96 On the observations of Comets. 



dB—dN' cos za'c'-{-d N cos zc'a' 



/cos N— cos Dcos N'\ /cosN'— cosDcosN\ 



=d N (^ ^iiTD'^hTN' ) "^"^^ [ ihi D~sTirr ) 



^d W (^ -^'-^7 -cot D cot N') 

 Vsin D sm N / 



+d N ( ■ ""^^' f^ -cot D cot n) 

 \ sin D sin D / 



f^N'cosNcosceN' 4-<?NcosN^cosecN ^dN'cotN' 4-^Ncot_N^ 

 sin D tan D 



and calling a the altitude of the comet, r its refraction, A the 

 altitude of the star, and R its refraction, the preceding for- 

 mula becomes 



Rsinossec A+r sin Asec« RtanA+rtana 



~ sin D tan D 



Having corrected the distances, find in the triangle pah 

 (fig. 1,) the angle ^a6 and the side ab. Then, knowing the 

 three sides of the triangle cab, find the angle cab, from which 

 subtracting pab, the angle pac will remain ; lastly, in the 

 triangle cap, having the sides pa and ca, and the angle pac, 

 the side cp and the angle cpa will be ascertained, and there- 

 fore the declination, and the right ascension of the comet. 

 In order to determine the effect produced in the positions 

 of the comet by an error in the distances, let cb=a, ca=h. 

 nb—c, and cab=x. Then we have 

 cos a — cos 6 cos c 



cos.r= — — ; — r— • 



sin sm c 



cos a' — cos b' cos c 



cos X'=- : TT~- ' 



sin 6 sin c 

 and on account of the little difference between sin 6 and sin 6', 

 we may suppose 



cos a — cos a'+(cos 6'— cos b) cos c 



cos X — cos x'=- ^— T — : 1 



sin o sm c 



-^m-i\{x-\-x')sm\{x—x') = 

 - 1s\n\{a-k-a')B\x\\ {a — a') -2 sini (6^+6) sini (6^ -6) cose 



sin 6 sin c 

 making x — x'=dx, a — a'=da, and b' — b=db ; and sup- 

 posing these are equal to their sines, we have 



,^ irfasini(a+a')4- i^&sini(6'+6)cosc 

 Idx sini(:r+^0= ^sCT^ifT^ ' 



da sin a-^db sin b cos c , 



dxr=z — ^ — i—. nearly ; 



sm b sm c sin a: 



