346-348] Moving Clusters 375 
will in this case be approximately equal, so that the value of £ a , rf, £ 2 in the 
expanded cluster will be inversely proportional to A + a, A +b, C + c. 
If the star-density of the cluster is small in comparison with that of the 
field of stars in which it moves, a, b and c may be neglected in comparison 
with A and C. The semi-axes of the cluster are now proportional to 
A~\ A C~^, so that, as C is greater than A, the cluster flattens out in 
the galactic plane; its cross section in this plane remaining circular. With 
the relation c = 6*31 A already used, the ratio of the axes of the cluster would 
be 2‘51: 2’51:1. 
If the system of the 5-stars discussed by Charlier* is treated as a moving 
cluster, or rather as a cluster at rest, we see that it ought to form a spheroidal 
system, flattened in the galactic plane, the ratios of its axes being as just 
stated. This is approximately what Charlier finds, except that he gives the 
ratio of the axes as 2 - 8 : 2‘8 :1. 
348. In only one known cluster, the Taurus cluster, is the star-density of 
the cluster itself other than small in comparison with that of the field of 
stars. Rasmuson f estimates the star-density in this cluster to be about one. 
star per 8 cubic parsecs, which is greater than the density of the galaxy 
itself. The shape of this cluster is accordingly determined more by the values 
of a, b and c arising from the field of the cluster itself, than from the values 
of A, A and C. If SI X *, Sly, SI Z 2 were all equal, then the ratio of the axes 
would be 251 to unity if the ratio were determined solely by A, A and C, 
and would be one of equality if the ratio were determined by a, b and c. 
Detailed calculation shews that the actual star-densities require theoretically 
an intermediate ratio of flattening of about 1*5 to unity, and this is almost 
precisely the degree of flattening found by Rasmuson from observation. 
In addition to this flattening in the plane of the galaxy, Rasmuson finds 
the axes in this plane to be slightly unequal; the velocity of the cluster in 
space is ample to account for this. 
Apart from the Taurus cluster, Rasmuson has studied the space distri 
bution of four other principal moving clusters. Three of these (Perseus, 
Scorpio-Centaurus and 61 Cygni) exhibit a distinct flattening parallel to the 
galactic plane, the degree of flattening being in each case less than the 
maximum permitted by theory, in which the ratio of the axes is 251 to 1. 
The greatest observed ratio is 2’34 to 1 in the Scorpio-Centaurus cluster. 
The fourth cluster, the Ursa Major cluster, does not shew galactic 
flattening, but is flattened at right angles to its direction of motion, as also 
in varying degrees are the three clusters just mentioned. If a cluster is 
bombarded by stars moving relatively to it in directions mainly along the 
axis of x, it is readily found that Sly 1 and SI Z will be equal, while SI X 2 will 
* Lund, Meddelanden, Series II, No. 14- (1916). f Ibid. No. 26 (1921).