ON THE CAPTDKE OF COMETS BT PLANETS. 519 



direction of Jupiter's quit.' The place of encounter with Jupiter will be 

 near an apse of the comet's resulting oi-bit about the sun, The comet 

 leaves the planet with the relative velocity v^, so that if s < 1 the motion 

 about the sun in the new orbit will be direct ; if s > 1 the motion ia the 

 new orbit will be retrograde. That is, by (4) when w < ^tt the resulting 

 motion is direct ; when m > ^ir the resulting motion is retrograde. 



In the second case the angle 2a, being greater than 180°, stands for 

 the angle between the asymptotes exterior to the orbit. Hence the 

 comet passing behind the planet will be turned forward and will leave 

 the planet in the direction of Jupiter's goal, and have a velocity that will 

 send it permanently out of the solar system. 



21. The results of Art. 20 assume that w is given. To find for what 

 value of u) the period of the resulting orbit is the shortest possible we 

 may put As^ = mr and 1 — s^ == 2.s cos 6 in (15), so that 



^ 1 - s2 ± 2s 



To find the minimum for @ place -^ = in this equation. This gives 



s = db 1, in wbich result, since s is inherently positive, only the positive 

 sign is used. But when s = 1, @ ^ ^r, /i = mr, and w = ^tt. Hence the 

 greatest effect of perturbation of a planet moving in a circular orhit in 

 shortening the periodic time of a comet originally moving in a parabola is 

 obtained if the comet's original orbit actually intersects the planet's orbit at 

 an angle of 45°, and if the comet is due first at the point of intersection at 

 the instant when the planet's distance therefrom is equal to the plaiiefs 

 distance from the sun multiplied by the ratio of the m.ass of the planet to the 

 inass of the sun. 



The relative velocity of the comet on leaving the planet's sphere of 

 action would be equal to, and directly opposite, the planet's velocity 

 (s ^ 1), and the comet would be left entirely at rest to fall to the sun. 

 This case could not happen for planets like the earth where mr is less 

 than the semi-diameter of the planet. In the case of the earth mr is less 

 than 300 miles, and actual collision would result. But for Jupiter mr is 

 greater than the distance of the second satellite from the planet. The 

 nearest approach of the comet to the planet would be mr (y 2 — 1), which 

 is more than four times the radius of Jupiter. Hence this case of 

 maximum diminution of major axis could occur near Jupiter. 



22. Isergonal ellipse for o) = 10°. — If we make <d = 10° the vanishing 

 points of the isergonal ellipses will be (Table II.) at d = 0, h = 01250 

 and d = 0,h= - -151 74. In fig. 2 let OE and OH be the axes of d and h 

 respectively. The vanishing points will be on the axis OH at distances 

 h' and h" above and below O. Upon this diagram are shown the halves 

 of four isergonal ellipses. The scales used for d and h are not equal to 

 each other, since the use of the same scale for both coordinates would 

 make the figures of inconvenient shape. In this and in all the figures 

 2-18, the unit in d is to the unit in A as 1 to sin (o. But to indicate 

 more clearly this scale, and at the same time to give a kind of shading 

 to a part of the area, there are drawn above the radical axis ae lines 

 parallel to OE, and parallel to OH, at intervals of "01 ; that is, the sides 



' The goal and the quit of a moving body are those two points on the celestial 

 sphere towards which and /row which the body is moving. 



