the Orbits of Comets, 47 



•of the comel, and the letter C at its projection in the plane 

 of the ecliptic; we shall have the angle STC, on taking 

 the difference of the geocentric longitudes of the sun and 

 the comet ; by afterwards adding the logarithm of the co- 

 sine of this angle with that of the cosine of the geocen- 

 tric latitude 3 of the comet, we shall have the logarithm of 

 the cosine of the angle STC; we shall therefore know ixi 

 the triangle STC, the side ST or R, the side S'C or r, and 

 the aniile STC ; we shall thus have by rectilinear trigono- 

 metry the angle CST ; we shall afterwards have the helio- 

 centric latitude or of the comet by means of the equation 

 sin 9 'sin CST 



sni ■ST = : — n^Fc~- 



sin CTb 



The angle TSC is the side of a spherical rectangular 

 triangle, the hypothenuse of which is the angle TSC, and 

 one of the sides of which is the angle cr; from thence we 

 shall easily extract the angle TSC, and consequently the 

 heliocentric lonijitude /3 of the comet. 



We shall have in the same manner ro-', jS', ru" and /3", and 

 the values of /3, /3', /3" will show whether the motion of 

 the comet be direct or retrograde. 



If we conceive the two arcs of latitude cr and zs' united 

 at the pole of the ecliptic, they will there form an angle 

 equal to /3' — /3; and in the spherical triangle formed by 

 this angle, and by the sides 90° — tcr and 90° — ot', the side 

 opposite to the angle /3' — /3 will be the angle at the sun 

 compreliended between the two vector radii r and r'. We 

 shall easilv determine it by the known analogies of spheri- 

 cal trigoni)metry, or by the following formula: 



COS ^= cos (|S' — /3).cos OT.cos ro-' + sin -n^.sinW, 

 in which ^^ represents this angle. 



By calling F' in a similar manner the angle formed by 

 the two vector radii r and r", we shall have 



cos F' = cos (/3" — /3).cos cr. cos w"— sin ro'.sin ts". 

 Now if the perihelion distance and the instant of the 

 passage of the comet by this point were exactly determined, 

 we shall have 



F = U and F' = U", 

 but as that will almost never happen, we shall suppose 



m=:U- F', n= V- F. 

 We shall here observe that the calculation of the tri- 

 angle STC, gives for the angle CST, two difTertnt values, 

 viz. CST and 180°; £dly, STC-CST. We shall thus 

 have two difl'crent values for each of the quantities /3, S7, 

 /3', ro', j3", ct". Most frequently the nature of the motion 



of 



