EQUATIONS IN FINITE DIFFERENCES. 103 



+ B,v;' {1 ^ F,v,+ F^vihcY 

 + C,v^{\ + F.,v^kQ..Y 

 + B,v,'{\ +&C.}' 

 + &c. 

 and comparing like powers of Va, we have 



:\B. = F„ ^C, = F, + 2F,B„ 



AD,= F, + {F,' + 2F,).B. + 3F,.a, 

 &c. = &c. 



whence integrating, so that each integral may vanish when .v = 0, as 

 lias been proved to be necessary, we have 



B, = F,. X, C = F,x + Fi . X {x - 1), 



Z>.= F,x + {F.f + SF.F,).^'^-^-^ + jp. /^(^-'i-)i^-^) ^ 



2 3 



&c. = &c. 



and », = y„ + B, »„' + C^Vo^ + D,Vo\ &c. is completely known. 

 Ex. 



sin.-sin.-sin.-j.timesiof(0)=e + ^ . a' 4- jf^] - ^^^ -^). ,= &e. 



Before leaving this class of equations, we may remark a curious relation 

 between the equations 



and i)^+, = /(p{vj)i 



which is such that the solution of one leads to that of the other, for 

 if we put J'iih) = v^, we have 



■ ./(Mx + i) =/(p{vJ), 



or «>,+ , =f(p{V:.); 

 if then we determine ?<„ so that Vo=f(fh), v,. wiU be readily found from «,. 





