without the Equivalence Hypothesis. 105 



found in Monthly Notices II. A. S., for Nov. 1917. The latter is espe- 

 cially interesting because it does without the " world-matter," but has 



v 



on the other hand ^ 44 =cos 2 ^> instead of 1, to suit the gravitational 



field equations slightly amplified by Einstein. At any rate both authors 

 are under the strange impression that the world cannot be infinite. 



Finally, notice that in the case of a homaloidal world 

 the theorem expressed by (5) gives [i/aicX) = 0, as it should 

 be, this being the well-known necessary and sufficient con- 

 dition for ds 2 to be reducible to a form with all constant 

 coefficients. 



3. Mathematical Supplement to the preceding section. 



In order to obtain the promised support for (1) as the 

 expression for the line-element of a four-manifold of constant 

 curvature, take g u =g 24: =g u = and measure w 1? w 2 , u s along 

 the principal axes of the three-dimensional linear vector 

 operator g lK (1, 2 V 3). This operator (which itself is no 

 relativistic entity, of course), being self-conjugated, has 

 always such orthogonal axes, and three corresponding prin- 

 cipal values, say, g 1} g 2 , g z . Thus, it 1? u 2 , w 3 being - in general 

 curvilinear coordinates, the expression for the line-element 

 will become 



ds 2 =g 1 du{ 2 + g 2 du 2 2 + g 3 du z 2 + g i du 4 2 , . . (6) 



and det.gij=g l g 2 g$g4, so that the components of the contra- 

 variant tensor will be, simply, 



7 ii= =— and nil for i^j. 



gi J 



As space-coordinates of this kind can be employed con- 

 veniently the polar coordinates ?*, </>, 6 or any other orthogonal 

 curvilinear coordinates known since the times of Lame. 

 Now, what has been repeatedly called the curvature of that 

 world which is given by the above differential form is itself 

 an invariant (one of many) of the differential form, viz. 

 proportional to 



that is, in our case, to 

 where 



=4.B* (7) 



9' 



B .._ U« I i 03 { _ f n 1 f «/8l , A . I i* I _ A fit I n a) 



