94 EEPOET 1891. 



«i, s^, .... he tlie corresponding generalised momenta. Let p^ be taken 

 as a type of the controllable coordinates which define the configuration of 

 the external system, ^j^ as a type of the uncontrollable coordinates which 

 define the positions of the molecules in the body. Since the energy of the 

 external system is assumed to be wholly potential, 



.: s„=^ =0 (16) 



O'la 



With the present notation the two general forms of the equation 

 expressing Hamilton's principle are 



Sr(T-U)tZ^=r2s8pT-Q8t . . . (17a) 



and 



sr2T(^^=r^s8p r+*sQ . . . (in) 



Of these the latter form must be used. Assume i to be so chosen as to 

 satisfy the relation 



which, since s„=0, may also be written 



[SVpJ=0; .... (18) 



a relation identical with that assumed in equation (13) and justifiable in 

 a similar manner. 



Equation (17b} now becomes, on introducing mean values, 



8(2iT)=t-8Q, 



giving, as before, equation (14), 



8Q _ 



-^=S2log(iT). 



It might at first sight appear as if the assumption as to the conserva- 

 tive nature of the external forces imposed a serious limitation on the 

 generality of the theorem, and, in fact, prevented its application to cyclical 

 processes. But this is really not the case. To remove the limitation it is 

 only necessary to suppose that the external system contains certain connec- 

 tions by which periodic motion of the body is converted into progressive 

 motion of some of the external coordinates, as exemplified in the crank 

 of a steam-engine. In other words, the external energy W must be a 

 multiple valued function of the controllable coordinates of the body. 

 From equation (18), i depends only on_the state of the bodv, not on that 

 of the external system, and evidently T depends only on the state of the 

 body. Hence, if the initial and final states of the body be the same, 

 although the initial and final states of the external system may be different, 

 we must have 



f|=0 ..... (19) 



