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1919]         ON THE RESULTANT  OF  A NUMBER OF  UNIT  VIBRATIONS           635
are distributed along a circular arc not constituting an entire circle. When the circle is complete the solution has already been given*, and the same solution obviously applies when the circular arc covers any number of complete revolutions. All phases of the resultant are then equally probable, and the only question relates to the probability of various amplitudes, or intensities. But if the arc over which the representative points are distributed is not a multiple of 2ir, all values of the resultant phase are not equally probable and the question is in many respects more complicated.
There is an obvious relation between the question of the resultant of random vibrations and that of the position of the centre of gravity of the representative points of the components. For if Q denote the phase of a unit component, the intensity of the resultant is given by
E2 = (2cos0)2+(Zsin0)2.
If we suppose unit masses placed at angles 6 round the circular arc of radius unity, the rectangular coordinates of the centre of gravity are
x — (2 cos 0)/n,       y — (% sin d)/n;
and r, the distance of the centre of gravity from the centre of the circle, is related to R according to r = R/n. And in like manner the phase of the resultant corresponds with the angular position of the centre of gravity.
The analogy suggests that a mechanical arrangement might be employed to effect vector addition. A disk, supported after the manner of a compass-card, would carry the loads, and the resulting deflexion from the horizontal would be determined by mirror reading. Perhaps there would be a difficulty in securing adequate delicacy.
To return to the theoretical question, if we suppose the circular arc to be ' very small, we see that the probability of various phases of the resultant, within the narrow limits imposed, follows the laws determined for the centre of gravity of points distributed at random along a straight line. In this case the amplitude of the resultant is n to a high degree of approximation, n being the number of unit components.
But when the circular arc (a) is so large that sin a deviates appreciably from a, the question is materially altered. We may, however, frame an argument on the lines followed in equations (1) and (2). Thus with a replacing a and {3 replacing b, we have for the resultant whose amplitude is R arid phase (reckoned from the middle) ®,
R sin 0 = sin /3. (£ - £_,) + sin 2/3. (f a - £_2) -f... + sin s/3. (&-£-,). ' - - .(20)
......(21)
V a    ' -    ' 7                   '     \ a        --     - -j
* Phil. Mag. Vol. x. p. 73 (1880); Scientific Papers, Vol. I. p. 491.    See also Phil. Mag. Vol. xxxvn. p. 321 (1919).   [This Volume, p. 604.]ary to consider the integral obtained by "the next following operation" (s =p + f - %ri). It would seem then that the above considerations are sufficient to justify the differentiation by which 0,t is obtained (jp = 0, s- -f), and a fortiori that for <fe (# = 0, s=-2), etc. W..IY.S.]elative index is then much restricted, the