without the Equivalence Hypothesis. 103 



natural components g ± , ..»g± given in (2), viz., by the 

 general transformation rule already quoted, 



«= ssrss ^ (4) 



the office of ( ) being to remind us that the values (2) are 

 meant, if a? l5 # 2 , x z , # 4 stand for r, <£, #, ct. Similarly we 

 shall have for the contra variant tensor gv = y;j in any 

 w-system, remembering that (</**) =1/ (</;), 



to be summed over k=1 to 4, as before. Thus, whenever 



required, it will be easy to pass from the above to any other 



system of reference. 



Even taking for granted that (1) does represent a world 



of constant curvature and is thus equivalent to a generally 



covariant way of defining that manifold, yet the reader may 



feel formally unsatisfied by seeing the fundamental tensor 



gij thus to assume a variety of forms in different reference 



systems. It will be well, therefore, to give here already 



certain properties of that tensor which do preserve even 



their outward form in all systems of coordinates. In fact, 



let, in the very old notation, (l/xkX) be the four-index 



symbols of Rieinann belonging to the general quadratic 



form (3), certain differential expressions in gy to be quoted 



later on. These "symbols" are themselves the constituents 



of a tensor of rank four ; in the case of n dimensions there 



n 2 

 are in the most general case only — ^ [n 2 — 1) linearly 



independent Riemann symbols *, which makes 20 for the 

 four-dimensional world. Now, by a most remarkable, 

 although half forgotten, theorem of general geometry |, the 

 necessary and sufficient condition for a manifold to be deve- 

 lopable upon a " sphere," i. e. to have constant Riemannian 

 curvature, is that all the Riemann symbols (ifi/cX) should 

 bear a constant ratio to the expressions giKg^iX — giXg^ K ; that 

 ratio being precisely what we have called the curvature of 

 the manifold in question. 



Thus, in our case, (3) being only (1) transformed, we have 

 in any reference system, natural or not, 



{tfi/cX)= jgigiKgpx-gotffiK) (5) 



* Cf. Wright, I. c. pp. 11 & 23. t Killing, I. c. p. 232. 



