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



the earth per second in feet per second; 3 is the density of 

 the matter of the group if distributed through the whole 

 space considered, the mean density of the earth being 

 unity ; x is the velocity, the earth's velocity in its orbit being 

 unity ; and P is the radius of a cylindrical stream of meieor- 

 oids, the unit being 4056 miles. If the unit for P be taken 



equal to 4056 ( 1+ — j- ) miles, and we use common logarithms, 



the value of p becomes 



15-7S 



7. Equation (7) may be thus expressed as a theorem. 



If a cylindrical stream of small bodies evenly distributed 

 and all moving in the same direction with common velocity 

 shall move past the earth supposed to be in the axis of the 

 cylinder, the small bodies by the law of universal gravitation 

 shall communicate to the earth a velocity along the axis in 

 each unit of time ; 



(a) that shall be proportional to the density of the group; 



(b) that shall decrease as the velocity increases varying very 

 nearly inversely as the square of the velocity ; 



(c) that shall increase with the radius of the cylinder, and 

 shall be proportional to the logarithm of that radius, measured 

 by a unit that differs from the earth's radius by a small quan- 

 tity which is a function of the velocity. 



Strictly taken, if the cylinder have an infinite radius the 

 velocity communicated is also infinite; but if the radius be 

 comparable with measurable stellar distances, the velocity com- 

 municated is not large, since log 10 P would then not exceed 12. 



8. If we assume the bodies at rest, and the earth in motion 

 through them with a velocity v, p expresses the resistance of 

 the system to the earth's motion. Hence, if infinite space were 

 filled evenly with discrete material bodies at rest, and the law of 

 universal gravitation holds true ad infinitum, there would then be 

 exerted an infinite resistance to the motion of a planet in a con- 

 tinuous right line through the system. Moreover, a finite system 

 of such discrete bodies at rest constitutes a truly though feebly 

 resisting medium by reason of gravity alone and independently 

 of their impact with the moving planet. 



If the small bodies however should constitute an elastic 

 medium the above reasoning does not apply. For equation 

 (6) depends upon the property that the small body enters the 

 sphere of the earth's action in one direction, describes a hyper- 

 bolic orbit, and leaves it in another. This may not be asserted 

 of particles of an elastic fluid filling space. 



Again, if the planet moves in its orbit about the sun, instead 



