14 



PROFESSOR HORACE LAMB ON THE PROPAGATION OF 



k K 



e 



Of, ft 



Fig. 1. 



at points of the axis of f, the consistent system of values indicated in fig. 1,* we 

 find, for the various parts of the first-mentioned side,t 





F'(-K) 





where the terms with F'( K) and F'(/c) in the denominator are due to the small 

 semicircles about the points (i K, 0). Equating the sum of these expressions to 

 zero, we find, since F'( K) = F'(K), 



= _ . H cos 



, , x , , 



- 



r 



- - 2.VH cos KX - 



* The function under the integral sign in (66) is uniquely determined (by continuity) within and on 

 the contour when once the values of the radicals v /(f 2 - A 2 ) and ^/(f 2 - k 3 ) at some one point are 

 assigned. The convention implied in the text is that the radicals are both positive at the point ( + , 0). 



It will be noticed that over the portion of the axis of between - k and - h the function in (66) 

 differs from that involved in the value of w as given by (52). This is allowed for in the second member 

 of (67). Corrections, or rather adjustments, of this kind occur repeatedly in the transformations of this 

 paper. 



t The symbol $ is used to distinguish the " principal value " of an integral (with respect to a real 

 variable) to which it is prefixed. 



