February i8, 19 15] 



NATURE 



675 



a gravitating spherical mass of gas in isothermal 

 equilibrium. The result represents the density of 

 the cluster satisfactorily near the centre, but in 

 the outer regions the cluster is less dense than the 

 theory requires. 



The physical conception thus introduced sug- 

 gested other possibilities. -A sphere of gas in 

 adiabatic, instead of isothermal, equilibrium might 

 be chosen as the standard of comparison. A series 

 of states exists, depending on the constant ratio y 

 of the specific heats of the gas, which have been 

 extensively studied by Lord Kelvin and others. 

 Emden's " Gaskugeln " j is. a work dealing ex- 

 haustively with the subject. In general, the law 



Fig. 2. — M 13 Herculi*. Expxure iih. (3 nights). 



of density cannot be expressed in finite terms. But 

 there are exceptional cases in which the differential 

 equation possesses a very simple solution. One of 

 these, discovered by Schuster, corresponds to the 

 value 7 = \2. Here the law expressing the density 

 at the distance r from the centre takes the form : 



where X is the total mass or number of stars. 

 This is finite, although the distribution extends to 

 infinitv. If a finite boundary be expected it is 

 impossible to fix one by the counts, and attempts 

 to do so have been proved illusory by the occur- 



XO. 2364, VOL. 94] 



rence of characteristic variable stars beyond the 

 supposed limit. However this may be, a compari- 

 son of the law with Bailey's counts of the « Cen- 

 tauri cluster showed immediately an agreement 

 within the limits within which radial symmetry is 

 observed. I next compared the law with Picker- 

 ing's curve of the projected densities, based on the 

 clusters ^ Centauri, M13 and 47 Tucanae (bright 

 and faint stars treated separately). The accord- 

 ance was again excellent, and left little doubt that 

 the law represented much more than a mere 

 formula of interpolation. When, however, 

 V. Zeipel's counts of M3 were examined, the outer 

 region was found to conform with the law, while 

 the inner revealed a higher density than 

 was to be expected. .As v. Zeif>el had, 

 on the other hand, succeeded in repre- 

 senting the central distribution by the 

 isothermal law, it was suggested that 

 the true standard of comparison was a 

 central isothermal core surrounded by an 

 adiabatic envelope, a comp>osite state of 

 equilibrium actually contemplated by 

 writers on the thermodynamics of the 

 subject. .Afterwards, by the use of 

 similar methods, Prof. Stromgren 

 proved that M5 (Serpentis) f)ossesses a 

 structure which, whatever the cause, is 

 identical with that of M3. v. Zeipel re- 

 marked that the excessive central con- 

 densation was more marked among the 

 bright than among the faint stars. 



The problem has again been discussed 

 by V. Zeipel in an elaborate memoir, 

 using in this instance counts of the stars 

 in M2 (Aquarii), M3, M13 and M15 

 (Pegasi). He first finds solutions corre- 

 sponding to these values of 7 : 



(M2) I-200, (.M3) 1156, (M13) 1183, 

 (.Mi5)ii79 



Thus M2 conforms with the same 

 simple law, which I had found to hold 

 so f>erfectly for o Centauri. On the 

 other hand, M3 is again seen to depart 

 from it, and even with the new value of 

 7 the representation is far from good. 

 The law of density here contemplated is 

 a solution of the equation : 





+rp=o, 



and satisfies a physical condition in being regular 

 at the centre. The general solution, however, 

 possesses a singularity at this f>oint, and contains 

 an additional arbitrary constant. Thus the parti- 

 cular law given above is only a sf)ecial case of the 

 general solution for 7 = 1*2, which, as v. Zeij>el 

 shows, can be expressed in elliptic functions. -Ac- 

 cordingly, he abandons the central condition, and 

 introduces the additional constant which is to be 

 determined, together with 7, for each case. With 

 this modification of the theory the values of 7 

 became : 



^\z) 1 194, (M3) 1198, (M13) i-203, (Ml 5) I 197, 



