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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
« + 4«1W'),   ......... (14)
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
=    /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