34 
The Rev. Professor Graves communicated the following 
method of solving a large class of linear differential equations 
by the application of certain theorems in the calculus of ope- 
rations :— 
1. If g and yf be any functions whatsoever of x, and mand 
r any numbers, positive or negative, whole or fractional, the 
symbolic equation 
(D+ 9+) y-y (D+ 9+ C2P*) 
holds good for any subject which we may conceive operated 
on by its two members. 
it will be convenient to put 
te pI) pr 
© it 
so that the preceding equation may be written in the form 
A yr” = PW Ansr- 
And operating on this again with the symbol {-"( _) ¥", we 
get 
PrAn = Amir po" 
2. It is easy to show that, for \ and wany functions of 2, 
(D +X) (D+ w)-(D+p)(D+rA) =H -X. 
Therefore, if be any function of x, and m any number, 
AD) (Dey) Aa eE (5), 
whence 
AvA(D+ x)={ (D+ y) Art 9-7() bbe Ac 6): 
If we now suppose that 
Xx-p=ep’, (1) 
where ¢ is some constant, this becomes 
A,A,(D+ x) =(D + x) ArAot {2eprr-r(3) } do 
whence again, 
