Prof. Thomson on the Dynamical Theory of Heat. 425 



were raised to t + dt, the volume remaining unchanged. Then 

 by equation (3) of § 21 of my former paper, derived from Clau- 

 sius's extension of Carnot^s theory, we have 



«=rl (»)*' 



where ^ denotes Carnot^s/M«c/?oM, the same for all substances 

 at the same temperatui'e. 



Now let the substance expand from any volume V to V, and, 

 being kept constantly at the temperature t, let it absorb a quan- 

 tity, H, of heat. Then 



banical woi 

 '^=f}dv (,), 



But if W denote the mechanical work which the substance does 

 in expanding, we have 



and therefore 



^=1.^ i'i)' 



This formula, established without any assumption admitting of 

 doubt, expresses the relation between the heat developed by the 

 compression of any substance whatever, and the mechanical work 

 which is required to effect the compression, as far as it can be de- 

 termined without hjTJothesis by purely theoretical considerations. 

 64. The preceding formula leads to that which I formerly gave 

 for the case of fluids subject to the gaseous laws; since for such 

 we have 



P^=P<Po{\+'E>t) (Ij^ 



from which we deduce, by (c), 



W=i)o«o(l+E01og|-' (2), 



and 



^=Epo^o.log^ = ^-^^W . . (3); 

 and therefore, by {d), 



^=Mmo^ (4). 



which agrees with equation (11) of § 49 of the fomer paper. 



* Throughout this paper, formula; which involve no hypothesis whatever 

 are marked with itahe letters; formula; which involve Boyle's and Daltou's 

 kw8 ^e marked with iVrabic numerals; and formula; involving, besides. 

 Mayer s hypothesis, are marked with lloman numerals. 



