MR. IVORY ON THE THEORY OF ASTRONOMICAL REFRACTIONS. 173 



As i and a are both very small numbers, if we put 



m = i — i^f 



n = i — a — P -{- 06 i 5-9 



the two last equations will become 



sin (p = sin d — m sin d, 



sin (<p -{-10) = sin0 — n sin d : 



and by employing the usual formula for deducing the variation of the arc from the 

 variation of the sine, we get 



(p z=z 6 — m tan & + -^ tan^ &, 



consequently 

 that is, 



<p -\.ld z= 6 ^ n tan & -\- -^ tan^ d : 



10 z=z {jn ^ n) tan & 3 — • tan^ d ; 



l& =z f a _ 2 a -I- ^ j tan ^ — (/a — ^j tan^ ; 



or, which is the same thing, 



^^ = atan^(n-a-^^)=a(l+a)tan^(l - '^). ^ 



agreeing exactly with Laplace's formula employed in computing the first part of the 

 Table of mean refractions published by the French Board of Longitude. 



2. The publication of Newton's Principia enabled geometers to take a more en- 

 larged view of the astronomical refractions, and one approaching nearer to nature. 

 According to Cassini, the atmosphere is a spherical stratum of air, uniform in its den- 

 sity throughout, diffused round the earth to the height of about five miles ; in reality 

 the density decreases gradually in ascending, and is hardly so much attenuated as to 

 be ineffective to refract the light at the great elevation of fifty miles. The path de- 

 scribed by the light of a star in its passage through the atmosphere is therefore not a 

 straight line, as it would be in the hypothesis of Cassini, but a curve more and more 

 inflected towards the earth's centre by the successive action of air of increasing den- 

 sity. Now in the Principia there is found whatever is necessary for determining the 

 nature of this curve, and consequently for solving the problem of the astronomical 

 refractions, which consists in ascertaining the difference between the direction of the 

 light when it enters the atmosphere, and its ultimate direction when it arrives at the 

 earth's surface. In the last section of the first book of his immortal work, Newton 

 teaches in what manner the molecules of bodies act upon the rays of light and refract 

 them ; and as the atmosphere must be uniform in its condition at all equal altitudes, 

 its action upon light can only be a force directed to the centre of the earth ; so that 



