420 Dr. H. Bateman on some 



If u and v are the component velocities, we have 



u — ir==f(X + iY, T)=/(a, t), 

 and the paths of the particles of fluid are obtained by writing 



_dX _dY 



u " dT ' v ~ dT ' 



Hence the paths of the particles are given by the equations 



which are the same as those obtained above. 



Since the equations a = const., /S = const., r = const. repre- 

 sent a point which moves along a straight line with the 

 velocity of light, it follows that a line of electric force in 

 the present type of electromagnetic field can be regarded as 

 made up of a system of particles which are projected from 

 the different positions of the moving point charge in direc- 

 tions which vary with t according to a certain law. These 

 particles are supposed to travel along straight lines with the 

 velocity of light, and their positions at any instant give a line 

 of electric force at this instant. This geometrical description 

 of the motion of the lines of force has already been given in 

 the simple case when /=0, i. e. when the moving point 

 (£, 7), f, t) is the only real singularity *. The only difference 

 in the present case is that the law for the variation of the 

 direction of projection is different. 



In the simple case a point with coordinates (X . Y) in the 

 plane II corresponds to one line of force, and if we associate 

 a time T with the point and consider the line of force at one 

 instant t, there will be just one point P on this line of force 

 which was projected from the moving point charge at time 



T = T. 



In the more complicated case we have the path of a 

 particle of fluid 



X=A(T), Y=B(T), 



as the image of a line of force, and we may obtain a line of 

 force from the lines of force in the simple case by connecting 

 up the points on these lines which correspond to the different 

 positions of our particles of fluid. 



In this more complicated case there is still a constant 

 electric charge 47r associated with the point charge 



* Bulletin of the American Mathematical Society, February a 1915 • 

 ' American Journal of Mathematics/ April 1915. 



