480 The Rev. B. Bronwin on the Integration of 



in the same manner ; and similarly in the first case, or 



by changing u into Dm and eliminating Du from the second 

 member. 



The equations (1) and (2) may be put under the forms 



X,=M + <^(7r)^rM4-^i(7r)A+ .... 



Then it will sometimes happen that by (A) and (C) they may 

 further be put under the forms 



'X,iiBU-\-a<f){7r)xu-\-b<l){'7r)a:(f){7r)xU'^ .... 



Xi=w + «<^(«')DM4-ft</>WI></>(w)DM+ .... 



Make <j^('7r)a?=^, and also </)('bt)D = ^, and both take the form 



Xi=M-fa^M + 6^M+ 



or 



Xi = (l--Jfc^)(l-^i^....M; 



which may be treated in the manner explained by Mr. Boole. 



We can, however, rarely integrate when the second member 

 contains more than two terms. Let then 

 X=f{7r)u+f,{7r)w^u. 



We may consider /(tt) and/j(7r) as integral functions of ir, and 

 therefore as factorial functions. 

 Assume 



«={7r + A)(7r + Ai) ('7r + A«)Mi, 



or 



u-(7r + h)-'(7r + h^)-' .... (7r + A„)- V 

 as the case may require. Then by (A) we have 



x^u={7r + h—r) .... (7r-f ^„~r)a?'*Mi, 

 er 



a?^M = (9r + A— r)-^ . . . (tt + A^— r)~^a7*'Mi. 



The constants h, h^, &c. being suitably chosen, after substi- 

 tuting the assumed value of u in the given equation, we must 

 operate on both members with the inverse of all those factors 

 which are common to the two terms of the second member. When 

 the method succeeds, the result will be an equation of an order 

 lower than the given one. 



But if the given equation be 



X=/Ww+/,(tir)D% 

 make 



tt=(«r + /0 {nr + h„)ut, 



or 



