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Full text of "Scientific Papers - Vi"

1912]             ELECTRICAL VIBRATIONS  ON  A THIN  ANCHOR-KING                 115
and on the right we may replace sc by its first approximate value. Referring to (2) we see that the negative sign must be chosen for a and x, so that os= — 2. The imaginary term on the right is thus
ITT
~ {j; (2)-J-3 (:>)} = 0-70336^.
For the real term Pocklington calculates 0'485, so that, L being written for log (8a/e),                                                                         '
a = ± [i + (0-243 + 0-352i)/Z}.
.(20)
" Hence the period of the oscillation is equal to the time .required for a free wave to traverse a distance equal to the circumference of the circle multiplied by 1 — 0'243/i, and the ratio of the amplitudes of consecutive vibrations is 1 : e~2'zl/L or 1 - 2'21/Z."
For the general value of m (19) is replaced by
(a9«9 - m2) L = ^^ {/2m_i (2m) - J2m+, (2m)} + R,...... ..(21)
where R is a real finite number, and finally
771 -     -I      I        •*•"        .      ^""    |    T          If)       \        T          /Q       \\                       Sf)i)\
The ratio of the amplitudes of successive vibrations is thus
1 : l-7,-»i/9JB-l(2m)-/nB+I(2»i)}/2A' ...............(23)
in which the values of «/2m_1(2)n)-/2,H+1(2m) can be taken from the tables (see Gray and Mathews).    We have as far as m equal to 12:
• ' m ......	^(2«H)-JM(2»0	J?l                 "^am— 1 (^"v ~ ^2m+l (2m)
1 2 3	0-448, 0-298 0-232	7                          0-136 j           8                           0-125 1           9                           0-116
4 ' 5	0-194 0-169	10                           0-108 11            "    '        (M02
6	"   0-150	12         :                 0-096
It appears that tne damping during a single vibration diminishes as m increases, viz., the greater the number of subdivisions of the circumference.
An approximate expression for the tabulated quantity when m is large may be at once derived from • a formula due .to Nicholson*, who shoes' that
* Phil. Magf-1908, Vol. xvi, pp. 276. 2?7.ht have investigated the problem upon this basis.