ON THE CAPTURE OF COMETS BY TLAXETS. 531 



then if c?oj" be made 15° the numbers in the second or third columns of 

 Table III. indicate how many quits are in the several zones of 15° on the 

 celestial sphere. These may be distributed at smaller intervals than 15° 

 by known processes. All the quits that lie in the luno between two semi- 

 circles drawn through S, so as to make angles of 30° with QS, will evi- 

 dently have orbits inclined less than 30° to Jupiter's orbit. From m" = 

 to to" = 30° all the quits are included in the lune. From co" = 30° to 

 (o" = 90° we compute r] from the equation sin ■>] = cot w" cot 30° ; then 

 the portion of the quits in any elemental zone that fall in the lune is to 

 the whole number of quits in that elemental zone as this value of r) is to 

 90°. These may be summed by finite summation, and the result is that 

 among the 839 comets 257 would move in orbits inclined less than 30° to 

 the orbit of Jupiter. 



42. If a like summation be made for the equal lune that contains 

 Jupiter's goal we find 51 to be the number out of the 839 comets which 

 move in orbits inclined more than 150° to Jupiter's orbit. That is, some- 

 what more than five times as mamj of these comets move in direct orhits 

 inclined less than 30° to Jupiter's orhit as move in retrograde orhits inclined 

 lets than 30° to Jupiter's orhit. 



43. The comet has been thus far considered as approaching Jupiter 

 while moving in a parabolic orbit about the sun. If the comet, however, 

 is moving in any other orbit, and it passes near to the planet, the result 

 of the planet's perturbing action will in general be quite similar to the 

 result when the orbit is parabolic, the other circumstances of the approach 

 being assumed to be alike in the two cases. 



44. These are perturbations during one transit past the planet. But 

 the comet, unless the orbit is further changed by another planet, must 

 return at each revolution to the place where it encountered Jupiter. At 

 some time Jupiter will be nigh that place nearly at the same time as the 

 comet, and the comet will suffer a new, and perhaps a large perturbation. 

 Its period will again be changed, being shortened or lengthened according 

 as the comet passes before or behind the planet. This process will be 

 repeated again and again, since after any number of encounters the new 

 orbit of the comet will still pass near to the orbit of the planet. 



This repeated action makes it possible to have an orbit shortened in 

 period by several passages near to Jupiter instead of its being done at 

 one passage. A much larger proportion of comets than 839 ont of 

 1,000,000,000 might therefore have their periodic times reduced below 

 the period of Jupiter. 



45. If the comet's orbit is largely inclined to the ecliptic, and hence 

 its motion makes a large angle with that of Jupiter, the diagrams figs. 

 10-18 show that there is nearly an even chance that the velocity will be 

 increased or diminished. A considerable fractional part of the whole 

 number of such comets will at each passage be thrown out of the solar 

 system altogether, or thrown into such long orbits that they will return 

 only at very great intervals of time. This class of comets cannot be there- 

 fore regarded as permanent members of the family of short period comets, 

 except such of them as happen to come so near to other planets as to 

 have their orbits changed in such wise that they do not ha^'e thereafter 

 the near approach to Jupiter's orbit. Bat when an orbit is greatly in- 

 clined to the plane of the solar system the comet passes through the 

 plane in general at a considerable angle, and the chance of coming close 

 to another planet is relatively small. 



M SI 2 



