220 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1955 



average, 12 nebulae per cubic megaparsec (55) . The average mass per 

 nebula, including intergalactic matter, can be estimated from the 

 internal motions in clusters of galaxies according to the "virial 

 theorem" (mean kinetic energy per unit mass proportional to the po- 

 tential of gi'avitation) ; this, of course, depends upon the assumption 

 that the clusters are held together by gravitation. The assumption can 

 nowadays hardly be subjected to doubt, considering that otherwise, 

 with the velocities observed, the clusters would have dispersed long 

 ago ; on the contrary, they are gathered so closely together that numer- 

 ous interpenetrations or "collisions" of the member galaxies of a 

 cluster must have happened during the lifetime of the universe (43). 

 Kepeated collisions must have led to "statistical equilibrium" in the 

 distribution of velocities of the member galaxies; the similarity be- 

 tween the radial density distribution of nebulae in these clusters and 

 that of an isotheimal gas sphere (56) supports this assumption and the 

 validity of the virial theorem. For the Virgo cluster a mass of 500,000 

 million suns per nebula results with Hubble's scale of distances (5), 

 and four times as much with the corrected scale. These data lead to a 

 world density of 2.5X10~^^ gm./cm.^ or 64 percent of the critical 

 density. If the result is taken literally, this would mean negative 

 curvature, an open and infinite space into which the universe is 

 irreversibly expanding. 



However, the calculations are not exact enough to warrant un- 

 reserved acceptance of such a conclusion. The estimate has come 

 astonishingly close to the critical density, and therefore, within the 

 limits of uncertainty in the data, the alternative case of closed space 

 and limited expansion followed by collapse is also possible. Indeed, 

 Zwicky (57) finds considerable amounts of matter in the space be- 

 tween the galaxies, and favors a world density about 25 times that of 

 our estimate, which would bring it far above the critical value. How- 

 ever, Zwicky's value is a very rough estimate, not based on the virial 

 theorem. Our estimate of 2 million million suns per nebula would 

 ascribe 90 percent of the mass to intergalactic matter (that between 

 the galaxies) and only 10 percent to the galaxies themselves; this 

 figure seems to be more realistic than Zwicky's, which would set the 

 percentages at 99.5 and 0,5, respectively. 



It is, perhaps, permissible to speculate on the closeness of the world 

 density to its critical value, and to suggest an intrinsic reason for 

 this near equality of the kinetic energy of expansion and the absolute 

 value of the gravitational potential. The reason should be sought in 

 the past history of the world. For example, an oscillating universe 

 whose maximum world radius greatly exceeds the present value would 

 lead to the above-mentioned near equality except when close to the 

 phase of greatest expansion (which should be far ahead of present 



