526 BEPOET — 1891. 



The element cZS may be taken to be the elemental zone between the two 

 small circles whose common pole is the planet's quit, and whose distances 

 from the planet's quit are w and w + du). Then dS = 'ivrsin co doi. The 

 number of comets entering S' in a unit of time with quits within that 

 elemental zone will be 



hiv^r'''' X 27r sin w dw = ^ ' (3 — 2 V 2 cos w)* sin w dta. 



The integral of this, 



^!^ ["(3 _ 2 y 2 cos 0))^ sin w c?w = livnvr''^, 

 2^2 Jo 



expresses the total number of comets that, under the hypotheses that have 

 been made, would in a unit of time enter the sphere S'. 



31. If we compare the two expressions obtained in Arts. 27 and 30 

 we find that the number of comets which, in a given period of time, come 

 nearer to the sun than r is to the number that (unperturbed) come nearer 

 to the planet than r' as ^f- is to 7*"'^. The factor | expresses the increase 

 of numbers caused by the planet's motion in its circular orbit. The value 

 of r', as has been said, must not be too small, nor yet must it be very 

 large. 



32. In order to determine the number N of comets which in a unit of 

 time will have their periodic times reduced below a given period we may 

 make use of the isergonal curves represented in figs. 2-18. Although 

 the diao-rams wei-e not constructed to exhibit the motions of the bodies, 

 yet they may be utilised for that purpose. Let OH be the tangent to the 

 planet's orbit, O the place of the planet considered at rest, and let the 

 plane HOE contain the shortest line d between the two orbits. This d 

 will be the abscissa of the point at which the comet's unpertui-bed orbit 

 will cut the plane. The ordinate of the same point, produced if necessary, 

 will be the projection of the comet's path upon the plane HOB, and the 

 comet's path makes with the plane the angle 6. The velocity of the 

 comet perpendicular to the plane will be y^ sin 0. By reason of the 

 hypothesis that the comets are equably distributed, the points of intersec- 

 tion with the plane HOE will be equably distributed over the plane. 

 Hence the number of comets whose quits are in the element rfS of the 

 celestial sphere and that will pass the planet in a unit of time in such a 

 way as to have their periodic times reduced below a given period will be 

 equal to the area inclosed in the corresponding isergonal curve multiplied 

 by the velocity perpendicular to the plane, v^ sin Q, and by the factor 



!— . If @ is the semi-major axis of the orbit for the limiting periodic 

 time, the. area of the corresponding isergonal curve will be (Art. 17) 



For cZS we may, as before, take I-k sin w cZw, and we shall then have 

 ,x ■"■"f • r4»i-(§^ /2)7?@cos6 wrN^l, 



The integration must extend through the positive values of the 

 quantity in square brackets beginning at cu = 0. [In case co = gives a 



