330 THEORY OF A SYSTEM OF ELECTRIFIED CONDUCTORS. 



,! by ubMituting these valuee of the X'B in (3), we obtain a third expression 



v t -iT*M.(,) (5). 



f-l r I 



Tb 6t and last of these expressions for V are distinguished by the 

 MiftixoH JT and (, in order to shew that the first is expressed in terms of the 



.hies * and the last in terms of the variables The coefficients A and a 

 nmy be functions of any number, m, of variables. Let y t be one of 

 ramble*. 



Since F,, V, and V t are three different expressions for the same quant i 



r 2(x r f r ) (r,). 



T 1 



If we now suppose the three sets of variables ar, f and y to vary in any 

 consistent manner, and remember that V, is a function of the x's and y's, I 

 tin- fs and y's, and V of the x's and fs, we find 



The three sets of variations Sx, 8 and Sy are not independent of each 

 nt her, for if the variations Sy and either of the other sets be given, the other 

 set may be determined. Hence we cannot immediately deduce any definite 

 results from this equation. But we know, from equation (2), that the coeffici 

 of the variations 8x vanish of themselves, and the remaining variations Sf and S// 

 are all independent of each other, so that we may equate the coefficient of 

 tif them to zero. We thus obtain two sets of equations, 



_dV t 

 '~' 



and + = ................................... (9). 



In this purely algebraical theory of quadratic functions, of the two sets of 

 variables x and (, either may be taken as the primary set. In the physical 

 .i| -plications of the theory, however, the variables form two classes which are not 

 interchangeable. 



Thus, in kinetics, the variables are the components of velocity and tli<- 

 imimentum; in the theory of elasticity, they are stresses and strains; in eld 



