288 Mr Dixon, On Lies Solution of a Partial 



particular complete primitive. In such a case however the 

 solution does not fail as may be seen. The failure in general is 

 due to the fact that the first derivatives of the proposed solution 

 do not give determinate values of the derivatives of z, but must 

 be taken in combination with the second derivatives of the 

 proposed solution. From these, one or more equations have to 

 be formed involving only the first derivatives of z, and these 

 equations are not generally satisfied by the values p l} p 2 ...p n . 



If however for particular values of the constants an equation 

 of the form A, 2 /* = can be formed from the equations 



/=0, u 1 = a l , u 2 = a 2 ...u n = a n (12), 



then, on the supposition that X = 0, the relations found from the 

 second derivatives of \ 2 /j, = will be simply repetitions of those 

 found by removing the factor \ from the first derivatives. The 

 effect will be the same as that of removing the factor X from the 

 final equations of § 7, and thus the results 



~=Pr 0=1, 2.. M), 



hold under these conditions, and there is no failure. 



§11. Another possibility is that the condition (8) may be 

 satisfied identically or in virtue of the equation f=0 only. Since 

 u x ... u n are supposed independent, it is not possible for the 

 conditions (10) to be satisfied identically, and hence (11) must 

 hold good. 



In this case the equations (12) give more relations than one 

 connecting z, x x , x 2 ... x n and therefore the supposition (a), (§ 6), 

 cannot be entertained. Let k be the number of independent 

 relations not involving p 1} p 2 ... p n that can be deduced from (12) ; 

 then in (6) we must suppose m not to be greater than n — k + 1 

 or it will be possible to deduce more relations than one among 

 z, x 1} x 2 ... x n . Also the relations (7) will reduce to n — m — k + 1 

 in number, k—1 being consequences of the rest, since the deter- 

 minants of order n — k + 2 formed from the matrix (11) vanish. 



§ 12. The equations (4 a) and (5 a) now give, in virtue of (6) 

 and (7), 



^(d^_ \ d(fu u u 2 ...u n _ k ^ = Q 



r^i \dx r J d{Z, p r , p k+1 ...p n ) 



and other such equations, each being linear and homogeneous in k 

 of the quantities 



^-p r (r = l,2...n). 



