Fundamental Concepts of Electrical Theory. 417 



of special fields is such that there is incidence between each 

 pair, then when all the fields are superposed the total field is 



The question naturally arises whether the fields of a 

 number of moving point charges can be mutually incident. 

 If we represent the path of a moving point charge by a 

 curve in a space of four dimensions as in the preceding- 

 paragraph, any two of the curves must lie on either a plane 

 or a sphere. A system of mutually incident fields is obtained 

 when the paths of the charges are represented by straight 

 lines or circles related to one another in a suitable way. The 

 following cases are of some interest, the paths being repre- 

 sented by 



(1) A system of straight lines through a point in the 



space of four dimensions ; 



(2) A system of circles through two points ; 



(3) A system of straight lines in a plane, or curves in a 



plane ; 



(4) A system of curves on a sphere. 



Another case of some interest occurs when all the fields of 

 one system are incident with all the fields of another system 

 but are not incident with one another. Paths represented by 

 the two systems of generators of a quadric surface give rise 

 to such a system. 



The systems of moving particles given by cases (1) and (2) 

 have already been studied in connexion with the theory of 

 time *. 



§ 8. The lines of electric force in a certain type of electro- 

 magnetic field. — If we adopt the customary idea of the field 

 of a moving point charge and imagine it to be specified by 

 Lienard's potentials, it is clear from the foregoing analysis 

 that the fields of a number of moving point charges can onlv 

 be incident in certain rather special cases. Moreover, the 

 idea of incidence loses its importance to some extent when 

 the tubes of force issuing from a single point charge are 

 supposed to fill the whole of space. Sir Joseph Thomson 

 has already suggested that this may not be the case, and 

 has used this idea in some of his physical theories. The 

 mathematical analysis which will now be given may perhaps 

 elucidate matters a little. 



Using the same notation as in § 6, let us write 



* l Messenger of Mathematics,' Nov. 1915. 

 Phil. Mag. S. 6. Vol. 34. No. 203. J\ 7 ov. 1917. 2 G 



