126 Mr. J. J. Sylvester on Polynomial Functions. 



given, that when U, a function of (n) variables, becomes express- 

 ible as a function of (n—r) orders, these orders may be taken 

 respectively any independent linear functions of the linear deri- 

 vatives of U, which remark completes the theory of functions, 

 subject to the loss of one or more orders. It is obvious (and I 

 am indebted to my esteemed friend Mr. Cayley for the remark), 

 that the conditions furnished as above by the (m— l)th, i. e, 

 linear derivatives, are identical with and may be more elegantly 

 replaced by those involved in the assertion of the existence of 

 linear relations between the 1st or (m—l)th degreed derivatives, 

 and we have then this very simple rule; if <t>, a function of 

 Xj, x^ . . . Xn, is expressible as a function 0/ n— r linear functions 

 ofx^, Xg . . . Xn, it is necessary and sufficient that (r) independent 

 linear relations shall exist between 



d<l> d<l> d<f} 



dx^ dx^ * * * dxn ' 



This rule itself also, it is evident, is capable of an independent 

 and immediate demonstration by means of integrating the partial 

 differential equation or equations by which it admits of being 

 expressed. The above theory may readily be extended to func- 

 tions of several systems of variables. Thus, for instance, the 

 determinant 



a b c 



a' V (/ 



vanishing will be indicative of the function 



{ 



being linearly equivalent to a function of the form 



+ C2/V + D2/VJ 



t. c. losing an order in respect of each of the two systems x,y,M; 

 u,v,w; and so in general. 



7 New Square, Lincoln's- Inn, 

 January 7,1863. 



{ 



