236 



NA TURE 



[July 9, 1908 



tude five must have absolute magnitude 4-5. In the box, 

 therefore, belonging to the fifth apparent magnitude, eighth 

 shell, all the stars are of absolute magnitude 4-5. In 

 the ninth shell a star must already have the absolute 

 magnitude 3-5 in order to shine as a fifth apparent magni- 

 tude at this greater distance, and so on. In this way the 

 absolute magnitudes were found which in our figure have 

 been inscribed on the lids of the boxes. 



We are now able to derive at once the Hii.vfiirc /nw, i.e. 

 the proportions in which stars of different absolute magni- 

 tude are mi.xed in the universe. For in one and the same 

 shell (eleventh) we find two stars of absolute magnitude 

 — 1-5, as against three of magnitude —0-5, fifteen of absolute 

 magnitude 0-5, seventy-six of absolute magnitude 1-5, &c. 



That is, our results for the eleventh shell furnish us with 

 the proportion in which stars of absolute magnitude —1-5, 

 — 0-5, &c., to 4-5, are mixed in space. The tenth shell 

 gives the proportions for all the absolute magnitudes 



between 



together give the proportion 



All the shells 

 nitudes 



By photometric measures it was found that the sun, 

 placed at a distance of 326 light-years, would shine as a 

 star of magnitude 10-5. In other words, the sun's absolute 

 magnitude is 10-5. .A star of absolute magnitude 95 will, 

 therefore, have 2-5 times the light-power — that is, 25 times 

 the luminosity of the sun. A star of absolute magnitude 

 8-5 will again have a luminosity which is 25 times 

 greater, and so on. 



Such results evidently enable us to transform our absolute 

 magnitudes into luminosities. Thus translated, 1 found 

 the results shown in the following table. 



Luminosity Table. 



Within a sphere having a radius of 555 light-years, there 

 must exist : — 



to too.oootiints more iumlnous than the sun 

 ,, 10,000 ,, ,, ,, ,, 



,, 1,000 „ ,, ,, ,, 



,1 too ,, ,, „ ,, 





3j3n 



z— 2^ 



X 

 IX 



J kzJ \iiC I 



~5. 



me. 





rjjj 



813 



S/7 



3ZS 



vii3Ea 



LJ2J W. 



270 



ISO 



iv[5j3 ""^ 





nj3_ 



sx 



^lana.!,' 



0ia'0[3i3 



— 1-5 to I4'5, that is, for a range of not less than sixteen 

 magnitudes. Not only that, but most of the proportions 

 are determined independently by the data of quite a number 

 of shells. So, for instance, the proportion of the stars 

 of absolute magnitude 4-5 to those of absolute magnitude 

 5-5. Each of the six shells from the fifth to the tenth 

 furnishes a determination of this proportion. All of them 

 are not equally trustworthy. If we take this into account, 

 we find that the agreement of the several determinations 

 is fairly satisfactory. By a careful combination of all the 

 results, a table representing the law of the mixture of the 

 stars of different absolute inagnitude w.^s finally obtained. 

 Rather than show you the direct result, however, I will 

 first replace the absolute magnitudes by luminosities ex- 

 pressed in the total light of our sun as a unit. This will 

 have the advantage of presenting a more vivid image of 

 the real meaning of our numbers. 



KG. 2019, VOL. 78] 



This table represents what, up to the present 

 time, we know about the mi.xture law. 

 The fainter the stars, the more numerous. 

 The rate at which the numbers increase with 

 the faintness is particularly noticeable for the very 

 bright stars. 



Passing to the fainter stars, this rate gradually 

 diminishes, and it looks as if we must expect 

 no further increase in number for stars the 

 luminosity of^ which falls below one-hundredlh of 

 that of the sun. Meanwhile, this is simply a 

 surmise. For stars of this older of faintness 

 data begin to fail. Here, as in nearly e\'ery in- 

 vestigation about the structure of the stellar 

 system, the want of data for stars below the ninth 

 apparent magnitude inakes itself very painfully 

 felt. 



But let us come back to our Fig. 4. I will 

 first remark that, knowing the mixture law, we 

 can predict the number of stars that we shall 

 get in the empty boxes belonging to the ninth, 

 tenth, &c., magnitude, as soon as continued 

 astronomical observations will permit us to in- 

 clude these stars in our discussion. For the 

 mixture law, as derived just now, shows that in 

 our universe the stars of absolute magnitude 55 

 are 3-5 tiines as numerous as the stars of absolute 

 magnitude 4-5. 



Now as in the eleventh shell the number of 

 stars of the absolute magnitude 4-5 is 5400 (see 

 Fig. 4), there must be 3-5 times 5400, that is, 

 18,000 stars of absolute magnitude 5-5 in this 

 shell. These belong all in the box of the ninth 

 apparent magnitude of this shell. In the saine 

 way we obtain the number of stars to be expected 

 in the boxes of the tenth, eleventh, &c., af>parent 

 magnitude for all our shells down to the eleventh. 

 There is exception only for the boxes belonging 

 to the lower shells, for which the absolute magni- 

 tude would exceed i4'5. 



It is evident, however, that the number of 

 stars in these exceptional boxes must be small, 

 and for what follows they are of little import- 

 ance. 



Star-iensity . 

 In the second place, our boxes now also lead to the 

 determination of the star-densities. For the volumes of the 

 consecutive shells are perfectly known ; they are in the 

 proportion of i : 308. For ttie sake of convenience, let 

 us say that the volume of each shell is exactly four times 

 that of the next preceding one. Now, to take an example 

 of the determination of the densities, consider the ninth 

 and tenth shells (see Fig. 4V In the ninth there are forty- 

 nine stars of absolute magnitude 2-5. Therefore, if in 

 the tenth the stars were as thickly crowded as in the ninth, 

 there would occur in this shell four times forty-nine, that 

 is ic)6 stars of this absolute magnitude 2-5. 



In reality we find but 140 of these stars. The con- 

 clusion evidently must be that the star-density in the tenth 

 shell is about 140/196, that is, about two-thirds of that 

 in the ninth shell. ."V similar conclusion is obtained by 



:r' J2 



