18 MEMOlllS OF THE NATIONAL ACADEMY OF SCIENCES. 



If now w be the augle wliich the comet's unperturbed motion is making with tlie planet's 

 motion, and if v, or its equal, ^)/^/3, be the planet's velocity in its orbit about tlie sun tlien 



y-^rS -2 V 2 cos ffi>]. The element fZS may be taken to be the elemental zone between the two 

 ircles, whose common pole is the planet's quit, and whose distances from the planet's quit 

 and <i]+doj. Then dS='27i sin oo da. 

 with quits within that elemental zone will be 



small circles, 



in a unit ot time 



:^)(('„r'^x27r sin m dc 

 The integral of this. 



'V2 



; 1 



(3—2 ■v/2 cos &j) - sin i^doo. 



2V2" 



(3—2 v'2cos <.;)"sin w dc^ = ^Tnicr''^. 



expresses the total number of comets that, under the hypotheses that have been made, would, iu a 

 uuit of timej enter the sphere S'. 



31. If we compare the two expressions obtained in articles 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 6r' is to 7r'\ The factor i expresses the in- 

 crease of numbers caused by the planet's motion in its circular orbit. The value of >•', as has been 

 said, must not be too small, nor yet must it be very large. 



32. Iu order to determine the number N of comets which, in a unit of time will have their 

 periodic times reduced below a giveu period, we may make use of the isergoual curves represented 

 in Figs. 2-18. Although the diagrams were not constructed to exhibit the motions of the bodies, 

 yet they may be utiUzed 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 tlie point at which the comet's unperturbed orbit \^^ll 

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

 Hie comet's path upon the pUme HOB, and the comet's path makes with the plane the angle d. 

 The velocity of the comet perpendicular to the plane will be Vo sin 0. By reason of the hypothesis 

 that the comets are equably distributed, the points of intersection with the plane HOE will be 

 equably distributed over the plane. Hence the nund)er of comets whose quits are iu the element 

 (1^ of the celestial spliere and that will pass the planet in a unit of time in such a way as to have 



