610 ON THE PROBLEM OF RANDOM VIBRATIONS, AND [441 Two Dimensions. If there is but one stretch of length I, the only possible value of r is of course I. When there are two stretches of lengths ^ and la, r may vary from 12 - ^ to k + k, and then if 6 be the angle between them - r2 = Zx2 + /22 - 2^ cos 0, ........................ (17) and Biii0dd = rdr/I1l2 ............................ (18) Since all angles 0 between 0 and TT are deemed equally probable, the chance of an angle between 6 and 6 + cW is ddj-jr. Accordingly the chance that the resultant r lies between r and r + dr is rdr ........................... (19) Trlib sin 6 ' ........................... or if with Prof. Pearson* we refer the probability to unit of area in the plane of representation, fc^S^sin* - 7 02 (^2) dA denoting the chance of the representative point lying in a small area dA at distance r from the origin. If the stretches ^ and 12 are equal, (20) reduces to Prof. Pearson's expression, applicable when r < 2£. When r > 2/1, <jf>2 O"2) = 0. When there are three equal stretches (n = 3), <£3 (r2) is expressible by elliptic functions f with a discontinuity in form as r passes through /. For values of n from 4 to 7 inclusive, Pearson's work is founded upon the general functional relation J (22) Putting r = 0, he deduces the special conclusion that as is indeed evident a priori. * Drapers'1 Company Research Memoirs, Biometric Series III., London, 1906. f Pearson (loc. cit.) attributes this evaluation to G. T. Bennett. J Compare Theory of Sound, § 42 a.alwri.swt- of coii.set'utive i.solated pointH JH i!/, ;->'» lhat. ii'(/,r be a large multiple of/, we may take