504 Lord Rayleigh on the Resultant 



as soon as 6 becomes sensible, so that the in te oral comes to 



depend upon that part of the range where is very small. 



We may then replace sin \6 by \Q, and taking -fy=6\s, we 



find 



1 p 2o-^sirrNr 

 - 1 COS —r^-cfy .... (11) 



as the equivalent of (10) when s becomes infinite. This is 

 the probability which attaches to a single integral value of /, 

 or to a change da, where dcr=aj2s. Thus the probability 

 that a lies between a and a-\-da may be written 



Ua C m 2<nfrsin*<f ,, 



1 cos — T - ftr-df, . . . (12) 



tt7rj a -^ 



which is the required result for a continuous distribution 

 and is applicable to any value of n. In our I'onner notation, 



7rj a ^ 



in which, however, a now represents the sum of the distances 

 from the centre of the line, instead of from one end of it. 

 If?i = l, (13) reduces to 



, . , If 00 sin (i + 2<x/a)^+ sin fl-2o-/a)^ . 

 *iW=-J o ^ *<% 



which is unity when a lies between +^a, but otherwise 

 vanishes. 



Again, if ?i = 2, we find that (f> 2 ! (o-)= + l/a, if cr lies 

 between -jra, and otherwise vanishes, and so on. 



More generally, the sequence formula may be deduced 

 from (11), but to obtain it in the original form (6), where 

 the distances are measured from the end of the line, we 

 must write <r— \na for a in (11). Then we I ave 



2 C"C* 2^, 1 , sin^r 



-1 1 cos— (<r — ina) . ~ aYaa/a, 



^Jo Jo- a <^ ^ 



in which 



f* 2-f , , . , 2iW n + 1 \ sin^r 



1 cos — T -[a — pM)d<r= cos— H (J ^— a). — ~, 



Ja-a a a \ 2 /l/r 



so that (11) is verified. 



We may now examine the form assumed by (f> n in (13) 7 

 when n is very large. The process is almost the same as 



