112
ELECTRICAL VIBRATIONS ON A THIN ANCHOR-RING
[36^
in which P, Q, R denote the components of electromotive intensity, 27T/.P 1 the period'of the disturbance, and 27r/a the wave-length corresponding in free aether to this period. At a great distance p from the source, we hav. from (1)
***-*?(-?.-£i-5)..................(3)
The resultant is perpendicular to p, and in the plane containing p and { Its magnitude is
aVa» . (A\ --------sw.%, .................................\*)
where % is the angle between p and £.
The required solution is obtained by a distribution of elementary vibratoi of this kind along the circular axis of the ring, the axis of the vibrate being everywhere tangential to the axis of the ring and the coefficient c intensity proportional to cosra<£', where m is an integer and $' defines point upon the axis. The calculation proceeds in terms of semi-pola coordinates z, OT, <f>, the axis of symmetry being that of z, and the origi being at the centre of the circular axis. The radius of the circular axis is < and the radius of the circular section is e, e being very small relatively to < The condition to be satisfied is that at every .point of the surface of th ring, where (-GT — a)2 + z2 = e2, the tangential component of (P, Q, R) shu vanish. It is not satisfied absolutely by the above specification; bi Pocklington shows that to the order of approximation required the spec fication suffices, provided a be suitably chosen. The equation detenninin a expresses the evanescence of that tangential component which is paralL to the circular axis, and it takes the form
d<f> II0 cos TO$ (m- — a2a2 cos <£) = 0,
where
nn =
V [e2 -f
.(5) .(6)
In (5) we are to retain the large term, arising in the integral when is small, and the finite term, but we may reject small quantities. Tin Pocklington finds
(a2a2 cos <}> — m2) cos m</> d<f>
~
= 0,
the condition being to this order of approximation the same at all points a cross-section.of <j>1} and we might have investigated the problem upon this basis.