286 Mr Dixon, On Lie's Solution of a Partial 



vanish unless those of the matrix 



.(11) 



are all zero. 



There are therefore two kinds of exceptional cases, which we 

 shall now examine*. 



§ 9. The equations (7) may be written 



\% +2% |^=0, ( 8 = i t 2...n). 



op s i=i dp s 



Thus if (7) and (8) are both true it follows that 



xy + 2*^-0, 



dz 



dz 



unless the determinants (11) vanish. In the same way from (9), 

 unless the determinants (11) vanish, 



* The proof also fails if 



d(0i. 02 ■■• <Pn) _ » 

 9(m x , u 2 ...u n ) 



that is to say, since m+1 , ... <p n may be chosen arbitrarily, if 



du x ' du 2 '" du n 



3mj ' 



3itj 



du„ 



= 0. 



These equations are ?i-m + l in number, and as «!...«„ already satisfy the 

 m equations (6) they will generally be inconsistent. If however any values satisfy 

 them they will also satisfy (7). If the values are isolated then these relations 

 afford a solution which is included in the complete primitive, since the values of 

 Ui...u n are given. If the values are not isolated then these relations are not 

 numerous enough to enable us to eliminate ^-...ZV 



