the calculus of functions. 
417 
to one of the form of 
F (<r, -tyx, x, . . -p n xj = 0. 
Assume -px — A* (p A x, the equation by this substitution 
becomes 
F | x, A' (p A x, A 1 <p* A a, x, . . . A (p n A v x j = 0 
find by Problem VI. such a value Ax that it shall not change 
by the substitution of ax, fix, yx, See. vx, put for Ax the quan- 
tity y, and the equation becomes 
F jA ‘y, A\ py, A l (p*y, ... A'(p n yj =0 
which is an equation of the required form. 
Problem XIX. 
Required the solution of the equation 
F | x, -px, x, . . -ty n x | = 0 
Assume x — (p f(px, then -p n x = (p f n (px, and the equa- 
tion becomes 
F | x, tip f (p x, (p f* (p x, . . (p t n (px | =0 
for x substitute (p‘ x, then 
F {<p‘ x, 9 fv, $'/* x, . . tp'/ s *•} —0 (a) 
which is an equation of the first order relative to CD* and may 
be solved by the methods in the beginning of this Paper, or, 
by means of the method given by Mr. Herschel, to which 
we have already alluded. 
With respect to the function f it is arbitrary, there are 
however, some observations respecting it, which require 
notice ; as without an attention to them we might fall into 
error. In the first place, it is evident, that we must not 
mdcccxv. 3 H 
