§3° Mr, Babbage’s essay towards 
equations, this step is supplied by the integrations which are 
necessary, and we thus arrive at their general solutions. 
Problem XXXI. 
Given the equation 
F | X, yj/ X, 4 / « 00 | = 0 
and also cd x = x 
Find -tyx in terms of oc and $ ax ; let it be 
\f/ < 3 ? = F 1 37, ^ Oi X I 
put ax for x; then it becomes 
•ty a x =2 Y^uX, <37 j = F | a x, *4/ x I 
put this value of i ] tax in the former equation, and we have 
ifrx = F | x, F (ux, if/ x ) j 
from which equation ipx may be found in terms of x. 
If a 3 x = x instead of odx = x, we should find 
4/<3? = F j <37, F ja , 37 , F | a 2 <37, J/ a? j 
Ex. i. Take the equation (4/a?)* . 4 / (a— x) = a?’ 1 where 
ax ~a — ce 
then 
and 4/ f a — x 
this substituted in the former gives 
+(*-*) = 
tyx : 
from which we find 
fyx : 
-fl ■ 1 1 -1 
n P 
l (a— *> ^ J 
which will be found on trial to satisfy the equation. 
