416 Olbers' Essay on Comets. 



iii. An Essay on the easiest and most convenieiit Method of calcu- 

 lating the Orbit of a Comet from Observations. By William 

 Olbers, M. D. 8vo. Weimar, 1797. 



[Continued from Vol. IX. p. 149.] 



Section II. 



On some Equations cf the First and Second Order , which have 



been proposed for determining the Equations of Comets. 



§ 20. 

 The suppositions which have been made the foundation of 

 the approximate solutions, § 11, lead, when geometrically con- 

 sidered, further than the conclusions which have been drawn 

 from them. Assuming that the path of the comet is a right 

 line, described with an equable velocity, the distances of the 

 comet from the earth may be found by equations of the first 

 degree. The supposition, that the chord is divided by the re- 

 volving radius, in the proportion of the times, leads to equations 

 of the second degree, from which these distances may be de- 

 termined. These equations require so much the more a par- 

 ticular investigation, as they have been recommended, not only 

 by their inventors, but by other mathematicians, far beyond 

 their real merits ; and have been condemned, on the other hand, 

 by those who have justly rejected them, upon grounds not alto- 

 gether satisfactory. 



^ 21. 



The problem of finding a line, which shall be cut by three 

 others in a given proportion, is of an indeterminate nature. It 

 is known that the condition is fulfilled by any of the tangents 

 of a parabola, of which the three given lines, together with one 

 line which is cut by them in the required ratio, are also tangents, 

 so that the curve is completely ascertained by its four tangents. 

 But the problem is only indeterminate, so long as the three lines 

 remain in one plane. When they are not in one plane, there 

 is only a §ingle position, in which a line passing through a 

 given point in one of them, will intersect the two others. If we 

 add the condition, that this line must be divided by them in a 

 given proportion, the points through which it must be drawn 

 are all given, and by an equation simply linear. In this manner 

 BouGUER thought it possible to determine the distances from 



