Mr. Ivory on the Astronomical Refractions. 423 



In order to place the proofs of the accuracy of my Table 

 in a clear point of view, I shall now put 



e 



6 = t'— t; 

 then fl will be the depression of the thermometer at any 

 height. The equations of the atmosphere, when the heat is 

 supposed to decrease uniformly, will now become, 



(It x ( A ) 



1-T+57 = (1— ) 4 



T+£7 = i«+/2 «> 2 + & c -; 



and the equations from which the Table is constructed will be, 



4==|(l-0) + |(l-«) 9 



i, (B) 



r+^' = i <°- 



These equations very clearly point out what is common to 

 all the atmospheres, and what causes them to agree in giving 

 the same refractions. The first term of the development of 



(16 



the function 1 , „ , is the same in them all and equal to \ w. It 



is this new element which I have introduced into the problem of 

 the refractions : and upon the exactness of the co-efficient \ 

 the accuracy of my Table will entirely depend. I have proved, 

 in my paper, that the usual formula for barometrical mea- 

 surements with its equation for heat, is true in both the at- 

 mospheres defined by the equations (A) and (B). With 

 respect to the same atmospheres I have likewise shown, by a 

 comparison with actual experiments extended to the greatest 

 heights to which we have been able to ascend, that the pres- 

 sures and densities are the same at like distances above the 

 earth's surface, as in the real atmosphere. To all this is to 

 be added the very exact agreement of my refractions with ob- 

 servations to a great distance from the zenith. Of this I am 

 now in possession of one proof, derived from a comparison 

 with actual observation, that will not be disputed. I humbly 

 hope that the examination of other unbiassed astronomers 

 will lead to a similar conclusion ; and to their determination, 

 to which I have ever appealed, I shall very cheerfully submit. 

 I shall now add some additional proofs of a different descrip- 

 tion. Dalton has suggested a very remarkable law relative 

 to the temperature of the atmosphere*. He supposes that a 



• Chcm. Phil. Part I. chap. 1. § 8. 



given 



