Criterion of Potential Energy. 699 



So long as this condition is satisfied, we may treat K as 

 the potential energy of the system. 



19. The condition (19), (19a) or (19 ft) necessarily implies 

 (14), which must in any case be fulfilled if the energy K is 

 to be treated as potential ; in general, however, it includes 

 more than is strictly demanded. But in most of the examples 

 which readily present themselves, and in which K behaves 

 as potential energy, the condition in question is in fact 

 satisfied. If (19 b) is to hold good without any restriction 

 being imposed on tlic velocities yfr, <f>, . . ., we shall have 



20M = 0, 2CN=0 (20) 



20. The C's being constants, it is evident that (14) will be 

 satisfied provided 



SCM, SON, . . . are all independent of yfr, <j>; . (21) 



which relations, though in general expressing more than the 

 requisite conditions (14), impose less restriction than do (20). 



21. Special Class ii. A simple case, illustrating the con- 

 dition (20), is when the M's, N's, . . . all vanish, or in other 

 words when, for given values of the working coordinates 

 -v/r, <£, #, . . . the velocities X, X 7 , . . . of the ignored coor- 

 dinates are determined solely by the momenta 0, C . . ., 

 independently of the velocities i^, <£>, 0. . . . 



22. Special Class iii. Another simple case is when the state 

 of the system, as defined by the working coordinates i/r, <£, Q, . . . , 

 is one of continued rest. For when i/r, <p, 6, . . . are all 

 constantly zero, (14) are satisfied. In any system, therefore, 

 for which (9 a) hold good, the energy K — due to the momenta 

 of the ignored coordinates — may be treated as potential 

 energy in computing what forces, corresponding to the 

 working coordinates, must be applied to the system to 

 maintain it " at rest." If, however, the system is moved 

 from one configuration to another, even infinitely slowly, 

 although at each instant the generalized forces required to 

 maintain (infinitely nearly) equilibrium will only differ infi- 

 nitesimally from ^K/^, dK/d$, . . . yet the time-integrals 

 of the forces in question will in general be finitely different 

 from those of dK/3^r, BK/d</>, . . . unless (20), or failing 

 that (21), or in any case unless (14) are satisfied. 



23. Though the rotational energy of the governor repre- 

 sented in fig. 1 can no longer be treated as potential when 

 the frame B B is permitted to turn without restriction of 

 direction, it is easy to devise a pair of governors carried by a 

 single frame, and so connected or so set in motion that their 



