apprehended, on other occasions, from this use of the
letter~${\rm S}$, and if the abridged word ${\rm Scal.}$ should
be thought too long, the sign $\mathop{\rm S}\limits_{\rm c}$
might be employed.)  Eliminating the four symbols
${\rm b}_1$,~${\rm c}_1$,~${\rm d}_1$,~${\rm d}$,
between the first equation (59), the first equation (60), and the
three equations (61), we obtain the result
$${\rm S} \left( {{\rm c} \over {\rm a}} + {{\rm b} \over {\rm a}} \right)
   =  {\rm S} {{\rm c} \over {\rm a}}
    + {\rm S} {{\rm b} \over {\rm a}}
   \eqno (62);$$
in which, by the foregoing article,
$\displaystyle {{\rm b} \over {\rm a}}$ and
$\displaystyle {{\rm c} \over {\rm a}}$
may represent any two geometrical fractions: so that we may write
generally
$${\rm S} \left( {{\rm h} \over {\rm g}} + {{\rm f} \over {\rm e}} \right)
   =  {\rm S} {{\rm h} \over {\rm g}}
    + {\rm S} {{\rm f} \over {\rm e}}
   \eqno (63),$$
and may enunciate in words the same result by saying, that the
{\it scalar\/} of the sum of any two such fractions is equal to
the {\it sum of the scalars}.  In like manner, the three other
projections
${\rm b}_2$,~${\rm c}_2$,~${\rm d}_2$,
being each perpendicular to ${\rm a}$, the three other partial
quotients, which enter into the second equation (60), are what we
have already called {\it vectors\/} in this paper, or more fully
they are the vector parts of the three quotients in the first
equation (59); so that we may write
$${{\rm b}_2 \over {\rm a}} = {\rm V} {{\rm b} \over {\rm a}},\quad
  {{\rm c}_2 \over {\rm a}} = {\rm V} {{\rm c} \over {\rm a}},\quad
  {{\rm d}_2 \over {\rm a}} = {\rm V} {{\rm d} \over {\rm a}}
   \eqno (64),$$
${\rm V}$ being used, as in a former article, for the
characteristic of the operation of {\it taking the vector
part\/}; we have, therefore,
$${\rm V} \left( {{\rm c} \over {\rm a}} + {{\rm b} \over {\rm a}} \right)
   =  {\rm V} {{\rm c} \over {\rm a}}
    + {\rm V} {{\rm b} \over {\rm a}}
   \eqno (65),$$
$${\rm V} \left( {{\rm h} \over {\rm g}} + {{\rm f} \over {\rm e}} \right)
   =  {\rm V} {{\rm h} \over {\rm g}}
    + {\rm V} {{\rm f} \over {\rm e}}
   \eqno (66),$$
and may assert that the {\it vector of the sum\/} of any two
geometrical fractions is equal to the {\it sum of the vectors}.
These formul{\ae} (63) and (66) are important in the present
system; they are however, as we see, only symbolical expressions
of those very simple geometrical principles from which they have
been derived, through the medium of the equations (58); namely,
the principles that, {\it whether on a line or on a plane}, the
{\it projection of a sum\/} of lines is equal to the {\it sum of
the projections}, if the word {\it sum\/} be suitably
interpreted.  The analogous interpretation of a
{\it difference\/} of lines, combined with similar
considerations, gives in like manner the formul{\ae}
$${\rm S} \left( {{\rm h} \over {\rm g}} - {{\rm f} \over {\rm e}} \right)
   =  {\rm S} {{\rm h} \over {\rm g}}
    - {\rm S} {{\rm f} \over {\rm e}}
   \eqno (67),$$
$${\rm V} \left( {{\rm h} \over {\rm g}} - {{\rm f} \over {\rm e}} \right)
   =  {\rm V} {{\rm h} \over {\rm g}}
    - {\rm V} {{\rm f} \over {\rm e}}
   \eqno (68);$$
that is to say, the {\it scalar and vector of the difference\/}
of any two geometrical fractions are respectively equal to the
{\it differences of the scalars and of the vectors\/} of those
fractions; precisely as, and because, the {\it projection of a
difference\/} of two lines, whether on a line or on a plane, is
equal to the {\it difference of the projections}.

\bigbreak

\centerline{\it Addition and Subtraction of Vectors by their
Indices.}

\nobreak\bigskip

10.
We see, then, that in order to combine by addition or subtraction
any two geometrical fractions, it is sufficient to combine
separately their scalar and their vector parts.  The former
parts, namely the scalars, are simply {\it numbers}, of the kind
called commonly real; and are to be added or subtracted among
themselves according to the usual rules of algebra.  But for
effecting with convenience the combination of the latter parts
among themselves, namely the vectors, which have been shown in a
former article to be of a kind essentially distinct from all
stages of the progression of real number from negative to
positive infinity (and therefore to be rather
{\it extra-positives\/} than either positive or
{\it contra\/}-positive numbers), it is necessary to establish
other rules: and it will be found useful for this purpose to
employ the consideration of certain connected {\it lines}, namely
the {\it indices}, of which each is determined by, and in its
turn completely characterises, that vector quotient or fraction
to which it corresponds, according to the construction assigned
in the 7th article.  If we apply the rules of that construction
to determine the indices of the vector parts of any two fractions
and of their sum, we may first, as in recent articles, reduce the
two fractions to a common denominator; and may, for simplicity,
take this denominator line~${\rm a}$ of a length equal to that
assumed unit of length which is to be employed in the
determination of the indices.  Then, having projected, as in the
last article, the new numerators ${\rm b}$ and ${\rm c}$, and
their sum~${\rm d}$, on a plane perpendicular to ${\rm a}$, and
having called these projections
${\rm b}_2$,~${\rm c}_2$,~${\rm d}_2$,
as before; we may conceive a right-handed rotation of each of
these three projected lines, through a right angle, round the
line~${\rm a}$ as a common axis, which shall transport them
without altering their lengths or relative directions, and
therefore without affecting their mutual relation as summands and
sum, into coincidence with three other lines
${\rm b}_3$,~${\rm c}_3$,~${\rm d}_3$,
such that
$${\rm d}_3 = {\rm c}_3 + {\rm b}_3
   \eqno (69);$$
and these three new lines will be the three indices required.
For a right-handed rotation through a right angle, round the
line~${\rm b}_3$ as an axis, would bring the line~${\rm a}$ into
the direction originally occupied by ${\rm b}_2$; and the length
of ${\rm b}_2$ is to the length of ${\rm a}$ in the same ratio as
the length of ${\rm b}_3$ to the assumed unit of length;
therefore ${\rm b}_3$ is, in the sense of the 7${}^{\rm th}$
article, the index of the vector quotient
$\displaystyle {{\rm b}_3 \over {\rm a}}$,
that is, the index of the vector part of the fraction
$\displaystyle {{\rm b} \over {\rm a}}$, or
$\displaystyle {{\rm f} \over {\rm e}}$;
and similarly for the indices of the two other fractions, in the
first equation (59).  We may therefore write, as consequences of
the construction lately assigned, and of the equations (49) and
(52),
$${\rm b}_3 = {\rm I} {{\rm f} \over {\rm e}};\quad
  {\rm c}_3 = {\rm I} {{\rm h} \over {\rm g}};\quad
  {\rm d}_3
   =  {\rm I}
         \left(
            {{\rm f} \over {\rm e}} + {{\rm h} \over {\rm g}}
         \right)
   \eqno (70);$$
if we agree for the present to prefix the letter~${\rm I}$ to the
symbol of a geometrical fraction, as the characteristic of the
operation of {\it taking the index of the vector part}.
Eliminating now the three symbols
${\rm b}_3$,~${\rm c}_3$,~${\rm d}_3$
between the four equations (69) and (70), we obtain this general
formula:
$${\rm I} \left( {{\rm h} \over {\rm g}} + {{\rm f} \over {\rm e}} \right)
   =  {\rm I} {{\rm h} \over {\rm g}}
    + {\rm I} {{\rm f} \over {\rm e}}
   \eqno (71),$$
which may be thus enunciated: the {\it index of the vector part
of the sum\/} of any two geometrical fractions is equal to the
{\it sum of the indices\/} of the vector parts of the summands.
Combining this result with the formula (63), which expresses that
the scalar of the sum is the sum of the scalars, we see that the
complex {\it operation of adding any two geometrical fractions},
of which each is determined by its scalar and by the index of its
vector part, may be in general {\it decomposed into two\/} very
simple but {\it essentially distinct operations\/}; namely,
{\it first}, the operation of adding together {\it two numbers},
positive or negative or null, so as to obtain a third number for
their sum, according to the usual rules of elementary algebra;
and {\it second}, the operation of adding together {\it two
lines\/} in space, so as to obtain a third line, according to the
geometrical rules of the composition of motions, or by drawing
the diagonal of a parallelogram.  In like manner the operation of
{\it taking the difference\/} of two fractions may be decomposed
into the two operations of taking separately the difference of
two numbers, and the difference of two lines; for we can easily
prove that
$${\rm I} \left( {{\rm h} \over {\rm g}} - {{\rm f} \over {\rm e}} \right)
   =  {\rm I} {{\rm h} \over {\rm g}}
    - {\rm I} {{\rm f} \over {\rm e}}
   \eqno (72);$$
or, in words, that the {\it index\/} (of the vector part) {\it of
the difference\/} of any two fractions is equal to the {\it
difference of the indices}.  And because it has been seen that
not only for numbers but also for lines, considered amongst
themselves, any number of summands may be in any manner grouped
or transposed without altering the sum; and that the sum of a
scalar and a vector is equal to the sum of the same vector and the
same scalar, combined in a contrary order; it follows that the
{\it addition\/} of any number of geometrical fractions is an
{\it associative\/} and also a {\it commutative\/} operation: in
such a manner that we may now write
$${{\rm h} \over {\rm g}} + {{\rm f} \over {\rm e}}
   =  {{\rm f} \over {\rm e}} + {{\rm h} \over {\rm g}};\quad
{{\rm k} \over {\rm i}}
    + \left( {{\rm h} \over {\rm g}} + {{\rm f} \over {\rm e}} \right)
   =  \left( {{\rm k} \over {\rm i}} + {{\rm h} \over {\rm g}} \right)
    + {{\rm f} \over {\rm e}}
   =  {{\rm f} \over {\rm e}}
    + {{\rm h} \over {\rm g}}
    + {{\rm k} \over {\rm i}},
   \hbox{ \&c.}
   \eqno (73),$$
whatever straight lines in space may be denoted by
${\rm e}$, ${\rm f}$, ${\rm g}$, ${\rm h}$, ${\rm i}$, ${\rm k}$,
\&c.  We may also write, concisely
$${\rm S} {\textstyle\sum} = {\textstyle\sum} {\rm S};\quad
  {\rm V} {\textstyle\sum} = {\textstyle\sum} {\rm V};\quad
  {\rm I} {\textstyle\sum} = {\textstyle\sum} {\rm I}
   \eqno (74);$$
$${\rm S} \Delta = \Delta {\rm S};\quad
  {\rm V} \Delta = \Delta {\rm V};\quad
  {\rm I} \Delta = \Delta {\rm I}
   \eqno (75);$$
using $\sum$, $\Delta$ as the characteristics of sum and
difference, while ${\rm S}$, ${\rm V}$ and ${\rm I}$ are still
the signs of scalar, vector, index.

\bigbreak

\centerline{\it Separation of the Scalar and Vector Parts of the
Product of any two}
\centerline{\it Geometrical Fractions.}

\nobreak\bigskip

11.
The definitions (46), (47) of addition and multiplication of
fractions, namely
$${{\rm c} \over {\rm a}} + {{\rm b} \over {\rm a}}
   =  {{\rm c} + {\rm b} \over {\rm a}},\quad
  {{\rm c} \over {\rm a}} \times {{\rm a} \over {\rm b}}
   =  {{\rm c} \over {\rm b}},$$
give obviously, for any 4 straight lines
${\rm a}$, ${\rm b}$, ${\rm c}$, ${\rm a}'$, the formula
$$    \left( {{\rm c} \over {\rm a}} + {{\rm b} \over {\rm a}} \right)
      \times {{\rm a} \over {\rm a}'}
   =  {{\rm c} + {\rm b} \over {\rm a}'}
   =  \left( {{\rm c} \over {\rm a}} \times {{\rm a} \over {\rm a}'} \right)
    + \left( {{\rm b} \over {\rm a}} \times {{\rm a} \over {\rm a}'} \right)
   \eqno (76);$$
and this other formula of the same kind,
$${{\rm a}' \over {\rm a}} \times
      \left( {{\rm c} \over {\rm a}} + {{\rm b} \over {\rm a}} \right)
   =  {{\rm a'} \over \displaystyle
         {{\rm a} \over {\rm c} + {\rm b}} \times {\rm a}}
   =  \left( {{\rm a}' \over {\rm a}} \times {{\rm c} \over {\rm a}} \right)
    + \left( {{\rm a}' \over {\rm a}} \times {{\rm b} \over {\rm a}} \right)
   \eqno (77),$$
may be proved without difficulty to be a consequence of the same
definitions; the operation of multiplying a line, by the quotient
of two others with which it is co-planar, being interpreted by
the definition (23), so as to give, in the present notation,
$${{\rm e} \over {\rm a}} \times {\rm a} = {\rm e}
   \eqno (78).$$
In fact, if we assume, as we may, seven new lines
${\rm d} {\rm b}' {\rm c}' {\rm d}' {\rm b}'' {\rm c}'' {\rm d}''$,
so as to satisfy the seven conditions
$$\left. \eqalign{
{\rm c} + {\rm b} = {\rm d},\quad
{{\rm b} \over {\rm a}} = {{\rm a} \over {\rm b}'},\quad
{{\rm c} \over {\rm a}} = {{\rm a} \over {\rm c}'},\quad
{{\rm d} \over {\rm a}} = {{\rm a} \over {\rm d}'},\cr
{{\rm b}'' \over {\rm a}'} = {{\rm a}' \over {\rm b}'},\quad
{{\rm c}'' \over {\rm a}'} = {{\rm a}' \over {\rm c}'},\quad
{{\rm d}'' \over {\rm a}'} = {{\rm a}' \over {\rm d}'},\cr}
   \right\}
   \eqno (79),$$
we shall have the first member of the formula (77) equal to
$\displaystyle
      {{\rm a}' \over {\rm a}} \times {{\rm a} \over {\rm d}'}
   = {{\rm a}' \over {\rm d}'}
   =$
the second member of that formula; it will therefore be equal to
$\displaystyle {{\rm d}'' \over {\rm a}'}$,
and consequently will be shown to be
$\displaystyle
   =  {{\rm c}'' \over {\rm a}'} + {{\rm d}'' \over {\rm a}'}
   =  {{\rm a}' \over {\rm c}'} + {{\rm a}' \over {\rm b}'}
   =$
the third member of that formula, if we can show that the
conditions (79) give the relation
$${\rm d}'' = {\rm c}'' + {\rm b}''
   \eqno (80).$$

Now those conditions show that the line ${\rm a}$ is common to
the planes of ${\rm b}$,~${\rm b}'$, and ${\rm c}$,~${\rm c}'$,
and that it bisects the angle between ${\rm b}$ and ${\rm b}'$,
and also the angle between ${\rm c}$ and ${\rm c}'$; therefore
the mutual inclination of the lines ${\rm b}'$ and ${\rm c}'$ is
equal to the mutual inclination of ${\rm b}$ and ${\rm c}$; while
the lengths of the two former lines are, by the same conditions,
inversely proportional to those of the two latter.  An on
pursuing this geometrical reasoning, in combination with the
definitional meanings of the symbolic equations (79), it appears
easily that the mutual inclinations of the lines
${\rm b}''$,~${\rm c}''$,~${\rm d}''$,
are equal to those of
${\rm b}'$,~${\rm c}'$,~${\rm d}'$,
and therefore to those of
${\rm b}$,~${\rm c}$,~${\rm d}$;
while the lengths of
${\rm b}''$,~${\rm c}''$,~${\rm d}''$
are inversely proportional to those of
${\rm b}'$,~${\rm c}'$,~${\rm d}'$,
therefore directly proportional to the lengths of
${\rm b}$,~${\rm c}$,~${\rm d}$:
since then the line~${\rm d}$ is the symbolic sum of ${\rm b}$
and ${\rm c}$, or the diagonal of a parallelogram described with
those two lines as adjacent sides, it follows that the
line~${\rm d}''$ is similarly related to ${\rm b}''$ and
${\rm c}''$, or that the relation (80) holds good.  The formula
(77) is therefore shown to be true: and although we have not
{\it yet\/} proved that the multiplication of two geometrical
fractions is {\it always\/} a {\it distributive\/} operation, we
see at least that either factor may be distributed into two
partial factors, and that the sum of the two partial products
will give the total product, whenever either total factor and the
two parts of the other factor are {\it co-linear\/}; that is,
whenever the planes of these three fractions are {\it parallel to
any common line}, such as the line~${\rm a}$ in the formul{\ae}
(76) (77): the {\it plane\/} of a geometrical fraction being one
which contains or is parallel to the numerator and denominator
thereof.  A {\it scalar\/} fraction, being the quotient of two
parallel lines, of which either may be transported without
altering its direction to any other position in space while both
may revolve together, may be regarded as having an entirely
{\it indeterminate plane}, which may thus be rendered parallel to
any arbitrary line; we shall therefore always satisfy the
condition of {\it co-linearity}, by distributing either or both
of two factors into their scalar and vector parts, and may
consequently write,
$$\eqalignno{
{{\rm h} \over {\rm g}} \times {{\rm f} \over {\rm e}}
   &= \left(
         {\rm V} {{\rm h} \over {\rm g}} \times {{\rm f} \over {\rm e}}
      \right)
    + \left(
         {\rm S} {{\rm h} \over {\rm g}} \times {{\rm f} \over {\rm e}}
      \right) \cr
   &= \left(
         {{\rm h} \over {\rm g}} \times {\rm V} {{\rm f} \over {\rm e}}
      \right)
    + \left(
         {{\rm h} \over {\rm g}} \times {\rm S} {{\rm f} \over {\rm e}}
      \right) \cr
   &= \left(
         {\rm V} {{\rm h} \over {\rm g}}
         \times {\rm V} {{\rm f} \over {\rm e}}
      \right)
    + \left(
         {\rm V} {{\rm h} \over {\rm g}}
         \times {\rm S} {{\rm f} \over {\rm e}}
      \right)
    + \left(
         {\rm S} {{\rm h} \over {\rm g}}
         \times {\rm V} {{\rm f} \over {\rm e}}
      \right)
    + \left(
         {\rm S} {{\rm h} \over {\rm g}}
         \times {\rm S} {{\rm f} \over {\rm e}}
      \right)
   & (81);\cr}$$
or more concisely,
$$(\beta + b) (\alpha + a)
   =  \beta \alpha + \beta a + b \alpha + b a
   \eqno (82),$$
if we denote, as in a former article, vectors by greek and
scalars by italic letters, and omit the mark of multiplication
between any two successive letters of these two kinds, or between
sums of such letters, when those sums are enclosed in
parentheses.  But the multiplication of scalars is effected, as
we have seen, by the ordinary rules of algebra; and to multiply a
vector by a scalar, or a scalar by a vector, is easily shown, by
the definitions already laid down, to be equivalent to
multiplying by the scalar, on the plan of the sixth article,
either the index or the numerator of the vector, without altering
the denominator of that vector: thus, in the second member of
(82), the term $ba$ is a known scalar, and the terms $b \alpha$,
$\beta a$ are known vectors, if the partial factors $a$, $b$,
$\alpha$, $\beta$ be known: in order therefore to apply the
equation (82), which in its form agrees with ordinary algebra, to
any question of multiplication of any two geometrical fractions,
it is sufficient to know how to interpret generally the remaining
term $\beta \alpha$, or the product of one vector by another.
For this purpose we may always conceive the index
${\rm I} \beta$ of the vector~$\beta$ to be the sum of two other
indices, which shall be respectively parallel and perpendicular
to the index ${\rm I} \alpha$ of the other vector~$\alpha$, as
follows:
$${\rm I} \beta'  \parallel {\rm I} \alpha,\quad
  {\rm I} \beta'' \perp     {\rm I} \alpha,\quad
  {\rm I} \beta'' + {\rm I} \beta' = {\rm I} \beta
   \eqno (83);$$
and then the vector~$\beta$ itself will be, by the last article,
the sum of the two new vectors $\beta'$ and $\beta''$, and the
planes of these two new vector fractions will be respectively
parallel and perpendicular to the plane of the vector
fraction~${\rm \alpha}$; consequently, the three fractions
$\beta'$,~$\beta''$,~$\alpha$ will be co-linear, and we shall
have, by the principle (76),
$$\beta \alpha
   =  (\beta' + \beta'') \alpha
   =  \beta' \alpha + \beta'' \alpha
   \eqno (84).$$

The problem of the multiplication of {\it any two\/} vectors is
thus decomposed into the two simpler problems, of multiplying
first {\it two parallel}, and secondly {\it two rectangular,
vectors\/} together.  If then we merely wish to separate the
scalar and the vector parts, it is sufficient to observe that if,
in the general formula (47), for the multiplication of any two
fractions, we suppose the factors to be parallel vectors, then
the line~${\rm a}$ is perpendicular to both ${\rm b}$ and ${\rm
c}$, and is also co-planar with them, so that they are
necessarily parallel to each other, and the product
$\displaystyle {{\rm c} \over {\rm b}}$
is a scalar; but if, in the same general formula, we suppose the
factors to be rectangular vectors, then the three lines
${\rm a}$,~${\rm b}$,~${\rm c}$
are themselves mutually rectangular, and the product of the
fractions is a vector.  Thus, in the formula (84), the partial
product $\beta' \alpha$ is a scalar, but the other partial
product $\beta'' \alpha$ is a vector; and we may write
$${\rm S} \mathbin{.} \beta \alpha
   =  \beta' \alpha;\quad
  {\rm V} \mathbin{.} \beta \alpha
   =  \beta'' \alpha
   \eqno (85).$$

We may therefore, more generally, under the conditions (83),
decompose the formula of multiplication (82) into the two
following equations:
$$\left. \eqalign{
{\rm S} \mathbin{.} (\beta + b) (\alpha + a)
   &= \beta' \alpha + b a;\cr
{\rm V} \mathbin{.} (\beta + b) (\alpha + a)
   &= \beta'' \alpha + \beta a + b \alpha \cr}
   \right\}
   \eqno (86).$$

Or we may write, for abridgment,
$$c = \beta' \alpha + ba;\quad
  \gamma = \beta'' \alpha + \beta a + b \alpha
   \eqno (87);$$
and then we shall have this other equation of multiplication,
$$\gamma + c = (\beta + b) (\alpha + a)
   \eqno (88).$$

And thus the general {\it separation of the scalar and vector
parts\/} of the product of any two geometrical fractions may be
effected.  But it seems proper to examine more closely into the
separate meanings of the two partial products of vectors, denoted
here by the two terms $\beta' \alpha$ and $\beta'' \alpha$; which
will be done in the two following articles.

\bigbreak

\centerline{\it Products of two Parallel Vectors; Geometrical
Representations of the}
\centerline{\it Square Roots of Negative Scalars.}

\nobreak\bigskip

12.
It was shown, in the last article, that the product of any two
parallel vectors, such as $\alpha$ and $\beta'$, that is, the
product of any two vectors of which the planes or the indices are
parallel, is equal to a scalar.  By pursuing the reasoning of
that article, it is easy to show, farther, that this {\it scalar
product of two parallel vectors\/} is equal to the {\it product
of the numbers\/} which express the lengths of the two parallel
indices; this numerical product being taken with a {\it
negative\/} or with a {\it positive\/} sign, according as these
indices are {\it similar\/} or {\it opposite\/} in direction.  In
fact, in the general formula
$\displaystyle
      {{\rm c} \over {\rm a}} \times {{\rm a} \over {\rm b}}
   =  {{\rm c} \over {\rm b}}$,
we have now ${\rm b} \perp {\rm a}$, ${\rm c} \parallel {\rm b}$;
the length of ${\rm c}$ is to the length of ${\rm b}$, in a ratio
compounded of the length of ${\rm c}$ to that of ${\rm a}$, and
of the ratio of the length of ${\rm a}$ to that of ${\rm b}$; and
the direction of ${\rm c}$ is opposite or similar to that of
${\rm b}$, according as the two quadrantal rotations in one
common plane, from ${\rm b}$ to ${\rm a}$, and from ${\rm a}$ to
${\rm c}$, are performed right-handedly round the same index, or
round opposite indices.

We know then perfectly how to interpret the product of any two
parallel vectors; and, as a case of such interpretation, if we
agree to say that the product of any two equal fractions is the
{\it square\/} of either, and to write
$${{\rm b} \over {\rm a}} \times {{\rm b} \over {\rm a}}
   =  \left( {{\rm b} \over {\rm a}} \right)^2
   \eqno (89),$$
whatever two lines may be denoted by ${\rm a}$ and ${\rm b}$, we
see that, in the present system, the {\it square of a vector is
always a negative scalar}, namely the negative of the square of
the number which denotes the length of the index of the vector;
in such a manner that, for any vector~$\alpha$, we shall have the
equation
$$\alpha^2 = - \overline{\alpha}^2
   \eqno (90),$$
if we agree to denote by the symbol $\overline{\alpha}$ that
positive or absolute number which expresses the {\it length of
the index\/} ${\rm I} \alpha$.  We have then, reciprocally,
$$\overline{\alpha}^2 = - \alpha^2
   \eqno (91);$$
and may therefore write
$$\overline{\alpha} = \surd ( -\alpha^2 )
   \eqno (92),$$
$-\alpha^2$being here a positive number (because $\alpha^2$ is
negative), and $\surd (- \alpha^2)$ being its positive or
absolute {\it square root}, which is an entirely {\it
determined\/} (and real) {\it number}, when the vector~$\alpha$,
or even when the length of its index, is determined.  But
although we might be led to write, in like manner, from (90), the
equation
$$\alpha = \surd ( - \overline{\alpha}^2 )
   \eqno (93),$$
yet the same principles prove that this expression, which may
denote generally any {\it square root of a negative number}, by a
suitable choice of the positive number~$\overline{\alpha}$, is
equal to a {\it vector\/} $\alpha$, of which the index
${\rm I} \alpha$ has indeed a {\it determined length}, but has an
entirely {\it undetermined direction\/}; the symbol in the second
member of the equation (93) may therefore receive (in the present
system) infinitely many different geometrical representations, or
constructions, though they have all one common character; and it
will be a little more consistent with the analogies of ordinary
algebra to write the equation under the form
$$\alpha = (- \overline{\alpha}^2 )^{1 \over 2}
   \eqno (94),$$
using a fractional exponent which suggests a certain degree of
indeterminateness, rather than a radical sign which it is often
convenient to restrict to one determined value.  Thus, for
example, the symbol $(-1)^{1 \over 2}$ or the {\it square root of
negative unity}, will, in the present system, denote, or be
geometrically constructed by, {\it any vector of which the index
is equal to the unit of length\/}; that is, any geometrical
fraction of which the numerator and the denominator are lines
equal to each other in length, but perpendicular to each other in
direction.  And we see that the geometrical principle, on which
this conclusion ultimately depends, is simply this: that {\it two
successive and similar quadrantal rotations, in any arbitrary
plane, reverse the direction\/} of any straight line in that
plane.  Mr.~Warren, confining himself to the consideration of
lines in {\it one fixed plane}, has been led to attribute to his
geometrical representations of the square roots of negative
numbers, {\it one fixed direction}, or rather axis, perpendicular
to that other axis on which he represents square roots of
positive numbers.  And other authors, both before and since the
publication of Mr.~Warren's work,\footnote*{{\it Treatise on the
Geometrical Representation of the Square Roots of Negative
Quantities}, by the Rev.\ John Warren, Cambridge, 1828.  See also
Dr.~Peacock's Treatises on Algebra, and his Report to the
British Association, containing references to other works.}
seem to have been in like manner disposed to represent positive
or negative numbers by lines in some one direction, or in the
direction opposite, but symbols of the form $a \surd (-1)$ by
lines perpendicular thereto.  Such is at least the impression on
the mind of the present writer, produced perhaps by an
insufficient acquaintance with the works of those who have
already written on this class of subjects.  It will however be
attempted to show, in a future article of this paper, that the
geometrical fractions which have been called {\it vectors}, in
the present and in former articles, may be symbolically equated
to their own indices; and that thus {\it every straight line
having direction in space\/} may properly be looked upon {\it in
the present system\/} as {\it a geometrical representation of a
square root of a negative number\/}; while positive and negative
numbers are in the same system regarded indeed as belonging to
one common {\it scale\/} of progression, from $-\infty$ to
$+\infty$, but to a scale which is not to be considered as having
any one direction rather than any other, in tridimensional space.


