OF LINEAR DIFFERENTIAL EQUATIONS. 233 



both inclusive^ the linear differential equation, or, as it is 

 proposed to call it, the " differential resolvent/' is of the 

 lorm 



and I completely determined the constants a^, «,,... «n-i 

 for all the cases up to and including n = ^. 



I found, moreover, that this result, in itself sufficiently 

 remarkable, might be put under a still more simple and 

 striking form by following a process of transformation 

 proposed by Prof. Boole in his "Memoir on a General 

 Method in Analysis,'^ which appeared in the Philosophical 

 Transactions for 1844, ^^^^ H* I found in fact that, 



writing e^ for <27, and D for ^ -j- or -r^, the differential re- 

 solvent of the trinomial equation (I) may be made to take 

 the form 



D(D-i)(D-2). . . {D-n + 2)y 



• ♦ • {p ^)e(-0^2/ = o. ... (A) 



the case n=2 being an exception. In this exceptional 

 case the sum of the roots (Zy) is not, as in the other cases, 

 equal to zero, and the differential resolvent must therefore 

 contain a term independent of y. This term written on 



the dexter = -e^, and the terms on the sinister follow the 

 2 ' 



law above indicated. 



Using the ordinary factorial notation, that is to say, 



representing 



{u){u—i){u — 2) . . . (u—r+i) 



