53, 54] WORK DONE BY FORCES. 193 



P ar 



p *~ 3; 



the change due to the motion is 



Of this the part due to the change in the cyclic velocities is 

 205) tP. 



and the work done by these forces is 



206) S-, A ~. 9? P.Sq, - - 



Now we have for any motion 



207) 

 and in an adiabatic motion this is .zero., so that 



208) 



Substituting this in the double sum 206) r we get 

 209) SA 



But this expression represents [..36, 35)] twice the energy of a 

 possible motion in which the velocities would be dq t ', and must 

 therefore be positive for all values of dqj, dg r '. 



Accordingly d- A > 0. 



The interpretation of this theorem for electrodynamics is known 

 as Lenz's Law 1 ), namely, ato, electrical current being represented by 

 a cyclic velocity, and the shape and relative position of the . circuits 

 by positional coordinates, if in any system of conductors carrying 

 currents, the relative positions of the conductors are changed, the 

 induced currents due to the motion of the conductors are so directed 

 as by their magnetic action to oppose the motion. 



54. Examples of Cyclic Systems. Let us consider the example 

 of equation 184) as illustrating the previous theorems. 

 We have for the momenta 



dT dT 



1) These Theorems are all given by Hertz, Prinzipien der MechaniJc, 

 568 583. 

 WEBSTER, Dynamics. 13 



