C>24 Mr. H. Bateman on the Relation 



Ifc has been shown that the fundamental equations of the 

 theory of electrons simpty describe the properties of an 

 arbitrary view of the universe*. 



We may pass from one view of the universe to another by 

 means of a transformation which transforms a representative 

 sphere into a corresponding representative sphere f. It has 

 been shown that the fundamental equations of the theory of 

 electrons are covariant for all transformations of" this kind. 



This group of transformations possesses the remarkable 

 property that the lines of curvature on the wave surface 

 enveloped by a system of representative spheres are trans- 

 formed into the lines of curvature on the corresponding 

 wave surface. The group of transformations is in fact 

 identical with that studied by Sophus Lie |. A particular 

 transformation due to Ribaucour § which has been called by 

 Laguerre || ")a transformation par directions reciproques" is 

 easily seen to be identical with the transformation used by 

 Lorentzlf, Larmor**, and Einstein ft, to pass from the views 

 obtained by one observer to the views obtained by another 

 observer moving with uniform velocity relative to the first. 



2. The late Russian mathematician Minkowski of Got- 

 tingen has made considerable use of a representation in 

 which a particle which is at the point (#, ?/, z) at time t is 

 represented by a point whose coordinates are (#, y, z, ict) 

 in a space of four dimensions J J. 



The group of Lorentzian transformations for which the 

 electron equations are covariant is then represented by the 

 group of transformations of rectangular axes in the space of 

 four dimensions. The more extensive group of spherical 

 wave transformations for which the electron equations are 



* See a paper by the author " On the Transformation of the Electro- 

 dynamical Equations," Proc. Lond. Math. Soc. (1910). 



"I" A simple transformation may be obtained by increasing" or decreasing 

 the radii of the spheres by the same amount. Other typical transfor- 

 mations are displacements, magnifications, and inversions. 



+ Mat heynatische Annalen, vol. v. Gottinger Nachrichten (1871). 



§ Comptes Rendus, t. lxx. p. 332 (1870). 



|[ Ibid. t. xcii. p. 71 (1881). See also Darboux's ThSorie des Surfaces, 

 t. i. p. 253. 



51 Amsterdam Proceedings (1904). The covariance of Maxwell's 

 equations was established by Voigt , Gottinger Nachr. 1887, p. 41. 



** ^Ether and Matter, 1900. 



ft Annalen der Physik, Bd. xvii. (1905). 



IX Gottinger Nachrichten, 1908. Physikalische ZeiUchrift, 1909, 

 pp. 104, 216. The transition from Minkowski's representation to our 

 representation gives rise to a very interesting correspondence between 

 the spheres in space and the points of a space of four dimensions. This 

 correspondence has been studied by Davboux, A?inales de VEcole Normale, 

 1872. 



