418 A Theorem on the Dissipation of Energy. 



by the condition that the kinetic energy of the system moving 

 with these velocities q' 1 ...q' r ... q' n , and no others, is mini- 

 mum, given q\ . . . cf r . That is, if p\ . . .p' n be the components 

 of momentum corresponding to q\ . . . q' n , %p'q' is minimum, 

 given q\ . . . q' r . The solution of which i= 

 // r+1 =0.../„=0. 

 But p\ . . ,p' n being the components of momentum produced 

 impulsively by the force Felt, each component of momentum 

 is equal to the corresponding component of force. That is, 

 for each coordinate 



But, as we have seen, 



therefor< 



i? ~ dq 



1 dq 

 Therefore, since |/,. +1 =0 &c.,p' n = Q, 



dq r+i '"dq n 

 And therefore H is minimum, given q x . . . q r . 



As an example of the theorem we may consider the distri- 

 bution of an electric current in multiple arc. Let a set of 

 wires, OC^P, OC 2 P, . . . OCJ>, be joined at and P, and a 

 constant current be maintained outside the junction at 0. 

 Let g-i . . . q n denote the currents in the several wires, p x . . ,p n 

 the corresponding components of electrokinetic momentum. 

 Then the current at is Sq, and is to be maintained constant 

 notwithstanding the resistance of the wires. 



Let q\ . . . q' n be the diminutions of the several currents 

 which would be caused by resistance in the infinitely small 

 time dt. Then "Zq* is the diminution of the current at due 

 to resistance in time dt. In order to maintain ^q constant, 

 we may suppose an impulsive electromotive force applied at 

 at the end of the interval dt to produce the current %q'. 



The distribution among the wires of this current Hq', at the 

 instant of its creation, for however short a time such distribu- 

 tion may continue, is determined by the laws of induction, 

 and will, by Thomson's theorem, be such as to make the elec- 

 trokinetic energy of the impulsive currents minimum, gixen'Zq'. 



corresponding to q ', ^LpJef is minimum, given 2^'. That i 



p'i=p'2 = .-"=p': 

 But if E be the component of electromotive force correspond- 





