566 REPORT — 1901. 



centre of this spliere would lose only 1/36-9 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 centimetres diameter per 360,000,000 cubic kilometres of space, in 

 our supposed universe reduced to globules of 2 centimetres 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 ' uniformly ' 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, contain equal total masses, reckoned gravita- 

 tionally, 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 experiences very approximately the same force as if the whole matter were 

 infinitely fine-grained, that is to say, utterly homogeneous. 



Let us therefore begin with a uniform sphere of matter of density p, gravita- 

 tional 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 to the inverse square of the distance from a fixed point. Let 



Xf, be the initial distance, _^ .17/ the attracting mass, v and x the velocity and 



o 



distance from the centre at time f. The solution of the problem for the time 



during which the particle is falling towards the centre is 



and 





where B denotes the acute angle whose sine is a / ' . This shows that the time 



'' \/l; 



of falling through any proportion 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 isi^rA /^ . Hence in our problem of homogeneous gravitational 



V onp 

 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 

 densitj'. 



To" apply this result to the supposed spherical universe of radius 3'09.10"' 

 kilometres, and mass equal to a thousand million times the mass of our sun, we 

 find the gravitational attraction on a body at its surface ^ives acceleration of 

 1*37. IC^^^ kilometres per second per second. This therefore is the value of 



—^ Xn, with one second as the unit of time and one kilometre as the unit of 



3 °' 



distance; and we find T = 62*8.10" seconds = 16"8 million years. Thus our 

 formulas become 



i«^ = L37.10-"r„(-J^'-l) 



givmg 



. = 5-23.10-^y,.,(^._l) 



