460 Prof. Donkin on the Specific Heat of Gases. 



c 



1 -{-0= — fJ which can only be objected to on two grounds — viz. 



that some (or all) of the quantities assumed as constant are not 

 rigorously constant, and that the usual mechanical theory of 

 sound is not rigorously correct; and both objections would be 

 irrelevant to the present purpose. 



Lastly, the assertion that " the two capacities are necessarily 

 equal," appears at first sight to mean that c and c' are neces- 

 sarily equal. But as the author cannot have intended to deduce 

 this conclusion from a fact which proves that c and c' are neces- 

 sarily unequal, it is probable that he only meant to say that when 

 heat is spent upon a given quantity of gas, how much of it goes 

 to make the gas hot depends only upon the change of tempera- 

 ture. If this be so, the language used is very inaccurate, and 

 almost certain to mislead an ordinary student. 



But, whatever Jamin may have meant, it is certain that Ganot 

 really did mean to say that Regnault had proved experimentally 

 the equality (or near equality) of c and d. And neither he nor 

 his translator, nor the author of the f Second Step/ have 

 noticed that to assert this equality is to deny the conservation 

 of energy. 



It may be conjectured that Ganot had somewhere met with 

 the statement " les deux capacites sont egales entre elles," in- 

 tended in the non-natural sense suggested above, and confirmed 



this must be equal to the whole energy spent in the operation, viz. — pdV ; 

 for if the gas were now allowed to expand freely to its former volume, it 

 would retain its new temperature, while nothing would have been spent 

 except the work of compression. Hence —pdV=Ac'dt; and combining 

 this equation with (1), we obtain by eliminating dt, 



dp __&-{- Ac' dV , /o\ 



J~ ~Ac T ~T~ [ } 



Next, let the gas (in its original condition) be heated, and at the same time 

 allowed to expand under constant pressure p, until its volume becomes 

 V-f dV and its temperature t-{-dt. The increment of actual heat will be 

 c'8t, and the whole energy required for the operation will therefore be 

 Acdt+pdY. But, by the definition of c, the whole energy required is 

 Ac8t. Hence, since in this case (1) gives pd\ T =ccbt (for dp = 0), we have 

 {Ac'-\-a)dt=Acdt, or Ac' + ct=Ac; hence from (2), 



dp _ _ c dV _ c dp 



p c' V c' p 



c 

 of which, according to the ordinary theory of sound, the equation 1 + 0= ~, 



is a consequence. 



It is hardly necessary to add that this demonstration is only given for 

 convenience of reference, and not as containing anything new. 



