304 



SCIENCE 



[N. S. Vol. LIV. No. 1396 



From the point of view of the future his- 

 torian this would serve as a large labor-saving 

 device, especially in view of the fact that 

 human knowledge is ever becoming more 

 specialized. 



It might be well to call attention to the 

 fact that a bibliography of relativity has also 

 been in progress in England,- namely, the 

 International Catalogue of Scientific Litera- 

 ture, under the direction of Dr. H. Forster 

 Morley. Dr. Morley has made a selected 

 chronological bibliography of relativity and 

 related problems from 1886 to the end of 

 1920. 



The recent visit of Dr. Albert Einstein has 

 not alone stimulated interest among scientific 

 men, but he has strengthened his theory by his 

 own clear presentation of relativity. 



Of course the theory has yet to receive its 

 final verification, before the whole can be ac- 

 cepted, and Dr. Einstein has expressed con- 

 fidence in the final answer. 



Not since the doctrine of evolution was 

 promulgated, has any advance of intellectual 

 progress, either of philosophic or scientific im- 

 portance, caused such profound interest, popu- 

 lar or scientific, as the theory of relativity. 

 And like all epoch-making ideas, the syn- 

 thetical character of the theory of relativity 

 will mark off a period of great importance in 

 the history of science. Hence the value of a 

 bibliogi-aphy of a subject in relation to the 

 history of science is in direct proportion to 

 the importance of the subject itself. 



Frederick E. Brasch 



The John Ckerar Libkart, 

 Chicago, Illinois 



SPECIAL ARTICLES 



EINSTEIN'S COSMOLOGICAL EQUATIONS 



In two earlier notes published in Science 

 (Vol. 52, p. 413, Vol. 53, p. 238) I gave certain 

 geometrical theorems connected with Einstein's 

 original (1914) equations of gravitation, 

 (?ji. = (in space free from matter). I shall 

 now extend some of the results so as to apply 

 to the modified equations employed in Ein- 



2 Dr. H. Forster Morley, Nature, 106, 811- 

 13, Feb. 17, 1921. 



stein's cosmological speculations. These he 

 first wrote (191Y) in the form, Gijc — kgoe 

 = 0; but more recently (1919) he has em- 

 ployed the form (tj^ — ^ Qiic G^O, which in- 

 cludes the previous form and which, when the 

 energy impulse tensor Tj^ is introduced in 

 the right hand member, has the advantage 

 of being possibly applicable to the microcosm 

 (atoms and electrons) as well as to the macro- 

 cosm (the stellar universe). Here Goc is the 

 contracted curvature tensor and G is the 

 scalar curvature. 



For hrevity we shall term any four dimen- 

 sional manifold which obeys the last equations, 

 a cosmological solution. 



I. The only cosmological solutions which 

 have the same light rays as the euclidean or 

 Minkowski world are those which have con- 

 stant curvature in the sense of Eiemann. In 

 other words, if a cosmological world is to 

 admit conformal representation on a euclidean 

 world, it must be of spherical (or pseudo- 

 spherical) character. This result is analogous 

 to the earlier result for (ru- ^ 0, that the only 

 manifolds having the Minkowski light equa- 

 tion are flat (zero curvature). Both results 

 are obviously valid also for geodesic represen- 

 tation (same equation of orbits). 



II. Here we discuss four-dimensional 

 curved manifolds which can be regarded as 

 imbedded in a flat space of flve dimensions. 

 Our result is that for the cosmological equa- 

 tions, there are two distinct possibilities. 



(a) In the first case at every point of the 

 manifold the four principal curvatures are 

 equal, that is ^^ = ^2 = ^3 = ^^, so that 

 every point is umbilical. The manifold is 

 then simply a hypersphere. 



(h) In the second case K,^^ E„=^ — -£"3 

 = — K^, that is, the four principal curvatures 

 are numerically equal, but two are positive 

 and two are negative. Such manifolds may be 

 regarded as a generalization of ordinary 

 minimal surfaces (where -ETj^^ — -^2)' ^^^ 

 may be described as hyperminimal spreads. 

 (It would be interesting to find an actual ex- 

 ample in finite form of such a spread.) 



It will be recalled that for our previous dis- 

 cussion of (rijt = 0, no solution in five dimen- 



