16 



d'" = 



M. Falk, On the Integration of partial 



I, r 



^«—1 ,0 ' ^n — \ ,0 



and A — 



* — i It — 



(»■) 



Su,«— 1 ' >i,n— 1 



Now differentiating F=Ç) and F,—Ç) partially with respect to x^ we get 



Q=n-\ 



t' + S L.^,„ ^„-„, = 



e=ru 



Adding these multiplied respectively by ^ ^ and — ^ ^ .^ , 

 z„„ will dissappear, and we get 



0=H- 1 





(31) 



Differentiating F=0 and i^,. = with respect to ?/ and eliminating 

 Zo„, we arrive at the equation 



a;; + s A"\-,„ =0 (32) , 



where q — 1 is substituted for q . 



If now in (31) and (32) we put successively r= 1, 2, ... (n — 1), 

 we shall get all distinct equations, that can be constructed by differentia- 

 ting (30) and eliminating xr„ „ and z„„. These will be 2n — 2 equations, 

 involving the (n — 1) remaining differential coefficients of the n'* order, viz. 

 ~,i-] i> ^n 2,2 5 ••• ~i «-1 1 the elimination of which, therefore, must lead to 

 ()i — 1) equations of condition. These relations we may easily obtain in 

 the following manner. Taking all the equations, involved in (31), together 

 with the equation (32) unaltered, and eliminating the remaining differential 

 coefficients of the n"" order, the result will be 



Af, x>f , D^K ... , n<;:i 



n("-i) r)(n-i) r)(n-i) r)(»-t) 



-'-Al 1 -*-'l 1 ■'-'■1 1 • • • 1 -^'n — 1 



A^ , Ar' , A<;' , . . . , a;;1i 



= . . . (33, 



f 



i 