\bigbreak

\centerline{\it Products of two Rectangular Vectors;
Non-commutativeness of the Factors, in the general}
\centerline{\it Multiplication of two
Geometrical Fractions.}

\nobreak\bigskip

13.
The reasoning by which it was shown, in the 11th article, that
the {\it product $\beta'' \alpha$ of any two rectangular
vectors}, $\alpha$ and $\beta''$, is {\it itself a vector}, may
be continued so as to show that the number expressing the length
of the index of this vector product is the product of the numbers
which express the lengths of the indices of the factors; or that,
in a notation similar to one employed in the last article,
$$\overline{\beta'' \alpha}
   =  \overline{\beta''} \overline{\alpha},
   \quad \hbox{when, }
  {\rm I} \beta'' \perp {\rm I} \alpha
   \eqno (95);$$
and therefore that, by the principle (92), for the same case of
{\it rectangular vectors}, we have the formula
$$\surd \{ - ( \beta'' \alpha )^2 \}
   = \surd ( - \beta''^2 ) \surd ( - \alpha^2 )
   \eqno (96).$$
Also in the general formula of multiplication
$\displaystyle
   {{\rm c} \over {\rm a}} \times {{\rm a} \over {\rm b}}
   =  {{\rm c} \over {\rm b}}$,
the three lines ${\rm a}$,~${\rm b}$,~${\rm c}$ compose here a
rectangular system; and therefore the {\it index of the product\/}
is parallel to the line~${\rm a}$, and is consequently {\it
perpendicular to the indices of the two factors\/};
${\rm I} \mathbin{.} \beta'' \alpha$
is therefore perpendicular to both ${\rm I} \beta''$ and
${\rm I} \alpha$; a conclusion which may be extended by (83) and
(85) to the multiplication of {\it any two vectors}, so that we
may write generally,
$${\rm I} \mathbin{.} \beta \alpha \perp {\rm I} \beta;\quad
  {\rm I} \mathbin{.} \beta \alpha \perp {\rm I} \alpha
   \eqno (97).$$
Again, we are allowed to suppose, in applying the same general
formula of multiplication to the same case of rectangular
vectors, that the index ${\rm I} \alpha$ of the multiplicand
$\displaystyle {{\rm a} \over {\rm b}}$
is not only parallel to the line~${\rm c}$, but similar (and not
opposite) in direction to that line; in such a manner that the
rotation round ${\rm c}$ from ${\rm b}$ to ${\rm a}$ is positive:
and then the rotation round ${\rm b}$ from ${\rm a}$ to ${\rm c}$
is positive, and so is the rotation round ${\rm a}$ from
${\rm c}$ to ${\rm b}$, and also that round $- {\rm a}$ from
${\rm b}$ to ${\rm c}$; therefore the index ${\rm I} \beta''$ of
the multiplier is similar in direction to $+ {\rm b}$, and the
index ${\rm I} \mathbin{.} \beta'' \alpha$ of the product is
similar in direction to $- {\rm a}$; consequently {\it the
rotation round the index of the product, from the index of the
multiplier to that of the multiplicand, is positive}.  And
although this last result has only been proved here for the case
of two rectangular vectors, yet it may easily be shown, by the
principles of the 11th article, to extend to the multiplication
of two general fractions.  For, in the notation of that article,
$\gamma$ denoting the vector part of the product of any two such
fractions, we have, by (87)
$${\rm I} \gamma
   =  {\rm I} \mathbin{.} \beta'' \alpha
       + a {\rm I} \beta + b {\rm I} \alpha
   \eqno (98);$$
${\rm I} \gamma$ is therefore the symbolic sum of
${\rm I} \mathbin{.} \beta'' \alpha$
and of two other lines which are respectively parallel to the
indices of the vector parts of the two factors, and which
consequently have their sum co-planar with those indices, and
therefore also co-planar, by (83), with ${\rm I} \beta''$ and
${\rm I} \alpha$; consequently ${\rm I} \gamma$ and
${\rm I} \mathbin{.} \beta'' \alpha$
both lie at the same side of the plane of ${\rm I} \alpha$ and
${\rm I} \beta''$; and therefore the rotation round
${\rm I} \gamma$, like that around
${\rm I} \mathbin{.} \beta'' \alpha$,
from ${\rm I} \beta''$ to ${\rm I} \alpha$, and consequently from
${\rm I} \beta$ to ${\rm I} \alpha$, is positive.  Hence also the
rotation round ${\rm I} \beta$ from ${\rm I} \alpha$ to
${\rm I} \gamma$ is positive; that is to say, in the
multiplication of two general geometrical fractions, {\it the
rotation round the index of the vector part of the multiplier,
from that of the multiplicand to that of the product, is
positive\/}; from which may immediately be deduced a remarkable
consequence, already alluded to by anticipation in the 8th
article, namely---that the {\it multiplication of two general
geometrical fractions is not a commutative operation}, or that
the {\it order of the factors is not in general indifferent\/};
since the index of the vector part of the product lies at one or
at another side of the plane of the indices of the vector parts
of the two factors, according as those factors are taken in one
or in the other order.  We have, for example, by the present
article, a relation of {\it opposition\/} of signs between the
products of two {\it rectangular\/} vectors, taken in two
opposite orders; which relation may be expressed by the following
{\it equation of perpendicularity},
$$\alpha \beta'' = - \beta'' \alpha,
   \quad \hbox{when, }
  {\rm I} \beta'' \perp {\rm I} \alpha
   \eqno (99).$$
But in the case where the indices of the vector parts $\alpha$
and $\beta$ of two fractional factors are {\it parallel\/} which
includes the case where either of those indices vanishes, the
corresponding factor becoming then a scalar), the part $\beta''$
of the vector $\beta$ vanishes, and the latter vector reduces
itself by (83) to its other part $\beta'$; so that in {\it
this\/} case, by the results of the last article, the orders of
the factors is indifferent, and the operation of multiplication
is commutative: and thus we may write, as the {\it equation of
parallelism\/} between two vectors,
$$\alpha \beta' = \beta' \alpha,
   \quad \hbox{when, }
  {\rm I} \beta' \parallel {\rm I} \alpha
   \eqno (100).$$

It is easy to infer hence, by (84) and (77), that in the more
general case of the multiplication of any two vectors $\alpha$ and
$\beta$, we may write, instead of (85), the following formul{\ae}
for the separation of the scalar and vector parts of the product:
$$\left. \eqalign{
{\rm S} \mathbin{.} \beta \alpha
   &= {\textstyle {1 \over 2}} (\beta \alpha + \alpha \beta)
    = \mathbin{\phantom{-}} {\rm S} \mathbin{.} \alpha \beta \cr
{\rm V} \mathbin{.} \beta \alpha
   &= {\textstyle {1 \over 2}} (\beta \alpha - \alpha \beta)
    = - {\rm V} \mathbin{.} \alpha \beta \cr}
   \right\}
   \eqno (101),$$
with corresponding formul{\ae} instead of (86), which give
$$(\beta + b) (\alpha + a) - (\alpha + a) (\beta + b)
   =  \beta \alpha - \alpha \beta
   \eqno (102),$$
the second member of this last equation being a vector different
from $0$, unless it happen that the planes (or the indices) of
the vectors $\alpha$ and $\beta$ are parallel to each other.
Finally, we may here observe that in virtue of the principles and
definitions already laid down, {\it the length of the index\/}
(${\rm I} \mathbin{.} \beta \alpha$) {\it of the vector part of
the product of any two vectors bears to the unit of length the
same ratio which the area of the parallelogram under the
indices\/} (${\rm I} \beta$ and ${\rm I} \alpha$) {\it of the
factors bears to the unit of area\/} the {\it direction\/} of
this index of the product being also (as we have seen) {\it
perpendicular to the plane\/} of the indices of the factors, and
therefore to the plane of the parallelogram under them; and being
changed to its own {\it opposite\/} when the order of the factors
is inverted, which {\it inversion\/} of their order may be
considered as corresponding to a {\it reversal of the face\/} of
the parallelogram: for all which reasons, there appears to be a
propriety in considering this index of the vector part of a
product of any two vectors as a symbolical representation of this
parallelogram under the indices of the factors, and in writing
the symbolical equation
$${\rm I} \mathbin{.} \beta \alpha
   = \parallelogram ({\rm I} \beta, {\rm I} \alpha)
   \eqno (103).$$

It will be remembered that the indices
${\rm I} (\beta + \alpha)$, ${\rm I} (\beta - \alpha)$,
of the sum and difference of the same two vectors, are
symbolically equal to two different diagonals of the same
parallelogram, by former articles of this paper.

\bigbreak

\centerline{\it On the Distributive Character of the Operation of
Multiplication, as performed generally}
\centerline{\it on Geometrical Fractions.}

\nobreak\bigskip

14.
We are now prepared to extend the formul{\ae} (76), (77),
respecting the multiplication of sums of geometrical fractions;
and to shew that similar results hold good, even when the
conditions of colinearity, assumed in those two formul{\ae}, is
no longer supposed to be satisfied.  That is, the two equations
$$    \left(
         {{\rm h} \over {\rm g}}
       + {{\rm f} \over {\rm e}}
      \right)
      \times
      {{\rm k} \over {\rm i}}
   =  \left(
         {{\rm h} \over {\rm g}}
         \times
         {{\rm k} \over {\rm i}}
      \right)
      +
      \left(
         {{\rm f} \over {\rm e}}
         \times
         {{\rm k} \over {\rm i}}
      \right)
   \eqno (104),$$
$$    {{\rm k} \over {\rm i}}
      \times
      \left(
         {{\rm h} \over {\rm g}}
       + {{\rm f} \over {\rm e}}
      \right)
   =  \left(
         {{\rm k} \over {\rm i}}
         \times
         {{\rm h} \over {\rm g}}
      \right)
      +
      \left(
         {{\rm k} \over {\rm i}}
         \times
         {{\rm f} \over {\rm e}}
      \right)
   \eqno (105),$$
can both be shown to be true, whatever may be the lengths and
directions of the six lines ${\rm e}$, ${\rm f}$, ${\rm g}$,
${\rm h}$, ${\rm i}$, ${\rm k}$; although, by the general
non-commutativeness of geometrical fractions as factors, which
was pointed out in the last article, the expressions contained
in these two equations are not to be confounded with each other.

Making for this purpose
$$\left. \matrix{
\displaystyle
   {{\rm f} \over {\rm e}} = \beta_1 + b_1,\quad
   {{\rm h} \over {\rm g}} = \beta_2 + b_2,\quad
   {{\rm k} \over {\rm i}} = \alpha + a,\cr
   \noalign{\vskip 3pt}
{\rm I} \beta_1'  \parallel {\rm I} \alpha,\quad
{\rm I} \beta_1'' \perp     {\rm I} \alpha,\quad
{\rm I} \beta_1'' + {\rm I} \beta_1' = {\rm I} \beta_1,\cr
   \noalign{\vskip 3pt}
{\rm I} \beta_2'  \parallel {\rm I} \alpha,\quad
{\rm I} \beta_2'' \perp     {\rm I} \alpha,\quad
{\rm I} \beta_2'' + {\rm I} \beta_2' = {\rm I} \beta_2,\cr
   \noalign{\vskip 3pt}
\beta_2'  + \beta_1'  = \beta',\quad
\beta_2'' + \beta_1'' = \beta'',\quad
\beta_2   + \beta_1   = \beta,\quad
b_2 + b_1 = b,\cr}
   \right\}
   \eqno (106),$$
the conditions (83) will be satisfied; and if we still assign to
$\gamma$ and $c$ the meanings (87), the equation (88) will hold
good, and $\gamma + c$ will be an expression for the first member
of (104).  Making also, in imitation of (87),
$$\left.
\eqalign{
c_1 &= \beta_1' \alpha + b_1 a,\cr
c_2 &= \beta_2' \alpha + b_2 a,\cr}
   \quad
\eqalign{
\gamma_1 &= \beta_1'' \alpha + \beta_1 a + b_1 \alpha,\cr
\gamma_2 &= \beta_2'' \alpha + \beta_2 a + b_2 \alpha,\cr}
   \right\}
   \eqno (107),$$
the second member of the same equation (104) becomes, by the
principles of the 11${}^{\rm th}$ article,
$(\gamma_2 + c_2) + (\gamma_1 + c_1)$;
and the equation resolves itself into the two following,
$$c = c_2 + c_1,\quad
  \gamma = \gamma_2 + \gamma_1
   \eqno (108);$$
which are easily seen to reduce themselves to these two,
$$(\beta_2' + \beta_1') \alpha
   =  \beta_2' \alpha + \beta_1' \alpha;\quad
  (\beta_2'' + \beta_1'') \alpha
   =  \beta_2'' \alpha + \beta_1'' \alpha
   \eqno (109);$$
the one being an equation between scalars, and the other between
vectors.  In like manner the equation (105) may be shown to
depend on the two following equations, less general than itself,
but of the same form,
$$\alpha (\beta_2' + \beta_1')
   =  \alpha \beta_2' + \alpha \beta_1';\quad
  \alpha (\beta_2'' + \beta_1'')
   =  \alpha \beta_2'' + \alpha \beta_1''
   \eqno (110).$$
And since, by (101), the three scalar products in the equations
(110) are respectively equal, and the three vector products are
respectively opposite (in their signs) to the corresponding
products in the equations (109), it is sufficient to prove either
of these two pairs of equations; for example, the pair (110).
Now the first equation of this pair is true, because the scalars
denoted by the three products
$\alpha \beta_1'$, $\alpha \beta_2'$,
$\alpha (\beta_2' + \beta_1')$,
are proportional, both in their magnitudes and in their signs, to
the indices of the parallel vectors $\beta_1'$, $\beta_2'$,
$\beta_2' + \beta_1'$; and the second equation of the same pair
is true, because the indices of the vectors denoted by the three
other products
$\alpha \beta_1''$, $\alpha \beta_2''$,
$\alpha (\beta_2'' + \beta_1'')$
are formed from the indices of the three coplanar vectors
$\beta_1''$, $\beta_2''$, $\beta_2'' + \beta_1''$,
by causing the three latter indices to revolve together, as one
system, in their common plane, round the index ${\rm I} \alpha$,
their lengths being at the same time changed (if at all) in one
common ratio, namely, in that of $\overline{\alpha}$ to $1$.
The formul{\ae} (104) (105) are therefore proved to be true; and
the same reasoning shows, that in any multiplication of two
geometrical fractions, either of the fractions may be {\it
distributed\/} into {\it any number\/} of parts, and that the sum
of the partial products will be equal to the total product: so
that we may write, generally,
$$\left( {\textstyle\sum} {{\rm k} \over {\rm i}} \right)
      \times \left( {\textstyle\sum} {{\rm f} \over {\rm e}} \right)
   =  {\textstyle\sum}
      \left(
         {\textstyle\sum} {{\rm k} \over {\rm i}}
         \times
         {\textstyle\sum} {{\rm f} \over {\rm e}}
      \right)
   \eqno (111).$$

The {\it multiplication of geometrical fractions\/} is therefore
a {\it distributive operation;} although it has been shown to be
{\it not\/}, in general, a {\it commutative\/} one.

\bigbreak

\centerline{\it On the Associative Property of the Multiplication
of Geometrical Fractions.}

\nobreak\bigskip

15.
Proceeding now, with the help of the distributive property
established in the last article, and of the principle that a
product is multiplied by a scalar when any one of its factors is
multiplied thereby, to prove that the multiplication of
geometrical fractions is general an {\it associative\/}
operation, or that the formula
$${{\rm k} \over {\rm i}}
      \times
      \left(
         {{\rm h} \over {\rm g}}
         \times
         {{\rm f} \over {\rm e}}
      \right)
   =  \left(
         {{\rm k} \over {\rm i}}
         \times
         {{\rm h} \over {\rm g}}
      \right)
      \times
      {{\rm f} \over {\rm e}}
   \eqno (112),$$
holds good for {\it any three fractions\/} (with other
formul{\ae} of the same sort for more fractional factors than
three), it will be sufficient to prove that the formula is true
for {\it any three vectors\/}; or that we may write generally
$$\gamma \times \beta \alpha = \gamma \beta \times \alpha
   \eqno (113),$$
the vector~$\gamma$ being not here obliged to satisfy the
equation (87); we may even content ourselves with proving that
the equation (113) is true in the two following cases, namely
first, when any two of the three vectors are parallel; and
secondly, when all three are rectangular to each other.  The
first case may be expressed by the three following equations as
its types---
$$\beta \times \beta \alpha = \beta \beta \times \alpha
   \eqno (114),$$
$$\beta \times \alpha \beta = \beta \alpha \times \beta
   \eqno (115),$$
$$\alpha \times \beta \beta = \alpha \beta \times \beta
   \eqno (116);$$
and the second case may be expressed by the equation
$$\alpha \beta \times \beta \alpha
   = (\alpha \beta \times \beta) \times \alpha,
      \quad\hbox{when } \beta \perp \alpha
   \eqno (117);$$
because, under this last condition, $\alpha \beta$ is, by
Art.~13, a vector, rectangular to both $\alpha$ and $\beta$.
Under the same condition we may, by (99), change $\alpha \beta$
to $- \beta \alpha$; therefore the first member of the equation
(117) may be equated to $- (\beta \alpha)^2$, and consequently, by
(96), to
$(-\beta^2) \times (-\alpha^2)
   =  \beta^2 \times \alpha^2
   = \beta^2 \alpha \times \alpha
   = (\alpha \times \beta \beta) \times \alpha$,
because $\beta^2$ or $\beta \beta$ is, by Art.~12, a scalar; thus
we may make (117) depend on (116), which again depends on (114),
and on the following equation,
$$\beta \times \beta \alpha = \alpha \beta \times \beta
   \eqno (118).$$

Equations (118) and (115) may both be proved by observing that,
by Art.~13, whatever two vectors may be denoted by $\alpha$ and
$\beta$, we have the expressions
$$\left. \eqalign{
\beta \alpha
   &= {\rm S} \mathbin{.} \beta \alpha
       +  {\rm V} \mathbin{.} \beta \alpha,\cr
\alpha \beta
   &= {\rm S} \mathbin{.} \beta \alpha
       -  {\rm V} \mathbin{.} \beta \alpha,\cr}
   \right\}
   \eqno (119),$$
with the relations
$$\left. \eqalign{
\beta \times {\rm S} \mathbin{.} \beta \alpha
      - {\rm S} \mathbin{.} \beta \alpha \times \beta
   =  0,\cr
\beta \times {\rm V} \mathbin{.} \beta \alpha
      + {\rm V} \mathbin{.} \beta \alpha \times \beta
   =  0,\cr}
   \right\}
   \eqno (120).$$

It remains then to prove the equation (114); and it is sufficient
to prove this for the case when $\alpha$ and $\beta$ are two
rectangular vectors.  But, in this case, $\beta \alpha$ is a
vector formed from $\alpha$ by causing its index to revolve
right-handedly through a right angle round the index ${\rm I} \beta$,
to which it is perpendicular, changing at the same time in
general the length of this revolving index from
$\overline{\alpha}$ to
$\overline{\beta} \times \overline{\alpha}$;
and the repetition of this process, directed by the symbol
$\beta \times \beta \alpha$,
conducts to a new vector, of which the index is in direction
opposite to the original direction of ${\rm I} \alpha$, and in
length equal to
$\overline{\beta}^2 \times \overline{\alpha}$:
this new vector may therefore be otherwise denoted by
$- \overline{\beta}^2 \times \alpha$,
or by $\beta^2 \times \alpha$, and the eqaution (114) is true.
The equations (113) and (112) are therefore also true; and since
the latter formula may easily be extended to any number of
fractional factors, we are now entitled to conclude what it was
in the beginning of the present article proposed to prove;
namely, that {\it the multiplication of geometrical fractions is
always an associative operation\/}: as the addition of fractions,
and the addition of lines, have in former articles been shown to
be.  In other words, any number of successive fractional factors
may be {\it associated\/} or grouped together by multiplication
(without altering their order) into a single product, and this
product substituted as a single factor in their stead; a result
which constitutes a new agreement (the more valuable on account
of the absence of identity in some other important respects),
between the {\it rules of operation\/} of ordinary algebra, and
those of the present Symbolical Geometry.

\bigbreak

\centerline{\it Other forms of the Associative Principle of
Multiplication.}

\nobreak\bigskip

16.
By the principles already established respecting the
transformation of geometrical fractions, any three such
fractions,
$\displaystyle {{\rm f} \over {\rm e}}$,
$\displaystyle {{\rm h} \over {\rm g}}$,
$\displaystyle {{\rm k} \over {\rm i}}$,
may be so prepared that the numerator of the first shall be in the
plane of the second, and that the numerator of the second shall
coincide with the denominator of the third; we may, therefore,
without diminishing the generality of the theorem expressed by
the formula (112), suppose that the line ${\rm i}$ is equal to
${\rm h}$, and that the fourth proportional to
${\rm g}$,~${\rm h}$,~${\rm f}$, is a new line~${\rm l}$; and
with this preparation the associative principle of
multiplication, established in the foregoing article, may be put
under the following form, in which the mark of multiplication
between two fractional factors is omitted for the sake of
conciseness:
$$\hbox{if }
{{\rm h} \over {\rm g}} = {{\rm l} \over {\rm f}},
   \quad\hbox{then }
{{\rm k} \over {\rm h}} {{\rm l} \over {\rm e}}
   = 
{{\rm k} \over {\rm g}} {{\rm f} \over {\rm e}}
   \eqno (121);$$
that is to say, {\it the product of any two geometrical fractions
will remain unaltered\/} in value, or will still continue to
represent the same third fraction, {\it if the denominator of the
multiplier and the numerator of the multiplicand be changed to
any new lines to which they are proportional}, or with which they
form a {\it symbolic analogy}, including a relation between
{\it directions\/} as well as a proportion of lengths, of the
kind considered in Mr.~Warren's work, (and earlier by Argand and
Fran\c{c}ais,) and in the seventh article of this paper.
Reciprocally, by the associative principle, the former of the two
equations (121) is in general a consequence of the latter, that
is, if the product of two geometrical fractions be equal to the
product of two other fractions of the same sort, and if the
multipliers have a common numerator, and the multiplicands a
common denominator, then the numerators of the two multiplicands
and the denominators of the two multipliers are the antecedents
and consequents of a symbolical proportion or analogy, of the
kind considered in the seventh article: for we may write
$${{\rm h} \over {\rm g}}
   =  {{\rm h} \over {\rm k}}
      \left(
         {{\rm k} \over {\rm g}}
         {{\rm f} \over {\rm e}}
      \right)
      {{\rm e} \over {\rm f}},\quad
{{\rm h} \over {\rm k}}
      \left(
         {{\rm k} \over {\rm h}}
         {{\rm l} \over {\rm e}}
      \right)
      {{\rm e} \over {\rm f}}
   =  {{\rm l} \over {\rm f}};$$
so that the first equation (121) may be obtained from the second,
by suitably grouping or associating the factors.

Again, the same associative principle shows that
$$\hbox{if }
{{\rm c} \over {\rm c}'}
   =  {{\rm b}' \over {\rm b}} {{\rm a}' \over {\rm a}},
   \quad\hbox{then }
{{\rm c} \over {\rm b}'}
   =  {{\rm c}' \over {\rm a}} {{\rm a}' \over {\rm b}},
   \eqno (122);$$
for the first equation (122) may be replaced by the system of the
three following equations,
$${{\rm a}' \over {\rm a}} = {{\rm b}'' \over {\rm a}''},\quad
  {{\rm b}' \over {\rm b}} = {{\rm c}'' \over {\rm b}''},\quad
  {{\rm c}' \over {\rm c}} = {{\rm a}'' \over {\rm c}''}
   \eqno (123);$$
of which the two last give, for the first member of the second
equation (122), the expression
$${{\rm c} \over {\rm b}'}
   =  {{\rm c}' \over {\rm a}''} {{\rm b}'' \over {\rm b}},$$
which is equal to the second member of the same second equation
(122), by the first of the three equations (123), and by the
theorem (121): whenever, therefore, we meet an equation between
one geometrical fraction and the product of two others, we are at
liberty to {\it interchange the denominator of the product and
the numerator of the multiplier}, provided that we at the same
time {\it interchange the denominators of the two factors\/}; no
change being made in the numerators of the product and the
multiplicand.  Conversely, this assertion respecting the liberty
to make these interchanges, and the formula (122), to which the
assertion corresponds, are modes of expressing the associative
principle of multiplication; for by introducing the equations
(123) we find that the theorem (122) conducts to the following
relation, or {\it identity between the two ternary products of
three fractions}, associated in two different ways, but with one
common order of arrangement,
$${{\rm c}' \over {\rm a}''}
      \left(
         {{\rm a}'' \over {\rm a}}
         {{\rm a}' \over {\rm b}}
      \right)
   =  \left(
         {{\rm c}' \over {\rm a}''}
         {{\rm a}'' \over {\rm a}}
      \right)
      {{\rm a}' \over {\rm b}}
   \eqno (124);$$
in which last form, as in (112), the three factors multiplied
together may represent any three geometrical fractions.  We may
also present the same principle under the form of the following
theorem---
$$\hbox{if }
      {{\rm c}' \over {\rm c}}
      {{\rm b}' \over {\rm b}}
      {{\rm a}' \over {\rm a}}
   =  1,
   \quad\hbox{then }
      {{\rm c}' \over {\rm a}}
      {{\rm a}' \over {\rm b}}
      {{\rm b}' \over {\rm c}}
   =  1
   \eqno (125);$$
and may derive from it, with the help of (123), the following
value of a certain product of six fractional factors,
$$    {{\rm a}'' \over {\rm c}''}
      {{\rm c}   \over {\rm a}  }
      {{\rm b}'' \over {\rm a}''}
      {{\rm a}   \over {\rm b}  }
      {{\rm c}'' \over {\rm b}''}
      {{\rm b}   \over {\rm c}  }
   =  1
   \eqno (126):$$
which must hold good whenever the three lines ${\rm a}$,
${\rm b}$, ${\rm c}$ are respectively coplanar with the three
pairs ${\rm a}'' {\rm b}''$, ${\rm b}'' {\rm c}''$,
${\rm c}'' {\rm a}''$.  Finally, it may be stated here, as a
theorem essentially equivalent to the associative principle of
multiplication, although not expressly involving the product of
two or more fractions, that {\it in the system of the six
equations\/} of which those marked (123) are three, and of which
the others are the three following analogous equations,
$${{\rm a} \over {\rm c}'} = {{\rm a}''' \over {\rm c}'''},\quad
  {{\rm b} \over {\rm a}'} = {{\rm b}''' \over {\rm a}'''},\quad
  {{\rm c} \over {\rm b}'} = {{\rm c}''' \over {\rm b}'''}
   \eqno (127);$$
{\it any five equations of the system include the sixth}.

\bigbreak

\centerline{\it Geometrical Interpretation of the Associative
Principle: Symbolic Equations between Arcs}
\centerline{\it upon a Sphere: Theorem of the two
Spherical Hexagons.}

\nobreak\bigskip

17.
If we attended only to the {\it lengths\/} of the various lines
compared, the associative principle of multiplication, under all
the foregoing forms, would be nothing more than an easy and known
consequence of a few elementary theorems respecting compositions
of ratios of magnitudes.  On the other hand it is permitted, in
the present symbolical geometry, to assume at pleasure the
{\it situations\/} of straight lines denoted by small roman
letters, provided that the lengths and directions are preserved.
The general theorem or property of multiplication, which has been
expressed in various ways in the two foregoing articles, may
therefore be regarded as being essentially a {\it relation, or
system of relations, between the directions of certain lines in
space}.

