

410 H. A. Newton — Effect upon the earth's velocity 



2. Assume in the first place that the earth is at rest and that 

 a group of small bodies each weighing m' pounds are evenly 

 distributed through space ; that they are all moving parallel to 

 each other so as to have a common velocity v relative to the 

 earth on entering the sphere of the earth's sensible attraction ; 

 and that there are at first n of these bodies in each cubic unit 

 of space. Let the space considered be a cylinder whose axis 

 has the direction of the motion of the bodies and passes 

 through the earth. Each of these bodies will describe a hyper- 

 bolic orbit about the earth, and leave the sphere of the earth's 

 action with the same velocity v with which it entered. By 

 reason of the smallness of the bodies, and their even distri- 

 bution, the action of the bodies on each other may be dis- 

 regarded. 



3. Because the bodies are assumed to be evenly distributed 

 they will have no resultant action upon the earth at right 

 angles to the original direction of motion. But in that direc- 

 tion of motion the momentum of the whole system of earth 

 and meteoroids will be the same before as after the passage of 

 the bodies. What the meteoroids lose the earth gains. Let 

 the asymptotes of the hyperbolic orbit of one of these bodies 

 make an angle a with its conjugate axis. The momentum of 

 the body on entering the earth's sphere of action will be m'v 

 in the direction of motion. Its momentum in the same direc- 

 tion upon leaving the sphere of action will m'v cos 2a, since 

 the direction of motion has changed 2a. The loss of momen- 

 tum will be 77^(1 — 0082 a), and the earth gains by reason of 

 the transit of the body an equal' momentum in the same 

 direction. 



4. Let the perpendicular distance from the earth to the orig- 

 inal line of motion of a meteoroid be p. The number of 

 bodies that pass the earth in a unit of time and that have such 

 distance greater than p and less than p+dp will be 2npnvdp. 

 The momentum communicated to the earth by the whole 

 group in a unit of time will be the same as that communicated 

 through their whole orbits by those which pass the earth in a 

 unit of time, and this will be the integral 



flnpnv^m'^l—- cos2a) dp 

 taken between the proper limits. 



5. The factor 1 — cos2a is a function of p and v. To find its 

 value let p and v be the distance and velocity of the meteoroid 

 at perigee, let r be the earth's radius, and g=32fa feet. Using 

 feet-second units we have 



By conservation of energy v*— w 9 = -^— , (1) 



Po 



By conservation of areas v Po =v P-> ( 2 ) 



And by nature of the hyperbola p=p (tsm a + sec a) (3) 



