384 Astronomical and Nautical Colkclions. 



cos j; cos y- cos (r+y) _ p-a ^^^ ^ _ jp^b_ . ^^^^^ -^ 

 sin X aes'm x be sin x 



the same manner sin : = J—, — cos z — J—, — -. Now sin 

 ae sin x ce sin x 



p — a 



(r — y) = sin z cos y — sin y cos ;: = cos y cos z — 



-b 



P ^ cos y — -JL—— cos z/ cos z + -^-^ cos z = -i- 



iesin X be sin a: 

 cos z — ^~^ cosy, consequently?- — ^sin(z— y) = ^- 



uc sin .c 



sin and f- sin (z — w) — ^ sin z + -^ sin y =: sin (z— y) 

 •i b c 



, sin ?/ + sin (z — ?/) — sin z 



— sin z + sin y ; whence p = -: — : ■, — ■ = — ' : > 



sin y sin (z' — y) sin z 



c a 



b 



which is Dr. Olbers's formula. Tu.] 



§. 81. 

 It will seldom or never occur to us to compute the elliptic orbit 

 for the sake of any material utility or advantage. The portion 

 of the orbit, which is in the neighbourhood of the sun, may almost 

 always be so accurately determined by the parabolic hypothesis, 

 that we can represent and predict and judge of its course, and 

 its distance from the earth and the sun, and identify it upon 

 a second appearance with sufficient precision. And this appears 

 to me to be the whole object of the calculation of a comet, since 

 the determination of an elliptic orbit can never ascertain the 

 period of revolution with any certainty, the deviations from the 

 parabolic orbit being always complicated with the errors of ob- 

 servation : and these errors are certainly in many cases greater 

 than could have been imagined, partly from the nature of the 

 light and from the form of the comet, and partly from the imper- 

 fections of our catalogues of fixed stars. The comet of 1770, 

 indeed, seems to afford a remarkable and well known exception 

 to the universality of this observation. But without wishing to 

 decide upon this point, we may at least remark, first, that the 