In this view of the subject no essential loss of generality (or
at least none which cannot easily be supplied by known and
elementary principles) will be sustained by supposing all the
straight lines
${\rm a} {\rm b} {\rm c}$,
${\rm a}' {\rm b}' {\rm c}'$,
${\rm a}'' {\rm b}'' {\rm c}''$,
${\rm a}''' {\rm b}''' {\rm c}'''$,
${\rm e} {\rm f} {\rm g} {\rm h} {\rm i} {\rm k} {\rm l}$,
of the two last articles to be {\it radii of one sphere}, setting
out from one {\it common origin\/} or centre~${\sc o}$, and
terminating in points upon one {\it common spheric surface},
which may be denoted respectively by the symbols
${\sc a} {\sc b} {\sc c}$,
${\sc a}' {\sc b}' {\sc c}'$,
${\sc a}'' {\sc b}'' {\sc c}''$,
${\sc a}''' {\sc b}''' {\sc c}'''$,
${\sc e} {\sc f} {\sc g} {\sc h} {\sc i} {\sc k} {\sc l}$.
In order more conveniently to study and express relations between
points so situated, we may agree to say that two {\it arcs upon
one sphere}, such as those from ${\sc g}$ to ${\sc h}$ and from
${\sc f}$ to ${\sc l}$, are {\it symbolically equal}, when they
are {\it equally long and similarly directed portions of the
circumference of one great circle\/}; and may denote this {\it
symbolical equality between arcs}, so called for the sake of
suggesting that (like the symbolical equality between straight
lines considered in the second article) it involves a relation of
{\it identity of directions}, as well as a relation of equality
of lengths, by writing any one of the three formul{\ae},
$$\left. \eqalign{
\frown {\sc l} {\sc f} &= \frown {\sc h} {\sc g},\cr
\frown {\sc f} {\sc l} &= \frown {\sc g} {\sc h},\cr
\frown {\sc l} {\sc h} &= \frown {\sc f} {\sc g},\cr}
   \right\}
   \eqno (128);$$
of which the second may be called the {\it inverse}, and the
third the {\it alternate\/} of the first.  Any one of these three
formul{\ae} (128) will thus express the {\it same relation between
the directions of the four coplanar radii}, namely the four lines
${\rm f} \, {\rm g} \, {\rm h} \, {\rm l}$, as that expressed by
the first equation (121), or by its inverse, or its alternate
equation; that is, by any one of the three following equations
between geometrical fractions,
$${{\rm l} \over {\rm f}} = {{\rm h} \over {\rm g}},\quad
  {{\rm f} \over {\rm l}} = {{\rm g} \over {\rm h}},\quad
  {{\rm l} \over {\rm h}} = {{\rm f} \over {\rm g}}
   \eqno (129).$$

The formul{\ae} (128) express also the same relation between the
same four directions, as that which would be expressed in a
notation of a former article, by any one of the three following
{\it symbolic analogies\/} between the same four lines,
$${\rm l} : {\rm f} :: {\rm h} : {\rm g},\quad
  {\rm f} : {\rm l} :: {\rm g} : {\rm h},\quad
  {\rm l} : {\rm h} :: {\rm f} : {\rm g}
   \eqno (130);$$
although it must not be forgotten that any one of the six latter
formul{\ae}, (129) and (130), expresses at the same time a
proportion between the lengths of four straight lines, not
generally equal to each other, which is not expressed by any one
of the three former symbolical equations (128), between pairs of
arcs upon a sphere.  In this notation (128), the last form of the
associative principle of multiplication which was assigned in the
foregoing article, so far as it relates to directions only, may
be expressed by saying that {\it any one of the six following
symbolical equations between arcs is a consequence of the other
five},
$$\left. \eqalign{
\frown {\sc a}' {\sc a} &= \frown {\sc b}'' {\sc a}'',\cr
\frown {\sc b}' {\sc b} &= \frown {\sc c}'' {\sc b}'',\cr
\frown {\sc c}' {\sc c} &= \frown {\sc a}'' {\sc c}'',\cr}
   \right\}
   \eqno (131);$$
$$\left. \eqalign{
\frown {\sc b} {\sc a}' &= \frown {\sc b}''' {\sc a}''',\cr
\frown {\sc c} {\sc b}' &= \frown {\sc c}''' {\sc b}''',\cr
\frown {\sc a} {\sc c}' &= \frown {\sc a}''' {\sc c}''',\cr}
   \right\}
   \eqno (132).$$

Regarding {\it any six points\/} upon a spheric surface, in
{\it any one order\/} of succession, as the {\it six corners of a
spherical hexagon\/} (which may have re-entrant angles, and of
which two or more sides may cross each other without being
prolonged), we may speak of the arcs joining {\it successive
corners\/} as the {\it sides\/}; those joining {\it alternate\/}
corners, as the {\it diagonals\/}; and those joining
{\it opposite\/} corners, as the {\it diameters\/} of this
hexagon: the first side, first diagonal, and first diameter,
respectively, being those three arcs which are drawn from the
first corner to the second, third and fourth corners of the
figure.  With this phraseology, the form just now obtained for
the result of the two foregoing articles may be expressed as a
relation between two spherical hexagons,
${\sc a} {\sc a}' {\sc b} {\sc b}' {\sc c} {\sc c}'$,
${\sc a}'' {\sc a}''' {\sc b}'' {\sc b}''' {\sc c}'' {\sc c}'''$,
and may be enunciated in words as follows:  {\it If five
successive sides of one spherical hexagon be respectively and
symbolically equal to five successive diagonals of another
spherical hexagon, the sixth side of the first hexagon will be
symbolically equal to the sixth diagonal of the second hexagon}.
This theorem of spherical geometry, which may be called, for the
sake of reference, the {\it theorem of the two hexagons}, is
therefore a consequence, and may be regarded as an interpretation
of the associative principle of multiplication: and conversely,
in all applications to spherical geometry, and generally in all
investigations respecting relations between the directions of
straight lines in space, the associative principle of
multiplication may be replaced by the theorem of the two
spherical hexagons.

\bigbreak

\centerline{\it Other Interpretation of the Associative Principle
of Multiplication: Theorem}
\centerline{\it of the two Conjugate Transversals of a
Spherical Quadrilateral (which are}
\centerline{\it the Cyclic Arcs of a circumscribed
Spherical Conic).}

\nobreak\bigskip

18.
The theorem of the two hexagons gives also the following theorem:
If upon each of the four sides of a spherical quadrilateral, or
on that side prolonged, a portion be taken commedial with the
side (two arcs being said to be {\it commedial\/} when the have
one common point of bisection); and if four extreme points of the
four portions thus obtained be ranged on one transversal arc of a
great circle, in such a manner that the part of this arc
comprised between the first and third sides is commedial with the
part comprised between the second and fourth: then the four other
extremities of the same four portions will be ranged on another
great circle; and the parts of this second or {\it conjugate\/}
transversal, which are intercepted respectively by the same two
pairs of opposite sides of the quadrilateral, will be in like
manner commedial with each other.

For let the corners of the quadrilateral be denoted by the
letters ${\sc a}$,~${\sc b}$,~${\sc c}$,~${\sc d}$, and let the
side from ${\sc a}$ to ${\sc b}$ be cut in two points ${\sc a}'$
and ${\sc b}''$, while the three other sides are cut in three
other pairs of points, which may be called ${\sc b}'$ and ${\sc
c}''$, ${\sc c}'$ and ${\sc d}''$, and ${\sc d}'$ and ${\sc a}''$
respectively.  Then, if the arcs from ${\sc a}'$ to ${\sc c}'$
and from ${\sc b}'$ to ${\sc d}'$ be commedial portions of one
common great circle, or of a first transversal arc, the arcs from
${\sc a}'$ to ${\sc b}'$ and from ${\sc d}'$ to ${\sc c}'$ will
be {\it symbolically equal arcs}, in the sense of the preceding
article; and therefore, in the notation of that article, we may
now write the equation
$$\frown {\sc b}' {\sc a}' = \frown {\sc c}' {\sc d}'
   \eqno (133).$$
In like manner the conditions, that the four portions of the
sides of the quadrilateral shall be commedial with the sides
themselves, give the four other equations of the same kind,
$$\left.
\eqalign{
\frown {\sc a}' {\sc a} &= \frown {\sc b} {\sc b}'';\cr
\frown {\sc c}' {\sc c} &= \frown {\sc d} {\sc d}'';\cr}
   \quad
\eqalign{
\frown {\sc b}' {\sc b} &= \frown {\sc c} {\sc c}'';\cr
\frown {\sc d}' {\sc d} &= \frown {\sc a} {\sc a}''.\cr}
   \right\}
   \eqno (134).$$
Hence, by alternation and inversion, we find that the five
successive sides
$$\frown {\sc a} {\sc b}'',\quad
  \frown {\sc d}' {\sc a},\quad
  \frown {\sc c}' {\sc d}',\quad
  \frown {\sc c} {\sc c}',\quad
  \frown {\sc c}'' {\sc c},$$
of the spherical hexagon
${\sc b}'' {\sc a} {\sc d}' {\sc c}' {\sc c} {\sc c}''$
are respectively and symbolically equal to the five successive
diagonals
$$\frown {\sc a}' {\sc b},\quad
  \frown {\sc d} {\sc a}'',\quad
  \frown {\sc b}' {\sc a}',\quad
  \frown {\sc d}'' {\sc d},\quad
  \frown {\sc b} {\sc b}',$$
of the other hexagon
${\sc b} {\sc a}'' {\sc a}' {\sc d} {\sc b}' {\sc d}''$;
and therefore, by the theorem of the two hexagons, the sixth side
of the former figure must be symbolically equal to the sixth
diagonal of the latter; that is, we may write the symbolical
equation,
$$\frown {\sc b}'' {\sc c}'' = \frown {\sc a}'' {\sc d}''
   \eqno (135).$$
But this expresses a relation equivalent to the following, that
the two arcs from ${\sc a}''$ to ${\sc c}''$ and from ${\sc b}''$
to ${\sc d}''$ are commedial portions of one common great circle,
or second transversal arc, which was the thing to be proved.

Reciprocally, the associative principle of geometrical
multiplication, in so far as it relates to the directions of
straight lines in space, may be expressed by the assertion that
the symbolical equation between arcs (135) is a consequence of
the five other equations of the same kind (133) and (134); this
principle of symbolical geometry may therefore be so interpreted
as to coincide with the foregoing {\it theorem of the two
conjugate transversals\/} of a spherical quadrilateral, instead
of the theorem of the two spherical hexagons.  It is easy to see
that to a given quadrilateral correspond infinitely many such
pairs of conjugate transversal arcs; and those readers who are
familiar with the theory of {\it spherical
conics\/}\footnote*{The plane of the first side of the
quadrilateral, or the plane of ${\sc o} {\sc a} {\sc b}$, if
${\sc o}$ denote the centre of the sphere, is cut by the plane of
the first transversal arc in the radius ${\sc a}' {\sc o}$, and
by the plane of the second transversal arc in the radius
${\sc b}'' {\sc o}$.  Thus the four plane faces of the
tetrahedral angle, of which the four edges are the four radii from
${\sc o}$ to the four corners
${\sc a}$,~${\sc b}$,~${\sc c}$,~${\sc d}$
of the quadrilateral, are cut by any secant plane parallel to the
plane of the first transversal arc in four indefinite straight
lines, which are respectively parallel to the four other radii
${\sc a}' {\sc o}$, ${\sc b}' {\sc o}$, ${\sc c}' {\sc o}$,
${\sc d}' {\sc o}$
of the sphere; and consequently, in virtue of the equation (133),
between the arcs which these last radii include, these four new
lines in one common secant plane have the angular relation
required for their being the (prolonged) sides of a (plane)
quadrilateral inscribed in a circle; therefore the four edges of
the same tetrahedral angle are cut by the same secant plane in
points which are on the circumference of a circle; therefore
they are edges or sides of a cone which has this circle for its
base, and has its vertex at the centre of the sphere.  But the
intersection of such a cone with such a concentric sphere is
called a {\it spherical conic\/}; a plane through its vertex,
parallel to its circular base, is called a {\it cyclic plane\/};
and the intersection of this latter plane with the sphere has
received the designation of a {\it cyclic arc}.  Therefore the
first transversal arc ${\sc a}' {\sc b}' {\sc c}' {\sc d}'$ is
(as asserted in the text) a cyclic arc of a spherical conic
circumscribed about the quadrilateral
${\sc a} {\sc b} {\sc c} {\sc d}$:
and by a reasoning of exactly the same kind it may be proved,
that the second transversal
${\sc a}'' {\sc b}'' {\sc c}'' {\sc d}''$
is another cyclic arc of the same conic, or that its plane is a
second cyclic plane, being parallel to the plane of another (or
{\it subcontrary\/}) circular section.}
will recognise in these conjugate transversals,
${\sc a}' {\sc b}' {\sc c}' {\sc d}'$,
${\sc a}'' {\sc b}'' {\sc c}'' {\sc d}''$,
the two {\it cyclic arcs\/} of such a conic, circumscribed about
the proposed quadrilateral ${\sc a} {\sc b} {\sc c} {\sc d}$; but
it suits better the plan of this communication on symbolical
geometry to pass at present to another view of the subject.

It may however be noticed here, that in the first of the two
hexagons already mentioned, {\it any two pairs of opposite sides
intercept commedial portions of either of the two sides
remaining\/}; and that the associative principle asserts that
{\it if \/} a spherical hexagon have {\it five\/} of its sides
thus {\it cut commedially}, the {\it sixth\/} side also will be
cut in the same way.  Or, because the two sets of alternate
diagonals of the second hexagon are sides of two triangles, which
have for their corners the alternate corners of this hexagon, we
may in another way eliminate this second hexagon, and may express
the same principle of spherical geometry by saying, that {\it if
one set of alternate sides of a\/} (first) {\it spherical
hexagon}, taken in their order, (as first, third, and fifth),
{\it be respectively and symbolically equal to the three
successive sides of a triangle}, then the {\it other set of
alternate sides of the same hexagon will be\/} in like manner
{\it symbolically equal to the sides of another triangle}.  This
last interpretation of the associative principle is even more
immediately suggested than any other, by the forms of the
equations (131) (132); in the notation of the present article,
{\it the two triangles\/} are ${\sc b} {\sc a}' {\sc b}'$ and
${\sc a}'' {\sc d} {\sc d}''$, which may be considered as having
their {\it bases\/} ${\sc a}' {\sc b}'$ and ${\sc a}'' {\sc d}''$
{\it on the two cyclic arcs\/} above alluded to, while their
{\it vertical angles\/} at ${\sc b}$ and ${\sc d}$ may be said to
be {\it angles in the same segment\/} (or in alternate segments)
{\it of the spherical conic\/}: since, by (134), the two arcual
sides ${\sc b} {\sc a}'$, ${\sc b} {\sc b}'$ of the one angle
intersect respectively the two sides ${\sc d} {\sc a}''$,
${\sc d} {\sc d}''$ of the other angle, in the points ${\sc a}$
and ${\sc c}$, which points of intersection, as well as the
vertices ${\sc b}$ and ${\sc d}$, are corners of the
quadrilateral inscribed in that spherical conic.

\bigbreak

\centerline{\it Symbolical Addition of Arcs upon a Sphere;
Associative and Non-commutative Properties}
\centerline{\it of such Addition.}

\nobreak\bigskip

19.
The foregoing geometrical interpretations of the associative
principle or property of the multiplication of geometrical
fractions, may assist us in forming and applying the conception
of the symbolical addition of arcs of great circles upon a
sphere, and in establishing and interpreting an analogous
principle or property of such symbolical addition.

As it has been already proposed in the third article of this
paper, and also in the works of other writers on subjects
connected with the present, to adopt, for the {\it addition of
straight lines having direction}, a rule expressed by the formula
$${\sc c} {\sc b} + {\sc b} {\sc a} = {\sc c} {\sc a}
   \eqno (7),$$
in whatever manner the three points ${\sc a} {\sc b} {\sc c}$ may
be situated or related to each other; so it seems natural to
adopt now, for the analogous {\it addition of arcs upon a sphere},
when directions as well as lengths are attended to, the
corresponding formula,
$$\frown {\sc c} {\sc b} + \frown {\sc b} {\sc a}
   = \frown {\sc c} {\sc a}
   \eqno (136).$$

Admitting this latter formula as {\it the definition of the
effect of the sign~$+$ when inserted between two such symbols of
arcs}, and granting also that it is permitted, in any such
formula, to substitute for any arcual symbol another which is
{\it equal\/} thereto, we shall have, by the two first and two last
equations (134) respectively, the two following other equations,
$$\left. \eqalign{
\frown {\sc b}'' {\sc c}''
   &= \frown {\sc a} {\sc a}' + \frown {\sc b}' {\sc c} \cr
\frown {\sc a}'' {\sc d}''
   &= \frown {\sc a} {\sc d}' + \frown {\sc c}' {\sc c} \cr}
   \right\}
   \eqno (137).$$
The two sums in these second members will therefore be
symbolically equal if we have the equation
$$\frown {\sc a}' {\sc d}' = \frown {\sc b}' {\sc c}'
   \eqno (138),$$
because (135) has been seen to follow from (133) and (134).  But
by (136) and (138), we have the expression
$$\frown {\sc b}' {\sc c}
   =  \frown {\sc a}' {\sc d}' + \frown {\sc c}' {\sc c}
   \eqno (139);$$
consequently the associative principle of multiplication,
considered in several recent articles, when combined with the
{\it formula of arcual addition\/} (136), conducts to the
following formula,
$$\frown {\sc a} {\sc a}'
   + ( \frown {\sc a}' {\sc d}' + \frown {\sc c}' {\sc c} )
   =  ( \frown {\sc a} {\sc a}' + \frown {\sc a}' {\sc d}' )
   + \frown {\sc c}' {\sc c}
   \eqno (140),$$
or, as it may be more concisely written,
$$\frown''' + ( \frown'' + \frown' )
   = ( \frown''' + \frown'' ) + \frown'
   \eqno (141):$$
which in its form agrees with ordinary algebra, and may be said
to express the {\it associative principle of the symbolical
addition of arcs\/}; since the three arcs added in (140) or (141)
may be any three arcs of great circles upon one common spheric
surface. It is remarkable that so much geometrical meaning should
be contained in so simple and elementary a form; for this form
(141), which is {\it apparently an algebraic truism}, and has
been here deduced from the associative principle of
multiplication of geometric fractions, may reciprocally be
substituted for it, and therefore includes in its interpretation,
{\it if we adopt the symbolical definition\/} (136) of the effect
of~$+$ between two symbols of arcs, all those theorems respecting
spherical great circles, triangles, quadrilaterals, hexagons, and
conics, which have been deduced or mentioned as geometrical
results of the associative principle in the two foregoing
articles.  And this encouragement to adopt the foregoing very
simple definition (136) of the meaning of a symbol such as
$\frown'' + \frown'$, is the more worthy of attention, because
the {\it same definition\/} conducts to a {\it departure from the
ordinary rules of symbolical addition\/} in {\it another\/}
important point; since, when combined with the {\it definition of
symbolical equality between arcs\/} assigned in the 17th article,
it shews that {\it addition of arcs is in general a
non-commutative operation}.  For if we conceive two arcs of
different great circles on one sphere, from ${\sc a}$ to
${\sc b}$ and from ${\sc c}$ to ${\sc d}$, to bisect each other
in a point~${\sc e}$, we shall then have the two symbolical
equations
$$\frown {\sc a} {\sc e} = \frown {\sc e} {\sc b},\quad
  \frown {\sc c} {\sc e} = \frown {\sc e} {\sc d}
   \eqno (142);$$
and therefore, whereas by (136),
$$\frown {\sc a} {\sc e} + \frown {\sc e} {\sc d}
   =  \frown {\sc a} {\sc d}
   \eqno (143),$$
the result of the addition of the same two arcs, taken in a
different order, will be
$$\frown {\sc e} {\sc d} + \frown {\sc a} {\sc e}
   =  \frown {\sc c} {\sc b}
   \eqno (144).$$
And although the two {\it sum-arcs}, $\frown {\sc a} {\sc d}$ and
$\frown {\sc c} {\sc b}$, thus obtained, connecting two opposite
pairs of extremities of the two commedial arcs
$\frown {\sc a} {\sc b}$ and $\frown {\sc c} {\sc d}$,
are {\it equally long}, yet they are in general {\it parts of
different great circles}, and therefore {\it not symbolically
equal\/} in the sense of the 17th article.  This result, which
may at first sight seem a paradox, illustrates and is intimately
connected with the analogous result obtained in the 13th article,
respecting the general non-commutativeness of geometrical
multiplication; for we shall find that there exists a species of
{\it logarithmic connexion\/} between arcs situated in different
great circles on a sphere and fractional factors belonging to
different planes, which is analogous to, and includes as a
limiting case, the known connexion between ordinary imaginary
logarithms and angles in a single plane.  It may be here
remarked, that with the same definition (136) {\it in any
symbolical addition of three successive arcs, the two partial
sum-arcs,
$$\frown'' + \frown ' \hbox{ and } \frown''' + \frown''
   \eqno (145),$$
are portions of the cyclic arcs of a certain spherical conic,
circumscribed about a quadrilateral which has\/}
$$\frown',\enspace
  \frown'',\enspace
  \frown''',
   \hbox{ and }
  \frown''' + \frown'' + \frown'
   \eqno (146),$$
that is, {\it the three proposed summand-arcs and their total
sum-arc, for portions of its four sides}, or of those sides
prolonged; as will appear by supposing that the three summands
$\frown'$,~$\frown''$,~$\frown'''$, coincide respectively with
the arcs $\frown {\sc c} {\sc c}'$, $\frown {\sc b}' {\sc c}$,
$\frown {\sc a}{\sc a}'$, in the notation of the preceding
article.

\bigbreak

\centerline{\it Symbolical Expressions for a Cyclic Cone;
Relations of such a Cone, and of its Cyclic}
\centerline{\it Planes, to a Product of Two Geometrical Fractions.}

\nobreak\bigskip

20.
It is evidently a determinate\footnote*{The evident and known
determinateness of this problem, corresponding to that of the
elementary problem of circumscribing a circle about a given plane
trangle, was tacitly assumed, but might with advantage have been
expressly referred to, in the outline of a demonstration which was
given in the note to Art.~(18).  The reasoning, towards the end
of that note, would then stand thus:---If ${\sc d}$ be any fourth
point on the determined spherical conic, which passes through the
three points ${\sc a}$, ${\sc b}$, ${\sc c}$, and has the arc
${\sc a}' {\sc b}'$ for a cyclic arc, it is also a fourth point
on the determined spherical conic which passes through the same
three points and has the arc ${\sc b}'' {\sc c}''$ for a cyclic
arc; therefore the two conics, determined by these two sets of
conditions, coincide one with the other: or, in other words, the
arc ${\sc b}'' {\sc c}''$ is a {\it second\/} cyclic arc of the
{\it same\/} spherical conic, of which the arc
${\sc a}' {\sc b}'$ is a {\it first\/} cyclic arc.}
problem to construct a {\it cyclic cone}, that is, a cone with
circular base (called usually a cone of the second degree), when
three of the {\it sides\/} (or generating straight lines) of the
cone are given in position, and when the plane of the base is
parallel to a given {\it cyclic plane}, which passes through the
vertex.  To treat this problem, which may be regarded as a
fundamental one in the theory of such cones, by a method derived
from the principles of the foregoing articles, let the three
given sides be denoted by the letters
${\rm a}$,~${\rm b}$,~${\rm c}$; and let the two known lines, in
which the given cyclic plane is cut by the planes of the two
pairs, ${\rm a} {\rm b}$ and ${\rm b} {\rm c}$, be denoted by
${\rm a}'$ and ${\rm b}'$; also let ${\rm d}$ denote any fourth
side of the sought cyclic cone, and ${\rm c}'$,~${\rm d}'$ the
lines of intersection of the given cyclic plane with the variable
planes of ${\rm c} {\rm d}$ and ${\rm d} {\rm a}$; then, if
suitable lengths be assigned to these straight lines, of which
the relative {\it directions\/} in space are the chief object of
the present investigation, the following equality between two
products of certain geometrical fractions will exist, and may be
regarded as a form of the {\it equation of the cone\/}:
$${{\rm c} \over {\rm b}'} {{\rm a}' \over {\rm a}}
   =  {{\rm c} \over {\rm c}'} {{\rm d}' \over {\rm a}}
   \eqno (147).$$
That is to say, when this equation is satisfied, the two lines
which are the respective intersections of the planes of the
fractional factors of these two equal products, namely the
intersection ${\rm b}$ of the planes ${\rm a} {\rm a}'$ and
${\rm b}' {\rm c}$, and the intersection~${\rm d}$ of the planes
${\rm a} {\rm d}'$ and ${\rm c}' {\rm c}$, are two sides of a
cyclic cone, which has for two other sides the lines ${\rm a}$
and ${\rm c}$, and which has for one cyclic plane the common
plane of the four lines ${\rm a}'$,~${\rm b}'$,~${\rm c}'$ and
${\rm d}'$; these eight lines
${\rm a}$, ${\rm b}$, ${\rm c}$, ${\rm d}$,
${\rm a}'$, ${\rm b}'$, ${\rm c}'$, ${\rm d}'$,
being here supposed to diverge from one common origin, namely the
vertex (or centre) of the cone. This may easily be shown to be a
consequence of what has been already established, respecting the
connexion of the cyclic arcs of a spherical conic with the
symbolic sums of certain other arcs.  Or, without introducing any
sphere, we may observe that, by (121) and its converse, the
equation (147) may be abridged to the following:
$${{\rm a}' \over {\rm b}'} = {{\rm d}' \over {\rm c}'};
   \quad\hbox{or, }
  {{\rm a}' \over {\rm b}'} {{\rm c}' \over {\rm d}'} = 1
   \eqno (148);$$
which shows, in virtue of the notation here employed, that
besides a certain proportionality of lengths, not necessary now
to be considered, there exists an equality between the angles of
rotation, in one common plane, which would transport the lines
${\rm b}'$ and ${\rm c}'$, respectively, into the directions of
${\rm a}'$ and ${\rm d}'$.  But the four lines
${\rm a}'$,~${\rm b}'$,~${\rm c}'$,~${\rm d}'$
are respectively parallel to the four symbolic differences,
${\rm b} - {\rm a}$, ${\rm c} - {\rm b}$, ${\rm d} - {\rm c}$,
${\rm a} - {\rm d}$, or to the four straight lines
${\sc b} {\sc a}$, ${\sc c} {\sc b}$, ${\sc d} {\sc c}$, ${\sc a} {\sc d}$,
that is to the successive sides of the plane quadrilateral
${\sc a} {\sc b} {\sc c} {\sc d}$,
if we now suppose the lines
${\rm a}$, ${\rm b}$, ${\rm c}$, ${\rm d}$
to terminate, in the points
${\sc a}$, ${\sc b}$, ${\sc c}$, ${\sc d}$,
on a transversal plane parallel to the plane of
${\rm a}'$, ${\rm b}'$, ${\rm c}'$, ${\rm d}'$.
We may therefore present the relation (148) under either of the
two forms:
$${{\rm b} - {\rm a} \over {\rm c} - {\rm b}}
      {{\rm d} - {\rm c} \over {\rm a} - {\rm d}}
   =  x';
   \quad\hbox{or, }
  {{\sc b} {\sc a} \over {\sc c} {\sc b}}
      {{\sc d} {\sc c} \over {\sc a} {\sc d}}
   =  x
   \eqno (149);$$
in which $x$ is a positive or negative scalar; or, using the
characteristic~${\rm V}$ of the operation of taking the vector
part, we may write
$${\rm V} \mathbin{.}
      {{\rm b} - {\rm a} \over {\rm c} - {\rm b}}
      {{\rm d} - {\rm c} \over {\rm a} - {\rm d}}
   =  0;
   \quad\hbox{or }
  {\rm V} \mathbin{.}
      {{\sc b} {\sc a} \over {\sc c} {\sc b}}
      {{\sc d} {\sc c} \over {\sc a} {\sc d}}
   =  0
   \eqno (150).$$
When the scalar~$x$ is positive, then, by considering the two
rotations above mentioned, we may easily perceive that the two
points ${\sc b}$ and ${\sc d}$ are at one common side of the
straight line ${\sc a} {\sc c}$, and that this line subtends
equal angles at those two points; being in one common plane with
them, as indeed the second equation (149) sufficiently expresses,
since it gives
$${\rm V} {{\sc b} {\sc a} \over {\sc c} {\sc b}}
   =  x {\rm V} {{\sc d} {\sc a} \over {\sc c} {\sc d}}
   \eqno (151);$$
so that the two triangles ${\sc a} {\sc b} {\sc c}$,
${\sc a} {\sc d} {\sc c}$, on the common base ${\sc a} {\sc c}$,
have one common perpendicular to their planes, which must
therefore coincide with each other.  In the contrary case, namely
when $x$ is negative, the equation (151) still shows that the four
points are (as above) coplanar with each other; and while the
points ${\sc b}$ and ${\sc d}$ are now at opposite sides of the
line~${\sc a} {\sc c}$, the angles which this line subtends at
those two points are now not equal but supplementary.  In each
case, therefore, the four points
${\sc a} {\sc b} {\sc c} {\sc d}$
are on the circumference of one common circle; the four lines
${\rm a}$,~${\rm b}$,~${\rm c}$,~${\rm d}$
are consequently sides of a cyclic cone; and the plane of the
four other lines
${\rm a}'$,~${\rm b}'$,~${\rm c}'$,~${\rm d}'$
is a cyclic plane of that cone.

