478 Dynamical Theory of Currents [CH. xvi 



accessible parts of the mechanism. Then throughout any imaginary motion 

 of the accessible parts of the system we shall have a knowledge of T and W 

 at every instant, and hence shall be able to calculate the value of 



T- W)dt ....(497). 



o 



We can imagine an infinite number of motions which bring the system 

 from one configuration A at time t = to a second configuration B at time t = r, 

 and we can calculate the value of the integral for each. Equation (496) shews 

 that those motions for which the value of the integral is stationary would be 

 the motions actually possible for the system. Having found which these 

 motions were, we should have a knowledge of the changes in the accessible 

 parts of the system, although the concealed parts remained unknown to us, 

 both as regards their nature and their motion. 



547. Equation (496) has been proved to be true only for a system con- 

 sisting of discrete material particles. At the same time the equation itself 

 contains, in its form, no reference to the existence of discrete particles. It 

 is at least possible that the equation may be the expression of a general 

 dynamical principle which is true for all systems, whether they consist of 

 discrete particles or not. We cannot of course know whether or not this 

 is so. W T hat we have to do in the present chapter is to examine whether 

 the phenomena of electric currents are in accordance with this equation. 

 We shall find that they are, but we shall of course have no right to deduce 

 from this fact that the ultimate mechanism of electric currents is to be found 

 in the motion of discrete particles. Before setting to work on this problem, 

 however, we shall express equation (496) in a different form. 



Lagranges Equations. 



548. Let 15 2 , ... 6 n be a set of quantities associated with a mechanical 

 system such that when their value is known, the configuration of the system 

 is fully determined. Then 0,, 2 , ... 6 n are known as the generalised coordi- 

 nates of the system. 



The velocity of any moving particle of the system will depend on the values 

 of -^ , ~^~ , etc. Let us denote these quantities by 6 lt 2 , etc. Let x be a 



Cartesian coordinate of any moving particle. Then by hypothesis x is a 

 function of B lt # 2 > ., say 







so that by differentiation, 



