BowditcJi on the elements of the orbit of a comet. 



67 



nd then without any proof, they erroneously p 



G — c 



G 



m 



which ou£;ht not to be done except 





9 



particular case 



> 



re the true inclination of the orbit is accidentally usee 

 observation. It being very evident that if by simplj 

 5 the elements K. I into K + m P, I, the quantities T. G 



first 



the 

 ng. 



/ 



s.c 





/ obse 



men- 



true ones, consequently 



tioned elements K +?n P, T, must be the 

 the true inclination I must have been used at the first observation. 

 This is the main source of the error of their demonstrations. The 

 same process is used with the fust and third operations, using the 

 quantities n, r, y instead of m, f, g, and by the same erroneous 



..^ 



T 



method they get these equations ,^ — ^ = ^h and ri =m. By mul- 



1 



G 



tiplying these four equations by the denominators of the terms in 

 the left hand side they get 



T 



S 



m(T 



0, T-.Sz=n (T— ; G— C=m (G— j), G— C=u (G— y). 



The sum of the two former is 



2 T— 2 S=m (T— + n (T— r) ; 2 G— 2 C=fli (G— -) + n (G— y) 



which are the erroneous equations given in the Principia. 



SECTION SECOND. 



A method of correcting the elements of the orbit of a comet is 

 given by La Place, in Page 225 kc. Tol. 1 of his ^'Jlecanique 

 Cele'ste^^^ in which he first compules, in an approxi^native mr^nner, 

 the perihelion distance D of the comet and the time T of passing 

 the perihelion. These elements were selected with the expecta- 

 tion that they would afford a more 8imple calculation than any 

 other comhination, by avoiding superfluous operations. This meth- 



od, however^ when a great number of observations are combined, 

 leads to very laborious calculations; anil the simplicity of the com- 



