ON THE CONNEXION OF QUATERNIONS

WITH CONTINUED FRACTIONS AND QUADRATIC EQUATIONS

By

William Rowan Hamilton

(Proceedings of the Royal Irish Academy, 5 (1853), pp. 219–221, 299–301)

Edited by David R. Wilkins On the connexion of quaternions with continued fractions and quadratic equations.

By Sir William R. Hamilton.

[Proceedings of the Royal Irish Academy, vol. v (1853), pp. 219–221, 299–301.]

[Communicated December 8th, 1851]

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.

1. If we write

u_{x} = \frac{b_{1}}{a_{1}+} \frac{b_{2}}{a_{2}+} … \frac{b_{x}}{a_{x}} = \frac{N_{x}}{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,

\begin{aligned}N_{x+1} & = N_{x}a_{x+1} + N_{x−1}b_{x+1}, \\ D_{x+1} & = D_{x}a_{x+1} + D_{x−1}b_{x+1}. \end{aligned}

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.

2. Supposing a and b to be two constant quaternions, these equations in differences are satisfied by supposing

\begin{aligned}N_{x} & = Cq_{1}^{x} + C′q_{2}^{x}, \\ D_{x} & = Eq_{1}^{x} + E′q_{2}^{x}, \\ C + C′ & = 0, Cq_{1} + C′q_{2} = b, \\ E + E′ & = 1, Eq_{1} + E′q_{2} = a; \end{aligned}

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.

3. By pursuing this track it is found, with little or no difficulty, that

2u_{x}^{−1} + q_{1}^{−1} + q_{2}^{−1} = \frac{q_{1}^{x} + q_{2}^{x}}{q_{1}^{x} − q_{2}^{x}} \frac{q_{1} − q_{2}}{b};

where

u_{x} = (\frac{b}{a+})^{x} 0, \frac{q_{1} − q_{2}}{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.

4. Let the continued fraction in quaternions be

u_{x} = (\frac{j}{i+})^{x} 0;

then the quadratic equation becomes

q^{2} = qi + j :

and two unequal roots of it are the following:

\begin{aligned}q_{1} & = \frac{1}{2}(1 + i + j − k), \\ q_{2} & = \frac{1}{2}(−1 + i − j − k). \end{aligned}

Substitution and reduction give hence these two expressions:

(\frac{j}{i+})^{2n} 0 = \frac{sin \frac{2nπ}{3}}{i sin \frac{2nπ}{3} − k sin \frac{(2n − 1)π}{3}} ;

\frac{2 ÷ (\frac{j}{i+})^{2n−1} 0}{i − k} = 1 − \frac{sin \frac{(2n − 1)π}{3}}{sin \frac{2(n − 1)π}{3} + j sin \frac{2nπ}{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.

5. It may have appeared strange that Sir William R. Hamilton should have spoken of two unequal quaternions, as being among the roots, or two of the roots, of a 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 such a quadratic equation (if of the above-written form) has generally six roots: whereof, however, two only are real quaternions, while the other four may, by a very natural and analogical extension of received language, be called imaginary quaternions. But the theory of such imaginary, or partially imaginary quaternions, in short, the theory of what Sir William R. Hamilton has ventured to name “Biquaternions,” in a paper already published, appears to him to deserve to be the subject of a separate communication to the Academy.

[Communicated May 24th, 1852]

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 imaginary roots of the equation

q^{2} = qi + j;

and showed that these were as well adapted as the two real roots assigned in his former communication, to furnish the real quaternion value of the continued fraction,

(\frac{j}{i+})^{x} 0.

He also showed that when the continued fraction

u_{x} = (\frac{b}{a+})^{x} 0

converges to a limit,

u = u_{∞} = (\frac{b}{a+})^{∞} 0,

the two quaternions a and b being supposed to be given and real, then this limit is equal to that one of the two real roots of the quadratic equation in quaternions,

u^{2} + ua = b,

which has the lesser tensor; and gave geometrical illustrations of these results.

The two real quaternion roots of the quadratic equation, q^{2} = qi + j, being, as in the abstract of December, 1851,

q_{1} = \frac{1}{2}(1 + i + j − k), q_{2} = \frac{1}{2}(−1 + i − j − k),

it is now shown that the four imaginary roots are

q_{3} = \frac{i}{2}(1 + √ − 3) − k, q_{4} = \frac{i}{2}(1 − √ − 3) − k,

q_{5} = \frac{1}{2}(i + k) + \frac{1}{2}(1 − j)√ − 3, q_{6} = \frac{1}{2}(i + k) − \frac{1}{2}(1 − j)√ − 3;

but that in whatever manner we group them, two by two, even by taking one real and one imaginary root, the formula

u_{x} = (1 − v_{x})^{−1}(v_{x}q_{1} − q_{2}), or \frac{u_{x} + q_{2}}{u_{x} + q_{1}} = v_{x},

where v_{x} = q_{2}^{x}v_{0}q_{1}^{−x}, v_{0} = \frac{u_{0} + q_{2}}{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

(\frac{j}{i+})^{x} 0, which agree with those formerly found, and may be collected into the following

period of six terms,

u_{0} = 0, u_{1} = k, u_{2} = \frac{1}{2}(k − i), u_{3} = k − i, u_{4} = −i, u_{5} = ∞, u_{6} = 0, u_{7} = k, &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 circulating, converges to a limit, the general value of

u_{x} = (\frac{10j}{5i+})^{x} c

was assigned for any arbitrary quaternion c, by the help of the quadratic equation

q^{2} = 5qi + 10j;

and it was shown that with only one exception, namely, the case when c = (2k − 4i), the limit in question was (for every other value of c),

u = (\frac{10j}{5i+})^{∞} c = 2k − i.