252 G. PLARR ON THE ELIMINATION OF a, 8, y, ETC. 
themselves also in other questions, and then to make a study of the properties 
of these functions and of the relations which may exist between them. Having 
succeeded in establishing some of the relations which exist a priori between 
these functions, namely, dependently of the conditions of integrability, we 
were enabled to carry out the elimination by taking into account concurrently 
(1st) the @ priori relations, and (2d) the conditions of integrability, these 
latter ones giving, so to say, the special data of the problem. 2 
Our definitions of those quaternion functions of a, 8, y, comprise expressions 
depending on two different operators, the one < being the common one, and 
the other > being in certain respects conjugate. 
Also we have made use of the property of both these operators, according 
to which the result, which they work out, remains unaffected in form, by any 
change in the direction of the axes of the Carthesian co-ordinates, and of the 
corresponding unit-vectors by which the operators are defined. 
In a final paragraph we have reintroduced the representation of a, B, y, by 
gig, 99d, Q~'kq, and given the expressions of the above quaternion 
functions by the help of gq. 
$1. Preliminaries. 
We establish by definition 
; 2 =tin tia tag 
2 ae Paes oe, i 
bY = ant + Gd + ae . 
The first of these operators is the one which has been designated by y. 
For altering this sign into <| we invoke the authority of page 610 of HAmiLTon’s 
“ Lectures,” where the symbol <4 was introduced with its present meaning. 
The symbol > for the second operator is then, so to say, a consequence of the 
first. Ifa denomination was wanted the first might be called the left handed, 
and the second the right handed operator. 
Both operators give the same result when applied to a scalar, as for ex- 
ample w— 
bw =a, 
because the place of 2,7, & is of no influence in respect to the scalars 
dw dw dw . 
dz’? dy? dz’ v 
Applied to a vector w, the operators give results satisfying to 
(2) bo = Kao, 
: dw 
because the derivates 7 , etc., are vectors, like w, and therefore 
te a 

