36 M. Falk, On the Integration of partial 



dx dy dz dz^^ dz 



Sl,0 ^0,1 \^1,0 l.O'O.l 01'' ^ 



or, according the usual notation, 



dx dy dz dp dq 



si^ = ^F - ( ^F iF\ - iF iF - iF iF ' 

 -d^y —dq -{P^/'^Tq) dx^PTz dy^^d^ 



which is the auxiliary system of Charpit in its symmetrical form. 



2) Auxiliai-y system of the diferential equation of the second order: 



F(x, y,z, .-1,0 , 2^0,1 ) ^2.0) ^1,1 , ^0,2) = . . . (64). 

 Here the equation (60) assumes the form 



r m^ — r ni -^ r =o, 



^2,0 ^1,1 '^0,2 ' 



whence m = ^v — . 



"^ ^2, o 



By means of this value of m we get 



Wo = <„, Tri=.<,,+/3C,o-'»<,o-/3C..o4.^^ -+^^V~^^^ — > 



TJ7 _1.^^ _ ß ^l. 1 "^ ^1,1 ^2, 'so , 2 



^ ~ m ~ 2 



Now m having one of the above values , let ^ denote the other ; 

 then these equations may be written 



>^o = ^C,o' ^1 = C.o(/3 + ^»0, W, = /3mC^^„, 

 and the equation (58) becomes 



Co [«^'2,0 ^^(/3 + Ä»0^\i + ßrndz^^ ^] + (cet + /3»?)(?^^ = 0. 

 The auxiliary system of (64) is, therefore, transformed into 



dy — mdx = (65) , 



C.,o(^^--2,o + y^'^-'x,i) + ^^^'= 0,1 





