OP ARTS AND SCIENCES. 197 



true sun, taking ac and dh as a pair of rectangular axes, and putting 

 ai = s, glz=:iy^, ho = y^, aig = (9, am = d^, ain = d.^, we have 



w, = s cos ^, — s cos ^, -r-^i 



y^=. S cos 0.^ S cos ^2 ;77r 



(66) 



in which — is the first derivative of the refraction, taken at a point 



midway between m and / ; and -— ^ is the same derivative taken at a 



dii 

 point midway between n and o. Adding these two equations, we 



obtain 



^1 + ^2 = 5 (cos d^ + cos d^ —scosdj^ ^ — « cos 02 flf^ (67) 



and, as the factors for converting the cosines of 6^ and 6^ into the 

 cosine of can never differ much from unity, it will be quite accurate 

 to write 



d'^r 



2/i + ^2 = « (cos ^1 + COS d.^) 4- s"- cos^ d — (68) 



in which ^ is the second derivative of the refraction, taken at the 



centre of the true sun. 



If we put i7 = a?p and bear in mind that on account of refraction the 

 horizontal diameter of the sun is contracted by a constant ratio, jw, 

 then we have 



Xj^ = fis sin 5^ (69) 



and 



tan 5 = -^= tan (9,-^^ (70) 



or 



tan /, dr,\ ^„,^ 



tan^, = -^(l-^) (71) 



dr 



Regarding 6^^ and ^ as variable, and differentiating, we get 



d (tan 01 ) tan 6 d^r^^ 



dC ~ ^^1 



(72) 



