Muvthy 



- gZ+— [v$] + $ = constant (1 -6) 



where the constant on the right-hand side is independent of the space 

 variables and, as is usually done, may be set equal to zero, it being 

 understood that <|) is suitably adjusted. 



Now, $ £ can be expressed in a manner similar to the other 

 derivatives in (1-3) as 



<£ = ( o>'y - V ) 4> - cox $ + $ (1 -7) 



where the speed V of the ACV in its course is an arbitrary function 

 of the time for accelerating motion. 



The relation between the pressure p (x, y, z;t) and the velocity 

 potential 4> (x, y, z;t) may therefore be written 



12 

 — - gz + -=■ ( v $) + (wy - V(t) ) $ -wx<J> + <*> =0 (1-8) 

 p 2 x yt 



II. 4 Conditions on Boundary Surfaces 



If F (X, Y, Z : t) E f(x, y, z;t) = 



is a boundary surface, which may be fixed or moving, the kinematic 

 condition on such a surface is 



— = 6 F +<J> F + & F + F =0 

 dt V X X ^Y Y *Z Z t 



Using the relations (1-3) and (1-7) the corresponding condition in the 

 (x, y, z) system becomes 



$f+<J>f+<I>f + (coy - V) f - cox f + f =0 (1-9) 



xxyyzz x yt 



The free surface of water given by the equation 



z - f(x,y;t) = 



is a boundary surface, fluctuating with respect to time, and the kine- 

 matic condition on this surface may therefore be written 



* f +* r + # ? +(coy-V)f v -coxf+f=0 (1-10) 

 xxyyzz ■*• yt 



112 



