94 M. R. Clausius on a modified Form of the second 



that the transformations which occur must exactly cancel each other, 

 so that their algebraical sum is zero. 



For were this not the case, then we might conceive all the 

 transformations divided into two parts, of which the first gives 

 the algebraical sum zero, and the second consists entirely of 

 transformations having the same sign. By means of a finite or 

 infinite number of simple circular processes, the transformations 

 of the first part must admit of being made in an opposite 

 manner, so that the transformations of the second part would 

 alone remain without any other change. Were these trans- 

 formations negative, i.e. from heat into work, and the trans- 

 mission of heat from a lower to a higher temperature, then of 

 the two the first could be replaced by transformations of the 

 latter kind, and ultimately transmissions of heat from a lower to 

 a higher temperature would alone remain, which would be com- 

 pensated by nothing, and therefore contrary to the above principle. 

 Further, were those transformations positive, it would only be 

 necessary to execute the operations in an inverse manner to 

 render them negative, and thus obtain the foregoing impossible 

 case again. Hence we conclude that the second part of the 

 transformations can have no existence. 



Consequently the equation 



I 



'^=0 (II) 



is the analytical expression of the second fundamental theorem 

 in the mechanical theory of heat. 



The application of this equation can be considerably extended 

 by giving to the magnitude t involved in it a somewhat differ- 

 ent signification. For this purpose, let us consider a circular 

 process consisting of a series of changes of condition made by a 

 Dody which ultimately returns to its original state, and for sim- 

 plicity, let us assume that all parts of the body have the same 

 temperature ; then in order that the process may be reversible, 

 the changing body when imparting or receiving heat can only 

 be placed in communication with such bodies as have the same 

 temperature as itself, for only in this case can the heat pass in 

 an opposite direction. Strictly speaking, this condition can 

 never be fulfilled if a motion of heat at all occurs ; but we may 

 assume it to be so nearly fulfilled, that the small differences of 

 temperature still existing may be neglected in the calculation. 

 In this case it is of course of no importance whether t, in the 

 equation (II), represents the temperature of the reservoir of 

 heat just employed, or the momentaiy temperature of the 

 changing body, inasmuch as both are equal. The latter signi- 

 ^cation being once adopted, however, it is easy to see that any 



