ox THE CAPTURE OF COMETS BY PLANETS. 529 



38. This expresses the elemental number of comets corresponding to 

 the elemental area Te. The integral of this expression, that is, ^nvf(f>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 

 @ <?• and u)' <BAS. When the elemental area does not extend from the 

 arc DS to the line BA, the area of another appropriate isergoual curve 

 is to be used in determining <^. 



By Art. 17 we have 



^=^ra^-^4@^- ^ @-^-@-y i 



For the elemental areas of the surface AFDS which end on the arc 

 DS we make @=r, and let </)o be the resulting value of (j> ; then 

 <j}Q=irm^r^(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,--i-v'- — 2v'v, cos (j!)"^V^''=rS-Vi-. 



Again by the laws of gravitation, 



\ @J ' 



Hence 



«-=3-- -2 . / 2- 



@ 



^='-i-^^^-^<^o.^ 



@_ 3 — S- — 2 cos- a)"±2 cos w"(s-— sin^ w")* 

 r — 9-8cos2(o"-6s2 + s4 ' 



Let <^' and ^" be the t^vo values of ^ obtained by substituting in ^ these 

 values of @, (^" representing the value for the point nearer to A. 



39. If a>"=90°, and therefore cos oj"=0, we have along the limitino- 

 line the two values of @ equal ; hence 



^-3--,and<^'=^^g.-^^, 



so that the number of comets having quits less than 90° from Jupiter's 

 quit and @<r is 



_Trnvm-r- ,fy ,,-,. .^7A^o ^^ ■> 

 = — — (7+v-) = v012 irnvm-r^. 



Sincethe whole number of such comets is (Art. 33) equal to '925 irnvmh^, 

 the number of comets the distance of whose quits from Jupiter's quit is 

 between 90° and 180° is -224 Trnvmh-'^. The number of the comets for 

 which @<r that have inclmations to the ecliptic less than 90° is to the 

 number that have inclinations greater than 90° as 701 is to 224. Or'the 

 1891. M u 



