530 



EEPOET — 1891. 



839 comets spoken of in Art. 36, 203 vnll after perturhation have retrograde 

 motions, and 636 ivill have direct motions. 



40. If m" is less than 90° the expression to be integi'ated in oi'der to 

 cover the area SAFD will be 



Tsiu tu" C'2 sin Iw" [*! 



Uof^s + ((^0 - </>') ds + U"cls. 



J V"^— 1 J sin w" J sin w" 



If oj" is greater than 90° the corresponding expression becomes 



Ci sin w h 



\4>„ds-U 

 J v^-i J 1 



I sin \u 



I <^Qcis — I (ji"ds 



As the valne of @ introduces into <^' and ^" only one radical in s, and 

 that a radical of the second degree, these integrations are possible. 

 Finite summation is however more convenient. Computing the values 

 for each interval of 15° we construct the following table. The first 

 column indicates the interval in values of w" ; the second column gives 

 that coefficient of ^Trnvm^r^ that must be used to obtain the number of 

 comets which in a unit of time will pass perihelion nearer than Jupiter's 

 distance to the sun, shall also have their periodic times reduced to be 

 less than Jupiter's period, and ohall also leave Jupiter's vicinity so that 

 the distance between the quits of the two bodies is between the two 

 values in column 1 ; the third column indicates the distribution of the 

 839 comets of Art. 36 through the twelve zones. 



Table III. 



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

 under consideration 267 have quits less than 45° from Jupiter's quit, 

 while only thirty-eight of them have quits within 45° of Jupiter's goal. 



41. Table III. gives the distribution of the comet quits relative ta 

 Jupiter's quit. It may also be used to determine how many of the 

 comets whose orbits are thus changed shall have an inclination 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 quit, and S the sun's position as seen from Jupiter. Then 

 in the triangle QQ'S put to" for QQ', the distance of the quits. The side 

 QS = 90°, and QSQ' will be the inclination of the orbits. Represent 

 this angle by i and the angle Q'QS by rj. Then sin rj = cot w" cot i. 



Let two small circles be drawn about Q at distances m" and to" + dtn" ; 



