628 



NATURE 



[October 24, iQor 



and a (the sun's radius) =7.10' kms. ; we had a = 3"87. lo~''. 

 Hence n ="] kms. gives a' = 3'87.io~' ; and (2"=icm. 

 gives a" = i/36'9. Hence if the whole mass of our supposed 

 universe were reduced to globules of density i '4 (being 

 the sun's mean density), and of 2 cms. diameter, distri- 

 buted uniformly through a sphere of 309. 10"' kms. radius, 

 an eye at the centre of this sphere would lose only i 36'9 

 of the light of a luminary outside it ! The smallness of 

 this loss is easily understood when we consider that there is 

 only one globule of 2 cms. diameter per 360,000,000 cubic 

 kilometres of space, in our supposed universe reduced to 

 globules of 2 cms. diameter. Contrast with the total 

 eclipse of the sun by a natural cloud of water spherules, 

 or by the cloud of smoke from the funnel of a steamer. 



Let now all the matter in our supposed universe be 

 reduced to atoms (literally brought back to its probable 

 earliest condition). Through a sphere of radius r let 

 atoms be distributed uniformly in respect to gravitational 

 quality. It is to be understood that the condition "uni- 

 formly" is fulfilled if equivoluminal globular or cubic 

 portions, small in comparison with the whole sphere, but 

 large enough to contain large numbers of the atoms, con- 

 tain equal total masses, reckoned gravitationally, whether 

 the atoms themselves are of equal or unequal masses, or 

 of similar or dissimilar chemical qualities. As long as 

 this condition is fulfilled, each atom e.xperiences very 

 approximately the same force as if the whole matter 

 were infinitely finegrained, that is to say, utterly homo- 

 geneous. 



Let us therefore begin with a uniform sphere of matter 

 of density p, gravitational reckoning, with no mutual 

 forces except gravitation between its parts, given with 

 every part at rest at the initial instant ; and let it be 

 required to find the subsequent motion. Imagining the 

 whole divided into infinitely thin concentric spherical 

 shells, we see that every one of them falls inwards, as if 

 attracted by the whole mass within it collected at the 

 centre. Hence our problem is reduced to the well-known 

 students' exercise of finding the rectilinear motion of a 

 particle attracted according the inverse square of the dis- 

 tance from a fixed point. Let .r(, be the initial distance, 



-^ iv' the attracting mass, v and x the velocity and dis- 



3 

 tance from the centre at time /. The solution of the 

 problem, for the time during which the particle is falling 

 towards the centre is 



3 ' \-v -n)/' 

 and 



t= /-^-f^-e-i-isinaeV /c- f' 



:in 2fl\"[ 



where denotes the acute angle whose sine is . /- 



V .r„. 



This shows that the time of fallmg through any propor- 

 tion of the initial distance is the same whatever be the 

 initial distance ; and that the time (which we shall denote 



by T) of falling to the centre is in- /^ . Hence in 



" V Sttp 

 our problem of homogeneous gravitational matter given 

 at rest within a spherical surface and left to fall inwards, 

 the augmenting density remains homogeneous ; and the 

 time of shrinkage to any stated proportion of the initial 

 radius is inversely as the square root of the density. 



To apply this result to the supposed spherical universe 

 of radius3-o9. io>'' kilometres, and mass equal to a thousand 

 million times the mass of our sun, we find the gravita- 

 tional attraction on a body at its surface gives acceleration 

 of I ■37. 10"'' kms. per second per second This therefore 



is the value of "''^''.v,,, with one second as the unit of 



3 

 time and one kilometre as the unit of distance ; and we 



NO. 1669, VOL. 64] 



find T = 52'8.io'3 seconds = i6'8 million years, 'j. Thus our 

 formulas become 



z)^=i-37.io-" j-„( "is -I V 



giving 



and 



= 5-23io-"^/.»-„(-^-i); 



..52-8.to.^[.--(.-^'^-)]; 

 whence, when sin 6 is very small, 



Let now for example .v,j = 3'o9. 10"' kms., and '-=10^; 



.1' 

 and therefore sin ^ = 5 = 3"i6.iq"* ; whence <' = 29i,ooo 

 kms. per second, and /=T-«7oSo seconds =T 

 - 2 hours approximately. 



By these results it is most interesting to know that our 

 supposed sphere of perfectly compressible fluid, beginning 

 at rest with density r6i.io~-"> of that of water, and of any 

 magnitude large or small, and left unclogged by ether to 

 shrink under the influence of mutual gravitation of its 

 parts, would take nearly seventeen million years to reach 

 •016 1 of the density of water, and about two hours 

 longer to shrink to infinite density at its centre. It is 

 interesting also to know that if the initial radius is 

 3'09. 10"' kilometres the inward velocity of the surface is 

 291,000 kilometres per second at the instant when its 

 radius is 309.10'-' and its density '0161 of that of water. 

 If now, instead of an ideal compressible fluid, we go back 

 to atoms of ordinary matter of all kinds as the primitive 

 occupants of our sphere of 309. 10"' kms. radius, all these 

 conclusions, provided all the velocities are less than the 

 velocity of light, would still hold ; notwithstanding the 

 ether occupying the space through which the atoms 

 move. This would, I believe,' exercise no resistance 

 whatever to uniform motion of an atom through it ; but 

 it would certainly add quasi inertia to the intrinsic 

 Newtonian inertia of the atom itself moving through 

 ideal space void of ether ; which, according to the 

 Newtonian law, would be exactly in proportion to the 

 amount of its gravitational quality. The additional 

 quasi inertia must be exceedingly small in comparison 

 with the Newtonian inertia, as is demonstrated by the 

 Newtonian proofs, including that founded on Kepler's 

 laws for the groups of atoms constituting the planets, 

 and movable bodies experimented on at the earth's 

 surface. 



In one thousand seconds of time after the density 

 •0161 of the density of water is reached, ihe inward 

 surface velocity would be 305,000 kilometres per second, 

 or greater than the velocity of light ; and the whole 

 surface of our condensing globe of gas or vapour or 

 crowd of atoms would Isegin to glow, shedding light 

 inwards and outwards. All this is absolutely realistic 

 e.xcept the assumption of uniform distribution through a 

 sphere of the enormous radius of 3'09. 10"' kilometres, 

 which we adopted temporarily for purposes of illustration. 

 The enormously great velocity (291,000 kms. per second) 

 and rate of acceleration (137 kms. per second per second) 

 of the boundary inwards, which we found at the instant 

 of density '0161 of that of water, are due to greatness 

 of the primitive radius and the uniformity of density in 

 the primitive distribution. 



To come to reality according to the most probable 

 judgment present knowledge allows us to form, suppose 

 at many millions, or thousands of millions, or millions of 

 millions of years ago, all the matter in the universe to 



1 "On the Motion produced in an Infinite Elastic Solid by the Motion 

 through the Space occupied by it of a Bjdy acting on it only by Attraction or 

 Repulsion." Cong. International de Physique, P.iris Vol. of Report. (_Phil. 

 Mn<:. August, 1900). 



