34 M. Falk, On tue Intégration of partial 



may be written 





! = 



the same notation being used as in §. 5. 



Multiplying- tiiese equations by a. and /3 respectively and adding, 

 we get 



i=n-l \ 



,=0 I ' " ' 



+ /3 C,„^-o.„+i + ^1 + /3^= ) 



which is linear with respect to the differential coefficients of z of the high- 

 est order. To integrate this equation, we have by § 3 the auxiliary 

 equations 



^ Co '^^'"' + s (-1)"{* C-,-x,Hi + ^L,..h"-' + (-1)""' /3C„,„= 0...(56), 



i=0 



dr/ — mdx = . . . . 

 the coefficients W satisfying the conditions 



(57), 

 (58), 



W.. = 



m 



and 



1=0 



i+i,i-— i~i 



m' *) 



^«0 



The equation (5G) may evidently be written 



/3 

 The root m = ^ of this equation gives 



(59) . 



TT 



^C„_,^ (^• = 0,l,2, ... «), 



*) As soon as either of tbe iudices of C becomes = 0, the t, to which that in- 

 dex belongs, must be regarded as vanishing. 



