H, A. Newton— Origin of Comets. 175 
due to Jupiter’s motion. Let v be the comet’s relative velocity 
on entering the sphere of Jupiter's action, and py the perpen- 
dicular from Jupiter on its relative path at that time. From 
Kepler’s law 7°d@ = pvt, where @ is the angle in the plane of 
the comet’s relative orbit defining the place of the comet. Dur- 
ing the time of the comet’s transit past Jupiter his motion may 
be regarded as in a straight line. Project that line on the plane 
of the comet’s relative orbit, let 8 be counted from this projection, 
and call the angle of projection 6. Then cos gj)= cos é cos 8, 
and we have, 
pw dP = —2mfa*v, cos 6 cos 6 dé. 
Denoting the total change of P by 4, and the first and last 
values of @ by 6’ and @”’, we have, 
PY, 4 = 2mfa’v, cos 6 (sin #’— sin #” 
= 4mfa’v, cos 6 cos $ (6+ 6) sin 4 (7— 6). 
But 6”— @’ is the change of direction of the comet's radius 
vector, and is approximately that angle of the asymptotes of the 
hyperbolic orbit which encloses the curve. Denote it by 2a, 
observing that sin @ will in general be not much different from 
unity. Again 4(6’+ 6”) is the angle defining the perijove of 
the comet's orbit, and cosé cos 4(0’+ 0”) = cosg, where ¢ is the 
angle between the direction of Jupiter’s motion and the direction 
from Jupiter of the transverse axis of the comet’s orbit. Hence, 
pt 4mfa'r, 
Po 
cos p sin a. 
That is, the total decrease in the kinetic energy of a comet caused by 
the perturbing action of a planet during the transit of the comet past 
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the planet (mv,), the cosine of half the angle through which the comet's 
direction is changed by the planet (sin a), the cosine of the angle at 
the planet between the direction of the planet’s motion and the trans- 
verse axis of the comet's relative orbit (cos g), and the reciprocal of 
the constant area described in the unit of time by the comet in its 
relative orbit (2+ pov, 
motions are directed when they are near 2 
e — and S the direction ee the ne me 
m the planet. Ifthe planet be rega 
as describing a circle cy ie ecliptic, then ah 
SP is a quadrant, and CSP is the incli- § 
hation of the comet’s orbit. Denote CSP 
by 4, CP by w, and CPS by &. CP will =a 
be greater or less than a quadrant according as S or @ is aya 
or less than a right angle, and by trigonometry tan i=sin f tan @. 
