222 
Mr. Babbage's essay towards 
Problem XXV. 
Given the equation 
F {$ 2 >*(x,y), xj = F [a, j 
making the substitution <p /((pa^Cpy) for $ (x, y) and in the 
result putting <p x for x, and <p*y for y, we shall find 
F {(p / 2 > 2 (x,y), (p x ]■ = F | a, (p ? / 2 > 1 (x,y ) j 
which will become identical if we select such a value for J (x,y) 
that (p / 2 > 2 (x,y) = a, and a lso/ 2 > 1 (#, y) = x such a value 
i$f(x,y)=z(pa -J - y — x\ hence the general solution of the 
Problem is 
4j(x,y)=z<p l ((pa + Qy __ (p 
the more general equation 
fa 2 >*(x,y) — x}F^x,y, 4> (x,y) &c.} = J/~(x,y) . 
F ^x,y,^(x,y),&c.} 
may be solved nearly in the same manner, its solution will be 
+ (#,y) = Q (<Py — Qx) 
Problem XXVI. 
Given the equation 
F {#>y> ^(x,y)A l ’ z (x,y), M ,1 {^>y)> 6cc.| = o 
Assume -p (x,y) = <p f (<p x, Cpy) then we have ip 2 * 1 (x,y) = 
(p l f 2 > 1 (<px,<py), and ^ 1}2 (x,y) = (p* J 1>z {x,y) and generally 
■p n > m (x,y) == <p f n > m (<p a?, <py ) 
Substituting these values the equation becomes 
T\ai,y>T fQw,<ty)i ?f'’ 2 0«, <py),<? y®- 1 & c - } = o 
