208 | MR W. J. M. RANKINE ON THE ECONOMY OF 
That is to say: when no heat is employed in producing variations of tempera- 
ture, the ratio of the heat received to the heat emitted by the working body of an 
expansive machine, is equal to that of the absolute temperatures of reception and 
emission, each diminished by the constant x, which is the same for all substances. 
Hence let 
m= —Q,—Q,=H,—-H, 
denote the maximum amount of power which can be obtained out of the total 
heat H,, in an expansive machine working between the temperatures 7, and 7,. 
Then 

Se ay aOR al AE SY oy SE Ae REY 
being the law which has been enunciated in Article 42, and which is deduced 
entirely from the principles already laid down in the Introduction and First 
Section of this paper. 
The value of the constant x is unknown; and the nearest approximation to 
accuracy which we can at present make, is to neglect it in calculation, as being 
very small as compared with +. 
(44.) This approximation haying been adopted, I believe it will be found that 
the formula (66.), although very different in appearance from that arrived at by 
Professor Tomson, gives nearly the same numerical results. For example: let the 
machine work between the temperatures 140° and 30° centigrade : then 7,=414°6, 
7,=304"6, and 
tl 
i =0:2653 
Professor THomson has informed me, that for the same temperatures he finds 
this ratio to be 0:2713.* 
(45.) To make a steam-engine work according to the conditions of maximum 
effect here laid down, the steam must enter the cylinder from the boiler without 
diminishing in pressure, and must be worked expansively down to the pressure 
and temperature of condensation. It must then be so far liquefied by conduc- 
tion alone, that on the liquefaction being completed by compression, it may be 
restored to the temperature of the boiler by means of that compression alone. 
These conditions are unattainable in steam-engines as at present constructed, and 
different from those which form the basis of the formulze and tables in the Fourth 
Section of this paper; hence it is found, both by experiment and by calculation 
* Prom information which I have received from Professor Tomson subsequently to the com- 
pletion of this paper, it appears that his formula becomes identical with the approximate formula here 
proposed, on making the function called by him w= ap J being Jouxe’s equivalent. 
T 
Mr Jowxe also, some time since, arrived at this approximate formula in the particular case of 
a perfect gas. ; 

