194 On Differential Equations 



result. I find that Mr. Harley's extension of the theorem 

 spoken of in the former part of this memoir is printed at 

 p. 199 of Boole's (posthumous) Supplementary Volume, 

 which reached me by the last (March) mail The extension 

 as there printed is erroneous to an extent which I have 

 already pointed out. 



V. In general, let 



^=/(2/) (J^) 



y = f{iv) - - (n) 



w=^ {az -\- hx)F{y) + {ax -\-hz)7r {y) -\- x (jj) (o) 



in which x and are independent, and F, tt, and % are func- 

 tional symbols, and let it be required to expand u in a series 

 of ascending powers of x. 



VI Put 

 («. + ,.)^) + («.+ ,.)^-^H.'|x|^^ (P) 



and we have 



In like manner 



and therefore, putting (q) and (r) under the forms 



dy r^ TTdir(w)\ d^l^ (w) fj T^, . .0 , . 



p(l-H ^^ 1 = ^^ (aFiy) + b.{y)] (t) 

 dz V dw J dw K ^"^^ ^"^V ^ ^ 



and dividing (s) by (t) we have, after a slight reduction, 



dy_^hF(y)-{-a7r(y)} dy 



dx~\aF{y) + h7r{y)^ dz ' ' " ' ^ '' 



VII. For simplicity, divide both the numerator and deno- 

 minator of the bracketed factor of the dexter of (u) by F (y), 

 then (u) becomes 



dx~\a-\-h(p(y)}dz~^^^^ dz ' " " ^^ 



