Jstronomical and Nautical Collections. 37o 



present, because it has been so fully explained by Laplace 

 himself in the Memoirs of the Academy for 1780, and after him 

 by Pingre in his Cometographie. 



[Note of the Editor Von Zach. The work of Laplace Oti 

 the Motion of the Planets being rare, the editor thinks it advis- 

 able to insert his formulas, in order that all the methods o- 

 correcting the elements of the orbit may be found together 

 and it will give room for remarking the utility of constant loga- 

 rithms which are of use in repeating the computation upon 

 various hypotheses. "We find (1) the true anomalies ip', <p", <p"', 

 from the time and distance of the perihelium, for the respective 

 observations, by means of Barker's table; as well as the dis- 

 tances from the sun, r, r", r" . (2) Then making cos « =r cos ^ 

 cos (A — a) we find 



1st constant L. = Log R + log sin ^ 

 lid == Log sin Q — log sin k 



Hid — Log R + log sin (A - «) 



Quantities which are obviously independent of the time and dis- 

 tance of the perihelium, and therefore retain the same values 

 notwithstanding their changes. We then make (3) 



Log sin K = I c. 1 . — log r 



Angle S = K + z, or rather 180° - K — >; 



Log sin X -^L log sin £ + II c . 1 



Log sin angle at the comet i= IIIc . 1 . — log (r cos a) 



C = a ± this angle = hel. long, comet. 



(4) Reckoning x, x^ ^'^d % between the 1st and 2d, 1st and 

 3d, and 2d and third observations, we have 



Cos X = cos (C" — C) cos a' cos >." + sin a' sin a" 



Cos x — cos (C" — C) cos a' cos x"' -f sin x' sin x"' 



Cos x''— cos (C" — C") cos x" cos x'" + sin x" sin a"' ; two 



only of these formulas being wanted, and the signs and cosines 



having been taken out before. 



(5) Putting now x! — (<?" — <P') ~ ^' «"d x - (<P''' — <?') = ". 

 we must have, if the time and distance of the perihelium are cor. 

 lect, p. = 0, and v = 0. But as this will seldom happen, we must 

 alter first the time of the passage of the perihelium only, and then 

 llie distance ; and by comparing liie three values of jw, and v thus 



