227] EFFECT OF A LINE OF PRESSURE. 397 



pti 



/; 



positive portion of the axis of m, together with an infinite quadrant. We 

 thus find 



which is equivalent to 



p -mx 



- \ dk= -r-rdm .................. (iv). 



- 



This is for x positive. In the case of K negative, we must take as our 

 contour the negative portions of the axes of , m, and an infinite quadrant. 

 This leads to 



r e -ikx / e mx 



, f . .dk = \ - - -dm .................. (v), 

 m *+(K-Hpi) J Q m + ^-iK 



as the transformation of the second member of (iii). 



In the foregoing argument /z x is positive. The corresponding results for 

 the integral 



are not required for our immediate purpose, but it will be convenient to state 

 them for future reference. For x positive, we find 



/ e ikx r e~** r e~ mx 



I TT-, - \ dk = \ T-r-f - ^dk= \ - -dm ... (vii): 

 /o *-(-*l) 7o^ + (-^i) Jo m+^ + iK 



whilst, for x negative, 



= _ 2,^^^+ r . &quot;* . N&amp;lt;ft 



J *+5-53 



&quot; ^^ dm ...... (viii). 



-- 



The verification is left to the reader. 



If we take the real parts of the formulae (ii), (iv), and (iii), (v), respectively, 

 we obtain the results which follow. 



The formula (18) is equivalent, for x positive, to 

 ^ . y = - **-*&amp;gt; sin , + /&quot; (t + * ) ( % 



~ (19), 



(k + fc) cos A;^ yu,i sin A;a? , 

 -7 dk 



and, for x negative, to 

 7TC 2 



Q y ~ (20) - 



