154 



KNOWLEDGE 



[July 1, 1899. 



out of one thousand one hundred and seventy-seven stars 

 three hundred and eighty-five were rated at 9'5 magnitude. 

 Now, it is very unlikely that a third of a thousand stars 

 taken at random should be within a tenth of a magnitude 

 of 9-5. It certainly looks as if all stars observed fainter 

 than 9-4 were indiscriminately put down as 9'5. In order 

 to put the question of the supposed rapid increase of faint 

 stars to a practical test, I took six gaugings in zone — 0°, 

 R.A. .5h. 30m. to 6h., with apertures of two and a-quarter 

 inches and three and three-quarter inches, with the 

 following result : — 



Stars visible with aperture of 21 inches ... 64 



Additional do. 



do. 



do. 



57 



According to the Vuichmtisterung there are in the same 

 region : — 



Stars of 8-9 magnitude and over 25 



Do. of 9-0 to 9'5 magnitude 126 



The increase of aperture from two and a-quarter inches 

 to three and three-quarter inches represents about one 

 magnitude, so that, if the magnitudes in the Diuchnius- 

 teninii were correctly estimated, we should have expected 

 to see about ten times as many with the larger aperture 

 instead of barely double the number. 



With a view to throwing some fresh light on the question 

 of the increase in the number of faint stars, I determined 

 to make a series of gaugings in the following manner : A 

 three and three-quarter inch refractor was provided with a 

 series of five diaphragms of diameters -15, 1'50, 2-25, 3'00, 

 and 3-75 inches respectively. The diameter of the field was 

 approximately sixty-four minutes, and each set of obser- 

 vations consisted in counting the number of stars visible 

 with each successive aperture. One advantage of this 

 method is that there is a direct relationship between the 

 aperture and the space-penetrating power, which is simply 

 proportional to it. Another is that, as each set of obser- 

 vations is made consecutively with the same instrument 

 on the same evening, the result is free from errors due to 

 changes in the atmosphere or the use of different instru- 

 ments. Although the photometric magnitude of a star 

 will be different under different conditions for the same 

 aperture, yet the difference of magnitude for two diflerent 

 apertures remains constant. 



As has been already remarked, the stars seen in a tele- 

 scope are contained in a cone whose vertical angle is the 

 angular diameter of the field. If we assume that all stars 

 are of equal intrinsic brightness, the height of the cone is 

 proportional to the aperture, and its volume to the cube 

 of the aperture. The following table shows the relation- 

 ship of the various apertures we are dealing with and the 

 volume of the space penetrated : — 



On the hypothesis of the equal distribution of stars in 

 space, the third line gives the relative number of stars 

 that ought to be visible with each aperture, and the last 

 line gives the ratio of the additional stars seen with the 

 larger aperture to those already in the field with lower 

 aperture. 



The following table gives the actual result of one hundred 

 gaugings made in the manner above described ; — 



T/c 2 



Stars of equal intrinsic brightness as actually seen. 



Ninety of the gaugings above tabulated were taken on 

 the sixteenth meridian between the pole and the equator, 

 one to each degree. The results of these ninety gaugings 

 are shown in Fig. 2, which has been constructed on the 

 same principle as Fig. 1. The observations for each five 

 degrees of the quadrant were added up, and the total, 

 multiplied by a co-efScient, was plotted on the correspond- 

 ing compartment in the diagram. These co-efScients, as 

 has been explained, are required in order that the density 

 may be truly represented on a plane ; they are 1, J, -f^, 

 -.]j, and Jy for the successive apertures. Fig. 2 con- 

 sequently represents an ideal section through the stellar 

 universe at R.A. 16h., showing how stars of equal intrinsic 

 brightness would have to be distributed in order to appear 

 to an observer on the earth as actually seen. 



A comparison of these two figures will show at a glance 

 that they indicate two totally distinct systems of stellar 

 distribution. In the one case there is a continuous 

 thinning out of stars as we proceed outwards, while in the 

 other case there is a decided increase in density of the 

 stars in the more remote parts of space. 



That faint stars are really much less numerous than 

 they would be on the hypothesis of equal distribution 

 in space, may be shown by comparing the actual number 

 observed with the number calculated from the numbers of 

 superior magnitudes, on the assumption that number and 

 magnitude continue to be related by the same law. The 

 following statement shows the result of such a comparison : 



Number of stars of the 17th magnitude that 

 ought to be seen within a circle of 1 degree 

 in diameter if the relationship of number and '■ 180,0(X) 

 magnitude, indicated by the Durchmusterung, 

 continued to the 17th magnitude. 



