22 Prof. F. D. Murnaghan on the 



\m 9P i - 2gq ^ p2 + 2 ^ dt%2 4 ^ 2 ^- j %?2 ^ a ^ 

 4 &#? (.\d#?/ d# P c^>) ""*" 4#, t#v Sa> d^ #* d#« B^j ' 



where p and q are distinct non-umbral symbols and r, s are 

 different fromp and q. In order to obtain the differential 

 equations G rs = 0, which Einstein assumes the $vmust satisfy, 

 it is necessary to contract this four-index tensor. Thus 



4 



G ps ^%{pq, qs} = and G PP ~%\pq, qp\ = 

 q=l 9 =i 



are the ten differential equations. Now in the expression 

 for ^pq, qs^ it will be noticed that differentiation with 

 respect to x s occurs in every term, telling us at once that 

 \pq, q£ j- ^ since none of the coefficients g r involve # 4 .. 

 Hence the results 



Gi 4 = ; G^ 4 = "0 ;, G = 



are a consequence of the assumed properties of a statical 

 field. 



In the remaining calculations it will simplify matters 



4 



to observe that in the summation X \pq, qs\ it is sufficient 



9=1 



to give q the two values different from p and s. For, 

 from the definition of the four-index symbols of the first 

 kind, [_pp, qs~] = (p not umbral) and, in our system of 

 orthogonal coordinates, 



\PP> 9 s } — g pk [pk, qs n ] ... (k umbral) 



— 9 pp [.PPi y s ~\ — ®y as * s similarly |p<?, ss^. 



(In this argument p, q, s take any numerical values distinct 

 or not.) Thus G ]2 = {13, 32} + {U, 42}. We shall now 

 suppose that one set of coordinate lines, x x varying say, are 

 geodesies through the space-time point to be specified and 

 the gravitating centre with the same value of #4, and we may 

 conveniently take x l as the arc distance from the gravitating 

 centre, which is, then, a singularity of the coordinate system : 

 for the knowledge of merely two coordinates x l = and its x± 

 is sufficient to determine it — the other two coordinates x 2 and 

 # 3 being indeterminate (for the sake of an analogy x 1 may 

 be compared with the axial distance in ordinary cylindrical 

 coordinates) . The equations of the geodesies in any space 



