32] 
213 
32. 
ON SOME ANALYTICAL FORMULAS, AND THEIR APPLICATION 
TO THE THEORY OF SPHERICAL COORDINATES. 
[From the Cambridge and Dublin Mathematical Journal, vol. i. (1846), pp. 22—33.] 
Section 1. 
The formulae in question are only very particular cases of some relating to the 
theory of the transformation of functions of the second order, which will be given in 
a following paper. But the case of three variables, here as elsewhere, admits of 
a symmetrical notation so much simpler than in any other case (on the principle that 
with three quantities a, b, c, functions of b, c; of c, a; and of a, b, may symmetrically 
be denoted by A, B, C, which is not possible with a greater number of variables) 
that it will be convenient to employ here a notation entirely different from that made 
use of in the general case, and by means of which the results will be exhibited in 
a more compact form. There is no difficulty in verifying by actual multiplication, any 
of the equations here obtained. 
It will be expedient to employ the abbreviation of making a single letter stand 
for a system of quantities. Thus for instance, if 8 = 9, this merely means that 
(8) is to stand for <i> (6, cf), 'Jr), &8 for led, &</>, kyjr, &c. 
Suppose then 
® V > £» (1); 
*>' = r, V, 
Q = A, B, C, F, G, H (2), 
W (w, ft)', Q) = A& + B vv ' + Off + F « + v'O + & (&' + + H (£/ + ?V) • • • (3), 
the function W satisfies a remarkable equation, as follows: