Relativity of Field and Matter. 805 



unaffected by our ignorance of the laws of matter. When 

 for instance we ask why Einstein's equations of gravitation 

 should be chosen rather than one of the more complex 

 systems of covariant equations which are possible, we have 

 to seek an answer, not in the structure of the field, but 

 in the structure of the electron. It depends on whether 

 the form of the electron corresponds to the locus of 

 curvature or to some more complicated absolute locus in 

 the world. Every thing points to the locus of curvature 

 as the more likely ; but until the precise mode of formation 

 of the electron is decided, other solutions may remain as 

 a perhaps far-fetched possibility. 



Appendix. 



Proof that the equations Q{ iV =tyuv express the condition 

 that the radius of curvature is the same at all points in 

 all directions. 



A four-dimensional manifold with general Biemannian 

 geometry can be represented as a surface in Euclidean 

 space of ten dimensions. At any point the tangent and 

 the normal give five dimensions, which osculate the surface 

 to an order sufficient for the calculation of the curvature. 



Let then the equation of this portion of the surface, 

 referred to rectangular Euclidean coordinates (# l9 X2, x 3 , 

 #4, z) along the lines of curvature and the normal, be 



2z = k ± 05^ + k 2 x<? + k 3 x£ + k 4 a'4 2 



so that hi, k 2 , k%, k± are the reciprocals of the principal radii 

 of curvature. We have, by Euclidean geometry, 



-ds 2 = dJ + dasf + dxf + daf + dxt. 



Since four coordinates are sufficient to specify a point on 

 the surface, we eliminate z, obtaining 



- ds 2 = (1 + kfxf)dx-? + . . . 4- %k 1 k 2 a? 1 a?2dM 1 d®2 + > • . 

 so that 



9W = -(* + VV)» fy* = -*m*vVv ( not summed). 

 Accordingly the first derivatives of the </'s vanish at the 



