35 
Ay A,A,(D + x) = { (D+ ) Aer tep-r- 2(¥ |} ards 
+Aby { 2eyr* - (2) } 4. 
But if we further suppose that 
k being some constant, the last equation assumes the simpler 
form, 
A.A, A.(D + x) =(D+ x) AwArAo+ (8¢-3rk)P7A,Ao. 
And continuing the same process, we should find generally 
AnrAn-1r yom Agate (D+ x) = (D+ x) (Ann Atnetyy ...ApAo 
+ {(n+l)e- slash rh) b-" Aim-ayp «Ay Ao 
Or, since by the theorem in § 1 the variable part of the 
last term is equal to 
Any Acn-1)r shay's A, Arp", 
n(n+1 
fee... ... A TAD y)— (Cas lc - ae vr] 
= (D 3P x) Ac A n-ayr see A,rAas 
3. This last formula enables us to effect the solution of the 
linear differential equation 
[D+ 9)(D+x) — {es De ee a ar] y— x, @) 
whenever the conditions (1) and i 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 
y= A} Ay... Ay (Dryyt An... AgpA,X. 
4. As regards the conditions (1) and (2), it will be ob- 
served, that the latter limits the nature of the function y, whilst 
the former makes the difference between e and y to depend 
upon that same function. ‘ 
oo 
D2 
