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% School of Mathematics, Trinity College, Dublin 2, Ireland
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% Trinity College, 2001.

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\centerline{\Largebf ON SOME EXTENSIONS OF QUATERNIONS}

\vskip12pt

\centerline{\Largebf By}

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\centerline{\Largebf William Rowan Hamilton}

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\centerline{\vbox{\halign{\largerm #\hfil\cr
(Philosophical Magazine (4th series):\cr
   \qquad vol.~vii (1854), pp.\ 492--499,\cr
   \qquad vol.~viii (1854), pp.\ 125--137, 261--9,\cr
   \qquad vol.~ix (1855), pp.\ 46--51, 280--290.)\cr}}}

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

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\centerline{\largerm 2001}

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\centerline{\Largebf NOTE ON THE TEXT}

\bigskip

The paper {\it On some Extensions of Quaternions}, by Sir William
Rowan Hamilton, appeared in 5 instalments in volumes vii--ix of
{\it The London, Edinburgh and Dublin Philosophical Magazine and
Journal of Science} (4th Series), for the years 1854--1855.
Each instalment (including the last) ended with the words
`{\it To be continued}'.

The articles of this paper appeared as follows:---

\bigskip

\hskip\parindent\hbox{\vbox{\halign{#\hfil &&\quad #\hfil\cr

   section~i.    &articles 1--6   &Supplementary   1854
      &vol.~vii    (1854), pp.\ 492--499,\cr
   section~ii.   &articles 7--16  &August          1854
      &vol.~viii   (1854), pp.\ 125--137,\cr
   section~iii.  &articles 17--25 &October         1854
      &vol.~viii   (1854), pp.\ 261--269,\cr
   section~iv.   &articles 26--29 &January         1855
      &vol.~ix     (1855), pp.\ 46--51,\cr
   section~v.    &articles 30--36 &April           1855
      &vol.~ix     (1855), pp.\ 280--290.\cr}}}

\bigbreak

(Articles 1--6 appeared in the supplementary number of the
{\it Philosophical Magazine\/} which appeared in the middle
of 1854.)

\bigbreak

Some errata noted by Hamilton have been corrected.

The headings `{\it Section\/}~I.', `{\it Section\/}~II.' and
`{\it Section\/}~III.' were not included for the first three
sections of the original text, but analogous headings were
included for the final two sections.

\bigbreak

The paper {\it On some Extensions of Quaternions}, is included in
{\it The Mathematical Papers of Sir William Rowan Hamilton},
vol.~iii (Algebra), edited for the Royal Irish Academy
by H. Halberstam and R. E. Ingram (Cambridge University Press,
Cambridge, 1967).

\bigbreak\bigskip

\line{\hfil David R. Wilkins}

\vskip3pt

\line{\hfil Dublin, March 2000}

\vskip12pt

Edition corrected---February 2001.

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\pageno=1

\null\vskip36pt

\noindent
{\largeit On some Extensions of Quaternions\/}\footnote*{See the
work entitled, ``Lectures on Quaternions,'' by the present
writer.  (Hodges and Smith, Dublin, 1853.)}.
{\largeit By\/} {\largerm Sir}
{\largesc William Rowan Hamilton,} {\largeit LL.D., M.R.I.A.,
F.R.A.S., Corresponding Member of the French Institute, Hon.\ or
Corr.\ Member of several other Scientific Societies in British
and Foreign Countries, Andrews' Professor of Astronomy in the
University of Dublin, and Royal Astronomer of
Ireland\/}\footnote\dag{Communicated by the Author.}.

\bigbreak

\vskip 12pt

\centerline{[{\it The London, Edinburgh and Dublin Philosophical
Magazine and Journal of Science,}}
\centerline{4th series, vol.~vii (1854), pp.\ 492--99,
   vol.~viii (1854), pp.\ 125--37, 261--9,}
\centerline{vol.~ix (1855), pp.\ 46--51, 280--90.]}

\bigskip

\centerline{{\largeit Section\/} {\largerm I.}}

\nobreak\bigskip

[1.]
Conceive that in the polynomial expressions,
$$\left. \eqalign{
{\rm P}
   &= \iota_0 x_0 + \iota_1 x_1 + \ldots + \iota_n x_n
    = \Sigma \iota x,\cr
{\rm P}'
   &= \iota_0 x_0' + \iota_1 x_1' + \ldots + \iota_n x_n'
    = \Sigma \iota x',\cr
{\rm P}''
   &= \iota_0 x_0'' + \iota_1 x_1'' + \ldots + \iota_n x_n''
    = \Sigma \iota x'',\cr}
   \right\}
   \eqno (1)$$
the symbols $x_0 \, \ldots \, x_n$, which we shall call the
{\it constituents\/} of the polynome~${\rm P}$, and in like
manner that the constituents $x_0' \, \ldots \, x_n'$ of
${\rm P}'$, and $x_0'' \, \ldots \, x_n''$ of ${\rm P}''$, are
subject to all the usual rules of algebra, and to no others; but
that the other symbols $\iota_0 \, \ldots \, \iota_n$, by which
those constituents of each polynome are here symbolically
multiplied, are not all subject to all those usual rules: and
that, on the contrary, these latter symbols are subject, {\it as
a system}, to some {\it peculiar laws}, of comparison and
combination.  More especially, let us conceive, in the first
place, that these $n + 1$ symbols, of the form~$\iota_f$, are and
must remain unconnected with each other by any {\it linear\/}
relation, with ordinary algebraical coefficients; whence it will
follow that an {\it equality\/} between any {\it two\/}
polynomial expressions of the present class requires that
{\it all\/} their {\it corresponding constituents\/} should be
{\it separately\/} equal, or that
$$\hbox{if}\quad {\rm P}' = {\rm P},
   \quad\hbox{then}\quad
   x_0' = x_0,\enspace
   x_1' = x_1,\ldots \enspace
   x_n' = x_n:
   \eqno (2)$$
and therefore, in particular, that the {\it evanescence\/} of any
{\it one\/} such polynome~${\rm P}$ requires the vanishing of
{\it each\/} constituent separately; so that
$$\hbox{if}\quad {\rm P} = 0,
   \quad\hbox{then}\quad
   x_0 = 0,\enspace
   x_1 = 0,\ldots \enspace
   x_n = 0.
   \eqno (3)$$

In the second place, we shall suppose that all the usual rules of
{\it addition\/} and {\it subtraction\/} extend to these new
polynomes, and to their terms; and that the symbols~$\iota$, like
the symbols~$x$, are {\it distributive\/} in their operation;
whence it will follow that
$${\rm P}' \pm {\rm P}
    =  \iota_0 (x_0' \pm x_0) + \ldots + \iota_n (x_n' \pm x_n),
   \eqno (4)$$
or that
$$\Sigma \iota x' \pm \Sigma \iota x
   = \Sigma \iota (x' \pm x):
   \eqno (5)$$
and as a further connexion with common algebra, we shall conceive
that each separate symbol of the form~$\iota$ may combine
commutatively as a factor with each of the form~$x$, and with
every other algebraic quantity, so that $\iota x = x \iota$, and
that therefore the polynome~${\rm P}$ may be thus written,
$${\rm P}
    = x_0 \iota_0 + x_1 \iota_1 + \ldots + x_n \iota_n
    = \Sigma x \iota.
   \eqno (6)$$

But, third, {\it instead\/} of supposing that the symbols~$\iota$
combine thus in general {\it commutatively, among themselves}, as
factors or as operators, we shall {\it distinguish generally\/}
between the two {\it inverted\/} ({\it or opposite\/})
{\it products}, $\iota \iota'$ and $\iota' \iota$, or
$\iota_f \iota_g$ and $\iota_g \iota_f$; and shall conceive that
all the $(n + 1)^2$ {\it binary\/} products ($\iota \iota'$),
{\it including squares\/} ($\iota^2 = \iota \iota$), of the
$n + 1$ symbols~$\iota$, are {\it defined\/} as being each equal
to a certain {\it given\/} or originally {\it assumed polynome},
of the general form~(1), by $(n + 1)^2$ equations of the
following type,
$$\iota_f \iota_g
   =  (fg0) \iota_0 + (fg1) \iota_1 + \ldots
       + (fgh) \iota_h + \ldots
       + (fgn) \iota_n;
   \eqno (7)$$
the $(n + 1)^3$ coefficients, or constituents, of the form
$(fgh)$, which we shall call the ``constants of multiplication,''
being so many given, or assumed, algebraic constants, of which
some may vanish, and which we do not here suppose to satisfy
{\it generally\/} the relation, $(fgh) = (gfh)$.  And thus the
{\it product of any two given polynomes}, ${\rm P}$ and
${\rm P}'$, of the form~(1), combined in a {\it given order\/}
as factors, becomes equal to a {\it third given
polynome},~${\rm P}''$, of the same general form,
$${\rm P}'' = {\rm P} {\rm P}'
   =  \Sigma x_f \iota_f \mathbin{.} \Sigma x_g' \iota_g
   =  \Sigma x_h'' \iota_h;
   \eqno (8)$$
the summations extending still from $0$ to $n$, and the
constituent~$x_h''$ of the product admitting of being thus
expressed:
$$x_h'' = \Sigma (fgh) x_f x_g'.
   \eqno (9)$$
As regards the subjection of the symbols~$\iota$ to the
{\it associative\/} law of multiplication, expressed by the
formula,
$$\iota \mathbin{.} \iota' \iota''
   =  \iota \iota' \mathbin{.} \iota'',$$
we shall make no supposition at present.

\bigbreak

[2.]
As a first simplification of the foregoing very
general\footnote*{Some account of a connected conception
respecting {\it Sets}, considered as including Quaternions, may
be found in the Preface to the Lectures already cited.}
conception, let it be now supposed that
$$\iota_0 = 1;
   \eqno (10)$$
the $n$ other symbols, $\iota_1, \iota_2,\ldots \, \iota_n$,
being thus the only ones which are not subject to all the
ordinary rules of algebra.  Then because
$$\iota_0 \iota_g = \iota_g,\quad
  \iota_f \iota_0 = \iota_f,
   \eqno (11)$$
it will follow that if either of the two indices $f$ and $g$ be
$= 0$, the constant of multiplication $(fgh)$ is either $= 1$ or
$= 0$, according as $h$ is equal or unequal to the other of those
two indices; and we may write,
$$(0fh) = (f0h) = 0,
   \quad\hbox{if}\quad h \gtlt f;
   \eqno (12)$$
$$(0ff) = (f0f) = 1.
   \eqno (13)$$
With this simplification, the {\it number\/} of the arbitrary or
disposable constants of the form $(fgh)$, which are not thus
known already to have the value~$0$ or $1$, is reduced from
$(n+ 1)^3$ to $(n + 1) n^2$; because we may now suppose that $f$
and $g$ are each $> 0$, or that they vary only from $1$ to $n$.
For we may write,
$${\rm P} = p + \varpi,\quad
  {\rm P}' = p' + \varpi',
   \eqno (14)$$
where
$$\left. \multieqalign{
p &= \iota_0 x_0 = x_0, &
\varpi
   &= \iota_1 x_1 + \ldots + \iota_f x_f + \ldots
       + \iota_n x_n,\cr
p' &= \iota_0 x_0' = x_0', &
\varpi'
   &= \iota_1 x_1' + \ldots + \iota_g x_g' + \ldots
       + \iota_n x_n';\cr}
   \right\}
   \eqno (15)$$
and then, by observing that $p$ and $p'$ are symbols of the usual
and algebraical kind, shall have this expression for the product
of two polynomes:
$${\rm P}'' = {\rm P} {\rm P}'
   =  (p + \varpi) (p' + \varpi')
   =  p p' + p \varpi' + p' \varpi + \varpi \varpi';
   \eqno (16)$$
where the last term, or partial product, $\varpi \varpi'$, is now
the only one for which any peculiar rules are required.

\bigbreak

[3.]
When the polynome~${\rm P}$ has thus been decomposed into two
parts, $p$ and $\varpi$, of which the one ($p$) is subject to all
the usual rules of algebraical calculation, but the other
($\varpi$) to peculiar rules; and when these two parts are thus
in such a sense {\it heterogeneous}, that an {\it equation\/}
between two such polynomes resolves itself immediately into
{\it two\/} separate equations, one between parts of the one
kind, and the other between parts of the other kind; so that
$$\hbox{if}\quad {\rm P} = {\rm P}',
   \quad\hbox{or}\quad p + \varpi = p' + \varpi',
   \quad\hbox{then}\quad p = p,
   \quad\hbox{and}\quad \varpi = \varpi';
   \eqno (17)$$
we shall call the former part~($p$) the {\it scalar part}, or
simply {\sc the scalar}, of the polynome~${\rm P}$, and shall
denote it, as such, by the symbol ${\rm S} \mathbin{.} {\rm P}$,
or ${\rm S} {\rm P}$; and we shall call the latter
part~($\varpi$) the {\it vector part}, or simply
{\sc the vector}, of the same polynome, and shall denote this
other part by the symbol ${\rm V} \mathbin{.} {\rm P}$, or
${\rm V} {\rm P}$: these {\it names\/} (scalar and vector), and
these {\it characteristics\/} (${\rm S}$ and ${\rm V}$), being
here adopted as an extension of the phraseology and notation of
the Calculus of Quaternions\footnote*{See Lectures,
{\it passim}.},
in which such scalars and vectors receive useful geometrical
interpretations.  From the same calculus we shall here borrow
also the conception and the sign of {\it conjugation}; and shall
say that any two polynomes (such as those represented by
$p + \varpi$ and $p - \varpi$) are {\sc conjugate}, if they have
{\it equal scalars\/} ($p$), but {\it opposite vectors\/}
($\pm \varpi$): and if either of these two polynomes be denoted
by ${\rm P}$, then the symbol ${\rm K} \mathbin{.} {\rm P}$, or
${\rm K} {\rm P}$, shall be employed to represent the other;
${\rm K}$ being thus used (as in quaternions) as the
{\it characteristic of conjunction}.  With these notations, and
with the recent significations of $p$ and $\varpi$,
$$p = {\rm S} (p + \varpi),\quad
  \varpi = {\rm V} (p + \varpi),\quad
  p - \varpi = {\rm K} (p + \varpi);
   \eqno (18)$$
or, writing ${\rm P}$ and ${\rm P}'$ for $p + \varpi$ and
$p - \varpi$,
$${\rm P}' = {\rm K} {\rm P},
   \quad\hbox{if}\quad
  {\rm S} {\rm P}' = {\rm S} {\rm P},
   \quad\hbox{and}\quad {\rm V} {\rm P}' = - {\rm V} {\rm P};
   \eqno (19)$$
and generally, for any polynome~${\rm P}$, of the kind here
considered,
$${\rm P} = {\rm S} {\rm P} + {\rm V} {\rm P},\quad
  {\rm K} {\rm P} = {\rm S} {\rm P} - {\rm V} {\rm P}.
   \eqno (20)$$
We may also propose to call the $n$ symbols
$\iota_1 \, \ldots \, \iota_n$
by the general name of {\sc vector-units}, as the
symbol~$\iota_0$ has been equated in (10) to the
{\sc scalar-unit}, or to $1$; and may call that equation~(10) the
{\sc unit-law}, or more fully, the {\it law of the primary unit}.

