176 H, A, Newton— Origin of Comets. 
If we suppose a large number of comets to approach the planet 
so that the value of w is constant but so that the points from 
which their motions are directed are equably distributed on the 
circumference of the small circle whose pole is P, and whose 
distance from P is w, then cased the mean value of tan is 
1] 
a f* tan w sin Pdf, that is — * tan w.* Therefore the greater 
9 
the algebraic value of tan a, ate greater that of the mean value 
of tanz, Hence, also, the greater the value of w between 0° and 
180° the greater the mean value of ¢, between the same limits. 
Therefore we may conclude that wien the perturbing action of 
a planet upon neighboring comets increases the angle between the 
directions of the motions of the comets and of the planet, that uction 
tends likewise in the mean to increase the tnclinahon gt the comets’ 
absolute motion makes with that of the planet on entering the 
sphere by @, and on leaving the sphere by w’, Let also 
2V=v+v’', and 2d=v—v’. Then by composition of motions, 
oe: ta aa —2QWv V COS =D, *+-(V+d)’?—2v,(V+d) COs @ 
=v-+y"—20,0' cos c= 24(V—d) —2v,(V—d) cos w’. 
Hence reducing, we have 
cos @— cos G’= =ry- at yale’ Cos @). 
If now the action of the planet is to diminish v, both V—d and 
d are pclae and the sign of cos w—cos w’ depends upon that 
of V~zx,cosw. Hence @ is increased by that action when 
nag positive. 
If w>90° this quantity is positive, and at the same time 
#>90° (27). Since the action of the planet is to increase @, the 
resulting value of 7 exceeds 90°. Therefore, when the inclination 
of a comet's orbit is greater than 90°, and the velocity of the comet is 
reduced by the teers 2 action of a 1 planet near to which it comes, 
the orbit after the perturbation has an inclination greater than 90° ; 
and, if a large number of cases be considered, the mean effect of the 
urbations is to increase the inclinations. 
in, for a comet moving ina parabolic orbit we have 
v=v,/2. Hence, ters nad cos 45°, and esi Legh tip 
positive when w>45°. e, whenever any comet 
wn a parabolic orbit bie near to a planet and by the panels 
* The integration should not extend to the second semicircle, since the comets 
corresponding thereto belong to the other node. 
