54 Conway—A New Foundation for Electrodynamics. 
which the force is to be estimated from the charge. (F, G, H) denotes a vector 
potential, and is equal to e(%, %, v;)/7 where (21, 22, 23) is the velocity of the 
charge, V being the velocity of radiation. These expressions include the laws of 
currents, but without modification cannot be extended to include radiation. The 
only successful attempt that has been made, as far as I am aware, to embrace all 
the facts of electricity is that of Levi-Civita,* who makes use of the fact that 
all these effects are propagated with velocity V, so that the quantities e¢, w, 2, w 
should not have their values at the time /, but at the time ¢—7/V. The electric 
force then satisfied the equation (V? V’?- 0°/d¢?)(X, Y, Z)=0. This scheme of 
Levi-Civita’s, although complete, is open to criticism. The old school of action- 
at-a-distance directed attention solely to the places at which the charges were 
situated. Maxwell, upholding the ideas of Faraday, located everything in the 
medium—the ether. In the above scheme we look at regions jived in the ether, 
and regard electric charge as something which may appear or disappear, increase 
or decrease, in this region. If, however, we are to regard as all-important—as 
we are compelled to do—the electric carrier, whether we call it ion, electron, or 
corpuscle, then we must look upon an electric charge as possessing an individuality 
of its own, and must propose a scheme which will enable us to follow each 
particle in its wanderings. I pass on, then, to a modification of Helmholtz’s 
theory, which will accomplish this. Going back to the ideas of B. Riemann, the 
quantities which are regarded as being propagated with velocity V are potentials 
—scalar and vector. The scalar potential of a point-charge e at rest is Ae/r, 
where A is a constant depending on the units employed. Suppose, now, that it 
is moving through a point A, and that, after a small interval of time, dé, it is 
passing through B, then, at any subsequent time, the disturbance generated will 
be enclosed between two spheres, one having its centre at A and its radius at Vé, 
where V is the velocity of the radiation, the other having its centre at B and its 
radius V(¢—d?t). The thickness of this shell at any point will vary, as 
V/(V—v cos @), where v is the velocity of the point-charge, when the ethereal 
disturbance was starting from A, and @ is the angle between the direction of 
motion of the particle and the line joining A to the point of the shell. We are 
thus led to take for the scalar potential the expression A Ve/r(V —v cos 0). Again, 
if there was question of finding the potential at any point (z, y, 2), at a time ¢ 
after starting, it is clear that the position of the particle at some previous time 7’ 
* Jl Nuovo Cimento, ser. 4, t. vi., 1897. 
