Sir W. R. Hamilton on a Symbolic and Biquadratic Equation. 127 



the current ; in the first place, because these arguments appear 

 to us to be beyond the reach of experimental control ; and in 

 the second place, because we think it can be of but slight use to 

 science to speculate on the properties of a current hypotheti- 

 cally detached from the material substratum without which we 

 are unable really to conceive of its existence. 



We have the honour to remain, &c, 



J. G. S. van Breda, 

 Teylerian Laboratory, Haarlem, W. M. LoGEMAN. 



May 1862. 



XVIII. On the Existence of a Symbolic and Biquadratic Equation 

 which is satisfied by the Symbol of Linear or Distributive Opera* 

 tion on a Quaternion. By Sir Willli am Rowan Hamilton, 

 LL.D. $c* 



1. AS early as the year 1846, I was led to perceive the exist- 

 -^*_ ence of a certain symbolic and cubic equation, of the 

 form 



= m— m'</> + m"<£ 2 — <£ 3 , (1) 



in which cf> is used as a symbol of linear and vector operation on 

 a vector, so that (f>p denotes a vector depending on p, such that 



<p(p+p')=<pp+4>p', (2) 



if p and p' be any two vectors ; while m, m 1 , and m" are three 

 scalar constants, depending on the particular form of the linear 

 and vector function cf>p, 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 



m(/>- 1 = m , -m"</> + (/) 2 , (3) 



— the required assignment of the inverse function, </> -1 /o, 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+i')=k+h', W 



where q and q' are any two quaternions, a symbolic and biqua- 

 dratic equation, of the form 



= n -. n if+ n '!f^ n !nf3 + f4 f .... (5) 



in which n, n', n", and n ,n are four scalar constants, depending on 



* Communicated by the Author. 



