18 



M. Falk, On the Ii^tegration of partial 



or, by a known property of determinants , wliere all terms in a certain 

 row or column, except one, are zero, and after transposition ot the rows 



(A'^'Ar — A'-'Aro- 



D[, Z); . . . . , d:,_, 



JJi) -r)(2) 7^(2) 



--= 



(3G) 



r)(n-i) r)(K~i) 7^(«-i) 



Now we have to shew, that the n first integrals (30) satisfy the 

 equations (36) by making their first factors all vanish, that is to say: the 

 first integrals of (12) satisfy all the equations, wiiich, for a certain value i 

 < n , are obtained from the formula 



D'"' ^''^ _ !>''■' A,<^' = (37) 



by making successively r = 1 , 2 , 3 , . . . , (u — 1). 

 We may in fact write the equations (27) thus 



D'"' I + A*'' I m = Q^ 



« i ^d.n-l T ^,- Sî-l.o r I ^oo\ 



} {^o). 



U. C^ + A C m = 



As m is the value of ü' satisfying the equations (16), these equa- 

 te a; 



tions (16), divided by dœ and added, after the first has been multiplied 

 l>y — /'(^), will give 





c ii-l 



an equation, that would have been immediately obtained by dift'erentiating 



the equation F=0 totally and afterwards putting in for '-^ . 



dx 



In the same manner the equation F^ = gives a similar relation 

 involving m,. Now substituting in these equations the values of m ami m,. 

 from (38), we get the two equations: 



e =»-1 





e=0 f=0 



~e . c "-C 1 e_ 



0. 



