89 



(2) If the density of the air decrease uniformly, it wiU 

 be proved that 



s == llLzJL ^ IL nearly 

 if) « 



R^»o^ /• ^fl'^'t""-'^ _ /• 3?btan.3Q f (f)—p) Y I' 

 ncnLt,J T^~Q y~~ COS.' 6 ^ if) ^ a' 



= [from ? = to ^ = a] 'M^tlx^ = [ia seconds] 



2 (j«— 1 ) ; » fare. 3 fl 



a ' COS. * 9 si)i. l" 



Taking ^ = 80° 45' this quantity = r',73. Consequently 

 the true value of fUhlI1^<^ is between 2",60 and i",73 



c/ 4 COS. * 80° 45' ' " 



and therefore the mean cannot err quite half a second from 

 the truth, and so the following formula may be considered 

 as giving the refraction as far as 80° 45' true to less than half 

 a second, viz. 



Refraction = Q - S'!!ii}lL^±± + lin-zllLl''±lI, (7) 



a COS. « e sin. 1" 2 a * cos. * 8 sin. V ^' ^ 



The third term is insensible when ^ is less than 74° and the 

 second and third insensible when is less than 40° 



It is evident that the two first terms must be derived from 

 assuming 0711/ law of variation of density, and then investi- 

 gating the quantity of refraction as far as these terms. The 

 following investigations in different hypotheses of density may 

 be considered useful. 



VOL. XII. o 



V + %) .UHi UiH 



