ON THE SEICHES OF LOCH EAftN. 



507 



+1 



Or, since | c?mQ'„(/*) = in all cases, 



3£=^S(2v+1)Q;(m>) I dnQ'Jji) cos n u {a{\ + ^jv -t) 



where 



. (32). 



If, therefore, we denote the y-nodal component of the disturbance by 3£„ , we have 



d£ y = (A'„ cos n v t + B'„ sin n„t)Q\(w) .... (33), 



2A',/(2v+ l)d P = rf/*Q',(/*) cos -^(1 +/*) 



ra„a. 



2B',/(2^+ l)3p= rf/nQ',^) sin ^-(1 + /*) 



• (34). 



Next suppose vT<2a. Then the pressure disturbance does not reach farther than 

 x = vT — a, that is w = t>T/« — 1 ; and we have 



/(/*, = 0, if #>T, /x>wT/a-l; 



/(/, , = ^ , if a(l + f*)/«T , - l<X»T/a - 1 . 



Hence (31) now gives 

 3jt> 



i/T/a-1 



3£ = ^ 2(2v + 1)Q» | ^Q'„(/x) | dT«, sin n,(T - , 



1 Ja(l+H-)/v 



•vT/a-l 



= ^%(2v+l)Q' v (w) \df* :,! ,.(/ )|o» [,7.(l+/.),.- /]--•-.. /,.(■:.-•/)] 



It follows that, if 0>T, then 



9£„ = (A'„ cos n v t+ B'„ sin w„tf)Q'„(«>) 



where now 



pflT/o-1 •. 



2A',/(2v + l)3p= d/*Q',(p) cos n ^ 1 *^ - Q,^ - l) cos n>T, 



/Vf/a-1 



2BV(2v + 1)923 = d/*Q',(/*) sin K, a ( 1+ ^ - qY— - l) sin «„T 



(32'). 



(33'), 



(34'). 



Superposing now the initial motion given by (6) and (8), we get for the v-nodal 

 component of £ 



L = [{K cos n v T v + A'„} cos n v t + {/f„ sin w„t„ + B'„} sin n v t]Q,\(w)^ 

 = k' y cos n-,(i -X^)QU W )> 

 where 



/,;7 = A:„ 2 + 2fc,(A , cos re^,, + B'„ sin »,t,) + A'„ 2 + B'„ 2 ; 

 k„ sin m„t„ + B'„ , 



tan w„x„ = 

 tan ra„( x * - T *) = 



k„ cos n„T„ + A'„ ' 



B'„ cos m„t„ — A' sin w„t„ 



(35). 



k„ + A'„ cos n v T v + B'„ sin n„T„ 

 The values of A'„ and B'„ are given by (34) or (34'), according as t'T>or<2a. 



