10 M. Falk, On the Integration of partial 



When i = r the corresponding terms in (6) vanish ; in order to ex- 

 clude such terms, Ave must transform the expression (6). To this end we 

 use the identity 



SS = SS+SS ....... (8). 



i= r—» i=l I.=:IJ 



whence the vanishing terms are already removed. 



The first terra in the right member of this identity undergoes no 

 change, wlien we interchange r and i\ thus tliis term becomes 



r^n-\ î-ï'-l 



O OL"^«-!-'.' ^ni-rr ^„.\-i A '^ n-\-,:r ^ „r.r ^n-\-i ,H y ■ • ■ (9). 



7-:z=l î=(» 



This double sum may be regarded as extended over the surface of a 

 rightangled triangel; and if we sura over the same surface in reverse 

 order, it will be seen, that the expression (9) becomes 



i=n-'^ ■)'=n-\ 



\^ \^\ n-\-i.i ^n-\-rr ^nA-i . i^n-\-r .r pK-r,r -^«-1-1,1+1 



t=:[| 7"=î + l. 



Thus the expression (6) becoraes 



ï=n-2 r=n-\ 



O O I «-'■'.' n-l-r.r ^ ii-X-i i ^ ii-\-v ,r ^n-i ,i ^n-lr .r+i ^u-y.r^n-\-i .i H ••{^^J 



!=n j-ii+l 



Now as the equation (7) is to be linear with respect to the ditferen- 

 tial coefficients of ,- of the n''' order, we raust have 



y y 7' 7 



from 7' = i -\- 1 to r = n — 1 and from i= to i =■- n — 2 . 



The last equations also may be written 

 7 7' 



7 — 7' 



H- (11). 



where we have successively to put r = 1 , 2 , 3 , ... {n — 1) ; also, of 

 course, it is supposed, that neither .^„_i,o nor Z^_^ ^ vanislies. 



Now the differential equation (7) will become 



S Z7;.„_„ = V (12), 



1 = 



supposing the following notations used, viz.: 



