4 116 ELECTRICAL VIBRATIONS ON A THIN ANCHOR-RING [3( when n and z are large and nearly equal, Jn (z) is related to Airy's integr; In fact, cos \ws + (n — z} (-} w\ dw . o so that Jr2m-1(2m)-J2rn+a(2m) = .................. (25) If we apply this formula to TO =10, we get O'lll as compared with t tabular 0108*. It follows from (25) that the damping in each vibration diminish without limit as m increases. On the other hand, the damping in a giv time varies as m'5r and increases indefinitely, if slowly, with m. We proceed to examine more in detail the character at a great distance the vibration radiated from the ring. For this purpose we choose axes x and y in the plane of the ring, and the coordinates (x, y, z) of any poi may also be expressed as r sin 6 cos (j), r sin 6 sin (/>, r cos 9. The contribute of an element adfi at <£' is given by (4). The direction cosines of tl element are sin <£', — cos </>', 0 ; and those of the disturbance due to it a taken to be I, m, n. The direction of this disturbance is perpendicular tc and in the plane containing r and the element of arc adfi. The fii condition gives lac + my + nz = 0, and the second gives I . z cos <p' + m . z sin <j> — n (ss cos <£' + y sin <£') = 0 ; so that _ I _ _ -m _ n (zz + y'1) siri <£' + ocy cos <£' (zz + «2) cos <^' + osy sin <f> zy cos (j)' — zoo sin </>' ' ............ (26) The sum of the squares of the denominators in (26) is r2 [z2 - (y sin $ + x cos </>')2}. Also in (4) qin'-v-l (fl! ^ $ ~ y C°S ^ -Z* + (X COS sm%-i -- — — - - and thus ra . I, sin ^ = (& + y-} sin 0' + #y cos - r2 . wi sin x = (z- + #2) cos </>' -f asy sin $', ............... (28) rs.n sin^; = To these quantities the components P, Q, R due to the element adcf>' a proportional. * Iog10r(-|) = 0-13166.en m is large may be at once derived from • a formula due .to Nicholson*, who shoes' that