I 



Growth of a Train of Waves. 21 



particle of the liquid by (175), without farther integration. 

 The formula (184) also shows how by successive integrations 

 by parts we can reduce 



W-ff)^V(*,*,i) . . . (185) 



to the primary integral S(#, z, t), as expressed in (171). 



§ 135. Going back now to §§ 128, 127, 118 : to make the 

 applied forcive a sinusoidally varying pressure put 



(3QS 



/('-£) = Bin •('-?) • • • • ( 186 ); 

 which, by (173), makes 



n(*,l,*) = -|^artV(*,l,0) . . (187). 



And now let us arrange to fully work out our problem for 

 two cases of surface distribution of pressure, corresponding 

 to the two initiational forms <£, y, described in §§ 96-113 

 above. For this purpose take, with the notation of § 101, 



Y(x, z, t) = <f>(x, z, t) ; 



or V(«, z, t) =+(*, s , t) = - -L |*<^, z, t) (188). 



For brevity we shall call these two cases case <f> and case i/r. 

 Thus, in these cases (171) and (175), expressing respectively 

 the velocity-potential at, and the vertical component dis- 

 placement of, any point of the fluid at any time, become 



S (.r, z, = | dqfa»(t-q)4>(a: 9 z, g) ; 



%, o 







i r* cos 



&(*, 2-, 0= -J ^ sin *>(>-?) t(^ *> ?) - (190). 



" J 



§ 136. The illustrations in figs. 36, 37, 38 are time-curves 

 in which the ordinates have been calculated by continuous 

 quadrature from one or other of the four formulas (189), 

 (190). 



§ 137. The curves in fig. 39, being space curves in which 

 the ordinates are vertical component displacements of the 

 water-surface, are therefore pictures of the water-surface 



(189); 



