1919] ON THE RESULTANT OF A NUMBER OF UNIT VIBBATIONS 641 It still remains to consider the case where 37r/2<a<27r. From 0 = 0 to 6 = \ (a TT), r (as before) ranges from 0 to 2. From 6 = \ (a TT) to 0 = ^(377- a)} r ranges from 2 cos (^ a 6) to 2. At this point (Fig. 6) a second range enters for r. From 6 % (3?r a) to 9 = £ a, the first range is, as before, from 2 cos (^ a 0) to 2, and the second range is from 0 to 2cos(27r £« 0). Lastly, from # = |-a to # = TT, the first range of r disappears, while the second continues to be from 0 to 2cos(2-7r ^a 6}. The probabilities of various 0's being positive and lying within specified ranges can be obtained as before. For the range from 0 = 0 to 0 = \ (a TT) we get the expression (33), and from 0 = \ (a TT) to 8 \ (3?r a) we get (34). For the third range from 0 = ^ (3?r a) to 0 = ^a, we get ' /2e08(a.r-ia-0r] rlr ^f 3^ 2COS(ia-0) and from 6 \ a. to 0 = TT, V(4 - r2) a2 .(39) If the integrations with respect to # are effected over the appropriate ranges and the results added, we get \ , as was to be expected. Finally, for the probabilities of various r's when 9 is left open, we get for r between 0 and 2 cos (TT £a) two ranges for 6, viz. from 0 to £ct cos"1 (^-r), and again, from 0 = 27r ^a cos~1(^r) to # = 7r, making altogether (a TT). Thus for these values of r the probability is that expressed in (36). When r lies between 2 cos (TT -£0) and 2, we recover in like manner (37). And as before we may verify the results by showing that when the second integrations are carried out over the appropriate ranges and the integrals added, we recover £. It may be remarked that the latter results may be applied to the complete circle. by making a = 2-rr (Fig. 7). The second range for r then disappears, and for the whole range now extending for all values of d from r = 0 to r = 2 we get ............................... (40) which needs to be doubled in order to take account of negative values of 9. This completes the investigation for an arbitrary a (less than 2-Tr), when n 2. Since even for the complete circle (« = 2?r) the case n = 3 leads to elliptic integrals, there is no encouragement to try an extension to other values of a. E. VI.w covered.