404 Integration Theorems of Four- Dimensional Analysis. 



Gibbs has shown that a rotation may be expressed by 



cj>p = i'(i P )+f(j P ) + kXkp), 



where (ip) = the scalar product of i and p ; and we may also 

 express a rotation in four-vectors similarly, 



< f,p = -(H>V ap+/3'Y /3 P + y f Y yp + 8'V 8p), 



t'j'k', ijk. are unit vectors and each group is mutually 

 perpendicular. Similarly a ft y' 8' and a ft y 8 are unit 

 mutually perpendicular vectors. 



The Einstein transformation is a particular case of this in 

 which 



8 , = k(ivi 1 + i i ), 



where k= (1— v 2 ) ~i, i= s/ — 1 ; v the arbitrary constant in (p 

 is the velocity and may be said to define the strain. 



The notation suggests that the restricted principle of 

 Relativity may be summed up by stating that a fundamental 

 four-dimensional medium exists which may b^ subjected to 

 a strain of the type (13*1) with the values of the vectors 

 given above and that phenomena described by the unstrained 

 vectors will be described by the strained vectors in exactly 

 the same way. 



It is natural to generalize <f> and give up the demand that 

 it should be linear. 



The strain is then heterogeneous and the principle becomes 

 more general. 



For small regions there is still a linear relation, for if a is 

 a function of p we have : 



da = —Y dpD 1 . cr = (f)dp ; 



but 4> now 'contains p in its constitution and the linear 

 relation is true only for small changes dp in the neigh- 

 bourhood of p. 



This appears to correspond to the " naturalness " of small 

 regions in the theory of Relativity. Space-time is Galilean 

 for infinitesimal regions. In addition (da) 2 is invariant or 



