without the Equivalence Hypothesis. 97 



this need — of studying properties intrinsically connected 

 with the manifold itself and of developing appropriate 

 methods — clearly and with much emphasis expressed in such 

 purely mathematical tracts as that* of Prof. Wright? But 

 a strong tendency of that kind, together with great expec- 

 tations for the future, manifests itself even in the old book 

 of Lame on curvilinear coordinates (1859) ; see the con- 

 cluding paragraph, p. 367, of these memorable Lessons. 



2. Space-time (world) of any constant curvature. 



What is here called curvature is a certain invariant of the 

 manifold and, as such, an intrinsic property of the manifold, 

 as real as and possibly more real than the mass of a lump of 

 matter. Whatever its value, nil, positive, or negative, it 

 cannot be settled either by mere reasoning or by convention, 

 but has to be found out by experiment or observation. 

 Being ignorant as to its sign or amount, the best way is to 

 leave it undetermined and to develop all formulae with the 

 corresponding degree of generality. Its evaluation is the 

 task of the future physicist or, more likely, the astronomer. 

 On the other hand, the reason why it is enough to limit one- 

 self to constant curvature, i. e. the same through all times 

 and everywhere, will be readily seen. Again, as concerns 

 the mathematical technicalities, it is almost as easy to study 

 a four-manifold of any constant curvature as a non-curved 

 or homaloidal one (from SfjuaXos = even ; an old name for 

 flat or Euclidean space, of any number of dimensions). To 

 deprive ourselves of generality would thus be a badly 

 compensated sacrifice. 



Let #!, # 2 , x 3 , #4, the first three space-like, and the fourth 

 time-like, be any coordinates fixing a world-point, and let 

 the invariant line-element, determining the metric properties 

 of the four-dimensional manifold, be given by the quadratic 

 differential form 



ds 2 = XX gijdxidxj) 



where gij—gji are, in general, some functions of all the 

 coordinates. If we pass to any other system a?/, then the 

 new gij will be linear homogeneous functions of the gij, 

 viz. in the usual abbreviated notation f , 



* Cambridge Tracts, No. 9 ; cf. especially pp. 3-4. 



t In which the sum signs are omitted, the tacit prescription being 

 that the sum, from 1 to 4, is to be taken over each term in which a suffix 

 occurs at least twice. 



Phil. Mag. S. 6. Vol. 36. No. 211. July 1918. H 



