Il.—On a Process in the Differential Calculus, and its application to the Solu- 
tion of certain Differential Equations. By the Rev. P. Ketianp, J.A., 
F.RSS.L. & E., F.CP.S., late Fellow of Queen’s College, Cambridge ; Professor 
of Mathematics, §c., in the University of Edinburgh. 
(Read 17th December 1849.) 
The facilities which are afforded by the introduction of the function /~ into 
certain classes of Differential expressions ‘are well known. This function has 
effected the combination and generalization of Problems, which, although found 
to be capable of solution in particular cases, were regarded rather as isolated and 
exceptional forms, than as integral parts of some comprehensive expression. 
But the subject is far from exhausted. Some of the most important Differential 
Equations have never as yet been solved by a general method. The present 
Memoir is intended to supply this defect. The process employed differs little 
from that which I have previously exhibited; but the range of Problems which 
it embraces is much more extensive, and the Problems themselves are of a 
more important character. 
Section I. PRELIMINARY THEOREMS. 
1. Let yoe*, 
1 d 
then (sigs) 9= 279, 
from which it follows, since the operation reproduces the function itself, that 
ea > 
= I as) “y=(-ar)‘y (1.), whatever pe May be. 

It is necessary to observe, that if u be negative, the above equation takes no 
cognizance of functions of integration, which would be introduced by means of 
the added arbitrary constants; and this remark applies to all our processes. 
In the equation sf e~ OT a0=ih, 

let 0=a2"> 
es r 
then i, eW%® gh—lir(n—1) or J a=/n 
|n a n—-1 ,—acx” 
or alae a e da, 
x 
VOL, XX. PART I. L 
