2 -J 2 ETHEE AND GRAVITATIONAL MATTER. 



Hence, for a spherical surface. (2) gives 



Q-Itt 3l /,\ 



= Y rp= ^ 



This shows that the average normal component force over the surface 

 S is infinitely great, if p is finite and r is infinitely great, which suffi- 

 ces to prove section 11. 

 Sec. 1 I. For example, let 



r=150.10\206.10 6 =3'09.10 16 km (5). 



This is the distance at which a star must be to have parallax one one- 

 thousandth of a second: because the mean distance of the earth from 

 the sun is L50,000,000 km-., and there are 206,000 seconds of angle in 

 the radian. Let us try whether there can be as much matter as a 

 thousand-million times the sun's mass, or, as we shall say for brevity, 

 a thousand-million suns, within a spherical surface of that radius (5). 

 The sun"- mass is 324,000 times the earth's mass, and therefore our 

 quantity of matter on trial is 3*24. 10" times the earth's mass. Hence 

 if we denote by g terrestrial gravity at .the earth's surface, we have 

 by (4) 



Q=3 '- 4 - 1,,1 ^!;o^i^) ;/=1 *' !7 - 1,rn - / • • • ,,;) - 



Hence if the radial force were equal over the whole spherical surface. 

 its amount would be 1'37 . 10 -11 of terrestrial surface-gravity; and 

 every body on or near that surface would experience an acceleration 

 toward the center equal to 



1 •:->7 . 1< |1: ' kms. per second per second . . . (7), 



because g is approximately L,000 cms. per second per second, or "01 

 km. per second per second. If the normal force is not uniform. 

 bodies on or near the spherical surface will experience centerward 

 acceleration, some at more than that rate, some less. Al exactly that 

 rate, the velocity acquired per year (thirty-one and a half million sec- 

 onds) would be 4*32. 10~ 6 kms. per second. With the same rate of 

 acceleration through five million years the velocity would amount to 

 21*6 kms. per second, if the body started from rest at our spherical 

 surface: and the space moved through in five million years would be 

 •17. L0 16 kms., which is only '055 of r (5). This i> so small that the 

 force would vary very little, unless through the accident of near 

 approach to some other body. With the same acceleration constant 

 through twenty-five million years the velocity would amount to 108 

 kms. per second: hut the -pace moved through in twenty-five million 

 years would he 4*25 . 10 16 kms.. or more than the radius /■, which shows 

 that the rate of acceleration could not he approximately constant for 

 nearly as long a time as twenty -five million years. It would, in fact. 



