On Ship-Waves, and on Waves in Deep Water. 49 



where p denotes the pressure, p the density of: the fluid, at any 

 point (x, y, z), and £ denotes the vertical component displace- 

 ment of the particle of fluid whose equilibrium position is at 

 point (#, y, z). When the upper surface of the liquid is 

 free from applied pressure equation (2) takes the form 





(3) 



for all points on the free surface. If the velocity- potential 

 <b satisfies equation (3) at all points of the fluid, each surface 

 which is level when the fluid is undisturbed is a surface of 

 constant pressure in the motion corresponding to this velo- 

 city-potential. Equations (2), and (3), also each involve the 

 assumptions that the motion is small and ir rotational. The 

 first of these requires that the squares of velocities of the 

 fluid particles should be negligible, and the latter is evidently 

 fulfilled in all the cases of motion to be considered, since in 

 each case the motion is produced from rest by pressures 

 applied to the boundary. 



When a particular motion is such as could be produced 

 from rest by impulsive pressures applied to the boundary of 

 the fluid there is a relation between </>(#, y, z, t), the velocity- 

 potential of the motion at any instant, and the impulsive 

 pressure IT(#, y) which caused the motion. This relation is 

 expressed by the equation 



IK>\ y) = - P (j>{x,y,z, t), . . . 



• • • (4) 



with z = 0, and £ = 0, if we exclude from consideration pres- 

 sures which are uniform over the whole free surface. An 

 application of this relation, which is of special importance in 

 connexion with the type of problem with which we are 

 dealing, is to the case where a finite pressure p(x, y) is 

 applied to the surface for an infinitesimal period of time 

 dr. The impulsive pressure is in this case p(x, y)dr, and 

 the velocity-potential of the motion to which it gives rise is 

 — (l/p)p(x,y,z,t)dT, where p(x, y, z, t) must be deter- 

 mined to satisfy equations (1) and (3) in addition to the 

 equation 



p(x, y, 0,0) =p(x,y) (5) 



As we may regard any continuous application of pressure to 

 the surface as equivalent to a series of impulsive pressures 

 delivered in consecutive infinitesimal intervals dr^, dr 2 ,dT 3 , 

 &c.,it is clear that a summation of velocity-potentials, similar 

 to that expressed by — (ljp).p(x,y,z,t)dr for the interval 

 Phil. May. S. 6. Vol. 36. No. 211. July 1918. E 



