j 
. 

DYNAMICAL THEORY OF HEAT. 269 
A, and B,. It is worthy of remark that these propositions are rigorously true, 
being demonstrable consequences of the fundamental principle of the dynamical 
theory of heat, which have been discovered by JouLs, and illustrated and verified 
most copiously in his experimental researches. 
19. Both the fundamental propositions may be applied in a perfectly rigorous 
manner to the second of the known methods of producing mechanical effect from 
thermal agency. This application of the first of the two fundamental propositions 
has already been published by RANKINE and Criaustus; and that of the second, 
as CLAusius shewed in his published paper, is simply CArNnot’s unmodified inves- 
tigation of the relation between the mechanical effect produced and the thermal 
circumstances from which it originates, in the case of an expansive engine work- 
ing 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 CarNnor’s axiom regarding the permanence of heat, which was given in my 
former paper,* required to make it express, not CARNOT’S axiom, but JOULE’s. 
20. Let us suppose a mass} of any substance, occupying a volume v, under a 
pressure p uniform in all directions, and at a temperature ¢, to expand in volume 
to v + dv, and to rise in temperature tot + dt. The quantity of work which it 
will produce will be 
par; 
and the quantity of heat which must be added to it to make its temperature rise 
during the expansion to ¢ + dt may be denoted by 
Mdv+N dt. 
The mechanical equivalent of this is : 
J (Mdv+N dd), 
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 
(p-IM) dv—JINdte. 
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, 
Jie -3M) ae-IN ag; 
where we must suppose ¢ 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 per- 
formed between limits corresponding to the initial and final volumes. Now if, at 
any subsequent time, the volume and temperature of the medium become what 
* « Account of Carnor’s Theory,” foot-note on § 26. 
+ This may have parts consisting of different substances, or of the same substance in different 
states, provided the temperature of all be the same. See below Part III., §§ 53-56. 
VOL. XX. PART II. AD 
