% This paper has been transcribed in Plain TeX by
% David R. Wilkins
% School of Mathematics, Trinity College, Dublin 2, Ireland
% (dwilkins@maths.tcd.ie)
%
% Trinity College, 2000.

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\centerline{\Largebf ON THE CONNEXION OF QUATERNIONS}

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\centerline{\Largebf WITH CONTINUED FRACTIONS AND}

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\centerline{\Largebf QUADRATIC EQUATIONS}

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\centerline{\Largebf By}

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\centerline{\Largebf William Rowan Hamilton}

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\centerline{\largerm (Proceedings of the Royal Irish Academy,
   5 (1853), pp.\ 219--221, 299--301)}

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

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\centerline{\largerm 2000}

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\centerline{\largeit On the connexion of quaternions with continued
fractions and}
\centerline{\largeit quadratic equations.}

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\centerline{{\largeit By\/}
{\largerm Sir} {\largesc William R. Hamilton.}}

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\centerline{[{\it Proceedings of the Royal Irish Academy},
   vol.~v (1853), pp.\ 219--221, 299--301.]}


\bigbreak

\centerline{[Communicated December 8th, 1851]}

\nobreak\bigskip

The Secretary, in the absence of Sir W.~R. Hamilton, read the
following remarks on the connexion of Quaternions with continued
fractions and quadratic equations.

\bigbreak

1.
If we write
$$u_x
   =  {b_1 \over a_1 + } {b_2 \over a_2 + } \, \ldots \, {b_x \over a_x}
   =  {N_x \over D_x},$$
it is known (see Sir J.~F.~W. Herschel's Treatise on Finite
Differences) that the numerator and denominator of the resultant
fraction satisfy two differential equations in differences, which
are of one common form, namely,
$$\eqalign{
  N_{x+1} &= N_x a_{x+1} + N_{x-1} b_{x+1},\cr
  D_{x+1} &= D_x a_{x+1} + D_{x-1} b_{x+1}.\cr}$$
And by the nature of the reasoning employed, it will be found
that these equations in differences, thus written, hold good for
quaternions, as well as for ordinary fractions.

\bigbreak

2.
Supposing $a$ and $b$ to be two constant quaternions, these
equations in differences are satisfied by supposing
$$\eqalign{
  N_x &= C q_1^x + C' q_2^x,\cr
  D_x &= E q_1^x + E' q_2^x,\cr
  C + C' &= 0,\quad C q_1 + C' q_2 = b,\cr
  E + E' &= 1,\quad E q_1 + E' q_2 = a;\cr}$$
$C$, $C'$, $E$, $E'$ being four constant quaternions, determined
by the four last conditions, after finding two other and unequal
quaternions, $q_1$ and $q_2$, which are among the roots of the
quadratic equation,
$$q^2 = qa + b.$$

\bigbreak

3.
By pursuing this track it is found, with little or no difficulty,
that
$$2 u_x^{-1} + q_1^{-1} + q_2^{-1}
   = {q_1^x + q_2^x \over q_1^x - q_2^x} {q_1 - q_2 \over b};$$
where
$$u_x = \left( {b \over a +} \right)^x 0,\quad
  {q_1 - q_2 \over b} = q_1^{-1} - q_2^{-1};$$
$q_1$, $q_2$, being still supposed to be two unequal roots of the
lately written quadratic equation in quaternions,
$$q^2 = qa + b.$$

\bigbreak

4.
Let the continued fraction in quaternions be
$$u_x = \left( {j \over i +} \right)^x 0;$$
then the quadratic equation becomes
$$q^2 = qi + j:$$
and two unequal roots of it are the following:
$$\eqalign{
  q_1 &= {\textstyle {1 \over 2}} (1 + i + j - k),\cr
  q_2 &= {\textstyle {1 \over 2}} (-1 + i - j - k).\cr}$$
Substitution and reduction give hence these two expressions:
$$\left( {j \over i + } \right)^{2n} 0
   =  {\displaystyle \sin {2n \pi \over 3}
         \over \displaystyle
            i \sin {2n \pi \over 3}
          - k \sin {(2n - 1) \pi \over 3}};$$
$${\displaystyle 2 \div \left( {j \over i + } \right)^{2n-1} 0
      \over i - k}
   =  1 -
      {\displaystyle \sin {(2n - 1) \pi \over 3}
         \over \displaystyle
              \sin {2 (n - 1) \pi \over 3}
          + j \sin {2n \pi \over 3}};$$
which may easily be verified by assigning particular values to
$n$.  No importance is attached by the writer to these particular
results: they are merely offered as examples.

