250 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



Let x=e^ and suppose u expanded in the form w=2fl„ a;"" : also write D for 



— : then 



V''«/ x" « 





— n 

 X 





:(-ir.«„i:^.-"^b.y(A) 



/-D 



=(-iri:^.».(C) 



As a particular case of formula (B) we have 



,t -D + fJ. l-B + fx + r ,g 



-D l-B + r ^ ' 



These four theorems will be found of the utmost importance in reducing dif- 

 ferential equations. Formulae somewhat analogous have been applied to the so- 

 lution of common differential equations by M. Cauchy, Exercices, vol. i., p. 163, 

 and Extircices fT Analyse, ii., 343 ; by Mr Gregory, Cambridge Mathematical Jour- 

 nal, i., 22, &c.: and by Mr Boole, Philosophical Transactions, 1844, 225. Under 

 the different heads in which we shall arrange differential equations, we shall 

 solve only the most simple examples, our object being to illustrate the method of 

 proceeding rather than to exhibit its power. 



Class I. Equatinna which are capable of solution ivithout transformation. 



6. Ex. I. ^ - a^ ^ = 0. 



By writing d for t— , this equation becomes 



hi I i -1 



(rf-a)y = or^=(rf--a') .0. 



