ETHER AND GRAVITATIONAL MATTER. 223 



have nuiny chances of being much greater than los kms. i)er .second, 

 and many chances also of being considerabl}^ less. 



Sec. 15. Without attempting to solve the problem of finding the 

 motions and velocities of the 1,000,000,000 bodies, we can see that if 

 they had been given at resf^ twenty-five million years ago distri))uted 

 uniformlj^ or nonuniformly through our sphere (5) of 3 '09. 10'" kms. 

 radius, a very large proportion of them would now have velocities not 

 less than 20 or 30 kms. per second, while many would have velocities 

 less than that; and certainly some would have velocities greater than 

 108 kms. per second; or if thousands of millions of years ago they 

 had been given at rest, at distances from one another veiy great in 

 comparison with r (5), so distributed that they should temporarily 

 now be equably spaced throughout a spherical surface of radius r (5), 

 their mean velocit}^ (reckoned as the square root of the mean of the 

 squares of their actual velocities), would now be 50*1: kms. per second.** 

 This is not very unlike what we know of the stars visible to us. 

 Thus it is quite possible, perhaps probable, that there ma}^ be as much 

 matter as a thousand million suns within the distance corresponding 

 to jjarallax one one-thousandth of a second (3*09. lO''' kms.). But it 

 seems perfectly certain that there can not be within this distance as 

 much matter as 10,000,000,000 suns; because if there were we should 

 find much greater velocities of visible stars than observation shows, 

 according to the following tables of results and statements from the 

 most recent scientific authorities on the subject. 



*"The potential energy of gravitation may be in reality the ultimate created 

 antecedent of all the motion, heat, and light at present in the universe." See 

 Mechanical Aniecedants of Motion, Heat, and Light. Art. LXIX of my Collected 

 Mathematical and Physical Papers, Vol. II. 



••To prove this, remark that the exhaustion of gravitational energy 



f+co p+x p+co 

 I I R^dx' dy dz, Thomson and Tait's Natural Philosophy, Part 

 -co .1 -00 J -co 

 II, section 549) when a vast number, N, of equal masses come from rest at infinite 

 distances from one another to an equably spaced distribution through a sphere of 

 radius r is easily found to be 3/10 Fr, where F denotes the resultant force of the 

 attraction of all of them on a material point, of mass equal to the sum of their 

 masses, placed at the spherical surface. Now, this exhaustion of gravitational energy 

 is spent wholly in the generation of kinetic energy; and therefore we have 



:2hn''= 1 Fr, and by (7) F=l-37 . 10-":2wi; whence 

 2 10 ^ 



2.m 5 

 which, for the case of equal masses, gives, with (5) for the value of r, 



V 



^=^ (I 1-37 . 10-13. 3.Q9 _ 1016) =50.4 kms. per second. 



