406 SURFACE WAVES. [CHAP. IX 



Now in the case of (iii) we have, 



......... (vii), 



whence, putting a = 2&, the expression (vi) takes the form 



(viii), 

 where JB S = - -& - -j- ....................... (ix). 



The theorem in question then gives 



L 



O ixu/h / & ixulh 



- -du+ 



u coth u gh/c 2 J Q u coth u gh/c 2 



If in the former integral we write - u for u, this becomes 



o u coth u - 

 The surface-form is then given by 



It appears that the surface-elevation (which is symmetrical with respect 

 to the origin) is insensible beyond a certain distance from the seat of disturb 



When, on the other hand, c 2 &amp;lt; gk, the equation (iv) has a pair of real roots 

 ( + a, say), the lowest roots ( + /3 ) of (v) having now disappeared. The integral 

 (ii) is then indeterminate, owing to the function under the integral sign 

 becoming infinite within the range of integration. One of its values, viz. the 

 principal value, in Cauchy s sense, can however be found by the same method 

 as before, provided we exclude the points = +a from the contour by drawing 

 semicircles of small radius e round them, on the side for which v is positive. 

 The parts of the complex integral (iii) due to these semicircles will be 



f(Y 



where / (a) is given by (vii) ; and their sum is therefore equal to 



where A= , a , ... (xiv). 



2 _gh(gh L 



The equation corresponding to (xi) now takes the form 



j ( a ~ e + f 1 ^^...du^-nA *maxlh + n^B 8 e-Wh. ..(xv), 

 (Jo Ja+J ucothu-gh/c 2 *\ ^ h 



