56 Conwayr—Llectro-magnetic Mass. 
If y denotes the distance of the electron e at the time ¢ from the point 
(x, y, 2), we can easily verify from the equation 
Vip = V0$/o, 
that 
Page (el IL Gp 1 oe 2 
AVE 7 ome ap AA OES I. 
or to the second power of = 
1th ag ore 
va el= + orm ap 
To the same approximation 
(FZ; G, H) = =e ay 
where (#, #1, 2) 1s the velocity of the electron. When there is a great number 
of electrons moving over small paths, it is more convenient to refer to fixed volume 
elements.* In this case the scalar potential due to a volume element dz is pdz/r, 
where p is the density at the time ¢-7/V, whilst the vector potential is 
(21, %, %3) dz7/ Vr, where (%, %, ¢3) denotes the current. 
In the first case let us consider the mutual forces of two portions of electricity 
A and B moving with the same velocity. On rejecting the parts of the mutual 
forces which balance, we have left only a scalar potential 
Gf _ & _ 
NOt Bg eM Bay WP AH Be) 0 
and from the vector potential we get components of force ~ 
— é (4, iy 2,)/r. 
We thus get for the electric force two parts: the one opposite in direction to 
the acceleration f and equal to ef/r, and the other at right angles to the line 
joining A to B and equal to efcos¢/r, where ¢ is the angle which AB makes 
with the acceleration. 
In the second case if, as in the case of conduction currents, the electrical 
density is zero, the only portion of the force to be considered is éf|r. 
* Proc. London Mathematical Soc., Joc. cit. 
