Royal Society. 289 



Curve of no Transmission of Heat. For every such curve a certain 

 function of pressure, volume and actual heat, called a Thermo-dy- 

 namic Function (F), has a constant value (F^) proper to the parti- 

 cular curve under consideration ; vi'hose equation is therefore 



A curve whose coordinates represent the relation between pressure 

 and volume when the substance is absolutely destitute of heat, may 

 be called the Curve of Absolute Cold. It is at once an isothermal 

 curve and a curve of no transmission, and is an asymptote to all the 

 other curves of both those kinds, which approach it indefinitely as 

 the substance expands without limit. 



The whole theory of the expansive action of heat is comprehended 

 in the geometrical properties and mutual relations of those two 

 classes of curves ; and all those properties and relations are the con- 

 sequences of and are virtually expressed by the two following theo- 

 rems : — 



Theorem I. — The mechanical equivalent of the heat absorbed or 

 given out by a substance in passing from one given state as to pressure 

 and volume to another given state, throuyh a series of states repre- 

 sented by the coordinates of a given curve on a diagram of energy, 

 is represented by the area included between the given curve and two 

 curves of no transmission drawn from its extremities, and indefinitely 

 prolonged in the direction representing increase of volume. 



Theorem II. — If across any pair of curves of no transmission on a 

 diagram of energy there be drawn any series of isothermal curves at 

 intervals corresponding to equal differences of actual heat, the series 

 of quadrilateral areas thus ctit off from the space between the curves of 

 no transmission will be all equal to each other. 



These two propositions are the geometrical representation of the 

 application, to the particular case of heat and expansive power, of 

 two axioms respecting Energy in the abstract, viz. — 



Axiom I. — The sum of energy in the Universe is unalterable. 

 Axiom II. — The effect, in causing transformation of energy, of the 

 whole quantity of actual energy present in a substance, is the sum of 

 the effects of all its parts. 



The application of these axioms to heat and expansive power in- 

 volves the following 



Definition. — Expansive Heat is a species of actual Energy, the 

 presence of which in a substance affects, and in general increases, its 

 tendency to expand ; — 



and this definition, arrived at by induction from experience, is the 

 foundation of the theory of the expansive action of heat. 



The first section of the paper is occupied chiefly with the demon- 

 stration of the first of the theorems quoted and its immediate con- 

 sequences, which are applicable to all substances, homogeneous and 

 heterogeneous. 



The second section relates to the theory of the expansive action 

 of heat in homogeneous substances. 



From the second tlieorem above quoted, it is deduced, that the 

 area of any quadrihiteral bounded above and below by any two iso - 

 thermal curves, and laterally by two curves of no transmission, is 



