1919]         ON THE  RESULTANT OF  A  NUMBER OF  UNIT VIBRATIONS           633
We may now examine the form assumed by &lt;f&gt;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
&lt;�� (a) d&lt;r/a = � -      cos (2&lt;7^/a) e~H^6d^ a TT J o
= *J(6/mr)e-^na*d&lt;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 &lt;f&gt;n by &lt;3&gt;, we obtain from (13) and (14)
*�-* + 6Wi&lt;" - 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 '
Here (j)n (&lt;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
1919]         ON THE  RESULTANT OF  A  NUMBER OF  UNIT VIBRATIONS           633
We may now examine the form assumed by &lt;f&gt;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
&lt;�� (a) d&lt;r/a = � -      cos (2&lt;7^/a) e~H^6d^ a TT J o
= *J(6/mr)e-^na*d&lt;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 &lt;f&gt;n by &lt;3&gt;, we obtain from (13) and (14)
*�-* + 6Wi&lt;" - 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 '
Here (j)n (&lt;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