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The problem of the universe is essentially an application of the law of gravitation to a region of extremely low density. The mean density of matter up to a distance of some ten millions of light years from us is of the order of 10-30 gr./cm.3; if all the atoms of the stars were equally distributed through space there would be about one atom per cubic yard, or the total energy would be that of an equilibrium radiation at the temperature of liquid hydrogen. The theory of relativity points out the possibility of a modification of the law of gravitation under such extreme conditions. It suggests that, when we identify gravitational mass and energy, we have to introduce a constant. Everything happens as though the energy in zacuo would be different from zero. In order that absolute motion, i.e., motion relative to vacuum, may not be detected, we must associate a pressure p = -pc2 to the density of energy pC2 of vacuum. This is essentially the meaning of the cosmical constant X which corresponds to a negative density of vacuum po according to 'XC2 Po = -G 10-27 gr./cm.3(1) 47rG Let us consider the motion of matter symmetrically distributed round some fixed point 0. The classical equation of motion under the action of the modified gravitational field is 2Gm +(2) where m is the mass inside the sphere of radius r and center 0. The condition that the system expands, remaining similar to itself, is that h and m have to be proportional, respectively, to r2 and r3. This classical motion is a good approximation of the relativistic equations when r is small enough. When r is great, some geometrical modifications become important and the classical model must be interpreted as a map in euclidean space. This map is like an orthogonal projection: lengths perpendicular to the radius vector are not altered, but along the radius vector they are represented at a scale 1 -h/c2(3) and the scale vanishes at the boundary of the map where h = c2. If we write 12 PROC. N. A. S

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