'118 ELECTRICAL VIBRATIONS ON A THIN ANCHOR-RING [< Whether m be odd or even, the three components are in the same phi On the same scale the intensity of disturbance, represented by P2 4- Q2 -f-is in terms of 9, <j> cos28(S2+C^ + sm20(Ccos<f) + Ssm<t>')2, ............ (36 an expression whose sign should be changed when m is even. Introducing • values of G and S in terms of ® from (31), (33),' we find that P2 + Q- + is proportional to cos" 6 {©m+i2 + ®}n.-i2 + 2©TO+i ®m_a cos 2m0} + sin2 6 cos2 m<f) (@m+1 + ®w_! ...... (37 From this it appears that for directions lying in the plane of the ri (cos 6 = 0) the radiation vanishes with cos m<j£>. The expression (37) may a be written 2 + 20m+1®,ll_1 cos 2m<}> - % sin2 6 (0m+1 - B,^)2 (1 - cos ...... (38; or, in terms of J"'s, by (34), (35), 7T'2 [Jm+i2 + Jm-i - 2Jm+iJm-i cos 2m</> - £ sin2 & (Jm+i + ^m-i)2 (1 - cos 2m</> ...... (39) and this whether m be odd or even. The argument of the J"'s is aa sin 9. Along the axis of symmetry (Q = 0) the expression (39) should independent of <p. That this is so is verified when we remember that Jn vanishes except n = 0. The expression (39) thus vanishes altogether witl unless m = 1, when it reduces to IT- simply*. In the neighbourhood of t axis the intensity is of the order 6-m~~z. In the plane of the ring (sin 6 = 1) the general expression reduces to 7T2 (Jm+i — Jm-if cos2 m<£, or 4?r2/m/2 cos2 m0 .......... (40) It is of interest to consider also the mean value of (39) reckoned 01 angular space. The mean with respect to (/> is evidently 7T2 [/m+l2 + /m-!2 + \ S^2 0 (Jm+l + Jm-^] ............. (41 ) By a known formula in Bessel's functions (42) For the present purpose £2=a2a2sin20 = m-sin20; and (41) becomes (43) * [June 20. Reciprocally, plane waves, travelling parallel to the axis of symmetry a incident upon the ring, excite none of the higher modes of vibration.]1 are both odd and S and 0 are both pure imaginaries. But when m is odd, S and (7 are both real.