Olbers' Essay on Comets. 423 



,' = fg + V ^.- 1) = Q + ^/(Q«-4SP) . This is cs- 



seotially the same with the formula of Dust' jour, but the way 

 of obtaining it appears to be much easier and shorter than that 

 of this great mathematician : and a quadratic equation for h 

 may be derived from the same equations much more conve- 

 niently than in Professor Hennert's manner. 



§29. 



Pin ore' has attempted to make computations according to 

 both these methods, and has found the results of both extremely 

 erroneous. The coefficients S, Q, P, were always very small, 

 and therefore the smallest errors of the observations had a great 

 influence on the magnitudes to be determined ; so great indeed, 

 that he considers Hennert*s solution as quite useless. The 

 same must, however, be true of that of Dus^jour, for both de- 

 pend on the same equations. 



§ 30. 



It will be worth our while to examine more particularly the 

 cause of this insufficiency, Now it is obvious that the solution 

 would be mathematically correct, if the observations were per- 

 fect, and the proportions, expressed by M and N, truly ascer- 

 tained. But observations can never be free from all error : and 

 the proportions are obtained from a supposition not precisely 

 accurate. In Duse'jour's formulas, the value of §' depends 

 only on the apparent curvature of the orbit of the comet, or in 

 its deviation from a great circle : for if the three places lay in a 

 great circle, the coefficient of ^'2, or S, would be = 0, since, 

 in this case, tang. ^" sin. (a" — «') — tang. ^' sin. («'" — «") 

 — tang. 0'" sin. (»" — a) zi ; which may be shown by making 

 ^ the distance of the comet in longitude from the intersection 

 of the great circle in question with the ecliptic, and its inclina- 

 tion to the ecliptic (a. ; so that tang. = tang. /* sin. <Py tang. 

 |S"= tang. /A sin. {(p + «" — <»') and tang. $'" zz tang, {a sin. 

 [p 4- ec" — a) ; and, substituting these values in the equation, 

 we obtain sin. [<p + »" — a') sin. («" — a) — sin. q> sin. («'" — 

 »"j — sin (9 4- a" — «') sin. a" — »'), which is obviously zz 0, 



