420 Proceedings of Royal Society of Edinburgh. [sess. 
By using (179) we may, instead of (183), take the following as 
equally comprehensive : — 
{RS} or {RD}(a 
+ B d\£ l 
dx/dt J(z + lx) 
-g * 8 
4 ( 2 + 1 *) 
= Y(x , z , t), when i is even : 
= W(x , z , t), when i is odd. 
( 183 '). 
§ 134. Going hack to (171) and (175), remark that integration 
by parts gives 
j‘dqf(t - q)fv(x ,Z,t) =/( 0)V(* ,Z,t) -f(t)V (x ,Z, 0) 
+ / dgf(t-q)V(x, z, q) . (184). 
*'6 
This shows that if by quadrature or otherwise we have calculated 
the velocity-potential S(^, 2 ,^), as given by (171), we can find 
the vertical component displacement of any particle of the liquid 
by (175), without farther integration. The formula (184) also 
shows how by successive integrations by parts we can reduce 
j o dqf(t-q)~V(x,z,q) .... (185) 
to the primary integral S(a? ,z t t) t as expressed in (171). 
§135. Going back now to §§ 128, 127, 118: to make the 
applied forcive a sinusoidally varying pressure put 
= ( 186 ); 
which, by (173), makes 
n(a , 1 , 0 = - > 1 > 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 <j> , if/ , described in §§96-113 above. For this 
purpose take, with the notation of § 101, 
1 d 2 
Y(x,z,t) = cf>(x,z,t) ; or Y (x i z,t) = if/(x, z,t)= ^#(«,M). (188).. 
For brevity we shall call these two cases case <£ and case t }/. 
Thus, in these Cases (171) and (175), expressing respectively the 
