37(J Jbtronnmical and Nautical Co/lections. 



K the heliocentric distance of the comet from the node, 

 T K its heliocentric distance from the earth, K C the angle at 

 the comet, and lastly the supplement of T C the geocentric dis- 

 tance of the comet from the sun. [In order that C may represent 

 the geocentric place of the comet, it must be that point of the 

 sphere which is indicated by a line drawn from the sun, sup- 

 posed to be in its centre, parallel to the direction of the comet 

 from the earth ; and this line will make an angle with the re- 

 volving radius of the comet, equal to the alternate angle at the 

 comet, which will therefore be represented by K C . Tii.] Nov/ 

 it is obvious that all these distances may be found by the solution 

 of two spherical triangles. 



(1) In the right-angled triangle A C T we have T A, the differ- 

 ence of the earth's longitude and the geocentric longitude of 

 the comet, and A C the observed latitude of the comet : whence 

 we have (i.) cos T C := cos T A cos A C ; and (ii.) cot A T C =: 

 cot A C sin T A. 



(2) In the oblique angled triangle S? K T, we have Q, T the 

 difference of the longitude of the node and of ihe earth, the angle 

 T g^ K the inclination of the orbit, and the angle J? T K = 

 .'V T C ; hence we find Q K and T K by the formulas (iii.) tang J 



Ov,ta„,U.K-TK) = !l;ili.|ll^L||l un, J . T. 



We have then KC=:TC— TK, and R being the distance of 

 the earth from the sun, and r that of the comet, we have (v.) r. = 

 R sin T C 

 sin KC" ■ 



§.72. 



If we compare these formulas with those which have been 

 hitherto employed, we shall be aware of their great convenience 

 especially in the correction of the elements of a comet's orbit. 

 EuLER, for example, employs in his Recherches sur la viaie 

 orbite elKpfique de la comete de 1769, eight proportions, in- 

 stead of the five here laid down. The whole eight he is obliged 

 to compute for each of the hypotheses, which he assumes for 



