I 
24,0 Mr, Babbage's essay towards 
until the last equation will only contain x, ->px and their diffe- 
rentials ; this equation must be integrated, and it will deter- 
mine the value of -fyx in terms of x. 
The same method may be employed for the solution of the 
much more general equation 
X, $X, juX, . . 
provided also, that aP x — x. 
By substituting successively for x the quantities ax, odx, 
&c. a.P— l x, we shall havep equations containing the functions 
tyx, $ax, and ipap—*x and their differentials. 
Let each of these be differentiated as often as may be 
required, and we shall have two sets of equations by means 
of which all the quantities except x and $x, and their diffe- 
rentials may be eliminated, the result is a common differential 
equation whose integral will afford the value of -tyx in terms 
of x. If after satisfying the conditions of the Problem, there 
remain any arbitrary constants, we may suppose them func- 
tions of x, and new equations will thence arise by which they 
may be determined. 
It might occur (when there are several arbitrary quanti- 
ties) that, by assigning particular values to some of them, 
the others might remain in a certain degree arbitrary, should 
this be the case, we should obtain general solutions. 
. />*— 1 d n iS/ax 
■fya. X, 
dx n 
d m ^/a?x 
dx m 3 
&C. 1 = 
} = 0 
Problem XXXVI. 
Given the equation 
yx= d -¥- ■ 
T dx 
Assume $x= <p'f<px, then the equation becomes 
-1 dZ'fp* 
<p /«>*= — 
