III
129
DISTRIBUTION OF ELECTRICITY ON MOVING CONDUCTORS
external normal, and a, b, c denote the angles which n, makes
with the axes. From (1) and (2) we get
dv²p
dt
=
4π▼²д, or V²µ= (V²µ)。e¯***¸.
K
Hence, if the density inside is not zero initially, it still
continually approaches this value, and cannot be again pro-
duced by electrostatic influences.
Hence we have here
V² = 0
(5).
(6),
Further, from (1) and (3),
or by using equation (4)
dh
K-
dt
=
pi
ani
x d (api ape
K
+
4π dt an; ane.
афі
=
-
Ini
The equations (5) and (7) involve o alone.
(7).
Equation (5)
must be satisfied throughout space; equation (7) at the surfaces
of all conductors. is determined for all time by these equations
-which no longer involve a reference to any particular system
of coordinates-together with the well-known conditions of
continuity and the initial value of 4. In the differential co-
efficient dh/dt, h relates to a definite element of the surface;
if the velocities of this element relative to any system of co-
ordinates be a, B, y, then the above equations will refer to
these coordinates, provided dh/dt be replaced by
ah ah Əh
+a +B
θε да მყ
Əh
+7
az
We get for the heat generated in time St
SW = St√x(u² + v² + w²)dτ,
=
1
- 2 de foameds,
KC
- Sp8hds,
= -
where ds denotes an element of surface, and the integrals are
to be taken, the first throughout the interior, the others over
M. P.
K