\bigbreak

[4.]
Already, from these few definitions and notations, a variety of
symbolical consequences can be deduced, which have indeed already
occurred in the Calculus of Quaternions, but which are here taken
with enlarged significations, and without reference to
interpretation in geometry.  For example, in the general
equations~(20), we may {\it abstract from the operand}, that is,
from the polynome~${\rm P}$, and may write more briefly (as in
quaternions),
$$1       = {\rm S} + {\rm V},\quad
  {\rm K} = {\rm S} - {\rm V};
   \eqno (21)$$
whence
$${\rm S} = {\textstyle {1 \over 2}} (1 + {\rm K}),\quad
  {\rm V} = {\textstyle {1 \over 2}} (1 - {\rm K});
   \eqno (22)$$
or more fully,
$${\rm S} {\rm P}
   = {\textstyle {1 \over 2}} ({\rm P} + {\rm P}'),\quad
  {\rm V} {\rm P}
   = {\textstyle {1 \over 2}} ({\rm P} - {\rm P}'),
   \quad\hbox{if}\quad {\rm P}' = {\rm K} {\rm P}.
   \eqno (23)$$
Again, since (with the recent meanings of $p$ and $\varpi$),
$$\left. \eqalign{
   &
{\rm S} p = p,\quad
{\rm V} p = 0,\quad
{\rm K} p = p,\quad
{\rm S} \varpi = 0,\quad
{\rm V} \varpi = \varpi,\quad
{\rm K} \varpi = - \varpi,\cr
   &
{\rm S} (p - \varpi) = p,\quad
{\rm V} (p - \varpi) = - \varpi,\quad
{\rm K} (p - \varpi) = p + \varpi,\cr}
   \right\}
   \eqno (24)$$
we may write
$$\left. \eqalign{
   &
{\rm S} {\rm S} {\rm P} = {\rm S} {\rm P},\quad
{\rm V} {\rm S} {\rm P} = 0 = {\rm S} {\rm V} {\rm P},\quad
{\rm V} {\rm V} {\rm P} = {\rm V} {\rm P},\cr
   &
{\rm S} {\rm K} {\rm P} = {\rm S} {\rm P}
   =  {\rm K} {\rm S} {\rm P},\quad
{\rm V} {\rm K} {\rm P} = - {\rm V} {\rm P}
   =  {\rm K} {\rm V} {\rm P},\quad
{\rm K} {\rm K} {\rm P} = {\rm P};\cr}
   \right\}
   \eqno (25)$$
or more concisely,
$$\left. \eqalign{
   &
{\rm S}^2 = {\rm S},\quad
{\rm V} {\rm S} = {\rm S} {\rm V} = 0,\quad
{\rm V}^2 = {\rm V},\cr
   &
{\rm S} {\rm K} = {\rm K} {\rm S} = {\rm S},\quad
{\rm V} {\rm K} = {\rm K} {\rm V} = - {\rm V},\quad
{\rm K}^2 = 1.\cr}
   \right\}
   \eqno (26)$$
The operations, ${\rm S}$, ${\rm V}$, ${\rm K}$ are evidently
{\it distributive},
$${\rm S} \Sigma = \Sigma {\rm S},\quad
  {\rm V} \Sigma = \Sigma {\rm V},\quad
  {\rm K} \Sigma = \Sigma {\rm K};
   \eqno (27)$$
and hence it is permitted to multiply together any two of the
equations (21), (22), or to square any one of them, as if
${\rm S}$, ${\rm V}$, ${\rm K}$ were ordinary algebraical
symbols, and the results must be found to be consistent with
those equations themselves, and with the relations (26). Thus,
squaring and multiplying the equations~(21), we obtain
$$\left. \eqalign{
1^2
   &= ({\rm S} + {\rm V})^2
    = {\rm S}^2 + {\rm V}^2 + 2 {\rm S} {\rm V}
    = {\rm S} + {\rm V}
    = 1,\cr
{\rm K}^2
   &= ({\rm S} - {\rm V})^2
    = {\rm S}^2 + {\rm V}^2 - 2 {\rm S} {\rm V}
    = {\rm S} + {\rm V}
    = 1,\cr
1 {\rm K}
   &= ({\rm S} + {\rm V}) ({\rm S} - {\rm V})
    = {\rm S}^2 - {\rm V}^2
    = {\rm S} - {\rm V}
    = {\rm K};\cr}
   \right\}
   \eqno (28)$$
and the equations (22) give similarly,
$$\left. \eqalign{
{\rm S}^2
   &= {\textstyle {1 \over 4}} (1 + {\rm K})^2
    = {\textstyle {1 \over 4}} (1 + {\rm K}^2 + 2 {\rm K})
    = {\textstyle {1 \over 2}} (1 + {\rm K})
    = {\rm S};\cr
{\rm V}^2
   &= {\textstyle {1 \over 4}} (1 - {\rm K})^2
    = {\textstyle {1 \over 4}} (1 + {\rm K}^2 - 2 {\rm K})
    = {\textstyle {1 \over 2}} (1 - {\rm K})
    = {\rm V};\cr
{\rm S} {\rm V}
   &= {\rm V} {\rm S}
    = {\textstyle {1 \over 4}} (1 + {\rm K}) (1 - {\rm K})
    = {\textstyle {1 \over 4}} (1 - {\rm K}^2)
    = {\textstyle {1 \over 4}} (1 - 1)
    = 0.\cr}
   \right\}
   \eqno (29)$$
Again, if we multiply (22) by ${\rm K}$, we get
$$\left. \eqalign{
{\rm K} {\rm S}
   &= {\textstyle {1 \over 2}} {\rm K} (1 + {\rm K})
    = {\textstyle {1 \over 2}} ({\rm K} + {\rm K}^2)
    = {\textstyle {1 \over 2}} ({\rm K} + 1)
    = {\rm S},\cr
{\rm K} {\rm V}
   &= {\textstyle {1 \over 2}} {\rm K} (1 - {\rm K})
    = {\textstyle {1 \over 2}} ({\rm K} - {\rm K}^2)
    = {\textstyle {1 \over 2}} ({\rm K} - 1)
    = - {\rm V};\cr}
   \right\}
   \eqno (30)$$
all which results are seen to be symbolically true, and other
verifications of this sort may easily be derived, among which the
following may be not unworthy of notice:
$$({\rm S} \pm {\rm V})^{2m} = 1,\quad
  ({\rm S} \pm {\rm V})^{2m+1} = {\rm S} \pm {\rm V},\quad
  \left( {1 \pm {\rm K} \over 2} \right)^m
   = {1 \pm {\rm K} \over 2},
   \eqno (31)$$
where $m$ is any positive whole number.

\bigbreak

[5.]
As a second simplification of the general conception of polynomes
of the form (1), which will tend to render the laws of their
operations on each other still more analogous to those of the
quaternions, let it be now conceived that the choice of the
``constants of multiplication,'' $(fgh)$, is restricted by the
following condition, which may be called the ``Law of
Conjugation:''
$${\rm K} \mathbin{.} \iota \iota' = \iota' \iota,
   \quad\hbox{or}\quad
  {\rm K} \mathbin{.} \iota_f \iota_g = \iota_g \iota_f;
   \eqno (32)$$
namely the condition that, ``opposite (or inverted) products of
any two of the $n$ symbols $\iota_1,\ldots \, \iota_n$, shall
always be conjugate polynomes.''  The indices $f$ and $g$ being
still supposed to be each $> 0$, the constants of multiplication
$(fgh)$, which had remained arbitrary and disposable in [2.],
after that first simplification which consisted in supposing
$\iota_0 = 1$, come now to be still further reduced in number,
from $(n + 1) n^2$ to ${1 \over 2} n (n^2 + 1)$.  For we have
now, by operating with ${\rm S}$ on the equation~(32), the
following formula of relation between those constants,
$$(fg0) = (gf0);
   \eqno (33)$$
and by comparing coefficients of $\iota_h$, this other formula is
obtained,
$$- (fgh) = (gfh),
   \quad\hbox{if}\quad h > 0;
   \eqno (34)$$
whence
$$(ffh) = 0,
   \quad\hbox{if}\quad h > 0.
   \eqno (35)$$
Writing, for conciseness,
$$(fg0) = (fg),\quad (ff) = (f),
   \eqno (36)$$
the {\it squares},~$\iota^2$, of the $n$
{\it vector-units\/}~$\iota$, will thus reduce themselves to so
many constant {\it scalars},
$$\iota_1^2 = (1),\quad
  \iota_2^2 = (2),\quad \ldots \quad
  \iota_f^2 = (f),\quad \ldots \quad
  \iota_n^2 = (n);
   \eqno (37)$$
and besides these, we shall have
$\displaystyle (n + 1) \times {n (n - 1) \over 2}
   = {\textstyle {1 \over 2}} (n^3 - n)$
{\it other\/} scalars, as constants of multiplication; namely the
constituents $(fgh)$ of the polynomial expansions of all the
binary products $\iota \iota'$ or $\iota_f \iota_g$, of unequal
vector-units, taken in any one selected {\it order}, for instance
so that $g > f$; it being unnecessary now, on account of the
formul{\ae} of relation (33), (34), to attend {\it also\/} to
the {\it opposite\/} order of the two factors, if the object be
merely to determine the {\it number\/} of the independent
constants, which number is thus found to be
$n + {1 \over 2} (n^3 - n) = {1 \over 2} (n^3 + n)$,
as above stated.  Such then is the number of the constants of
multiplication, including $n$ of the form~$(f)$, and
${1 \over 2} n (n - 1)$ of the form $(fg)$, besides others of
the form $(fgh)$, which remain still arbitrary, or disposable,
after satisfying, first, the {\it Unit-Law}, $\iota_0 = 1$, and,
second, the {\it Law of Conjugation},
${\rm K} \mathbin{.} \iota \iota' = \iota' \iota$.

\bigbreak

[6.]
From this law of conjugation, (32), several general consequences
follow.  For, {\it first}, we see from it that ``the
{\it square\/} of every {\it vector\/} is a {\it scalar},'' which
may be thus expanded:
$$\left. \eqalign{
\varpi
   &= (\iota_1 x_1 + \ldots + \iota_n x_n)^2
    = (1) x_1^2 + (2) x_2^2 + \ldots + (n) x_n^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
       + 2 (12) x_1 x_2
       + 2 (13) x_1 x_3
       + \ldots
       + 2 (fg) x_f x_g
       + \ldots;\cr}
   \right\}
   \eqno (38)$$
that is, more briefly,
$$(\Sigma \iota x)^2
   = \Sigma (e) x_e^2 + 2 \Sigma (fg) x_f x_g,
   \eqno (39)$$
the summations extending to values of the indices $> 0$ and $g$
being $> f$.  In the {\it second\/} place, and more generally,
``{\it inverted products of any two vectors\/} are equal to
{\it conjugate polynomes\/};'' or in symbols,
$$\varpi' \varpi = {\rm K} \mathbin{.} \varpi \varpi',
   \eqno (40)$$
whatever two vectors may be denoted by $\varpi$ and $\varpi'$.
In fact, these two products have (according to the
definition~[3.]\ of conjugates) one {\it common scalar part}, but
{\it opposite vector parts},
$$\left. \eqalign{
{\rm S} \mathbin{.} \varpi' \varpi
   &= {\rm S} \mathbin{.} \varpi \varpi'
    = \Sigma (e) x_e x_e' + \Sigma (fg) (x_f x_g' + x_g x_f');\cr
- {\rm V} \mathbin{.} \varpi' \varpi
   &= {\rm V} \mathbin{.} \varpi \varpi'
    = \Sigma (fgh) (x_f x_g' - x_g x_f') \iota_h:\cr}
   \right\}
   \eqno (41)$$
whence also we may write, as in quaternions,
$${\rm S} \mathbin{.} \varpi \varpi'
   = {\textstyle {1 \over 2}}
         (\varpi \varpi' + \varpi' \varpi),\quad
  {\rm V} \mathbin{.} \varpi \varpi'
   = {\textstyle {1 \over 2}}
         (\varpi \varpi' - \varpi' \varpi).
   \eqno (42)$$
And, {\it thirdly}, the result (40) may be still further
generalized as follows: ``The {\it conjugate of the product of
any two polynomes\/} is equal to the {\it product of their
conjugates}, taken in an {\it inverted order\/};'' or in symbols,
$${\rm K} \mathbin{.} {\rm P} {\rm P}'
   = {\rm K} {\rm P}' \mathbin{.} {\rm K} {\rm P}.
   \eqno (43)$$
In fact, we have now, by (16), (24), (27) and (40),
$$\eqalignno{
{\rm K} {\rm P}''
   &= {\rm K} \mathbin{.} {\rm P} {\rm P}'
    = {\rm K} \mathbin{.} (p + \varpi) (p' + \varpi') \cr
   &= {\rm K} (p p' + p \varpi' + p' \varpi + \varpi \varpi') \cr
   &= p p' - p \varpi' - p' \varpi + \varpi' \varpi \cr
   &= (p' - \varpi') (p - \varpi)
    = {\rm K} {\rm P}' \mathbin{.} {\rm K} {\rm P},
   &(44)\cr}$$
as asserted in (43).  It follows also, {\it fourthly}, that
``the product of any two conjugate polynomes is a {\it scalar},
independent of their order, and equal to the difference of the
squares of the scalar and vector parts of either of them;'' for,
$$\hbox{if}\quad {\rm P}' = {\rm K} {\rm P}
   \quad\hbox{then}\quad
   {\rm P} {\rm P}' = (p + \varpi) (p - \varpi) = p^2 - \varpi^2;
   \eqno (45)$$
where $\varpi^2$ is, by (38) or (39), a scalar.  And if we
agree to call the {\it square root\/} (taken with a suitable
sign) of this scalar product of two conjugate polynomes,
${\rm P}$ and ${\rm K} {\rm P}$, the common {\sc tensor} of each,
and to denote it by the symbol ${\rm T} {\rm P}$; if also we give
the name of {\sc versor} to the quotient of {\it a polynome
divded by its own tensor}, and denote this quotient by the symbol
${\rm U} {\rm P}$: we shall then be able to establish several
{\it general formul{\ae}}, as extensions from the theory of
quaternions.  For we shall have
$${\rm T} {\rm P} = {\rm T} {\rm K} {\rm P}
   =  \surd ( {\rm P} {\rm K} {\rm P} )
   =  \{ ({\rm S} {\rm P})^2
           - ({\rm V} {\rm P})^2 \}^{1 \over 2};
   \eqno (46)$$
$${\rm T}(p \pm \varpi) = (p^2 - \varpi^2)^{1 \over 2};\quad
  {\rm T} p = (p^2)^{1 \over 2},\quad
  {\rm T} \varpi = (- \varpi^2)^{1 \over 2};
   \eqno (47)$$
$${\rm U} {\rm P}
   =  {{\rm P} \over \surd ( {\rm P} {\rm K} {\rm P} )},\quad
  {\rm U} (p \pm \varpi)
   =  {p \pm \varpi \over (p^2 - \varpi^2)^{1 \over 2}};
   \eqno (48)$$
$${\rm P}
   =  {\rm T} {\rm P} \mathbin{.} {\rm U} {\rm P}
   =  {\rm U} {\rm P} \mathbin{.} {\rm T} {\rm P};
   \eqno (49)$$
$${\rm T} {\rm U} {\rm P} = {\rm U} {\rm T} {\rm P} = 1;\quad
  {\rm T} {\rm T} {\rm P} = {\rm T} {\rm P};\quad
  {\rm U} {\rm U} {\rm P} = {\rm U} {\rm P}:
   \eqno (50)$$
with some other connected equations.  But, although the chief
{\it terms\/} (such as scalar, vector, conjugate, tensor,
versor), and the main {\it notations\/} answering thereto (namely
${\rm S}$, ${\rm V}$, ${\rm K}$, ${\rm T}$, ${\rm U}$), of the
calculus of quaternions, along with several general
{\it formul{\ae}\/} resulting, come thus to receive extended
significations, as applying to certain {\it polynomial\/}
expressions which involve $n$ vector-units, and for which as many
as ${1 \over 2} (n^3 + n)$ constants of multiplication are still
left arbitrary and disposable; yet it must be observed, that we
have not hitherto established any {\it modular property\/} of
either of the two functions, which have been called above the
{\it tensor\/} and {\it versor\/} of a polynome; nor any
{\it associative law}, for the multiplication of three such
polynomes together.

\nobreak\bigskip

Observatory of Trinity College, Dublin,

\nobreak

\qquad June 6, 1854.

\vfill\eject

\centerline{{\largeit Section\/} {\largerm II.}}

\nobreak\bigskip

[7.]
Let us now consider generally the {\it associative\/} law of
multiplication, which may be expressed by the formula already
mentioned but reserved in [1.],
$$\iota \mathbin{.} \iota' \iota''
   =  \iota \iota' \mathbin{.} \iota'';
   \eqno (51)$$
or by this other equation,
$$\iota_e \mathbin{.} \iota_f \iota_g
   = \iota_e \iota_f \mathbin{.} \iota_g:
   \eqno (52)$$
and let us inquire into the conditions under which this law shall
be fulfilled, for any 3 unequal or equal symbols of the
form~$\iota$.

If the conception of the polynomial expression
$${\rm P} = \Sigma \iota x
   =  \iota_0 x_0 + \iota_1 x_1 + \ldots + \iota_n x_n,
   \qquad (1)$$
be no further restricted than it was in [1.], then {\it each\/}
of the three indices $e$,~$f$,~$g$, in the equation (52), may
receive any one of the $n + 1$ values from $0$ to $n$; so that
there are in this case $(n + 1)^3$ associative conditions of this
form (52), whereof {\it each}, by comparison of the coefficients
of the $n + 1$ symbols~$\iota$, breaks itself up into $n + 1$
separate equations, of the ordinary algebraical kind, making in
all no fewer than $(n + 1)^4$ algebraical relations, to be
satisfied, if possible, by the $(n + 1)^3$ constants of
multiplication, of the form~$(fgh)$: respecting which constants,
it will be remembered that the general formula has been
established,
$$\iota_f \iota_g
   =     (fg0) \iota_0 + \ldots
       + (fgh) \iota_h + \ldots
       + (fgn) \iota_n.
   \qquad (7)$$
We may therefore substitute, in (52), the expressions,
$$\left. \multieqalign{
  \iota_f \iota_g &= \Sigma_h (fgh) \iota_h, &
  \iota_e \iota_f &= \Sigma_h (efh) \iota_h,\cr
  \iota_e \iota_h &= \Sigma_k (ehk) \iota_k, &
  \iota_h \iota_g &= \Sigma_k (hgk) \iota_k;\cr}
   \right\}
   \eqno (53)$$
and then, by comparing coefficients of $\iota_k$, this
associative formula~(52) breaks itself up, as was just now
remarked, into $(n + 1)^4$ equations between the $(n + 1)^3$
constants, which are all included in the
following\footnote*{This formula~(54) may be deduced from the
equation~(214) in p.~239 of the writer's ``Researches respecting
Quaternions'', published in the Transactions of the Royal Irish
Academy, vol.~xxi, part~2, by changing there the letters
$r \, s \, t \, r' \, s'$ to $f \, h \, g \, e \, k$, and
substituting the symbol~$(fgh)$ for $n_{g,f,h}$.  Or the same
formula~(54) may be derived from one given in page~(30) of the
Preface to the same author's Lectures on Quaternions, (Dublin,
Hodges and Smith, 1853), by writing $g \, f \, e \, k$ instead of
$f \, g \, g' \, h'$, and changing each of the two symbols
$1_{g,f,h}$, $1_{g,f,h}'$, to $(fgh)$.  But the {\it general
reductions\/} of the present paper have not been hitherto
published.}:
$$\Sigma_h (fgh) (ehk) = \Sigma_h (efh) (hgk);
   \eqno (54)$$
where the {\it four\/} indices $e \, f \, g \, k$ may each
separately receive any one of the $n + 1$ values from $0$ to $n$,
and the summations relatively to $h$ are performed between
the same limits.

\bigbreak

[8.]
Introducing next the simplification (10) of article~[2.], or
supposing $\iota_0 = 1$, which has been seen to reduce the number
of the constants of multiplication from $(n + 1)^3$ to
$(n + 1) n^2$, we find that the number of equations to be
satisfied by them is reduced in a still greater ratio, namely
from $(n + 1)^4$ to $(n + 1) n^3$.  For, if we suppose the
index~$g$ to become $0$, and observe that each of the constants
$(f0h)$ and $(0fh)$ is equal, by (12) and (13), to $0$ or to
$1$, according as $h$ is unequal or equal to $f$, we shall see
that the sum in the left-hand member of the formula~(54) reduces
itself to the term $(efk)$: but such is also in this case the
value of the right-hand sum in the same formula, because in
calculating that sum we need attend only to the value $h = k$, if
$g$ be still $= 0$.  In like manner, if $f = 0$, each sum reduces
itself to $(egk)$; and if $e = 0$, the two sums become each
$= (fgk)$.  If then any one of these three indices,
$e$,~$f$,~$g$, be $= 0$, the formula~(54) is satisfied: which
might indeed have been foreseen, by observing that, in each of
these three cases, one factor of each member of the equation
(52) becomes $= 1$.  We may therefore henceforth suppose that
each of the three indices, $e$,~$f$,~$g$, varies only from $1$ to
$n$, or that
$$e > 0,\quad f > 0,\quad g > 0;
   \eqno (55)$$
while $k$ may still receive any value from $0$ to $n$, and $h$
still varies in the summations between these latter limits: and
thus the number of equations, supplied by the formula~(54),
between the constants $(fgh)$, is reduced, as was lately stated,
to $(n + 1) n^3$; while the number of those constants themselves
had been seen to be reduced to $(n + 1) n^2$, by the same
supposition $\iota_0 = 1$.

