67, 68] CYCLIC SYSTEMS. 133 



and for the cyclic coordinates 



<A p-<L(W\-*fc 



^ s ~dt(dq s ')~ dt ' 



A motion in which there are no forces tending to change the 

 cyclic coordinates is called an adiabatic motion, since in it no 

 energy enters or leaves the system through the cyclic coordinates. 

 (It may do so through the positional coordinates.) Accordingly 

 in such a motion the cyclic momenta remain constant. The case 

 of the gyrostat worked out above was such a motion. 



In adiabatic motions the cyclic velocities do not generally 

 remain constant. In the above example, for instance, the cyclic 

 velocity <' was given by 



A motion in which the cyclic velocities remain constant is 

 called isocyclic. 



In such a motion the cyclic momenta do riot generally remain 

 constant, but forces have to be applied. 



If the motion is isocyclic, the only variables appearing in T are 

 the q's, the positional coordinates. The positional forces, (3), are 

 then derivable from a force-function W T*, so that even if the 

 system possessed no potential energy, it would appear to possess 

 an amount of potential energy T. If the motion on the other 

 hand is adiabatic, the energy in the form T p again contains only 

 the coordinates q s , and the positional forces are now derivable 

 from the force-function T p + W, so that in this case a system 

 without potential energy would appear to contain the amount of 

 potential energy +T P . In this manner we are enabled to explain 

 potential energy as kinetic energy of concealed cyclic motions, 

 thus adding materially to our conceptions of the nature of force. 

 For it is to be noted that kinetic energy is an entity depending 

 only on the property of inertia, which is possessed by all bodies, 

 while potential energy is a term only employed to disguise our 

 ignorance of the nature of force. Accordingly when we are able 

 to proceed to an explanation of a static force by means of kinetic 



* The reason for the appearance of W with the positive sign is that, as ex- 

 plained in 62, end, P, denotes the external impressed forces, which in the case of 

 equilibrium, are equal and opposite to the internal forces given by W. 



