618 ON THE PROBLEM OF RANDOM VIBRATIONS, AND [441 Thus 1 / 48r2 18r4 2r6 r8 \] V ~ ~T + ~^~ ~ ss + 16WJ / 6r= 15r4 7r6 3r8 6- T 4s2 12s3 ' 128«V j ' ......(45) in agreement (so far as it goes) with Pearson, whose o-2 is equal to our s. The leading term is that given in 1880. Three Dimensions. We may now pass on to the corresponding problem when flights take place in three dimensions, where we shall find, as might have been expected, that the mathematics are simpler. And first for two flights of length ^ and lz. If p, be the cosine of the angle between ^ and lz and r the resultant, giving rdr~ — lilzd/^ ............................... (46) The chance of r lying between r and r + dr is the same as the chance of /u. lying between ^ and /j, + dp, that is — ^d/j,, since all directions in space are to be treated as equally probable. Accordingly the chance of a resultant between r and r + dr is The corresponding volume is 4i7rr2dr, so that in the former notation Zi and lz being supposed equal. It will be seen that this is simpler than (21). It applies, of course, only when r < 2L When r > 21, <£2 = 0. In like manner when ^ and lt differ, the chance of a resultant less than r is zero, when r falls short of the difference between 12 and llt say Za — ^. Between lz—lj and 12 + 1T the chance is When r has its greatest value (Z, -f Zj), (49) becomes (2, + ^-(2,-^ """"4U ............................ (50) The " chance " is then a certainty, as also when r > ^ + Z2.hed; the subject may be commended to those better versed in pure mathematics. Probably what is required is a better criterion as to the differentiation under the integral sign*.