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

OPERATION ON A QUATERNION

By

William Rowan Hamilton

(Philosophical Magazine, 24 (1862), pp. 127–128.)

Edited by David R. Wilkins On the Existence of a Symbolic and Biquadratic Equation which is satisfied by the Symbol of Linear or Distributive Operation on a Quaternion. By Sir William Rowan Hamilton, LL.D. &c.*

[The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, 4th series, vol. xxiv (1862), pp. 127–128.]

1. As early as the year 1846, I was led to perceive the existence of a certain symbolic and cubic equation, of the form

0 = m − m′ϕ + m′′ϕ^{2} − ϕ^{3},  (1)

in which ϕ is used as a symbol of linear and vector operation on a vector, so that ϕρ denotes a vector depending on ρ, such that

ϕ(ρ + ρ′) = ϕρ + ϕρ′,  (2)

if ρ and ρ′ be any two vectors; while m, m′ and m′′ are three scalar constants, depending on the particular form of the linear and vector function ϕρ, or on the (scalar or vector) constants which enter into the composition of that function. And I saw, of course, that the problem of inversion of such a function was at once given by the formula

mϕ^{−1} = m′ − m′′ϕ + ϕ^{2},  (3)

—the required assignment of the inverse function, ϕ^{−1}ρ, being thus reduced to the performance of a limited number of direct operations.

2. Quite recently I have discovered that the far more general linear (or distributive) and quaternion function of a quaternion can be inverted, by an analogous process, or that there always exists, for any such function fq, satisfying the condition

f(q + q′) = fq + fq′,  (4)

where q and q′ are any two quaternions, a symbolic and biquadratic equation, of the form

0 = n − n′f + n′′f^{2} − n′′′f^{3} + f^{4},  (5)

in which n, n′, n′′, and n′′′ are four scalar constants, depending on the particular composition of the linear function fq; and that therefore we may write generally this Formula of Quaternion Inversion,

nf^{−1} = n′ − n′′f + n′′f^{2} − f^{3}.  (6)

* Communicated by the Author.

3. As it was in the Number of the Philosophical Magazine for July 1844 that the first printed publication of the Quaternions occurred (though a paper on them had been previously read to the Royal Irish Academy in November 1843), I have thought that the Editors of the Magazine might perhaps allow me thus to put on record what seems to myself an important addition to the theory, and that they may even allow me to add, in a Postscript to this communication, so much as may convey a distinct conception of the Method which I have pursued.

Observatory of Trinity College, Dublin,

June 27, 1862.