\bigbreak

21.
In the foregoing article, the coplanarity of each of the four
sets of three lines,
${\rm a}' {\rm a} {\rm b}$,
${\rm b}' {\rm b} {\rm c}$,
${\rm c}' {\rm c} {\rm d}$,
${\rm d}' {\rm d} {\rm a}$,
allows us to suppose that four other lines
${\rm b}''$,~${\rm c}''$,~${\rm d}''$,~${\rm a}''$,
in the same four planes respectively, and all, like the eight
former lines, diverging from the vertex of the cone, are
determined so as to satisfy the four equations:
$${{\rm b}'' \over {\rm b}} = {{\rm a} \over {\rm a}'};\quad
  {{\rm c}'' \over {\rm c}} = {{\rm b} \over {\rm b}'};\quad
  {{\rm d}'' \over {\rm d}} = {{\rm c} \over {\rm c}'};\quad
  {{\rm a}'' \over {\rm a}} = {{\rm d} \over {\rm d}'}
   \eqno (152);$$
and then, since these equations, combined with (148), give, by
the associative property of the multiplication of geometrical
fractions, this other equation,
$${{\rm b}'' \over {\rm c}''} = {{\rm a}'' \over {\rm d}''}
   \eqno (153),$$
it follows that these four new lines are in one common plane; and
also that the rotations in that plane, from ${\rm b}''$ and
${\rm c}''$ to ${\rm a}''$ and ${\rm d}''$, respectively, are
equal.  And this new plane is evidently a
{\it second\/}\footnote*{See the remarks made in the note to the
foregoing article.}
{\it cyclic plane of the same cone\/}; for we may now write,
instead of (147), the analogous equation:
$${{\rm c} \over {\rm c}''} {{\rm b}'' \over {\rm a}}
   =  {{\rm c} \over {\rm d}''} {{\rm a}'' \over {\rm a}}
   \eqno (154);$$
the two members being here equal respectively to the reciprocals
of the two members of the first equation (148): nor is it
necessary to retain the restriction that the lines
${\rm a}$,~${\rm b}$,~${\rm c}$,~${\rm d}$
should terminate in one common plane.  In like manner, the two
members of the equation (147) are respectively equal to the
reciprocals of the two members of the equation (153); a
geometrical (like an arithmetical) fraction being said to be
changed to its {\it reciprocal}, when the numerator and
denominator are interchanged.  We have therefore this
theorem:---{\it A cyclic cone is the locus of the intersection of
the planes of two geometrical fractions, of which the product is
a constant fraction, while the numerator of the multiplier and
the denominator of the multiplicand are constant lines.  These
two lines are two fixed sides of the cone; the plane of the two
other and variable lines}, which enter as denominator and
numerator into the expressions of the same two fractional
factors, {\it is one cyclic plane of that cone\/}; and {\it the
plane of the constant product is the other cyclic plane}.  The
investigation in the last article shows also that the condition
for four points ${\sc a} {\sc b} {\sc c} {\sc d}$ being {\it
concircular\/} or {\it homocyclic}, that is, for their being
corners of a quadrilateral inscribed in a circle, is expressed by
the second equation (150); which may therefore be called the
{\it equation of homocyclicism}.  The same investigation shows
that if we only know that ${\sc a} {\sc b} {\sc c} {\sc d}$ are
four points on one common plane, we may still write an equation
of the form (151); which may for that reason be said to be a
{\it formula of coplanarity}.

\bigbreak

\centerline{\it Symbolical Expressions and Investigations of some
Properties of Cyclic Cones, with}
\centerline{\it reference to their Tangent Planes.}

\nobreak\bigskip

22.
If the side~${\rm b}$ of the cyclic cone be conceived to approach
to the side~${\rm a}$, and ultimately to coincide with it, the
first equation (152) will take this limiting form:
$${{\rm b}'' \over {\rm a}} = {{\rm a} \over {\rm a}'}
   \eqno (155);$$
which expresses the known theorem that the side of
contact~${\rm a}$ bisects the angle between the traces ${\rm a}'$
and ${\rm b}''$ of the tangent plane on the two cyclic planes;
bisecting also the vertically opposite angle between the traces
$- {\rm a}'$ and $- {\rm b}''$, but  being perpendicular to the
bisector of either of the two other angles, which are
supplementary to the two already mentioned, namely the angle
between the traces ${\rm a}'$ and $- {\rm b}''$, and that between
$- {\rm a}'$ and ${\rm b}''$.  And if in like manner we conceive
the side~${\rm d}$ to approach indefinitely to the side~${\rm
c}$, the plane of these two sides will tend to become another
tangent plane to the cone; of which plane the traces ${\rm c}'$
and ${\rm d}''$ on the two cyclic planes will satisfy an equation
of the same form as that last written, namely the following,
which is the limiting form of the third equation (152):
$${{\rm d}'' \over {\rm c}} = {{\rm c} \over {\rm c}'}
   \eqno (156).$$
At the same time, the two secant planes ${\rm b} {\rm c}$ and
${\rm d} {\rm a}$ will tend to coalesce in one secant plane,
containing the two sides of contact ${\rm a}$ and ${\rm c}$, with
which the two other sides ${\rm b}$ and ${\rm d}$ tend to
coincide; so that the traces ${\rm d}'$ and ${\rm a}''$ of the
latter secant plane, on the two cyclic planes, will ultimately
coincide with the traces ${\rm b}'$ and ${\rm c}''$ of the former
secant plane on the same two cyclic planes; and the equations
(148) (153) become:
$${{\rm a}' \over {\rm b}'} = {{\rm b}' \over {\rm c}'};\quad
  {{\rm b}'' \over {\rm c}''} = {{\rm c}'' \over {\rm d}''}
   \eqno (157);$$
which expresses that the traces ${\rm b}'$ and ${\rm c}''$ of the
one remaining secant plane bisect respectively the angles between
the pairs of traces, ${\rm a}'$,~${\rm c}'$, and
${\rm b}''$,~${\rm d}''$, of the two tangent planes, on the two
cyclic planes.  And the two remaining equations (152) concur in
giving this other equation:
$${{\rm c}'' \over {\rm c}} = {{\rm a} \over {\rm b}'}
   \eqno (158):$$
expressing that the rotations in the secant plane from ${\rm b}'$
to ${\rm a}$ and from ${\rm c}$ to ${\rm c}''$, that is to say
from one trace to one side, and from the other side to the other
trace, are equal in amount, and similarly directed; in such a
manner that these two traces ${\rm b}'$ and ${\rm c}''$, of the
secant plane on the two cyclic planes, are equally inclined to
the straight line which bisects the angle between these two sides
${\rm a}$ and ${\rm c}$, along which the plane cuts the cone: all
of which agrees with the known properties of cones of the second
degree.

\bigbreak

23.
The eight straight lines ${\rm a}$, ${\rm c}$, ${\rm a}'$,
${\rm b}'$, ${\rm c}'$, ${\rm b}''$, ${\rm c''}$, ${\rm d}''$,
being supposed to be equally long, the first of them, which has
been seen to coincide in direction with the bisector of the angle
between the third and sixth, can differ only by a scalar (or real
and numerical) coefficient from their symbolic sum; because the
diagonals of a plane and equilateral quadrilateral figure (or
rhombus) bisect the angles of that figure.  We have therefore, by
(155), and by the supposition of the equal lengths of the eight
lines,
$${\rm a}' + {\rm b}'' \parallel {\rm a};
   \quad\hbox{or,}\quad
  {\rm a}' + {\rm b}'' = l {\rm a}
   \eqno (159),$$
$l$ being a numerical coefficient, and the sign of parallelism
being designed to include the case of coincidence.

In like manner, by (156), we have
$${\rm d}'' + {\rm c}' \parallel {\rm c};
   \quad\hbox{or,}\quad
  {\rm d}'' + {\rm c}' = l' {\rm c}
   \eqno (160),$$
$l'$ being another scalar coefficient.  Again, by (157),
$$\left.
\eqalign{
   {\rm c}' + {\rm a}'   &\parallel {\rm b}'; \cr
   {\rm b}'' + {\rm d}'' &\parallel {\rm c}''; \cr}
  \quad
\eqalign{
{\rm c}' + {\rm a}'   &= m  {\rm b}'; \cr
{\rm b}'' + {\rm d}'' &= m' {\rm c}''; \cr}
   \right\}
   \eqno (161),$$
$m$ and $m'$ being two other scalars.  But, by (158),
$${{\rm c}'' \over {\rm c}} {{\rm b}' \over {\rm a}} = 1
   \eqno (162);$$
therefore
$${{\rm b}'' + {\rm d}'' \over {\rm d}'' + {\rm c}'}
      {{\rm c}' + {\rm a}' \over {\rm a}' + {\rm b}''}
   = {m' \over l'} {m \over l}
   = {\rm V}^{-1} 0
   \eqno (163);$$
this symbol ${\rm V}^{-1} 0$ denoting generally, in the present
system, {\it any geometrical fraction of which the vector part is
zero}, and therefore any positive or negative number (including
zero).  (Compare the definition and remarks in the 7th article).

By comparing this equation (163) with the first form (150), we
see that the four straight lines,
$$- {\rm b}'',\enspace {\rm d}'',\enspace - {\rm c}',\enspace {\rm a}'
   \eqno (164),$$
which have been supposed to diverge from one common origin, namely
the vertex of the cone, have their terminations on the
circumference of one common circle.  But these four lines, by
supposition, are also equally long; they must therefore be four
sides of a new cone, which is not only cyclic, as having a
circular base, but is also a {\it cone of revolution}.  The axis
of revolution of this new cone is perpendicular to the plane of
the circle in which the four lines (164) terminate; and this
plane is parallel to the plane of the symbolic differences of
those four lines, namely, the following,
$${\rm d}'' + {\rm b}'',\enspace
  - {\rm c}' - {\rm d}'',\enspace
  {\rm a}' + {\rm c}',\enspace
  - {\rm b}'' - {\rm a}'
   \eqno (165);$$
but these have been seen to be parallel respectively to the four
lines ${\rm c}''$, ${\rm c}$, ${\rm b}'$, ${\rm a}$, which are
contained in the secant plane of the former cone; consequently
the axis of revolution of the new cone is perpendicular to this
secant plane.  We arrive therefore, by this symbolical process,
at a new proof of the known theorem, discovered by
M.~Chasles,\footnote*{See the Translation of Two General Memoirs
by M.~Chasles, on the General Properties of Cones of the Second
Degree, and on the Spherical Conics; which Translation was
published, with an Appendix, by the Rev.\ Charles Graves, in
Dublin, 1841.}
that two planes, touching a cyclic cone along any two sides,
intersect the two common cyclic planes in four right lines, which
are sides of one common cone of revolution, whose axis of
revolution is perpendicular to the plane of the two sides of
contact of the former cone.

\bigbreak

24.
If we conceive the first and fourth of the sides (164) of the
cone of revolution to tend to coincide with each other, then the
fourth of the sides (165) of the plane quadrilateral inscribed in
the circular base of that cone will tend to vanish; consequently
the direction of this last mentioned side
$- {\rm b}'' - {\rm a}'$, or the opposite direction of
${\rm a}'+ {\rm b}''$, will become at last tangential to this
circular base; and the plane of the two sides previously
mentioned, namely $- {\rm b}''$ and ${\rm a}'$, which plane has
been seen to touch the cyclic cone along the side~${\rm a}$, will
become ultimately tangential also to the cone of revolution,
touching it along the line~${\rm a}'$, which becomes one trace of
the second cyclic plane on the first cyclic plane; the opposite
line, $- {\rm a}'$, being of course also situated in the
intersection of those two planes, so that it may be regarded as
the opposite trace of one cyclic plane on the other.  Thus, at
the limit here considered, the equation (155) and the second
equation (157) are replaced by the equations
$${- {\rm a}' \over {\rm a}} = {{\rm a} \over {\rm a}'},\quad
  {- {\rm a}' \over {\rm c}''} = {{\rm c}'' \over {\rm d}''}
   \eqno (166);$$
of which the first expresses that the side~${\rm a}$ is equally
inclined to the two opposite traces, ${\rm a}'$ and $- {\rm a}'$;
while the numerical coefficient~$l$ vanishes, and the formula
(163) is replaced by this other,
$${\rm V} \mathbin{.} {{\rm d}'' - {\rm a}' \over {\rm d}'' + {\rm c}'}
      {{\rm c}' + {\rm a}' \over {\rm a}}
   =  0
   \eqno (167).$$

We see also that the two rectangular but equally long lines
${\rm a}$,~${\rm a}'$, of which the former is a side of the
cyclic cone, while the latter is part of the line of intersection
of the two cyclic planes of that cone, are such that their plane
is a common tangent to both the cyclic cone and the cone of
revolution; which latter cone has also, as sides of the same sheet
with ${\rm a}'$, the two other of the four lines (164), namely
the lines $- {\rm c}'$ and ${\rm d}''$.  Indeed, the formula
(167) is sufficient to show, by comparison with the first formula
(150), that if the three straight lines ${\rm a}'$, ${\rm d}''$,
$- {\rm c}'$ be still supposed to diverge from one common origin,
the circle passing through the three points in which they
terminate is touched, at the termination of the line~${\rm a}'$,
by a straight line parallel to the line~${\rm a}$; and therefore
that the cone of revolution, having these three equally long
lines ${\rm a}'$, $- {\rm c}'$, ${\rm d}''$ for sides of one
common sheet, is touched along the side~${\rm a}'$ by the plane
which contains the two rectangular lines ${\rm a} {\rm a}'$; so
that we may regard this formula (167) as containing the
symbolical solution to the problem, to draw a tangent plane,
along any proposed side, to the cone of revolution which passes
through that side and through two other sides also given, and
belonging to the same sheet as the former.  Now if three such
sides be connected by three planes, forming three faces of a
triangular pyramid, inscribed in a single sheet of a cone of
revolution, and having its vertex at the vertex of that cone,
while the sheet is touched by a fourth plane along one edge of
the pyramid, it follows from the most elementary principles of
solid geometry, that the difference between the two exterior
angles which the faces meeting at that edge make with the tangent
plane to the cone is equal to the difference of the two interior
angles which the same two faces make with the third face of the
pyramid; the greater exterior angle being the one which is the
more remote from the greater interior angle; as may be shown by
conceiving three planes to pass through the three edges
respectively, and through the axis of revolution of the cone.
The same equality between the differences of these two pairs of
angles between planes, will become still more evident if, without
making use of any formula of spherical trigonometry, we consider
a spherical triangle inscribed in a small circle on the sphere,
which small circle is touched at one corner of the triangle by a
great circle, while arcs are drawn to that and to the two other
corners from a pole of the small circle; the only principles
required being these: that the base angles of a spherical
isosceles triangle are equal, and that the arcs from the pole of a
small circle are all perpendicular to its perimeter.  If then we
denote by the symbol $\angle ({\rm a}, {\rm b}, {\rm c})$ the
acute or right or obtuse dihedral or spherical angle, at the
edge~${\rm b}$, between the planes ${\rm a} {\rm b}$ and
${\rm b} {\rm c}$, in such a manner as to write, generally,
$$\angle ({\rm a}, {\rm b}, {\rm c})
   =  \angle ({\rm c}, {\rm b}, {\rm a})
   =  \angle (- {\rm a}, {\rm b}, - {\rm c})
   =  \angle ({\rm a}, - {\rm b}, {\rm c})
   =  \pi - \angle ({\rm a}, {\rm b}, - {\rm c})
   \eqno (168);$$
$\pi$ being the symbol for two right angles, we shall have, in
the present question, the equation
$$\angle ({\rm a}', {\rm d}'', - {\rm c}')
       - \angle ({\rm a}', - {\rm c}', {\rm d}'')
   = \angle (- {\rm a}, {\rm a}', - {\rm c}')
       - \angle ({\rm a}, {\rm a}', {\rm d}'')
   \eqno (169);$$
and therefore, by subtracting both members from $\pi$,
$$\angle ({\rm a}', {\rm d}'', {\rm c}')
       + \angle ({\rm a}', {\rm c}', {\rm d}'')
   = \angle (- {\rm a}, {\rm a}', {\rm c}')
       + \angle ({\rm a}, {\rm a}', {\rm d}'')
   \eqno (170).$$

We have also here the relation
$$\angle ({\rm c}', {\rm a}', {\rm d}'')
   =  \angle ({\rm a}, {\rm a}', {\rm c}')
       + \angle ({\rm a}, {\rm a}', {\rm d}'')
   \eqno (171),$$
because the plane ${\rm a} {\rm a}'$ is intermediate between the
planes ${\rm a}' {\rm c}'$ and ${\rm a}' {\rm d}''$, or lies {\it
within\/} the dihedral angle $({\rm c}', {\rm a}', {\rm d}'')$
itself, and not within either of the two angles which are
exterior and supplementary thereto; which again depends on the
circumstance that both the cyclic planes are necessarily exterior
to each sheet of the cyclic cone.  Adding therefore the equations
(170) and (171), member to member, and subtracting $\pi$ on both
sides of the result, we find for the {\it spherical excess\/}
of the new triangular pyramid $({\rm a}', {\rm c}', {\rm d}'')$,
or for the excess of the sum of the mutual inclinations of its
three faces ${\rm a}' {\rm c}'$, ${\rm a}' {\rm d}''$,
${\rm c}' {\rm d}''$, above two right angles, the expression:
$$\angle ({\rm a}', {\rm d}'', {\rm c}')
       + \angle ({\rm a}', {\rm c}', {\rm d}'')
       + \angle ({\rm c}', {\rm a}', {\rm d}'')
       - \pi
   =  2 \angle ({\rm a}, {\rm a}', {\rm d}'')
   \eqno (172).$$

This spherical excess therefore remains unchanged, while the two
lines ${\rm c}'$, ${\rm d}''$, move together on the two cyclic
planes, in such a manner that their plane, always passing through
the vertex of the cone, continues to touch that cyclic cone;
${\rm a}'$ being still a line situated in the intersection of the
two cyclic planes, and ${\rm a}$ being still a side of contact of
the cone with a plane drawn through that intersection.  And
hence, or more immediately from the equation (170), the known
property of a cyclic cone is proved anew, that the sum of the
inclinations (suitably measured) of its variable tangent plane to
its two fixed cyclic planes is constant.

\bigbreak

\centerline{\it Condition of Concircularity, resumed.
New Equation of a Cyclic Cone.}

\nobreak\bigskip

25.
The equation (150) of {\it homocyclism}, or of
{\it concircularity}, which was assigned in the 20th article,
and which expresses the condition requisite in order that four
straight lines in space, ${\rm a}$, ${\rm b}$, ${\rm c}$,
${\rm d}$, diverging from one common point ${\sc o}$, as from an
origin, may terminate in four other points ${\sc a}$, ${\sc b}$,
${\sc c}$, ${\sc d}$, which shall all be contained on the
circumference of one common circle, may also, by (149), be put
under the form
$${{\rm b} - {\rm a} \over {\rm c} - {\rm b}}
   = x {{\rm d} - {\rm a} \over {\rm c} - {\rm d}}
   \eqno (173),$$
where $x$ is a scalar coefficient.  It gives therefore the two
following separate equations, one between scalars, and the other
between vectors:
$${\rm S} {{\rm b} - {\rm a} \over {\rm c} - {\rm b}}
   = x {\rm S} {{\rm d} - {\rm a} \over {\rm c} - {\rm d}};\quad
  {\rm V} {{\rm b} - {\rm a} \over {\rm c} - {\rm b}}
   = x {\rm V} {{\rm d} - {\rm a} \over {\rm c} - {\rm d}}
   \eqno (174);$$
of which the latter is only another way of writing the equation
(151).  If then we agree to use, for conciseness, a new
characteristic of operation,
$\displaystyle {{\rm V} \over {\rm S}}$,
of which the effect on any geometrical fraction, to the symbol of
which it it prefixed, shall be defined by the formula
$${{\rm V} \over {\rm S}} \mathbin{.} {{\rm b} \over {\rm a}}
   = {\rm V} {{\rm b} \over {\rm a}} \div {\rm S} {{\rm b} \over {\rm a}}
   \eqno (175);$$
so that this new characteristic
$\displaystyle {{\rm V} \over {\rm S}}$,
which (it must be observed) {\it is not a distributive symbol},
is to be considered as directing to {\it divide the vector by the
scalar part\/} of the geometrical fraction on which it operates;
we shall then have, as a consequence of (173), this other form of
the {\it equation of concircularity\/}:
$${{\rm V} \over {\rm S}} \mathbin{.}
         {{\rm b} - {\rm a} \over {\rm c} - {\rm b}}
   = {{\rm V} \over {\rm S}} \mathbin{.}
         {{\rm d} - {\rm a} \over {\rm c} - {\rm d}}
   \eqno (176).$$

Conversely we can return from this latter form (176) to the
equation (173); for if we observe that, in the present system of
symbolical geometry, {\it every geometrical fraction is equal to
the sum of its own scalar and vector parts}, so that we may write
generally (see article~7),
$${\rm S} {{\rm b} \over {\rm a}} + {\rm V} {{\rm b} \over {\rm a}}
   =  {\rm V} {{\rm b} \over {\rm a}} + {\rm S} {{\rm b} \over {\rm a}}
   =  {{\rm b} \over {\rm a}}
   \eqno (177),$$
or more concisely,
$${\rm S} + {\rm V} = {\rm V} + {\rm S} = 1
   \eqno (178);$$
and, if we add the identity,
$${\rm S} {{\rm b} - {\rm a} \over {\rm c} - {\rm b}}
      \div {\rm S} {{\rm b} - {\rm a} \over {\rm c} - {\rm b}}
   = {\rm S} {{\rm d} - {\rm a} \over {\rm c} - {\rm d}}
      \div {\rm S} {{\rm d} - {\rm a} \over {\rm c} - {\rm d}}
   \eqno (179),$$
of which each member is unity, to the equation (176), attending
to the definition (175) of the new characteristic lately
introduced, we are conducted to this other formula,
$${{\rm b} - {\rm a} \over {\rm c} - {\rm b}}
         \div {\rm S} {{\rm b} - {\rm a} \over {\rm c} - {\rm b}}
   =  {{\rm d} - {\rm a} \over {\rm c} - {\rm d}}
         \div {\rm S} {{\rm d} - {\rm a} \over {\rm c} - {\rm d}}
   \eqno (180);$$
which allows us to write also
$${{\rm b} - {\rm a} \over {\rm c} - {\rm b}}
         \div {{\rm d} - {\rm a} \over {\rm c} - {\rm d}}
   =  {\rm S} {{\rm b} - {\rm a} \over {\rm c} - {\rm b}}
         \div {\rm S} {{\rm d} - {\rm a} \over {\rm c} - {\rm d}}
   \eqno (181),$$
where the second member, being the quotient of two scalars, is
itself another scalar, which may be denoted by $x$; and thus the
equation (173) may be obtained anew, as a consequence of the
equation (176).  We may therefore also deduce from the
last-mentioned equation the following form,
$${{\rm c} - {\rm d} \over {\rm d} - {\rm a}}
   =  x {{\rm c} - {\rm b} \over {\rm b} - {\rm a}}
   \eqno (182);$$
and thence also, by a new elimination of the scalar
coefficient~$x$, performed in the same manner as before, may
derive this other form,
$${{\rm V} \over {\rm S}} \mathbin{.}
         {{\rm c} - {\rm d} \over {\rm d} - {\rm a}}
   = {{\rm V} \over {\rm S}} \mathbin{.}
         {{\rm c} - {\rm b} \over {\rm b} - {\rm a}}
   \eqno (183).$$
Indeed, the geometrical signification of the condition (176)
shews easily that we may in any manner transpose, in that
condition, the symbols ${\rm a}$, ${\rm b}$, ${\rm c}$, ${\rm
d}$; since if, {\it before\/} such a transposition, those symbols
denoted four diverging straight lines (not generally in one
common plane), which terminate on the circumference of one common
circle, then {\it after\/} this transposition they must still
denote four such diverging lines.  We may therefore interchange
the symbols ${\rm a}$ and ${\rm c}$, in the condition (176),
which will thus become
$${{\rm V} \over {\rm S}} \mathbin{.}
         {{\rm b} - {\rm c} \over {\rm a} - {\rm b}}
   = {{\rm V} \over {\rm S}} \mathbin{.}
         {{\rm d} - {\rm c} \over {\rm a} - {\rm d}}
   \eqno (184);$$
but also, as in ordinary algebra, we have here,
$${{\rm b} - {\rm c} \over {\rm a} - {\rm b}}
   =  {{\rm c} - {\rm b} \over {\rm b} - {\rm a}};\quad
  {{\rm d} - {\rm c} \over {\rm a} - {\rm d}}
   =  {{\rm c} - {\rm d} \over {\rm d} - {\rm a}}
   \eqno (185);$$
the equation (183) might therefore have been in this other way
deduced from the equation (176), as another form of the same
condition of concircularity: and it is obvious that several other
forms of the same condition may be obtained in a similar way.

