254 REPORTS ON THE STATE OF SCIENCE. 



for simplicity two dimensions only, we can represent each stellar 

 motion by a clot whose co-ordinates represent the two components of the 

 velocity. All the theories agree that the dots would be arranged in an 

 elongated distribution — elongated along the line already referred to 

 along which the stars move in greater numbers, one stream towards 

 one end, and the other towards the other end. But the different theories 

 do not agree in the detailed arrangement of the dots. We can sum- 

 marise the dot-diagram in a more convenient way ; we can draw contour 

 lines to specify the density of the dots— i.e., mark out places on the 

 diagram where the dots are equally thick. That is perhaps the most 

 illuminating way of stating a frequency law. In the diagram I com- 

 pare the two-drift and ellipsoidal theories by their contour lines — lines 

 which indicate what values of the (linear) stellar velocity occur with 

 equal frequency. For the ellipsoidal theory there are similar ellipses 

 (fig. A). Those for the two-drift theory are less simple curves, and 

 rather curiously they may take two different forms (figs. B and C), 

 according to the precise values of the arbitrary constants. There may 

 be two points of maximum frequency or only one point. The central 

 figure B is especially interesting, because it shows that even if there 

 are two distinct streams we may find no indication of a separation of 

 the dots into two groups. Actually the values of the drift motions are 

 such that it is uncertain whether we should find the contours of the 

 form B or of form C ; they are near the critical values. The especial 

 matter to which I would call attention is that A and B represent the 

 distribution of velocities for different theories of the universe ; A treats 

 the universe as a single system and B divides it into two. 11 



To determine these contour lines from the observations is a mathe- 

 matical problem of some difficulty. Without going into detail, it comes 

 to solving a certain integral equation. The equation in question is 

 one that is theoretically soluble, and a quite similar one occurring 

 in another branch of stellar statistics has been elegantly treated by 

 Schwarzschild. 12 As applied to the present problem, however, there 

 are difficulties in obtaining a convergent solution which have not yet 



11 Some further information as to these diagrams may be of interest. For both 

 A and B the contours are drawn for the same values of the density, viz., 0-96, 0-83, 

 0-64, 0-38, 0-13, taking the density at the centre as unit. A is for a Schwarzschild 

 ellipse in which the ratio of the axes is 0-77. B is for two equal drifts whose relative 



velocity is equal to the quantity — , which in the mathematical theory gives a measure 



of the average individual velocities. C is for two equal drifts whose relative velocity 



2 



is %- ■ for convenience C has been drawn on half the scale of B, so that the relative 



h 

 drift velocity is represented by the same length in both figures ; the contours are 

 drawn for the relative densities 0-9, 0-75, 0-4, 0-1, 0001. The critical case arises 

 when the relative velocity of the two drifts is equal to \Zi'/A, if the drifts are equal ; 

 if, as actually occurs, the number of stars in the two drifts is unequal this will be 

 modified. The actual distribution in the sky is decidedly more elongated than that 

 shown in A and B ; in selecting constants for B, I wished not to approach too near the 

 critical case. It may be noted that A and B are practically indistinguishable when 

 drawn on this scale ; the only difference that can be detected is that the outermost, 

 ellipse of B is slightly less eccentric than that of A. 

 i 2 Ast. Nach., No. 4422. 



