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Full text of "Scientific Papers - Vi"

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 CS ^ -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