1912]
ELECTRIC CURRENTS IN A THIN ANCHOR-RING
107
coordinates, or rather velocities, by which the kinetic energy of the system is defined.
For the present application we suppose that the distribution of current round the circumference of the section is represented by
{«„ + «! cos 0! + «2 cos 20! + ...} -/-- ,
.(31)
so that the total current is «„. The doubled energy, so far as it depends upon the interaction of the ring currents, is
«i cos 0! + «2 cos
cos
, (32)
where if]2 has the value given in (2), simplified by making p: and p2 both equal to p. To this has to be added the double energy arising from the interaction of the ring currents with the primary current. For each element of the ring currents (31) we have to introduce a factor proportional to the area of the circuit, viz., TT (a + p cos 0a)2. This part of the double energy may thus be taken to be
I (Ufa (a + p cos 0i)2 («„ + «i cos 0X + «2 cos 20X + . . .),
J
that is
a2 + ijt-p2) a0 + aP«i +
aa, etc., not appearing. The sum of (33) and (32) is to be made a minimum by variation of the a's.
We have now to evaluate (32). The coefficient of ot02 is the quantity already expressed in (16). For the other terms it is not necessary to go further than the first power of p in (2). We get
47m [«„" {log — (l + £) - 2J + i («•> + K2 + -K + • • •) LI P \ *a l )
(34)
Differentiating the sum of (33), (34), with respect to a0, «i, etc., in turn, we find
H(a2 + -|p2) + 4aa0 jlog — (l + ^~J - 21 + p«! flog —' - ^J = °> (35)
O« 1 \ O., ^
:0, ..................................(36)
.(37)ircuits where no independent electromotive force acts. If #x be regarded as given, the corresponding values of Xz, x3i ... are to be found by making T a minimum. Thus