\bigbreak

[9.]
Additional reductions are obtained by introducing the law of
conjugation~(32), or by supposing
${\rm K} \mathbin{.} \iota_f \iota_g = \iota_g \iota_f$,
with the consequences already deduced from that law or equation
in [5.]. Using $\Sigma'$ to denote a summation relatively to $h$
from $1$ to $n$, and taking separately the two cases where
$k = 0$ and where $k > 0$, we have, for the first case, by~(54),
$$\Sigma' (efh) (gh) = \Sigma' (fgh) (eh);
   \eqno (56)$$
and for the second case,
$$(ef) (g0k) - (fg) (e0k)
   =  \Sigma' \{ (efh) (ghk) + (fgh) (ehk) \}.
   \eqno (57)$$
No new conditions would be obtained by interchanging $e$ and $g$;
but if we cyclically change $efg$ to $fge$, each of the two sums
(56) is seen to be equal to another of the same form; and two
new equations are obtained from (57), by adding which thereto we
find,
$$0 = \Sigma' \{ (efh) (ghk) + (fgh) (ehk) + (geh) (fhk) \};
   \eqno (58)$$
and therefore,
$$(fg) (e0k) - (ef) (g0k) = \Sigma' (geh) (fhk).
   \eqno (59)$$
When $e = f$, the equations (56) and (59) become, respectively,
$$0 = \Sigma' (fh) (fgh),
   \eqno (60)$$
and
$$(fg) (f0k) - (f) (g0k) = \Sigma' (gfh) (fhk);
   \eqno (61)$$
which are identically satisfied, if we suppose also $f = g$; the
properties~[5.]\ of the symbols $(fgh)$ being throughout attended
to: while, by the earlier properties~[2.], the symbol $(e0k)$ or
$(0ek)$ is equal to $0$ or to $1$, according as $e$ and $k$ are
unequal or equal to each other.  And no equations distinct from
these are obtained by supposing $e = g$, or $f = g$, in (56) and
(59).  The associative conditions for which $k = 0$ are,
therefore, in number, $n(n - 1)$ of the form~(60), and
${1 \over 3} n (n - 1) (n - 2)$ of the form~(56); or
${1 \over 3} (n^3 - n)$ in all.  And the other associative
conditions, for which $k > 0$, are, in number,
$n^2 (n - 1)$ of the form~(61), and
${1 \over 2} n^2 (n - 1) (n - 2)$ of the form~(59), or
${1 \over 2} (n^4 - n^3)$ in all.  It will, however, be found
that this last number admits of being diminished by
${1 \over 2} (n^2 - n)$, namely by one for each of the symbols of
the form $(fg)$; and that if, before or after this reduction, the
associative equations for which $k > 0$ be satisfied, then those
other ${1 \over 3} (n^3 - n)$ conditions lately mentioned, for
which $k = 0$, are satisfied also, as a necessary consequence.
The {\it total number\/} of the {\it equations of association},
included in the formula~(54), will thus come to be reduced to
$${\textstyle {1 \over 2}} (n^4 - n^3)
       - {\textstyle {1 \over 2}} (n^2 - n),
   \quad\hbox{or to}\quad
  {\textstyle {1 \over 2}} n (n - 1) (n^2 - 1);$$
but it may seem unlikely that even so large a number of
conditions as this can be satisfied {\it generally}, by the
${1 \over 2} n (n^2 + 1)$ constants of multiplication~[5.].  Yet
I have found, not only for the case $n = 2$, in which we have
thus 5 constants and 3 equations, but also for the cases $n = 3$
and $n = 4$, for the former of which we have 15 constants and 24
equations, while for the latter we have 34 constants and 90
equations, that all these associative conditions can be
satisfied: even in such a manner as to leave some degree of
indetermination in the results, or some constants of
multiplication disposable.

\bigbreak

[10.]
Without expressly introducing the symbols~$(fgh)$, results
essentially equivalent to the foregoing may be deduced in the
following way, with the help of the characteristics~[3.]\ of
operation, ${\rm S}$,~${\rm V}$,~${\rm K}$.  The formula of
association~(51) may first be written thus\footnote*{There is
here a slight departure from the notation of the Lectures on
Quaternions, by the suppression of certain points, which
circumstance in the present connexion cannot produce ambiguity.}:
$$\iota \, {\rm S} \iota' \iota''
       + \iota {\rm V} \iota' \iota''
   =  {\rm S} \iota \iota' \mathbin{.} \iota''
       + {\rm V} \iota \iota' \mathbin{.} \iota'';
   \eqno (62)$$
in which the symbols ${\rm S} \iota \iota'$ and
${\rm V} \iota \iota'$ are used to denote concisely, without a
point interposed, the scalar and vector parts of the product
$\iota \iota'$, but a point is inserted, after those symbols, and
before $\iota''$, in the second member, as a mark of
multiplication: so that, in this abridged notation,
${\rm S} \iota \iota' \mathbin{.} \iota''$ and
${\rm V} \iota \iota' \mathbin{.} \iota''$
denote the products which might be more fully expressed as
$({\rm S} \mathbin{.} \iota \iota') \times \iota''$ and
$({\rm V} \mathbin{.} \iota \iota') \times \iota''$;
while it has been thought unnecessary to write any point in the
first member, where the factor~$\iota$ occurs at the left hand.
Operating on (62) by ${\rm S}$ and ${\rm V}$, we find the two
following equations of association, which are respectively of the
scalar and vector kinds:
$${\rm S} ( \iota \, {\rm V} \iota' \iota''
       - \iota'' \, {\rm V} \iota \iota' )
   =  0;
   \eqno (63)$$
$${\rm V} ( \iota \, {\rm V} \iota' \iota''
       + \iota'' \, {\rm V} \iota \iota' )
   =  \iota'' \, {\rm S} \iota \iota'
       - \iota \, {\rm S} \iota' \iota'';
   \eqno (64)$$
because the law~(32) of conjugation,
$\iota' \iota = {\rm K} \iota \iota'$, gives, by (41),
$${\rm S} \varpi' \varpi = + {\rm S} \varpi \varpi',\quad
  {\rm V} \varpi' \varpi = - {\rm V} \varpi \varpi'.$$
For the same reason, no essential change is made in either of the
two equations, (63), (64), by interchanging $\iota$ and
$\iota''$; but if we cyclically permute the three vector-units,
$\iota \, \iota' \, \iota''$, then (63) gives
$${\rm S} (\iota \, {\rm V} \iota' \iota'')
   =  {\rm S} (\iota' \, {\rm V} \iota'' \iota)
   =  {\rm S} (\iota'' \, {\rm V} \iota \iota');
   \eqno (65)$$
and there arise three equations of the form~(64), which give, by
addition,
$${\rm V} (
         \iota   \, {\rm V} \iota'  \iota''
       + \iota'  \, {\rm V} \iota'' \iota
       + \iota'' \, {\rm V} \iota   \iota' )
   = 0;
   \eqno (66)$$
and therefore conduct to three other equations, of the
form\footnote*{This formula is one continually required in
calculating with quaternions (compare page~li of the Contents,
prefixed to the author's Lectures).}
$${\rm V} (\iota \, {\rm V} \iota' \iota'')
   =  \iota'' \, {\rm S} \iota \iota'
    - \iota'  \, {\rm S} \iota'' \iota.
   \eqno (67)$$
Equating $\iota''$ to $\iota$, the two equations~(65) reduce
themselves to the single equation,
$${\rm S} (\iota \, {\rm V} \iota \iota') = 0;
   \eqno (68)$$
and the formula~(67) becomes
$${\rm V} (\iota \, {\rm V} \iota \iota')
   = \iota^2 \iota' - \iota {\rm S} \iota \iota':
   \eqno (69)$$
both which results become identities, when we further equate
$\iota'$ to $\iota$.  And no equations of condition, distinct
from these, are obtained by supposing $\iota'' = \iota'$, or
$\iota' = \iota$, in (65) and (67).  The number of the
symbols~$\iota$ being still supposed $= n$, and therefore by
[5.]\ the number of the constants which enter into the
expressions of their $n^2$ binary products (including squares)
being $= {1 \over 2} (n^3 + n)$,
these constants are thus (if possible) to be made to satisfy
${1 \over 3} (n^3 - n)$
associative and scalar equations of condition, obtained through
(63), from the comparison of the scalar parts of the two ternary
products, $\iota \mathbin{.} \iota' \iota''$ and
$\iota \iota' \mathbin{.} \iota''$; namely, $n (n - 1)$ scalar
equations of the form~(68) and
${1 \over 3} n (n - 1) (n - 2)$
such equations, of the forms~(65).  And the same constants of
multiplication must also (if the associative law is to be
fulfilled) be so chosen as to satisfy
${1 \over 2} (n^3 - n^2)$
vector equations, equivalent each to $n$ scalar equations, or in
all to
${1 \over 2} (n^4 - n^3)$
scalar conditions, obtained through (64) from the comparison of
the vector parts of the same two ternary products~(51); namely,
$n (n - 1)$ vector equations of the form (69), and
${1 \over 2} n (n - 1) (n - 2)$
other vector equations, included in the formula~(64).  This new
analysis therefore confirms completely the conclusion of the
foregoing paragraph respecting the general existence of
${1 \over 2} (n^4 - n^3) + {1 \over 3} (n^3 - n)$
associative and scalar equations of condition, between the
${1 \over 2} (n^3 + n)$
disposable constants of multiplication, when the general
conception of the polynomial expression~${\rm P}$ of [1.]\ is
modified by the suppositions, $\iota_0 = 1$ in [2.], and
$\iota' \iota = K \iota \iota'$ in [5.].  At least the analysis
of the present paragraph [10.]\ confirms what has been lately
proved in [9.], that the number of conditions of association can
be {\it reduced so far\/}; but the same analysis will also admit
of being soon applied, so as to assist in proving the existence
of those {\it additional\/} and {\it general\/} reductions which
have been lately mentioned without proof, and which depress the
number of conditions to be satisfied to
${1 \over 2} (n^4 - n^3) - {1 \over 2} (n^2 - n)$.
Meanwhile it may be useful to exemplify briefly the foregoing
general reasonings for the cases $n = 2$, $n = 3$, that is, for
trinomial and quadrinomial polynomes.

\bigbreak

[11.]
For the case $n = 2$, the two distinct symbols of the
form~$\iota$ may be denoted simply by $\iota$ and $\iota'$; and
the equations of association to be satisfied are all included in
these two,
$$\iota \mathbin{.} \iota \iota' = \iota^2 \iota',\quad
  \iota' \mathbin{.} \iota' \iota = \iota'^2 \iota;
   \eqno (70)$$
which give, when we operate on them by ${\rm S}$ and ${\rm V}$,
two scalar equations of the form~(68), and two vector equations
of the form~(69), equivalent on the whole to six scalar
equations of condition, between the five constants of
multiplication, (1) (2) (12) (121) (122), if we write, on
the plan of the preceding articles,
$$\iota^2 = (1),\quad
  \iota'^2 = (2),\quad
  {\rm S} \iota \iota' = (12),\quad
  {\rm V} \iota \iota' = - {\rm V} \iota' \iota
       = (121) \iota + (122) \iota'.
   \eqno (71)$$
From (68), or from (60), or in so easy a case by more direct
and less general considerations, we find that the comparison of
the scalar parts of the products~(70) conducts to the two
equations,
$$0 = (121) (1) + (122) (12) = (122) (2) + (121) (12).
   \eqno (72)$$
From (69), or (61), we find that the comparison of the vector
parts of the same products~(70) gives immediately four scalar
equations, which however are seen to reduce themselves to the
three following:
$$(121) (122) = - (12);\quad
  (122)^2 = (1);\quad
  (121)^2 = (2);
   \eqno (73)$$
the first of these occurring twice.  And it is clear that the
equations~(72) are satisfied, as soon as we assign to (1), (2)
and (12) the values given by (73).  If then we write, for
conciseness,
$$(121) = a,\quad
  (122) = b,
   \eqno (74)$$
we shall have, for the present case ($n = 2$), the values,
$$(1) = b^2,\quad
  (2) = a^2,\quad
  (12) = - ab.
   \eqno (75)$$
And hence, (writing $\kappa$ instead  of $\iota'$,) we see that
the {\it trinome}\footnote*{I am not aware that this trinomial
expression (76), with the formula of multiplication (77),
coincides with any of the triplet-forms of Professor De Morgan,
or of Messrs.\ John and Charles Graves: but it is given here
merely by way of illustration.},
$${\rm P} = z + \iota x + \kappa y,
   \eqno (76)$$
where $x \, y \, z$ are ordinary variables, will possess all the
properties of those {\it polynomial expressions\/} which have
been hitherto considered in this paper, and especially the
associative property, if we establish the formula of
multiplication,
$$(\iota x + \kappa y) (\iota x' + \kappa y')
   =  (bx - ay) (bx' - ay')
       + (a \iota + b \kappa) (x y' - y x');
   \eqno (77)$$
wherein $a$ and $b$ are any two constants of the ordinary and
algebraical kind.  In this trinomial system,
$$z'' + \iota x'' + \kappa y''
   = (z + \iota x + \kappa y) (z' + \iota x' + \kappa y'),
   \eqno (78)$$
if
$$\left. \eqalign{
x'' &= z x' + z' x + a (x y' - y x'),\cr
y'' &= z y' + z' y + b (x y' - y x'),\cr
z'' &= z z' + (bx - ay) (bx' - ay');\cr}
   \right\}
   \eqno (79)$$
we have therefore the two {\it modular relations},
$$\left. \eqalign{
z'' + b x'' - a y'' &= (z + bx - ay) (z' + bx' - ay'),\cr
z'' - b x'' + a y'' &= (z - bx + ay) (z' - bx' + ay');\cr}
   \right\}
   \eqno (80)$$
that is to say, the functions $z \pm (bx - ay)$ are two
{\it linear moduli\/} of the system.  A general theory with which
this result is connected will be mentioned a little further on.
Geometrical interpretations (of no great interest) might easily
be proposed, but they would not suit the plan of this
communication.

\bigbreak

[12.]
For the case $n = 3$, or for the {\it quadrinome\/}
$${\rm P} = x_0 + \iota_1 x_1 + \iota_2 x_2 + \iota_3 x_3,
   \eqno (81)$$
we may assume
$$\left. \multieqalign{
  \iota_1^2 &= a_1, &
  \iota_2^2 &= a_2, &
  \iota_3^2 &= a_3,\cr
  {\rm S} \iota_2 \iota_3 &= b_1, &
  {\rm S} \iota_3 \iota_1 &= b_2, &
  {\rm S} \iota_1 \iota_2 &= b_3,\cr}
   \right\}
   \eqno (82)$$
and
$$\left. \eqalign{
{\rm V} \iota_2 \iota_3
   &= - {\rm V} \iota_3 \iota_2
    = \iota_1 l_1 + \iota_2 m_3 + \iota_3 n_2,\cr
{\rm V} \iota_3 \iota_1
   &= - {\rm V} \iota_1 \iota_3
    = \iota_2 l_2 + \iota_3 m_1 + \iota_1 n_3,\cr
{\rm V} \iota_1 \iota_2
   &= - {\rm V} \iota_2 \iota_1
    = \iota_3 l_3 + \iota_1 m_2 + \iota_2 n_1;\cr}
   \right\}
   \eqno (83)$$
and then the ${1 \over 2} (n^4 - n^3) = 27$ scalar equations of
condition, included in the vector form,
$${\rm V} ( \iota \mathbin{.} \iota' \iota'' )
   =  {\rm V} ( \iota \iota' \mathbin{.} \iota'' ),
   \eqno (84)$$
are found on trial to reduce\footnote*{The reason of this
reduction is exhibited by the general analysis in [14.].}
themselves to 24; which, after elimination of the 6 constants of
the forms here denoted by $a$ and $b$, or previously by $(f)$ and
$(fg)$, furnish 18 equations of condition between the 9 other
constants, of the forms here marked $l$,~$m$,~$n$, or previously
$(fgh)$; and these 18 equations may be thus
arranged\footnote\dag{For it is found that each of the three
constants $(eff) + (fgg)$ must give a null product, when it is
multiplied by any one of the constants $(e' f' g')$, or by
any one of these other constants
$(e'' f'' f'') - (e'' g'' g'')$;
if each of the three systems, $efg$, $e' f' g'$, $e'' f'' g''$,
represent, in some order or other, but not necessarily in one
common order, the system of the three unequal indices,
$1$,~$2$,~$3$.}:
$$\left. \eqalign{
0  &= l_1 (n_1 - m_1)
    = l_2 (n_2 - m_2)
    = l_3 (n_3 - m_3),\cr
0  &= l_2 (n_1 - m_1)
    = l_3 (n_2 - m_2)
    = l_1 (n_3 - m_3),\cr
0  &= l_3 (n_1 - m_1)
    = l_1 (n_2 - m_2)
    = l_2 (n_3 - m_3);\cr}
   \right\}
   \eqno (85)$$
$$\left. \eqalign{
0  &= n_1^2  - m_1^2
    = n_2^2  - m_2^2
    = n_3^2  - m_3^2,\cr
0  &= (n_2 + m_2) (n_1 - m_1)
    = (n_3 + m_3) (n_2 - m_2)
    = (n_1 + m_1) (n_3 - m_3),\cr
0  &= (n_3 + m_3) (n_1 - m_1)
    = (n_1 + m_1) (n_2 - m_2)
    = (n_2 + m_2) (n_3 - m_3);\cr}
   \right\}
   \eqno (86)$$
they are therefore {\it satisfied}, without any restriction on
$l_1 \, l_2 \, l_3$, by our supposing
$$n_1 = m_1,\quad
  n_2 = m_2,\quad
  n_3 = m_3;
   \eqno (87)$$
but if we do {\it not\/} adopt this supposition, they
{\it require\/} us to admit this {\it other\/} system of
equations,
$$0 = l_1 = l_2 = l_3 = n_1 + m_1 = n_2 + m_2 = n_3 + m_3.
   \eqno (88)$$
Whichever of these two suppositions, (87), (88), we adopt,
there results a corresponding system of values of the six
recently eliminated constants, of the forms $a$ and $b$, or $(f)$
and $(fg)$; and it is found\footnote*{This fact of
calculation is explained by the general analysis of [15.].  The
values of $a$ and $b$ may be deduced from the formul{\ae},
$a_1 = m_1^2 - l_2 l_3$, $b_1 = l_1 m_1 - m_2 n_3$,
with others cyclically formed from these.}
that these values satisfy, without any new supposition being
required, the
${1 \over 3} (n^3 - n) = 8$
scalar equations, included in the general form
$${\rm S} ( \iota \mathbin{.} \iota' \iota'' )
   =  {\rm S} ( \iota \iota' \mathbin{.} \iota'' ),
   \eqno (89)$$
which are required for the associative property.

