the calculus of functions. 403 
Supposing a, / 3 , 7, &c. v, were to become «, of , , . ct n , the 
equation ( 1 ) would be changed into 
px + Ax x (pccx B x x (pet* x + &c. + Nr x <pa * - 1 x = 0 (2) 
after further if * x, is such a function of x, that u n -x = x, if we 
are acquainted with one .particular solution of ( 2 ) , we may 
easily determine the general one thus : 
Assume p x —fx x % £x,etx,ec'x> . .a w —‘ a:} 
here we must observe, that since % is symmetrical relative to 
all the quantities contained within the brackets, it is immate- 
rial in what order they are placed, and from the condition 
that ot n x = x, it follows that if we substitute a k x for x, ( k be- 
ing successively equal to 1, 2, 3, 4, and »— 1, we shall always 
have these values x, a x, 0? x ... a n ~'x only differently 
arranged, from these considerations the equation (2) will 
become 
0 — ( fx -j- Ax xfccx -j-Bx %fux-\~ &c. -j-N x xfcc n —' 1 x) 
X {x,«r,ax, . . . « w -\r } ; 
this equation may be satisfied by making the factor which 
multiplies % equal to nothing, and this is always the case 
when/ is a particular solution, hence 
(p X —fx x X { x, « X, cxl X, . . a? 1 —' 1 X | . 
Problem VI. 
To find a function of x, such that if instead of x we succes- 
sively substitute «x, fix, yx , &c. vx, the results shall all be equal 
to the original function ; or in other words, to determine <px 
from the equations 
p X = p aX = p fi X = &c. = p v X 
3 F 2 
