18 Prof. Thomson on the Dynamical Theory of Heat. 



perfectly rigorous manner to the second of the known methods 

 of producing mechanical effect from thermal agency. This ap- 

 plication of the fii-st of the two fundamental propositions has 

 ah-eady been published by Rankine and Clausius ; and that of 

 the second, as Clausius showed in his pubUshed paper, is simply 

 Carnot's unmodified investigation of the relation between the 

 mechanical effect produced and the thermal circumstances from 

 which it originates, in the case of an expansive engine working 

 within an infinitely small range of temperatures. The simplest 

 investigation of the consequences of the first proposition in this 

 application, which has occurred to me, is the following, being 

 merely the modification of an analytical expression of Carnot's 

 axiom regarding the permanence of heat, which was given in my 

 former paper*, requii-ed to make it express, not Carnot's axiom, 

 but Joule's. 



20. Let us suppose a massf of any substance, occupying a 

 volume V, under a pressure p uniform in all directions, and at a 

 temperature t, to expand in volume to i; + dv, and to rise in tem- 

 peratm-e to t + dt. The quantity of work which it will produce 

 will be 



vdvj 

 and the quantity of heat which must be added to it to make its 

 temperature rise during the expansion to t + dt may be denoted by 



Mdv + 'Ndt. 

 The mechanical equivalent of this is 



3{Mdv + 'Ndt), 

 if J denote the mechanical equivalent of a unit of heat. Hence 

 the mechanical measure of the total external effect produced 

 in the circumstances is 



(j)-JM)dv-J'Ndt. 

 The total external effect, after any finite amount of expansion, 

 accompanied by any, continuous change of temperature, has 

 taken place, will consequently be, in mechanical terms, 



/{{p-ni)dv-J'Ndt} ; 

 where we must suppose t to vary with v, so as to be the actual 

 temperature of the medium at each instant, and the integration 

 with reference to v must be performed between limits correspond- 

 ing to the initial and final volumes. Now if, at any subsequent 

 time, the volume and temperature of the medium become what 

 they were at the beginning, however arbitrarily they may have 

 been made to vary in the period, the total external effect must, 



* Account of Camot's Theorj', foot-note on § 26. 



t This may have parts consisting of different substances, or of the same 

 substance in different states, ))rovided the temperature of all be the same. 

 See below, part 3, § 53-56. 



