294 A Principle resulting from Hamilton's Theory of Motion. 

 Therefore, introducing the function V, 



dt Ubt\ u ' 

 and the equation of the action can be written 



But since 



dt oc k 





we have 



W=j 2Tdt=2tT, 



dw=mT+%fdt, 



2tdT + %Tdt + *S |£ dc k -td$ m ; 



and from this 

 or 



dE-2 l^<fc*=2rfT + 2Trflog/, 



dE-2^<fc fc =2T<nog(*T). .... (15) 



Now this equation, which has already been given by Szily* 

 for the special case in which no c k are present, the paths are 

 closed, and the periods are the same for all the points, leads im- 

 mediately to the second proposition of the mechanical theory of 

 heat. For that purpose let us consider, first, that the left-hand 

 side of it is nothing else but the energy which, with the change of 

 the molecular motion, is communicated to the body as heat from 

 without ; and therefore, in the usual notation of the theory of heat, 



it is ~ dQ. If we then make use of the assumption that T is 



proportional to the absolute temperature ®, we immediately 

 obtain 



§■=#, • (16) 



understanding by r/S a complete differential. 



Thus the Second Proposition is derived, like the First, from 

 a general mechanical principle. But the above representation 

 permits us to perceive for the two propositions not merely this 



* Pogg. Ann. vol. cxlv. p. 295. Phil. Mag. S. 4. vol. xliii. p. 339, 



