332 Sir Joseph Larmor 



this extent taken over a small region would have a stationary 

 value for the flat, changing in the same direction on both sides 

 of it. Cf. supra, p. 329. Thus for a spherically symmetrical field 

 the constitution of the fourfold must be determined in polar 

 coordinates by the equation 



Sct2 - {c/c'f 8/2 + {rSdf + (r sin O^f - c'^Si, 



showing that the positional part of the extension is very slightly 

 non-uniform and so not quite Euclidean. It appears to be this 

 secondary feature, not the energy-momentum-stress tensor con- 

 ditions, that modifies gravitation from the Newtonian law. 



The expositions of relativity do not mention an extended 

 fourfold, which would be foreign to the cardinal idea that space 

 is constructed from physical origins, only in so far as it is needed — 

 even though it has to be implied that it is reproduced unerringly 

 each time. But the instrument of such construction or continua- 

 tion of a metric space is an infinitesimal linear measuring rod 

 supposed to have complete free mobility without change of in- 

 trinsic length : and it would seem to be a tenable view that such 

 a mobile apparatus must determine an underlying flat space of 

 higher dimensions* in which the physical system may be supposed 

 imbedded. 



It is to be noted here that a surface defined intrinsically in the 

 Gaussian manner by the distance relation on it 



Ss^ =fSp^ + 2gSpSq + hSq^, 



remains the same surface when the coordinate quantities p, q are 

 changed to others p , q' which are any assigned functions of them 

 both, so that 



Ss^=.f'8p'-^ + 2g'8p'8q' + h'8q'^, 



provided 8s is measured by the same infinitesimal unchanging 

 measuring rod extraneous to the surface in both cases. These two 

 equations represent the same surface, only the generalised co- 

 ordinates of the same point on it are changed from {p, q) to {p, q'). 

 The intrinsic curvatures are the same from whichever form they 

 be calculated: if one form represents a flat, so does the other. On 

 this definition by an intrinsic differential relation surfaces are 

 indistinguishable, if one can be bent to fit the other without 

 stretching. So in the Riemann theory of spaces of more than two 

 dimensions it is the functional forms of the coefficients in the 

 quadratic function of differentials and the mobile absolute mea- 

 suring rod that determine the nature of the space; any transforma- 

 tion of coordinates changes the coefficients (or potentials in the 

 gravitational formulation) but so that the space remains un- 



* For a radial field it need be of onlj' one more dimension. 



