51 



motions arc all reversed, it produces as much mechanical effect as 

 can be produced by any thermo-dynamic engine with the same tem- 

 peratures of source and refrigerator, from a given quantity of heat. 



The second of these propositions was first enunciated by Car- 

 not, and demonstrated by him on the assumption of his principle of 

 the permanence of heat. It was first enunciated and demonstrated, 

 without making that assumption, upon the true principles of the 

 dynamical theory, by Clausius, in the second part of his paper* 

 (published in May 1850), who founds it on an axiom substantially 

 equivalent to that quoted above. The author of the present paper 

 gives the demonstration, which is closely analogous to Carnot's 

 original demonstration, and the axiom on which it is founded, just as 

 they occurred to him at a time when he was only acquainted with 

 the first part (published in April 1850) of Clausius"* paper, and was 

 not aware that the proposition had been either enunciated or de- 

 monstrated except by Carnot. 



From the establishment of the second proposition, on the princi- 

 ples of the dynamical theoiy, and an axiom that cannot pro- 

 bably be denied, it is shown that all the conclusions obtained by 

 Carnot and others who have followed him and adopted his princi- 

 ples, which depend merely on the fundamental equation expressing 

 "Carnot's function," in terms of certain physical properties of any 

 substance whatever, require no modification. 



But the Theory of the motive power of heat through finite ranges 

 of temperature requires most important alterations which form the 

 subject of the second part of the present paper. The following ex- 

 pressions are given for the amount of work (VV) derivable from a 

 unit of heat introduced into an engine at the temperature S, if 

 the coldest part of the engine is at the temperature T ; in terms 

 of the portion (1— R) of the unit of heat which is converted into 

 work, and for the remainder, (R,) which is emitted as waste into 

 the refrigerator. 



W^J (l-R); 



where J denotes the " mechanical equivalent" of a unit of heat 

 determined by Joule. 



* roggendorfF's Annalen, 1850. 