\bigbreak

[13.]
In this manner I have been led to the {\it two\/} following
systems of {\it associative quadrinomials}, which may be called
systems (A) and (B); both possessing all those general properties
of the polynomial expression~${\rm P}$, which have been
considered in the present paper; and one of them including the
quaternions.

For the system~(A), the quadrinomial being still of the
form~(81), or of the following equivalent form,
$${\rm Q} = w + \iota x + \kappa y + \lambda z,
   \eqno (90)$$
where $w \, x \, y \, z$ are what were called in [1.]\ the
{\it constituents}, the laws of the {\it vector-units\/}
$\iota \, \kappa \, \lambda$ are all included in this formula of
multiplication for {\it any two vectors}, such as
$$\rho = \iota x + \kappa y + \lambda z,\quad
  \rho' = \iota x' + \kappa y' + \lambda z':
   \eqno (91)$$
$$\eqalignno{
\hbox{(A)}\ldots\qquad
\rho \rho'
   &=    (m_1^2 - l_2 l_3) x x'
       + (l_1 m_1 - m_2 m_3) (y z' + z y') \cr
   &\mathrel{\phantom{=}} \mathord{}
       + (m_2^2 - l_3 l_1) y y'
       + (l_2 m_2 - m_3 m_1) (z x' + x z') \cr
   &\mathrel{\phantom{=}} \mathord{}
       + (m_3^2 - l_1 l_2) z z'
       + (l_3 m_3 - m_1 m_2) (x y' + y x') \cr
   &\mathrel{\phantom{=}} \mathord{}
       + (\iota   l_1 + \kappa  m_3 + \lambda m_2)
               (y z' - z y') \cr
   &\mathrel{\phantom{=}} \mathord{}
       + (\kappa  l_2 + \lambda m_1 + \iota   m_3)
               (z x' - x z') \cr
   &\mathrel{\phantom{=}} \mathord{}
       + (\lambda l_3 + \iota   m_2 + \kappa  m_1)
               (x y' - y x');
   &(92)\cr}$$
and it is clear that {\it Quaternions\/}\footnote*{See the
author's Lectures, or the Philosophical Magazine
for July, 1844, in which the first printed account of quaternions
was given.}
are simply that particular {\it case} of such {\sc quadrinomes}~(A),
for which the {\it six\/} arbitrary constants
$l_1 \, \ldots \, m_3$ and the {\it three\/} vector-units
$\iota \, \kappa \, \lambda$ receive the following values:
$$l_1 = l_2 = l_3 = 1,\quad
  m_1 = m_2 = m_3 = 0,\quad
  \iota = i,\quad
  \kappa = j,\quad
  \lambda = k.
   \eqno (93)$$

For the other associative quadrinomial system~(B), which we may
call for distinction {\sc tetrads}, if we retain the expressions
(90), (91), we must replace the formula of
{\it vector-multiplication\/}~(92) by one of the following form:
$$\eqalignno{
\hbox{(B)}\ldots\qquad
\rho \rho'
   &=    (l x + m y + n z) (l x' + m y' + n z') \cr
   &\mathrel{\phantom{=}} \mathord{}
       + (\kappa  n - \lambda m) (y z' - z y')
       + (\lambda l - \iota   n) (z x' - x z') \cr
   &\mathrel{\phantom{=}} \mathord{}
       + (\iota   m - \kappa  l) (x y' - y x');
   &(94)\cr}$$
involving thus only {\it three\/} arbitrary constants,
$l \, m \, n$, besides the {\it three\/} vector-units,
$\iota \, \kappa \, \lambda$; and apparently having no connexion
with the {\it quaternions}, beyond the circumstance that one
common analysis [12.]\ conducts to both the
{\it quadrinomes\/}~(A) and the {\it tetrads\/}~(B).

As regards certain {\it modular properties\/} of these two
quadrinomial systems, we shall shortly derive them as
consequences of the general theory of polynomes of the
form~${\rm P}$, founded on the principles of the foregoing
articles.

\bigbreak

[14.]
In general, the formula~(59) gives, by [2.], the two following
equations, which may in their turn replace it, and are, like it,
derived from the comparison of the {\it vector parts\/} of the
general associative formula, or from the supposition that $k > 0$
in (54):
$$(fg) = \Sigma (geh) (fhe),
   \quad\hbox{if}\quad e \gtlt g;
   \eqno (95)$$
$$0 = \Sigma (geh) (fhk),
   \quad\hbox{if}\quad k \gtlt e, \enspace k \gtlt g;
   \eqno (96)$$
the summation extending in each from $h = 1$ to $h = n$.
Interchanging $f$ and $g$ in (95), we have
$$(gf) = \Sigma (feh) (ghe),
   \quad\hbox{if}\quad e \gtlt f;
   \eqno (97)$$
and making $g = f$, in either (95) or (97), we obtain the
equation,
$$(f) = \Sigma (feh) (fhe),
   \quad\hbox{if}\quad e \gtlt f.
   \eqno (98)$$
For each of the $n$ symbols~$(f)$, there are $n - 1$ distinct
expressions of this last form, obtained by assigning different
values to $e$; and when these expressions are equated to each
other, there result $n (n - 2)$ equations between the symbols of
the form $(fgh)$.  For each of the ${1 \over 2} n (n - 1)$
symbols of the form $(fg)$, where $f$ and $g$ are unequal, there
are $n - 1$ expressions~(95), and $n - 1$ other expressions of
the form~(97), because, by (33) and (36), $(gf) = (fg)$; and
thus it might seem that there should arise, by equating these
$2n - 2$ expressions for each symbol~$(fg)$, as many as $2n - 3$
equations from each, or
${1 \over 2} n (n - 1) (2n - 3)$
equations in all between the symbols~$(fgh)$.  But if we observe
that the sums of the $n - 1$ expressions (95) for $(fg)$, and of
the $n - 1$ expressions~(97) for $(gf)$ are, respectively,
$$(n - 1) (fg) = \Sigma_e \Sigma_h (geh) (fhe),\quad
  (n - 1) (gf) = \Sigma_e \Sigma_h (feh) (ghe);
   \eqno (99)$$
where the summations may all be extended from $1$ to $n$, because
$(ffh)$ and $(ggh)$ are each $= 0$, by (35), since $h > 0$; and
that these two double sums (99) are equal; we shall see that the
formula
$$(gf) = (fg),
   \eqno (100)$$
though true, gives no information respecting the symbols~$(fgh)$:
or is not to be counted as a new and distinct equation, in
combination with the $n - 1$ equations~(95), and the $n - 1$
equations~(97).  In other words, the comparison of the
sums~(99) shows that we may confine ourselves to equating
separately to each other, for each pair of unequal indices $f$
and $g$, the $n - 1$ expressions (95) for the symbol~$(fg)$, and
the $n - 1$ other expressions~(97) for the symbol~$(gf)$,
without proceeding afterwards to equate an expression of the one
set to an expression of the other set.  We may therefore
suppress, as unnecessary, an equation of the form~(100), for
each of the ${1 \over 2} n (n - 1)$ symbols of the form~$(fg)$,
or for each pair of unequal indices $f$ and $g$, as was stated by
anticipation towards the close of paragraph~[9.].  There remain,
however, $2(n - 2)$ equations of condition, between the
symbols~$(fgh)$, derived from each of those
${1 \over 2} n (n - 1)$ pairs; or as many as $n (n - 1) (n - 2)$
equations in all, obtained in this manner from (95) and (97),
regarded as separate formul{\ae}.  Thus, without yet having used
the formula~(96), we obtain, with the help of (98), by
elimination of the symbols $(f)$, $(fg)$, $(gf)$, through the
comparison of $n - 1$ expressions for each of those $n^2$
symbols, $n^2 (n - 2)$ equations of condition, homogeneous and of
the second dimension, between the symbols of the form~$(fgh)$.
And without any such elimination, the formula~(96) gives
immediately ${1 \over 2} n^2 (n - 1) (n - 2)$ other equations of
the same kind between the same set of symbols; because after
choosing any pair of unequal indices $e$ and $g$, we may combine
this pair with any one of the $n$ values of the index~$f$, and
with any one of the $n - 2$ values of $k$, which are unequal both
to $e$ and to $g$.  There are therefore, altogether,
${1 \over 2} n^2 (n + 1) (n - 2)$ homogeneous equations of the
second dimension, obtained by comparison of the {\it vector
parts\/} of the general formula of association, to be satisfied
by the ${1 \over 2} n^2 (n - 1)$ symbols of the form~$(fgh)$.

\bigbreak

[15.]
To prove now, generally, that when the {\it vector parts\/} of
the associative formula are thus equal, the {\it scalar parts\/}
of the same formula are necesarily equal also, or that the system
of conditions~(56) in [9.]\ is included in the system (57) or
(59); we may conveniently employ the notations ${\rm S}$ and
${\rm V}$, and pursue the analysis of paragraph~[10.], so as to
show that the system of equations~(65), including (68), results
from the system~(67), including (69); or that if the
formula~(84) be satisfied for every set of three unequal or
equal vector-units, $\iota \, \iota' \, \iota''$, then, for every
such set, the formula~(89) is satisfied also.  For this purpose,
I remark that the formula of {\it vector-association\/}~(67),
when combined with the {\it distributive\/} principle of
multiplication~[1.], and of operation with ${\rm S}$ and
${\rm V}$ [5.], gives generally, as in quaternions, the
transformation
$${\rm V} \rho \, {\rm V} \sigma \tau
   =  \tau \, {\rm S} \rho \sigma - \sigma \, {\rm S} \rho \tau;
   \eqno (101)$$
where $\rho$, $\sigma$, $\tau$ may denote {\it any three
vectors}, and the symbol ${\rm V} \rho \, {\rm V} \sigma \tau$ is
used to signify concisely the vector part of the product
$\rho \times {\rm V} (\sigma \tau)$; whence also we may derive by
(41) this other general transformation,
$${\rm V} ( {\rm V} \sigma \tau \mathbin{.} \rho )
   =  \sigma \, {\rm S} \rho \tau - \tau \, {\rm S} \rho \sigma.
   \eqno (102)$$
If then we write
$${\rm V} \sigma \tau   = \rho',\quad
  {\rm V} \tau   \rho   = \sigma',\quad
  {\rm V} \rho   \sigma = \tau',
   \eqno (103)$$
and introduce another arbitrary vector~$\varpi$, we shall have
$${\rm V} \rho' \varpi
   =  \sigma   \, {\rm S} \tau   \varpi
    - \tau     \, {\rm S} \sigma \varpi;
   \eqno (104)$$
and therefore
$${\rm V} \rho \, {\rm V} \rho' \varpi
   =  \tau'    \, {\rm S} \tau   \varpi
    + \sigma'  \, {\rm S} \sigma \varpi;
   \eqno (105)$$
but also
$${\rm V} \rho \, {\rm V} \rho' \varpi
   =  \varpi   \, {\rm S} \rho \rho'
    - \rho'    \, {\rm S} \rho \varpi;
   \eqno (106)$$
whence
$$\varpi \, {\rm S} \rho \rho'
   =  \rho'    \, {\rm S} \rho    \varpi
    + \sigma'  \, {\rm S}  \sigma \varpi
    + \tau'    \, {\rm S}  \tau   \varpi,
   \eqno (107)$$
and consequently
$${\rm S} \rho \rho'
   =  {\rm S} \sigma \sigma'
   =  {\rm S} \tau \tau';
   \eqno (108)$$
but this is precisely by (103) the formula of
{\it scalar-association\/} (65), stated in its most general form.
The general dependence of (65) on (67), or of (56) on (57), is
therefore proved to exist; and the
${1 \over 3} (n^3 - n)$ associative conditions, for which $k = 0$
in (54), are seen to be consequences of
${1 \over 2} (n^4 - n^3)$ other conditions for which $k > 0$; or
even of those conditions diminished in number by
${1 \over 2} (n^2 - n)$, according to what was stated by
anticipation in [9.], and has been proved by the analysis of
[14.].  This result is the more satisfactory, because otherwise
the {\it conditions of association\/} would essentially involve a
system of homogeneous equations of the {\it third\/} dimension
relatively to the symbols~$(fgh)$, obtained by substituting in
(56) the expressions~(95) or (97) for the symbols of the
form~$(fg)$, including the values~(98) of the symbols~$(f)$.
But we see now (as above stated) that the {\it total\/} number of
{\it distinct\/} conditions may be reduced to
${1 \over 2} (n^4 - n^3) - {1 \over 2} (n^2 - n)$,
between the total number ${1 \over 2} (n^3 + n)$ of constants of
multiplication; or finally, after the {\it elimination\/} of the
${1 \over 2} (n^2 + n)$ symbols of the forms $(f)$ and $(fg)$, to
a {\it system of homogeneous equations of the second dimension},
namely, those determined in [14.], of which the number amounts
(as in that paragraph) to
$${\textstyle {1 \over 2}} (n^4 - n^3) - n^2
   =  {\textstyle {1 \over 2}} n^2 (n + 1) (n - 2),
   \eqno (109)$$
between the symbols~$(fgh)$, whereof the number is
$${\textstyle {1 \over 2}} (n^3 + n)
       - {\textstyle {1 \over 2}} (n^2 + n)
   =  {\textstyle {1 \over 2}} n^2 (n - 1).
   \eqno (110)$$

\bigbreak

[16.]
For example, when $n = 2$, the {\it two\/} constants (121) and
(122) have been seen in [11.]\ to be unresticted by {\it any\/}
condition.  When $n = 3$, we have 9 constants, lately denoted by
$l_1 \, l_2 \, l_3$ $m_1 \, m_2 \, m_3$ $n_1 \, n_2 \, n_3$,
wherewith to satisfy 18 homogeneous equations of the second
dimension, namely those marked (85) and (86) in [12.]; which it
has been seen to be possible to do, in two distinct ways (A) and
(B), and even so as to leave some of the constants arbitrary, in
each of the two resulting systems, of {\it associative
quadrinomes and tetrads}.  A similar result has been found by me
to hold good for the case $n = 4$, or for the case of
{\it associative quines}, such as
$${\rm P} = w + \iota x + \kappa y + \lambda z + \mu u,
   \eqno (111)$$
involving {\it four\/} vector-units
$\iota \, \kappa \, \lambda \, \mu$,
which obey the laws of conjugation~(32), and of
association~(51).  For although there are in this case only
$24 = {1 \over 2} n^2 (n - 1)$
constants of the form $(fgh)$, to satisfy
$80 = {1 \over 2} n^2 (n + 1) (n - 2)$
homogeneous equations of the second dimension, yet I have found
that the {\it forms\/}\footnote*{The subject may be illustrated
by the very simple remark, that although the four equations
$tx = 0$, $ty = 0$, $ux = 0$, $uy = 0$, are such that
{\it no three\/} of them {\it include\/} the {\it fourth}, since
we might (for example) satisfy the three first alone by supposing
$t = 0$, $x = 0$, yet they can {\it all four\/} be
{\it satisfied\/} together by supposing either $x = 0$, $y = 0$,
or $t = 0$, $u = 0$.  Compare the equations (85) or (86), which
are of the forms $tx = 0$, $ty = 0$, $tz = 0$, $ux = 0$,
$uy = 0$, $uz = 0$, $vx = 0$, $vy = 0$, $vz = 0$.  In the theory
of {\it quines}, however, the forms are not quite so simple.}
of these equations are such as to allow this to be done in
various ways, and even without entirely determining the
constants.  And it appears not impossible that similar results
may be obtained for higher values of $n$; or that
{\it associative\/}\footnote\dag{The {\it octaves}, or
{\it octonomial expressions}, which Mr.~Cayley published in the
Philosophical Magazine for March 1845, and which had been
previously but privately communicated to me by Mr.~J.~T. Graves
about the end of 1843, after my communication to him of the
quaternions, are {\it not associative polynomes}.  Thus in
Mr.~Cayley's notation, the four following of his seven
{\it types}, $(123)$ $(624)$ $(176)$ $(734)$, give
$\iota_1 \mathbin{.} \iota_2 \iota_4 = \iota_1 \iota_6
   =  - \iota_7$,
but
$\iota_1 \iota_2 \mathbin{.} \iota_4 = \iota_3 \iota_4
   =  + \iota_7$;
or with Mr.~Graves's symbols, the triads $i \, j \, k$,
$i \, o \, n$, $j \, l \, n$, $k \, l \, o$, give
$i \mathbin{.} jl = in = -o$, but
$ij \mathbin{.} l = kl = +o$.  See note to page~(61) of the
Preface to my Lectures.  It was my perceiving this latter
property of Mr.~Graves's symbols in 1844, which chiefly
discouraged me from pursuing the study of those {\it octaves}, as
a species of {\it extensions of the quaternions}, which
Mr.~Graves as well as Mr.~Cayley had designed them to be, and
which in one sense no doubt they are.}
polynomes of higher orders than {\it quines\/} may be discovered.