\bigbreak

26.
From the fundamental importance of the {\it circle} in
geometry, it is easy to foresee that these various forms of the
condition of concircularity must admit of a great number of
geometrical applications, besides those which have already been
given in some of the preceding articles of this essay on
Symbolical Geometry.  For example, we may derive in a new way a
solution of the problem proposed at the beginning of the 20th
article, by conceiving that the symbols ${\rm a}$, ${\rm b}$,
${\rm c}$ denote three given sides of a {\it cyclic cone},
extending from the vertex to some given plane which is parallel
to that one of the two {\it cyclic planes\/} which in the problem
is supposed to be given; for then the equation (183) may be
employed to express that the variable line~${\rm d}$ is a fourth
side of the same cyclic cone, drawn from the same vertex as an
origin, and bounded by the same given plane, or terminating on
the same circumference, or circular base of the cone, as the
three given sides, ${\rm a}$, ${\rm b}$, ${\rm c}$.  Or we may
change the symbol~${\rm d}$ to another symbol of the form
$x {\rm x}$, and may conceive that ${\rm x}$ denotes a variable
side of the cone, still drawn as before from the vertex, but not
now terminating on any one fixed plane, nor otherwise restricted
as to its length; while $x$ shall denote a scalar coefficient or
multiplier, so varying with the side or line~${\rm x}$ as to
render the product-line $x {\rm x}$ a side of which the extremity
is (like that of ${\rm d}$) concircular with the gven extremities
of ${\rm a}$, ${\rm b}$, ${\rm c}$; and we may express these
conceptions and conditions by writing as the equation of the cone
the following:
$${{\rm V} \over {\rm S}} \mathbin{.}
      {{\rm c} - x {\rm x} \over x {\rm x} - {\rm a}}
   = \beta
   \eqno (186);$$
where $\beta$ is a given geometrical fraction of the {\it
vector\/} class, namely, that vector which is determined by the
equation
$${{\rm V} \over {\rm S}} \mathbin{.}
      {{\rm c} - {\rm b} \over {\rm b} - {\rm a}}
   = \beta
   \eqno (187).$$

The {\it index\/} ${\rm I} \beta$ of this vector~$\beta$ is such that
$${\rm I} \beta \parallel {\rm I} {{\rm c} - {\rm b} \over {\rm b} - {\rm a}}
   \eqno (188);$$
it is therefore (by the principles of articles 7 and 10) a line
perpendicular to each of the two lines represented by the two
symbolical differences ${\rm c} - {\rm b}$, ${\rm b} - {\rm a}$,
and therefore also perpendicular to the line denoted by their
symbolical sum, ${\rm c} - {\rm a}$; so that we may establish the
three formul{\ae},
$${\rm I} \beta \perp {\rm c} - {\rm b};\quad
  {\rm I} \beta \perp {\rm b} - {\rm a};\quad
  {\rm I} \beta \perp {\rm c} - {\rm a}
   \eqno (189),$$
and may say that ${\rm I} \beta$ is a line {\it perpendicular to
the plane\/} in which all the three lines ${\rm a}$, ${\rm b}$,
${\rm c}$ all terminate.  This constant index ${\rm I} \beta$,
connected with the equation (186) of the cyclic cone just now
determined, as being the {\it index of the constant vector
fraction\/}~$\beta$, to which the first member of that equation
is equal, is therefore perpendicular also to the given cyclic
plane of the same cone, and may be regarded as a symbol for one
of the two {\it cyclic normals\/} of that conical locus of the
variable line~${\rm x}$ lately considered.  In the particular
case when the three given lines ${\rm a}$, ${\rm b}$, ${\rm c}$
are all {\it equally long}, so that the cyclic cone (186) becomes
a {\it cone of revolution\/}, then the index ${\rm I} \beta$,
which had been generally a symbol for a cyclic normal, becomes a
symbol for the {\it axis of revolution\/} of the cone.  Other
forms of equations of such cyclic and other cones will offer
themselves when the principles of the present system of
symbolical geometry shall have been more completely unfolded; but
the forms just given will be found to be sufficient, when
combined with some of the equations assigned in previous
articles, to conduct to the solution of some interesting
geometrical problems: to which class it will perhaps be permitted
to refer the general determination of the {\it curvature of a
spherical conic}, or the construction of the cone of revolution
which {\it osculates\/} along a given side to a given cyclic
cone.

\bigbreak

\centerline{\it Curvature of a Spherical Conic, or of a Cyclic
Cone.}

\nobreak\bigskip

27.
To treat this problem by a method which shall harmonise with the
investigations of recent articles of this paper, let the symbols
${\rm a}'$, ${\rm c}'$, ${\rm d}''$, be employed with the same
significations as in article~24, so as to denote three equally
long straight lines, of which ${\rm a}'$ is a trace of one cyclic
plane on the other, while ${\rm c}'$ and ${\rm d}''$ are the
traces of a tangent plane on those two cyclic planes; and let
${\rm c}$ (still bisecting the angle between ${\rm c}'$ and
${\rm d}''$) be still the equally long side of contact of that
tangent plane with the given cyclic cone.  We shall then have, by
(156), the symbolic analogy,
$${\rm d}'' : {\rm c} :: {\rm c} : {\rm c}'
   \eqno (190),$$
which, on account of the supposed equality of the lengths of the
lines ${\rm c}$, ${\rm c}'$, ${\rm d}''$, gives also the two
following formul{\ae}, of parallelism and perpendicularity,
$${\rm d}'' + {\rm c}' \parallel {\rm c};\quad
  {\rm d}'' - {\rm c}' \perp     {\rm c};\quad
   \eqno (191);$$
of which indeed the former has been given already, as the first of
the two formul{\ae} (160).  Conceive next that through the side
of contact~${\rm c}$ we draw two secant planes, cutting the same
sheet of the cone again in two known sides
${\rm c}_1$,~${\rm c}_2$, and having for their known traces on
the first cyclic plane (which contains the trace~${\rm c}'$ of
the tangent plane) the lines ${\rm c}_1'$,~${\rm c}_2'$, but for
their traces on the second cyclic plane (or on that which
contains ${\rm d}''$) the lines ${\rm d}_1''$,~${\rm d}_2''$;
these lines,
${\rm c} {\rm c}_1 {\rm c}_2 {\rm c}_1' {\rm c}_2' {\rm d}_1'' {\rm d}_2''$,
being supposed to be all equally long.  We may then write (in
virtue of what has been shewn in former articles) at once the two
new symbolic {\it analogies},
$${\rm d}_1'' : {\rm c}_1 :: {\rm c} : {\rm c}_1';\quad
  {\rm d}_2'' : {\rm c}_2 :: {\rm c} : {\rm c}_2'
   \eqno (192);$$
the two new {\it parallelisms},
$${\rm d}_1'' + {\rm c}_1' \parallel {\rm c}_1 + c;\quad
  {\rm d}_2'' + {\rm c}_2' \parallel {\rm c}_2 + c
   \eqno (193);$$
and the two new {\it perpendicularities},
$${\rm d}_1'' - {\rm c}_1' \perp {\rm c}_1 + c;\quad
  {\rm d}_2'' - {\rm c}_2' \perp {\rm c}_2 + c
   \eqno (194):$$
we shall have also these two other formul{\ae} of parallelism,
$${\rm d}_1'' - {\rm c}_1' \parallel {\rm c}_1 - c;\quad
  {\rm d}_2'' - {\rm c}_2' \parallel {\rm c}_2 - c
   \eqno (195).$$

Now if we conceive a cone of revolution to contain upon one sheet
the three equally long lines ${\rm c}$,~${\rm c}_1$~${\rm c}_2$,
which are also (by the construction) three sides of one sheet of
the given cyclic cone, we may (by the last article) represent a
line in the direction of the {\it axis\/} of this cone of
revolution by the symbol,
$${\rm I} {{\rm c}_2 - {\rm c} \over {\rm c} - {\rm c}_1}
   \eqno (196);$$
or by this other symbol, which denotes indeed a line having an
opposite direction, but still one contained upon the indefinite
axis of the same cone of revolution, if drawn from a point on
that axis,
$${\rm I} {{\rm c}_2 - {\rm c} \over {\rm c}_1 - {\rm c}}
   \eqno (197).$$

On account of the parallelisms (195) we may substitute for the
last symbol (197) this other of the same kind,
$${\rm I} {{\rm d}_2'' - {\rm c}_2' \over {\rm d}_1'' - {\rm c}_1'}
   \eqno (198);$$
which expression, when we add to it another, which is a symbol of
a null line (because in general the index of a scalar vanishes),
namely the following,
$${\rm I} {{\rm c}_1' - {\rm d}_1'' \over {\rm d}_1'' - {\rm c}_1'}
   = 0
   \eqno (199),$$
takes easily this other form,
$${\rm I} {{\rm d}_2'' - {\rm c}_2' \over {\rm d}_1'' - {\rm c}_1'}
   =  {\rm I} {{\rm d}_2'' - {\rm d}_1'' \over {\rm d}_1'' - {\rm c}_1'}
    + {\rm I} {{\rm c}_1' - {\rm c}_2' \over {\rm d}_1'' - {\rm c}_1'}
   \eqno (200).$$

The sought axis of the cone of revolution through the sides
${\rm c} {\rm c}_1 {\rm c}_2$ of the cyclic cone, or a line in
the direction of this axis, is therefore thus given, by the
expression (200), as the symbolic {\it sum\/} of two other lines;
which two new lines, by comparison of their expressions with the
form (188), are seen to be in the directions of the axes of
revolution of two new or {\it auxiliary cones\/} of revolution;
one of these auxiliary cones containing, upon a single sheet, the
three lines
$${\rm c}_1',\enspace {\rm d}_1'',\enspace {\rm d}_2''
   \eqno (201),$$
so that it may be briefly called the cone of revolution
${\rm c}_1'$~${\rm d}_1''$~${\rm d}_2''$; while the other
auxiliary cone of revolution, which may be called in like manner
the cone ${\rm c}_2'$~${\rm c}_1'$~${\rm d}_1''$, contains on one
sheet this other system of three straight lines,
$${\rm c}_1',\enspace {\rm d}_1'',\enspace {\rm c}_2'
   \eqno (202).$$

The symbolic {\it difference\/} of the same two lines, namely,
that of the lines denoted by the symbols
$${\rm I} {{\rm d}_2'' - {\rm d}_1'' \over {\rm d}_1'' - {\rm c}_1'},\quad
  {\rm I} {{\rm c}_1' - {\rm c}_2' \over {\rm d}_1'' - {\rm c}_1'}
   \eqno (203),$$
which lines are thus in the directions of the axes of these two
new cones of revolution, may easily be expressed under the form
$${\rm I} {{1 \over 2} ({\rm d}_2'' + {\rm c}_2')
      - {1 \over 2} ({\rm d}_1'' + {\rm c}_1')
   \over {1 \over 2} ({\rm d}_1'' + {\rm c}_1') - {\rm c}_1'}
   \eqno (204);$$
it is therefore (by the same last article) a line perpendicular
to the plane in which the three following straight lines
terminate, if drawn from one common point, such as the common
vertex of the four cones,
$${\rm c}_1',\quad
  {\textstyle {1 \over 2}} ({\rm d}_1'' + {\rm c}_1'),\quad
  {\textstyle {1 \over 2}} ({\rm d}_2'' + {\rm c}_2')
   \eqno (205).$$
This plane contains also the termination of the
line~${\rm d}_1''$, if that line be still drawn from the same
vertex; because, in general, whatever may be the value of the
scalar~$x$, the three straight lines denoted by the symbols
$${\rm c}_1',\quad
  (1 - x) {\rm d}_1'' + x {\rm c}_1',\quad
  {\rm d}_1''
   \eqno (206),$$
all terminate on one straight line, if they be drawn from one
common origin; and this last straight line is situated in the
first secant plane, and connects the extremities of the two
equally long lines ${\rm c}_1'$,~${\rm d}_1''$, which are the
traces of that second plane on the two cyclic planes.  The
remaining line,
${1 \over 2} ({\rm d}_2'' + {\rm c}_2')$,
of the system (205), if still drawn from the same vertex as
before, bisects that other straight line, situated in the same
secant plane, which connects the extremities of the two equally
long traces ${\rm c}_2'$,~${\rm d}_2''$, of that other secant
plane on the same two cyclic planes.  And these two connecting
lines, thus situated respectively in the first and second secant
planes do not generally intersect each other; because they cut
the line of mutual intersection of those two secant planes,
namely the side~${\rm c}$ of the given cyclic cone, in points
which are in general situated at different distances from the
vertex.  It is therefore in general a determinate problem, to
draw through the first of these two connecting lines a plane
which shall bisect the second: and we see that the plane so
drawn, being that in which the three lines (205) terminate, is
perpendicular to the line (204), that is to the symbolic
difference,
$${\rm I} {{\rm d}_2''- {\rm d}_1'' \over {\rm d}_1'' - {\rm c}_1'}
      - {\rm I} {{\rm c}_1'- {\rm c}_2' \over {\rm d}_1'' - {\rm c}_1'}
   \eqno (207),$$
of the two lines (203), of which the symbolic sum (200) has been
seen to be a line in the direction of the axis (197) of the first
cone of revolution considered in the present article; while the
two lines (203), of which we have thus taken the symbolic sum and
difference, have been perceived to be in the directions of the
axes of the two other and auxiliary cones of revolution, which we
have also had occasion to consider.  But in general, by one of
those fundamental principles which the present system of
symbolical geometry has in {\it common\/} with other systems, the
symbolical sum and difference of two adjacent and coinitial sides
of a parallelogram may be represented or constructed
geometrically by the two diagonals of that figure; namely the sum
by that diagonal which is intermediate between the two sides, and
the difference by that other diagonal which is transversal to
those sides: and every other transversal straight line, which is
drawn across the same two sides in the same direction as the
second diagonal, is bisected by the first diagonal, because the
two diagonals themselves bisect each other.  We may therefore
enunciate this theorem:---{\it If across the axes\/} (203)
{\it of the two auxiliary cones of revolution, which contain
respectively the two systems of straight lines\/} (201)
{\it and\/} (202), (each system of three straight lines being
contained upon a single sheet), {\it we draw a rectilinear
transversal, perpendicular to the plane which contains the first
and bisects the second of the two connecting lines, drawn as
before in the two secant planes; and if we then bisect this
transversal by a straight line drawn from the common vertex of
the cones: this bisecting line will be situated on the axis of
revolution\/} (197) {\it of that other cone of revolution, which
contains upon one sheet the three given sides of the given cyclic
cone.}  (The drawing of this transversal is possible, because the
preceding investigation shews that the plane of the axes of
revolution of the two auxiliary cones is perpendicular to that
other plane which is described in the construction.)

\bigbreak

28.
Since, generally, in the present system of symbolical geometry,
the vector part of the quotient of any two parallel lines, and
the scalar part of the quotient of any two perpendicular lines,
are respectively equal to zero, we may express that {\it three\/}
straight lines ${\rm a}$,~${\rm b}$,~${\rm c}$, if drawn from a
common origin, all {\it terminate on one common straight line},
by writing the equation
$${\rm V} {{\rm c} - {\rm a} \over {\rm b} - {\rm a}}
   = 0
   \eqno (208);$$
and may express that {\it two\/} straight lines,
${\rm a}$,~${\rm c}$, are {\it equally long}, or that they are
fit to be made adjacent sides of a rhombus (of which the two
diagonals are mutually rectangular), by this other formula:
$${\rm S} {{\rm c} + {\rm a} \over {\rm c} - {\rm a}}
   = 0
   \eqno (209).$$

If then we combine these two conditions, which will give
$${\rm S} {{\rm c} + {\rm a} \over {\rm b} - {\rm a}}
   = 0
   \eqno (210),$$
and therefore
$${\rm S} {{\rm c} \over {\rm b} - {\rm a}}
   = - {\rm S} {{\rm a} \over {\rm b} - {\rm a}},\quad
  {\rm V} {{\rm c} \over {\rm b} - {\rm a}}
   = {\rm V} {{\rm a} \over {\rm b} - {\rm a}}
   \eqno (211),$$
we shall thereby express that the chord or secant of a circle or
sphere, which passes through the extremity of one given
radius~${\rm a}$, and also through the extremity of another given
and coinitial straight line~${\rm b}$, meets the circumference of
the same circle or the surface of the same sphere again at the
extremity of the other straight line denoted by ${\rm c}$, which
will thus be another radius.  But with the same mode of
abridgment as that employed in the formula (178), we have, by
(211),
$$({\rm V} + {\rm S}) {{\rm c} \over {\rm b} - {\rm a}}
   =  ({\rm V} - {\rm S}) {{\rm a} \over {\rm b} - {\rm a}}
   \eqno (212),$$
and therefore
$${\rm c}
   =  ({\rm V} - {\rm S}) {{\rm a} \over {\rm b} - {\rm a}}
         \mathbin{.} ({\rm b} - {\rm a})
   \eqno (213).$$

This last is consequently an expression for the second
radius~${\rm c}$, in terms of the first radius~${\rm a}$, and of
the other given line~${\rm b}$ from the same centre, which
terminates at some given point upon the common chord or secant,
connecting the extremities of the two radii.  If therefore we
write for abridgment
$${\rm m} = {\textstyle {1 \over 2}} ({\rm d}_2'' + {\rm c}_2')
   \eqno (214),$$
so that ${\rm m}$ shall be a symbol for the last of the three
lines (205); and if we employ the two following expressions,
formed on the plan (213),
$$\left. \eqalign{
{\rm m}'
   &= ({\rm V} - {\rm S}) {{\rm c}_1' \over {\rm m} - {\rm c}_1'}
         \mathbin{.} ({\rm m} - {\rm c}_1') \cr
{\rm m}''
   &= ({\rm V} - {\rm S}) {{\rm d}_1'' \over {\rm m} - {\rm d}_1''}
         \mathbin{.} ({\rm m} - {\rm d}_1'') \cr}
   \right\}
   \eqno (215),$$
the symbols ${\rm c}_1'$, ${\rm d}_1''$ retaining their recent
meanings; then the four straight lines,
$${\rm c}_1',\enspace
  {\rm d}_1'',\enspace
  {\rm m}',\enspace
  {\rm m}''
   \eqno (216),$$
all drawn from the given vertex of the cones, will be equally
long, and will terminate in four concircular points; or, in other
words, their extremities will be the four corners of a certain
quadrilateral inscribed in a circle: of which plane
quadrilateral the two diagonals, connecting respectively the ends
of ${\rm c}_1'$,~${\rm m}'$, and of ${\rm d}_1''$,~${\rm m}''$,
will intersect each other at the extremity of the line~${\rm m}$,
which is drawn from the same vertex as before.  It may also be
observed respecting this line~${\rm m}$, that in virtue of its
definition (214), and of the second parallelism (193), it bisects
the angle between the two equally long sides ${\rm c}$,~${\rm c}_2$
of the given cyclic cone.  Thus {\it the four lines\/} (216) {\it
are four sides of one common sheet of a new cone of revolution, of
which the axis is perpendicular to the plane described in the
construction of the foregoing article\/}; because these four
equally long lines (216) terminate on the same plane as the three
lines (205), that is on a plane perpendicular to the line (204)
or (207), which latter line has thus the direction of the axis of
revolution of the new auxiliary cone. It is usual to say that
four diverging straight lines are {\it rays of an harmonic
pencil}, or simply that they are {\it harmonicals}, when a
rectilinear transversal, parallel to the fourth, and bounded by
the first and third, is bisected by the second of these lines: so
that, in general, any four diverging straight lines which can be
represented by the four symbols
$${\rm a},\enspace {\rm a} + {\rm b},\enspace
  {\rm b},\enspace {\rm a} - {\rm b},$$
or by the symbols which are obtained from these by giving them
any scalar coefficients, have the {\it directions\/} of four such
harmonicals.  We are then entitled to assert that {\it the
fourth harmonical to the axes of the three cones of revolution\/}
$$({\rm c}_1' {\rm d}_1'' {\rm d}_2''),\enspace
  ({\rm c} {\rm c}_1 {\rm c}_2),\enspace
  ({\rm c}_2' {\rm c}_1' {\rm d}_1'')
   \eqno (217),$$
which three axes have been already seen to be all situated in one
common plane, {\it is the axis of that new or fourth cone of
revolution\/} $({\rm c}_1' {\rm d}_1'' {\rm m}' {\rm m}'')$,
{\it which contains on one sheet the four straight lines\/}
(216).  And if we regard the four last-mentioned lines as
{\it edges of a tetrahedral angle}, inscribed in this new cone of
revolution, we see that {\it the two diagonal planes\/} of this
tetrahedral angle {\it intersect each other along a straight
line~${\rm m}$, which bisects the plane angle
$({\rm c}, {\rm c}_2)$ between two of the edges of the trihedral
angle $({\rm c} {\rm c}_1 {\rm c}_2)$}; which latter angle is at
once inscribed in the given cyclic cone, and also in that cone of
revolution which it was originally proposed to construct.

\bigbreak

29.
Conceive now that this original cone of revolution
$({\rm c} {\rm c}_1 {\rm c}_2)$
comes to {\it touch\/} the given cyclic cone along the
side~${\rm c}$, as a consequence of a gradual and unlimited
approach of the second secant plane $({\rm c} {\rm c}_2)$, to
coincidence with the given tangent plane
$({\rm c}' {\rm c} {\rm d}'')$, which touches the given cone
along that side; or in virtue of a gradual and indefinite
tendency of the side~${\rm c}_2$ to coincide with the given
side~${\rm c}$.  The line~${\rm m}$, bisecting always the angle
between these two sides ${\rm c}$, ${\rm c}_2$, will thus itself
also tend to coincide with ${\rm c}$; and the diagonal planes of
the tetrahedral angle
$({\rm c}_1' {\rm d}_1'' {\rm m}' {\rm m}'')$,
which planes still intersect each other in ${\rm m}$, will tend
at the same time to contain the same given side. But that
side~${\rm c}$ is (by the construction) a line in the plane of
one face of that tetrahedral angle, namely in the plane of
${\rm c}_1'$ and ${\rm d}_1''$, which was the first secant plane
of the cyclic cone; consequently the tetrahedral angle itself,
and its circumscribed cone of revolution, tend generally to
flatten together into coincidence with this secant plane, as
${\rm c}_2$ thus approaches to ${\rm c}$: and the axis of the
cone
$({\rm c}_1' {\rm d}_1'' {\rm m}' {\rm m}'')$
coincides ultimately with the normal to the first secant plane
$({\rm c}_1' {\rm d}_1'')$.  At the same time the traces
${\rm c}_2'$ and ${\rm d}_2''$, of the second secant plane on the
two cyclic planes, tend to coincide with the traces ${\rm c}'$
and ${\rm d}''$ of the given tangent plane thereupon.  We have
therefore this new theorem, which is however only a limiting form
of that enunciated in article~27:---If through a given
side~$({\rm c})$ of a given cyclic cone, we draw a tangent plane
$({\rm c}' {\rm c} {\rm d}'')$, and a secant plane
$({\rm c}_1' {\rm c} {\rm c}_1 {\rm d}_1'')$; and if we then
describe three cones of revolution, the first of these three
cones containing on one sheet the two traces
$({\rm c}_1', {\rm d}_1'')$ of the secant plane, and one
trace~$({\rm d}'')$ of the tangent plane; the second cone of
revolution touching the cyclic cone along the side of
contact~$({\rm c})$, and cutting it along the side of the
section~$({\rm c}_1)$; and the third cone of revolution
containing the same two traces $({\rm c}_1', {\rm d}_1'')$ of the
secant plane, and the other trace~$({\rm c}')$ of the tangent
plane: {\it the fourth harmonical to the axes of revolution of
these three cones will be perpendicular to the secant plane}.

\bigbreak

30.
Finally, conceive that the remaining secant plane
${\rm c}_1' {\rm d}_1''$ tends likewise to coincide with the
tangent plane ${\rm c}' {\rm d}''$; the cone of revolution which
lately {\it touched\/} the given cyclic cone along the given
side~${\rm c}$, will now come to {\it osculate\/} to that cone
along that side: and because a line in the direction of the
mutual intersection of the two cyclic planes has been already
denoted by ${\rm a}'$, therefore the first and third of the three
last-mentioned cones of revolution tend now to touch the planes
${\rm a}' {\rm d}''$ and ${\rm a}' {\rm c}'$, respectively, along
the lines ${\rm d}''$ and ${\rm c}'$.  The theorem of article~27,
at the limit here considered, takes therefore this new
form:---{\it If three cones of revolution be described, the first
cone cutting the first cyclic plane $({\rm a}' {\rm c}')$ along
the first trace~$({\rm c}')$ of a given tangent plane
$({\rm c}' {\rm c} {\rm d}'')$ to a given cyclic cone, and
touching the second cyclic plane $({\rm a}' {\rm d}'')$ along the
second trace~$({\rm d}'')$ of the same tangent plane; the second
cone of revolution osculating to the same cyclic cone, along the
given side of contact~$({\rm c})$; and the third cone of
revolution touching the first cyclic plane and cutting the second
cyclic plane, along the same two traces as before: then the
fourth harmonical to the axes of revolution of these three cones
will be the normal to the plane $({\rm c}' {\rm d}'')$ which
touches at once the given cyclic cone, and the sought osculating
cone, along the side~$({\rm c})$ of contact or of osculation.}

\bigbreak

31.
To deduce from this last theorem an {\it expression\/} for a
line~${\rm e}$ in the direction of the axis of the osculating
cone of revolution, by the processes of this symbolical geometry,
we may remark in the first place, that when any two straight
lines ${\rm a}$,~${\rm b}$, are equally long, we have the three
equations following:
$${\rm S} {{\rm a} \over {\rm b}}
   =  {\rm S} {{\rm b} \over {\rm a}},\quad
  {\rm V} {{\rm a} \over {\rm b}}
   =  - {\rm V} {{\rm b} \over {\rm a}},\quad
  {\rm I} {{\rm a} \over {\rm b}}
   =  - {\rm I} {{\rm b} \over {\rm a}}
   \eqno (218),$$
from the two former of which it may be inferred that the relation
$${{\rm V} \over {\rm S}} \mathbin{.} {{\rm a} \over {\rm b}}
   =  - {{\rm V} \over {\rm S}} \mathbin{.} {{\rm b} \over {\rm a}}
   \eqno (219) $$
holds good, not only when the two lines ${\rm a}$,~${\rm b}$, are
thus equal in length, but generally for any two lines: because if
we multiply or divide either of them by any scalar coefficient,
we only change thereby in one common (scalar) ratio both the
scalar and vector parts of their quotient, and so do not affect
that other quotient which is obtained by dividing the latter of
these two parts by the former.  We may also obtain the equation
(219), as one which holds good for any two straight lines
${\rm a}$,~${\rm b}$, under the form
$${\rm S} {{\rm b} \over {\rm a}} {\rm V} {{\rm a} \over {\rm b}}
      + {\rm V} {{\rm b} \over {\rm a}} {\rm S} {{\rm a} \over {\rm b}}
   = 0
   \eqno (220),$$
by operating with the characteristic~${\rm V}$ on the identity,
$${\rm S} {{\rm b} \over {\rm a}} \mathbin{.} {{\rm a} \over {\rm b}}
      + {\rm V} {{\rm b} \over {\rm a}} \mathbin{.} {{\rm a} \over {\rm b}}
   = {{\rm b} \over {\rm b}}
   = 1
   \eqno (221);$$
while if we operate on the same identity (221) by the
characteristic~${\rm S}$, we obtain this other general formula,
which likewise holds good for any two straight lines
${\rm a}$,~${\rm b}$, whether equal or unequal in length, and
will be useful to us on future occasions,
$${\rm S} {{\rm b} \over {\rm a}} {\rm S} {{\rm a} \over {\rm b}}
      + {\rm V} {{\rm b} \over {\rm a}} {\rm V} {{\rm a} \over {\rm b}}
   = 1
   \eqno (222).$$

Again, if there be three equally long lines,
${\rm a}$,~${\rm b}$,~${\rm c}$,
then since the principle contained in the third equation (218)
gives
$${\rm I} {{\rm b} - {\rm a} \over {\rm c}}
   =  {\rm I} {{\rm b} \over {\rm c}}
       - {\rm I} {{\rm a} \over {\rm c}}
   =  {\rm I} {{\rm c} \over {\rm a}}
       - {\rm I} {{\rm c} \over {\rm b}}
   \eqno (223),$$
which last expression is only multiplied by a scalar when the
line~${\rm c}$ is multiplied thereby; while the index of a
geometrical fraction is (among other properties) a line
perpendicular to both the numerator and denominator of the
fraction; we see that the symbol
$\displaystyle {\rm I} {{\rm c} \over {\rm a}}
       - {\rm I} {{\rm c} \over {\rm b}}$
denotes generally a line perpendicular to both ${\rm c}$ and
${\rm b} - {\rm a}$, if only the two lines ${\rm a}$ and
${\rm b}$ have their own lengths equal to each other, without any
restriction being thereby laid on the length of~${\rm c}$: this
symbol denotes therefore, under this single condition, a straight
line contained in a plane perpendicular to~${\rm c}$, and having
equal inclinations to ${\rm a}$ and ${\rm b}$.  Thus, under the
same condition, the symbol
$\displaystyle {\rm I} {{\rm c} \over {\rm a}}
       - {\rm I} {{\rm c} \over {\rm b}}$
may represent the axis~${\rm d}$ of a cone of revolution, which
contains upon one sheet the two equally long lines ${\rm a}$ and
${\rm b}$, while the third line~${\rm c}$ is in or parallel to
the {\it single\/} cyclic plane of this {\it monocyclic cone}, or
the plane of its circular base, or of one of its circular
sections; or coincides with or is parallel to some tangent to
such circular base or section.  If then we know any other line
${\rm a}'$, contained in the plane which touches this monocyclic
cone along the side~${\rm a}$, we may substitute for~${\rm c}$,
in this symbol
$\displaystyle {\rm I} {{\rm c} \over {\rm a}}
       - {\rm I} {{\rm c} \over {\rm b}}$,
that part or component of this new line~${\rm a}'$ which is
perpendicular to the side of contact~${\rm a}$; and therefore may
write with this view,
$${\rm c}
   =  {\rm V} {{\rm a}' \over {\rm a}} \mathbin{.} {\rm a}
   =  {\rm a}' - {\rm S} {{\rm a}' \over {\rm a}} \mathbin{.} {\rm a}
   \eqno (224),$$
which will give
$${\rm d}
   =  {\rm I} {{\rm a}' \over {\rm a}}
       - {\rm I} {{\rm a}' \over {\rm b}}
       + {\rm S} {{\rm a}' \over {\rm a}} {\rm I} {{\rm a} \over {\rm b}}
   \eqno (225),$$
as a general expression for a line~${\rm d}$ in the direction of
the axis of a cone of revolution which is touched by the plane
${\rm a} {\rm a}'$ along the side of contact~${\rm a}$, and
contains on the same sheet the equally long side~${\rm b}$.  We
may also remark that because the normal plane to a cone of
revolution, drawn along any side of that cone, contains the axis
of revolution, so that the plane containing the axis and the side
is perpendicular to the tangent plane, we have a relation between
the three directions of ${\rm a}$,~${\rm a}'$,~${\rm d}$, which
does not involve the direction of ${\rm b}$, and may be expressed
by any one of the three following formul{\ae}:---
$$\angle ({\rm a}', {\rm a}, {\rm d}) = {\pi \over 2},\quad
  {\rm d} \perp {\rm V} {{\rm a}' \over {\rm a}} \mathbin{.} {\rm a},\quad
  {\rm S} {{\rm a}' \over {\rm d}}
   =  {\rm S} {{\rm a}' \over {\rm a}} {\rm S} {{\rm a} \over {\rm d}}
   \eqno (226);$$
in each of which it is allowed to reverse the direction of
${\rm d}$, or to change ${\rm d}$ to $-{\rm d}$.  (Compare the
formul{\ae} (168), for the notation of dihedral angles.)  It may
indeed by easily proved, without the consideration of any cone,
that any one of these three formul{\ae} (226) involves the other
two; but we see also, by the recent reasoning, that they may all
be deduced when an expression of the form (225) for ${\rm d}$ is
given; or when this line~${\rm d}$ can be expressed in terms of
${\rm a}$, ${\rm a}'$, and of another line~${\rm b}$ which is
supposed to have the same length as ${\rm a}$, by any symbol
which differs only from the form (225) through the introduction
of a scalar coefficient.

