284 Mr Dixon, On Lies Solution of a Partial 



and the products added. We thus have on the right 

 z \ 



and on the left 



(dz _ \ d(f, u ly u 2 ...u n ) 



va^i / d(z,p 1> p 2 ...p n )' 



du x du 2 da 3 du n 



dx x ' dx x ' dx x ' ' dx x 



3/ dih du 2 du n 



dp x ' dp x ' dp x dpi 



df du x du n 



dp 2 ' dp 2 dp 2 



dpn 



In like manner 



dz 

 dx r 



Pi 



d(f, u x ...u n ) 



d{z,p x ...p n ) 



= the same determinant with x r written for x x in the first row. 



In these equations we may substitute for u lt u 2 ... u n any n 

 independent functions of them, say fa, <f> 2 ... </>„, the effect being to 

 multiply each equation by the Jacobian 



9($1, (f) 2 ...<f> 7l ) % 



dp n ' dp r 



* If the relations among u x , v 2 ... u n expressed by (6) are such that they cause 

 this Jacobian to vanish for arbitrary forms of 4>m+i> •••0)t> then the equations (6) 

 must be satisfied doubly by these relations, and this state of things can always be 

 avoided. 



