between JElectromagnetism and Geometry. 625 



covariant is represented by the group of conformal trans- 

 formations of the space of four dimensions *. 



Minkowski, Born f, and Herglotz J endeavour to represent 

 the paths of a system of connected particles by means of the 

 orthogonal trajectories of a system of go' hyperplanes in the 

 space of four dimensions. The sections cut out on the dif- 

 ferent hyperplanes by a tube of orthogonal trajectories may 

 be derived from one another by means of displacements (i.e. 

 transformations of rectangular axes) in the space of four 

 dimensions, and so the corresponding views of the system of 

 particles are derived from one another by means of trans- 

 formations for which the electron equations are covariant. 



This result may be generalized by considering the ortho- 

 gonal trajectories of a system of hyperspheres (or spheres) 

 in the space of four dimensions. It has been proved that the 

 sections of these hyperspheres (or spheres) by a tube of 

 orthogonal trajectories, may be derived from one another 

 by conformal transformations of the space of four dimen- 

 sions §. The corresponding views of the connected system 

 of particles are consequently derived from one another by 

 means of transformations for which the electron equations 

 are covariant. 



3. The space-time vectors introduced by Minkowski || admit 

 of simple representations by means of our representative 

 spheres (or spherical waves). 



If we take a particular sphere A as the sphere of obser- 

 vation, its relation to a second sphere B may be specified by 

 a space-time vector (AB) of the first kind which has the 

 effect of displacing the sphere A so that it becomes con- 

 centric with B and at the same time of increasing or 

 diminishing its radius so that it becomes equal to that of B^[. 

 The vector, in fact, is exactly analogous to a displacement 

 vector from one point to another. 



Now just as there are different physical quantities which 

 may be represented by vectors, so there are different physical 

 vectors which may be specified by means of space-time vectors 

 of the first kind. 



* See papers by E. Cunningham and the author, Proc. London Math. 

 Soc. 1910. 



t Ann. d. Physik, vol. xxx. (1909). Physik. Zeitschr. vol. x. p. 814 

 (1909). 



t Ann. d. Physik, vol. xxxi. Heft ? (1910). 



§ This is practically done by Darboux, Lecons sur les Systemes 

 orthogonanx, Paris, 1898, Ch. II. 



|| Gottinyer Nachr. 1908. Physik. Zeitschr. 1909, p. 104. 

 •jf The components of the displacement of the centre and the change 

 in radius may be taken as the four components of the space-time vector. 



Phil. Mag. S. 6. Vol. 20. No. 118. Oct. 1910. 2 T 



