128 SEE— DYNAMICAL THEORY [April .9, 



equations become : 



47rn = J J V^dS - 47r F' • a(/ - /„) - a\t - Q 



(25) 



where a and /?' are positive numerical coefificients in the form of 

 undetermined multipliers. The first of these negative terms depends 

 on appulses due to close approach, the second on the general shrink- 

 age due to the mutual attraction of the stars of the cluster at all 

 distances. 



The action of the clustering power upon the figure of a cluster, 

 as Herschel remarked {Phil. Trans., 1789, p. 219), is analogous to 

 that of gravity on the figure of a planet. It might be compared also 

 to the well-known effect of surface tension on the figure of a drop 

 of dew or a drop of mercury, etc. In these last phenomena the sur- 

 face is made a minimuni, for a given volume, by the restriction of the 

 elastic layer constituting the outer boundary. A soap bubble is also 

 a good illustration of such niiniuial surfaces, the mathematical theory 

 of which has been placed on a strictly rigorous basis by the researches 

 of the late Professor Weierstrass, one of my most revered teachers 

 at the University of Berlin. In the case of the clusters, however, 

 we have not only a tendency to minimal surfaces, but also for such 

 an internal arrangement of the stars with increase of density towards 

 the center as will reduce the potential energy of the system to a mini- 

 mum. The theory of the clusters is therefore much more complex 

 than that of simple minimal surfaces, such as we see in drops of 

 dew or soap bubbles, to which the analysis of Weierstrass is applicable. 



In the case of the minimal surfaces of the type rigorously treated 

 by Weierstrass, the determination of the minimum is found by the 

 usual condition in the calculus of variations, 



hu=0, (26) 



where the function « = x(-r, y, -) represents the surface. 



• In the more general problem of clusters, the determination of the 

 minimum potential energy applies to every shell as well as the exter- 



