192 V. OSCILLATIONS AND CYCLIC MOTIONS. 



53. Work done by the Cyclic and Positional Forces. 



I. In an isocyclic motion, the work done l>y 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 



196) dT = \^l (qj dp s + p s dfr'), 



and in an isocyclic change, every dq s ' vanishing, 



197) ST- 2 *'** 

 But since 



198) ^ = P., dj>. = P s dt, and since qi = -~> ql dt = dq s , 

 and the above expression for the gain of energy becomes 



199) dT = 



But the work done by the cyclic forces is 



200) 8 A =? s P s dq s = 2dT. 



Therefore the last part of the theorem is proved. Again, in any 

 motion, 



sol) w 



and in an isocyclic motion, 



202) . 8T 



But since the work of the positional forces is 



203) 8 A = p. 8q , _ - dq. = - ST, 



the first part of the proposition is also proved. 



II. In an adiabatic motion, the cyclic velocities will in general 

 be changed. 



Then they change in such a way that the positional forces 

 caused by the change of cyclic velocities oppose the motion, that is, 

 do a positive amount of work. For since for any positional force 



