KNOWLEDGE. 



[January 2, 1893. 



surface of a sphere of unit rndius is 12. Hence, on the 

 surface of a sphere of double this radius, four times the 

 number, or -18 equidistant stars may be placed ; on a 

 sphere of three times the radius, nme times the number. 

 and so on. Now the sum of ten terms of this series 1-, 2", 

 3-, etc., is 385, and as the twelve stars which may be 

 placed on the first sphere nearly represent the number of 

 stars of the first magnitude and brighter visible in both 

 hemispheres, we have the total number of stars to the 

 sixth magnitude inclusive, 88.5 x 12 = 4620. 



Now the number of stars to the sixth magnitude in both 

 hemispheres as observed by Heis and Gould, is 4181, and 

 the number contained in tlie " Harvard Photometry" 

 and the rianometrin Ar(ientina is 37.*)5, so that the 

 number of stars computed on the above principle does not 

 differ widely from the number actually observed. 



Let us see, however, what the unit parallax would be 

 for the observed number of stars to the sixth magnitude, 

 assuming a parallax of 0009'' for stars of this magnitude. 

 Taking the number as 3735, we have by the tetrahedron 

 formula ) = \^|i|l=4-7187 times the average distance 

 between the stars, and since )•=''"—, we have the " unit 

 parallax " =c xO-OOn' ^-4-71H7x0009 ^0-042408; thatis, 

 the mean parallax of the nearest stars to the earth would 

 be 0-042". Excluding stars with a large parallax, this 

 may not perhaps be far from the truth. I find that in 

 31 binary stars brighter than the sixth magnitude (and 

 for which a parallax has not yet been determined), the 

 average " hypothetical pfaallax" — or the parallax on the 

 assumption that the mass of the system is equal to the mass 

 of the sun— is OOCH". If we assume the mass of each of 

 these systems to be, on an average, twice the sun's mass, we 

 must divide this by the cube root of 2. This gives for the 

 average parallax0054", which does not differ widely from the 

 unit parallax found above for stars of the sixth magnitude. 



But we are still confronted with the difficulty that with 

 a parallax of only 0-00!(", stars of the sixth magnitude 

 would be on the average considerably larger than the sun. 

 The same remark applies to stars of the sixteenth 

 magnitude, for which the parallax would be (with the light 

 ratio of 2'512) only 0-00009". Placed at this vast distance 

 the sun would, I find, be reduced to a star of magnitude 

 21-3, and would, therefore, be utterly invisible in the 

 largest telescopes yet constructed. It would be over 100 

 times fainter than a star of the sixteenth magnitude ! 



To reduce the sun to a star of the sixth magnitude, it 

 should be placed at a distance corresponding to a parallax 

 of 0-1". Unless, therefore, stars of the sixth magnitude 

 are, on the average, considerably larger than our sun, we 

 seem justified in thinking that their average parallax is 

 not less than one-tenth of a second. But, as has been 

 stated, this is about the average parallax of stars of the 

 first magnitude, and it is difficult to believe that these 

 bright stars are as far from the earth as the comparatively 

 faint stars which lie near the limit of naked eye vision. 

 There seems, however, no escape from the conclusion that 

 sixth magnitude stars are probably nearer to us than 

 their brightness might lead us to suppose, and to explain 

 the difficulty with reference to the bright stars, we may 

 perhaps assume that their brilliancy is due rather 

 to their great size than proximity to our system. From 

 the small parallax found for Arcturus, Vega, C'apella, 

 Canopus and other bright stars, we have good reason to 

 think that these stars are vastly larger than our sun. 

 Spectroscopic obsoivations of 'C Ursie Majoris indicate 

 that this second magnitude star has a mass about forty 

 times the mass of the sun, and possibly other bright stars 

 may have similarly large masses. Wirins and a Centauri, 

 however, form notable exceptions to this rule. 



Assuming an average parallax of 0-1" for stars of the 

 sixth magnitude, the parallax of a star at the " unit 

 distance " from the sun would — on the tetrahedron 

 formula — be 0-47". This is about the parallax found for 

 61 Cygni, and does not much exceed that of Sirius. There 

 are several other stars with a parallax of somewhat 

 similar amount, and possibly there may be others hitherto 

 undetected. 



With a parallax of 0-1" for a sixth magnitude star, the 

 parallax of an eleventh magnitude would be 0-01", and 

 that of a sixteenth magnitude 0-001". Now, with the 

 unit distance corresponding to a parallax of 0-47", let us 

 see what would be the number of equidistant stars con- 

 tained in a sphere of radins equal to the distance of a 

 sixteenth magnitude star. We have r=JJ-^, or 470 times 

 the distance between the adjacent stars. Hence n = 

 (470)' x35-548=3, 690,700,000, a number about 36 times 

 greater than the number of the visible stars, generally 

 assumed at 100 millions. According to Dr. Gould's 

 formula (sum of stars to //i"' magnitude inclusive = 

 1-00.51 X (3-9120) '"), the number of stars to the sixteenth 

 magnitude would be 3,024,057,632, or about 30 times the 

 number actually visible. 



Probably, however, we are not justified in assuming a 

 uniform distribution of stars to the sixteenth magnitude, 

 most of these faint stars belonging to the Milky Way. 

 Professor C'eloria found that, near the pole of the Galaxy, 

 a small telescope which showed stars to only the eleventh 

 magnitude revealed as many stars as Herschel's gauging 

 telescope of 18-8 inches apertux-e. Here, therefore, we 

 seem to have the extension of our sidereal system limited 

 to the distance of eleventh magnitude stars. Let us now 

 assume a uniform distribution of stars to the eleventh 

 magnitude. With a parallax of 0-01", and a " unit 

 parallax" of 0-47", we have (■=(47)3x35-548 = 3,690,700. 

 The number by Gould's formula is 3,283,876. Both ■ 

 results are largely in excess of the number actually visible, 

 and show, I think, that there is probably a " thinning 

 out " of the stars before we reach the eleventh magnitude 

 distance, at least in extra galactic regions. If we suppose 

 that of the 100 millions of visible stars, 50 millions are 

 scattered uniformly through a sphere, with a radius equal 

 to the distance assumed for stars of the eleventh magnitude 

 — the remaining 50 millions being included in the Milky 

 Way — we have an average " unit parallax '"" for these 50 

 millions of aboutO-ll', which seems to indicate a " thinning 

 out" of the stars towards the boundaries of the sidereal 

 system. If this be so, we may conclude that the stars with a 

 larger parallax than 0-11" are exceptions to the general rule 

 of stellar distribution, and form perhaps comparatively near 

 neighbours of our sun. These near stars seen from the 

 outskirts of the visible universe might perhaps form a small 

 open cluster. Thus the parallax of a Centauri being 

 0-76'', the distance of a sixteenth magnitude star would be 

 760 times the distance of a Centauri, and it follows that 

 the sun and a Centauri seen from a sixteenth magnitude 

 star — equally distant from both — would appear as two 

 faint stars about i^ minutes of arc apart. 



The above results are of course based on the assumption 

 that the faint telescopic stars lie at a distance indicated 

 by their brightness. Such, however, may not be the case. 

 Many of these small stars may be in reality absolutely 

 small. The apparently close connection between bright 

 and faint stars, as shown by photographs of the Milky 

 Way near a Cygni and a Crucis, suggests that bright naked 

 eye stars and faint telescopic objects may, ia some cases at 

 least, lie in the same region of space. If this be so, the 



* Tliat is, the panillnx of a sUii- as seen from its iioiii-est neighbour. 



