the Orbits of Comets, id 



Having thus the values oi x, y, and r, we shall form the 

 quantity 



{„, . , . , sin(^— a^l 

 (i?'-)).cos(^-a) -^-'S 



+ i?oj;.sin (^— a) + J?.(i?'— 1). 

 The perihelion distance D of the comet will be 



the cosine of the anomaly u of the comet will be given by 

 the equation j^ 





from which we shall conclude, by the table of the move- 

 ment of the comets, the time consumed in traversino- the 

 angle u*. In order to have the instant of the passage by 

 the perihelion, we must add this time to the epoch, if P is 

 negative, and subtract it if P is positive, because in the for- 

 mer case the comet approaches the perihelion, and in the 

 second case it removes from it. 



With respect to the second comet of 1781, the epoch 

 being fixed as above, on the 19th November, at 8*^ 29' 44", 

 we have at this epoch 



A = 57° 5f 4", 

 i? = 0,987248, R = 0,988820, 

 the equations (1), (2), (3) and (4) become thus 



r^ = 1,667387. x' - 0,7106137.x -f 0,974653 ...(l) 

 ^„ , 10,6iS4 

 y = - 11,0665 -i ^ — + 3,9927. :»;... (2) 



0,03931687 

 y = o, 771014. x-i -^ 0,0408605 3... (3) 



O = y- + 0,Ool 87057. a;\ 



-hXO,Sl69372.3/ — 3,691334-0;]'— l,8320446.y... (4). 



2 

 + 0,0324357.:c -}- 1,026006 . 



• Call, as above, D the perihelion distance of the comet, U'it% anomaly, 

 t. e. the angle formed by its vector radius with the axis of the parabola 

 which it describes, lastly t, the time passed since the perihelion passage. 

 This being done, according to the liws of parabolic motion, the time t and 

 the anomaly U are united log-ether by the following ties.- 



D'ir ( ) 



{l)....t = — -^i tang iU + i tang3} U\ , 



m which » IS the demi-circuiiiference, or 3,14159'2()5, and T the duration of 

 the sidereal revolution of the earth, or 365 days 256383. i/ being given, it 

 is easy to calculate / by this formula. But if / is given, tlie search for the 

 tang ? U requires the resolution of these equations of the 3d degree. In or- 

 der to a\-oid this difficulty, astronomers have forn:ed a table of valuesof t in 

 a parabola in which D will be equal to unity, and from this table when once 

 calculated, we may extract the values of U ; < being known. This is what 

 we call a table of the motion of comets. We may supply the place of this 

 table by resolving the equation (1) by some trials. _/^g 



