.,j Of KEPLER'.S PROBLEM. -^45 



20. In calculating; tjie place. Qf a.cp&etj asieenfrom.tlie icajitib, 

 the aftf onorner has ,occafi.on to, compute, not only the helipceji- 



trlc 



If we compare the motion of the comet in tlie eccentric ortit, immediately to 

 the motion of the body in the parabolic trajeftor-y, it is obvioup^.that the angular 

 velocity of the former is lefs than tlie 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 veftores in the ellipfe increafe at a flower rate than the radii veftores 

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

 than the angular velocity in the 'para^ol?,, ifx order,tl}at the areas defcribed:m the 

 fame time may preferve their juft proportion. Hence it is clear; that the angu- 

 lar velocity of the comet wUl, 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 vanilh, the two heliocentric places exaftly coinciding. 

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



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



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

 viz. 



which is eafily reduced to this, 



-^(-7iTT)4-;3-']-'{f-r*}-3>{f.^}-.... 



and, if we neglefl: the quantities multiplied by h and its powers, we Ihall have 

 fimply, 



I x' 1 x^ 



2 2 5 



whence x =J)Lll^ ^ = 0.8098 = tan 39°, nearly. 



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

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

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

 perihelion, the true anomaly in the ellipfe will be lefs tlian 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 underllood with the utmoft rigour ; for we 

 have arrived at it by neglefting the quantities multiplied by the fmall fraftion ^ 

 and its powers. 



Vol. v.— p. II. I i 



