November 20, 1903.] 



SCIENCE. 



653 



Clausius's derivation of the numerical 

 measure of entropy is based upon the idea 

 that degeneration and regeneration are 

 balanced in a reversible process. In this 

 derivation it is necessary to consider a 

 cyclic process (reversible) in order that no 

 outstanding change of state may be left as 

 a result of the process, so that one need 

 consider only the exchange of work and 



heat between the substance which is being 

 studied and external substances, and in 

 particular one is not under the necessity 

 of considering the amount of degeneration 

 or regeneration involved in the change of 

 state of a particular substance. 



Consider a fluid, which, starting from 

 the state P, Fig. 1, is made to pass through 

 a reversible cyclic process and return to 

 the state P by any combination whatever 

 of slow heating and cooling, expanding and 

 compressing. The closed curve represents 

 the cyclic process, and the moving point Q 

 as it moves along the process curve repre- 

 sents the changing state of the fluid. 



A clear idea of the external actions 

 which take place may be obtained by draw- 

 ing a series of adiabatic or isentropic lines 

 (an isentropic line represents the variation 

 of pressure with volume when the fluid 

 neither gives off nor receives heat). The 



fluid is receiving heat while Q is crossing 

 isentropic lines from small numbers to large 

 munbers in Fig. 1, and giving off heat 

 while Q is crossing isentropic lines from 

 large numbers to small numbers. The 

 fluid is expanding and doing external work 

 Avhen Q is moving to the right, and con- 

 tracting and having work done upon it 

 when Q is moving to the left. The fluid 

 is in general at high temperature for those 

 positions of Q where p and v are both large, 

 and at low temperatures for those positions 

 of Q where p and v are both small. 



Consider two portions, a and b, of the 

 given process curve which lie between 

 a pair of isentropic lines. Let T^ be 

 the high temperature of the fluid when 

 Q is passing along a, and let dff, be the 

 amount of heat taken in by the fluid while 

 Q is passing along a. Let T„ be the low 

 temperature of the fluid when Q is pa.ssing 

 along b, and let diZa be the amount of heat 

 given off by the fluid while Q is passing 

 along b. 



Consider the reversible cyclic process 

 which is represented by the two portions 

 a and b of the given process curve, together 

 with the isentropic lines between which a 

 and b lie. We will call this cyclic process 

 an elementary cyclic process to distinguish 

 it from the given process. The net result 

 of the elementary cyclic process would be 

 the taking in of the quantity dH^ of heat 

 at Ti, the conversion of a definite fraction 

 dW of this heat into work and the giving 

 off of the remainder, dn„, of the heat at 

 temperature 1\. Therefore, according to 

 Arts. 9 and 10 we have 



dH, 



(i) 



Now, d/f, is heat received by the fluid, 

 and dn.. is heat given off by the fluid, and 

 one or the other should be considered as 

 negative, say dH^, then equation (i) should 

 be written : 