\nobreak\bigskip

Observatory of Trinity College, Dublin,

\nobreak

\qquad July 4, 1854.

\vfill\eject

\centerline{{\largeit Section\/} {\largerm III.}}

\nobreak\bigskip

[17.]
The following remarks may be useful, as serving to illustrate and
develope the general analysis contained in some of the preceding
paragraphs, especially in [14.]\ and as adapted to give some
assistance towards any future study of associative polynomes,
such as quines, of an order higher than quadrinomes, but subject
like them to the law of conjugation~(32).

The expression~(98) may be thus more fully written:
$$(f) = (fee)^2 + (feg) (fge)
       + \Sigma^\backprime (feh) (fhe);
   \eqno (112)$$
where $e \, f \, g \, h$ are all supposed to be unequal; the
summation~$\Sigma^\backprime$ being performed relatively to $h$,
for all those $n - 3$ values of the latter, which are distinct
from each of the three former indices.  Interchanging $e$ and
$g$, and subtracting, we eliminate the symbol~$(f)$, and obtain
the following formula:
$$\hbox{I.}\ldots\qquad
  (fee)^2 - (fgg)^2
   =  \Sigma^\backprime \{ (fgh) (fhg) - (feh) (fhe) \};
   \eqno (113)$$
which type~I.\ includes generally $n(n - 2)$ distinct and
homogeneous equations, of the second dimension, with $2(n - 2)$
terms in each, between the
${1 \over 2} n^2 (n - 1)$ symbols of the form $(fgh)$.  Thus, for
the case of {\it quadrinomials\/} ($n = 3$), by writing, in
agreement with (82) and (83),
$$a_1 = (1),\quad
  b_1 = (23),\quad
  l_1 = (231),\quad
  m_1 = (313),\quad
  n_1 = (122),
   \eqno (114)$$
and suppressing the sum~$\Sigma^\backprime$, we have by (112)
the two expressions (compare a note to [12.]):
$$a_1 = m_1^2 - l_2 l_3 = n_1^2 - l_2 l_3;
   \eqno (115)$$
together with four others formed from these, by cyclical
permutation of the indices $1$,~$2$,~$3$; and we are thus
conducted, by elimination of the three symbols
$a_1$,~$a_2$,~$a_3$, to three equations of the form
$n_1^2 = m_1^2$; that is, to the 3~equations on the first line of
(86), involving each 2~terms.  For {\it quines\/} ($n = 4$), if
we make also, with the same permitted permutations,
$$\left. \eqalign{
   &
  a_4 = (4),\quad
  c_1 = (14),\quad
  p_1 = (234),\cr
   &
  r_1 = (141),\quad
  s_1 = (142),\quad
  t_1 = (143),\quad
  u_1 = (144),\cr}
   \right\}
   \eqno (116)$$
the index~$h$ receives one value under each sign of
summation~$\Sigma^\backprime$, and the resulting formul{\ae} may
be thus written:
$$\eqalignno{
  ( a_1 + l_3 l_2 + p_2 t_1 - s_1 p_3 = \, ) \,
      n_1^2 + p_2 t_1
  &=  m_1^2 - s_1 p_3
   =  u_1^2 + l_3 l_2;
  & (117)\cr
  ( a_4 - s_1 t_2 - s_2 t_3 - s_3 t_1 = \, ) \,
      r_1^2 - s_2 t_3
  &=  r_2^2 - s_3 t_1
   =  r_3^2 - s_1 t_2;
  & (118)\cr}$$
where the line~(117) is equivalent to three lines of the same
form: so that the elimination of $a_1 \, \ldots \, a_4$ conducts
here to 8 equations, of 4 terms each, between the 24 symbols of
the form $(fgh)$, or $l_1 \, \ldots \, u_3$, as by the general
theory it ought to do.  For polynomes of {\it higher\/} orders
($n > 4$), we have the analogous equations,
$$\eqalignno{
(f) - (feg) (fge) - (fgk) (fkg) - (fke) (fek)
      \hskip -12em \cr
   &= (fee)^2 - (fgk)
         (fkg) + \Sigma^\backprime (feh) (fhe) \cr
   &= (fgg)^2 - (fke)
         (fek) + \Sigma^{\backprime\backprime} (fgh) (fhg) \cr
   &= (fkk)^2 - (feg)
         (fge) + \Sigma^{\backprime\backprime} (fkh) (fhk);
   & (119)\cr}$$
where $h$, under $\Sigma^{\backprime\backprime}$, receives only
$n - 4$ values, being distinct from each of the four unequal
indices, $e \, f \, g \, k$.

\bigbreak

[18.]
By changing $e$ to $f$ in (95), and attending to the properties
of the symbols $(fgh)$, we obtain the expression
$$(fg) = \Sigma (fgh) (hff);
   \eqno (120)$$
where $f$ and $g$ are unequal, and the summation~$\Sigma$ extends
from $h = 1$ to $h = n$.  The term for which $h = f$ vanishes,
and the formula (120) may be thus more fully written:
$$(fg) - (fge) (eff)
   = (fgg) (gff) + \Sigma^\backprime (fgh) (hff);
   \eqno (121)$$
where the letters $e \, f \, g$ denote again some three unequal
indices, and the summation~$\Sigma^\backprime$ is performed as in
the foregoing paragraph.  But also, by (97) and (100),
$$(fg) - (fge) (eff)
   =  (fee) (gee) + \Sigma^\backprime (feh) (ghe);
   \eqno (122)$$
subtracting, therefore, (122) from (121), we eliminate the
symbol~$(fg)$, and obtain the type
$$\hbox{II.}\ldots\qquad
(fee) (gee) - (fgg) (gff)
   = \Sigma^\backprime \{ (fgh) (hff) - (feh) (ghe) \};
   \eqno (123)$$
which represents in general a system of $n (n - 1) (n - 2)$
distinct and homogeneous equations of the second dimension,
containing each $2 (n - 2)$ terms, and derived by eliminations of
the kind last mentioned, from the formul{\ae} (95), (97),
(100), in a manner agreeable to the analysis of paragraph~[14.].
Indeed, it was shown in that paragraph, that the equation
$$(gf) = (fg),
   \qquad (100),$$
though known from earlier and simpler principles to be true,
might be regarded as {\it included\/} in (95) and (97); but
this need not prevent us from {\it using\/} that equation in
combination with the others, whenever it may seem advantageous to
do so: and other combinations of them may with its help be
formed, which are occasionally convenient, or even sometimes
necessary, although the {\it general\/} results of the
elimination of the symbols~$(fg)$ are sufficiently represented by
the recent type~II., or by the formula (123).  For example, a
subordinate type, including only
${1 \over 2} n (n - 1) (n - 2)$ distinct equations, of $2(n - 2)$
terms each, between the symbols $(fgh)$, may thus be formed, by
subtracting (95) from (97), under the condition that
$e \, f \, g$ shall still denote some three unequal indices;
namely,
$$0 = \Sigma \{ (feh) (ghe) - (geh) (fhe) \};
   \eqno (124)$$
or more fully, but at the same time with the suppression of a few
parentheses, which do not appear to be at this stage essential to
clearness,
$$(fge) (eff + egg)
   = \Sigma^\backprime
         ( geh \mathbin{.} fhe - feh \mathbin{.} ghe ):
   \eqno (125)$$
this last formula admitting also of being obtained from (122),
by interchanging $f$ and $g$, and subtracting.  Again, a type
which is in general still more subordinate, as including only
${1 \over 2} n (n - 1)$ distinct equations, of $2 (n - 2)$ terms
each, may be derived by the same process from (120); namely the
type,
$$0 = \Sigma (fgh) (hff + hgg);
   \eqno (126)$$
or in a slightly more expanded form,
$$(fge) (eff + egg)
   =  \Sigma^\backprime (gfh) (hff + hgg);
   \eqno (127)$$
which may also be easily derived, in the same way, from (121).
It will, however, be found, by pursuing a little further the
analysis of [14.], that the equations of this last type, (126)
or (127), are always consequences of the equations of the
intermediate type, (124) or (125); the sum of the $n - 2$
equations of the form (125), which answer to the various values
of $e$ that remain when $f$ and $g$ have been selected, being in
fact equivalent to the formula (126).  It will also be found, by
the same kind of analysis, that the intermediate equations of the
type (124) or (125) are {\it generally\/} deducible from those
of the form (123).  But on the subject of these {\it general
reductions}, connected with the elimination of the symbols $(fg)$
or $(gf)$, it may be proper to add a few words.

\bigbreak

[19.]
Let us admit, at least as temporary abridgements, the notations
$$[fg] =  \Sigma (fgh \mathbin{.} hff);\quad
  [fge] = \Sigma (fhe \mathbin{.} geh);
   \eqno (128)$$
where $e$, $f$, $g$ are any three unequal indices, and $h$ varies
under~$\Sigma$, as before, from $1$ to $n$.  Then the
formula~(95) gives $n - 1$ distinct equivalents for the
symbol~$(fg)$, of which one is by (120) of the form $[fg]$, and
the $n - 2$ others are each of the form $[fge]$; in such a manner
that we may write, instead of (95), with these last notations,
the system of the two formul{\ae},
$$(fg) = [fg],\quad
  (fg) = [fge];
   \eqno (129)$$
whereof the latter is equivalent to a system of $n - 2$
equations: and of course, instead of (97), we may in like manner
write
$$(gf) = [gf],\quad
  (gf) = [gfe].
   \eqno (130)$$
The equations (99) may now be thus presented:
$$\left. \eqalign{
(n - 1) (fg)
   &= [fg] + \Sigma^\backprime [fge]
    = \Sigma \Sigma (fhe \mathbin{.} geh);\cr
(n - 1) (gf)
   &= [gf] + \Sigma^\backprime [gfe]
    = \Sigma \Sigma (feh \mathbin{.} ghe);\cr}
   \right\}
   \eqno (131)$$
where $e$ under the sign~$\Sigma'$ is distinct from each of the
two indices $f$ and $g$; but, under the double
sign~$\Sigma \Sigma$, both $e$ and $h$ may each receive any one
of the values from $1$ to $n$.  The two double sums are equal, as
in [14.], and therefore we must have, {\it identically},
$$[fg] + \Sigma' [fge] = [gf] + \Sigma' [gfe];
   \eqno (132)$$
the equation~(100) being at the same time seen again to be a
consequence, by simple additions, of the formul{\ae} (95) and
(97).  Thus, after assigning any two unequal values to the
indices $f$ and $g$, we see that the two symbols, $(fg)$,~$(gf)$;
the two others, $[fg]$,~$[gf]$; the $n - 2$ symbols,~$[fge]$; and
the $n - 2$ symbols,~$[gfe]$, are indeed all equal to each other:
but that the $2n - 1$ equations between these $2n$ equal symbols
are connected by a {\it relation}, such that any $2n - 2$ of
them, which are distinct among themselves, {\it include\/} the
remaining one: and that therefore, after the elimination of
$(fg)$ and $(gf)$, there remain only $2(n - 2)$ {\it distinct\/}
equations of condition, as was otherwise shown in [14.].  But, in
that paragraph, we proposed to form those resulting conditions on
a plan which may now be represented by the formul{\ae}
$$[fg] = [fge],\quad
  [gf] = [gfe];
   \eqno (133)$$
whereas we now prefer, for the sake of the convenience gained by
the disappearance of certain terms in the subtractions, to employ
that other mode of combination, which conducted in [18.]\ to the
formula (123), and may now be denoted as follows:
$$[fg] = [gfe],\quad
  [gf] = [fge].
   \eqno (134)$$
Summing these last with respect to $e$, we find
$$(n - 2) [fg] = \Sigma' [gfe],\quad
  (n - 2) [gf] = \Sigma' [fge];
   \eqno (135)$$
and therefore, by the identity (132),
$$(n - 3) [gf] = (n - 3) [fg].
   \eqno (136)$$
{\it If, then}, $n > 3$, we are entitled to infer, from (123) or
(134), the following formula, which is equivalent to (126),
$$[gf] = [fg];
   \eqno (137)$$
and therefore also by (134) this other type, equivalent to
(124),
$$[fge] = [gfe],
   \eqno (138)$$
which includes $n - 2$ equations, when $f$ and $g$ are given, and
conducts, reciprocally, by (132), to (137).  {\it In general},
therefore, if we adopt the type (134), we need not retain
{\it also\/} either of these two latter types, (137), (138).  But
in the particular {\it case\/} where $n = 3$, that is, in the
case of quadrinomes, the identity~(132) reduces the two equations
(134) to one, after $f$ and $g$ have been selected; and with this
one we must then combine either of the two equations (137) or
(138), which in this case become identical with each other.

\bigbreak

[20.]
In particular, for this case of {\it quadrinomials\/} ($n = 3$),
we have with the notations (114), (128), the four following
values for (23), or for $b_1$ (compare again a note to [12.]):
$$\left. \eqalign{
[23]  &= 231 \mathbin{.} 122 + 233 \mathbin{.} 322
       = l_1 n_1 - n_2 m_3;\cr
[231] &= 211 \mathbin{.} 311 + 231 \mathbin{.} 313
       = - m_2 n_3 + l_1 m_1;\cr
[32]  &= 321 \mathbin{.} 133 + 322 \mathbin{.} 233
       = l_1 m_1 - m_3 n_2;\cr
[321] &= 311 \mathbin{.} 211 + 321 \mathbin{.} 212
       = - n_3 m_2 + l_1 n_1;\cr}
   \right\}
   \eqno (139)$$
but, whether we equate the first to the fourth, or the second to
the third of these expressions for $b_1$, in conformity with the
type (134), we obtain only one common equation of condition,
$n_2 m_3 = n_3 m_2$, equivalent indeed by cyclical permutation to
three, namely to the following,
$$0 = n_2 m_3 - n_3 m_2 = n_3 m_1 - n_1 m_3 = n_1 m_2 - n_2 m_1;
   \eqno (140)$$
which evidently agree with certain simple combinations of the six
equations on the two last lines of (86).  If however we compare
either the first value (139) with the third, or the second of
those values with the fourth, according to the type (137) or
(138), we find by each comparison the common condition
$l_1 n_1 = l_1 m_1$, and thus recover the equations of the first
line of (85).  In this way then we may obtain the required
number of six distinct equations, with two terms each, between
the nine symbols $(fgh)$, or $l_1 \, \ldots \, n_3$, for the case
of quadrinomes, by elimination of the three symbols~$(fg)$, or of
$b_1$,~$b_2$,~$b_3$.

\bigbreak

[21.]
For the case of {\it quines\/} ($n = 4$), the general theory
requires that the corresponding elimination of the
$6 = {1 \over 2} n (n - 1)$ symbols of this form~$(fg)$, or
$b_1 \, \ldots \, c_3$, should conduct to
$24 = n (n - 1) (n - 2)$ distinct equations of condition, with
$4 = 2 (n - 2)$ terms each, between the
${1 \over 2} n^2 (n - 1) = 24$ symbols of the form $(fgh)$, or
$l_1 \, \ldots \, u_3$, each equation thus obtained being
homogeneous, and of the second dimension; and that all these 24
conditions should be included in the formula (134), or in the
single type (123).  And in fact we thus obtain, by comparison of
the six expressions for $b_1$, of which one is
$$b_1 = (23) = [23]
   =  \Sigma (23h \mathbin{.} h22)
   =  l_1 n_1 - n_2 m_3 - p_1 r_2,
   \eqno (141)$$
the four following equations of condition, included in that type
or formula:
$$\left. \eqalign{
  0 &= [23] - [321] = [32] - [231];\cr
  0 &= [23] - [324] = [32] - [234];\cr}
   \right\}
   \eqno (142)$$
that is, with the notations $l_1 \, \ldots \, u_3$,
$$\left. \eqalign{
  n_2 m_3 - m_2 n_3 &= p_3 s_3 - p_1 r_2 = p_1 r_3 - p_2 t_2;\cr
  n_2 m_3 + u_2 u_3 &= p_3 s_3 + l_1 m_1 = l_1 n_1 - p_2 t_2;\cr}
   \right\}
   \eqno (143)$$
while we have in like manner six expressions for $c_1$, of which
one is
$$c_1 = [41] = \Sigma ( 41h \mathbin{.} h44 )
   =  - ( r_1 u_1 + s_1 u_2 + t_1 u_3 ),
   \eqno (144)$$
and of which the comparison conducts to the four other distinct
conditions:
$$\left. \eqalign{
  r_1 u_1 - n_1 r_2
   &=  l_3 t_3 + n_3 t_1 = - l_2 s_2 - t_1 u_3;\cr
  r_1 u_1 + m_1 r_3
   &=  l_3 t_3 - s_1 u_2 = - l_2 s_2 - m_2 s_1;\cr}
   \right\}
   \eqno (145)$$
where cyclical permutation of indices is still allowed.  The
equations obtained from the types (137), (138) would be found
(as the theory requires) to be merely consequences of these; for
example, by making $e = 1$, $f = 2$, $g = 3$, $h = 4$, those two
types give only the conditions,
$$l_1 (n_1 - m_1) = p_1 (r_2 + r_3) = p_2 t_2 + p_3 s_3,
   \eqno (146)$$
which are obviously included in (143).

\bigbreak

[22.]
With respect to those other homogeneous equations of the second
dimension, between the symbols $(fgh)$, which are obtained
immediately, or without any elimination of the symbols
$(f)$,~$(fg)$, from the general conditions of association, and
are included in the formula~(96), they may now be developed as
follows.