\bigbreak

5.
It may have appeared strange that Sir William R. Hamilton should
have spoken of {\it two\/} unequal quaternions, as being
{\it among\/} the roots, or {\it two of the roots}, of a
{\it quadratic equation\/} in quaternions.  Yet it was one of the
earliest results of that calculus, respecting which he made (in
November, 1843) his earliest communication to the Academy, that
{\it such\/} a quadratic equation (if of the above-written form)
has generally {\it six roots\/}: whereof, however, {\it two
only\/} are {\it real quaternions}, while the other four may, by
a very natural and analogical extension of received language, be
called {\it imaginary quaternions}.  But the theory of such
{\it imaginary}, or {\it partially\/} imaginary quaternions, in
short, the theory of what Sir William R. Hamilton has ventured to
name ``{\it Biquaternions},'' in a paper already published, appears
to him to deserve to be the subject of a separate communication
to the Academy.

\bigbreak

\centerline{[Communicated May 24th, 1852]}

\nobreak\bigskip

6.
Sir William R. Hamilton read a supplementary Paper in
illustration of his communication of the 8th of December last, on
the connexion of Quaternions with continued fractions and
quadratic equations.

In this paper he assigned the four Biquaternions which are the
{\it imaginary\/} roots of the equation
$$q^2 = qi + j;$$
and showed that {\it these\/} were as well adapted as the two
{\it real\/} roots assigned in his former communication, to
furnish the real quaternion value of the continued fraction,
$$\left( {j \over i + } \right)^x 0.$$
He also showed that when the continued fraction
$$u_x = \left( {b \over a + } \right)^x 0$$
converges to a {\it limit},
$$u = u_\infty = \left( {b \over a + } \right)^\infty 0,$$
the two quaternions $a$ and $b$ being supposed to be given and
real, then this limit is equal to {\it that one of the two real
roots of the quadratic equation in quaternions},
$$u^2 + ua = b,$$
{\it which has the lesser tensor\/}; and gave geometrical
illustrations of these results.

The {\it two real\/} quaternion roots of the quadratic equation,
$q^2 = q i + j$, being, as in the abstract of December, 1851,
$$q_1 = {\textstyle {1 \over 2}} ( 1 + i + j - k),\quad
  q_2 = {\textstyle {1 \over 2}} (-1 + i - j - k),$$
it is now shown that the {\it four imaginary\/} roots are
$$q_3 = {i \over 2} (1 + \surd - 3) - k,\quad
  q_4 = {i \over 2} (1 - \surd - 3) - k,$$
$$q_5 = {\textstyle {1 \over 2}} (i + k)
          + {\textstyle {1 \over 2}} (1 - j) \surd - 3,\quad
  q_6 = {\textstyle {1 \over 2}} (i + k)
          - {\textstyle {1 \over 2}} (1 - j) \surd - 3;$$
but that in whatever manner we group them, {\it two by two},
{\it even\/} by taking {\it one\/} real and {\it one\/} imaginary
root, the formula
$$u_x = (1 - v_x)^{-1} (v_x q_1 - q_2),
   \quad\hbox{or}\quad
  {u_x + q_2 \over u_x + q_1} = v_x,$$
where $v_x = q_2^x v_0 q_1^{-x}$,
$\displaystyle v_0 = {u_0 + q_2 \over u_0 + q_1}$,
and which is at once simpler and more general than the equations
previously communicated, conducts still to values of the
continued fraction~$u_x$, or
$\displaystyle \left( {j \over i + } \right)^x 0$,
which agree with those formerly found, and may be collected into
the following period of six terms,
$$u_0 = 0,\enspace
  u_1 = k,\enspace
  u_2 = {\textstyle {1 \over 2}} (k - i),\enspace
  u_3 = k - i,\enspace
  u_4 = -i,\enspace
  u_5 = \infty,\enspace
  u_6 = 0,\enspace
  u_7 = k,\enspace\hbox{\&c.}$$
In general it may be remembered that $q_1,$~$q_2$, are roots of
the quadratic equation $q^2 = qa + b$.

As an example of a continued fraction in quaternions which,
instead of thus {\it circulating, converges\/} to a limit, the
general value of
$$u_x = \left( {10j \over 5i + } \right)^x c$$
was assigned for any arbitrary quaternion~$c$, by the help of the
quadratic equation
$$q^2 = 5 qi + 10j;$$
and it was shown that with only one exception, namely, the case
when $c = (2k - 4i)$, the limit in question was (for {\it every
other\/} value of $c$),
$$u = \left( {10j \over 5i + } \right)^\infty c = 2k - i.$$

\bye

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