640 ON THE KESULTANT OF A NUMBER OF UNIT VIJBEATIONS [442 limits coincide. From 6~\a to <9 = 3?r/2 ŁŤ, there are no corresponding values of r. At the latter limit a zero value of r enters, and from 6 37T/2 -|a to 0 = TT, r ranges from 0 to 2 cos (2?r Ła 0). The whole range from 6 = 0 to 6 = TT thus divides itself into four parts. In the first part from 6 = 0 to 6 = -J (a - TT), we get as the chance of (9 from (29) In the second part from 6> = -|-(a TT) to $ = -ija, the chance is f2 dr /0/1X (34) a v ' For the third part, from 6 = $a to 0 = 37r/2 - \a, there is no possibility. For the fourth part, from 9 3?r/2 Ła to (9 = TT, the chance for 6 is (35) If we integrate (33), (34), and (35) over the (positive) "ranges to which they apply and add the results, we get the correct value, viz. Ł. This part of the question might be treated more simply without introducing r at all. We have next to consider what in this case, viz. TT < a < 3?r/2, are the probabilities of various rs when Q is allowed to vary. When r is less than its value at 6 = ir, viz. 2 COS(TT ŁŤ), the corresponding range for 6 is made up of two parts, the first from #=() to 9 = \OL~- cos"1 (Ł?"), and the second from 6 = 2?r ^ a cos"1 (Łr) to 0 = TT, so that the whole range of 6 is Ła cos"1 (Łr) + TT [2-7T | a cos""1 (-|r)} = a TT. Thus from r = 0 to r = 2 cos (TT Ła) the chance of r lying between r and r + cZr is -Tr) 9 ............................... V ; When r lies between 2 cos (TT ^-a) and 2, the second part disappears arid we have only the one range of 9, equal to Ła cos"1 (|-r), so that the chance of r lying between r and r + dr is Expressions (36) and (37), obtained on the supposition that 6 is positive, are to be doubled when we allow for the equally admissible negative values of 0. When (36), (37), as they stand, are integrated over the ranges of r to which they apply and added, the sum is Ł, as it should be under the suppositions made.. 4, where a is supposed to be o?r/4.