14 



M. Falk, On the Integration of partial 



§• 4. 



Relations hebveen two and three first integrals of the equ. Cl'2). 



Let /M./ and m correspond to tlie first integral F = (p — f(ip) = 0^ 

 ^*'* and m,. to another F^=<f,. — /,(///,,)= . Then from (17) and (18) 

 we have evidently 



('■) 



/*„_, m = fA,'„L, m, , 



yM-, + ^,_, m = ^;'' + ^;' 



o-'i 



in,. 



(24). 



(1=1, 

 The equations (11) give 



, (H-1)). 



/^, 





K-. 



if 



■/w^,;->,o 



3i^' 



or 



/*' = 



^c. 



<^'n— 1,0 



(25) ; 



and in the same manner an analogous expression is obtained for jM.,"\ 

 Using these equations and putting ^ — '- = r"* , the equations (24) become 



r 



'u-l.O 



-t.J 



m 



n-1,0 



1»,. 



^B— 1— i,z _1 iJn— /,ï— 1 J,, ^_^ J»— 1 — z ,î [ S/;_,- ,-„1 



Hi, 



• (26), 



y • v- ur) ^(r) 



^/!— 1 ,0 ^n— 1.0 S)„_i,,) ^n-I,0 



in the last of which we have to put successively ^ = 1 , 2 , . . . , (n — 1). 

 Eliminating successively m aud m,. , we get from the two equations (26) 



r i> + r A. m,, = 0, 



^11. n -\ i ' ^«-1 ,0 ^-i, ' ' 



^„.-i ^,- -^ C,,., A,- m = 

 (/=1,2,..., (»—1) ) , 

 where the following notations are used , viz. : 



(27) 



rr = 



Q 



A'"l 



^FI — 1,1P ' ^U — 1 ,11 



a'" = 



(J. (J 





