29.3 Prof. J. J. Mullet on a Mechanical Principle 



Of the applications of the proposition of the action, those shall 

 be introduced here which can be made of it in the mechanical 

 theory of heat. If heat be conceived as molecular motion, the 

 application to it of the Energy proposition leads immediately to 

 the first main proposition of this doctrine. Corresponding to 

 this, we are now investigating what, on the same hypothesis, 

 results from the Action theorem. These molecular actions are 

 stationary motions of a system of points; and the simplest case 

 of such motions is obviously that in which all the points move 

 in closed paths, and with a period common to all of them. This 

 shall first be supposed. 



As, for closed paths, the two limits of the integral which 

 forms the action coincide, when the integration is extended over 

 an entire revolution we obtain 



£?< dt *1q* dt~~ dt La* J ' 



and hence the equation of the Action proposition is transformed 

 into 



_<£ ran 



dt + Ld/J ' 



or, written explicitly, 



If now, for one revolution, we name the mean value of the vis 

 viva T, and that of the force-function U, we obtain 



\t-TJ)dt = t(T-V), 



-H 



- dV = tdT - tdV + r £dt - Vdt, 



2 =r — dc k =\ 2 ^ — dc /c \dt=tl i ^r— dc k , 

 ^ c k Jo L %c k J dc* 



E=T+U; 



and if we insert this value in the above equation, we obtain 



tdT-tdV + ZTdt + t2,^-dc==0, 



from which 



dV-%^dc k =dT + 2Tdhgt, . . . (14) 



a well-known equation, already advanced by Clausius* for such 

 motions. 



If now we apply this or related equations to the molecular 

 * Pogg. Ann. vol. cxlii. p. 433. Phil. Mag. S. 4. vol. xlii. p. 161. 



