INTEGRATION OF DIFFERENTIAL EQUATIONS. 113 



Let 



~fk '/ * T7 i 1 ' vT 7 * \" ^n 



(10), 



_ _ ...... _ 



(tliere are ^ factors in both numerator and denominator) ; equa- 

 tion (9) becomes 



n (n-l)...(n m + 3) (n m + 2)(n m + 1) & + &5 M _, n = 0; 



and, as in the two preceding cases, we shall have 

 2b n x n = < (k) X, 



and l m **N.&4>(ty. 



(10) may be written thus, as^> = ^, 



(n m + 2\ /w m + 2 



- ...... -- q + l 



V m J \ m _ ^_ 



XT 

 Now 



mm 



ft~* f n 1 f7 q 



therefore ^. 



m J \ m - 1 - ' ' 



But if we make </> (&) = A; w , then 



Let us also put f(k) = k q m ; therefore 



'n - m + 2' 



) <KC. w -m 



-*! 



\ 



) - &c - 



y JVr^"^~ 



