ASTRONOMY: KAPTEYN AND ADAMS 
17 
preferred. If we use that developed by Kapteyn^ we obtain the com- 
parison shown in Table I. Except for some of the i^-type stars the agree- 
ment of the values is very satisfactory. The theory, of course, requires 
that p be proportional to 1 + cos^ X. 
The values of P2/P1 are no smaller for the stars having the very 
smallest proper motions than for the other stars in the list. This proves 
that the two star streams extend to the greatest distances for which we 
have means of judging. This fact has been doubted by Eddington^ 
who proposed an attractive explanation of the behavior of the helium 
stars, which show hardly a trace of the second stream, on the supposition 
that the greater part of them are beyond the region where we must 
admit the existence of two streams. Several objections have already 
been raised against this hypothesis,"^ and we believe that Eddington 
has abandoned it. It is interesting, however, to find in the radial 
motions of these stars so strong a proof of the great extent of the star 
streams. 
The low value of P2/P1 for some of the K stars, particularly for some 
of those of very small proper motion, seems difficult of explanation. A 
preliminary investigation of the proper motions for the stars having 
values between 0.026" and 0.039'^ indicates that members of the second 
stream are not wanting. Evidently the anomally will require much 
further study. 
The second conclusion indicated by Table I, that p increases with 
increase in proper motion, may be explained in any one of at least three 
ways. We may assume that: 
a. The real velocity of the stars decreases with the distance. 
h. The more luminous stars move more slowly than the fainter ones. 
c. The distribution of the velocities of the stars is not in accordance 
with Maxwell's law, the large velocities being in excess. 
The application of the first two of these explanations is evident. 
In the consecutive groups for each spectral type the average magnitude 
is roughly the same. Hence, with the average proper motions increas- 
ing regularly from one group to another, the average distances must 
decrease, and the luminosities become less. 
The possibility of the third explanation, c, may seem somewhat less evi- 
dent. Its consideration, however, is essential since it has already been 
shown by Schwarzschild^ that the distribution of velocities, using values 
relative to the sun, cannot agree with Maxwell's law, and that the larger 
values must be in excess. The radial velocities used here afford the 
data for the derivation of the velocity-law relative to the center of the 
stellar system. Such a derivation we have actually carried out in the 
