450 Proceedings of Philosophical Societies. [Dec. 



himself feels respecting the important conclusions which he deduces 

 from his observations. 



Another paper by Dr. Brinkley was read at the same time, con- 

 taining analytical investigations respecting astronomical refractions, 

 and the application thereof to the formation of convenient tables, 

 together with the results of observations of circumpolar stars, tend- 

 ing to illustrate the theory of refractions. 



The author deduces, by a very short method,, from the common 

 principle of the constant ratio of the sine of incidence and refrac- 

 tion, the same fluxional expression that Laplace has deduced (Mec. 

 Celest. torn. iv. p. 214). The approximate integration of the 

 fluxional expression is obtained by a method which affords a result 

 showing the effect of the spherical form of the atmosphere. The 

 formula derived consists of two parts : one showing the refraction 

 (p) that would take place were the surface of the earth a plane ; the 

 other the effect due to the spherical form. The latter at 80° zenith 

 distance amounts only to about 14", and at 40° zenith distance is 

 insensible. 



Let 9 = the zenith distance ; in : 1 the ratio of the sine of 

 incidence to the sine of refraction in air of the density of that at 

 the surface ; a — radius of the earth, and I = height of an uniform 

 atmosphere. Then 



_ „ . (m — 1)1 tan. 6 , 5 (m — 1 ) I 2 tan.3 6 



Refraction =-. p - Vct,s.M ^. r + 8a» cos.* * sin. l" "^ 



It is shown that as far as 80° zenith distance this formula cannot 

 err half a second, whatever be the law of variation of density in 

 the atmosphere. The third term is insensible at 74°, and at 80° 

 amounts only to 1 ±" '. At 40° zenith distance both the second and 

 third are insensible. When 9 is greater than about 80°, a knowledge 

 of the law of variation of density is necessary for the integrations ; 

 but as the approximate formula as far a? about 74° is independent of 

 the law of variation of density, it follows that whatever law of 

 variation of density be assumed, the same conclusion ought to be 

 deduced as far as about /4°. This is shown by direct investigation 

 by assuming different laws of variation of density which, besides 

 affording some conclusions useful in our inquiries on this subject, 

 • may be considered interesting. 



The author then proceeds to deduce a formula convenient for 

 computation and for tables. In this formula he substitutes the 

 value of m as deduced from the experiments of Biot and Arago on 

 the refractive force of air, and from the experiments of Dalton and 

 Gay-Lussac on the change of density by the variation of tempera- 

 ture. In this manner a; formula for refraction is deduced entirely 

 independent in astronomical observations; by which (barom. 29-(>0 

 and Fahrenheit's therm. 50) the refraction at 45° = 57-6", at 74 D 



= 198*6". 



The author gives his reasons for thinking that greater accuracy 



