Mechanical Theory of Heat to the Steam Engine. 199 



If on the other hand, we make the assumption, that the ex- 

 pansion is driven until the steam by the expansion has cooled 

 from the temperature of the boiler to that of the condenser, 

 which, it is true, it is not completely possible to do, but which 

 still forms the limiting case to which we must approximate as 

 closely as possible, we need only put T 2 = T 0 whereby we obtain 



(24) W'= i [m 1 r \ "* m 0 r & + Mc {T % «, T 0 )] . 



If we also eliminate from this m 0 r 0 by means of the before- 

 cited equation, in which also we must put T 2 = T Q1 we have 



(xi) W' = ^[ mi r, Sl^f + MciT, - T 0 + T 0 



22. If we write the foregoing equation in the following form, 

 (25) W'= », r x + Mc {T, -T 0 ). i(l +^ff- % ^) , 



the two products which occur herein, Mc(T x -T 0 ) and m, r„ 

 represent together the quantity of heat given out by the source 

 of heat, during a circular process. The first is namely the quan- 

 tity of heat which is necessary in order to heat the mass if which 

 comes from the condenser in the fluid state, with the temperature 

 T 0 up to T x , and the last represents the quantity of heat which 

 is required to convert the portion ra, at the temperature T x into 

 steam. As m , is little smaller than M, the last quantity of heat 

 is far greater than the first. 



We will bring the factor belonging to Mc(T x -T 0 ) into a 

 somewhat different form, in order to be able to compare with 

 each other more conveniently the two factors, with which these 

 two quantities of heat are multiplied in equation (25). If then, 

 for the sake of abbreviation, we introduce the letter z with the 

 signification 



T - T 



(26) * = 



* The foregoing equations, which represent the work under the two simplifying 

 suppositions mentioned at the conclusion of §19, had been developed by me a long 

 time since, and publicly brought forward in my lectures at the University of Berlin 

 in the summer of 1854. When later in the year 1855 the Philos. Trans, of the Roy. 

 Soc. of London for the year 1854 appeared, I found in them a memoir of Rankine 

 " On the geometrical representation of the expansive action of heat and the theory 

 of thermo-dynamic engines," and was astonished to find that Rankine had arrived 

 at the same time, quite independently and by a different process, at equations which 

 not only in their essential contents, but also in their form, corresponded almost 

 completely with mine, only that Rankine had not considered the circumstance that 

 a quantity of liquid is mixed with the steam at its entrance into the cylinder. By 

 the earlier publication of this paper the priority was lost for this part of my inves- 

 tigation, nevertheless the correspondence was in so far a gratification to me, as it 

 gave me a guarantee that the mode of considering the subject employed was really 

 a natural one. 



