On Atmospherical Refraction. 357 



pressions, the values derived from the various laws which may- 

 be supposed to govern the variations of temperature :■ observing 

 first, that in general 



m = 766, p = . 000 2825 = -^ : whence 

 3540 



-|- = 4.621, 4^ = 21.3536, J-^ = 16358, -L - 72052, 

 mp trfp^ mp^ m'f ' 



1 



— , = 57 907 320. 



7}ip'' 



7. (A) If the temperature were uniform, we should have 



- 1 J >- , w A y» « j<^" 1 4.621 



2j~z, dy=dz, ^= 1, ^'z=0, ^"=:0, and t-=: s = s ; 



dr mps s 



and when s=l, 3.621. 



d^v V 



df' mp-s^ 



d'v \ f I ] 2v' 



i~5 = — n\ — —s\ + — 7-,; or if v=iO, 16358 X 3.621. 



d'v f I V 6 (I \ \ 



ri = \ 1 *— 3 + 1 • -rr- 3.621 (6x57907320 



dr \mp ) 7np^ \mp j m^p* ^ 



X 3.621 + 16358^) = 5524050000; ^i^ of which is 7672300. 

 Hence, for s = l, we have the equation .0002825 = 1.8105 

 r= + 2467 r* + 7672300 r« + ... , in which, if we put r''^ 

 .000130, we shall have .0002825= .0002939 + ...; which is 

 too much: then taking r^ = .000120, we have .0002825= 

 .00021726 + .00003552 + .00001325 + [.00001647]: and 

 this is somewhat too great a remainder ; for the quotients of the 

 terms being 6, 3. . ., the remainder ought not to exceed the last 

 term; so that r* must be about .000121, and r = .0110, or 

 37' 50", which is too great by about one ninth By the assist- 

 ance of this series we might easily compute the refraction upon 

 the hypothesis of Professor Bcssel, who supposes the variation 

 of density to follow the same law as if the temperature were 

 uniform, but alters the value of m, so as to accommodate it to 

 the actual magnitude of tlie refraction in low altitudes. 

 (B) In Professr Leslie's hypothesis, we have 



45 

 7t = — = .09 

 " 500 



Vol. XI. 2 B 



