ON SnOOTING STARS. 



19 



^3 



By the supposition the absolute motions would evidently be directed from all parts of the 

 tial sphere in equal numbers. Consider first all those bodies that have a given absolute 

 velocity v' not less than the earth's velocity v. Let 0, figure 5, be the point 

 to which the earth is moving, and 00' a lune of the celestial sphere, thru a .< 

 meteor whose absolute motion is from some point of the area abed will have its 

 relative motion from an area ABCD nearer to 0. Let OA=OB = #, and v" 

 represent the meteor's relative velocity. Let n be the number of meteors 

 coming from all parts of the celestial sphere, v! the number whose relative 

 motions arc from OAB, and d <p be the angle 0, then we have, 



M /=^ (1 _ cos0 «). 



But by the law of composition of motion we also have, 



or 



and 

 Hence, 



and 



v' 2 = v 2 +v" 2 —2 v v" cos 0, 



v" = v cos ± (v l - v 2 sin 2 0) , 



v' cos Oa = »" cos 0— v. 



1 — cos Oa = l+^-siu 2 0=Fcos ° — ^siu 2 0) , 



v V U 



vth" ( v . „ , i> 2 . , ,* 1 



»'= — — < 1+- : sm 2 0±cos0(l s am 2 0) y 



4 - ( v' V ) 



Since v' is greater than v, or equal to it, the sign of the radical in the value of v" does not 

 change. 



Let now Z be the zenith of an observer, be the point to which the 

 earth is moving, ACA' be the horizon, OZ be «, AOC be <p, and AOB be 

 dtp. The number of meteors whose relative motions are from the two 

 vertically opposite triangles OAB and OA'B', is obtained by adding to I'l 

 the value of n' given above its value when is made tc — 0, which gives 



—J- (1 -4- — sin 2 0). Those visible by any observer must come from points 



above the horizon. Let it be assumed that the number of those that are actually visible is 

 proportional to the number of those coming from the visible hemisphere. If N represents 

 this latter number, then 



But tan #= cos a sec <p, since ACO is a right angle, and hence 



An 



Siu 2 — 



cos' a set- a 

 I + cos 2 a sue 2 v 



Substituting and integrating we have, 



"KT " It . " \ 



N = - (1-+-. cos a). 



(309) 



