Januaky 2, 1893.] 



KNOWLEDGE 



the tussocks very easily come out, though the tussocks 

 themselves look solid anil unyicldiug, and seem to be the 

 most appropriate parts for an enemy to seize upon ; but if 

 they be roughly seized, their silkiness causes them to slide 

 from the grasp, the caterpillar thus escaping: and leaving 

 with its would-be captor a little bundle of tickling hairs 

 as the only trophy. Mr. Poulton has pointed out that the 

 tuesocks may in this way be of some protective value. 

 He ofl'ered a "hop-dog" to a hungry green lizard. The 

 animal was apparently experienced enough to know the 

 deceptiveness of the tussocks, but, nevertheless, the 

 promptings of hunger were imperative and it resolved to 

 make a venture, but it carefully avoided the tempting tufts 

 and kept trying for some minutes to find a more suitable 

 point for attack ; it finally seized the caterpillar on the 

 back some distance behind the tussocks, whence the moral 

 seems to be that a less hungry assailant would probably 

 have left the hairy morsel alone. A somewhat similar 

 caterpillar, that of the vapourer moth (Onjijia nntiiiua), on 

 being offered to another and, as it proved, less cautious 

 lizard, was at once seized by the tussocks, but its assailant 

 had the mortification of losing its prey, and getting only a 

 mouthful of hairs for its pains; these were by no means to 

 its taste, and it thus learnt a lesson of prudence by the 

 experiment, and made no attempt to mend its fortune by 

 any other venture in the same direction. 

 (To be cuittiinieil.) 



THE NUMBER AND DISTANCE OF THE 

 VISIBLE STARS. 



By .J. E. GoKE, F.R.A.S. 



THAT the visible stars are not uniformly scattered 

 through space, and are not of uniform size and 

 intrinsic brightness, is clearly shown by modern 

 researches. Measures of stellar parallax show 

 that some small stars (that is, faint stars) 

 are actually nearer to our system than many of the 

 brighter stars, while the period of revolution of some 

 binary stars shows that their mass is relatively small 

 compared with the brilliancy of their luminous surfaces. 

 We may, however, perhaps assume that the stars out to 

 some limited distance in space are scattered with some 

 rough approach to uniformity. We can at least calculate 

 the average distance between the neighbouring stars, which 

 would give a certain number of stars in a sphere of given 

 radius. This is easily done by supposing the stars placed 

 at the angular points of a tetrahedron. A tetrahedron 

 is a solid figure bounded by four equal surfaces, each 

 surface being an equilateral triangle. It is clear that in 

 such a solid each of the angular points is equidistant from 

 the other three angular points of the figure. 



If e be the length of the side of each equilateral triangle 

 (or edge of the tetrahedron) it may be eas:ly shown, that the 

 volume of the tetrahedron is ^V ''' ^ '• ^^ ^^e make e=l 

 the volume is jV ^'-. Now it is clear that the number of 

 equidistant points contained in a sphere of given radius 

 will be equal to the number of these tetrahedrons which the 

 sphere contains. If '■ be the radius of a sphere, its volume 

 will be f IT J-*, and if n be the number of equidistant points 

 or stars contained in the sphere, we have J n r''=ii x Jj ^^-, 

 whence r = v'^^^, and n -= r x 35-548. From these 

 formuliB we can compute the radius of a sphere containing 

 a given number of equidistant stars, or the number of 

 stars contained in a sphere of given radius. 



This formula, however, only applies to a sphere of a 



radius large in comparison with the distance between the 

 stars distributed through it. For if we make e=i=l. or 

 the distance between tlie stars equal to the radius of the 

 sphere, the formula would give 35 equidistant stars in the 

 sphere of unit radius. This number is evidently too groat, 

 as the number of stars which can be placed on the surface 

 of a sphere of given radius, equidistant from each other 

 and from the centre of the sphere, is only 12. The 

 difference is clearly due to the fact that in this case the 

 volume of the tetrahedron is so large in proportion to the 

 volume of the sphere that the latter cannot be acci ritely 

 divided into tetrahedrons. In the case, however, of a 

 sphere whose volume is large in proportion to the volume 

 of the tetrahedron formed by four adjacent stars, 

 the formula will be approximately correct. Let us call 

 the distance between two adjacent stars the unit ilistcinrr. 



Now considering a Ccntauri, for which the largest 

 parallax has been found (about 0-70"), it is obvious at 

 once that this star cannot be at the " unit distance" from 

 the sun. For if a Centauri was at the " unit elistance " 

 we might expect to find some ten or eleven other stars 

 with a similar large parallax. Such is, however, probably 

 not the case, and we may therefore conclude that this star 

 is comparatively near our system, and forms an exception 

 to the general rule of stellar distance. 



To make this point clearer, let us see what number oi 

 stars should be visible to the naked eye — say to the sixth 

 magnitude inclusive — on the assumption that the distance 

 between the sun and a Centauri forms the " unit distance" 

 between two stars of the visible sidereal system, or at 

 least that portion of the system which is visible without a 

 telescope. To make this calculation it will of course be 

 necessary to assume some average distance for stars of 

 the sixth magnitude, based on actual measurement. Now 

 Peters found an average parallax of 0-102" for stars of the 

 first magnitude, Gylelen found 0-083", and Elkin 0-089". 

 These results are fairly accordant, and we may assume the 

 mean of these values, or 0-09" as the mean parallax of an 

 average star of the first magnitude. 



With a "light ratio" of 2-512, the light of a fir.=t 

 magnitude star is 100 times the light of a sixth, and hence 

 the distance of an average sixth magnitude star would be 

 ten times that of a first. Its parallax would therefore be 

 0-009." Hence the radius of a sphere containing all stars 

 to the sixth magnitude mclusive would be ^.l^^ or 84--1 limes 

 the distance of a Centauri. Hence we have ?^=(8i•4,3x 

 35-548=21,372,000, a number enormously greater than 

 the known number of stars to the sixth magnitude. 



This leads us to doubt whether the mean parallax of 

 sixth maguituele stars is so small as 0-009". I find that 

 the sun placed at the distance indicated by this paiallax 

 would shine only as a star of the eleventh magnitude ; 

 that is, a sixth magnitude star would be five magnitudes, 

 or 100 times brighter than the sun placed in the same 

 position. If of the same intrinsic brilliancy of surface, 

 this would imply that an average sixth magnitude star 

 has ten times the diameter of the sun, and therefore 1000 

 times its volume ! Some sixth magnitude stars may 

 possibly exceed our sun in size, but that the afeiw/e 

 volume of these small stars is 1000 times that of the sun 

 seems wholly improbable. Certainly the calculated masses 

 of those binary stars for which a parallax has been deter- 

 mined do not give any grounds for supposing that such 

 enormous bodies exist among stars of the sixth magnitude. 



Assuming, however, that the parallax of a first magnitude 

 star is 0-09", and that of a sixth magnitude is one-tenth of 

 this, or 0-009", let us see what number of stars should be 

 visible to the sixth magnitude. As already stated, the 

 number of equidistant stars which can be placed on the 



