ON THE EXISTENCE OF A SYMBOLIC AND BIQUADRATIC EQUATION, WHICH IS SATISFIED BY THE SYMBOL OF LINEAR

OPERATION IN QUATERNIONS

By

William Rowan Hamilton

(Proceedings of the Royal Irish Academy, 8 (1864), pp. 190–191)

Edited by David R. Wilkins

On the Existence of a Symbolic and Biquadratic Equation, which is satisfied by the Symbol of Linear Operation in Quaternions.

Sir William Rowan Hamilton.

Read June 23rd, 1862.

[Proceedings of the Royal Irish Academy, vol. viii (1864), pp. 190–191.]

1. In a recent communication (of June 9, 1862), I showed how the general Linear and Quaternion Function of a Quaternion could be expressed, under a standard quadrinomial form; and how that function, when so expressed, could be inverted.

2. I have since perceived, that whatever form be adopted, to represent the Linear Symbol of Quaternion Operation thus referred to, that symbol always satisfies a certain Biquadratic Equation, with Scalar Coefficients, of which the values depend upon the particular constants of the Function above referred to.

3. This result, with the properties of the Auxiliary Linear and Quaternion Functions which which it is connected, appears to me to constitute the most remarkable accession to the Theory of Quaternions proper, as distinguished from their separation into scalar and vector parts, and from their application to Geometry and Physics, which has been made since I had first the honour of addressing the Royal Irish Academy on the subject, in the year 1843.

4. The following is an outline of one of the proofs of the existence of the biquadratic equation, above referred to. Let

fq = r  (1)

be a given linear equation in quaternions; r being a given quaternion, q a sought one, and f the symbol of a linear or distributive operation: so that

f(q + q′) = fq + fq′,  (2)

whatever two quaternions may be denoted by q and q′.

5. I have found that the formula of solution of this equation (1), or the formula of inversion of the function, f, may be thus stated:

nq = nf^{−1}r = F r;  (3)

where n is a scalar constant depending for its value, and F is an auxiliary and linear symbol of operation depending for its form (or rather for the constants which it involves), on the particular form of f; or on the special values of the constants, which enter into the composition of the particular function, fq.

6. We have thus, independently of the particular quaternions, q and r, the equations,

F fq = nq, F fr = nr;  (4)

or, briefly and symbolically,

F f = fF = n.  (5)

7. Changing next f to f_{c} = f + c, that is to say, proposing next to resolve the new linear equation,

f_{c}q = fq + cq = r,  (6)

where c is an arbitrary scalar, I find that the new formula of solution, or of inversion, may be thus written:

f_{c}F_{c} = n_{c};  (7)

where

F_{c} = F + cG + c^{2}H + c^{3},  (8)

and

n_{c} = n + n′c + n′′c^{2} + n′′′c^{3} + c^{4};  (9)

G and H being the symbols (or characteristics) of two new linear operations, and n′, n′′, n′′′ denoting three new scalar constants.

8. Expanding then the symbolical product f_{c}F_{c}, and comparing powers of c, we arrive at three new symbolical equations, namely, the following:

fG + F = n′; fH + G = n′′; f + H = n′′′;  (10)

by elimination of the symbols, F, G, H, between which and the equation (5), the symbolical biquadratic,

0 = n − n′f + n′′f^{2} − n′′′f^{3} + f^{4},  (A)

is obtained.