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LXV. The Relation between Electromagnetism and Geometry. 

 By H. Bateman, Fellow of Trinity College, Cambridge, 

 and Reader in Mathematical Physics at the University of 

 Manchester*. 



1. TT> ECENT theoretical researches in electromagnetism 



X\j indicate that the science of electromagnetism is 

 closely connected with the geometry of a system of spheres. 

 According to the generalized form of the principle of 

 Huyghens, an electromagnetic disturbance at any point in 

 space can be regarded as the resultant of a large number 

 of elementary disturbances which are propagated in the form 

 of spherical waves. It should be profitable then to study 

 the geometrical properties of an aggregate of spherical 

 waves travelling inwards or outwards with the velocity of 

 light* 



Two distinct sets of properties must be dealt with. First 

 of all we must regard the spheres simply as geometrical 

 figures and study the geometrical properties in the usual 

 way, and secondly we must consider the relations between 

 the different spheres when various numbers are attached 

 to each. 



If ct denote the radius of a sphere which is contracting 

 with the velocity c, it will have contracted to a point at a 

 time t subsequent to the moment at which it was first con- 

 templated. Similarly, if it is expanding with the velocity 

 of light its radius must have been zero at a time t previous 

 to the moment when it was first contemplated. We shall 

 say in either case that the sphere is the representative sphere 

 of a particle which is at its centre at time + t. 



For some purposes it is convenient to study the kinematics 

 of a particle when different times are associated with its 

 different positions, and for other purposes it is convenient to 

 study the geometry of the system of representative spheres. 

 The advantage of using the second method is that we may 

 study the whole history of a particle by considering its 

 chain of representative spheres at a given moment of con- 

 templation t- 



A complex of go 3 representative spheres which are related 

 to one another in some way will be called a view of the 

 universe. It may be replaced by the corresponding system 

 of particles if each particle is considered at an appropriate 

 time determined by the radius of the representative sphere. 



* Communicated by the Author. 



t It should be noticed that if a particle is moving with a velocity less 

 thaa that of light, no two of its representative spheres with positive 

 radii intersect. 



