340-342] Star-Streaming 369 
Kapteyn found that, for his assumed distribution of density to be possible 
for stars moving in a state of steady motion with star-streaming of the kind 
already described, the velocity of star-streaming had to have a certain velocity 
at each point in the system, a result which is of course also clear from the 
general discussion of § 339. He calculated that in the galactic plane, the 
necessary velocity of star-streaming would be about 13’0 kilometres a second 
at 1010 parsecs from the centre, and that it would be fairly uniformly equal 
to 19 '5 kilometres a second at all distances greater than 2000 parsecs. This 
would give a relative velocity for the two streams of 39 kilometres a second. 
Kapteyn estimated the relative velocity of the observed star-streaming 
near the sun to be about 40 kilometres a second, so that his investigation 
would seem, at first glance, to shew that star-streaming could only be ex 
plained if the sun was something over 2000 parsecs from the centre of the 
system. This would however be antagonistic to the hypothesis from which the 
investigation started, that the sun is near the centre of the system. Correcting 
for this, Kapteyn found that the relative velocity of star-streaming would be 
39 kilometres a second at a distance of rather over 1000 parsecs, and would 
be 33 kilometres a second at 500 parsecs. Taking 35 kilometres a second to 
be the minimum velocity consistent with observation, Kapteyn concluded that 
the distance of the sun from the centre of the system must be greater than 
600 parsecs. 
As Kapteyn believed that the observed symmetry of brightness of the sky 
precluded a distance greater than 700 parsecs, he adopted 650 parsecs as the 
distance of the sun from the centre of the galactic system. 
342. An investigation of the problem of star-streaming which I published 
in the same year shewed* that Kapteyn’s assumption as to the distribution 
of velocities being Maxwellian was not only unjustified, but was also un 
necessary. As the discussion of § 339 has shewn, the problem is fully 
determinate when the gravitational field of the stars is given, so that any 
extraneous assumption is superfluous, and can only lead to erroneous results. 
My own investigation amounted in effect to examining what form of the 
distribution function/ in §339 would give rise to the field of star-densities 
estimated by Kapteyn. The data introduced were the axes of the Schwarzschild 
ellipsoid, at a point which was ultimately to be identified with the position of 
the sun. The corresponding mean velocity of peculiar motion is considerably 
greater than the 103 kilometres a second estimated by Kapteyn. As a con 
sequence the density of matter necessary to produce these velocities was found 
to be greater than Kapteyn’s value. My ultimate figure was 0’0451 x 4 - 8 x 10 33 
grammes per cubic parsec, which with ten stars per cubic parsec gives an 
average mass of 216 x 10 33 grammes per star, or T08 times the mass of the 
sun. 
* M.N. lxxxii. (1922), p. 122.