produced by small bodies passing near the earth. 413 



of in the assumed right line, the integration may not be ex- 

 tended ad infinitum. For the earth's motion to and fro would 

 for the very distant meteoroids develop resistances in opposite 

 directions which would cancel each other, the action of the 

 planet upon the more remote bodies not being instantaneous 

 but requiring long periods of time for its development. 



9. The action of one of these bodies on the earth may also 

 be looked upon as like an impact. For we may consider the 

 hyperbolic orbit to be replaced by its asymptotes and the whole 

 action of the earth in changing the body's motion to be concen- 

 trated at one point, namely, the center of the hyperbola. The 

 reaction of the body upon the earth will be of the nature of an 

 impact in the line drawn to the center of the hyperbola. The 

 combined impacts of all the bodies would have a resultant in 

 the direction of the motion of the bodies. 



Again the action may be regarded as though the earth was 

 in motion and the bodies at rest, and that the earth drew the 

 small bodies around as it passed them into its own wake where 

 they exert a greater attraction than they did in front of the 

 earth. This concentration would not take place if the bodies 

 formed an elastic medium. 



10. Thus far the small bodies have been assumed to be at 

 rest or moving in one direction with one velocity. Let us now 

 extend our hypothesis and assume that the bodies have all the 

 same absolute speed cu, but that their absolute velocities are 

 directed to points evenly distributed over the celestial sphere, 

 that the bodies are as before evenly distributed in space, and 

 that the earth moves with a velocity u through the system. 

 The velocities relative to the earth will not be uniform nor 

 their directions evenly distributed. 



To represent these velocities draw AB = u, and about B as a 

 center with the radius cu describe a spherical surface CD. AB 

 will represent in amount and direction the earth's velocity, CB 

 the meteoroid's absolute velocity and AC the meteoroid's rela- 

 tive velocity. There will be two cases, according as A is 

 within or without the sphere: in other words according as c is 

 less or greater than unity. The distinction between these will 

 be considered further on. The meteoroids may be supposed to 

 come from points evenly distributed over the spherical surface 

 CD. 



Let the angle ABC = 0, BAC = <p and AC=m Then 



x i = l+c i — 2c cos (9, 



xdx=cs'm ddd, 



l-c'+x 2 , 

 cos q)=z . 



11. The bodies which move in directions which make angles 



