68 70] CYCLIC SYSTEMS. 135 



an increase in the momentum p r at the same rate. For 



dp, _ _&r_ _dp,. 



'~'-'- 



III a. If an increase in one of the cyclic momenta p r , the 

 positional coordinates being unchanged, causes an increase in the 

 impressed force P s necessary to be applied to one of the positional 

 coordinates q s (in order to prevent its changing), then an 

 adiabatic increase of the positional coordinate q s will cause the 

 cyclic velocity q r ' to increase at the same rate. For 



_ 



dp r dp r dq s dq s ' 



III b. If an increase in one of the cyclic velocities /, the 

 positional coordinates being unchanged, causes an increase in the 

 impressed force P s necessary to be applied to one of the positional 

 coordinates q 8 (in order to prevent its changing), then an isocyclis 

 increase of the positional coordinate q s will cause the cyclic 

 momentum p r to decrease at the same rate. For 



70. Work done by the cyclic and positional forces. 



I. In an isocyclic motion, the work done by the cyclic forces 

 is double the work done by the system against the positional 

 forces. In such motions the energy of the system accordingly 

 increases by one-half the work done by the cyclic forces, the other 

 half being given out against the positional forces. For if we use 

 the energy in the form 



we have in any change 



(1) ST= 



and in an isocyclic change, every 8^/ vanishing, 



(2) 8r=iS.5.'8p.. 

 But since 



(3) ^ = P., &P, - P.$t, and since ?.'=, 5/S* = Sq s , 



and the above expression for the gain of energy becomes 



(4) ST = ^ s 



