396 SURFACE WAVES. [CHAP. IX 



Hence in (13) we must replace P by Q/jr.&k, where 



and integrate with respect to k between the limits and oo ; thus 



If we put =k + im, where k, m are taken to be the rectangular coordinates 

 of a variable point in a plane, the properties of the expression (18) are 

 contained in those of the complex integral 



:*. 



It is known (Art. 62) that the value of this integral, taken round the 

 boundary of any area which does not include the singular point (=c), is zero. 

 In the present case we have c = K + i^ , where K and /^ are both positive. 



Let us first suppose that x is positive, and let us apply the above theorem 

 to the region which is bounded externally by the line m = and by an infinite 

 semicircle, described with the origin as centre on the side of this line for 

 which m is positive, and internally by a small circle surrounding the point 

 (*, JAJ). The part of the integral due to the infinite semicircle obviously 



vanishes, and it is easily seen, putting c = re ld , that the part due to the small 

 circle is 



if the direction of integration be chosen in accordance with the rule of Art. 33. 

 We thus obtain 



dk _ V -MI)* = 



which is equivalent to 



rpikx C p~ikx 



, * . .dk = 2 7r ie l{K+t ^ x + 

 *-(K+ft) 7 



On the other hand, when x is negative we may take the integral (i) round 

 the contour made up of the line m = and an infinite semicircle lying on the 

 side for which m is negative. This gives the same result as before, with the 

 omission of the term due to the singular point, which is now external to the 

 contour. Thus, for x negative, 



p -ikx 



An alternative form of the last term in (ii) may be obtained by integrating 

 round the contour made up of the negative portion of the axis of &, and the 



