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Proceedings of the Royal Society, A, Vol. LXXXVII. pp. 193—202, 1912.]
ALTHOUGH much attention has been bestowed upon the interesting ubject of electric oscillations, there are comparatively few examples in vhich definite mathematical solutions have been gained. These problems jre much simplified when conductors are supposed to be perfect, but even hen the difficulties usually remain formidable. Apart from cases where •he propagation may be regarded as being in one dimension*, we have Sir J. Thomson's solutions for electrical vibrations upon a conducting sphere >r cylinderf. But these vibrations have so little persistence as hardly to leserve their name. A more instructive example is afforded by a conductor n the form of a circular ring, whose circular section is supposed small. There is then in the neighbourhood of the conductor a considerable store of energy which is more or less entrapped, and so allows of vibrations of •easonable persistence. This problem was very ably treated by Pocklington £ n 1897, but with deficient explanations§. Moreover, Pocklington limits his letailed conclusions to one particular mode of free vibration. I think I shall be doing a service in calling attention to this investigation, and in exhibiting the result for the radiation of vibrations in the higher modes. But I do not attempt a complete re-statement of the argument.
Pocklington starts from Hertz's formulas for an elementary vibrator at ;he origin of coordinates £, 77, f,
Q =
* Phil. Mag. 1897, Vol. XLIII. p. 125; 1897, Vol. XLIV. p. 199; Scientific Papers, Vol. iv. pp. 276, 327.
t Recent Researches, 1893, §§ 301, 312. [1913. There is also Abraham's solution for the sllipsoid.]
J Camb. Proceedings, 1897, Vol. ix. p. 324.
§ Compare W. M°F. Orr, Phil. Mag. 1903, Vol. vi. p. 667.  Under this condition I     M12dd>2 must J                                                                                    J-*         r