35 
A,,A,Ay(D + x) = {(D + x) Aap tep-" — Or (5) } 4rd. 
+A, { 2c" — r (=) } A, : 
But if we further suppose that 
(Fars (2) 
k being some constant, the last equation assumes the simpler 
form, 
A, A, A.(D +x) =(D+ x) AxwArdAot (8¢-3rh)p"A,Ao. 
And continuing the same process, we should find generally 
AnrAqa-yr.-»ArAo(D+ yx) =(D+ x) Anr Aner.» Ar Ao 
+ {(n+1)e- so rh} b-" Aig-ijp + AA: 
Or, since by the theorem in §1 the variable part of the 
last term is equal to 
Any Aqn-iyr alae A,A,w', 
We, As [Au(D+ y) = (at 1) ay 
= (D ar x) Ape (aay toe AA 
3. This last formula enables us to effect the solution of the 
linear differential equation 
[(D+4)(D+x)- (+ 1je-* AD ray yr] y=X, @) 
whenever the conditions (1) and (2) are satisfied; as it fur- 
nishes us in that case with the means of inverting the operator 
in the left-hand member. Thus we find 
yA} Al es Ay (Deyyt An.» «Ag A,X. 
4, As regards the conditions (1) and (2), it will be ob- 
served, that the latter limits the nature of the function x, whilst 
the former makes the difference between $ and y to depend 
upon that same function. 
D2 
