4 REPORT 1851. 



efficient having a specific value for each substance, depending on its chemical con- 

 stitution. — is the real specific heat. Let this be denoted by fe. 

 2 k 

 The expansive pressure of any substance is represented as follovys : — Let Vobe the 

 volume which unity of weight of the substance, if in the state of perfect gas, would 

 occupy at the absolute temperature t^ under the pressure unity ; r the actual abso- 

 lute temperature; V the actual volume of unity of weight; Aj, Aj, &c., and/ (V) 

 certain functions of the density ; then 



pressure=P=X2 . l{ i_^_^._^_&, | ^^^y,^ 



This formula represents perfectly the results of M. Regnault's experiments on the 

 pressure and expansion of gases (Trans. Roy. Soc. Edin. vol. xx.). 



The maximum pressure of a vapour in contact with its liquid is expressed by the 

 formula 



log P=«----,. 



», /3, '/ having specific values for each fluid (see Edin. New Philosophical Journal, 

 July 1849). 



If unity of weight of a substance be made to undergo the change of temperature 

 dr and the change of volume dY, the following is the quantity of heat which it 

 must receive, — 



+<iV.(,-.)g, 



of which ferf T alone remains in the body as heat ; the rest being transformed into 

 expansive power and molecular action. From this equation are deduced the known 

 laws of the apparent specific heat of gases, and the velocity of sound. 



d . Q— PdV is an exact difi'erential. This is the algebraical statement of the law 

 proved experimentally by Mr. Joule, of the equivalence of heat and mechanical 

 power. 



The total heat of evaporation of a liquid increases uniformly with temperature, and 

 the rate of its increase is equal to the apparent specific heat of the vapour under 

 constant pressure. 



The apparent specific heat of a vapour at saturation is negative ; that is to say, if 

 vapour at saturation be allowed to expand, it must receive heat from without, or a 

 portion will be liquified to supply the heat required to expand the rest. This result 

 has also been arrived at by Clausius. 



If a body be expanded by heat at a higher temperature r„ and condensed at a 

 lower Tq, a certain proportion of the heat employed in expanding it will be trans- 

 formed into expansive power ; and this proportion is a function of those two tem- 

 peratures alone, being represented by 



and is independent of the nature of the substance. This principle is known as 

 Carnot's Law ; and its consequences have also been investigated by Clausius and 

 Professor William Thomson. 



The results of these theoretical principles, as applied to the steam-engine, have 

 been developed in practical formula and tables, and compared with experiment, and 

 the agreement is in all cases most satisfactory (Trans. Roy. Soc. Edin. vol. xx.). 



On the Velocity of Soitnd in Liquid and Solid Bodies of Limited Dimen- 

 sions, especialhj along prismatic masses of liquid. By W. J.M. Rankine. 

 This paper is a sequel to one read at Edinburgh to the British Association in 



August 1850, and published in the Cambridge and Dublin Mathematical Journal for 



February 1851, on the Laws of Elasticitj"-. 



