1919] ON THE RESULTANT OF A NUMBER OF UNIT VIBRATIONS 633 We may now examine the form assumed by <f>n in (13), when n is very large. The process is almost the same as that followed in a recent paper*. By taking logarithms we find -n « + 4«1W'), ......... (14) where Retaining for the moment only the leading term, we get fj O /*OT <£ (a) d<r/a = - cos (2<7^/a) e~H^6d^ a TT J o = *J(6/mr)e-^na*d<T/a ...................... (16) In comparing this with (5), we must observe that there as denotes the mean of the distances of which a- is the sum, so that a- = nae, and thus the two results are in agreement. If we denote the leading term in <f>n by <3>, we obtain from (13) and (14) *-* + 6Wi<" - 6°* 3? + ^M'^' by means of which the approximation in powers of 1/w can be pursued. The terms written would suffice for a result correct to 1/n2 inclusive, but we may content ourselves with the term which is of the order l/?i in comparison with the leading term. We have nir = /fA V {n'-rr _ dn* V n'-rr [4 ?ia2 n2a4 j ' and accordingly Here (j)n (<r) dor/a expresses the probability that the sum of the distances, measured from the centre of the line, shall lie between a- and cr + da-. In terms of the mean (x) of the distances, we should have V VTT/ { 5n \4 a2 a4 J) ...... as. the probability that x shall lie between x and x + dec. It should be observed that in virtue of the exponential factor only moderate values of naP/a? need consideration. * Phil. Mag. Vol. xxxvn. p. 344 (1919), equations (65), (66), etc. [This Volume, p. 624.]ilr/ n + 1