These things being premised, if we change ${\rm a}$, ${\rm b}$,
${\rm d}$, in this form (225), to ${\rm c}'$, ${\rm d}''$,
${\rm n}'$, we find
$${\rm n}'
   =  {\rm I} {{\rm a}' \over {\rm c}'}
       - {\rm I} {{\rm a}' \over {\rm d}''}
       + {\rm S} {{\rm a}' \over {\rm c}'}
            {\rm I} {{\rm c}' \over {\rm d}''}
   \eqno (227),$$
as an expression for a line~${\rm n}'$ in the direction of the
axis of revolution of the cone which touches the first cyclic
plane~${\rm a}' {\rm c}'$ along the first trace~${\rm c}'$ of the
tangent plane, and cuts the second cyclic
plane~${\rm a}' {\rm d}''$ along the second trace~${\rm d}''$ of
the same tangent plane; that is to say, in the direction of the
axis of the third cone of revolution, described in the
enunciation of the theorem of article~30.  Again, if we change
${\rm a}$,~${\rm b}$,~${\rm d}$, in the same general formula
(225), to ${\rm d}''$, ${\rm c}'$, $- {\rm n}''$, and attend to
the third equation (218), we find
$${\rm n}''
   =  {\rm I} {{\rm a}' \over {\rm c}'}
       - {\rm I} {{\rm a}' \over {\rm d}''}
       + {\rm S} {{\rm a}' \over {\rm d}''}
            {\rm I} {{\rm c}' \over {\rm d}''}
   \eqno (228),$$
as an expression for another line~${\rm n}''$, in the direction
of the axis of another cone of revolution, which cuts the first
cyclic plane~${\rm a}' {\rm c}'$ along the trace~${\rm c}'$, and
touches the second cyclic plane~${\rm a}' {\rm d}''$ along the
other trace~${\rm d}''$ of the tangent plane; that is, in the
direction of the axis of revolution of the first of the three
cones, described in the enunciation of the same theorem of
article~30.  And since these expressions give
$${\rm n}'' - {\rm n}'
   =  \left(
         {\rm S} {{\rm a}' \over {\rm d}''}
       - {\rm S} {{\rm a}' \over {\rm c}'}
      \right)
      {\rm I} {{\rm c}' \over {\rm d}''}
   \eqno (229),$$
we have the two perpendicularities
$${\rm n}'' - {\rm n}' \perp {\rm c}',\quad
 {\rm n}'' - {\rm n}' \perp {\rm d}''
   \eqno (230);$$
so that a transversal drawn across the two axes of revolution
last determined, in the direction of this symbolic difference
${\rm n}'' - {\rm n}'$, is perpendicular to both the traces of the
tangent plane~${\rm c}' {\rm d}''$, and therefore has the
direction of the normal to that plane, or to the cyclic cone; or,
in other words, this transversal has the direction of the fourth
harmonical mentioned in the theorem.  But the lines ${\rm n}''$
and ${\rm n}'$, of which the symbolic {\it difference\/} has thus
been taken, have been seen to be in the directions of the first
and third of the same four harmonicals; and the axis of the
osculating cone, which axis we have denoted by ${\rm e}$, has (by
the theorem) the direction of the second harmonical: it has
therefore the direction of the symbolical {\it sum\/} of the same
two lines ${\rm n}''$,~${\rm n}'$, because it bisects their
transversal drawn as above.  Thus by conceiving the bisector to
terminate on the transversal, we find, as an expression for this
sought axis~${\rm e}$, the following,
$${\rm e}
   =  {\textstyle {1 \over 2}} ({\rm n}'' + {\rm n}')
   =  {\rm I} {{\rm a}' \over {\rm c}'}
       - {\rm I} {{\rm a}' \over {\rm d}''}
       + {\textstyle {1 \over 2}}
            \left( {\rm S} {{\rm a}' \over {\rm c}'}
               + {\rm S} {{\rm a}' \over {\rm d}''} \right)
            {\rm I} {{\rm c}' \over {\rm d}''}
   \eqno (231).$$

\bigbreak

32.
This symbolical expression for ${\rm e}$ contains, under a not
very complex form, the solution of the problem on which we have
been engaged; namely, {\it to find the axis of the cone of
revolution, which osculates along a given side to a given cyclic
cone}.  It may however be a little simplified, and its general
interpretation made easier, by resolving the line~${\rm a}'$ into
two others, which shall be respectively parallel and
perpendicular to the {\it lateral normal plane}, as follows:
$${\rm a}'
   =  {\rm a}^\backprime + {\rm a}^{\backprime\backprime};\quad
  {\rm a}^\backprime \perp {\rm d}'' - {\rm c}';\quad
  {\rm a}^{\backprime\backprime} \parallel {\rm d}'' - {\rm c}'
   \eqno (232);$$
so that
$${\rm a}^\backprime
   =  {\rm V} {{\rm a}' \over {\rm d}'' - {\rm c}'}
         \mathbin{.} ({\rm d}'' - {\rm c}');\quad
  {\rm a}^{\backprime\backprime}
   =  {\rm S} {{\rm a}' \over {\rm d}'' - {\rm c}'}
         \mathbin{.} ({\rm d}'' - {\rm c}')
   \eqno (233);$$
which will give, by (191) and (218), because
${\rm a}^{\backprime\backprime} \perp {\rm d}'' + {\rm c}'$,
$${\rm S} {{\rm d}'' \over {\rm a}^{\backprime\backprime}}
       + {\rm S} {{\rm c}' \over {\rm a}^{\backprime\backprime}}
   =  0;\quad
  {\rm S} {{\rm a}^{\backprime\backprime} \over {\rm c}'}
       + {\rm S} {{\rm a}^{\backprime\backprime} \over {\rm d}''}
   =  0
   \eqno (234);$$
also
$${\rm I} {{\rm d}'' \over {\rm a}^{\backprime\backprime}}
       - {\rm I} {{\rm c}' \over {\rm a}^{\backprime\backprime}}
   =  0;\quad
  {\rm I} {{\rm a}^{\backprime\backprime} \over {\rm c}'}
       - {\rm I} {{\rm a}^{\backprime\backprime} \over {\rm d}''}
  =  0
   \eqno (235);$$
and
$${\rm S} {{\rm d}'' \over {\rm a}^\backprime}
       - {\rm S} {{\rm c}' \over {\rm a}^\backprime}
   =  0;\quad
  {\rm S} {{\rm a}^\backprime \over {\rm c}'}
       - {\rm S} {{\rm a}^\backprime \over {\rm d}''}
  =  0
   \eqno (236).$$

For by thus resolving ${\rm a}'$, in (231), into the two
components ${\rm a}^\backprime$ and
${\rm a}^{\backprime\backprime}$, it is at once seen, by (234)
(235), that the latter component
${\rm a}^{\backprime\backprime}$ disappears from the result,
which reduces itself by (236) to the following simplified form,
$${\rm e}
   =  {\rm I} {{\rm a}^\backprime \over {\rm c}'}
       - {\rm I} {{\rm a}^\backprime \over {\rm d}''}
       + {\rm S} {{\rm a}^\backprime \over {\rm c}'}
            {\rm I} {{\rm c}' \over {\rm d}''}
   \eqno (237);$$
and this gives, by comparison with the forms (225) and (226), a
remarkable relation of rectangularity between two planes, of which
one contains the axis~${\rm e}$ of the osculating cone, namely
the planes ${\rm a}^\backprime {\rm c}'$ and ${\rm c}' {\rm e}$;
which relation is expressed by the formula,
$$\angle ({\rm a}^\backprime, {\rm c}', {\rm e})
   =  {\pi \over 2}
   \eqno (238).$$

In like manner, from the same expression (231), by the same
decomposition of ${\rm a}'$, we may easily deduce, instead of
(237), this other expression for the axis of the osculating cone,
$${\rm e}
   =  {\rm I} {{\rm a}^\backprime \over {\rm c}'}
       - {\rm I} {{\rm a}^\backprime \over {\rm d}''}
       - {\rm S} {{\rm a}^\backprime \over {\rm d}''}
            {\rm I} {{\rm d}'' \over {\rm c}'}
   \eqno (239);$$
and may derive from it this other relation, of rectangularity
between two other planes, namely the planes
${\rm a}^\backprime {\rm d}''$ and ${\rm d}'' {\rm e}$,
$$\angle ({\rm a}^\backprime, {\rm d}'', {\rm e})
   =  {\pi \over 2}
   \eqno (240).$$

Hence follows immediately this theorem, which furnishes a
remarkably simple {\it construction with planes}, for determining
generally a line in the required direction of the axis of the
osculating cone:---{\it If we project the line~${\rm a}'$ of
mutual intersection of the two cyclic planes
${\rm a}' {\rm c}'$, ${\rm a}' {\rm d}''$, of any given cyclic
cone, on the lateral normal plane which is drawn along any given
side~${\rm c}$; if we next draw two planes,
${\rm a}^\backprime {\rm c}'$, ${\rm a}^\backprime {\rm d}''$,
through the projection~${\rm a}^\backprime$ thus obtained, and
through the two traces, ${\rm c}'$, ${\rm d}''$, of the tangent
plane on the two cyclic planes; and if we then draw two new
planes, ${\rm c}' {\rm e}$, ${\rm d}'' {\rm e}$, through the same
two traces of the tangent plane, perpendicular respectively to
the two planes
${\rm a}^\backprime {\rm c}'$, ${\rm a}^\backprime {\rm d}''$,
last drawn: these two new planes will intersect each other along
the axis~${\rm e}$ of the cone of revolution, which osculates
along the given side~${\rm c}$ to the given cyclic cone.}

And by considering, instead of these cones and planes, their
intersection with a spheric surface described about the common
vertex, we arive at the following {\it spherographic
construction},\footnote*{This construction was communicated to
the Royal Irish Academy (see {\it Proceedings\/}), at its meeting
of November~30th, 1847, along with a simple geometrical
construction for generating a system of two reciprocal ellipsoids
by means of a moving sphere, as new applications of the author's
Calculus of Quaternions to Surfaces of the Second Order.  With
that Calculus, of which the fundamental principles and
formul{\ae} were communicated to the same Academy on the 13th of
November, 1843, it will be found that the present System of
Symbolical Geometry is connected by very intimate relations,
although the subject is approached, in the two methods, from two
quite different points of view: the {\it algebraical
quaternion\/} of the one method being {\it ultimately\/} the same
as the {\it geometrical fraction\/} of the other.}
for finding the {\it spherical centre of curvature of a given
spherical conic\/} at a given point, or the pole of the small
circle which osculates at that point to that conic:---{\it From
one of the two points of mutual intersection of the two cyclic
arcs let fall a perpendicular upon the normal arc to the conic,
which latter arc is drawn through the given point of osculation;
connect the foot of this (arcual) perpendicular by two other arcs
of great circles, with those two known points, equidistant from
the point upon the conic, where the tangent arc meets the two
cyclic arcs; draw through the same two points two new arcs of
great circles, perpendicular respectively to the two connecting
arcs: these two new arcs will cross each other on the normal arc,
in the pole of the osculating circle, or in the spherical centre
of curvature of the spherical conic}, which centre it was
required to determine.

\bigbreak

\centerline{\it On Elliptic Cones, and on their Osculating Cones of Revolution.}

\nobreak\bigskip

33.
With the same significations of ${\rm a}'$, ${\rm a}^\backprime$,
${\rm c}$, ${\rm c}'$, ${\rm d}''$, and ${\rm e}$, as symbols of
certain straight lines, connected with a given cyclic cone, as in
the last article of this Essay; and with the same use of the
sign~${\rm I}$, as the characteristic of the {\it index\/} of the
vector part of any geometrical fraction in general; if we now
write
$${\rm f} = {\rm I} {{\rm a}'  \over {\rm c}'};\quad
  {\rm g} = {\rm I} {{\rm d}'' \over {\rm a}'};\quad
  {\rm h} = {\rm I} {{\rm d}'' \over {\rm c}}
          = {\rm I} {{\rm c}   \over {\rm c}'}
   \eqno (241);$$
$${\rm i} = {\rm I} {{\rm a}^\backprime  \over {\rm c}'};\quad
  {\rm k} = {\rm I} {{\rm d}'' \over {\rm a}^\backprime};\quad
  {\rm l} =   {{\rm h}   \over {\rm c}} {\rm a^\backprime}
          \parallel {\rm I} {{\rm a}'  \over {\rm a}^\backprime}
   \eqno (242);$$
% CHECK (242) above
we shall thus form symbols for certain other straight lines,
${\rm f}$, ${\rm g}$, ${\rm h}$, and ${\rm i}$, ${\rm k}$, ${\rm
l}$, which may be conceived to be all drawn from the same common
origin as the former lines, namely from the vertex of the cyclic
cone.  And these new lines will be found to be connected with
{\it another\/} cone, which may be called an {\it
elliptic\/}\footnote\dag{The methods of the present Symbolical
Geometry might here be employed to prove that the {\it normal
cone}, here called {\it elliptic}, from its connexion with its
two focal lines, is itself {\it another cyclic cone\/}; being cut
in circles by two sets of planes, which are perpendicular
respectively to the two focal lines of the former cone.  But it
may be sufficient thus to have alluded to this well-known
theorem, which it is not necessary for our present purpose to
employ.  There is even a convenience in retaining, for awhile,
the two contrasted designations of {\it cyclic\/} and
{\it elliptic}, for these two reciprocal cones, to mark more
strongly the difference of the modes in which they here present
themselves to our view.}
{\it cone\/}; namely the cone which is {\it normal},
{\it supplementary}, or {\it reciprocal\/} to the former
{\it cyclic\/} cone.  They may also be employed to assist in the
determination of the {\it cone of revolution}, which
{\it osculates\/} along a given side to this new or elliptic
cone; as will be seen by the following investigation.

\bigbreak

34.
The lines ${\rm f}$ and ${\rm g}$ being, as is shewn by their
expressions (241), perpendicular respectively to the planes
${\rm a}' {\rm c}'$ and ${\rm a}' {\rm d}''$, which were the two
cyclic planes of the former or cyclic cone, are themselves the two
{\it cyclic normals\/} of that cone; and because the line
${\rm h}$ is, by the same system of expressions (241),
perpendicular to the plane ${\rm c}' {\rm d}''$ which touches
that cyclic cone along the side ${\rm c}$, it is the variable
normal of that former cone: or this new line ${\rm h}$ is the
{\it side\/} of the new or normal cone, which {\it corresponds\/}
to that old side~${\rm c}$.  The inclinations of ${\rm h}$ to
${\rm f}$ and ${\rm g}$, respectively, are given by the following
equations, which are consequences of the same expressions (241):
$$\left. \eqalign{
\angle ({\rm f}, {\rm h})
   &= \angle ({\rm a}', {\rm c}', {\rm c})
    = \angle ({\rm a}', {\rm c}', {\rm d}'') \cr
\angle ({\rm h}, {\rm g})
   &= \angle ({\rm a}', {\rm d}'', {\rm c})
    = \angle ({\rm a}', {\rm d}'', {\rm c}') \cr}
   \right\}
   \eqno (243);$$
and we have seen, in article~24, that for the cyclic cone an
equation which may now be thus written holds good:
$$\angle ({\rm a}', {\rm c}', {\rm d}'')
      + \angle ({\rm a}', {\rm d}'', {\rm c}')
   = 2a
   \eqno (244);$$
where $a$ is a constant angle: therefore for the cone of normals
to that cyclic cone, the following other equation is satisfied:
$$\angle ({\rm f}, {\rm h})
      + \angle ({\rm h}, {\rm g})
   = 2a
   \eqno (245);$$
$a$ being here the same constant as before.  The sum of the
inclinations of the variable side~${\rm h}$ of the new or
{\it elliptic\/} cone to the two fixed lines ${\rm f}$ and
${\rm g}$ is therefore constant; in consequence of which known
property, these two fixed lines are called the {\it focal
lines\/} of the elliptic cone.  And we see that these two
{\it focal lines\/} ${\rm f}$,~${\rm g}$, of the {\it normal\/}
cone, coincide, respectively, in their directions, with the two
{\it cyclic normals\/} (or with the normals to the two cyclic
planes) of the {\it original\/} cone: which is otherwise known to
be true.

\bigbreak

35.
Another important and well-known property of the elliptic cone
may be proved anew by observing that the expressions (241) give
$$\left. \eqalign{
\angle ({\rm f}, {\rm h}, {\rm c})
   &= \angle ({\rm f}, {\rm h}, {\rm c}')
      - \angle ({\rm c}, {\rm h}, {\rm c}')
    = {\textstyle {1 \over 2}} \pi - \angle ({\rm c}, {\rm c}') \cr
\angle ({\rm c}, {\rm h}, {\rm g})
   &= \angle ({\rm d}'', {\rm h}, {\rm g})
      - \angle ({\rm d}'', {\rm h}, {\rm c})
    = {\textstyle {1 \over 2}} \pi - \angle ({\rm d}'', {\rm c}) \cr} 
   \right\}
   \eqno (246);$$
and that we have, by (190),
$$\angle ({\rm d}'', {\rm c}) = \angle ({\rm c}, {\rm c}')
   \eqno (247);$$
for thus we see that
$$\angle ({\rm f}, {\rm h}, {\rm c})
   = \angle ({\rm c}, {\rm h}, {\rm g})
   \eqno (248);$$
that is to say, the lateral normal plane ${\rm h} {\rm c}$ to the
reciprocal or elliptic cone (which is at the same time the
lateral normal plane of the original or cyclic cone) bisects the
dihedral angle $\angle ({\rm f}, {\rm h}, {\rm g})$, comprised
between the two {\it vector planes}, ${\rm f} {\rm h}$,
${\rm h} {\rm g}$, which connect the side~${\rm h}$ of the
elliptic cone with the two focal lines ${\rm f}$ and ${\rm g}$.

Or, because the expressions (241) shew that these two vector
planes, ${\rm f} {\rm h}$, ${\rm h} {\rm g}$, of the elliptic
cone, are perpendicular respectively to the two {\it traces\/}
${\rm c}'$ and ${\rm d}''$ of the tangent plane to the cyclic
cone, on the two cyclic planes of that cone; which traces are, as
the formula (247) expresses, inclined equally to the side of
contact~${\rm c}$ of the original or cyclic cone, while that side
or line~${\rm c}$ is also the normal to the reciprocal or
elliptic cone; we might hence infer that the tangent plane to the
latter cone is equally inclined to the two vector planes: which
is another form of the known relation.

\bigbreak

36.
The expressions (242), combined with (241), shew that the two new
lines ${\rm i}$ and ${\rm k}$, as being perpendicular
respectively to the two traces ${\rm c}'$ and ${\rm d}''$, are
contained respectively in the two vector planes ${\rm f} {\rm h}$
and ${\rm h} {\rm g}$.  But each of the same two lines,
${\rm i}$, ${\rm k}$, is also perpendicular to the line
${\rm a}^\backprime$, to which the remaining new line~${\rm l}$
is also perpendicular, as the same expressions shew; they shew
too that ${\rm a}^\backprime$ is a line in the common lateral and
normal plane ${\rm c} {\rm h}$ of the two cones, while ${\rm l}$
is also contained in that plane: the plane ${\rm i} {\rm k}$
therefore cuts the plane ${\rm c} {\rm h}$ perpendicularly in the
line~${\rm l}$.  This latter line~${\rm l}$ is also, by the same
expressions, perpendicular to the line~${\rm a}'$ (that is to the
intersection of the two cyclic planes of the cyclic cone), which
is perpendicular to both ${\rm f}$ and ${\rm g}$; and therefore
${\rm l}$ can be determined, as the intersection of the common
normal plane ${\rm c} {\rm h}$ with the plane  of the two focal
lines ${\rm f} {\rm g}$; after which, by drawing through the
line~${\rm l}$, thus found, a plane ${\rm i} {\rm k}$
perpendicular to ${\rm c} {\rm h}$, the lines ${\rm i}$ and
${\rm k}$ may be obtained, as the respective intersections of
this last perpendicular plane with the two vector planes,
${\rm f} {\rm h}$, ${\rm h} {\rm g}$.  And we see that these
three new lines, ${\rm i}$, ${\rm k}$, ${\rm l}$, introduced by
the expressions (242), are such as to satisfy the following
conditions of dihedral perpendicularity:
$${\textstyle {1 \over 2}} \pi
    = \angle ({\rm h}, {\rm l}, {\rm i})
    = \angle ({\rm k}, {\rm l}, {\rm h})
   \eqno (249);$$
$${\textstyle {1 \over 2}} \pi
    = \angle ({\rm h}, {\rm i}, {\rm c}')
    = \angle ({\rm d}'', {\rm k}, {\rm h})
   \eqno (250);$$
$${\textstyle {1 \over 2}} \pi
    = \angle ({\rm a}^\backprime, {\rm c}', {\rm i})
    = \angle ({\rm a}^\backprime, {\rm d}'', {\rm k})
   \eqno (251);$$
with which we may combine the following relations:
$$\angle ({\rm f}, {\rm h}, {\rm i})
    = \angle ({\rm k}, {\rm h}, {\rm g})
    = 0;\quad
  \angle ({\rm f}, {\rm l}, {\rm g})
    = \pi;$$
$$\angle ({\rm f}, {\rm h}, {\rm l})
   = \angle ({\rm l}, {\rm h}, {\rm g})
   \eqno (252).$$

\bigbreak

37.
The positions of these three lines ${\rm i}$, ${\rm k}$, ${\rm
l}$, being thus fully known, by means of the expressions (242),
or of the corollaries which have been deduced from those
expressions, let us now consider, in connexion with them, the two
formul{\ae} of dihedral perpendicularity, (238), (240), which
were given in article~32, to determine the axis~${\rm e}$ of a
cone of revolution, which osculates along the side~${\rm c}$ to
the given cyclic cone, and which formul{\ae} may be thus
collected:
$${\textstyle {1 \over 2}} \pi
    = \angle ({\rm a}^\backprime, {\rm c}', {\rm e})
    = \angle ({\rm a}^\backprime, {\rm d}'', {\rm e})
   \eqno (253).$$
The comparison of (253) with (251) shews that the planes
${\rm c}' {\rm e}$, ${\rm d}'' {\rm e}$, must coincide
respectively with the planes ${\rm c}' {\rm i}$,
${\rm d}'' {\rm k}$; because they are drawn like them
respectively through the lines ${\rm c}'$, ${\rm d}''$, and are
like them perpendicular respectively to the planes
${\rm a}^\backprime {\rm c}'$, ${\rm a}^\backprime {\rm d}''$;
the line~${\rm e}$ must therefore be the intersection of the two
planes ${\rm c}' {\rm i}$, ${\rm d}'' {\rm k}$, which contain
respectively the two lines ${\rm i}$, ${\rm k}$, and are, by
(250), perpendicular to the two planes ${\rm i} {\rm h}$,
${\rm k} {\rm h}$, or (by what has been seen in the last article)
to the two vector planes ${\rm f} {\rm h}$, ${\rm g} {\rm h}$.
We can therefore construct the line~${\rm e}$ as the intersection
of the two planes ${\rm i} {\rm e}$, ${\rm k} {\rm e}$, which are
thus drawn through the lately determined lines
${\rm i}$,~${\rm k}$, at right angles to the two vector planes;
and we may write, instead of (253), the formul{\ae}
$${\textstyle {1 \over 2}} \pi
    = \angle ({\rm h}, {\rm i}, {\rm e})
    = \angle ({\rm h}, {\rm k}, {\rm e})
   \eqno (254).$$

\bigbreak

38.
Again, because this line~${\rm e}$ is (by Art.~32) the axis of a
cone of revolution which {\it osculates\/} to the given cyclic
cone, or which touches that cone not only along the
side~${\rm c}$ itself but also along another side infinitely near
thereto; while, in general, the lateral normal planes of a cone
of revolution all cross each other along the axis of that cone;
it is clear that ${\rm e}$ must be the line along which the
common lateral and normal plane~${\rm c} {\rm h}$ of the two
reciprocal cones is intersected by an infinitely near normal and
lateral plane to the first or cyclic cone, which is also at the
same time a lateral and normal plane to the second or elliptic
cone; consequently {\it the two cones of revolution which
osculate to these two reciprocal cones, along these two
corresponding sides}, ${\rm c}$ and ${\rm h}$, {\it have one
common axis},~${\rm e}$.  And it is evident that a similar result
for a similar reason holds good, in the more general case of
{\it any two reciprocal cones}, which have a common vertex, and
of which each contains upon its surface all the normals to the
other cone, {\it however arbitrary the form\/} of either cone may
be; any two such cones having always {\it one common system of
lateral and normal planes}, and {\it one common conical envelope
of all those normal planes\/}: which common envelope is thus the
{\it common conical surface of centres of curvature}, for the two
reciprocal cones.

