TRANSACTIONS OF THK SECTIONS. 555 



logism, if we break the total elevation into a sum of smaller 

 ones, as through a few hundred feet the average temperature 

 can differ but little from the half-sum of the extremes. Deriving 

 the densities from the barometer heights and temperatures, 

 using the expansion of dry air, we find very nearly 



1 -a;^' 



when a — 39534 feet and x is the decrease of density divided 

 by that at the surface. 



As the horizontal refraction is proportional to the density of 

 air at the vertex of the ray's trajectory, 



,-- (H)-H 



^- (H) • 



The quantity I is given with sufficient accuracy for this in- 

 quiry (see Mec. CeL, iv. 283.) by the equation 

 -2£H 

 I = e, 

 e being the base of Napier's logarithms, and e a coefficient de- 

 rived from the observed extinction of light in traversing the 

 atmosphere, which for H = (H) is supposed to give 



^~ "3746* 

 Substituting these expressions and putting 



and developing d h, we obtain 



diy = -^P^^^^,xfp-M-^-y + ^-f-,^^-] 



in which A = -y _^ a > ^^^ *^^ others B, C, &c., its suc- 

 cessive differential coefficients divided by 1, \'2, &c., which 

 are easily derived independently. This series is converging, 

 though but slowly, and 27 terms of it give for the integral be- 

 tween the limits y := + — ^^ — ^ — —. 



-\(\\ 93368 



D = — — 



/5'34377 - 0-02635\. 



Thus we find that for a circle 7 minutes radius round the axis 

 of the shadow, the illumination derived from this source must 



