302 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



Ex.9. A!«,, + aA*M^ + 6M, = X 



u,- 



1-(A- -a)-^(0 + X) -^ '■- -^^-^ 



a-/3 



a-/3 



(A*-/3)-^(0 + X) 



= A(l + ay + B(l + ^y + A^(^^^_ a)-i.X-^(A*- ^)-^ X 



a-/3 



«j. + a .r A*Mj, = X. 

 1 



Ex. 10. 



This equation gives «.-t j 



1 — a a; A^ 



X 



Now 



or 



=(l-aa;Ai)''.,r, where i.^ is determined by the equation 



v^ — a-xiJ^x A^ »j. = X 

 Aia7A*»^=a7Aw,, + ^»r + i (Prop. 3.) 



»,. — a^X^AtJ, — Pi-.(:» =X 



»x + l- 



■^ 2 ,r+r 



2 l + a2«2 



2X 



a2 ^ + 2a^- ■■^~ a'{x + 2x') 



which being solved by the ordinary method, i\ and therefore ?/.„ (provided a^ u^ can 

 be found) is known. 



Equations of Differences with two independent variables are not capable of 

 solution to any great extent. An example or two will suffice to illustrate our 

 process. 



Ex. 11 . ^x «r, ;, - ^;/ «.■■, y = *• 



1 



The solution is 



X ^y 



Treat a^ as a constant c, then the solution of {'^x-'') ^x, y=* is 



«..,, = (c + l)1A-* 



«j-, y = (^y + 1)' Vy - \ - 1 b 



='«u+x-f>y 

 Vy+x being an ai'bitrary function of y + a;. 



