16 Mr. W. J. M. Rankine on the Mechanical Action of Heat. 



quantity corresponding to the mechanical power (that is to say, 

 the heat), represented by the product of that distance by that 

 force, or by 



- 87rR2 J^ Q^Du^lr{u, D, r)du x S(Rw) 

 which, because -^ = — ^ . -^, and — 5 — =M, is equal to 



+ 



QM.^^^{u,J),ry{^ 3^-^)du. 



"We must suppose that the velocity of oscillation is equalized 

 throughout the atomic atmosphere, by a propagation of motion 

 so rapid as to be practically instantaneous. 



Then if the above expression be integrated with respect to du, 

 from M = to u = l, the result will give the whole increase of heat 

 in the atom arising from the condensation 8D ; and dividing that 

 integral by the atomic weight M, we shall obtain the correspond- 

 ing development of heat in unity of weight. This is expressed 

 by the following equation : — 



-SQ'=2Q ^J^^f^'du . «=t(«, D, t) 



-^f'du.uBu^|r{u,'D,T)\ .... (2) 



The letter Q' is here introduced to denote, when positive, that 

 heat which is consumed in producing changes of volume and of 

 molecular arrangement, and when negative, as in the above equa- 

 tion, the heat which is produced by such changes. 



The following substitutions have to be made in equation (1) 

 of this Section. 



For Q is to be substituted its value, according to equation XII. 

 of the Introduction ; or abbreviating Cw/ii into k, — 



«=2c|,(^-«)- •••■•. •'' 

 The value of the first integral in equation (2) of this Section is 



/I 1 



du . u"'\jr{u, D, t) = g. 



The value of the second integral. 



■3 / du .uSw^iiijjy^T), 

 ^ 



remains to be investigated. The first step in this inquiry is given 

 by the condition, that whatsoever changes of magnitude a given 



