336 Mv MeikXe m the Theory of 



where A and B are two arbitrary constants. Hence, when 



^.-1 



d^=o,dq « ^* dp ; but, by hypothesis, dq oc df oc dp, or 

 /? ec 1, which is absurd. Indeed as we shall soon see, (B) is 

 the only form, which constructed geometrically, can agree with 

 the forementioned law connecting p, ^ and t when q = o; and 

 which will make the specific heats independent of the actual den- 

 sity or pressure, as their constant ratio requires them to be. 



When, in equation (B), § = 1, gr varies as log p, that is, when 

 the heat varies equably, the pressure under a constant volume 

 varies in geometrical progression. If p be constant, the varia- 

 tions of q are as those of — log §, or of + log - ; that is as the 



variations in the logarithm of the volume. Hence, when the 

 quantity of heat varies in arithmetical progression, the volume 

 under a constant pressure varies in geometrical progression, or the 

 real temperatures are as the logarithms of those on the common 

 scale of an air-thermometer, reckoning from — 448° F. or — 266°.7 

 cent., and placing the new zero at — 447° F. or at ^ — 265°, 1 

 cent. The absolute zero might thus correspond with the first 

 two of these numbers, or with minus infinity, by the new scale ; 

 but this is a point which I do not pretend to decide. 



The divisions on the scale ought therefore to form a geome- 

 trical progression, increasing with the temperature, instead of 

 being, as at present, equal parts. 



When q=^o, p-^^ (C). 



Let T be the temperature when p and § are each equal unit ; 

 then, if these. vary while q^=.o, we have from equation (A), 



f = *(l + »0 = 4^^ = e-' (D). 



The change of temperature, by the common scale, produced by 

 the change of density from unit being i. 



Equations (C) and (D) are equivalent to M. Poisson's equa- 

 tions (5) ; or to Mr Ivory's equations (D), Phil. Mag. Ixvi. 9- 

 They form' the law which connects the pressure and density to- 

 gether, or with the temperature on the common scale, when the 

 heat in the air is constant. 



But the relations of the different quantities may be more 

 clearly exhibited by means of a geometrical diagram. This 

 will appear whilst I proceed with the construction, illustrating 

 it, at the same time, by a sort of example. 



