CHAPTER VII. 



THE POTENTIAL ENERGY OF A DISPLACEMENT. 



68. The Potential Energy of a Displacement. Suppose a 

 rigid body which possesses freedom of the n th order be 

 submitted to the influence of any system of forces in- 

 cluded within the restriction of 6. Let the symbol O 

 define a position of the body from which the forces would 

 be unable to disturb it. By a twist of amplitude 0' about 

 a screw belonging to the screw complex of the n th 

 order, which expresses the nature of the freedom, the 

 body may be displaced from O to an adjacent position P, 

 while the energy consumed in making the twist is denoted 

 by V. It appears from 7 that the same amount of energy 

 would be required, whatever be the route by which the 

 movement is made from O to P. It follows that Kcan 

 only depend on certain constants and on the position of 

 P with respect to O. The most natural co-ordinates by 

 which the position P can be specified with respect to O 

 are the co-ordinates of the twist ( 34) by which the 

 movement from O 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 complex 

 defining the freedom of the body, then ( 65) there will 

 be only n co-ordinates required, and these may be de- 

 noted by 0/, , 0/. 



The Potential V must therefore depend only upon 

 certain constants relating to the forces and upon the n 

 quantities 0/, . . . . , n ' ; and since these quantities are 

 small, it follows that V must be capable of development 

 in a series of ascending powers and products of the 

 co-ordinates, whence we may write 



