POTENTIAL ENERGY OF A DISPLACEMENT. 7 1 



The expression for the potential assumes the simple 

 form 



va + ...+ 



If the function Vbe proportional to the product of 

 the pitch of the displacement screw and the square 

 of the amplitude, then every displacement screw will 

 coincide with the screw about which the wrench is 

 evoked. 



73. Form of the Potential. The n principal screws of 

 the potential form a unique group, inasmuch as they 

 are co-reciprocal, as well as being conjugate screws of 

 the potential. They therefore fulfil 15 -f 15 = 30 conditions, 

 being the total number available in the selection of six 

 screws. 



We are now going to show that the expression of 

 the potential will consist of the sum of n square terms, 

 whenever it is referred to any set of n conjugate screws 

 of the potential. 



The energy consumed in giving a body a twist of 

 amplitude & from the position of equilibrum O to a new 

 position P, is equal to FvfW* (72), and TJ is the screw on 

 which the wrench is evoked. Suppose that now from the 

 position P the body receive a twist of amplitude $' about 

 a screw 0, it would generally not be correct to assert 

 that the energy consumed in the second twist was pro- 

 portional to the square of its amplitude. For, during 

 the second twist, either a portion of the energy will 

 be consumed in doing work against the wrench on TJ, or 

 the energy expended in the second twist will be ren- 

 dered less, in consequence of the assistance afforded by the 

 wrench on /. If, however, TJ be reciprocal to 0, then the 

 quantity of energy consumed in the twist about will be 

 unaffected by the presence of a wrench on ?j. Hence 

 if 9 and be two conjugate screws of the potential, 