Making $k = f$ in (96), and then interchanging $f$ and $e$, for
the sake of comparison with (123), we obtain the type
$$\hbox{III.}\ldots\qquad
fgg \mathbin{.} gee - gff \mathbin{.} fee
   =  \Sigma^\backprime (gfh \mathbin{.} hee);
   \eqno (147)$$
which includes generally ${1 \over 2} n (n - 1) (n - 2)$ distinct
equations, of $n - 1$ terms each.  For quines, we have thus 12
equations of 3 terms sufficiently represented by the following:
$$\left. \multieqalign{
n_1 n_2 - m_2 m_1 &= p_3 r_3, &
n_1 u_2 + m_2 u_1 &= - l_3 u_3; \cr
r_2 m_2 + u_2 r_1 &= s_2 n_3, &
r_2 n_2 - u_2 r_3 &= + t_2 m_1; \cr}
   \right\}
   \eqno (148)$$
the value~4 being attributed to the index $h$ or $e$, in forming
the equations on the first line, but to $f$ or $g$ for the second
line.  For quadrinomes, the corresponding equations are only
three, namely
$$0 = n_1 n_2 - m_1 m_2 = n_2 n_3 - m_2 m_3 = n_3 n_1 - m_3 m_1;
   \eqno (149)$$
which however are sufficient, in conjunction with the three
lately marked as (140), to reproduce the six equations of the
two last lines of (86).  In general, by adding and subtracting
the two types (123), (147), we obtain the formula,
$$(fee \pm fgg) (gee \mp gff)
  =  \Sigma^\backprime (fgh) (hff \mp hee)
       - \Sigma^\backprime (feh \mathbin{.} ghe);
   \eqno (150)$$
where, as a verification, if we take the lower signs, and
interchange $f$ and $g$, so as to recover the first member with
the upper signs, the comparison of the two expressions for that
member conducts to an equation between the two second members,
which may also be obtained by the comparison of (125) and
(127).

\bigbreak

[23.]
Again, making $f = e$ in (96), and then changing $k$ to $f$, we
obtain the formula,
$$\hbox{IV.}\ldots\qquad
0 = (egf) (eff + egg) + \Sigma^\backprime (egh \mathbin{.} ehf);
   \eqno (151)$$
where $e \, f \, g \, h$ are again unequal indices.  This IVth
type includes generally $n (n - 1) (n - 2)$ distinct equations,
with $n- 1$ terms each.  For the case $n = 3$ there arise thus 6
equations of 2 terms, namely the six on the two last lines of
(85); so that the 18 equations (85), (86), for associative
quadrinomials, have thus been completely reproduced, as
consequences of the general theory.  For the case of quines, the
type (151) gives 24 equations of 3 terms, which may be
represented as follows:
$$\left. \eqalign{
0 &= l_2 (n_1 - m_1) + p_2 s_1 = l_3 (n_1 - m_1) + p_3 t_1;\cr
0 &= s_2 (r_2 + r_3) + t_1 t_2 = t_3 (r_2 + r_3) + s_3 s_1;\cr}
   \right\}
   \eqno (152)$$
$$\left. \eqalign{
0 &= t_1 (u_1 - m_1) + l_3 s_1 = p_2 (u_1 - m_1) + l_2 p_3;\cr
0 &= s_1 (u_1 + n_1) - l_2 t_1 = p_3 (u_1 + n_1) - l_3 p_2;\cr}
   \right\}
   \eqno (153)$$
either $h$ or $e$ being $= 4$ in (152), and either $f$ or $g$
having that value in (153), while $1$,~$2$,~$3$ may still be
cyclically permuted.

\bigbreak

[24.]
Finally, by supposing, as in (119), that $e \, f \, g \, k$ are
four unequal indices, and that $h$ under
$\Sigma^{\backprime\backprime}$ is unequal to each of them, we
obtain from (96) one other type, including {\it generally\/}
${1 \over 2} n (n - 1) (n - 2) (n - 3)$ equations, of $n - 1$
terms each, but furnishing no new conditions of association for
quadrinomials: namely,
$$\hbox{V.}\ldots\qquad
efk \mathbin{.} gee
       + fgk \mathbin{.} egg
       + egk \mathbin{.} fkk
   =  \Sigma^{\backprime\backprime} (fhk \mathbin{.} geh).
   \eqno (154)$$
For quines, the sum $\Sigma^{\backprime\backprime}$ vanishes, and
we obtain twelve equations of three terms each, which may (with
the help of permutations) be all represented by the four
following:
$$\left. \eqalign{
0 &= l_1 r_2 - u_2 s_3 - n_3 t_2
   = l_1 r_3 - m_2 s_3 + u_3 t_2,\cr
0 &= l_1 r_1 + n_2 s_3 + m_3 t_2
   = n_1 p_1 - m_2 p_2 - u_3 p_3;\cr}
   \right\}
   \eqno (155)$$
where the index~$4$ has been made to coincide with $e$ or with
$g$ in the first line, but with $f$ or $k$ in the second.

\bigbreak

[25.]
In general, the number of distinct associative equations,
included in the three last types (147), (151), (154), or III.,
IV., V., which have been all derived from the formula~(96), and
have been obtained without elimination of $(f)$ or $(fg)$,
amounts in the aggregate to
$${\textstyle {1 \over 2}} n (n - 1) (n - 2)
       + n (n - 1) (n - 2)
       + {\textstyle {1 \over 2}} n (n - 1) (n - 2) (n - 3)
   =  {\textstyle {1 \over 2}} n^2 (n - 1) (n - 2);
   \eqno (156)$$
as, by the analysis of [14.], it ought to do.  And when we add
this number to the $n (n - 2)$ of the type~I., or (113), and to
the $n (n - 1) (n - 2)$ of the type~II., or (123), obtained by
such elimination, we have in all this other number,
$${\textstyle {1 \over 2}} n^2 (n - 1) (n - 2)
       + n^2 (n - 2)
   =  {\textstyle {1 \over 2}} n^2 (n + 1) (n - 2),
   \eqno (157)$$
of distinct and homogeneous equations of the second dimension,
between the ${1 \over 2} n^2 (n - 1)$ symbols of the
form~$(fgh)$: as, by the formul{\ae} (109), (110) of [15.], we
ought to have.  As regards the {\it signification\/} of the five
foregoing principal types, which it has been thought convenient
to distinguish among themselves, and to arrange according to the
various ways in which they involve the symbols of the
form~$(eff)$, it will be found, on reviewing the analysis
employed, that they all express {\it ultimately\/} only
consequences of that {\it one\/} very simple and useful formula,
$${\rm V} \iota \, {\rm V} \iota' \iota''
   =  \iota'' \, {\rm S} \iota \iota'
       - \iota' \, {\rm S} \iota'' \iota,
   \qquad (67)$$
which, with a slightly different notation, has been elsewhere
shown to be so important in the Calculus of
Quaternions\footnote*{See the author's Lectures on this subject,
cited in former notes to this paper.}.
In fact, the equations (95)~$\ldots$~(98), on which those five
separate types have been founded, may all be deduced from (67)
and (69), whereof the latter is a consequence of the former.

\nobreak\bigskip

Observatory of Trinity College, Dublin,

\nobreak

\qquad August~14, 1854.

\vfill\eject

\centerline{{\largeit Section\/} {\largerm IV.}}

\nobreak\bigskip

[26.]
For quines, the equations of condition between the 24 symbols
$l_1 \, \ldots \, u_3$ amount (as has already been remarked) to
80 in all; namely to 8, 24, 12, 24, and 12 equations, included
respectively in the five types last mentioned, and sufficiently
developed above, by the formul{\ae} (117) (118) (143) (145) (148)
(152) (153) (155): which also enable us, with the help of
(141), (144), to determine the values of the four symbols
$a_1 \, \ldots \, a_4$, and of the six other symbols
$b_1 \, \ldots \, c_3$, when values of $l_1 \, \ldots \, u_3$
have been found, which satisfy the eighty conditions.  And then,
if we denote the quine itself by the following expression
(compare~[1.]),
$${\rm P}
   =  x_0 + \iota_1 x_1 + \iota_2 x_2
          + \iota_3 x_3 + \iota_4 x_4,
   \eqno (158)$$
which is a little more symmetric than the form (111), the
{\it laws of multiplication of any two such quines},
${\rm P}$,~${\rm P}'$, will be sufficiently expressed by the
formul{\ae}
$$\left. \eqalign{
   &  \iota_1^2 = a_1,\quad
      \iota_4^2 = a_4,\quad
      {\rm S} \iota_2 \iota_3 = b_1,\quad
      {\rm S} \iota_1 \iota_4 = c_1,\cr
   &  {\rm V} \iota_2 \iota_3
         =  \iota_1 l_1 + \iota_2 m_3
          + \iota_3 n_2 + \iota_4 p_1,\cr
   &  {\rm V} \iota_1 \iota_4
         =  \iota_1 r_1 + \iota_2 s_1
          + \iota_3 t_1 + \iota_4 u_1;\cr}
   \right\}
   \eqno (159)$$
if we remember that $1$,~$2$,~$3$ may still be cyclically
permuted, and that the law of conjugation~(32) gives
$${\rm K} \iota' \iota = \iota \iota',\quad
  {\rm S} \iota' \iota = {\rm S} \iota \iota',\quad
  {\rm V} \iota' \iota = - {\rm V} \iota \iota'.
   \eqno (160)$$
For in this manner, by (41), if $\varpi$ denote, as in (14),
the vector part of ${\rm P}$, so that
$$\varpi
   =  \iota_1 x_1 + \iota_2 x_2 + \iota_3 x_3 + \iota_4 x_4,
   \eqno (161)$$
we shall have
$$\eqalignno{
{\rm S} \varpi \varpi'
   &= a_1 x_1 x_1' + a_2 x_2 x_2'
          + a_3 x_3 x_3' + a_4 x_4 x_4' \cr
   &\mathrel{\phantom{=}} \mathord{}
       + b_1 (x_2 x_3' + x_3 x_2') + \hbox{\&c.}
       + c_1 (x_1 x_4' + x_4 x_1') + \hbox{\&c.},
   &(162)\cr
{\rm V} \varpi \varpi'
   &= (\iota_1 l_1 + \iota_2 m_3 + \iota_3 n_2 + \iota_4 p_1)
         (x_2 x_3' - x_3 x_2') + \hbox{\&c.} \cr
   &\mathrel{\phantom{=}} \mathord{}
       + (\iota_1 r_1 + \iota_2 s_1 + \iota_3 t_1 + \iota_4 u_1)
         (x_1 x_4' - x_4 x_1') + \hbox{\&c.},
   &(163)\cr}$$
each ``\&c.'' representing terms obtained by the permutations
already mentioned; and if the constants
$a \, b \, c \, l \, m \, n \, p \, r \, s \, t \, u$
have been chosen so as to fulfil the conditions above developed,
we may then conclude (compare~(51)) that the following
{\sc equations of association} hold good, for the multiplication
of any {\it three\/} such {\it vector-units\/}~$\iota$, or
{\it quadrinomial vectors\/}~$\varpi$, or {\it quinquinomial
expressions\/}~${\rm P}$, whether equal or unequal among
themselves:
$$\iota \mathbin{.} \iota' \iota''
   =  \iota \iota' \mathbin{.} \iota'';\quad
  \varpi \mathbin{.} \varpi' \varpi''
   =  \varpi \varpi' \mathbin{.} \varpi'';\quad
  {\rm P} \mathbin{.} {\rm P}' {\rm P}''
   =  {\rm P} {\rm P}' \mathbin{.} {\rm P}'';
   \eqno (164)$$
which it has been the main object of our recent investigations to
establish.

\bigbreak

[27.]
Without pretending to do more, on the present occasion, than
merely to {\it exemplify the possibility of satisfying}, for
{\it quines}, the foregoing equations of association, I may here
remark that if we restrict the question by assuming (with the
usual permutations),

\vfootnote*{This line is lettered thus,
because it contains the conditions common to the two systems (A)
and (B) of associative quines, which are deduced a little further
on.}

$$\hbox{(A, B)*}\ldots\qquad
n_1 = m_1,\quad p_1 = 0,\quad u_1 = 0,
   \eqno (165)$$
then numerous simplifications take place, and the 80 equations
between the 24 symbols $l \, m \, n \, p \, r \, s \, t \, u$ are
found to reduce themselves to 44 equations between the 15 symbols
$l \, m \, r \, s \, t$, obtained from the five types I. to V. of
recent paragraphs, which may be thus denoted and arranged:
$$\vbox{\halign{#\hfil \quad &\hfill
      $\displaystyle #$\hfill &\quad \hfil $\displaystyle #$\cr
from type I.&
   m_1^2 = l_2 l_3,\quad
   r_2^2 - r_1^2 = s_3 t_1 - s_2 t_3;
   &(166)\cr
from II. and III.&
   m_1 l_1 = m_2 m_3,\quad
   m_1 r_1 = m_2 s_1 = m_3 t_1,
   &(167)\cr
and&
   m_1 r_2 = l_2 s_2,\quad
   m_1 r_3 = l_3 t_3,\quad
   m_1 (r_1 + r_2 + r_3) = 0;
   &(168)\cr
from IV.&
   s_2 (r_2 + r_3) = - t_1 t_2,\quad
   t_3 (r_2 + r_3) = - s_3 s_1,
   &(169)\cr
and&
   m_1 s_1 = l_2 t_1,\quad
   m_1 t_1 = l_3 s_1;
   &(170)\cr
and from V.&
   m_1 s_2 = l_3 r_2,\quad
   m_1 t_3 = l_2 r_3,\quad
   l_1 (r_1 + r_2 + r_3) = 0.
   &(171)\cr}}$$
Now these conditions may all be satisfied in each of two
principal ways, conducting to {\it two distinct systems of
associative quines}, which may be called Systems (A) and (B), but
which are {\it not the only possible systems\/} of such quines,
because we {\it need\/} not have commenced by assuming the
equations (165), although that assumption has simplified the
problem.  For first we may suppose that the constants $l$ and $m$
are different from zero, but that the constants $r$ are connected
by the relation
$$\hbox{(A)}\ldots\qquad
r_1 + r_2 + r_3 = 0;
   \eqno (172)$$
or secondly, we may reject this relation between the
constants~$r$, and suppose instead that the six constants $l$ and
$m$ all vanish, so that
$$\hbox{(B)}\ldots\qquad
l_1 = l_2 = l_3 = m_1 = m_2 = m_3 = 0.
   \eqno (173)$$
With the first supposition, (172), we are to combine the nine
relations between the fifteen constants $l \, m \, r \, s \, t$,
which are sufficiently expressed by the formula (167), or by the
following:
$$\hbox{(A${}_1$)}\ldots\qquad
l_1 = m_1^{-1} m_2 m_3,\quad
s_1 = m_2^{-1} m_1 r_1,\quad
t_1 = m_3^{-1} m_1 r_1;
   \eqno (174)$$
and then all the other conditions of association will be found to
be satisfied, if we equate each of the ten symbols $a \, b \, c$
to zero, or if we establish this other formula,
$$\hbox{(A${}_2$)}\ldots\qquad
a_1 = 0,\quad
b_1 = 0,\quad
c_1 = 0,\quad
a_4 = 0:
   \eqno (175)$$
while there will still remain {\it five arbitrary constants\/} of
the system, for instance $r_1 \, r_2 \, m_1 \, m_2 \, m_3$.  With
the second supposition, (173), we are to combine four distinct
relations between the nine constants $r \, s \, t$, contained in
the formula~(169), or in the following:

\vfootnote*{It must be
observed that these equations (176), which are part of the
{\it basis\/} of the system~(B), are {\it true\/} in the
system~(A) also, as corollaries from (174) and (172) which last
equation does not hold in (B); and which allows us to reduce, for
(A) but not for (B), the relation (177) to the simpler form
$r_1 r_2 r_3 = s_1 s_2 s_3$.}

$$\hbox{(B${}_1$)*}\ldots\qquad
r_1 + r_2 = - s_1^{-1} t_1 t_3,\quad
s_1 s_2 s_3 = t_1 t_2 t_3;
   \eqno (176)$$
which give also, as a consequence, this other relation:
$$(r_1 + r_2) (r_2 + r_3) (r_3 + r_1) = - s_1 s_2 s_3;
   \eqno (177)$$
and then the other conditions of association will all be
satisfied, if we make, instead of (175),
$$\hbox{(B${}_2$)}\ldots\qquad
a_1 = b_1 = c_1 = 0;\quad
a_4 = (r_1 + r_2 + r_3)^2:
   \eqno (178)$$
{\it this system also involving five arbitrary constants}, for
example $s_1 \, s_2 \, t_1 \, t_2 \, t_3$.  The assertion
respecting {\it quines}, which was made near the end of [16.],
has therefore been fully justified.

