22 MEMOIRS OF THE NATIONAL ACADEMY OF SCIENCES. 



We see also from the last column of this table that of the 839 comets under consideration 267 

 liave quits less than 45° from Jupiter's quit, while only 38 of them have quits within 45° of Jupi- 

 ter's goal. 



41. Table iii gives the distribution of the comet quits relative to Jupiter's quit. It may also 

 be used to determine how many of the comets whose orbits are thus changed shall have an incli- 

 nation to the plane of Jupiter's orbit less than a given angle. 



Let the angle be 30°. Let Q be Jupiter's quit on the celestial sphere, Q' the comet's (juit, and 

 S the sun's position as seen from Jupiter. Then iu the triangle QQ'S put gj" for QQ' the distance 

 of the quits. The side QS = 90°, aud the QS(J' will be the inclination of the orbits. Represent 

 this angle by i and the angle Q'QS by ij. Then .vm ;/= cot a>" cot i. 



Let two small circles be drawn about Q at distances co" and a)"+da)" then if d ,i" be made 

 15° the numbers in the second or third columns of Table ill indicate how many quits are in the 

 several zones of 15° on the celestial sphere. These may be distributed at smaUer intervals than 

 15° by known processes. All the quits that lie in the lune between two semicircles drawn through 

 S so as to make angles of 30° with QS will evidently have orbits inclined less than 30° to Jupiter's 

 orbit. From &>" = to &)" = 30° all the quits are included in the lune. From ca" = 30° to 

 co" = 90° we comj)ute ?/ from the equation sin ?/ = cot co" cot 30°; then the portion of the quits in 

 any elemental zone that fall iu the lune is to the whole number of quits in that elemental zone as 

 this value of t/ 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 3(P to the or))it 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, somewhat more than five times as many of these comets move in direct orbits inclined 

 less than 30° to Jupiter^s orbit as move in retrograde orbits inclined less than 30° to Jupiter''s orbit. 



43. The comet has been thus far considered as approaching Jupiter while moving in a para- 

 bolic orbit about the sun. If the comet, however, is moving in any otiier 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 paraboUc, 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 

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

 comet, aud the comet will suffer a new and perhaps a lai'ge perturbation. Its i)eriod will again 

 be changed, being shortened or lengthened according as the comet jiasses 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 iu period by several passages 

 near to Jupiter instead of its being done at one passage. A much larger proportion of comets 

 than 839 out 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 oi'bits that they will return only at very great intervals of time. This class of comets 

 can not be therefore regarded as permanent members of the family of short period comets, except 

 such of them as happen to come so near to other jjlauets as to have their orbits changed in such 

 wise that they do not have thereafter the near approach to Jupiter'o orbit. But when an orbit is 

 greatly inclined to the plane of the solar system the comet passes through tiie plane in general at 

 a considerable angle and the chance of coming close to another planet is relatively small. 



46. On the other hand all the comets which after perturbation are moving in orbits somewhat 

 but not greatly inclined to the ecliptic are liable to meet, in fact are sooner or later almost certain 

 to meet other planets in such a way as to suffer perturbations that will prevent future close 



