426 



Olbers' Essai^ on Comets. 



solution, which is indirect, but more easy and convenient than 

 could well have been imagined, considering the intricate nature 

 of the problem. 



§ 34. 



Let S be the sun. A, B, and 

 C, three places of the co- c 

 met not remote from each 

 other, and a, b, c the corre- 

 sponding places of the earth; 

 we shall assume that the middle 

 revolving radii SB, S6 divide 

 the chords AC, a c, in D and dj 

 in the proportion of the inter- 

 vals ; so that adidczz. AD 

 '."DC ::z t' : f ; a supposition 

 which is very near the truth 

 when the arcs are small ; for, ^ 



first, the difference between the sectors proportional to the 

 times, and the triangles, which are in the exact proportions 

 of the two segments of the chords, is very small, and of an 

 order higher than the sectors themselves ; secondly, the differ- 

 ences, or the portions of the segments contained between the 

 chords and the arcs, are greater as the sectors are greater, 

 though not generally in the same proportion; while, in the 

 third place, there is always one position of the revolving 

 radius, for each parabolic or elliptic arc, which divides the 

 chord in the precise proportion of the arcs or of the times. In 

 what cases this last circumstance occurs, for a parabolic curve, 

 has been investigated by Newton, by Gregory, and, more 

 recently, by Lambert, who have shown that the proportion 

 can never be very different from that which is here supposed, 

 unless the two intervals employed be very unequal. For the 

 earth, the difference must be the less considerable, when the 

 times are nearly equal, as the orbit approaches so nearly to 

 a circle. — Newton, Principia, III, lemm. viii. Gregory, Astr. 

 Phys. V. xviii. Lambert, Beytr'dge, III. Propr. ins. orb. com. 

 Astr. Jahrb. Bert. 1779. 



(To be continued.^ 



