Art. XV. ASTRONOMICAL AND NAUTICAL COL- 

 LECTIONS. No. VII. 



i. An Essay on the easiest and most convenient Method of 

 calculating the Orbit of a Comet from Observations. By 

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

 [Continued from Vol. XI. p. 182.] 

 Section II. 

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

 been proposed for determining the Equations of Comets. 



§ 40. 



Now the chord k" being (§ 7) == ^ {[x'" — x'}^ + Ql'" —y'Y 

 -\- (z" — z')^) ; if we develops this formula, and remember 

 that r'3 == a'* + j/'^ + ^''\ and r""^ == x"'® +?/'"- +z"' -, we shall 

 find K' — v' C*" ' + ''"^ ~ --'^ ^'" — %' y'" — 2^' ^" ) • '^""i since 

 we had x' = ^ cos a.' — R' cos A', ;/' = ^' sin a! — R' sin A' 

 z — g' tang ; and x'" =: M^' cos a'" — R'" cos A!",y"' = M ^ 

 sin ct" — R'" sin A"', z'" t^ M^' tang &"' ; we obtain x x" + 

 y y"' = R' R'" cos (A'" — A') — g' R'" cos (A'" — a') - M/ 

 R' cos (A' — a!") + M§'* cos (a'" — a' ;) and z' z'" = Mg'* 

 tang ^' tang /3"' ; so that the whole equation becomes k"^ = ?'* 

 + r"'« - 2 R' R'" cos (A"'- A') +2 5' R" cos (A"'-«') + 2M 

 §' R' cos (A' — a") - 2 M/2 cos (a'" — «') — 2 Mg'* tang ^' 

 tang B'" : for which we may write A = V (F + G^' + H^'^)- 

 §41. 



Now T being the time between the first and third observa- 

 tion, we have, from Lambert's very elegant theorem, 



T = j^ /[/•' + r" + k') ' - (/ + r" - k") \ But if we 



substituted, in this equation, the values of r', r"' and k", we should 

 arrive at an equation of enormous difficulty: it might, how- 

 ever, be reduced to the 12th degree by substituting for the 

 equation of Lambert the approximation of Duse'jour, 



T« =: i —-r ; and even to the 6th, by supposing 



r' -f- r'" r'* + r"'« 



— - — = ^ , which is, however, only at all aduiis- 



