between Electro-magnetism and Geometry. 627 



Two special space-time vectors or* the second kind are said 

 to be orthogonal when their corresponding lines are polar 

 lines with regard to the sphere of observation *. 



A space-time vector of the third kind may be regarded as 

 representing the relation of three spheres B, C, D to the 

 sphere of observation A. It may be represented by the 

 plane of similitude of the four spheres K, B, C, D, and may 

 be classified as spacial or temporal according as the plane 

 does or does not intersect the sphere. Two space-time vectors 

 of the third kind are said to be orthogonal when their repre- 

 sentative planes are conjugate with regard to the sphere of 

 observation. 



The application of these ideas to electromagnetism depends 

 upon the fact that the components of the magnetic induction 

 together with the components of the electric force must be 

 regarded as the six components of a space-time vector of the 

 second kind. In the case of the simplified equation of the 

 theory of electrons, the components of the convection current 

 together with the volume density of the electricity form the 

 four components of a space-time vector of the first kind f. 



The components of the electromagnetic vector potential 

 together with the electromagnetic scalar potential form the 

 four components of a space-time vector of the third kind J. 



The study of the properties of these vectors is facilitated 

 by considering integral forms of the type 



R x d(y, z) + Kd(z,x)+K z d(x,y) 



+ E/<>, o + ly%, t) + E z d(z, 0, 



p<o x d(y, s, +po>d(z, x, t) + pu?d(x, y, t) -pd{x, y, z), 

 A x d(y, z, t)A-K y d(z, x, t) +A g d(x, i h t) -<!><*(#, y, *), 



as in my paper on the transformation of the electrodynamical 

 equations. 



It should be remarked that the transformations which can 

 be used to transform a ■particular electromagnetic field into 

 another are not confined to the group of spherical wave 



* A vector of the second kind may be regarded as temporal when its 

 representative line meets the sphere of observation in real points, and as 

 spacial when the line does not meet the sphere in real points. 



"J" The principle of the conservation of energy is expressed by the fact 

 that the space-time vectors, whose four components are the three com- 

 ponent forces and the rate at which work is being done, is normal to the 

 space-time vector of the first kind mentioned above. 



\ There is a reciprocal relation between vectors of the first and third 

 kind. 



2T2 



