Fig. 19.

A

DETERMINATION

OF THE

Difference between the Motion of a Comet in anElliptic, arid a Parabolic Orbit.

jtOl$n»i;ET PNG be a parabola, and PBH a very excentricalE -L ellipsis, having the fame focus S, and vertex P withLjlJ the parabola; let moreover N and n be considered ascotemporary positions of two bodies, in these orbits,moving from the perihelion P at the same time, about the funin the focus 8. Make NBC perpendicular to PSO, and callPC, x ; PS, c ; and the greater axis of the ellipsis, a : then

the lesser axis will be — 2 ^/ c x a — r, the parameter — lf£ndff j

and the ordinate BCty of the ellipsis) =

f ac — cc x ax —;

— 2v-

(by the proper-

2C'X* XI-

2 I a 1

2 C'X* X

1 - - -nearly. This last taken from CN (— 2 c 2 x\

by the nature of the parabola) leaves c'x* x — -f- — = BN;

which being multiplied by x and the fluent found, we thence

have sbc* x — 4 - — for the measure of the area NPB, orZ-r 5 <r.

NP-u, very near : which subtracted from the area NSP (—

CN x jPC — CN x 7CS = 2c'x* X jx — 2 c^x* x ~x—~c —

c i x i x leaves c^x 1 x c -f- -{-x —-r= the area

2-PS. Moreover, the parameter of the parabola being 4 c, and

that