CHAPTER VIII. 



THE POTENTIAL. 



98. The Potential. 



Suppose a rigid body which possesses freedom of the ?tth order be sub 

 mitted to a system of forces. Let the symbol define a position of the 

 body from which the forces would be unable to disturb it. By a twist of 

 amplitude & about a screw belonging to the screw system, the body may be 

 displaced from to an adjacent position P, the energy consumed in making 

 the twist being denoted by the Potential V, and no kinetic energy being 

 supposed to be acquired. The same energy would be required, whatever be 

 the route by which the movement is made from to P. So far as we are 

 at present concerned V varies only with the changes of the position of P 

 with respect to 0. The most natural co-ordinates by which the position 

 P can be specified with respect to are the co-ordinates of the twist ( 32) 

 by which the movement from to P could be effected. In general these 

 co-ordinates will be six in number; but if n of the screws of reference be 

 selected from the screw system defining the freedom of the body, then ( 95) 

 there will be only n co-ordinates required, and these may be denoted by 

 0i, ... &n. 



The Potential V must therefore depend only upon certain quantities 

 independent of the position and upon the n co-ordinates #/, . . . n ; and since 

 these are small, it will be assumed that V must be capable of development 

 in a series of ascending powers and products of the co-ordinates, whence we 

 may write 



+ terms of the second and higher orders, 



where H, H 1} ... H n are constants, in so far as different displacements are 

 concerned. 



In the first place, it is manifest that H = ; because if no displacement 

 be made, no energy is consumed. In the second place, H l , ... H n must also 

 be each zero, because the position is one of equilibrium ; and therefore, 