\bigbreak

[28.]
Finally, as regards the system (A) of quines, it may be observed,
1st, that {\it in this system}, by (162) and (175), we have
generally,
$$\hbox{(A${}_3$)}\ldots\qquad
{\rm S} \varpi \varpi' = 0;
   \eqno (179)$$
or that ``the product of any two quadrinomial vectors
$\varpi$,~$\varpi'$, reduces itself to a pure vector;'' and
2nd, that, by (163), (165), ``this vector product,
$\varpi \varpi'$, is of trinomial form, involving no part with
$\iota_4$ for a factor.'' This product is therefore already seen
to be of the form
$$\varpi \varpi' = \iota_1 X_1 + \iota_2 X_2 + \iota_3 X_3;
   \eqno (180)$$
but I say, 3rd, that ``its three coefficients, or
{\it coordinates}, $X_1$,~$X_2$,~$X_3$ have constant ratios,'' or
that ``the product $\varpi \varpi'$ may be constructed by a right
line in space of which the direction though not the length is
fixed,'' and which may therefore be conceived to ``coincide in
position with one fixed axis~($\xi$) {\it of the system}.''  In
fact, by (163), (165), (174), we have
$$m_1 X_1 = m_2 X_2 = m_3 X_3 = X,
   \eqno (181)$$
and therefore
$$\hbox{(A${}_4$)}\ldots\qquad
\varpi \varpi' = X \xi,
   \eqno (182)$$
if we make for abridgement
$$\eqalignno{
   &
X =      m_2 m_3 (x_2 x_3' - x_3 x_2')
       + m_3 m_1 (x_3 x_1' - x_1 x_3')
       + m_1 m_2 (x_1 x_2' - x_2 x_1') \cr
\hbox{(A${}_5$)}\ldots\quad \vcenter to 0pt{\vss\hbox{$\left\{
   \vphantom{\vrule height6ex depth-6ex} \right.$}\vss}
   \,
   &\mathrel{\phantom{=}} \mathord{}
       + m_1 r_1 (x_1 x_4' - x_4 x_1')
       + m_2 r_2 (x_2 x_4' - x_4 x_2')
       + m_3 r_3 (x_3 x_4' - x_4 x_3'),
   &(183)\cr
   & \hbox{and } \xi
    = m_1^{-1} \iota_1 + m_2^{-1} \iota_2 + m_3^{-1} \iota_3.
   &(184)\cr}$$
In the 4th place, ``if any quadrinomial vector~$\varpi$ be
multiplied by or into the axis~$\xi$, the product vanishes;'' or
in symbols,
$$\hbox{(A${}_6$)}\ldots\qquad
\xi \varpi = 0,\quad \varpi \xi = 0;
   \eqno (185)$$
because by (172) the scalar coefficient~$X$ becomes $= 0$, if we
change either $x_1$, $x_2$, $x_3$, and $x_4$, or $x_1'$, $x_2'$,
$x_3'$, and $x_4'$, to $m_1^{-1}$, $m_2^{-1}$, $m_3^{-1}$, and
$0$, respectively.  This coefficient~$X$ vanishes also, when we
equate $x_1' \, x_2' \, x_3' \, x_4'$ to
$x_1 \, x_2 \, x_3 \, x_4$ respectively; and hence, or from
(179), we may infer, 5th, that ``in this system of quines~(A),
the square of every quadrinomial vector vanishes.'' And finally,
by an easy combination of the formul{\ae} (182), (185), or of
the 3rd and 4th of the foregoing properties of this system, we
see, 6th, that, in it, ``every product of three quadrinomial
vectors vanishes;'' or that
$$\hbox{(A${}_7$)}\ldots\qquad
\varpi \varpi' \mathbin{.} \varpi'' = 0,\quad
\varpi \mathbin{.} \varpi' \varpi'' = 0.
   \eqno (186)$$

\bigbreak

[29.]
The {\it associative property\/} (164) is therefore verified for
the system~(A), by showing that, {\it in it}, each of the two
ternary products of vectors, which ought to be equal, vanishes.
In the system~(B), it is easy to see that any such ternary
product must be itself a vector; because, in (B), no binary
product of vectors involves~$\iota_4$, nor does any such product
involve a scalar part, except what arises from $\iota_4^2$.  We
have, therefore, {\it here}, this new result,
$$\hbox{(B${}_3$)}\ldots\qquad
{\rm S} ( \varpi \varpi' \mathbin{.} \varpi'' )
   =  {\rm S} ( \varpi \mathbin{.} \varpi' \varpi'' )
   =  0.
   \eqno (187)$$
And when we proceed to develop these two ternary products, the
associative property of multiplication is again found to be
verified, under the form,
$$\hbox{(B${}_4$)}\ldots\qquad
\varpi \varpi' \mathbin{.} \varpi''
   =  \varpi \mathbin{.} \varpi' \varpi''
   =  a_4 (    \varpi x_4' x_4''
             - \varpi' x_4'' x_4
             + \varpi'' x_4 x_4' );
   \eqno (188)$$
where it is worth observing that, by the laws of the system in
question, the result may be put under this other and somewhat
simpler form:
$$\hbox{(B${}_5$)}\ldots\qquad
\varpi \varpi' \mathbin{.} \varpi''
   =  \varpi \mathbin{.} \varpi' \varpi''
   =     \varpi   \, {\rm S} \varpi'  \varpi''
       - \varpi'  \, {\rm S} \varpi'' \varpi
       + \varpi'' \, {\rm S} \varpi   \varpi'.
   \eqno (189)$$
Indeed, this last expression might have been {\it foreseen}, as a
consequence from the {\it general principles\/} of this whole
theory of {\it associative polynomes}\footnote*{It will
hereafter be proved {\it generally\/} that for {\it all\/}
associative polynomes which satisfy the law of conjugation
(though {\it not exclusively\/} for such {\it associative\/}
polynomes), the {\it tensor\/}, as defined in [6.], is also a
{\it modulus\/}; which theorem can be verified for the quines (A)
and (B), and for the quadrinomes and tetrads so lettered in
[13.], as well as for the trinomes~[11.].},
combined with the particular property (187) of the quines~(B).
For, by that property, each of the two ternary products is equal
to its own vector part; but by (101) we have, {\it generally},
in the present theory, as in the calculus of quaternions, the
following expression for the {\it vector part of the product of
any three vectors}, of any such associative polynomes as we are
considering:
$${\rm V} \mathbin{.} \rho \sigma \tau
   =    \rho   \, {\rm S} \sigma \tau
      - \sigma \, {\rm S} \tau   \rho
      + \tau   \, {\rm S} \rho   \sigma;
   \eqno (190)$$
which is a formula of continual application in Quaternions, and
in these extensions also is important.

\nobreak\bigskip

Observatory of Trinity College, Dublin,

\nobreak

\qquad August~16, 1854.

\vfill\eject

\centerline{{\largeit Section\/} {\largerm V.}}

\nobreak\bigskip

[30.]
In applying to associative {\it quines\/} the general theory of
the Third Section\footnote*{Comprising paragraphs [17.]\ to
[25.], and published in the Phil.\ Mag.\ for October, 1854.},
we may (as has been seen) omit each of the
signs~$\Sigma^\backprime$ as unnecessary, the index~$h$ receiving
only one value in the sum thereby indicated; and may suppress
each sum $\Sigma^{\backprime\backprime}$ as vanishing.  In this
manner the type~IV., or the formula (151), becomes,
$$\hbox{IV.}\ldots\qquad
(egf) (eff + egg) = geh \mathbin{.} ehf;
   \eqno (191)$$
while the equation~(127), already derived as a sub-type from II.,
gives, by interchanging $e$ and $f$,
$$(egf) (fee + fgg) = (geh) (hee + hgg).
   \eqno (192)$$
Multiplying the latter of these two equations by $eff + egg$, and
the former by $fee + fgg$, and subtracting, we eliminate the
symbol~$(egf)$, and find that
$$(geh) \{ (ehf) (fee + fgg) - (eff + egg) (hee + hgg) \} = 0;
   \eqno (193)$$
and a similar elimination of $(geh)$ gives the equation,
$$(egf) \{ (ehf) (fee + fgg) - (eff + egg) (hee + hgg) \} = 0.
   \eqno (194)$$
And because $(geh) = - (egh)$, by (34), we may make any separate
or combined interchanges, of $e$ with $g$, and of $f$ with $h$,
and so vary the expression within the $\{ \, \}$, without
introducing any new factor, distinct from $(egf)$ and $(egh)$,
outside them.  If, then, for any particular arrangement of the
four unequal indices, $e$,~$f$,~$g$,~$h$, as chosen from among
the four numbers $1$,~$2$,~$3$,~$4$, the {\it two\/} following
conditions are not {\it both\/} satisfied,
$$(egf) = 0,\quad
  (egh) = 0,
   \eqno (195)$$
we must have, for that arrangement of the indices, a system of
{\it four\/} other equations, whereof one is
$$\hbox{VI.}\ldots\qquad
(ehf) (fee + fgg) = (eff + egg) (hee + hgg),
   \eqno (196)$$
while the three others are formed from it, by the interchanges
just now mentioned.  And if we further suppose that the two sums,
$fee + fgg$ and $hee + hgg$, are for the same arrangement
different from zero, and write for abridgement, {\it as a
definition},
$$(ehf)_0 = (fee + fgg)^{-1} (eff + egg) (hee + hgg),
   \eqno (197)$$
the four equations furnished by the formula (196), which may be
regarded as a {\it sixth type for quines}, may be concisely
expressed as follows:
$$\left. \multieqalign{
  (ehf) &= (ehf)_0, &
  (ghf) &= (ghf)_0;\cr
  (efh) &= (efh)_0, &
  (gfh) &= (gfh)_0.\cr}
   \right\}
   \eqno (198)$$
With the notations $l_1 \, \ldots \, u_3$, for the symbols
$(efg)$, $(eff)$, we find thus that unless $l_1$ and $p_1$ both
vanish, we must have the four equations,
$$\left. \multieqalign{
t_2 (n_1 - m_1) &= (m_2 - n_2) (r_2 + r_3); &
s_3 (n_1 - m_1) &= (m_3 - n_3) (r_2 + r_3); \cr
p_3 (r_2 + r_3) &= (u_2 + n_2) (n_1 - m_1); &
p_2 (r_2 + r_3) &= (m_3 - u_3) (n_1 - m_1); \cr}
   \right\}
   \eqno (199)$$
and that unless $s_1 = 0$, $t_1 = 0$, then
$$\left. \multieqalign{
-l_2 (u_2 - m_2) &= (n_1 + u_1) (n_3 + u_3); &
 t_3 (u_2 - m_2) &= (r_1 + r_2) (n_3 + u_3); \cr
 l_3 (n_3 + u_3) &= (u_1 - m_1) (u_2 - m_2); &
 s_2 (n_3 + u_3) &= (r_1 + r_3) (u_2 - m_2). \cr}
   \right\}
   \eqno (200)$$

\bigbreak

[31.]
Supposing then that no one of the twelve symbols~$(efg)$
vanishes, and that each of the twelve sums $eff + egg$ is also
different from zero, the various arrangements of the four indices
$e \, f \, g \, h$ give us a system of twenty-four equations,
included in the new type~VI., or in any one of the four
formul{\ae} (198); which equations may, by (34), be arranged in
twelve pairs, as follows:
$$(ehf) = (ehf)_0 = - (hef)_0.
   \eqno (201)$$
It might seem that {\it twelve\/} equations between the twelve
symbols of the form~$(eff)$ should thus arise, by the comparison
of two expressions for each of the twelve symbols of the
form~$(efg)$; but if we write for abridgement
$$[g] = (fee + fgg) (fhh + fgg) \{ (ehf)_0 + (hef)_0 \},
   \eqno (202)$$
and observe that by the definition~(197) of the
symbol~$(ehf)_0$, we have then
$$[g] =  (eff + egg) (fhh + fgg) (hee + hgg)
       + (hff + hgg) (fee + fgg) (ehh + egg),
   \eqno (203)$$
we shall see that this quantity~$[g]$ is independent of the
arrangement of the three indices $e$,~$f$,~$h$; and therefore
that the twelve equations between the twelve symbols~$(eff)$,
obtained through (201), reduce themselves to the {\it four\/}
following relations,
$$[e] = 0,\quad
  [f] = 0,\quad
  [g] = 0,\quad
  [h] = 0;
   \eqno (204)$$
which are not even all distinct among themselves, since any
{\it three\/} of them include the fourth.  An easy combination of
the two first or of the two last of these four relations (204)
conducts to this other formula, which is equivalent to three
distinct equations:
$$\eqalignno{
(eff + ehh) (fee + fgg) (ghh + gee) (hgg + hff)
   \hskip -12em \cr
   &= (eff + egg) (fee + fhh) (ghh + gff) (hgg + hee);
   &(205)\cr}$$
and which may also be thus written,
$$(hef)_0 (egf)_0 = (ehf)_0 (gef)_0.
   \eqno (206)$$
With the notations $l_1 \, \ldots \, u_3$, the twenty-four
equations~(201) are sufficiently represented by the formul{\ae}
(199) and (200), if cyclical permutation of the indices be
employed; the four equations (204) take the forms,
$$\left. \eqalign{
(r_1 + r_2) (n_3 + u_3) (m_2 - n_2)
   &= (r_1 + r_3) (m_2 - u_2) (m_3 - n_3),\cr
(n_1 + u_1) (n_2 + u_2) (n_3 + u_3)
   &= (m_1 - u_1) (m_2 - u_2) (m_3 - u_3);\cr}
   \right\}
   \eqno (207)$$
whereof the equation on the second line may be obtained from the
product of the three represented by the first line: and the three
equations (205) or (206) are included in the following, which
is evidently a consequence of (207),
$$(r_1 + r_3) (n_1 + u_1) (n_2 + u_2) (n_3 - m_3)
   =  (r_1 + r_2) (u_1 - m_1) (n_2 - m_2) (u_3 - m_3).
   \eqno (208)$$

\bigbreak

[32.]
As regards the quotients and products of the symbols~$(efg)$,
which we shall continue to write occasionally without
parentheses, we have by type~VI., or by (197), (198),
$${ehf \over ghf} = {eff + egg \over gff + gee};
   \eqno (209)$$
$$ehf \mathbin{.} efh = (eff + egg) (ehh + egg);
   \eqno (210)$$
$$ehf \mathbin{.} gfh = (eff + egg) (ghh + gee);
   \eqno (211)$$
eliminating $(ehf)$ between the two last of which three
equations, we obtain a relation of the same form as the first.
Interchanging $g$ and $h$ in (210), and subtracting, we find
that
$$\hbox{I.}\ldots\qquad
ehf \mathbin{.} efh - egf \mathbin{.} efg
   =  (egg)^2 - (ehh)^2;
   \eqno (212)$$
but this is precisely what the type~I., or the formula (113),
becomes for quines, when we cyclically advance the four indices
in the order $f \, e \, g \, h$; the conditions (177), (118) of
that {\it first type\/} will therefore be satisfied, if we
satisfy those of the {\it sixth}.  Had we divided instead of
subtracting, we should have found
$${ehf \mathbin{.} efh \over egf \mathbin{.} efg}
   =  {eff + egg \over eff + ehh}.
   \eqno (213)$$
To interchange $f$,~$g$, and divide, would only lead by (210) to
another equation of this last form; but the same operations
performed on (211) conduct to the equation
$${ehf \over heg} = {ghh + gee \over fhh + fee};
   \eqno (214)$$
which, when we interchange $g$ and $h$, reproduces the
formula~(192); and shows thereby that the sub-type (127),
included under type~II., is satisfied by our new type~VI., which
indeed it had assisted to discover.  The same equation (192) may
also be derived from the formula (205), by dividing each member
of that formula by $(fee + fhh) (hff+ hgg)$, and attending to the
expressions given by type~VI., for $(egf)$ and $(geh)$
respectively.  To interchange $e$~$h$, in (211) and divide,
would only conduct to another equation of the same form as
(214).  Permuting cyclically the three indices $e$,~$f$,~$g$ in
(209), and multiplying together the two equations so obtained
therefrom, the product gives
$${fhg \mathbin{.} ghe \over ehg \mathbin{.} fhe}
   =  {gff + gee \over eff + egg};
   \eqno (215)$$
and if we multiply this equation by (209) itself, we find that
$$ehf \mathbin{.} fhg \mathbin{.} ghe
   =  ghf \mathbin{.} ehg \mathbin{.} fhe.
   \eqno (216)$$
In fact if we operate thus on the expression (197) for
$(efh)_0$, or for its equal $(ehf)$, or on the formula~(196), we
are led to this new equation,
$$ehf \mathbin{.} fhg \mathbin{.} ghe
   =  (hee + hgg) (hff + hee) (hgg + hff),
   \eqno (217)$$
of which the second member does not alter, when we interchange
any two of the three indices $e$,~$f$,~$g$.  Another
multiplication of three equations of the form~(209), with the
cycle~$e \, g \, h$, conducts to the equation $[f] = 0$ of
(204).  Interchanging $e$,~$h$ in (210), and substituting the
value so obtained for the product of the two extreme factors of
the second member of (217), we find this other expression,
$$ehf \mathbin{.} fhg \mathbin{.} ghe
   =  hef \mathbin{.} hfe \mathbin{.} (hee + hff);
   \eqno (218)$$
which is still seen to remain unaltered, by an interchange of $e$
and $f$.  Interchanging $f$,~$g$, and dividing, we obtain by
(216) an equation of the same form as (213); and if we divide
each member of (218) by $(hef)$, we are conducted to the formula
$$\hbox{IV.}\ldots\qquad
fhg \mathbin{.} hge = hfe \mathbin{.} (hee + hff),
   \eqno (219)$$
which is of the same form as the equation (191), or as the
type~IV., and may be changed thereto by cyclical permutation of
the four indices~$e \, f \, g \, h$.  The same relation (219)
may also be derived more directly from type~VI., by substitutions
of the values (198); for it will be found that the
definition~(197) gives this {\it identity},
$$(fhg)_0 (ghe)_0 = (hee + hff) (fhe)_0.
   \eqno (220)$$
The conditions of type~IV., like those of type~I., and of the
subtype (127) of II., are therefore all included in those of the
new type~VI.; which gives also in various ways this other formula
respecting products of four symbols of the form $(efg)$,
$$egh \mathbin{.} fhg \mathbin{.} gfe \mathbin{.} hef
   =  ehg \mathbin{.} fgh \mathbin{.} gef \mathbin{.} hfe:
   \eqno (221)$$
indeed it will be found that the members of this last equation,
taken in their order, are respectively equal by (196) to the
members of the equation~(205).

With the notations $l_1 \, \ldots \, u_3$, supposing that none of
the twelve constants $l \, p \, s \, t$ vanish, and that the
twelve combinations of the forms $n_1 - m_1$, $n_1 + u_1$,
$u_1 - m_1$, $r_1 + r_2$, are in like manner different from zero,
we find thus, or from the equations (199) (200), combined with
their consequences~(207), the following among other relations,
in which cylical permutation of the indices is still allowed:
$$\left. \multieqalign{
{l_1 \over t_2} &= - {n_3 + u_3 \over r_1 + r_3}, &
{t_2 \over s_3} &=   {m_2 - n_2 \over m_3 - n_3}, &
{s_3 \over l_1} &=   {r_1 + r_2 \over u_2 - m_2}, \cr
{p_1 \over p_2} &=   {m_2 - u_2 \over n_1 + u_1}, &
{l_1 \over p_1} &=   {r_2 + r_3 \over n_1 - m_1}, &
{s_1 \over t_1} &=   {n_3 + u_3 \over m_2 - u_2}; \cr}
   \right\}
   \eqno (222)$$
$$\left. \multieqalign{
  l_2 l_3 &= (n_1 + u_1) (m_1 - u_1), &
  s_2 t_3 &= (r_1 + r_2) (r_1 + r_3), \cr
  p_1 s_2 &= (m_2 - n_2) (m_2 - u_2), &
  p_1 t_3 &= (m_3 - n_3) (n_3 + u_3), \cr
- l_2 s_2 &= (n_1 + u_1) (r_1 + r_3), &
  l_3 t_3 &= (u_1 - m_1) (r_1 + r_2), \cr
  p_1 s_1 &= (m_1 - n_1) (n_3 + u_3), &
  p_1 t_1 &= (m_1 - n_1) (m_2 - u_2). \cr}
   \right\}
   \eqno (223)$$
The conditions (152), (153), of the {\it fourth type}, are
satisfied; and we have these other products, of which some have
occurred already, in (176), (177), in connexion with the
particular systems (A) and (B) of quines:
$$\left. \eqalign{
s_1 s_2 s_3
    = t_1 t_2 t_3
   &=  - (r_1 + r_2) (r_2 + r_3) (r_3 + r_1);\cr
s_1 p_2 l_3
    = t_1 l_2 p_3
   &=    (n_1 + u_1) (m_1 - u_1) (m_1 - n_1);\cr
s_2 s_3 l_2 p_3
   &= t_2 t_3 p_2 l_3;\cr}
   \right\}
   \eqno (224)$$
where the two members of the equation on the last line are easily
proved by (223) to be respectively equal to those of (208).

