PHYSICS: H. BATEMAN 
141 
less than the velocity of light the different spheres bearing electricity that exist 
at time t do not intersect and if the arbitrary function f(a) is never zero 
there will be a sphere through each point of space so that our elementary aethers 
will fill the whole of space; if however the function f(a) is sometimes zero, for 
axemple if it is zero when a is less than ao, then the elementary aethers will 
not fill the whole of space. 
If we subtract from the above field another one of the same type in which 
the unit vector function s has a different value we obtain a field in which 
pv and p are zero except in the neighbourhood of the concentrated electric 
charges, there is thus a cancelling of electricity when the two elementary 
aethers are superposed and we get an aether in the ordinary sense of the 
word. The field is now one in which concentrated charges of opposite signs 
are continually produced by a process of separation analogous to that described 
by Heaviside in 1901. The field thus obtained belongs to the type in which 
there is a rectilinear flow of energy and no accumulation of energy at any point 
of space : the energy in such a field may therefore be regarded as kinetic energy 
or energy of motion. 
The most general field that possesses the property just mentioned and the 
additional property that the volume charge and current in the aether are 
zero outside the singularities of the field is obtained by writing 
M = H + *E = cF(a,(3) (VaXV/3) =iF (a,(3) \~Va -^V^ 
where a and are defined by equations of type 
z - ct =f(a,(3) + (x + iy) 0 (a,(3), % + ct = g(a,(3) - *-f^L 
f, g and 6 being arbitrary functions of a and /3. It may be remarked that 
M.M = 0 and that 0 is a solution of the wave equation. 
In all the fields of the above type electricity or magnetism travels along 
straight lines with the velocity of light (the case of a plane wave of light 
is, however, an exception). To obtain fields in which electricity or mag- 
netism travels with a velocity less than that of light we must superpose fields 
of the above type in such a way that there is a cancelling of nearly all the 
concentrated electric or magnetic charges. It is fairly easy to prove that 
the field of an isolated electric pole moving with a velocity less than that of 
light can be regarded as the limit of two superposed radiant fields of the type 
obtained by subtracting two of our elementary solutions. According to this 
idea the electricity at an electric pole is continually being renewed, moreover, 
it is the electric charge itself which is directly responsible for the effects pro- 
duced at a distance, but to understand fully the production of these effects 
we must consider how this charge is constituted remembering how the field 
