1^8 Mr. Babbage's essay towards 
may in fact be solved by the methods of the first part ; fory 
may be considered as a constant quantity, and if in the solution 
of<p 2 < t~x (Probs. 9, 10, and 14, Part I.) we put arbitrary 
functions ofy instead of the constant quantities which occur, 
we shall have a solution of the given equation, thus a parti- 
cular solution of (p 2 oc = x is f [x ) == — — instead of c put % (y) 
then a solution of the given equation is 
+ (*>y) = 7 
b — x 
-xxy 
2, 1 
4' (a?,y) = %K4/a?,y),y) 
6 _ 
1 — 
/; — x 
l -^xy 
b — x 
— — X? 
x-bxxy _ 
’ / : Cl/ 
i-bxy 
We might also instead of 6 put any other arbitrary function 
ofy, and the result will be the same. The equations 
^'\x,y) =a(a?,y) and t]/* \x,y ) = a (a?,y) 
may be treated in a similar manner, in the first y must be con- 
sidered as constant, and x must be so treated in the latter. 
In general, when functions are taken only relative to one of 
the variables, the rules delivered in my former Paper are 
sufficient for their solution, such is the equation 
F [x,y)^^x,y')^ 2, \x,y),..^ u,l (^ } y)}—o 
It might however occur, that though the order of the function 
does not vary relative to the other variable, yet that that vari- 
able may occur in different forms in each function. An 
example will render this more evident a, (3 , &c. being known 
functions, let 
F { #,y, ^ (#,y)>+ 2 ' \%> ay), l ($> Qy), • • ^ l (p, *y) }—° 
here though the functions do not vary in order relative toy. 
