General Relativity and Einstein's Theorv. 409 



way that the total or Gaussian curvature k^^^ko specifies that of a 

 two dimensional surface. If the four dimensional space had been 

 homoloidal, each of the terms in (58) would have vanished, and 

 the nature of the representative space would have been too re- 

 stricted to be capable of representing a radial gravitational field. 

 Before converting (56) into a set of scalar relations between 

 the spaces derivatives of the ^'s, it must be recalled that the warp 

 we have given to space makes it impossible to represent even 

 ordinary three dimensional space in the neighborhood of a gravi- 

 tating body by rectangular coordinates (or by any coordinates, 

 such as polar coordinates, which are reducible to the rectangular 

 form). We may, however, use rectangular coordinates for an 

 approximate mapping out of that portion of space which is not 

 too close to the center of attraction. Consequently if a^^, Xo, x^ 

 in the linear element 



ds^ = g^^dx^^ + g^odxo^ + g3zdx/ + g^^dx^^ 

 are identified with the rectangular coordinates x, y, z, we are 

 justified in computing the g's to the first order of approximation 

 only. The coordinate x^ represents the time /, and evidently 



cr <T o" li^ 



611 .i>22 i»33 ''' > 



Take the origin at the center of attraction in the radial field 

 under discussion, and put 



A and v being functions of 



only, and very small compared to unity at all points not too near 

 the center of attraction. As the coordinate system assumed 

 holds only approximately, terms in (56) involving squares or 

 products of the first derivatives of A and v must be neglected as 

 compared with terms which are linear in the first or second 

 derivatives of these quantities. Hence the first and third terms 

 of this equation may be omitted, leaving 



B — —T> • \<Tii.,v] + T>' Xo^Vg^o. (59) 



Substituting their values for the symbols, it is found that 



^„ = ^'+'^ + y, { ^"+ V"- ^-^' I = ., (60) 



