TRANSACTIONS OF SECTION A. . 565 



If we compare the sunlight with the liglit from the thousand million 

 Stars, each being supposed to be of the same size and brightness as our sun, we 

 find that the ratio of the apparent brightness of the star-lit slvy to the bright- 

 ness of our sun's disc would be 3'87.10~'^. This ratio' varies directly with 

 the radius of the containing sphere, the number of equal globes per equal volume 

 being supposed constant ; and hence to make the sum of the apparent area of discs 

 3*87 per cent, of the whole sky, the radius must be 3*09.10-' kilometres. With 

 this radius light would take 3:|^.10'^ years to travel from the outlying stars to 

 the centre. Irrefragable dynamics proves that the life of our sun as a luminary 

 is probably between fifty and 100 million years ; but to be liberal, suppose each of 

 our stars to have a life of 100 million years as a luminary, and it is found that 

 the time taken by light to travel from the outlying stars to the centre of the 

 spliere is three and a quarter million times the life of a star. Hence it follows 

 that to make the whole sky aglow with the light of all the stars at the same time 

 the commencements of the stars must be timed earUer and earlier for the more and 

 more distant ones, so that the time of the arrival of the light of every one of 

 them at the earth may fall within the durations of the lights of all the others at 

 the earth. jNIy supposition as to uniform density is quite arbitrary ; but never- 

 theless I think it highly improbable that there can be enough of stars (bright or 

 dark) to make a total of star-disc area more than 10~''- or 10~'' of the whole sky. 



To help to understand the density of the supposed distribucion of 1,000 million 

 suns in a sphere of 3'09.10"' kilometres radius, imagine them arranged exactly in 

 cubic order, and the volume per sun is found to be 123'5.10'" cubic kilometres, 

 and the distance from one star to any one of its six nearest neighbours would be 

 4-98.10'" kilometres. The sua seen at this distance would probably be seen as a 

 star of between the first and second magnitude ; but supposing our 1,000 millioa 

 suns to be all of such brightness as to be stars of the first magnitude at distance 

 corresponding to parallax l"-0, the brightness at distance 3-09.10"' kilometres 

 would be one one-millionth of this ; and the most distant of our stars would be 

 seen through powerful telescopes as stars of the sixteenth magnitude. Newcomb 

 estimated from thirty to fifty million as the number of stars visible in modern 

 telescopes. Young estimated at 100 million the number visible through the Lick 

 telescope. This larger estimate is only one tenth of our assumed 1,000 million 

 mas3es equal to the sun, of which, however, 900 million might be either non- 

 luminous, or, though luminous, too distant to be seen by us at their actual 

 distances from the earth. Remark, also, that it is only for facility of counting 

 that we have reckoned our universe as 1,000 million suns ; and that" the meaning 

 of our i-eckoning is that the total amount of matter within a sphere of 3-09.10'" 

 kilometres radius is 1,000 million times the sun's mass. The sun's mass is 

 1*99.10-" metric tons, or 1-99.10^" grammes. Hence our reckoning of our sup- 

 posed spherical universe is that the ponderable part of it amounts to 1-99.10*- 

 graiTimes, or that its average density is 1-61.10-'-" of the density of water. 



Let us now return to the question of sum of apparent areas. The ratio of 

 this sum to 4ir, the total apparent area of the sky viewed in all directions, is given 



by the formula ' : a = '^ - ( . ) ' provided its amount is so small a fraction of 



unity that its diminution by eclipses, total or partial, may be neglected. In 

 this formula, N is a number of globes of radius a uniformly distributed within a 

 spherical surface of radius /-. For the same quantity of matter in N' globes of the 

 same density, uniformly distributed through the same sphere of radius r, we have 



J'=/''*,y and therefore ?' = -. With N = 10", r- 3-09.10"^ kilometres; and 

 IS \a / a a 



a (the sun's radius) =7.10' kilometi-es ; we had a = 3-87.10~'^. Hence 

 «' = 7 kilometres gives a' = 3-87.10"'; and a" -\ centimetre gives a" = 1/36-9. 

 Hence if the whole mass of our supposed universe were reduced to globules of 

 density l-i (being the sun's mean density), and of 2 centimetres diameter, dis- 

 tributed uniformly through a sphere of 3-09.10"' kilometres radius, an eye at the 



' Pidl. Mag., August 1901, p. 175. 

 1901. ' P p 



