HI, A, Newton— Origin of Comets. i77 
action has its velocity diminished, the angle between the directions 
of the motions of the two bodies about the sun is increased in all 
cases in which that angle is at first greater than 45°. 
80. Again, when is less than 45°, and the planet before 
disturbance moves in a parabola so that v=, 2, then 
V—v, cos w is positive for all values of w unless V<y, that is, 
unless v+v’<2v,, in other words 2d>2v,( /2—1)=0°8282,. 
Therefore, when a planet overtaken by a comet, the directions 9 
their motions differing less than 45°, hie its velocity apeiaio 
that difference of direction will be increased unless the comet comes 
so near to the planet as to lose by its perturbing action a part of as 
velocity at least equal to about gths of the planet's velocity 
3 erefore, it is only” in exceptional cases (8, 29, 30) 
that the shortening by a planet's action of the periodic time of 
a comet moving in an orbit of long period is not connected 
with an increase of the angle of divergence of the two motions, 
and a consequent tendency to increase the inclination of the 
two orbits. In the exceptional cases the comet overtakes the 
hoor , passes around close to and in front of it, and is thus left 
ehind with an absolute veldeit} and hence a periodic time 
much less than that of the planet, and with a direct motion 
32. On the other hand, for given values of % and io and 
w>90°, the smallest value of v’ corresponds to w’=180°, 
v= Uy—r. the comet approaches in a parabolic ‘orbit 
v=v,,/2, and we have for the smallest value of v’, 
(v*-+v,’—2uv, cos kage =i | ste sbe cos = ea 
at by a ‘planet’ attraction acting oe a single passage 
be red parabolic orbit to one whose periodic time 
is less yew oot of the planet. 
the planets at first a trifle 2 than if moving in a parabola. 
If one of them does not lose velocity, or if becsing hind a 
never to return, Bat if it passes in front: of a large p 
Within a moderate. distance of it, it loses velocity meng 
remain a permanent member of our s system. 
Most observed comets have on sia: hypothesis thus. lost 
