OF LINEAR DIFFERENTIAL EQUATIONS. 235 



or, what is the same thing. 



The deduction of the most general results from the 

 canonical resolvent is properly a subject for separate dis- 

 cussion, and I hope to treat of it in a future memoir ; but 

 there is a particular case so marked in character, and lying 

 so near at hand, that I am induced to present it here. The 

 differential resolvents for 



y''—nay+{n—i)bx = o (Ill) 



and 



y'^—nay''-^-\-{n—i)bx-=o (IV) 



where a and b are any constants, are 



e(«-0«2^=o (E) 



and 



= [?^— i]^-^«de^ (F) 



respectively. For by simply writing 



bx n 

 tor x 



n 



and 



^ for y, 



the equation (I) is transformed into the equation (III) ; 

 and these substitutions being made in the differential re- 

 solvent (A), the symbol D remaining for these substitu- 



