'60 Dr. G. Green on Ship- W aves, and on Waves in 



(31) and (32), we Lave, by one integration, 



qit—rfltf+h) 



■*•(*. **. ') = - fcj <* T e ~^~ ■ cos t? - 



. . . (49) 





J (t-ry_ slng{ t-r)* 



w>* -"• 4ot' 



. . . (50) 



These integrals are well known in connexion with the 

 ship- waves problem. To obtain the motion at any point of 

 space at any fixed time t we have now to consider for each 

 value of t two values of t at which the phase is stationary, 

 namely those given by 



x-vr = i[(vt-x)± i/(««-o:) 2 -8/] = i(X + R). (51) 



With Tx and r 2 to represent the values of t given by (51), our 

 final results for the motion due to the applied surface 

 pressure indicated in (c) of § 2 are 



■ _ g(t-r)\z+h) 

 _,/ T==Tl \ g$a Z 4^ (*-~t) 2 



V S7* 







sin(*+^), .... (53) 



dx 2 



4^' 



These results are valid in the regions well behind the mid 

 point of the moving pressure which is included by the planes 

 X 2 — 8?/ 2 = 0, but are not valid along, or in the immediate 

 neighbourhood of, these planes. Along each of these planes 

 the correct evaluations of the integrals in (49) and (50), 



