378 
The Galactic System of Stars [ch. xiv 
When such a mingling of clusters takes place, the dimensions of the 
resulting system may be far greater than those of the component systems. 
Let T lf fPi denote the kinetic and potential energies of a single cluster before 
mingling, and let J l denote its energy of translation relative to the centre of 
gravity of all the clusters. Then if T 0 and W 0 denote the kinetic and potential 
energy of the resulting single cluster, the equation of energy is 
T 0 + W r 0 = 2(r i +F 1 + JJ. 
If the constituent clusters were each in a state of steady motion before 
the mingling took place, Poincare’s theorem gives 
2 T,+ F x = 0 
for each, while by the time the final cluster has attained to a state of steady 
motion we must have 
2T 0 + W 0 =0. 
By simple algebra we obtain from these equations 
W 0 = XW 1 + 22/,. 
Thus in the act of combination an amount of energy 2%J ly which is double 
the total original energy of translation of all the clusters, is changed into 
potential energy, and so is spent in expanding the final cluster against its own 
gravitational attraction. With moderate initial velocities of translation, the 
ultimate expansion may be enormous. If we regard the galactic system as 
having been formed by the commingling of a number of clusters, or of the 
stellar products of a number of spiral nebulae, there is no difficulty in the 
circumstance that its dimensions are far greater than those of the constituent 
nebulae or clusters are likely to have been. 
351. The galactic system is too symmetrical in shape and shews too 
clearly defined a structure to have been formed merely out of a random 
conglomeration of moving clusters; if it was formed by moving clusters, 
something must have guided their motion into an ordered shape. No 
hypothesis as to the origin of the galactic system can be accepted which 
does not account for the very clearly marked galactic plane. 
This receives a simple and natural explanation in the hypothesis that the 
main part at least of the system represents the final stage of development of 
a single huge spiral nebula. Observation shews that the spiral nebulae retain 
their characteristic flattened shape throughout the greater part of their 
evolutionary history; the stars in their outer regions must move almost 
exactly in the plane of the nebula, or they would, after a comparatively short 
time, shew more scattering about this plane than is actually observed. 
The galactic plane accordingly admits of simple explanation as the plane 
of the parent nebula. The persistence of the stellar velocities in this plane