228 
Mr. Babbage’s essay towards 
This equation may be solved by nearly the same method as 
that employed in the last Problem. 
If the function occurs in different shapes or of various orders, 
this method is inapplicable, as in the following Problem. 
Problem XXX. 
Given the equation 
F | 2,I ^ 2 (x>y^) ) (x,y), x, yj = o 
The difficulties in this case appear to be much encreased from 
this circumstance, that the second function of -ty 2 * r (a?, a?) rela- 
tive to x is quite different from the second function of 2 (x, x) 
relative to the same quantity. The method of solution which I 
shall explain is equally applicable to all of this species, and con- 
sists in reducing them to a class which has been already solved. 
It may be observed, that whether we take the second func- 
tion of ^ 2 > 1 (x, x) relative to x, or whether we take the simul- 
taneous function of \p 2 > 1 ( x,y ) considered as a simple function, 
and in the result put x for y, the two expressions will be the 
same ; the first gives 
^ 2, I (^2, I Q V} ^ ^ 2, I ^ 
and the second is 
v / 2 > 1 1 (x,y), 1 (x,y)) 
‘which when y becomes equal to x is identical with the former; 
but 
^ 2 ,i(^ 2 , '(xj),$ 2 >'(x,y))=^ 2 > 1 [ty($(x,y),y),^(^(x,y),y') ]■ 
2, I, I, I, I 
I 2 I 2 I 
— ^ (>,>’) 
the lower line of indices denoting the quantities relative to 
which the operations are performed. In a similar manner it 
may be shown, that i, 2 
X, 2)3 I, 2, I, 2, 2 
^ — ^ (x>y) 
