104 INTEGRATION OF DIFFERENTIAL EQUATIONS. 



The principle of our analysis, it has already been remarked, 

 is contained in the equation 



n (n - 1) -p (p-l) = (n-p) (n+p-l); 

 and this consideration suggests an extension of it. 



For it is obvious that the coefficients of a n in 



\ d m ~'-* 



differ only in this, that where the first has the factors 



(n - m + s + 2) (n - m + s + 1 ) , 

 the second has p (p 1). 



Thus the same transformation applies; and if either p, or 

 p 1 is divisible by m, the solution of 



may be made to depend on that of 



and thus effected in finite terms. 



The formula of reduction in this case is a little more com- 

 plicated than those already given, and we will not dwell longer 

 upon it, our object being rather to point out the integrability of 

 certain classes of equations than actually to integrate them. 



The equation 



n (n - 1) -p (p - 1) = ( n -p) (n +p - I) 

 is a particular case of 



and the latter will give us various formulae of reduction accord 

 ing to the value of JJL. Thus 



