ON A NEW AND GENERAL METHOD OF INVERTING A LINEAR AND QUATERNION

FUNCTION OF A QUATERNION

By

William Rowan Hamilton

(Proceedings of the Royal Irish Academy, 8 (1864), pp. 182–183)

Edited by David R. Wilkins

On a New and General Method of Inverting a Linear and Quaternion Function of a Quaternion.

Sir William Rowan Hamilton.

Read June 9th, 1862.

[Proceedings of the Royal Irish Academy, vol. viii (1864), pp. 182–183.]

Let a, b, c, d, e represent any five quaternions, and let the following notations be admitted, at least as temporary ones:—

ab − ba = [ab], S[ab]c = (abc);
(abc) + [cb] Sa + [ac] Sb + [ba] Sc = [abc];
Sa[bcd] = (abcd);

then it is easily seen that

[ab] = −[ba]; (abc) = −(bac) = (bca) = &c.;
[abc] = −[bac] = [bca] + &c.;
(abcd) = −(bacd) = (bcad) = &c.;
0 = [aa] = (aac) = [aac] = (aacd), &c.

We have then these two Lemmas respecting Quaternions, which answer to two of the most continually occurring transformations of vector expressions:—

I… 0 = a(bcde) + b(cdea) + c(deab) + d(eabc) + e(abcd),
or I′… e(abcd) = a(ebcd) + b(aecd) + c(abed) + d(abce);
and II… e(abcd) = [bcd] Sae − [cda] Sbe + [dab] Sce − [abc] Sde;

as may be proved in various ways.

Assuming therefore any four quaternions a, b, c, d, which are not connected by the relation,

(abcd) = 0,

we can deduce from them four others, a′, b′, c′, d′, by the expressions,

a′(abcd) = f[bcd], b′[abcd] = −f[cda], &c.,

where f is used as the characteristic of a linear or distributive quaternion function of a quaternion, of which the form is supposed to be given; and thus the general form of such a function comes to be represented by the expression,

V … r = fq = a′ Saq + b′ Sbq + c′ Scq + d′ Sdq;

involving sixteen scalar constants, namely those contained in a′ b′ c′ d′.

The Problem is to invert this function f; and the solution of that problem is easily found, with the help of the new Lemmas I. and II., to be the following:—

\begin{aligned}VI … q(abcd)(a′b′c′d′) & = (abcd)(a′b′c′d′)f^{−1}r \\ & = [bcd](rb′c′d′) + [cda](rc′d′a′) + [dab](rd′a′b′) + [abc](ra′b′c′); \end{aligned}

of which solution the correctness can be verified, à posteriori, with the help of the same Lemmas.

Although the foregoing problem of Inversion had been virtually resolved by Sir W. R. H. many years ago, through a reduction of it to the corresponding problem respecting vectors, yet he hopes that, as regards the Calculus of Quaternions, the new solution will be considered to be an important step. He is, however, in possession of a general method for treating questions of this class, on which he may perhaps offer some remarks at the next meeting of the Academy.