244 ^ NEW mid UNIVERSAL SOLUT'ION 



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

 meter of the parabola to the fqiiare root of the parameter of the 

 ellipfe ; that is, as ^2p to ^u (i — i' ) ; or as ^2p to y/p { i + i) ; 

 or as ^1 to f/i + t; or, finally, as ^i + a to i. But the area, 

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

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

 fedlor of the parabola, cut off by the radius vedlor r : There- 

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

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

 cut off by the radius vedor r, in the proportion of ^i + x to i. 

 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 vedlores ; or, which is the fame thing, they are numbers 

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

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

 the given inftant of time, be diminifhed in the pi'oportion of 

 ^/i + A to I , w^e iliall 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 W^l+>^ 



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

 time, we muft apply to it the equation in the formula of Art. i8. 

 to have the true anomaly in the eccentric orbit f . 



5. In 



* fiJe Table Generale du Mouvement des Cometes, Aftrohomie de La LandEj 

 tom. iii. p. 335. 2d edit. 



+ It may be remarked, that the angle z is alWays lefs than the angle »» and 

 that the equation to be applied to z is always additive. 



If 



