174 H. A. Newton— Origin of Comets. 
tional to the amount of Jupiter’s motion normal to the average 
direction of the relative orbit. 
[This is more definitely shown, if desired, and the quantity 
of the change is found by solving the equations of motion for 
this case. Let a be the unit of distance, f the sun’s attraction 
at the unit of distance, v the velocity of the comet in its orbit 
about the sun, r and 79 the distances of the comet from the sun 
and Jupiter, and m the mass of J upiter (sun’s mass = 1): then if 
Whiade* st i Lal 
any change in P will evidently change Go i ns time of the 
comet in its elliptic orbit, or by diminution of P the orbit may 
change from an hyperbola to an ellipse. 
rs an x, y, z, be the codrdinates of the comet relative to 
the X,Y, %, of the planet relative to the sun, 2, Y%, %, O 
the Sana relative to the planet; so thata=at+am y=yH%t+y 
and z= 4+ % Neglecting the comet’s mass we have for ‘ie 
equations of motion, 
a6) 
bed al fd 
Fe en Sa’ (5+ at A 
minep nay by ay ie and 2dz, adding and observing that 
= dx, + dx, dy = dy, + dy, dz = dz, + dz, we have by re- 
roche 
dP __—- 2mfa (ux, da, . y, dy, , 4% d, 
a cd ee me ee 
a 2infa’ (a, de: --y, z, dz,.\ds, 
me (et ee 7, de,) dt’ 
where ds, is the element of J upiter’s orbit about the sun. The 
quantit 7s the parenthesis i is the cosine of the angle at the 
dlanet between the comet’s radius vector and the direction of 
upiter’s motion, which angle we may denote by gp. The factor 
ds, . 
oz iS the planet's velocity in its orbit, which may be denoted 
by v; Then we have, 
qdP_—_ 2mfa’v, 
Hx TE 008 9 
The integral for P for the time that the comet is within the 
sphere of ; Jupiter’s special action (that is, while the comet may 
be treated as moving in a hyperbolic orbit about Jupiter, and 
the sun only as a perturbing body), gives the change in P 
