106 Mr Meikle on a proposed Improvement 



gcthcr with the errors he complains of, are the offspring of his 

 own contradictory hypotheses, and do not proceed from the real 

 nature of the subject. This will be rendered evident from the 

 perfect consistency of the following plain view of the case, in 

 which no hypothetical work is introduced. 



Let t be the temperature, or rather the indication on the common scale of 

 an air thermometer, p the corresponding^ pressure, and ^ the density of a mass 

 of air ; then a being the expansion for one degree, and b another constant, we 

 have, from the law of Boyle, 



p = *j(H-aO (A.) 



Whilst the quantity of heat in any body undergoes the minute variation 

 dg, the corresponding variation dt in its temperature must obviously be in- 

 versely as its specific heat. Hence, 



the last term of which is the general expression for the specific heat of any 

 body ; especially if the volume and pressure do not vary at the same time, 

 for in that case, the variation of heat might not change the temperature. But 

 differentiating equation (A) with p constant, we obtain 



at =■ —— . do^ 



and substituting this value of dt in the general expression, the specific heat 

 of air under a constant pressure, relatively to a degree of the scale to which 

 t belongs, is 



d^ 1 + a^ 

 Differentiating, again, equation (A) with ^ constant, and supposing that the 

 mass of air undergoes the same variation dq in its quantity of heat as in the 

 former case, we obtain for the specific heat of air under a constant volume, 

 for the same degree of the thermometer, 



dp \ •\- at 

 Now, it is admitted by all parties, and corresponds with experiment, that 

 these specific heats have to each other an invariable ratio ; or, in other words, 

 that the relations of the differentials is of a known and determinate charac- 

 ter. Hence, they are of the fittest possible sort for integration. Calling this 

 constant ratio that of A; : 1, and we get 



dp 1 + at d^ i + at 



From the conditions under which we have obtained this equation, dq has 

 the same value in both terms. The degree of the common scale, considered 

 as a linear quantity, is constant, and is likewise the same in both terms. 

 Hence, dividing by the common factors, we obtain 



