300 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



X+0 X+0 



"^~ 1-(1 + A,y e''^^ 1-2(1 + A,ye' 



jx + l jx.+ l /a?+l 



Cor. In the same manner may the more general equation 



u, —aCr + iy u,. + 1 + 6 (*■ + 1)'' (j- + 2Yu^^2 + &c. = X, be solved. 



It is not necesary to solve such equations as M,^j + a«„ + &c.=X, since it 

 is evident that, by putting ^ = ^ y, this form of equation is reduced to 

 vy + av/^i + &c.=X, which has been already solved. 



We proceed then to the solution of equations involving fractional differences. 

 Here we must, at present, confine ourselves to very simple examples. 



Ex. 4. A- u^ —au^=0. 



Since a* e"* ■'■ = (e'" - 1) i e'" •'■• 



It is evident that if w = log (a- + 1) the solution of the equation is u,. = A e™ * 



^Ae'^^°s('^' + ^) =A(a^+lf 

 Ex.5. A^ Uj. — aUj. = ce'"' 



u^.=A{a-+iy + - 



or, if e'= 1 + b\ u=A (a' + If + j:^ (*- + 1)' 



CoK. If b=a, this solution fails. Put 6=a + /3 as in similar cases in Differ- 

 ential Equations : 



then ^(6=+l)' = ^(a= + l + 2«/3r = ^(«=+ir+^2a/3^(«2 + i)-i 



A, being an arbitrary constant. 



It may be interesting to verify this solution. 



A* (a^ + iy-a.(a^ + lY=:0 evidently, 



and Ai2acz{a' + ir-'^ = 2ac X aia^ + 1}'-'^ + i2a c (a^ + 1)'^ by Prop. 3. 



=2a^cx (aP + 1)'- 1 + c(a^ + ly 

 Ai .2acx ia' + iy-'^-a .2 acx {a^ + ir-'^ = c{a'' + ir = ce^ ' as it ought. 



Ex. 6. Generally, let A^u^-au^^X 

 then M,=:A(a2 + l)- + (Ai-o)-iX 



