INTEGRATION OF DIFFERENTIAL EQUATIONS. 99 

 We now proceed to the more general equation, 



As before, we shall get 



{n(n-l)-p(p-l)}a n + fa^ = .......... (12). 



Now n(n-l) -p (p-l) = (n-p) (n+p-l) ...... (13), 



which is the fundamental principle of our analysis, 



.-. (n-p)(n+p-i)a n +fa^ = ......... (14). 



Assume (n+p-l) a n = (n-p + 2) b n .... ........ (15), 



n , 



and (n -p + 2) (n+p -3)b n + fb^ z = ......... (16). 



Again, assume 



( n +p - 3) b n = (n -^ + 4) c n ............... (17), 



and so on successively. , Thus we shall get a series of equations, 

 of which 



(n-p + ri^+p-fjL-m + fl^O ......... (18), 



is the general type, where p is even. 



If p is even, let p = ^, .'. p p 1 = 1. 



If it is odd, let^> = p + 1, ,\ p + p = 1 : 

 and in "both cases (18) becomes 



n(n-l)l n + 2*1^ = 0, 

 and therefore 



2l n x n =Cain(2X+a) ..................... (19). 



Let {n-p+(n-<*)}(n+p-p + l}i n + fi n _ z = ^ 

 (n-p + fjL)(n+p-fj,-l')k n + flc^ = 0, 

 be any two consecutive equations ; then 



(w+p-A* + l). = (- d p + /A)^... ......... (20), 



n p + /JL = n+p fj, + l-2(p p) 1, 



_ I ....... (21), 



72 



