2 
in which A, B, &c., are any functions of x, being regarded as 
the result of the operation of 
Di AI A. 
upon y, all attempts to solve it by the method of the separa- 
tion of symbols must be directed to the transformation of this 
operator into a form in which it appears as the product of a 
series of operations, each of which admits of being inverted; 
and, conversely, if the result of a series of such operations be 
to produce a complex operation of the preceding form, we 
may apply it to the construction of soluble forms of differen- 
tial equations. 
Thus the operation 
(D+ $9) (D+), 
in which ¢ and y denote any functions of x, will be found on 
development to be equal to 
D+ (o +o) D+ Ob+H)s 
consequently, we may identify the general linear differential 
equation of the second order, 
Dy + ADy + By = X (1) 
with Dy + (p+ p) Dy + (gh + Wy = X. (2) 
And if we could succeed in solving the equations 
o+P=A, ob+ y= B, (3) 
and so obtaining finite values of @ and y in terms of A and B, 
we should be able to effect the solution of the general linear 
differential equation of the second order, at least in a symbolic 
form; for as 
(D+ oy) = eleerfejoae 
and (D +p)? = efvatfeSvex, 
we should have 
y =E]¥ [el(r-oiar (oleae x. 
Unfortunately it happens that, in trying to determine 
g and y from the equations (3), we either obtain a differential 
equation of the first order and second degree, or are led back 
again to the solution of the equation (1). 
