282 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



entiation and the multiplication by a constant in the same manner, regarding both 

 the one and the other as an operation. A similar process has been employed for 

 the solution of ordinary simultaneous equations by Mr Gregory in the Camh-idge 

 Mathi'tnntical Jownal, i. 173. 



Ex. 1. -rT+<»y=0, -4+6.r=0 



By taking the ^ differential of the first equation, we get — + « ^ = o ; which, 

 by virtue of the second, gives 



^^ -abx = 0; or.j- = Ae"'" .-. y= -X J-e"^' . 

 Kx. 2. -^ +ay +bx = (i, ^ + ay + b'x=0. 



These equations may be written 



whence, by eliminating «/, we obtain 



Let aK ^K be the roots of the equation 



(a + a') (z + b)—a(/=^0 



then (A _ a A (^ -i3^\x=0; which gives 



and 



Idi X b 

 adl^ a 



Ex.3. 



=-(r^:)----(?^yB^.. 



— +ay + bx=f{t) 

 —^+a'y + b'x=(l>(t) 



{ (^ ^0 (l)i ^*) -'^*'h= (^i^'^V'^'^-'^'^' 

 This coincides with Ex. -4, Class 1, and the solution is 



a^-Si\dti J ^ ai-l3^\dei '^ } ^ 



(t) 



