290 Royal Society. 



the product of the difference between the two quantities of actual 

 heat proper to the isothermal curves, by the difference between the 

 two thermo-dynamic functions proper to the curves of no transmis- 

 sion, being represented by an expression of this form, 



(Q-Q.)-(F,-FJ. 



While the area of a figure bounded above by the isothermal cur%'e 

 of Q|, and laterally by the indefinitely-extended curves of no trans- 

 mission corresponding to the thermo-dynamic functions F^^, F^, is 

 represented by Q (F — F ). 



The area of a diagram of energy of any figure is calculated by 

 conceiving it to be divided, by a network of isothermal curves and 

 curves of no transmission, into an indefinite number of stripes or 

 quadrilaterals, finding the area of each and adding them by summa- 

 tion or integration. By the aid of these principles various problems 

 are solved. 



In the third section the same principles are applied to determine 

 the efficiency of thermo-dynamic engines worked by the expansion 

 and contraction of permanent gases without and with the aid of 

 economisers or regenerators. The efficiency of a thermo-dynamic 

 engine is the proportion of the whole heat communicated to the 

 working substance which is converted into motive power. 



The maximwn theoretical efficiency of a thermo-dynamic engine 

 working between the limits of actual heat Q, and Q.,, whether with- 

 out a regenerator or with a perfect regenerator, is expressed by the 

 fraction Qj — Q, 



A theoretically perfect regenerator does not increase the maxi- 

 mum efficiency between given limits of actual heat, but merely 

 enables that efficiency to be attained with a smaller extent of expan- 

 sion, and consequently with a smaller engine. 



The fourth section treats of the relation between actual heat and 

 temperature, which must be known before the propositions of the 

 preceding sections can be applied to actual substances. Existing 

 experimental data are not yet adequate to the exact determination of 

 this relation ; but it is considered they are sufficient to show that a 

 relation deduced by the author from the Hypothesis of Molecular 

 Vortices (see Philosophical Magazine for December 1851, p. 510), is 

 at least near enough to the truth for all purposes connected with the 

 computation of the efficiency of thermo-dynamic engines. This re- 

 lation is expressed by the formula 



Q=fe(T-fTo). 

 where T is temperature, measured from the melting-point of ice ; 

 To, the height of the melting-point of ice above the point of total 

 privation of heat ; and fe, the mechanical value of the real specific 

 heat of the substance. According to computations made in 1852 

 by the author from experiments by Messrs. Thomson and Joule, 

 Tq=212\° Centigrade=490|^ Fahrenheit, a value which may be 

 considered sufficiently correct for practical purposes. 



The maximum theoretical efficiency of every conceivable thermo- 

 dynamic engine receiving heat at the temperature T,, and giving out 



