determining the Orbits of Comets. 205 



in place of which one may also use these : 



tan (45° + «/) = \/ (j) 



1 . ^, tan2eo' 



-r-sinF = 



Vq sm*;(v"-v) frr" 



1 F .". sec 2 a/ 



7^ C0S 7 cos | cV ' -v)#r r" 



co = | (©" + d)-2 F. 



For the determination of «/, it will, however, be advisable to 

 use logarithms of more than five places. 



If we now enter with the true anomalies v —• w 9 v" — w 9 or 

 cu — v and u) — v", into Barker's table, or any one of the other 

 tables for the parabolic motion of comets, we shall find the 

 time which has elapsed since the passage through the perihe- 

 lion, or which will elapse till that moment; and the double de- 

 termination of the time of passage will be the last proof of the 

 correctness of the calculation. 



For Barker's table we have, if M and M" stand for the 

 mean motions which correspond to the true anomalies, 



(VII) T = / + M«f = <' + M"«f, 



where the constant log n = 0*0398723, and the upper signs 

 are to be taken for a direct motion if v ~7 <*>, v" ~7 «>, or for a 

 retrograde motion if w^w,uV w ; and the lower signs for 

 the contrary cases. 



In order to be perfectly satisfied of the correctness of the 

 calculation, it will now be advisable to calculate the place of 

 the comet at the time of the second observation, in order to 

 compare it with the one actually obtained by observation. By 

 the nature of the proceeding, one will easily perceive that this 

 middle observation was only used for calculating M, and even 

 there only one new quantity (the angle X) was employed, de- 

 rived from the two data of the observation, as the distance <r' 

 disappears from the expressions both of M' and M". The ri- 

 gorously conducted calculation will, therefore, even when the 

 complete value of p" is applied, give accurately only this angle, 

 or the quantity 



sin (a' - 0') 



tan 2'=z 



tan 8' 



and the effect of the errors on the middle longitude and lati- 

 tude will consequently be thus : 



