210 Prof. Clausius on the Internal Work 



would algebraically have to be stated as higher. Instead of 

 equation (2), we must therefore write 



f iZ>-^- (2a) 



Applying this to equation (1), we obtain, instead of equation (5), 



dQ + dK 



T 



+ dZ>0. .... (5a) 



The further question now arises, what influence would it have 

 on the formulae, if a direct passage of heat took place between 

 parts of different temperature within the body in question. 



In case the body is not of uniform temperature throughout, 

 the differential expression occurring in equation (5a) must not 

 be referred to the entire body, but only to a portion whose tem- 

 perature may be considered as the same throughout ; so that if the 

 temperature of the body varies continuously, the number of parts 

 must be assumed as infinite. In integrating, the expressions which 

 apply to the separate parts may be united again to a single ex- 

 pression for the whole body, by extending the integral, not only 

 to the alterations of one part, but to the alterations of all the parts. 

 In forming this integral, we must now have regard to the passage 

 of heat taking place between the different parts. 



It must here be remarked that dQ, is an element of the heat 

 which the body under consideration gives up to, or absorbs from, 

 an external body which serves only as a reservoir of heat, and 

 that this element does not come into question now that we are 

 discussing the passage of heat between the different parts of the 

 body itself. This transfer of heat is mathematically expressed 

 by a decrease in the quantity of heat H in one part, and an 

 equivalent increase in another part ; and accordingly we require 



to direct our attention only to the term -~r in the differential 



expression (5a). If we now suppose that the infinitely small 

 quantity of heat dH leaves one part of the body whose tempera- 

 ture is T x , and passes into another part whose temperature is T 2 , 

 there result the two following infinitely small terms, 



dR 1 dK 



which must be contained in the integral ; and since T 1 must be 

 greater than T 2 , it follows that the positive term must in any 

 case be greater than the negative term, and that consequently 

 the algebraic sum of both is positive. The same thing applies 

 equally to every other element of heat transferred from one part 

 to another ; and the alteration which the integral of the whole 



