of a Mass of Matter. 211 



differential expression occurring in (5a) undergoes, on account 

 of this transfer of heat, can therefore only consist in the addition 

 of a positive quantity to the value which would else have been 

 obtained. But since, as results from equation (5 a), the first 

 value which would be obtained, without taking this direct trans- 

 fer of heat into consideration, cannot be less than nothing, this 

 can still less be the case when it has been increased by another 

 positive quantity. 



We may therefore write as a general expression, including all 

 the circumstances which occur in non-reversible alterations, the 

 following, instead of equation (II.) : — 



f^+*5 + Cra>o. .... (ii«) 



The theorem which in § 1 was enunciated in reference to 

 circular processes only, and was represented by the expression 

 (la), has thus assumed a more general form, and may be enun- 

 ciated thus : — 



The algebraic sum of all the transformations occurring in any 

 alteration of condition whatever can only be 'positive, or, as an 

 extreme case, equal to nothing. 



In my previous paper I have spoken of two transformations 

 with opposite signs, which neutralize each other in the algebraic 

 sum, as compensating transformations. The foregoing theorem 

 may therefore be enunciated still more briefly as follows :— 



Uncompensated transformations can only be positive. 



§ 12. In conclusion, we will submit the integral 



rdB 



J T 



which has been frequently used above, to a somewhat closer con- 

 sideration. We will call this integral, when it is taken from any 

 given initial condition to the condition actually existing, the equi- 

 valent value of the heat in the body (Korperwarme) calculated from 

 the given initial condition. That is, when in any way whatever 

 work is transformed into heaf, or heat into work, and the quan- 

 tity of heat present in the body thereby altered, the increment or 

 decrement of this integral gives the equivalent value of the trans- 

 formations which have taken place. Further, if transfers of heat 

 take place between parts of different temperature within the body 

 itself, or within a system of bodies, the equivalent value of these 

 transfers of heat is likewise expressed by the increment or decre- 

 ment of this integral, if it is extended to the whole system of 

 bodies under consideration. 



In order to be able actually to perform the integration which 

 has been indicated, we must know the relation between the quan- 



