288 Royal Society. 



Jan. 19.— Charles Wheatstone, Esq., V.P., in the Chair. 

 A paper was read, entitled " On the Geometrical Repiesentation 

 of the Expansive Action of Heat, and the Theory of Thermo-dynamic 

 Engines." By W. J. Macquorn Rankine, F.R.S.S.L. & E. &c. 



The author remarks, that if abscissa; be measured from an origin 

 of rectangular coordinates, representing the volumes assumed by an 

 elastic substance, and if ordinutes, at right angles to those abscissae, 

 be taken to denote the corresponding expansive pressures exerted by 

 the substance, then any succession of changes of pressure and volume 

 may be represented geometrically by the coordinates of a curve. If 

 such a curve have two extremities, the area included between the curve 

 and the ordiiiates let fall from its extremities will represent (when 

 positive) the expansive power given out by the substance during the 

 process represented by the curve. Should the curve be closed, return- 

 ing into itself, so as to denote a cycle of periodical changes of pressure 

 and volume, then M'ill the area, enclosed within the curve, represent 

 (when positive) the expansive power given out during one cycle of 

 changes. This area is positive when increase of volume takes place 

 on the whole at greater pressures than diminution of volume. The 

 area of such a closed curve represents also (when positive) the me- 

 chanical equivalent of the heat which permanently disappears, or is 

 converted into expansive power, during a cycle of changes, for were 

 it not so, the sum of energy in the universe would be changed, 

 which is impossible. 



As the principles of the expansive action of heat are capable of 

 being presented to the mind more clearly by the aid of diagrams of 

 energy than by means of v.'ords and symbols alone, such diagrams 

 are applied, in the present paper, partly to the illustration and de- 

 monstration of propositions previously proved by other means, but 

 chiefly to the solution of new questions, especially those relating to 

 the theory of thermo-dynamic engines. 



Throughout the whole of this paper, quantities of heat are ex- 

 pressed, not by units of temperature in an unit of weight of water, 

 but by equivalent quantities of mechanical power, stated in foot- 

 pounds according to the ratio established by Mr. Joule's experiments 

 on friction (Phil. Trans. 1850), that is to say, 772 foot-pounds per 

 degree of Fahrenheit, or 1389-6 foot-pounds per Centigrade degree, 

 applied to one pound of liquid water at atmospheric temperatures. 



A curve described on a diagram of energy, such that its ordinates 

 represent the pressures of a homogeneous substance corresponding 

 to various volumes of an unit of weight, while the total sensible or 

 actual heat (Q) present in an unit of weight of the substance, is 

 maintained at a constant value (Q,), may be called the Isothermal 

 Curve of Q, for the given substance. Its equation is 

 Q=Q,. 

 If an unit of weight of a substance be allowed to expand, under a 

 pressure equal to its own elasticity, without receiving or emitting 

 heat, its actual heat will diminish during the expansion, and its 

 pressure will diminish more rapidly than it would do if the actual 

 heat were maintained constant. A curve whose coordinates repre- 

 sent this mode of variation of pressure and volume may be called a 



