310 Mr PockliTigton, Standing waves parallel to a plane beach 



to {n + l)/2 terms, the coefficient of the last term being 1 instead 

 of 2. ^ 



That is, 



/? = exp (n - 1) 77^/4 + exp (n - 3) 7^^74 . cot a + etc. 



to n terms, the first term of the previous series being the sum of 

 the first and last of this, and so on, for 



cot {n — r) a = tan m. 

 Secondly, let n be even. Pairing the terms 

 /) = 2 cos (n - 1) 7r/4 + 2 cos {n - 3) 7r/4 . cot a + etc. to n/2 terms 



= exp (w - 1) m/4: + exp {n - 3) Tri/i . cot a + etc. to n terms, 

 which is of the same form as in the case of n odd. 



If w = cos Tr/n + i sin tt/ti, 



we have cot ra = - ^a>*■ (co^-*" - l)/(aj'' - 1), 



so that 



p = exp (»^ - 1) 7^^74 - exp {n - 3) m/i . tco (w^-i - l)/(a> - 1) 

 + exp (n - 5) 77^/4 . *2co3 (co^-i _ 1) (a>«-2 _ i)/(^ _ i) (^2 _ i) 

 — etc. to n terms, 

 the indices of co in i^oj' being the triangular numbers in order. Also 



i = exp 7^^■/2. 

 Hence* 



/> = (1 - w) (1 - co2) (1 - oj3) ... (1 - co^^-i) exp (n - 1) 77*74. 

 Being real this is equal to its conjugate 



(1 + co«-i) (1 + co^-2^ ...{1+co) exp (1 - n) m/i. 

 Multiplying, we find 



p^=(l~ a>2) (1 - co4) ... (1 _ oj^n-2^. ■ 



But a>2, a>4, etc. are the roots of {x'' - l)/(x - 1) = 0, whence p^ = n. 

 Hence the amplitude at the origin is ^/n times the maximum 

 amplitude at infinity (but this has been proved only for the case 

 of n integral). 



* Vide Todhunter, Theory of Equal ions (1885), p. 217 (ch. xxiv, § 292). l 