Eliminating therefore what belongs, in the present question, to
the original or cyclic cone, or confining ourselves to the
formul{\ae} (245), (249), (252), (254), we are conducted to the
following {\it construction, for determining the axis~${\rm e}$
of that new cone of revolution, which osculates along a given
side~${\rm h}$ to a given elliptic cone\/}; this latter cone
having ${\rm f}$ and ${\rm g}$ for its given focal lines, or
being represented by an equation of the form (245):---{\it Draw,
through the given side,~${\rm h}$, the normal
plane~${\rm h} {\rm l}$, bisecting the angle between the two
vector planes, ${\rm f} {\rm h}$, ${\rm g} {\rm h}$, and meeting
in the line~${\rm l}$ the plane ${\rm f} {\rm g}$ of the two
given focal lines; through the same line~${\rm l}$ draw another
plane~${\rm i} {\rm k}$, perpendicular to the normal
plane~${\rm h} {\rm l}$, and cutting the vector planes in two new
lines, ${\rm i}$ and ${\rm k}$; through these new lines draw two
new planes, ${\rm i} {\rm e}$,~${\rm k} {\rm e}$, perpendicular
respectively to the two vector planes ${\rm f} {\rm i}$,
${\rm g} {\rm k}$, or ${\rm f} {\rm h}$, ${\rm g} {\rm h}$: these
two planes will cross each other on the normal plane, in the
sought axis~${\rm e}$ of the osculating cone of revolution.}

\bigbreak

39.
Or if we prefer to consider, instead of cones and planes, their
intersections with a spheric surface described about the common
vertex, as its centre; we then arrive at the following {\it
spherographic construction, for finding the spherical centre of
curvature of a given spherical ellipse}, at any given point of
that curve, which may be regarded as being the {\it reciprocal\/}
of the construction assigned at the end of the 32nd article of
this essay:---Draw, from the given point~${\sc h}$, of the
ellipse, the normal arc ${\sc h} {\sc l}$, bisecting the
spherical angle ${\sc f} {\sc h} {\sc g}$ between the two vector
arcs ${\sc f} {\sc h}$, ${\sc g} {\sc h}$, and terminated at
${\sc l}$ by the arc ${\sc f} {\sc g}$ which connects the two
given foci, ${\sc f}$ and ${\sc g}$; through~${\sc l}$ draw an
arc of a great circle ${\sc i} {\sc k}$, perpendicular to the
normal arc ${\sc h} {\sc l}$, and cutting one of the two vector
arcs ${\sc h} {\sc f}$, ${\sc h} {\sc g}$, and the other of those
two vector arcs prolonged, in two new points, ${\sc i}$ and
${\sc k}$; through these two new points draw two new arcs of
great circles, ${\sc i} {\sc e}$, ${\sc k} {\sc e}$,
perpendicular respectively to the two vector arcs, or to the
arcs ${\sc h} {\sc i}$, ${\sc h} {\sc k}$: {\it the two new
arcs so drawn will cross each other on the normal arc
(prolonged), in a point~${\sc e}$, which will be the spherical
centre of curvature sought}, or the pole of the small circle
which osculates at the given point~${\sc h}$ to the given
spherical ellipse.

And since it is obvious (on account of the spherical right angles
${\sc h} {\sc i} {\sc e}$, ${\sc h} {\sc k} {\sc e}$, in the
construction), that the points ${\sc i}$,~${\sc k}$ are the
respective middle points of those portions of the vector arcs, or
of those arcs prolonged, which are comprised within this
osculating circle; so that the arc ${\sc i} {\sc k}$, which has
been seen to pass through the point~${\sc l}$, and which crosses
at that point~${\sc l}$ the arcual major axis of the ellipse
(because that axis passes through both foci), is the {\it common
bisector\/} of these two intercepted portions of the vector arcs,
which intercepted arcs of great circles may be called (on the
sphere) the two {\it focal chords of curvature\/} of the
spherical ellipse; we are therefore permitted to enunciate the
following {\it theorem},\footnote*{This theorem was proposed by
the present writer, in June 1846, at the Examination for Bishop
Law's Mathematical Premium, in Trinity College, Dublin; and it
was shewn by him in a series of Questions on that occasion, which
have since been printed in the Dublin University Calendar for
1847, (see p.~{\sc lxx}), among the University Examination Papers
for the preceding year, that this theorem, and several others
connected therewith (for example, that the trigonometric tangent
of the focal half chord of curvature is the harmonic mean between
the tangents of the two focal vector arcs), might be deduced by
{\it spherical trigonometry}, from the known constancy of the sum
of the two vector arcs, or focal distances, for any one spherical
ellipse.  But in the method employed in the present essay, no use
whatever has hitherto been made of any formula of spherical or
even plane trigonometry, any more than of the doctrine of
coordinates.}
which is in general sufficient for the determination of the
spherical centre of curvature, or pole of the osculating small
circle, at any proposed point of any such ellipse:---{\it The
great circle which bisects the two focal (and arcual) chords of
curvature of any spherical ellipse, for any point of osculation,
intersects the (arcual) axis major in the same point in which that
axis is cut by the (arcual) normal to the ellipse, drawn at the
point of osculation.}

\bigbreak

\nobreak\bigskip

\centerline{\it On the Tensor of a Geometrical Quotient.}

\nobreak\bigskip

40.
The equations (218) (222), of Art.~31, shew that for any two
equally long straight lines, ${\rm a}$,~${\rm b}$, the following
relation holds good,
$$\left( {\rm S} {{\rm b} \over {\rm a}} \right)^2
     - \left( {\rm V} {{\rm b} \over {\rm a}} \right)^2
   = 1
   \eqno (255);$$
or, more concisely, that
$${\rm T} {{\rm b} \over {\rm a}} = 1
   \eqno (256),$$
if we introduce a new characteristic~${\rm T}$ of operation on a
geometrical quotient, defined by the general formula,
$${\rm T} {{\rm b} \over {\rm a}}
   = \sqrt{\vphantom{\biggl\{}}
      \left\{
         \left( {\rm S} {{\rm b} \over {\rm a}} \right)^2
       - \left( {\rm V} {{\rm b} \over {\rm a}} \right)^2
      \right\}
   \eqno (257);$$
where it is to be observed, that the expression of which the
square root is taken is essentially a positive scalar, because
the square of {\it every\/} scalar is {\it positive}, while the
square of {\it every\/} vector is on the contrary a
{\it negative\/} scalar, by the principles of the 12${}^{\rm th}$
article.  Hence, generally, for {\it any two\/} straight lines
${\rm a}$,~${\rm b}$, of which the lengths are denoted by
$\overline{\rm a}$,~$\overline{\rm b}$, we have the equation,
$${\rm T} {{\rm b} \over {\rm a}}
   =  \overline{\rm b} \div \overline{\rm a}
   \eqno (258);$$
because the expression (257) is doubled, tripled, or multiplied
by any positive number, when the line~${\rm b}$ is multiplied by
the same number, whatever it be, while the line~${\rm a}$ remains
unchanged.  This geometrical signification of the expression
$\displaystyle {\rm T} {{\rm b} \over {\rm a}}$,
may induce us to name that expression the {\sc tensor} of the
geometrical quotient
$\displaystyle {{\rm b} \over {\rm a}}$,
on which the characteristic~${\rm T}$ has operated; because this
{\it tensor\/} is a number which directs us how to {\it extend\/}
(directly or inversely, that is, in what ratio to lengthen or
shorten) the denominator line~${\rm a}$, in order to render it
{\it as long\/} as the numerator line~${\rm b}$: and it appears
to the writer, that there are other advantages in adopting this
name ``tensor'', with the signification defined by the formula
(257).  Adopting it, then, we might at once be led to see, by
(258), from considerations of compositions of ratios between the
lengths of lines, that in any multiplication of geometrical
quotients among themselves, ``the tensor of the product is equal
to the product of the tensors.''  But to establish this important
principle otherwise, we may observe that by the equations (87),
(88), (99), (100), of Arts.\ 11, 13, the vector part~$\gamma$ of
the product $c + \gamma$ of any two geometrical quotients,
represented by the binomial forms $b + \beta$, $a + \alpha$, is
changed to its own opposite, $- \gamma$, while the scalar
part~$c$ of the same product remains unchanged, when we change
the signs of the vector parts $\beta$,~$\alpha$, of the two
factors, without changing their scalar parts $b$, $a$, and also
{\it invert}, at the same time, the {\it order\/} of those
factors; in such a manner that either of the two following
{\it conjugate equations\/} includes the other:
$$\left. \eqalign{
c + \gamma &= (b + \beta) (a + \alpha) \cr
c - \gamma &= (a - \alpha) (b - \beta) \cr}
   \right\}
   \eqno (259);$$
and these two conjugate equations give, by multiplication,
$$c^2 - \gamma^2 = (b^2 - \beta^2) (a^2 - \alpha^2)
   \eqno (260),$$
because the product $(a + \alpha) (a - \alpha) = a^2 - \alpha^2$
is scalar, so that
$$(a^2 - \alpha^2) (b - \beta) = (b - \beta) (a^2 - \alpha^2).$$

This product, $a^2 - \alpha^2$, of the two {\it conjugate
expressions}, or {\it conjugate geometrical quotients}, denoted
here by
$$a + \alpha,\quad a - \alpha
   \eqno (261),$$
is not only scalar, but is also {\it positive\/}; because we
have, by the principles of the 12${}^{\rm th}$ article, the two
inequalities,
$$a^2 > 0,\quad \alpha^2 < 0
   \eqno (262).$$

Making then, in conformity with (257),
$${\rm T} (a + \alpha) =  {\rm T} (a - \alpha) =  \surd (a^2 - \alpha^2)
   \eqno (263),$$
we see that either of the two conjugate equations (259) gives, by
(260),
$${\rm T} (c + \gamma)
   =  {\rm T} (b + \beta) \mathbin{.} {\rm T} (a + \alpha)
   \eqno (264);$$
or eliminating $c + \gamma$,
$${\rm T} (b + \beta) (a + \alpha)
   = {\rm T} (b + \beta) \mathbin{.} {\rm T} (a + \alpha)
   \eqno (265).$$

It is easy to extend this result to any number of geometrical
quotients, considered as factors in a multiplication; and thus to
conclude generally that, as already stated, {\it the tensor of
the product is equal to the product of the tensors\/}; a theorem
which may be concisely expressed by the formula,
$${\rm T} {\textstyle \prod} = {\textstyle \prod} {\rm T}
   \eqno (266).$$

\bigbreak

\centerline{\it On Conjugate Geometrical Quotients.}

\nobreak\bigskip

41.
It will be found convenient here to introduce a new
characteristic, ${\rm K}$, to denote the operation of passing
from any geometrical quotent to its {\it conjugate}, by
preserving the scalar part unchanged, but changing the sign of
the vector part; with which new characteristic of
operation~${\rm K}$, we shall have, generally,
$${\rm K} {{\rm b} \over {\rm a}}
   = {\rm S} {{\rm b} \over {\rm a}}
      - {\rm V} {{\rm b} \over {\rm a}}
   \eqno (267);$$
or,
$${\rm K} (a + \alpha) = a - \alpha
   \eqno (268),$$
if $a$ be still understood to denote a scalar, but $\alpha$ a
vector quotient.  The {\it tensors of two conjugate quotients are
equal to each other}, by (263); so that we may write
$${\rm T} {\rm K} {{\rm b} \over {\rm a}}
   = {\rm T} {{\rm b} \over {\rm a}},
      \hbox{ or briefly, }
  {\rm T} {\rm K} = {\rm T}
   \eqno (269);$$
and {\it the product of any two such conjugate quotients is equal
to the square of their common tensor},
$${{\rm b} \over {\rm a}} {\rm K} {{\rm b} \over {\rm a}}
   = \left( {\rm T} {{\rm b} \over {\rm a}} \right)^2
   \eqno (270).$$

By separation of symbols, we may write, instead of (267),
$${\rm K} = {\rm S} - {\rm V}
   \eqno (271),$$
and the characteristic~${\rm K}$ is a {\it distributive symbol},
because ${\rm S}$ and ${\rm V}$ have been already seen to be
such: so that the equations (74) (75), of Art.~10, give now the
analogous equations,
$${\rm K} {\textstyle \sum} = {\textstyle \sum} {\rm K},\quad
  {\rm K} {\textstyle \Delta} = {\textstyle \Delta} {\rm K},
   \eqno (272),$$
or in words, {\it the conjugate of a sum\/} (of any number of
geometrical quotients) {\it is the sum of the conjugates\/}; and
in like manner, the conjugate of a difference is equal to the
difference of the conjugates.  But also we have seen, in (178),
that
$$1 = {\rm S} + {\rm V},$$
because a geometrical quotient is always equal to the sum of its
own scalar and vector parts; we may therefore now form the
following {\it symbolical expressions for our two old
characteristics of operation, in terms of the new
characteristic\/}~${\rm K}$,
$$\left. \eqalign{
{\rm S} &= {\textstyle {1 \over 2}} (1 + K) \cr
{\rm V} &= {\textstyle {1 \over 2}} (1 - K) \cr}
   \right\}
   \eqno (273).$$

We may also observe that
$${\rm K} {\rm K} {{\rm b} \over {\rm a}}
   =  {{\rm b} \over {\rm a}},
      \hbox{ or }
  {\rm K}^2 = 1
   \eqno (274);$$
the {\it conjugate of the conjugate\/} of any geometrical
quotient being equal to that quotient itself.  Combining (273),
(274), we find, by an easy symbolical process, which the
formul{\ae} (272) shew to be a legitimate one,
$$\left. \eqalign{
{\rm K} {\rm S}
   &= {\textstyle {1 \over 2}} ({\rm K} + {\rm K}^2)
    = {\textstyle {1 \over 2}} ({\rm K} + 1)
    = + {\rm S} \cr
{\rm K} {\rm V}
   &= {\textstyle {1 \over 2}} ({\rm K} - {\rm K}^2)
    = {\textstyle {1 \over 2}} ({\rm K} - 1)
    = - {\rm V} \cr}
   \right\}
   \eqno (275);$$
and accordingly the operation of {\it taking the conjugate\/} has
been defined to consist in changing the sign of the vector part,
without making any change in the scalar part, of the quotient on
which the operation is performed.  From (273), (274), we may also
infer the symbolical equations,
$$\left. \eqalign{
{\rm S}^2
   &= {\textstyle {1 \over 4}} (1 + {\rm K})^2
    = {\textstyle {1 \over 2}} (1 + {\rm K})
    = {\rm S} \cr
{\rm V}^2
   &= {\textstyle {1 \over 4}} (1 - {\rm K})^2
    = {\textstyle {1 \over 2}} (1 - {\rm K})
    = {\rm V} \cr
{\rm S} {\rm V}
   &= {\rm V} {\rm S}
    = {\textstyle {1 \over 4}} (1 - {\rm K}^2)
    = 0 \cr}
   \right\}
   \eqno (276);$$
and in fact, after once separating the scalar and vector parts of
any proposed geometrical quotient, no farther separation of the
same kind is possible; so that the operation denoted by the
characteristic~${\rm S}$, if it be again performed, makes no
change in the scalar part first found, but reduces the vector
part to zero; and, in like manner, the operation~${\rm V}$
reduces the scalar part to zero, while it leaves unchanged the
vector part of the first or proposed quotient.  We may note here
that the same formul{\ae} give these other symbolical results,
which also can easily be verified:
$${\rm K} {\rm S} = {\rm S} {\rm K};\quad
  {\rm K} {\rm V} = {\rm V} {\rm K}
   \eqno (277);$$
and
$$({\rm S} + {\rm V})^n = {\rm S}^n + {\rm V}^n = {\rm S} + {\rm V} = 1
   \eqno (278);$$
at least if the exponent~$n$ be any positive whole number, so as
to allow a finite and integral development of the symbolic power
$$({\rm S} + {\rm V})^n = 1^n
   \eqno (279).$$

With respect to the {\it geometrical signification\/} of the
relation between conjugate quotients, we may easily see that if
${\rm c}$ and ${\rm d}$ denote any two equally long straight
lines, and $x$ any scalar coefficient or multiplier, then the two
quotients
$${x {\rm c} \over {\rm c} + {\rm d}},\quad
  {x {\rm d} \over {\rm c} + {\rm d}}
   \eqno (280) $$
will be, in the foregoing sense, {\it conjugate\/}; because their
{\it sum\/} will be a {\it scalar}, namely~$x$, but their
{\it difference\/} will be a {\it vector\/}, on account of the
mutual perpendicularity of the lines ${\rm c} - {\rm d}$ and
${\rm c} + {\rm d}$, which are here the diagonals of a rhombus,
and of which the latter bisects the angle between the sides
${\rm c}$ and ${\rm d}$ of the rhombus.  (Compare (209).)

Conversely, if the relation
$${{\rm b}' \over {\rm a}}
   =  {\rm K} {{\rm b} \over {\rm a}}
   \eqno (281),$$
be given, we shall have, by the definition (267) of ${\rm K}$,
$$0 = {\rm V} {{\rm b}' + {\rm b} \over {\rm a}}
    = {\rm S} {{\rm b}' - {\rm b} \over {\rm a}}
   \eqno (282);$$
whence it is easy to infer that, {\it if two conjugate
geometrical quotients or fractions be so prepared as to have a
common denominator\/} (${\rm a}$), {\it their numerators\/}
(${\rm b}$,~${\rm b}'$) {\it will be equally long, and will be
equally inclined to the denominator, at opposite sides thereof,
but in one common plane with it\/}; in such a manner that the
line ${\rm a}$ (or $- {\rm a}$) {\it bisects the angle\/} between
the lines ${\rm b}$ and ${\rm b}'$, if these three straight lines
be supposed to have all one common origin.  We are then
conducted, in this way, to a very simple and useful
{\it expression, for\/} (what may be called) {\it the
reflexion\/}~(${\rm b}'$) {\it of a straight line\/}~(${\rm b}$),
{\it with respect to another straight line\/}~(${\rm a}$), namely
the following:
$${\rm b}' = {\rm K} {{\rm b} \over {\rm a}} \times {\rm a}
   \eqno (283).$$
And whenever we meet with an expression of this form, we shall
know that the two lines ${\rm b}$ and ${\rm b}'$ are equally
long; and also that if they have a common origin, the angle
between them is bisected there by one of the two opposite lines
$\pm {\rm a}$, or by a parallel thereto.

Finally, we may here note that, by the principles of the present
article, and of the foregoing one, we have the following
expressions, which hold good for any pair of straight lines,
${\rm a}$ and ${\rm b}$:
$$\left. \eqalign{
{{\rm a} \over {\rm b}}
   &= \left( {\rm T} {{\rm a} \over {\rm b}} \right)^2
     {\rm K} {{\rm b} \over {\rm a}} \cr
{\rm S} {{\rm a} \over {\rm b}}
   &= \left( {\rm T} {{\rm a} \over {\rm b}} \right)^2
     {\rm S} {{\rm b} \over {\rm a}} \cr
{\rm V} {{\rm a} \over {\rm b}}
   &= - \left( {\rm T} {{\rm a} \over {\rm b}} \right)^2
     {\rm V} {{\rm b} \over {\rm a}} \cr}
   \right\}
   \eqno (284).$$

\bigbreak

\centerline{\it Equations of some Geometrical Loci.}

\nobreak\bigskip

42.
The equation
$${\rm S} {{\rm r} \over {\rm a}} = 0
   \eqno (285),$$
signifies, by what has been already shewn, that the straight
line~${\rm r}$ is perpendicular to ${\rm a}$; it is therefore the
equation of a {\it plane}, perpendicular to this latter line, and
passing through some fixed origin of lines, if ${\rm r}$ be
regarded as a variable line, but ${\rm a}$ as a fixed line from
that origin.  The equation
$${\rm S} {{\rm r} - {\rm a} \over {\rm a}} = 0,
   \quad{or}\quad
  {\rm S} {{\rm r} \over {\rm a}} = 1
   \eqno (286),$$
expresses, for a similar reason, that the variable line~${\rm r}$
terminates on another plane, parallel to the former plane, and
having the line~${\rm a}$ for the perpendicular let fall upon it
from the origin.  If ${\rm b}$ denote the perpendicular let fall
from the same origin upon a third plane, the equation of this
third plane will of course be, in like manner
$${\rm S} {{\rm r} \over {\rm b}} = 1
   \eqno (287);$$
and it is not difficult to prove, with the help of the
transformations (284), that this other equation
$${\rm S} {{\rm r} \over {\rm b}}
   =  {\rm S} {{\rm r} \over {\rm a}}
   \eqno (288),$$
represents a fourth plane, which passes through the intersection
of the second and third planes just now mentioned, namely, the
planes (286), (287), and through the origin.

In general, the equation
$${\rm S}
      \left(
         {{\rm r} \over {\rm a}}
       + {{\rm r} \over {\rm a}'}
       + {{\rm r} \over {\rm a}''}
       + \hbox{\&c.}
      \right)
   = a
   \eqno (289),$$
expresses that ${\rm r}$ terminates on a fixed plane, if it be
drawn from a fixed origin, and if the lines ${\rm a}$,
${\rm a}'$, ${\rm a}''$, \&c., and the number~$a$ be given.  It
may also be noted here that the equation of the plane which
perpendicularly bisects the straight line connecting the
extremities of two given lines, ${\rm a}$ and ${\rm b}$, may be
thus written:
$${\rm T} {{\rm r} - {\rm b} \over {\rm r} - {\rm a}} = 1
   \eqno (290).$$

\bigbreak

43.
On the other hand, the equation
$${\rm S} {{\rm r} - {\rm b} \over {\rm r} - {\rm a}} = 0
   \eqno (291),$$
expresses that the lines from the extremities of ${\rm a}$ and
${\rm b}$ to the extremity of ${\rm r}$ are perpendicular to each
other; or that the line ${\rm r}$ terminates upon a {\it spheric
surface}, in two diametrically opposite points of which
surface the lines ${\rm a}$ and ${\rm b}$ respectively terminate:
and this diameter itself, from the end of ${\rm a}$ to the end of
${\rm b}$, regarded as a {\it rectilinear locus}, is represented
by the equation
$${\rm V} {{\rm r} - {\rm b} \over {\rm r} - {\rm a}} = 0
   \eqno (292);$$
which may however be put under other forms. A transformation of
the equation (291) is the following:
$${\rm T}
      \left(
         {2 {\rm r} - {\rm b} - {\rm a} \over {\rm b} - {\rm a}}
      \right)
   =  1
   \eqno (293);$$
which expresses that the variable radius
${\rm r} - {1 \over 2} ({\rm b} + {\rm a})$
has the same length as the fixed radius
${1 \over 2} ({\rm b} - {\rm a})$.
For example, by changing $- {\rm a}$ to $+ {\rm b}$, in this last
equation of the sphere, we find
$${\rm T} {{\rm r} \over {\rm b}} = 1,
   \quad\hbox{or}\quad
  \left( {\rm S} {{\rm r} \over {\rm b}} \right)^2
      - \left( {\rm V} {{\rm r} \over {\rm b}} \right)^2
   =  1
   \eqno (294),$$
as the equation of a spheric surface described about the origin
of lines, as centre, with the line ${\rm b}$ for one of its
radii, so as to touch, at the end of this line~${\rm b}$, the
plane (287).  (Comp.~(255)).

And a small {\it circle\/} of this sphere (294), if it be
situated on a secant plane, parallel to this tangent plane (287),
which new plane will thus have for its equation,
$${\rm S} {{\rm r} \over {\rm b}} = x
   \eqno (295),$$
where $x$ is a scalar, numerically less than unity, and constant
for each particular circle, will also be situated on a certain
corresponding cylinder of revolution, which will have for its
equation
$$\left( {\rm V} {{\rm r} \over {\rm b}} \right)^2 = x^2 - 1
   \eqno (296);$$
where $x^2 - 1$ is negative, as it ought to be, by the
12${}^{\rm th}$ article, being equal to the square of a vector.
The sphere may be regarded as the locus of these small circles;
and its equation (294) may be supposed to be obtained by the
elimination of the scalar~$x$ between the equations of the plane
(295), and of the cylinder (296).

\bigbreak

44.
Conceive now that instead of cutting the cylinder (296)
{\it perpendicularly\/} in a {\it circle\/}, we cut it
{\it obliquely}, in an {\it ellipse}, by the plane having for its
equation
$${\rm S} {{\rm r} \over {\rm a}} = x
   \eqno (297),$$
where $x$ is the same scalar as before; so that this new plane is
parallel to the fixed plane (286), and cuts the plane of the
circle (295) in a straight line situated on that other fixed
plane (288), which has been seen to contain also the intersection
of the same fixed plane (286) with the tangent plane (287).  The
{\it locus of the elliptic sections}, obtained from the circular
cylinders by this construction, will be an {\it ellipsoid\/}; and
conversely, an ellipsoid may in general be regarded as such a
locus.  The equation of the {\it ellipsoid}, thus found, by
eliminating~$x$ between the equations (296), (297), is the
following:
$$\left( {\rm S} {{\rm r} \over {\rm a}} \right)^2
      - \left( {\rm V} {{\rm r} \over {\rm b}} \right)^2
   = 1
   \eqno (298);$$
and by some easy modifications of the process, it may be shewn
that a {\it hyperboloid}, regarded as a certain other locus of
ellipses, may in general be represented by an equation of the
form
$$\left( {\rm S} {{\rm r} \over {\rm a}} \right)^2
      + \left( {\rm V} {{\rm r} \over {\rm b}} \right)^2
   = \mp 1
   \eqno (299).$$
The upper sign belongs to a hyperboloid of {\it one sheet}, but
the lower sign to a hyperboloid of {\it two sheets\/}; while the
{\it common asymptotic cone\/} of these two (conjugate)
hyperboloids (299) is the locus of a certain other system of
ellipses, and is represented by the analogous but intermediate
equation,
$$\left( {\rm S} {{\rm r} \over {\rm a}} \right)^2
      + \left( {\rm V} {{\rm r} \over {\rm b}} \right)^2
   = 0
   \eqno (300).$$
These equations admit of several instructive transformations, to
some of which we shall proceed in the following article.

\bigbreak

\centerline{\it On some Transformations and Constructions of the
Equation of the Ellipsoid.}

\nobreak\bigskip

45.
The equation (298) of the ellipsoid resolves itself into factors,
as follows:
$$    \left(
         {\rm S} {{\rm r} \over {\rm a}}
       + {\rm V} {{\rm r} \over {\rm b}}
      \right)
      \left(
         {\rm S} {{\rm r} \over {\rm a}}
       - {\rm V} {{\rm r} \over {\rm b}}
      \right)
   = 1
   \eqno (301);$$
where the sum and difference, which when thus multiplied together
give unity for their product, are {\it conjugate expressions\/}
(in the sense of recent articles); they have therefore a
{\it common tensor}, which must itself be equal to unity; and
consequently we may write the equation of the ellipsoid thus,
$${\rm T}
      \left(
         {\rm S} {{\rm r} \over {\rm a}}
       + {\rm V} {{\rm r} \over {\rm b}}
      \right)
   = 1
   \eqno (302),$$
where the sign of the vector may be changed.  Substituting for
the characteristics of operation ${\rm S}$ and ${\rm V}$, their
symbolical values (273), we are led to introduce two new fixed
lines ${\rm g}$ and ${\rm h}$, depending on the two former fixed
lines ${\rm a}$ and ${\rm b}$, and determined by the equations
$${{\rm r} \over 2 {\rm a}} + {{\rm r} \over 2 {\rm b}}
   = {{\rm r} \over {\rm g}};\quad
  {{\rm r} \over 2 {\rm a}} - {{\rm r} \over 2 {\rm b}}
   = {{\rm r} \over {\rm h}}
   \eqno (303);$$
and thus the equation of the ellipsoid may be changed from (302)
to this other form
$${\rm T}
      \left(
         {{\rm r} \over {\rm g}}
       + {\rm K} {{\rm r} \over {\rm h}}
      \right)
   = 1
   \eqno (304);$$
which, by the principles (269), (272), (274), may also be thus
written,
$${\rm T}
      \left(
         {{\rm r} \over {\rm h}}
       + {\rm K} {{\rm r} \over {\rm g}}
      \right)
   = 1
   \eqno (305);$$
so that the symbols, ${\rm g}$ and ${\rm h}$, may be interchanged
in either of the two last forms of the equation of the ellipsoid.

