calculus of functions and other branches of analysis. 199 
teristic capable of assuming all varieties of form; then amongst 
these varieties we may have ^ x = o , and the expression 
becomes . 
0 
In order to ascertain its real meaning, let us suppose 
■tyx = VQX + v*<px + &c. 
s 
which if we suppose v — o gives -tyx=o 
then = v<p -j + v*<p -j + &c. 
and the expression becomes 
1 1 
( fh v— \ vi 1 [ fh ■*• --^1 1 7/ 2 1 for* 
1 I \ 1 l \ 
1 <?) r , f?'- I JL 1 * 
1 r x — V— v — f~j v 
v x 
A'* 
VQX + V^X + ScC. 
<px+v<px+8cc. 
which becomes when v — o or -ipx — o 
4,*— J,.! (px—<p~ 
X ___ X 
ifx ' (px 
where cpx is quite arbitrary. 
If we suppose ypx to become a symmetrical function of x 
and j,we have and our expression becomes ~ which 
actually does vanish in all cases except at the same time 
= 0. 
fx—fx .fax 
As a second example take the expression * ^ ----- which 
becomes — if the two following equations hold tru efx fax— 1 
and fx *~fx .fotx — o, a. being any given function, such that 
ux szzx. In order to ascertain its real value in that case, let 
us suppose 
fx becomes fx -\-vq>x 
1 x 
and fx becomes fx + V( P X 
D d 
MDCCCXVII. 
