270 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



If — =»; this gives 



•'^0, 



whence v^A.e"''^'' 



and y = vx-Axe'^''''. 



This equation may be integrated in the following manner. The equation 



, /-D + 1 i „ 



may be made to depend on the equation 



by the relation y=Vt CTB^l rrp «'> where 



Pi/(D)=/(D)/(D-i)/(D-l) &c 



_ D(D-i)(D-l) 



(D-^)(D-1) 



= Dc 



/_D « 



Now » + a -/- 1 p^ e' »=0 is equivalent, by (D), to 



« + <? -s/ — 1 e" , — ^ ~ « = 



or v — a- — tv=0 



whence i>= A, e"'''' 



y = A « e""'' the same result as before. 



This process, which is due to Mr Boole, is of great importance in the solu- 

 tion of certain classes of ordinary linear equations, but I have not, as yet, found 

 it very extensively applicable to equations with fractional indices. 



Ex. 8. More generally, to investigate the conditions of integrability of the equation 



