154 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1925 



permanence of star clusters, in particular, the great star cluster 

 or galaxy near the center of ^vhich our solar system finds itself. 

 Silberstein bases his calculations upon the four-dimensional space- 

 time relations of de Sitt«r, and from this starting point he last year 

 deduced a relation which could be evaluated in terms of observed 

 astronomical data in such a way as to give a numerical value for 

 an invariant characteristic of spacetime called by the mathemati- 

 cian the radius of curvature. This quantity, symbolized by R, has 

 the finite value of 10^- astronomical units, that is 10'- times the dis- 

 tance from earth to sun. This theoi'y, with the consequent value of 

 R, has not been universally accepted, but this does not detract from 

 the interest of the subsequent reasoning by which Silberstein deduces 

 a criterion of stability in terms of the total mass of a system of 

 material bodies (molecules or stars) and the radius of the system. 

 Associated with any given mass there is a critical distance. If a 

 star be at a greater distance than this critical value from the center 

 of gravity, its orbit will of necessity be a hyperbola. This means 

 that sooner or later it will desert the system forever. On the other 

 hand, if its distance from the mass center be less than the critical 

 value it will describe an elliptic orbit, thus remaining indefinitely 

 within the system. 



This criterion has been applied to those globular clusters far out 

 in space beyond our own galaxy, for which the astronomer has been 

 able to form estimates of their size and nuiss. They are found to be 

 considerably less massive than our galaxy and very much more 

 closely packed, so closely packed that the calculated critical radius 

 is much greater than the dimensions of the clusters, which may, 

 therefore, from the point of view of this theory, be considered as 

 stable aggregates of stars. 



The reverse is the case of our own galax3\ Mucli too widely scat- 

 tered for its mass, its radius exceeds the critical value for stability, 

 and therefore this theory predicts that it will suffer from what 

 Doctor Silberstein terms '' hyperbolic desertion " until its ranks be 

 reduced and its volume diminished to such an extent that the cri- 

 terion might perchance be satisfied. In its present form it is, like 

 the Roman Empire, far too inflated to be enduring. 



Densities can be treated in a similar manner. Silberstein evalu- 

 ates the critical density of matter in space in terms of his finite 

 curvature invariant R, the gravitational constant and the velocitj'^ 

 of light — three fundamental quantities in this complex universe. 

 Any aggregation of matter of less than this critical density will be 

 unstable and tend to dissipate, whereas any aggregation of density 

 exceeding this value will be in a state of stability. The galaxy of 

 stars in which our system finds itself is estimated to liave a density 

 fifty-two times too small to satisfy the conditions for permanence. 



