POTENTIAL ENERGY OF A DISPLACEMENT. 65 



O, being 0/ + S0/, . . . . n ' + S0 n ' ( 65). The potential V 

 of the position P is 



it being understood that S0/, . . . , n ' are infinitely small 

 magnitudes of a higher order than 0/, 0n'. 



The work done in forcing the body to move from Pto 

 P' is V - V. This must be equal to the work done in 

 the twists about the screws of reference whose am- 

 plitudes are S0/, . . . . , S0 n ', by the wrenches on the 

 screws of reference whose intensities are i^", . . . . , Tj n x/ . 

 As the screws of reference are co-reciprocal, this work 

 will be equal to ( 35) 



Since the expression just written must be equal to 

 V - Vfor every position P in the immediate vicinity of 

 P, we must have the coefficients of S0/, . . . , S0 B ' equal in 

 the two expressions, whence we have n equations, of 

 which the first is 





Hence, we deduce the following useful theorem : 



If a free or constrained rigid body be displaced from 

 a position of equilibrium by twists of small amplitudes, 

 0J 7 , ---- , n 7 , about n co-reciprocal screws of reference, 

 and if V denote the work done in producing this move- 

 ment, then the reduced wrench has, for components on the 

 screws of reference, wrenches of which the intensities are 

 found by dividing twice the pitch of the corresponding 

 reference screw into the differential coefficient of V 



F 



