Sou-ditch on the etements of the orhit of a comet, 6B 



tude of the lunh K + mP^ aod inclination I + w Q, whicli will 

 therefore be respectively 



T+itt.(/-T)+n.(r-T); 



and as the true longitude of the node is by hypotlicsis K + 7n P, 

 aHd tlie true inclination I-f 7i Q, the preceding values must corres. 

 pond to the true orbit. Therefore if we put the first of the precede 

 ing expressions equal to the time S actually elapsed between tlie 

 first and third observations^ as found by observation^ and the sec- 



C 



ond expression equal to — , the ratio of the observed times elapsed 



betw^een the first and second^ and the second and third observa- 

 tions, which, as is well known, expresses also the ratio of the are- 

 as described by the radii vectores in the same times we shall have 



S= T + m.{t^T) + n . (r— T) ; C - G + w . (^— G) + n . (y— G) 



which by transposition become 



G — C=:«i.(G— ^)+n.(G — y). ^ 



whence we may determine m, n, and thus obtain the corrected ele- 



* 



ments of the orbit. 



The equations of Newton for finding m, n, are 



2T-2S = m.(T-.0+«.CT-r),. . 



2G— 2C = m.(G— -)-Fn.(0 — v). 



and as the left hand sides of these equations arc double those of 

 the corresponding equations [2), it is evident that Newton's rule 

 will make the values of m -}- n double what they ought to be. 



The truth of the equations*(.^) which I have computed will al- 

 so be evident from the following simple case. Suppose that the 

 true orbit was obtained at the second operation, then the general 

 values of the longitude of the node, K + m P, and the inclination 

 I + 7iQ, will become K + P, and I ; that is K + mP = K + P. 





