Olbers on Comets. 151 



of the comet from the earth. Now three points not in a right 

 line determine the position of a plane ; consequently, two points 

 and the sun determine a plane which must pass through the 

 third point; whence we derive the first equation. The condi- 

 tion that the three points must be in a parabola, of which the 

 focus is the sun's centre, affords us the second equation; and 

 the comparison of the times with the revolving radii and the 

 chords, give us the two others. In general, for ?i obsei-vations, 

 and ti unknown quantities, we have 3 »i — 6 equations ; n — 2 

 being derived from the condition that the sun must be in the 

 plane of the orbit ; 7i — 2 from the properties of the parabola, 

 and n — 1 from the relation between the times and the distances 

 and chords. 



§ 6. 



With so great an abundance of equations, it might naturally 

 be supposed that a few observations would lead us pretty 

 readily to the direct determination of the elements, with geo- 

 metrical accuracy. But when we consider the equations them- 

 selves, we find them so intricate, that the utmost powers of al- 

 gebra, and the patience of the most indefatigable calculator, 

 might be exhausted on them in vain. The equations may be 

 represented in the most convenient form, by considering the 

 three curtate distances of the comet from the earth, or the pro- 

 jections of the distances on the plane of the ecliptic, as the un- 

 known quantities. 



§7. 



We may denote the quantities relating to the three different 

 observations by as many accents added to the respective letters ; 

 and we may make 

 A', A", A", the sun's longitude. 

 a, x", a,'", the comet's longitude. 



0", B", the comet's latitude. * 



R', R", R ', the earth's distance from the sun. 

 t', the time between the \st and 2d observation. 

 t", the time between the, '2d and 3(/, . 

 T = < + /,', between tiie Ist and '6d.: all these being given 



quantities. 



