December 27, 19 17] 



NATURE 



Z^l 



The latter law is artificial, but was used by Max- 

 well because it introduced considerable simplifica- 

 tion into the discussion. Where stars are con- 

 cerned it is necessary to distinguish between real 

 collisions and encounters. The latter occur when 

 two stars approach one another sufficiently closely 

 to produce a relative change in path without 

 actually colliding. The number of collisions will 

 naturally be considerably less than that of the 

 •encounters. The fundamental general equation 

 of statistical mechanics is formed, and the effect 

 ■of the collisions and encounters obtained. The dis- 

 cussion follows closely along normal lines. The 

 integration of the fundamental equation when the 

 •solution is a frequency-fujiction of type A is per- 

 formed, the solution being rather more compli- 

 cated than for Maxwell's law of repulsion. The 

 time of relaxation, which is a measure of the time 

 taken by the system to reach a steady state, is 

 found to be about 10^^ years. Jeans had pre- 

 viously obtained, by somewhat different reasoning, 

 a value of 10^'' years, which is of the same order 

 of magnitude. 



In (2) some of the results obtained in (i) are 

 applied to prove the law of equipartition of energy 

 for the stars. The proof is elementary and applies 

 only for translational velocities, any possible 

 energy of rotation not being taken into account. 

 As regards translational .energy, recent results 

 Indicate that the most massive stars have the 

 slowest velocities on the average, and vice versa, 

 which is in the sense required by equipartition. 

 •But whether there is anything like real equiparti- 

 tion, even for translational velocities, we do not 

 know ; still less do we know to what extent the 

 energy of rotation shares in the equipartition. In 

 any case, we should not expect equipartition to 

 hold unless the system had practically reached a 

 steady state, and other evidence must be adduced 

 to settle this point. 



In (3) the hydrodynamical analogy is used, the 

 average motion of a small group of stars under the 

 general attraction of the stellar svstem being con- 

 sidered, neglecting the effects of encounters and 

 collisions on the motion of individual stars. The 

 equation of motion for a steady state is derived 

 from (i) and integrated. The result is obtained 

 that in a star cluster, in which the stars are sym- 

 metrically distributed about an axis, in which 

 there Is hydrodyn^lmical equilibrium and ellio- 

 soidal velocity surfaces, these surfaces must be 

 spheroids with their axes of rotation perpendicular 

 to the radius vector from the centre of the cluster. 

 The same result had previously been obtained 

 otherwise by Jeans. It was proved by Schwarz- 

 schild that the velocity surfaces are approximately 

 spheroids with their rotation axes directed towards 

 the vertex. Jeans, through insuflficient evidence, 

 had concluded that this direction was not perpen- 

 dicular to the radius vector. On the other hand, 

 Prof. Charlier, on the evidence afforded by recent 

 investigations at Lund, concludes that the two 

 directions are perpendicular. Jeans has since 

 accepted the evidence on which Prof. Charlier 

 bases this conclusion. The result supports, but 

 NO. 2513, VOL. 100] 



does not prove, the supposition that our stellar 

 system is in such equilibrium, for there are other 

 factors to be taken into consideration. 



In (4) Prof. Charlier discusses and compares 

 what he calls the monistic and dualistic concep- 

 tions of the stellar universe. According to the 

 former, the universe can be considered as a single 

 system which, if it has not actually attained a 

 steady state, is on the way to doing so. By the 

 latter he means the hypothesis that there are two 

 intermingling star-streams, though it is doubtful 

 whether the originators of that hypothesis ever 

 conceived that there were two streams of stars 

 approaching and passing through one another. 

 Our knowledge of stellar motions is derived almost 

 entirely from the nearer stars, and it would be 

 dangerous to make so sweeping an assertion. 

 Reasons are advanced by Prof. Charlier for sup- 

 posing that the methods of statistical mechanics 

 as developed in (i) can be applied to the monistic 

 conception, and an endeavour is made to show that 

 the state of motion in our system is comparable 

 with the results given by the kinetic theory. The 

 time of relaxation obtained in (i) was thought by 

 Jeans to be too long for our system to be con- 

 sidered as yet in a steady state. Prof. Charlier 

 brings forward evidence to show that the velocities 

 of the stars are in qualitative agreement with the 

 requirements of the kinetic theory [see (2)], and 

 that red stars are more nearly in statistical equi- 

 librium than the younger blue stars. The results 

 obtained in (3) also supported the idea of a steady 

 state. To Eddington's difficulty of believing that 

 the evidence of scattered clusters of stars moving 

 with a common velocity, such as the Ursa Major 

 cluster, can be explained if the chance attractions 

 of stars passing in the vicinity have an appreciable 

 effect on stellar motions. Prof. Charlier replies that 

 it is possible that such clusters are but the rem- 

 nants of much larger clusters, most of the members 

 of which have succumbed to encounters with other 

 stars by the way. The sparseness of the stars in 

 these clusters may be held to support this view. 

 Furthermore, Jeans has shown that a compact 

 globular cluster moving through another mass of 

 stars will be spread out into a disc-like arrange- 

 ment, perpendicular to the direction of motion. The 

 conditions of Jeans's discussion cannot be exactly 

 reproduced in the stellar universe, but it is 

 interesting to note that Turner has shown that the 

 Ursa Major system has approximately this shape. 



The fifth paper is a valuable discussion of the 

 various methods which have been used for 

 analysing stellar motions, and forms a convenient 

 summary for purposes of reference. The analysis 

 on the simple hypothesis of a single star-stream, 

 on that of two star-streams developed by Kapteyn 

 and Eddington, on the ellipsoidal hypothesis of 

 Schwarzschild — all of which are based upon the 

 directions of the motions only — and that on the 

 correlation methods developed by Prof. Charlier 

 himself — in which both the magnitude and direc- 

 tion of the motions are taken into account — are 

 discussed and illustrated by application to one 

 particular region of the sky. H, S. Jones. 



