24^ Mr. Babbage's essay towards 
be changed by Prob. VIII. Part I. into any arbitrary function 
which does not vary when y becomes ay ; its general solution 
is therefore 
<py = C—y + «y) <p (J. «y) 
& 
and consequently the general solution of the equation of this 
Problem is 
^ ( x >y ) = f<p {y, ~*y)+7 {«y-y) <p (y, «F) 
I 
Ex. i. Take the equation 
4- ( x, 7 ) = 'Ato> 
in this case a.yz=a—~y, and the general solution is 
+ (x>y) ~£*<p{y, a—y) + £ (a — sy) <p (y, a—y) 
i 
Ex. s. Let the given equation be 
(x, J - ) 
-V / 
dx 
here ay = — and the general solution is 
+ ( x ’y)—?<t<iy’~)+e 
Problem XXXIX. 
Given the equation 
H*>y) = d± ^ } 
supposing u p y 
By substituting successively ay, a*y, &c. aP-'y for y, we have 
the following equations 
*{*.y) = - 
See. &c. 
^(x,*P-'y) = d -l^p± 
