Of KEPLER'S PROBLEM. 245 



20. In calculating the place of a comet, as feen from. the earth, 

 the aftronomer has occasion to, compute, not only the heliocen- 



trie 



If Ave compare the motion of the comet in the eccentric orbit, immediately to 

 the motion of the body in the parabolic trajectory, it is obvious, that the angular 

 velocity of the former is lefs than the angular velocity of the latter, at the peri- 

 helia of the curves : therefore, fuppofing the two bodies to pafs over the perihe- 

 lia together, the body in the parabola will advance before the comet. But, as 

 the radii veclores in the ellipfe increafe at a flower rate than the radii vecbores 

 in the parabola, the angular velocity in the ellipfe will increafe at a fafter rate 

 than the angular velocity in the parabola, in order that the areas defcribed in the 

 fame time may preferve their juit proportion. Hence it is clear, that the angu- 

 lar velocity of the comet will, in the firft place, become equal to the angular ve- 

 locity of the body in the parabola, after which the former body will gain upon 

 the latter ; the difference of the true anomalies will become lefs and lefs, and will 

 at laft vanifh, the two heliocentric places exactly coinciding. 



If we denote by v the true anomaly, common to both the ellipfe and parabola, 



when the heliocentric places coincide ; and if x = tan -, it will not be difficult 



2 



to deduce, from the reafoning above, the following equation for determining x, 

 viz. 



which is eafily reduced to this, 



and, if we neglect the quantities multiplied by h and its powers, we fhall have 

 fjmply, 



1 x l 2 a-* 



""22 c 



whence x = ,/* /l0 ^ $ = 0.8098 = tan 39 , nearly. 



> o 



Therefore v = 78 : and, whatever be the eccentricity of the orbit, provi- 

 ded it be very great, the heliocentric place in the ellipfe will, at this diftance from 

 the perihelion, coincide with the heliocentric place in the parabola : nearer the 

 perihelion, the true anomaly in the ellipfe will be lefs than the true anomaly in 

 the parabola : and, more remote from the perihelion, the true anomaly in the 

 ellipfe will be greater than the true anomaly in the parabola. I need not re- 

 mark, that this conclufion is not to be underftood with the utmofl rigour ; for we 

 have arrived at it by neglecting the quantities multiplied by the fmall fraction >. 

 and its powers. 



Vol. V.~ P. II. I i 



