DYNAMICAL PRINCIPLES TO PHYSICAL PHENOMENA. 
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{ xx}x 2 ; 
[yy}y* ; 
(zzjz 2 ; 
{uu}u 2 ; 
{ww} vr ; 
{xw}xw ; 
{yz}yz. 
§ 4. We must now proceed to examine the terms of these types in greater detail, 
and see what coordinates the coefficients {xx}, {yy}, &c., involve. 
Let us commence with the term {xx}x 2 , which corresponds to the expression for the 
kinetic energy in the ordinary dynamics of a rigid body. We have to consider what 
coordinates the quantity {xx} can be a function of. We know that it may be a 
function of the geometrical coordinates x, but we need not consider here the 
consequences of this, as they are fully investigated in treatises on ordinary dynamics. 
Next, {xx} may be a function of the electrical coordinates y, for in a paper published 
in the Philosophical Magazine for April, 1881, I have shown that the kinetic 
energy of a small sphere of mass m, charged with a quantity e of electricity, and 
moving with a velocity v, is 
.< 9 ) 
where a is the radius of the sphere, and p, the magnetic permeability of the medium 
surrounding the sphere. Thus {xx} may be a function of the electrical coordinates. 
The easiest way of finding the effects of the electrification on the motion of the body 
is to notice that by equation (9) these effects are exactly the same as those due to 
an increase 4 pe 2 /l5a in the mass of the sphere, so that whenever an electric charge is 
communicated to a moving sphere its velocity will be impulsively changed. If the 
sphere is in air there will, of course, be a limit to the quantity of electricity which 
can be accumulated on it, and so a limit to the apparent increase in mass. Taking 
Dr. Macfarlane’s value for the electric strength of air (Phil. Mag., Dec., 1880), 
viz., 75, as the intensity in electrostatic measure in O.G.S. units of the greatest force 
which a fairly thick layer of air can bear, it is easy to see from equation (9) that the 
ratio of the greatest apparent increase in mass to the mass of the sphere is of the 
order 3T0 ~ 19 /p where p is the density, measured in C.G.S. units, of the sphere which 
is supposed to be solid. If the charge is on the surface of a thin spherical shell, the 
ratio will be greater in the proportion of the radius of the shell to three times its 
thickness. 
Let us now go on to consider the electrical effects of this term. Since the potential 
