62$ Relation between Elect romag netism and Geometry. 



transformations. If we denote the velocity of light by unity? 

 it may be shown, for instance, that a transformation which is 

 such that 



\[dx* + dy 2 + dz 2 - dt*] + ix[y z dx + v y dy + v g dz - dt~\* 

 =dx* + dy' 2 + dz"-dt'*. 



is suitable for the purpose provided the vector v whose 

 components are (t^, v , v g ) is connected with the components 

 (E z , E , E z ) (H x , H , H z ) of the electric and magnetic force 

 by the relations 



It can be shown that an expression of the form 

 v[y x dx + vdy 4- vdz — dt] 



is an invariant for transformations of this kind and for the 

 whole group of spherical wave transformations. I have been 

 try ing to find a physical interpretation of this vector. 



The University, Manchester, 

 June 16th, 1910. 



[Note added Aug. 6th, 1910.] Since the electromagnetic 

 equations specify the properties of a view of a set of particles 

 and a view is represented by a hypersurface in the four- 

 dimensional space, it appears that a transformation from one 

 view to another for which the electromagnetic equations are 

 covariant need only give a conformal representation of one 

 hypersurface on the other, and not necessarily a conformal 

 transformation of the whole hyperspace. 



It is possible then that the motion of a connected system 

 of particles may be represented by a continuous conformal 

 transformation of a hypersurface or, in particular, by a con- 

 tinuous deformation without stretching. The path of a particle 

 is represented by the successive positions of a point on the 

 hypersurface in the successive deformations. The case in 

 which the hypersurface becomes torn during the deformation 

 is probably irrelevant for physics since a partic'e corre- 

 sponding to a point at which the tear originates would divide 

 into two. This case may, however, be of some biological 

 interest. 



