1919] ON THE RESULTANT OF A NUMBER OF UNIT VIBRATIONS 637 On continuing the integration the first part yields finally 8(?i-l)a-2sin2^a; while the remaining parts give the original terms over again with omission of those containing 0. Thus Expectation of intensity = n + 8 cr2 sin2 %a{n — l + n-2 + n-3 + ... + 1} = n + 4*ri (n - 1) cr2 sin2 £ a ...................... (27) If a = 0, this becomes n2, as was to be expected. If a = ZTT, or any multiple of 2-Tr, the expectation is n, as we knew. In general, when a. becomes great, so as to include many complete revolutions, the importance of the n- part decreases. In (27) n may have any integral value. In the case of n = 2, we may go further and find the expression for the probability of a given amplitude (r) taken always positive, and phase (6).. The amplitude of the components is unity, and the phases, measured from the centre of the arc, 6l and 0,,. The probability that these phases shall lie between 0l and 0l + d0i> #2 and 0.2 + d0z is a~2d01d02. We have now to replace the two variables 0lt <92 by r, 0, where r = 2 cos | (0, - 0a), 6 = £ (01 + 08). <>r 61=>0± cos"1 (£r), 0a = 0 I- cos"1 (-|-r), c^ _ ^ - _±J_ dll - + 1 dffs _ rW ~ ' dr ~ 7(4 - r'J) ' dr ~ 2 ' ~ A r i ia ,00. Accordingly — : - = -• —77 - ~. ......................... (28) h J a2 a2 a v y rrho interchange of 01 and ^a makes no difference to r and 6, so that we may take as the chance that the amplitude of the resultant shall He between r and r + dr and the phase between 0 and 0 + d0. In (29) a is supposed not to exceed %7r, AH a check, we may revert to the case where a = 2?r. The limits for 0 are then independent of the value of r, and are taken to be — TT and + TT. And 2 ^ M^ .........(30) _„ a TT represents the chance that r shall lie between r and r + dr independently of what 0 may be, in agreement with Pearson's expression*. Integrating again with respect to r, we find as should be, all cases being now covered. * Compare Phil. Mag. Vol. xxxvn. p. 328 (1919), equation (21). [This Volume, p. 610.} whose amplitude is R arid phase (reckoned from the middle) ®,