528 



EEPOBT — 1891. 



the snn, BA the j^lanet's velocity about the sun, and therefore HB the 

 comet's velocity Vq in its orbit about the planet before perturbation. About 

 B as centre describe the semi-circumference KHT. Since the relative 



Fig. 19. 



velocity after as well as before perturbation is equal to HB, therefore the 

 velocity of the comet about the sun after perturbation will evidently be 

 represented by a line drawn from some point in the semi-circumference 

 KHT to A. If the velocity is increased the new velocity will be repre- 

 sented by a line to A from some point in the arc KH, if diminished by a line 

 to A from some point in the arc HT. If the new velocity is less than the 

 planet's velocity, and so the new cometic period less than the planet's period, 

 the new velocity will be represented by a line to A from some jaoint in the 

 arc ET. If in a diagram constructed for w=BAH the isergonal curve 

 be drawn for @ = r, those comets for which d and h represent points 

 within that isergonal curve will after perturbation have velocities repre- 

 sented by lines drawn from points in ET to A, while comets for which 

 d and h represent points outside that isergonal curve will after pertur- 

 bation have directions of motion represented by lines drawn to A from 

 points in EHK. The number of comets having motions represented by 

 lines to A from points in ET will be proportional to the area of the 

 isergonal curve @=r. Let the angle BAS represent a limiting value w" 

 of distance of quits of comets from Jupiter's quit after perturbation. The 

 comets which are thus limited and at the same time have @<r will be 

 moving in lines directed to A from points in the area bounded by the 

 straight lines SA and AF, and the arcs FD and DS. Let u> receive an 

 increment c?to=H/i and let a new semi-circumference be drawn with B/i 

 as radius. To the elemental arc H/i will correspond the elemental area 

 along the semi-circumference KET. If ET lies wholly in SAFD the 

 number of comets that pass the planet in a unit of time having initial 

 angles of direction with Jupiter's motion between to and w + dm will be 

 equal to the area of the isergonal curve for @=r multiplied by the 

 elemental number V't sin (Ddo), and by the relative velocity v^ sin of the 

 comet perpendicular to the isergonal area. If the area of the isergonal 

 curve be represented by cjis^ sin 6, then this product will be 



<^ 



n sin wfZw nv , , 

 2-- = 4^^'' 



since \/2vo=ss?;, and ^2 sin uido}=sds. 



