Fundamental Concepts of Electrical Theory. 421 



(f, 77, f, r), and the lines of force at any instant start 

 from the position of the point charge at this instant. 

 The point charge is, however, not generally the only singu- 

 larity in the field, there are other singularities which move 

 along straight lines with the velocity of light. A good idea 

 of the nature of these singularities is obtained by con- 

 sidering the singularities in the associated two-dimensional 

 fluid motion, assuming that sources and sinks correspond 

 respectively to positive and negative charges travelling with 

 the velocity of light. 



Now in the theory of two-dimensional fluid motion singu- 

 larities are often avoided by the introduction of free surfaces 

 or boundaries made up of lines of flow, and a similar idea may 

 be adopted with advantage in the present case. Since a line 

 of flow corresponds to a line of electric force, the free surfaces 

 or boundaries in the electromagnetic field will be lines of 

 electric force ; this is just what has been suggested by 

 Sir Joseph Thomson. It should be noticed also that if 

 the boundary of the two-dimensional motion is stationary or 

 fixed, the corresponding boundary of the electromagnetic 

 field is made up of lines of force of the simple field, i. e. the 

 ordinary electromagnetic field associated with a moving 

 point charge. 



The idea that the tubes of electric force do not fill the 

 whole of space is a very natural one if we wish to retain 

 the principle that two tubes cannot occupy the same space 

 at the same time, for then some room must be left for the 

 tubes of force issuing from other charges. 



m • • • 



This principle cannot be regarded as an axiom at present 

 because it is not generally accepted. Sir Joseph Thomson 

 has, for instance, considered the work done when one 

 charged particle penetrates into a tube of force issuing from 

 another *. It is clear, then, that there are several directions 

 in which the theory of moving tubes of force may be de- 

 veloped. The following possibilities may be cited as worthy 

 of discussion : — 



(1) Permanent and mutual incidence of the lines of force 



of the different point charges. This need not apply 

 to all the lines of force. 



(2) Temporary incidence of lines of force of different 



point charges but no cutting of lines of force. The 

 analogy with a system of perfectly elastic moving 

 spheres may be useful in the study of this case. 



(3) Motion in which the lines of force can cut right 



through one another. 



* Phil. Mag. Oct. 1913. 



