24$ Mr. Babbage’s essay towards 
this substituted in the former, gives 
<py = — cpofy 
IJ 13 
whose general solution being found by the method in the first 
part gives 
$y— (~y + *y) <p (y, *y, ay, * 3 y) 
13 *3 
and consequently 
cpy = (— ccy -f o?y) q> (ay, ofy, d 9 y,y) 
14 3 
these values being respectively substituted, we have for the 
general solution of the Problem in this example, 
■y (x,y)=£* <p (y, a 3 y)+7(—y +*y— ay+**y)<^y7ay^y7^y) + 
a 
I S X I 
+(— y+*'y)${y, ay,ay,u 3 y)sinx+(--ay+a 3 y)(p(ay,ay,a 3 y,y)cos x 
3 3 
If the original equation had been 
4 ( x, y ) = 
the partial differential equation to be solved would have been 
H*.y 
This form is rather remarkable, the equation can always 
be integrated when np is a whole number ; let us suppose n 
to be a fraction and p a whole number, some multiple of the 
denominator of n. 
Ex. Let n — \,p — 2, then np = 1, and a*y=y, and the 
equation to be solved is 
I 
, , X d 2 -^ (s, «,y) 
= “IT" 
whose solution is ^ (sc, y) — f <py, or by assigning a proper 
I 
form to <py it becomes 
E 
>K x,y) (y,~«y) 
Not only may the index of differentiation become fractional, 
