20 



Lord Kelvin on the 



any arbitrary function of x. We assume, as is convenient 

 for our present purpose, that for large values of x 



Y{x, z, 0)=0, and W<tf, s, 0)=0 



(181) 



This implies that for all values of x and z_, large or small, 

 but for large values of q, 



Y(w,z,q)=0, and W(*,*,j)=0 



(182). 



§ 133. In the V-problem the initiational condition is : — 

 displacement zero and initiational velocity virtually given 

 throughout the fluid as the determinate result of an arbitrarily 

 distributed impulsive pressure on the surface. 



In the TV-problem the initiational condition is : — the fluid 

 held at rest with its surface kept to any arbitrarily prescribed 

 shape by fluid pressure, and then left free by sudden and 

 permanent annulment of this pressure. 



Without going into the question of a complete solution 

 of this (V, TV) problem for any arbitrary initiational data, 

 we find a class of thoroughly convenient solutions in a 

 formula originally given in the ' Proceedings of the Royal 

 Society of Edinburgh/ January 1887; republished in the 

 Phil. Mag., February 1887 ; and used in § 3 and § 99 above. 

 We may now write that formula in the following compre- 

 hensive realised expression for V or TV : — 



{RSfor {ED} 



d i+ i+ k 



1 



dt i dxidz k *S(z + ix) 



.Xj; 



+«r) — Y(%, z,t), when i is even; 

 = W(#, z, t), when i is odd 



By using (179) we may, instead of (183), take the following 

 as equally comprehensive : — 



i 



:l 



{RS}or{RD}^A + B 



d \ffi 1_ 



dxjdtf ^{z + uxj 





e 4( ~ +lx) = V(a?j #, t), when i is even ; 



= W(x, s, t), when i is odd.. 



§ 134. Going back to (171) and (175), remark that 

 integration bv parts gives 



Cdqf(t 

 Jo. 



q) ^V(»,' t, 0=/(0)V(^ z, t) -f(t)V(x, 



Ad 9i f{t-q)Y{x, Z , 



9) 



,0) 



(184). 



This shows that if by quadrature or otherwise we have 

 calculated the velocity-potential S (a?, z, t), as given bv (171), 

 we can find the vertical component displacement of any 