\bigbreak

[33.]
As yet we have only partially satisfied the conditions of
type~II., or of the formula (123), which may for quines be
written thus:
$$\hbox{II.}\ldots\qquad
fgh \mathbin{.} hff
   = fee \mathbin{.} gee - fgg
          \mathbin{.} gff + feh \mathbin{.} ghe.
   \eqno (225)$$
Substituting for the last product in this formula its value given
by (211), namely
$$feh \mathbin{.} ghe
   =  (fgg + fhh) (gee + gff),
   \eqno (226)$$
and writing for abridgement
$$v_f = fee + fgg + fhh,
   \eqno (227)$$
we are in this way led to establish the following {\it seventh
type\/}\footnote*{It will be shown that this {\it single type\/}
(the seventh) {\it includes all the others}, or is sufficient to
express {\it all\/} the {\it general conditions\/} of
association, between the 24 symbols of the forms $(efg)$ and
$(eff)$.  But the eliminations required for this deduction cannot
be conveniently described at this stage.}
for quines:
$$\hbox{VII.}\ldots\qquad
fgh \mathbin{.} hff
   =  v_f \mathbin{.} gee + fhh \mathbin{.} gff.
   \eqno (228)$$
Or since, by the sixth type, we have already
$$\hbox{VI.}\ldots\qquad
fgh \mathbin{.} (hee + hff) = (fee + fhh) (gee + gff),
   \eqno (229)$$
it is only necessary, for the purpose of satisfying the
conditions of type~II., or the equations (143), (145), to
suppose besides that
$$\hbox{III.}\ldots\qquad
fgh \mathbin{.} hee = fee \mathbin{.} gff - fgg \mathbin{.} gee;
   \eqno (230)$$
such being the expression which remains, when we subtract
(228) from (229).  But this last equation~(230) is precisely
what the type~III., or the formula~(147), becomes for quines; it
reproduces therefore the equations (148): and conversely, if we
retain that old type~III., it will not be {\it necessary},
although it may be convenient, to introduce the new type~VII., in
combination with type~VI.  And if in (230) we substitute for the
symbol~$(fgh)$ its value given by (229), and so combine types
III.\ and VI., we obtain the equation
$${fee + fhh \over hee + hff}
   =  {gff \mathbin{.} (fee + fgg) - fgg \mathbin{.} (gee + gff)
         \over hee \mathbin{.} (gee + gff)};
   \eqno (231)$$
that is, by (209),
$${fge \over hge} + {fgg \over hee}
   = {gff \over hee} \mathbin{.} {fhe \over ghe};
   \eqno (232)$$
or finally,
$$\hbox{V.}\ldots\qquad
hee \mathbin{.} fge = fgg \mathbin{.} ghe - gff \mathbin{.} fhe.
   \eqno (233)$$
But this is exactly what the type~V., or the formula~(154),
becomes for quines, when we suppress the
sum~$\Sigma^{\backprime\backprime}$, change $k$ to $h$, and
advance cyclically the three indices $e \, f \, h$; it includes
therefore the equations (155), which were the only remaining
conditions of association to be fulfilled.  If then we satisfy
the {\it two types}, III.\ and VI., we shall satisfy {\it all the
conditions of association for quines\/}: since we shall thereby
have satisfied {\it also\/} the {\it four other earlier types},
namely those numbered as I.~II.~IV.~V.  It only remains then
to consider {\it what new restrictions\/} on the
constants~$(eff)$ are introduced by the comparison of the values
which type~III.  gives for the other constants~$(efg)$, as
expressed in terms of them, with the values furnished by
type~VI.; or to discuss the consequences of the following general
formula, obtained by eliminating the symbol~$(fgh)$ between
(229) (230), and not essentially distinct from the recent
equations~(231):
$$\hbox{VIII.}\ldots\qquad
hee \mathbin{.} (fee + fhh) (gee + gff)
   =  (hee + hff) (fee \mathbin{.} gff - gee \mathbin{.} fgg);
   \eqno (234)$$
which contains all the old and new relations, subsisting between
the twelve constants of the form~$(eff)$, and may be regarded as
an {\it eighth type\/} for quines.

\bigbreak

[34.]
Denoting the first minus the second member of (234) by the
symbol $[efgh]$, we easily see that
$$\eqalignno{
[efgh]
   &= gee \mathbin{.} (v_f \mathbin{.} hee + fgg \mathbin{.} hff)
       + gff \mathbin{.} (fhh \mathbin{.} hee - hff
       \mathbin{.} fee) \cr
   &= hee \mathbin{.} (v_f \mathbin{.} gee + fhh \mathbin{.} gff)
       + hff \mathbin{.} (fgg \mathbin{.} gee - gff
       \mathbin{.} fee);
   &(235)\cr}$$
and therefore that we have, {\it identically},
$$[efgh] = [efhg];
   \eqno (236)$$
this last or eighth type (234) contains therefore, at most, only
a system of twelve equations.  Interchanging $f$ and $g$, and
attending to the notation (202), we see, by (203), (234), that
of the three equations
$$[efgh] = 0,\quad
  [egfh] = 0,\quad
  [e] = 0,
   \eqno (237)$$
any two include the third; if then we only seek what {\it new\/}
conditions, additional to those marked (204), are to be
satisfied by the symbols~$(eff)$, or rather by the eight
following ratios of those symbols,
$$eff : egg : ehh;\quad
  fee : fgg : fhh;\quad
  gee : gff : ghh;\quad
  hee : hff : hgg,
   \eqno (238)$$
we need only retain at most four new equations, suitably selected
from among those furnished by type~VIII., such as the four
following, which differ among themselves by the initial letters
within the brackets, and so belong to different groups,
$$[efgh] = 0,\quad
  [fghe] = 0,\quad
  [ghef] = 0,\quad
  [hefg] = 0;
   \eqno (239)$$
and then to combine these with any three of the four former
relations~(204), for example with the three first, namely
$$[e] = 0,\quad
  [f] = 0,\quad
  [g] = 0;
   \eqno (240)$$
from which the fourth equation $[h] = 0$ would follow, by means
of the {\it identity},
$$\eqalignno{
(ehf)_0 (fge)_0 (gfh)_0 (heg)_0
   &= (ehg)_0 (fgh)_0 (gfe)_0 (hef)_0 \cr
   &= (eff + egg) (fee + fhh) (gee + ghh) (hff + hgg).
   &(241)\cr}$$
It might seem however that the seven equations (239) and (240),
thus remaining, should suffice to determine seven of the eight
ratios~(238): whereas I have found that it is allowed to
{\it assume two pairs of ratios\/} arbitrarily, out of the four
pairs (238), and then to deduce the two other pairs from them.
For I find that it is sufficient to retain, instead of the
{\it twelve\/} equations included under type~VIII., or the
{\it seven\/} equations (239), (240), a system of only
{\it four\/} equations of the type just mentioned; namely
{\it two pairs}, selected from any two of the four groups, which
have (for each group, and also for each pair) a common initial
letter within the brackets; for instance, these two pairs of
equations:
$$[efgh] = 0,\quad
  [egfh] = 0,\quad
  [fegh] = 0,\quad
  [fgeh] = 0;
   \eqno (242)$$
which leave as many as {\it eight arbitrary constants\/} (for
example these eight, $eff$, $egg$, $ehh$, $fhh$, $ghh$, $hee$,
$hff$, $hgg$, from which all the rest can be determined) {\it in
the resulting system of associative quines}.  An outline of the
investigation by which this important reduction is effected, may
be presented in the following way.

\bigbreak

[35.]
The two first equations (242) connect the three last pairs of
ratios (238), in such a manner that when any two of those three
pairs are assumed, or known, the third can be determined.  Hence,
with the interpretation (197) of the symbol $(ghf)_0$, we easily
find that those two equations (242) give,
$$\hbox{({\it e})}\ldots\qquad
\left\{
\eqalign{
gee \mathbin{.} hgg - ghh \mathbin{.} hee
   &= (ghf)_0 \mathbin{.} fee; \cr
v_g \mathbin{.} hee + gff \mathbin{.} hgg
   &= (ghf)_0 \mathbin{.} fgg; \cr
- ( v_h \mathbin{.} gee + hff \mathbin{.} ghh )
   &= (ghf)_0 \mathbin{.} fhh; \cr}
   \right\}
   \eqno (243)$$
because we find that $fee$, $fgg$, $ghh$ are proportional to the
left-hand members of these last equations (243); and that the
sum of the two first of those left-hand members is identically
equal to the product $(gee + gff) (hee + hgg)$.  For the same
reason, the two first of these three equations (243) express
really only one relation, namely that which is contained in the
second equation (242), although they do so under different
forms, both of which it is useful to know; and it is convenient
to have ready also this other combination, obtained by adding the
three equations (243) together,
$$\hbox{({\it e})}\ldots\qquad
v_h \mathbin{.} gff - v_g \mathbin{.} hff
   = v_f \mathbin{.} (ghf)_0;
   \eqno (244)$$
which, like those equations (243) themselves, we shall consider
as belonging to the group~({\it e}), because they are all derived
from two of the three equations of that group, included under
type~VIII., which in the recent notation $[efgh]$ have $e$ for
their initial letter; and because the third equation of that
group, included under the same type, namely
$$\hbox{({\it e})}\ldots\qquad
[ehfg] = 0,
   \eqno (245)$$
may be derived from them, by the elimination of the
symbol~$(ghf)_0$ between the first and third of the equations
(243).  In like manner the two last equations (242) include a
third of the same type~VIII., and belonging to the same
group~({\it f}), namely
$$\hbox{({\it f})}\ldots\qquad
[fheg] = 0;
   \eqno (246)$$
because they conduct to the following system of expressions,
which may be formed from (243) (244) by cyclical permutation
of $efg$:
$$\hbox{({\it f})}\ldots\qquad
\left\{
\eqalign{
eff \mathbin{.} hee - ehh \mathbin{.} hff
   &= gff \mathbin{.} (ehg)_0; \cr
v_e \mathbin{.} hff + egg \mathbin{.} hee
   &= gee \mathbin{.} (ehg)_0; \cr
- ( v_h \mathbin{.} eff + hgg \mathbin{.} ehh )
   &= ghh \mathbin{.} (ehg)_0; \cr
v_h \mathbin{.} egg - v_e \mathbin{.} hgg
   &= v_g \mathbin{.} (ehg)_0. \cr}
   \right\}
   \eqno (247)$$
Multiplying then the equations (243) and (244) by $(ehg)_0$,
and observing that the identity (220) gives
$$(ehg)_0 \mathbin{.} (ghf)_0
   =  (hee + hff) (ehf)_0,
   \eqno (248)$$
we find, on substitution of the first for the second members of
(247), that the results are divisible by $hee + hff$; and that
thus the elimination of the third pair of ratios (238), between
(243), (244), (247), or between the four equations (242),
conducts to expressions of the recent forms, namely,
$$\hbox{({\it g})}\ldots\qquad
\left\{
\eqalign{
hee \mathbin{.} egg - ehh \mathbin{.} hgg
   &= fgg \mathbin{.} (ehf)_0; \cr
v_e \mathbin{.} hgg + eff \mathbin{.} hee
   &= fee \mathbin{.} (ehf)_0; \cr
- ( v_h \mathbin{.} egg + hff \mathbin{.} ehh )
   &= fhh \mathbin{.} (ehf)_0; \cr
v_h \mathbin{.} eff - v_e \mathbin{.} hff
   &= v_f \mathbin{.} (ehf)_0. \cr}
   \right\}
   \eqno (249)$$
A similar analysis may be applied to effect the elimination of
the fourth pair of ratios (238), with results entirely analogous.
On the whole then it is found, that the four equations (242)
express such connexions between the four pairs of ratios (238),
as to satisfy not only the two remaining equations, (245) and
(246), of their {\it own\/} groups, ({\it e}) and ({\it f}), but
also the six other equations of the two {\it other\/} groups,
({\it g}) and ({\it h}), included under type~VIII.; namely
$$\left. \eqalign{
\hbox{({\it g})}\ldots\qquad
[gefh] = 0,\quad
[gfeh] = 0,\quad
[ghef] = 0;\cr
\hbox{({\it h})}\ldots\qquad
[hefg] = 0,\quad
[hfeg] = 0,\quad
[hgef] = 0;\cr}
   \right\}
   \eqno (250)$$
for the first line is satisfied by the ratios (249), and the
second by the analogous ratios, which are found in a similar way.
Thus all the {\it twelve\/} equations of type~VIII.\ are
satisfied, if we satisfy only {\it four\/} suitably selected
equations of that type; for example, the equations (242): which
was what we proposed to demonstrate.

\bigbreak

[36.]
The {\it eighty\/} equations of association, assigned in the
Third Section, between the {\it twenty-four\/} constants
$l_1 \, \ldots \, u_3$, or $(efg)$,~$(eff)$, have therefore, by
the recent analysis, been ultimately {\it reduced to sixteen\/};
namely the {\it four\/} equations which thus remain from the last
type~VIII.; and the {\it twelve\/} others which were contained in
the type~III., established in that earlier Section: and which (as
was lately remarked) leave still no fewer than {\sc eight
constants arbitrary} in this theory of {\sc associative quines}.
We may indeed vary in many ways, consistently with the same
general theory, and by the assistance of the other recent
types~VI.\ and VII., the system of the sixteen equations of
condition which are to be satisfied, and the choice of the eight
constants which are to be regarded as still remaining arbitrary
and undetermined: and it may not be useless, nor uninteresting,
to make some remarks hereafter, upon the subject of such
selections.  But in the mean time it appears to be important to
observe, that if some of the recent results, especially the
formul{\ae} (210), (228), be combined with some of those
previously obtained, and more particularly with the equations
(112), (121), of Section~III., the following very simple
expressions are found, for the ten remaining {\it constants of
multplication}, the discussion of which had been reserved:
$$(f) = v_f^2;\quad
  (fg) = v_f \mathbin{.} v_g;
   \eqno (251)$$
or, with the notations $abc$, and with the usual cyclical
permutation of the indices $1$,~$2$,~$3$,
$$a_1 = v_1^2,\quad
  a_4 = v_4^2,\quad
  b_1 = v_2 v_3,\quad
  c_1 = v_1 v_4.
   \eqno (252)$$
If then we write for abridgement,
$$\left. \eqalign{
v &= v_1 x_1 + v_2 x_2 + v_3 x_3 + v_4 x_4,\cr
v' &= v_1 x_1' + v_2 x_2' + v_3 x_3' + v_4 x_4',\cr}
   \right\}
   \eqno (253)$$
the square of any quadrinomial vector~$\varpi$, and the scalar of
the product of two such vectors, will take these remarkably
simple forms:
$$\varpi^2 = v^2;\quad
  {\rm S} \varpi \varpi' = v \mathbin{.} v';
   \eqno (254)$$
this latter scalar thus decomposing itself into a
product\footnote*{A similar decomposition into linear factors
takes place for the {\it quadrinomes\/} (A) of par.~[13.], but at
the expense of one of the six arbitrary constants
$l_1 \, l_2 \, l_3 \, m_1 \, m_2 \, m_3$,
when we establish between those symbols the relation,
$$l_1 m_1^2 + l_2 m_2^2 + l_3 m_3^2
   = l_1 l_2 l_3 + 2 m_1 m_2 m_3.$$

In general, I find that it is possible to satisfy all the
conditions of association for {\it polynomes}, and at the same
time to secure a decomposition of ${\rm S} \varpi \varpi'$ into
linear factors, while yet preserving so many as $3n - 4$
constants of multiplication arbitrary.  (For quadrinomes,
$3n - 4 = 9 - 4 = 5$; for quines $3n - 4 = 12 - 4 = 8$.)}
of two linear functions of the constituents, namely those here
denoted by $v$ and $v'$.  And because it is easy to prove, from
what has been already shown, compare (244), that in the present
theory the constants~$v_e$ are connected by relations of the form
$$- v_e \mathbin{.} efe = v_e \mathbin{.} fee
   =  v_f \mathbin{.} eff
       + v_g \mathbin{.} efg
       + v_h \mathbin{.} efh,
   \eqno (255)$$
we find, by multiplying this equation by $v_g$, and attending to
(251), the following theorems for those {\it general associative
quines\/} which have been in this section considered:
$$\left. \eqalign{
   &0 = {\rm S} \iota_g \, {\rm V} \iota_e \iota_f
      = {\rm S} \iota_g \iota_e \iota_f;\cr
   &0 = {\rm S} \varpi \varpi' \varpi'';\quad
    0 = ({\rm V} \varpi \varpi')^2;\cr}
   \right\}
   \eqno (256)$$
results which may be compared with some of those obtained in
Section~IV., for the two particular quine-systems, (A) and (B).

\nobreak\bigskip

Observatory of Trinity College, Dublin,

\nobreak

\qquad February~14, 1855.

\nobreak\bigskip

\centerline{[To be continued.]}

\bye

.
