IjO Aslronomical and Nautical CuUections. 



tuted for it a variety of more or less perfect approximations. 

 It seems, therefore, to be interesting to examine the problem, 

 with all its difficulties, and to take a view of the methods which 

 have been proposed for its solution, before we proceed to at- 

 tempt any improvement in them. 



§2. 



In every observation we have to consider two triangles ; the 

 one lying between the comet, the sun, and the earth; the 

 other its orthographical projection, on the plane of the ecliptic : 

 one side is common to both, that is, the distance from the earth 

 to the sun ; and the observation gives the angles at the earth : 

 but another element is wanting to the complete determination 



of either triangle. 



§3. 



We may safely consider the small part of the orbit of every 

 comet, which is in the neighbourhood of the sun, as a parabolic 

 curve, having its focus in the centre of the sun, and conse- 

 quently situated in a plane which passes through the sun. Sup- 

 posing the situation of the plane to be given, the line of direc- 

 tion found by each observation determines a point in it ; and 

 two points, together with the focus, determine the parabola : 

 if three such directions are given, there can be only one inclina- 

 tion of the orbit for any given position of the node, in which 

 the points will be found in a parabola, and for a given inclina- 

 tion only one line of the nodes : and four such observations 

 leave neither the inclination nor the intersection undetermined, 

 even without any regard to the intervening time, 



§4. 

 Three observations would be sufficient, if we only assumed 

 that the times are proportional to the spaces described : but the 

 absolute times being also determined by the distances and the 

 chords, we have a superfluity of conditions, since we obtain 

 four equations for three unknown quantities. 



§5. 

 We may easily form a general idea of these four equa- 

 tions. The three unknown quantities may be the three distances 



