112 INTEGRATION OF DIFFERENTIAL EQUATIONS. 



i 

 We must, it is evident, make < (k) = k " l , and then 



Tp -rr 



Hence # = -77^ ^7 is the solution of the equation 



; 



for as before the factor ra* may be neglected. 



A particular case of this result is that in which 5 = 0. If 

 with this value of 5 we have m = 2, the equation to be integrated 

 takes the form 



and the solution is 



Equation (7) is the most general one in which the coefficient 

 of a n differs in one factor only from what it is in the case of 



But our method is applicable in other cases. 



Let us resume the equation discussed in the last number of 

 the Journal, 



^-*k-l), ............... (8). 



By the usual method of making y 'a n x n , we get 

 n (n - 1) ... ( - m + 3) {( - m + 2) (n - m + 1) -p . (p - l)j a 



or n (n - l)...(n - m + 3) {(n - m + 2 -jp) (n-m + 1 +^)) a n 



+ ka n _ m = ...... (9). 



It may be remembered that we found it necessary that p or 

 p 1 should be divisible by. m. Suppose then that p is so 

 divisible, and that the quotient is q. 



