MONS. [660 
661] 
155 
by w, and in 
divided by w 1} 
rotations are 
, w), such that 
? 2 > (7 2 , D 2 shall 
661. 
ON THE DOUBLE »-FUNCTIONS. 
% respectively. 
parameters of 
lines J and K 
eable with it; 
are directrices 
es: and these 
s corresponding 
3sition of the 
5 how we can 
[From the Proceedings of the London Mathematical Society, vol. ix. (1878), pp. 29, 30.] 
Prof. Cayley gave an account of researches * on which he is engaged upon the 
double ^-functions. In regard to these, he establishes in a strictly analogous manner 
the theory of the single ^-functions, the process for the single functions being in fact 
as follows:—Considering u, x as connected by the differential relation 
8u = 
8x 
fa — x.b — x.c — x.d — x 
then, if A, B, G, D, il denote functions of u, viz. for shortness, the single letters are 
used, instead of writing them as functional symbols, A (u), B (u), &c., then, by way of 
definition of these functions (called, the first four of them ^-functions, and the last 
an «-function), we assume 
A, B, C, D= Cl fa — x, Clfb — x, Clfc — x, Clfd — x 
respectively, together with one other equation, as presently mentioned. Without in any 
wise defining the meaning of il, we then obtain a set of equations of the form 
A8B — B8A = il 2 Vc — x . d — x 8u, 
(mere constant coefficients are omitted), or, what is the same thing, 
A8B- B8A = CD 8u, 
which are differential equations defining the nature of the ratio-functions A : B : G : D. 
If, proceeding to second differential coefficients, we attempt to form the expressions for 
A8*A — (êd.) 2 , &c., these involve multiples of fl3 2 il — (Sil) 2 ; in order to obtain a con- 
[* See paper, number 665.] 
20—2