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 ¢ and yp be any functions whatsoever of x, and mand 
r any numbers, positive or negative, whole or fractional, the 
symbolic equation 
(Di 9+ 7F) yay (D+9+ 9") 
holds good for any subject which we may conceive operated 
on by its two members. 
It will be convenient to put 
pe ged we 
wy 
so that the preceding equation may be written in the form 
Am yr rT YW" Amer ° 
And operating on this again with the symbol {"( ) ¥", we 
get 
Yo An = Arie Yr. 
2. It is easy to show that, for \ and wany functions of a, 
(D+) (D+ p)-(D+p)(D+A) =u -N. 
Therefore, ify be any function of x, and m any number, 
ALD sy @ sy ate e yon (5), 
whence 
Ay Ag(D + x) =4 (D+ x) Art x’ 9 (5) } At Ar(x- 9!) 
If we now suppose that 
X- ~ = ap", (1) 
where c is some constant, this becomes 
A, A,(D+ x) =(D+ x) ArAo+ {2eprr=r(F) } de 
whence again, 
