of a Mass of Matter. 89 



its value may approach to nought as nearly as we please. Hence 

 it may be seen that the case in which the transformations take 

 place reversibly is a limit which in reality is never quite reached, 

 but to which we can approach as nearly as we please. We may 

 therefore, in theoretical discussions, still speak of this case as one 

 which really exists ; indeed, as a limiting case it possesses special 

 theoretical importance. 



I will take this opportunity of mentioning another process in 

 which likewise this distinction is to be observed. In order for 

 one body to impart heat to another by conduction or radiation 

 (in the case of radiation, wherein mutual communication of heat 

 takes place, it is to be understood that we speak here of a body 

 which gives out more heat than it receives), the body which 

 parts with heat must be wanner than the body which takes up 

 heat ; and hence the passage of heat between two bodies of dif- 

 ferent temperature can take place in one direction only, and not 

 in the contrary direction. The only case in which the passage 

 of heat can occur equally in both directions is when it takes place 

 between bodies of equal temperature. Strictly speaking, how- 

 ever, the communication of heat from one body to another of the 

 same temperature is not possible; but since the difference of 

 temperature may be as small as we please, the case in which it 

 is equal to nothing, and the passage of heat accordingly rever- 

 sible, is a limiting case which may be regarded as theoretically 

 possible. 



§ 5. We will now deduce the mathematical expression for the 

 above theorem, treating in the first place the case in which the 

 change of condition undergone by the body under consideration 

 takes place reversibly. The result at which we shall arrive for 

 this case will easily admit of subsequent generalization, so as to 

 include also the cases in which an alteration occurs irreversibly. 



Let the body be supposed to undergo an infinitely small altera- 

 tion of condition, whereby the quantity of heat contained in it, 

 and also the arrangement of its constituent molecules, may be 

 altered. Let the quantity of heat contained in it be expressed 

 by H, and the alteration of this quantity by dH. Further, let 

 the work, both internal and external together, performed by the 

 heat in the change of arrangement be denoted by dh, a magni- 

 tude which may be either positive or negative according as the 

 active force of the heat overcomes the forces acting in the con- 

 trary direction, or is overcome by them. We obtain the heat 

 expended to produce this quantity of work by multiplying the 

 work by the heat-equivalent of a unit of work which we may 

 call A ; hence it is Adh. 



The sum ^H + Adh is the quantity of heat which the body 

 must receive from without, and must accordingly withdraw from 



