75 



The advantage arises from the simplicity of the conclusion, 

 in reducing the problem to the taking out of five versed sines. 

 As the angle M is always nearly 60°, it is obvious the table 

 of versed sines ' required by this method must extend to the 

 semicircle. 



5. The quantity ^°'„''°^'/^ , (cos A — cos D) may be com- 

 puted by the expansion of ^°^ ^^'^^."^^ ^, Because h = H + p 

 cos H — r and h' =H' + h' cos H' — r' we readily obtain by 

 Taylor's theorem. 



cosA ^ cos(H+;,cosH-r) ^ -^ j, , sin H— r tan H)— J 



cos H cos H \^ ' 



sin* 1' {p COS H — 7-)* &c. 



^, = I— sin r ip' sin ll'—r' tan H')— i^ 



cos H' ^ \X^ / -a 



sin* 1* {p' cos H'— r')"- &c. 



Hence omitting small terms depending on the sun or star 



cos h cos h' _ , . -^ r/sin H'— r'tan H' \ i_ • 2 i/zHp'tos H'-r'jM 



cos H cos H'"^""^"**^ l;,sinH-rtanH ;— ^^m 1 | ^^, | 



Now because the refraction varies nearl}' as the cotangent 

 of the altitude, and at 45" = 5T', 3* let 



r' = 57''', 3cot H' — <J' 



r = 57, 3 cot H — S 

 then 



'q^''''"^*' —1 aJn ^» <P'sinH'4-psinH— lU", 61 , • ^ ,, j (;■/ cm K'— </)« 2 



cub 11 cos H' — ' ="1-'^ ^-t-J-tanH'-t-JtanH \ 2="* ^A'i-'*' J 



This 



* M. Laplace has proved independently of any hypothesis on the constitution of the 

 atmosphere, that at all heights above 10°, the refraction varies accurately as the co- 

 tangent of the altitude (Mecanique Celeste, Tom. iv. p. 269). The tables at the end of 

 this paper are however computed from the tables of refraction, in the requisite tables. 



