246 Mr. Ivory on the Figure of the Planets, ^-c. 



the density of the air, and put di = co; then we have y = 



— , and the foregoing equation will become, 



^ = (X--1) ^^. (1) 



It must be] observed, that this equation expresses the rela- 

 tion of the differentials only for a small initial variation of the 

 mass of air, or when / = 0. What may be their general re- 

 lation, when i has acquired any finite value, cannot possibly 

 be duduced from this equation alone. M. Poisson derives 

 this integral from the expression ( 1 ), viz. 



1 + a^ + ai — / f \^~^ 



g' being the density at the initial temperature 9. There is no 

 doubt that the formula ( 1 ) is thus satisfied : for if we differen- 

 tiate the integral, and put z = in the differential, we shall 

 arrive at the formula in question. But there are innumerable 

 other integrals that will equally produce the same result in 

 the like circumstance : for instance, 



/3(A-- 1) 



i+as V f' / 



/3 being any arbitrary number. It appears therefore that 

 M. Poisson's integral is not the only one that will fulfil the 

 required conditions ; there is no evidence that it contains the 

 law of nature more than any other ; it is dependent on a hy- 

 pothesis, or an arbitrary assumption. 



We shall pursue this investigation on less exceptionable 

 principles in the manner following. Suppose that the air is 

 condensed by the diminution of its temperature, the pressure 

 remaining constant : then, r being the decrease of tempera- 

 ture when the density varies from g' to g, we shall have, by 

 the usual principles, 



J_ — ^ +'^^ (2) 



Differentiate this equation, and put x = in the differential ; 

 then, "'^'^ _ iiL. 



This equation expresses the relation of the small initial varia- 

 tions of temperature and density. If we combine it with the 



foi'mula ( 1 ), and put k — 1 = — , we shall get, 



di a 



The next question is, whether /c — 1, or -^, is a constant 



or 