\bigbreak

46.
Let $\overline{\rm r}$, $\overline{\rm g}$, $\overline{\rm h}$ be
conceived to be numerical symbols, denoting respectively the
lengths of the three lines ${\rm r}$, ${\rm g}$, ${\rm h}$; and
make, for conciseness,
$${\rm r} \div \overline{\rm r}^2 = {\rm r}';\quad
  {\rm g} \div \overline{\rm g}^2 = {\rm g}';\quad
  {\rm h} \div \overline{\rm h}^2 = {\rm h}'
   \eqno (306);$$
so that the symbols ${\rm r}'$, ${\rm g}'$, ${\rm h}'$ shall
denote three new lines, having the {\it same directions\/} as the
three former lines ${\rm r}$, ${\rm g}$, ${\rm h}$, but having
their {\it lengths\/} respectively {\it reciprocals\/} of the
lengths of those three former lines.  Then, by the properties of
conjugate quotients already established, we shall have the
transformations
$${{\rm r} \over {\rm g}} = {\rm K} {{\rm g}' \over {\rm r}'};\quad
  {\rm K} {{\rm r} \over {\rm h}} = {{\rm h}' \over {\rm r}'}
   \eqno (307);$$
whereby the equation (304) of the ellipsoid becomes
$${\rm T}
      \left(
         {{\rm h}' \over {\rm r}'}
       + {\rm K} {{\rm g}' \over {\rm r}'}
      \right)
   = 1
   \eqno (308).$$

Let ${\rm g}''$ be a new line, not fixed but variable, and
determined for each variable direction of ${\rm r}'$ or of
${\rm r}$ by the formula
$${\rm g}'' = {\rm K} {{\rm g}' \over {\rm r}'} \times {\rm r}';
   \hbox{ or }
  {\rm g}'' = {\rm K} {{\rm g}' \over {\rm r}} \times {\rm r}
   \eqno (309);$$
so that this new and variable line ${\rm g}''$ is, by what was
shewn respecting the expression (283), the {\it reflexion\/} of
the fixed line ${\rm g}'$ with respect to a line having the
variable direction just mentioned, of ${\rm r}'$ or of ${\rm r}$:
we may then write the equation (308) of the ellipsoid as follows,
$${\rm T} {{\rm h}' + {\rm g}'' \over {\rm r}'} = 1
   \eqno (310).$$
And by comparing this with the formula (256), we see that the
length of the line~${\rm r}'$, or {\it the reciprocal of the
length $\overline{\rm r}$ of the variable semidiameter ${\rm r}$
of the ellipsoid, is equal to the length of the line
${\rm h}' + {\rm g}''$}; which latter line is the symbolical sum
of one fixed line, ${\rm h}'$, and of the variable reflexion,
${\rm g}''$, of another fixed line, ${\rm g}'$; this
{\it reflexion\/} having been already seen to be performed with
respect to the variable radius vector or semidiameter, ${\rm r}$,
of the ellipsoid, {\it of which semidiameter the dependence of
the length on the direction admits of being thus represented, or
constructed, by a very simple geometrical rule.}

\bigbreak

47.
To make more clear the conception of this geometrical rule, let
${\sc a}$ denote the centre of the ellipsoid, which centre is the
origin of the variable line~${\rm r}$; and let two other fixed
points, ${\sc b}$ and ${\sc c}$, be determined by the symbolical
equations
$${\rm g}' = {\sc a} - {\sc c} = {\sc a} {\sc c};\quad
  {\rm h}' = {\sc b} - {\sc c} = {\sc b} {\sc c}
   \eqno (311):$$
these two notations, ${\sc a} {\sc c}$ and ${\sc a} - {\sc c}$,
(of which one has been already used in the {\it text\/} of the
first article\footnote*{It was for the sake of making easier the
transition to the notation ${\sc b} - {\sc a}$, which appears to
the present writer an expressive one, for the straight line drawn
{\it to\/} the point~${\sc b}$ {\it from\/} the point~${\sc a}$,
that he proposed to use, with the {\it same\/} geometrical
signification, the symbol ${\sc b} {\sc a}$, instead of
${\sc a} {\sc b}$: although it is certainly more usual, and
perhaps also more natural, when {\it direction\/} is attended
to, to employ the latter symbol ${\sc a} {\sc b}$, and not the
former ${\sc b} {\sc a}$, to denote the line thus drawn from
${\sc a}$ to ${\sc b}$.}
of this Essay on Symbolical Geometry, while the other was
suggested in a {\it note\/} to the same early article,) being
each designed to denote or signify a straight line drawn
{\it to\/} the point~${\sc a}$ from the point~${\sc c}$.  Let
${\sc d}$ be a new or fourth point, not fixed but variable, and
determined by the analogous equation
$${\rm g}'' = {\sc c} - {\sc d} = {\sc c} {\sc d}
   \eqno (312):$$
then because, in virtue of the relation (309), the lines
${\rm g}'$,~${\rm g}''$ are equally long, it follows that the
variable point~${\sc d}$ is situated on the surface of that fixed
and {\it diacentric sphere}, which we may conceive to be
described {\it round\/} the fixed point~${\sc c}$ as centre, so
as to pass {\it through\/} the centre~${\sc a}$ of the ellipsoid
as through a given superficial point of this diacentric sphere.
Again, in virtue of the same relation (309), or of the geometric
reflexion which the second formula so marked expresses, the
symbolic sum of the two lines ${\rm g}'$,~${\rm g}''$, has the
direction of the line~${\rm r}$, or the exactly contrary
direction; in fact, that relation (309) conducts to the following
scalar quotient
$${{\rm g}' + {\rm g}'' \over {\rm r}}
   = {{\rm g}' \over {\rm r}} + {\rm K} {{\rm g}' \over {\rm r}}
   = 2 {\rm S} {{\rm g}' \over {\rm r}}
   \eqno (313);$$
and this symbolic sum, ${\rm g}' + {\rm g}''$, may also, by
(311) (312), be thus expressed
$${\rm g}' + {\rm g}''
   =  ({\sc a} - {\sc c}) + ({\sc c} - {\sc d})
   =  {\sc a} - {\sc d}
   =  {\sc a} {\sc d}
   \eqno (314).$$
If then we denote by ${\sc e}$ that variable point on the surface
of the ellipsoid at which the line ${\rm r}$ terminates, so that
$${\rm r} = {\sc e} - {\sc a} = {\sc e} {\sc a}
   \eqno (315),$$
we shall have the relation
$${{\sc a} - {\sc d} \over {\sc e} - {\sc a}}
   =  {{\sc a} {\sc d} \over {\sc e} {\sc a}}
   =  2 {\rm S} {{\rm g}' \over {\rm r}}
   =  V^{-1} 0
   \eqno (316),$$
which requires that the three points, ${\sc a}$, ${\sc d}$,
${\sc e}$, should be situated on one common straight line.  We
know then the geometrical position of the auxiliary and variable
point~${\sc d}$, or have a simple construction for determining
this variable point~${\sc d}$, as corresponding to any particular
point~${\sc e}$ on the surface of the ellipsoid, when the
centre~${\sc a}$, and the two other fixed points, ${\sc b}$ and
${\sc c}$, are given; for we see that we have merely to seek the
{\it second intersection\/} of the semidiameter
${\sc e} - {\sc a}$ (or ${\sc e} {\sc a}$) of the ellipsoid, with
the surface of the diacentric sphere, the {\it first\/}
intersection being the centre~${\sc a}$ itself; since this second
point of intersection will be the required point~${\sc d}$.

\bigbreak

48.
But also, by (311) (312), we have
$${\rm h}' + {\rm g}''
   =  ({\sc b} - {\sc c}) + ({\sc c} - {\sc d})
   =  {\sc b} - {\sc d}
   =  {\sc b} {\sc d}
   \eqno (317):$$
this line ${\sc b} {\sc d}$ has therefore, by (310), the length
of the line ${\rm r}'$; which length is, by (306), (315) the
reciprocal of the length $\overline{\rm r}$ of the semidiameter
${\sc e} {\sc a}$ of the ellipsoid.  The lines
${\rm g}$,~${\rm h}$ have generally unequal lengths; and because,
by (304) (305), their symbols may be interchanged, we may choose
them so that the former shall be the longer of the two, or that
the inequality
$$\overline{\rm g} > \overline{\rm h}
   \eqno (318) $$
shall be satisfied; and then, by (306), the line~${\rm g}'$ will,
on the contrary, be shorter than the line~${\rm h}'$, or the
fixed point~${\sc b}$ will be {\it exterior\/} to the fixed
diacentric sphere.  Drawing, then, from this external
point~${\sc b}$, a tangent to this diacentric sphere, and taking
the length of the tangent so drawn for the unit of length, the
reciprocal of the length of the line~${\sc b} {\sc d}$, which is
considered in (317), will be the length of that other
line~${\sc b} {\sc d}'$, which has the same direction as
${\sc b} {\sc d}$, but terminates at another variable
point~${\sc d}'$ on the surface of the diacentric sphere; in such
a manner that this new variable point~${\sc d}'$, without
generally coinciding with the point~${\sc d}$, shall satisfy the
two equations,
$${{\sc d}' - {\sc b} \over {\sc d} - {\sc b}}
   = {\rm V}^{-1} 0;\quad
  {{\sc d}' - {\sc c} \over {\sc d} - {\sc c}}
   = {\rm T}^{-1} 1
   \eqno (319);$$
for then the two lines ${\sc d}' - {\sc b}$, ${\sc d} - {\sc b}$
(or ${\sc d}' {\sc b}$, ${\sc d} {\sc b}$) will be, in this or in
the opposite order, the whole secant and external part, while the
length of the tangent to the sphere has been above assumed as
unity.  Under these conditions, then, {\it the lengths of the
lines ${\sc d}' {\sc b}$ and ${\sc e} {\sc a}$ will be equal},
because they will have the length of the line ${\sc d} {\sc b}$
for their common reciprocal; so that we shall have the equation
$${{\sc e} - {\sc a} \over {\sc d}' - {\sc b}} = {\rm T}^{-1} 1
   \eqno (320);$$
or, in a more familiar notation,
$$\overline{{\sc a} {\sc e}} = \overline{{\sc b} {\sc d}'}
   \eqno (321).$$

It may be noted here that the new radius ${\sc d}' - {\sc c}$ of
the diacentric sphere admits (compare the formula (213)) of being
symbolically expressed as follows,
$${\sc d}' - {\sc c}
   =  {\rm K} {{\rm g}'' \over {\rm h}' + {\rm g''}}
         \mathbin{.} ({\rm h}' + {\rm g}'')
   \eqno (322);$$
and, accordingly, this last expression satisfies the two
conditions (319), because it gives
$${{\sc d}' - {\sc b} \over {\sc d} - {\sc b}}
   =  {\rm S} {{\rm h}' - {\rm g}'' \over {\rm h}' + {\rm g}''}
   \eqno (323),$$
and
$${{\sc d}' - {\sc c} \over {\sc d} - {\sc c}}
   =  {\rm K} {{\rm g}'' \over {\rm h}' + {\rm g}''}
         \mathbin{.} {{\rm h}' + {\rm g}'' \over - {\rm g}''}
   \eqno (324),$$
of which latter expression the tensor is unity.

\bigbreak

49.
The remarkably simple formula,
$$\overline{{\sc a} {\sc e}} = \overline{{\sc b} {\sc d}'}
   \eqno (321),$$
to which we have thus been conducted for the ellipsoid, admits of
being easily translated into the following rule for constructing
that important surface; which rule for the {\it construction of
the ellipsoid\/} does not seem to have been known to
mathematicians, until it was communicated by the present writer
to the Royal Irish Academy in 1846, as a result of his Calculus
of Quaternions, between which and the present Symbolical Geometry
a very close affinity exists.

{\it From a fixed point~${\sc a}$, on the surface of a given
sphere, draw a variable chord of that sphere, ${\sc d} {\sc a}$;
let ${\sc d}'$ be the second point of intersection of the spheric
surface with the secant~${\sc d} {\sc b}$, which connects the
variable extremity~${\sc d}$ of this chord~${\sc d} {\sc a}$ with
a fixed external point~${\sc b}$; and take the radius
vector~${\sc e} {\sc a}$ equal in length to ${\sc d}' {\sc b}$,
and in direction either coincident with, or opposite to, the
chord ${\sc d} {\sc a}$: the locus of the point~${\sc e}$, thus
constructed, will be an ellipsoid, which will have its centre
at the fixed point~${\sc a}$, and will pass through the fixed
point~${\sc b}$.}

The fixed sphere through ${\sc a}$, in this construction of the
ellipsoid, is the {\it diacentric sphere\/} of recent articles;
it may also be called a {\it guide-sphere}, from the manner in
which it assists to mark or to {\it represent the direction}, and
at the same time serves to {\it construct the length\/} of a
variable semidiameter of the ellipsoid; while, for a similar
reason, the points ${\sc d}$ and ${\sc d}'$ upon the surface of
this sphere may be said to be {\it conjugate guide-points\/}; and
the chords ${\sc d} {\sc a}$ and ${\sc d}' {\sc a}$ may receive
the appellation of {\it conjugate guide-chords}.  In fact, while
either of these two guide-chords of the sphere, for instance (as
above) the chord ${\sc d} {\sc a}$, coincides in
{\it direction\/} with a semidiameter~${\sc e} {\sc a}$ of the
ellipsoid, the distance $\overline{{\sc d}' {\sc b}}$ of the
extremity ${\sc d}'$ of the other or conjugate guide-chord,
${\sc d}' {\sc a}$, from the fixed external point~${\sc b}$,
represents, as we have seen, the {\it length\/} of that
semidiameter.  And that the fixed point~${\sc b}$, although
exterior to the diacentric sphere, is a superficial point of the
ellipsoid, appears from the construction, by conceiving the
conjugate guide-point~${\sc d}'$ to approach to coincidence with
${\sc a}$; for ${\sc e}$ will then tend to coincide either with
the point~${\sc b}$ itself, or with another point diametrically
opposite thereto, upon the surface of the ellipsoid.

\bigbreak

50.
Some persons may prefer the following mode of stating the same
geometrical construction, or the same fundamental property, of
the ellipsoid: which other mode also was communicated by the
present writer to the Royal Irish Academy in 1846.  {\it If, of a
rectilinear quadrilateral ${\sc a} {\sc b} {\sc e} {\sc d}'$, of
which one side~${\sc a} {\sc b}$ is given in length and in
position, the two diagonals ${\sc a} {\sc e}$, ${\sc b} {\sc d}'$
be equal to each other in length, and intersect\/} (in ${\sc d}$)
{\it on the surface of a given sphere\/} (with centre~${\sc c}$),
{\it of which sphere a chord~${\sc a} {\sc d}'$ is a side of the
quadrilateral adjacent to the given side~${\sc a} {\sc b}$, then
the other side~${\sc b} {\sc e}$, adjacent to the same given
side~${\sc a} {\sc b}$, is a chord of a given ellipsoid.}

Thus, denoting still the centre of the sphere by ${\sc c}$, while
${\sc a}$ is still the centre of the ellipsoid, we see that the
form, magnitude, and position, of this latter surface are made by
the foregoing construction to depend, according to very simple
geometrical rules, on the positions of the three points
${\sc a}$,~${\sc b}$,~${\sc c}$; or on the form, magnitude, and
position of what may (for this reason) be named the
{\it generating triangle\/} ${\sc a} {\sc b} {\sc c}$.  Two of
the sides of this triangle, namely ${\sc b} {\sc c}$ and
${\sc c} {\sc a}$, are perpendicular, as it is not difficult to
shew from the construction, to the two
{\it planes of circular section\/} of the ellipsoid; and the
third side~${\sc a} {\sc b}$ is perpendicular to one of the two
{\it planes of circular projection\/} of the same ellipsoid: this
third side~${\sc a} {\sc b}$ being the axis of revolution of a
circumscribed circular cylinder; which also may be proved,
without difficulty, from the construction assigned above.  (See
Articles 52, 53.)  The length~$\overline{{\sc b} {\sc c}}$ of the
side~${\sc b} {\sc c}$ of the triangle, is (by the construction)
the semisum of the lengths of the greatest and least
semidiameters of the ellipsoid; and the
length~$\overline{{\sc c} {\sc a}}$ of the side~${\sc c} {\sc a}$
is the semidifference of the lengths of those extreme
semidiameters, or principal semiaxes, of the same ellipsoid:
while (by the same construction) these greatest and least
{\it semiaxes}, or their prolongations, intersect the surface of
the diacentric sphere in points which are situated, respectively,
on the finite side~${\sc c} {\sc b}$ of the triangle
${\sc a} {\sc b} {\sc c}$ itself, and on that
side~${\sc c} {\sc b}$ prolonged through ${\sc c}$.  The
{\it mean\/} semiaxis of the ellipsoid, or the semidiameter
perpendicular to the greatest and least semiaxes, is (by the
construction) equal in length (as indeed it is otherwise known to
be) to the radius of the enveloping cylinder of revolution, or to
the radius of either of the two diametral and circular sections:
the length of this mean semiaxis is also constructed by the
portion~$\overline{{\sc b} {\sc g}}$ of the axis of the enveloping
cylinder, or of the side~${\sc b} {\sc a}$ of the generating
triangle, if ${\sc g}$ be the point, distinct from ${\sc a}$, in
which this side~${\sc b} {\sc a}$ meets the surface of the
diacentric sphere.  And hence we may derive a simple geometrical
signification, or property, of this remaining
side~${\sc b} {\sc a}$ of the triangle ${\sc a} {\sc b} {\sc c}$,
as respects its length~$\overline{{\sc b} {\sc a}}$; namely, that
this length is a fourth proportional to the three semiaxes of the
ellipsoid, that is to say, to the mean, the least, and the
greatest, or to the mean, the greatest, and the least of those
three principal and rectangular semiaxes.

\bigbreak

\centerline{\it On tbe Law of the Variation of the Difference
of the Squares of the Reciprocals of the}
\centerline{\it Semiaxes of a Diametral Section.}

\nobreak\bigskip

51.
To give a specimen of the facility with which the foregoing
construction serves to establish some important properties of the
ellipsoid, we shall here employ it to investigate anew the known
and important law, according to which the difference of the
squares of the reciprocals of the greatest and least
semidiameters, of any plane and diametral section, varies in
passing from one such section to another.  Conceive then that the
ellipsoid itself, and the auxiliary or diacentric sphere which
was employed in the foregoing construction, are both cut by a
plane ${\sc a} {\sc b}' {\sc c}'$, passing through the
centre~${\sc a}$ of the ellipsoid, and having ${\sc b}'$ and
${\sc c}'$ for the orthogonal projections, upon this secant
plane, of the fixed points ${\sc b}$ and ${\sc c}$.  The
auxiliary or guide-point~${\sc d}$ comes thus to be regarded as
moving on the circumference of a circle, which passes through
${\sc a}$, and has its centre at ${\sc c}'$: and since the
semidiameter~${\sc e} {\sc a}$ of the ellipsoid, as being equal
in length to ${\sc d}' {\sc b}$, by the formula (321) of Art.~48,
(or because these are the two equally long diagonals of the
quadrilateral ${\sc a} {\sc b} {\sc e} {\sc d}'$ of Art.~50),
must vary inversely as ${\sc d} {\sc b}$ (by an elementary
property of the sphere), we are led to seek the difference of the
squares of the greatest and least values of ${\sc d} {\sc b}$, or
of ${\sc d} {\sc b}'$, since the square of the
perpendicular~${\sc b}' {\sc b}$ is constant for the section.
But the shortest and longest straight lines,
${\sc d}_1 {\sc b}'$, ${\sc d}_2 {\sc b}'$, which can be thus
drawn to the circumference of the auxiliary circle round
${\sc c}'$ (namely the section of the diacentric sphere), from
the fixed point~${\sc b}'$ in its plane, are those drawn to the
extremities ${\sc d}_1$,~${\sc d}_2$ of that diameter
${\sc d}_1 {\sc c}' {\sc d}_2$ which passes through, or tends
towards this point~${\sc b}'$; in such a manner that the four
points ${\sc b}' {\sc d}_1 {\sc c}' {\sc d}_2$ are situated on one
straight line.  Hence the difference of the squares of
${\sc d}_1 {\sc b}'$, ${\sc d}_2 {\sc b}'$, is equal to four
times the rectangle under ${\sc d}_1 {\sc c}'$, or
${\sc a} {\sc c}'$, and ${\sc b}' {\sc c}'$; that is to say, under
the projections of the sides ${\sc a} {\sc c}$,
${\sc b} {\sc c}$, of the generating triangle, on the plane of
the diametral section.  {\it It is, then, to this rectangle,
under these two projections of two fixed lines, on any variable
plane through the centre of the ellipsoid, that the difference of
the squares of the reciprocals of the extreme semidiameters of
the section is proportional}.  Hence, in the language of
trigonometry, this difference of squares is proportional (as
indeed it is well known to be) to the product of the sines of the
inclinations of the cutting plane to two fixed planes of circular
section; which latter planes are at the same time seen to be
perpendicular to the two fixed sides ${\sc a} {\sc c}$,
${\sc b} {\sc c}$, of the generating triangle in the
construction.

It seems worth noting here, that the foregoing process proves at
the same time this other well-known property of the ellipsoid,
that the greatest and least semidiameters of a plane section
through the centre are perpendicular to each other; and also
gives an easy geometrical rule for {\it constructing the semiaxes
of any proposed diametral section\/}; for it shews that these
semiaxes have the {\it directions\/} of the two rectangular
guide-chords ${\sc d}_1 {\sc a}$, ${\sc d}_2 {\sc a}$; while
their {\it lengths\/} are equal, respectively, to those of the
lines ${\sc d}_1' {\sc b}$, ${\sc d}_2' {\sc b}$.

\bigbreak

\centerline{\it On tbe Planes of Circular Section and Circular
Projection.}

\nobreak\bigskip

52.
It may not be uninstructive to state briefly here some simple
geometrical reasonings, by which the line~${\sc b} {\sc g}$ of
Art.~50 may be shewn to have its length equal to that of the
radius of an enveloping cylinder of revolution, as was asserted
in that article; and also to the radius of either of the two
diametral and circular sections of the ellipsoid.  First, then,
as to the cylinder: the equation
$\overline{{\sc a} {\sc e}} = \overline{{\sc b} {\sc d}'}$
shews that the rectangle under the two lines ${\sc a} {\sc e}$ and
${\sc b} {\sc d}$ is constant for the ellipsoid, because the
rectangle under ${\sc b} {\sc d}'$ and ${\sc b} {\sc d}$ is
constant for the sphere; and the point~${\sc d}$ has been seen to
be situated on the straight line~${\sc a} {\sc e}$ (prolonged if
necessary).  Hence the double area of the triangle
${\sc a} {\sc b} {\sc e}$, or the rectangle under the fixed
line~${\sc a} {\sc b}$, and the perpendicular let fall thereon
from the variable point~${\sc e}$ of the ellipsoid, is always
less than the lately mentioned constant rectangle; or than the
square of the tangent to the diacentric sphere from ${\sc b}$;
or, finally, than the rectangle under the same fixed
line~${\sc a} {\sc b}$ and its constant part~${\sc g} {\sc b}$:
except at the limit where the angle ${\sc a} {\sc d} {\sc b}$ is
right, at which limit the double area of the triangle
${\sc a} {\sc b} {\sc e}$ becomes equal to the last mentioned
rectangle.  The ellipsoid is therefore entirely enveloped by
that cylinder of revolution which has ${\sc a} {\sc b}$ for axis,
and $\overline{{\sc g} {\sc b}}$ for radius; being situated
entirely {\it within\/} this cylinder, except for a certain
limiting curve or system of points, which are {\it on\/} (but not
outside) the cylinder, and are determined by the condition that
${\sc a} {\sc d} {\sc b}$ shall be a right angle.  This limiting
condition determines a {\it second spherical locus\/} for the
guide-point~${\sc d}$, besides the diacentric sphere; it serves
therefore to assign a {\it circular locus\/} for that point,
which circle passes through the centre~${\sc a}$ of the
ellipsoid, because this centre is situated on each of the two
spherical loci.  And hence by the construction we obtain an
{\it elliptic locus\/} for the point~${\sc e}$, namely the
ellipse of contact of the ellipsoid and cylinder; which ellipse
presents itself here as the intersection of that enveloping
cylinder of revolution with the plane of the circle which has
been seen to be the locus of ${\sc d}$.---It may also be shewn,
geometrically, by pursuing the same construction into its
consequences, that the ellipsoid is enveloped by {\it another\/}
(equal) cylinder of revolution, giving a {\it second\/} diametral
{\it plane of circular projection\/}; the first such plane being
(by what precedes) perpendicular to the line~${\sc a} {\sc b}$:
and that the axis of this second circular cylinder, or the normal
to this second plane of circular projection of the ellipsoid, is
parallel to the straight line which touches, at the
centre~${\sc c}$ of the diacentric sphere, the circle
circumscribed about the generating triangle
${\sc a} {\sc b} {\sc c}$.

\bigbreak

53.
Again, with respect to the diametral and circular sections of the
ellipsoid, considered as results of the construction: if we
conceive that the guide-point~${\sc d}$, in that construction,
approaches in any direction, on the surface of the diacentric
sphere, to the centre~${\sc a}$ of the ellipsoid, the conjugate
guide-point~${\sc d}'$ must then approach to the point~${\sc g}$,
because this is the second point of intersection of the
side~${\sc b} {\sc a}$ of the triangle with the surface of the
diacentric sphere, if the point~${\sc a}$ itself be regarded as
the first point of such intersection.  Thus, during this approach
of ${\sc d}$ to ${\sc a}$, the semidiameter~${\sc e} {\sc a}$ of
the ellipsoid, having always (by the construction) the direction
of $\pm {\sc d} {\sc a}$, and the length of ${\sc d}' {\sc b}$,
must tend to touch the diacentric sphere at ${\sc a}$, and to
have the same fixed length as the line ${\sc b} {\sc g}$, or as
the radius of the cylinder.  And in this way the construction
offers to our notice a {\it circle\/} on the ellipsoid, whose
radius $= \overline{{\sc b} {\sc g}}$, and whose plane is
perpendicular to the side~${\sc a} {\sc c}$ of the generating
triangle; which side is thus seen to be a {\it cyclic normal\/}
of the ellipsoid, by this process as well as by that of the
51${}^{\rm st}$ article.

Finally, with respect to that {\it other\/} cyclic plane which is
perpendicular to the side~${\sc b} {\sc c}$ of the triangle
${\sc a} {\sc b} {\sc c}$, it is sufficient to observe that if we
conceive the point~${\sc d}'$ to revolve in a small circle on the
surface of the diacentric sphere, from ${\sc g}$ to ${\sc g}$
again, preserving a constant distance from the fixed external
point~${\sc b}$, then the semidiameter~${\sc e} {\sc a}$ of the
ellipsoid will retain, by the construction, during this
revolution of ${\sc d}'$, a constant length
$= \overline{{\sc b} {\sc g}}$; while, by the same construction,
the guide-chord~${\sc d} {\sc a}$, and the
semidiameter~${\sc e} {\sc a}$ of the ellipsoid, will at the same
time revolve together in a diametral plane perpendicular to
${\sc b} {\sc c}$: in which {\it second cyclic plane}, therefore,
the point~${\sc e}$ will thus trace out a {\it second circle\/}
on the ellipsoid, with a radius equal to the radius of the former
circle; or to that of the {\it mean sphere\/} (constructed on the
mean axis as diameter, and containing both the circles hitherto
considered); or to the radius of either of the two enveloping
cylinders of revolution.---It is evident that if the
guide-point~${\sc d}$ describe any other circle on the diacentric
sphere, parallel to this second cyclic plane, the conjugate
guide-point ${\sc d}'$ will describe another parallel circle,
leaving the length
$\overline{{\sc b} {\sc d}'} = \overline{{\sc e} {\sc a}}$
unaltered; whence the known theorem flows at once, that if the
ellipsoid be cut by a concentric sphere, the section is a
spherical ellipse;\footnote*{This easy mode of deducing, from the
author's construction of the ellipsoid, the known spherical
ellipses on that surface, was pointed out to him in 1846, by a
friend to whom he had communicated that construction, namely by
the Rev.\ J.~W. Stubbs, Fellow of Trinity College, Dublin.
Several investigations, by the present author, connected with the
same construction of the ellipsoid, have appeared in the
{\it Proceedings} of the Royal Irish Academy, (see in particular
those for July 1846); and also in various numbers of the (London,
Edinburgh and Dublin) {\it Philosophical Magazine\/}: in which
magazine several articles on Quaternions have been already
published by the writer, and are likely to be hereafter
continued, which may on some points be usefully compared with the
present Essay on Symbolical Geometry.}
and also that the concentric cyclic cone which rests thereon
(being the cone described by the guide-chord~${\sc d} {\sc a}$ in
the construction) has its two cyclic planes coincident with the
two cyclic planes of the ellipsoid.

\bye

.
