INTEGRATION OF DIFFERENTIAL EQUATIONS. 115 



The change which this will introduce in the process, is not 

 difficult to perceive. We must assume 



n m+1 nm+1 



a n =/(&) - I 



J ^ ' n-m + 2 n m + 2 



1-1 \-q 



m m 



as before, the transformed equation will be 

 n (n m + 2) (n m + l}\ 



Now 



m 



.. 



therefore 



l\ n m + 1 





\ 

 +) 



m m 



I -mfl 



If then we make /(&) = tf m and <j>(Jc) = k m ,WQ get 



fn-m + l\ & 



I / "*" tt-wi+2 



\ m / _ Z.~~ 



/n _ m + 2 v "* 



I -- 1-1 ... &c. 

 V m J 



Hence, finally, 



As an illustration, let us take the case of the equation which 

 occurs in the theory of the figure of the earth, 



Here m = 2, p = 3, p 1 = 2 : hence q = 1, and as p is not 

 a multiple of m, the formula (13) is to be used. 



82 



