2 8 8 
Mr. Babbage’s essay towards 
4 
p-r 
'CL 
X X : 
(dxx) n 
&C. 
&c. 
d n 4^“ z * 
(dJ- 2 x) n 
A n \ P~ l 
a oj /ol l x 
(doiP 1 xp 
but 'tya.tx = -tyop and combining all these equations we have 
d. n d. n d n -\,x 
(da.P~~ Z x) H (dx) H 
\ptz f OC = ypcV 
(da.P 1 x) n 
which is a differential equation of the pn tb order and putting 
= z we have 
A 
% 
1 x.daP 2 x. . . . dax. dx) n 
this being integrated gives the value of z or rpx. 
Problem XXXV. 
Given the equation 
P j •£> vf/x, fyotx, ~ |=o also of x = x 
Find the value of -^ax from this equation and substitute in 
it ax, cfx, &c. oiP— 1 x for x, then we have 
■i/ax— F 
>Kz= F {zx, tyux, 
&c. &c. 
JpaPx=F( CL P ~ l X, ^cf~ l x, ^ 1 
* 1 dJ-'x J 
In each of these equations for ffi** 5 -= ■ &c. * 
l X 
d-^ctx 
\ dxx r, 
d^/a^X 
1 da z X \ 
dx 
l dx! 
dx 
Y^l 9 
dx 
dx 
