[From the Proceedings of the London Mathematical Society, VoL IV.] 



LVI. On the Theory of a System of Electrified Conductors, and other Physical 

 Theories involving Homogeneous Quadratic Functions. 



[Read April 10th, 1873.] 



THE theory of homogeneous functions of the second degree is useful in 

 several parts of natural philosophy. 



The general form of such a function may be written 



V=^(A u x 1 3 + 2A^c l x 3 + &c.) ........................... (1), 



r=n i=n 



or more concisely Vx = \ 2 2 (A r jcjc t ) ............................ (l*), 



in which each term consists of a product of two out of n variables x u ... ar n , 

 which may or may not be different, and of a coefficient A n belonging to that 

 pair of variables. 



Differentiating V with respect to each of the n variables in succession, we 

 get n new quantities ... of the form 



(2). 



Multiplying each by its corresponding x, and taking half the sum of the 

 products, we obtain a second expression for V, 



F=iT(av) ............ ...................... (3). 



rl 



Since each of the n quantities is a linear function of the n variables x, 

 we may, by solving the n equations of the form (2), obtain expressions for 

 each x in terms of the fs 



r=n 



x,= 2 (a^fc) .................................... (4); 



T"l 



VOL. it. 42 



