20 MEMOIKS OF THE NziTIOiJ^AL ACADEMY OF SCIENCES. 



the relative 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 semicircumference KHT to A. If the velocity is increased, the new velocity 

 will be represented by a liue to A from some point in the arc KH; if diminished, by a line to A 

 fi-om some point in the arc IIT. If the new velocity is less than the planet's velocity, and so the 

 new coinetic period less than the planet's period, the new velocity will be represented by a line to 

 A from some point in the arc ET. If in a diagram constructed for (y= 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 represented by lines drawn from points in ET to A, while 

 comets for which d and h represent points outside that isergonal curve will, after perturbation, 

 have du-ections 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 'Z=r. Let the angle BAS represent a limitingValue oj" 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 wiU 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 oj receive an increment da='Sh 

 and let a new semicircumference be drawn with Bh as radius. To the elemental arc H^ will cor- 



Fig. 19. 



respond the elemental area along the semicircumference 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 

 Jupitei's motion between gj and a)+d&} will be equal to the area of the isergonal curve for 'S=r 

 multiplied by the elemental Jiumber |« sin &)fZm, and by the relative velocity Vg sin of the 

 comet perpendicular to the isergonal area. If the area of the isergonal curve be represented by 

 ^/s^ sin 6, then this product will be 



$ . ,, n sin cjd'M nv ^ , 

 s^iiO • ■'" «^" ^- 2 ^T"^'^* 



since v'2!'o=s!', and \/2 sin codoo—sds. 



38. This expresses the elemental number of comets corresponding to the elemental area Te. 

 The integral of this expression, that is, \nrf'l>ds, so taken as to cover the area AFDS will give 

 the number of comets which in a unit of time will pass the planet in such a way as to have '5 <»• 

 and &)'< BAS. When the elemental area does not extend from the arc DS to the line BA, the 

 area of another appropriate isergonal curve is to be used in determining <P. 



By article 17 we have 



For the elemental areas of the surface AFDS which end on the arc DS we make ®=r, and 

 let ^o he the resulting value of <?; then 0„=;7-mV'(4-s^). 



For elemental areas that end on the radius AS the values of® on that line are functions of s. 

 To compute them let v' be the comet's velocity in its orbit about the sun, and hence equal to the 

 distance of the point on AS from A; then by the triangle of velocities 

 v''-\-v''^—2v'v. cos a)"=Vo^=sH^. 



$= nm^ 



