190 V. OSCILLATIONS AND CYCLIC MOTIONS. 



The motion of a particle relatively at rest upon the surface of 

 the earth is isocyelic, taking account of the earth's rotation. 



In such a motion the cyclic momenta do not generally remain 

 constant, but forces have to be applied. 



In the example of the bead on the revolving rod if r varied 

 forces would have to be applied to the rod to keep the rotation <p' 

 constant. 



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

 the #'s, the positional coordinates. The positional forces , 187), are 

 then derivable from a force -function W T 1 ), 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 as variables 

 only the coordinates q sj 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 employed only 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 phenomena, 

 we have made a distinct advance in our knowledge of the subject. 

 A striking example is furnished by the kinetic theory of gases, by 

 means of which we are enabled to pass from the bare statement that 

 all gases press against their confining vessels to the statement that 

 this pressure is due to the impact of the molecules of the gas against 

 the walls of the vessel. 



52. Properties of Cyclic Systems. Reciprocal Relations. 



Since by the properties of the kinetic energy we have three different 

 kinds of quantities represented by partial derivatives of one or the 

 other of two functions, 



189) P. ||, *-| ' 



applying the principle that a derivative by two variables is independent 

 of the order of the differentiations we obtain six reciprocal theorems. 

 We shall throughout suppose that there is no potential energy. 



1) The reason for the appearance of W with the positive sign is that as 

 explained in 37, P g denotes the external impressed forces, which in the case 

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



