652 



SCIEkCE. 



[N. S. Vol. XVIII. No. 464. 



If the operation of the engine involves 

 sweeping processes of any kind then the 

 degeneration iiH. exceeds the regeneration 

 mW or 



or, using the value (H^ — W) for Fo, and 

 solving for W we have 



w< 



in + 



;^i' 



(3) 



in which m and n have the same values as 

 in equation (1). Comparing this with 

 equation (2)6 it follows that any irre- 

 versible engine working between given 

 temperatures T^ and T^ has less efficiency 

 than a reversible engine working between 

 the same temperatures. 



10. THERMODYNAMIC DEFINITION OF THE 

 KATIO OF TWO TEMPERATURES. 



According to article 9 the ratio, HJH^, 

 of the heat taken from the boiler to the 

 heat given to the condenser by any re- 

 versible engine working between the given 

 temperatures Tj and T, is invariable, and 

 it can be easily shown that this ratio ap- 

 proaches unity as T^ and T„ approach 

 equality. Therefore, the ratio of the two 

 temperatures TJT^ may be defined as the 

 ratio of the two heats ILJIL„. That is: 



(4) 



11 . ENTROPY. THERMODYNAMIC DEGENERA- 

 TION. 



The statement of the second law of 

 thermodynamics can scarcely be looked 

 upon as complete until a precise and com- 

 plete nimierical measure of thermodynamic 

 degeneration has been established. This 

 numerical measure of thermodynamic de- 

 generation is called entropy. The notion 

 of entropy may be completely developed 

 by consideration of steady sweeps. I will 

 give this development first and I will give 

 Clausius 's development afterwards in order 



to point out an error in Clausius 's discus- 

 sion. 



Referring to article 9 we may write the 

 expression for the regeneration mM in the 

 form /(Ti) -W inasmuch as m is a func- 

 tion of Ti only. 



The degeneration nB.^ may be written 



inasmuch as the degeneration associated 

 with the transfer, of the heat Hj from T^ 

 to T. may be thought of as (a) the regen- 

 eration of H^ from temperature T^ to work, 

 and (&) the degeneration of this resulting 

 work to heat at temperature !„ ; in which 

 case the regeneration (a) is fiT^) -H^ ^^^ 

 the degeneration (&) is /{T^) -H^. 



Therefore, equation (2) a may be written 



f{T,) ■W=lfiT,)—f{T,)}H,. 



Using equation (1) and equation (4) we 

 have 



fiT,) T, 



From which the function / is to be de- 

 termined. Differentiating with respect to 

 T, we have 



1 



f{T,) T./ l^' 



T, 



and, therefore, since T^ and T„ are inde- 

 pendent of each other we have 



That is to say, the thermodynamic de- 

 generation associated with the conversion 

 of an amount of work W into heat at tem- 

 perature Tj is equal to W/T^, and the 

 thermodynamic degeneration associated 

 with the transfer of an amount of heat 

 II2 from temperature T^ to temperature 

 T„ is 



T-2 Ti' 



