244 A tftfW *«<* UNIVERSAL SOLUTION 



by the comet, in the fame time as the fquare root of the para- 

 meter of the parabola to the fquare root of the parameter of the 

 ellipfe ; that is, as y/ip to y/a (i — e* ) • or as y/zp to ^/p ( i + i) ; 

 or as y/2, to y/i + £ ; or, finally, as y/i + a to r. But the area, 

 defcribed by the comet in the eccentric ellipfe, that is, the fee- 

 tor of the ellipfe cut off by the radius vector g>, is equal to the 

 fector of the parabola, cut off by the radius vector r : There- 

 fore, the fector cut oif by a radius vector drawn to the body, 

 defcribing the parabola by the folar force, will be to the fector 

 cut off by the radius vector r, in the proportion of ^/i + \ to J. 

 Now, in the table of the motion in a parabola, the arguments 

 of the true anomaly are no other than the areas cut off by the 

 radii vectores ; or, which is the fame thing, they are numbers 

 proportional to thofe areas : Therefore, if the argument of the 

 true anomaly of the body in the parabolic trajectory, found for 

 the given inftant of time, be diminifhed in the proportion of 

 ^/i + \ to i , we fhall have the argument, which correfponds in 

 the table, to the angle z required. 



We have, therefore, this rule for finding the angle z at any 

 given time, by means of the table of the motion in a parabola * : 

 Let d denote the interval between the given time and the paf- 

 fage over the perihelion, expreffed in days ; then z will be the 



angle in the table that correfponds to the argument, — ■ x — — -. 



pT i/l-j-X 



Having thus found the angle z in the parabola for the given 

 time, we mufl apply to it the equation in the formula of Art. 1 8. 

 to have the true anomaly in the eccentric orbit f. 



5- In 



* Vide Table Generale du Mouvement des Cometes, Aftronomie de La. Lande, 

 torn. iii. p. 335. 2d edit. 



f It may be remarked, that the angle % is altoays lefs than the angle v, and 

 that the equation to be applied to 2 is always additive, 



If 



