406 SURFACE WAVES. [CHAP. IX 



Now in the case of (iii) we have, 



a [c \ c 

 whence, putting a = i/3 a , the expression (vi) takes the form 



ZnB 8 e-^ h ................................. (viii), 



where B 8 = - & ^ .. ... (ix). 



The theorem in question then gives 



/: 



jxulh /< a ixulh 



-f- , 



_oo u coth u - gk/c 2 J o u coth u - gh/c 2 o 



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



f. 



o u coth u- gk/c 2 o 



The surface-form is then given by 



. (xii). 



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- 

 ance. 



When, on the other hand, c 2 < gh, 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 f round them, on the side for which v is positive. 

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



/'()' 

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



2rrA sin axjh ( x iii)> 



where A= -7-JU ~ (xiv). 



The equation corresponding to (xi) now takes the form 



j [ a " e + [" I _5S/* d. _, 



(Jo J a+e j u coth u-gh/c 2 



