Studies in Wave Resistance. 5738 
distance to have a uniform velocity ¢ in the negative direction of Oz, the 
velocity potential is taken as cvu+qd; the squares of the velocity due to the 
disturbance ¢ are to be neglected. At the surface z= 0, the kinematical 
condition 1s 4 
Pall ; 
oct, (l) 
where € is the surface cee 
The condition for constant pressure at z = 0 gives cd¢d/dx—gt = 0, or 
ep _ 0b 
SoD = 2 
gant be 2 
At the bottom of the water 0¢/0z = 0; in what follows we shall assume 
the water to be of infinite depth. The remaining boundary condition is that 
dp /oy = 0 when y = 0, except over the surface of the ship; in the latter case, 
with pv as the normal, 
2 (e449) = 0. 
If the inclination of the ship’s surface to the plane y = 0 is everywhere 
small, the latter condition reduces to 
Ob _ 5 
f= ot = of 2), (3). 
where 7 = f(a, 2) is the equation to the ship’s surface; to the same order the 
condition (3) may be taken to hold at y = 0 over the median plane of the 
ship. 
A potential function to satisfy these conditions may be built up by a 
summation of simple harmonic terms in the co-ordinates ; it is sufficient here 
to state Michell’s expression, namely, 
Err cos (nz—e) cos (n§—e) 
ne 7 Lane PE $) (m? + 2)? 
cos {m (E—2) fev m+) dE do dm dn 
2c3 (> meme? (2+9)/9 
Salo | i AG: ®) (mPct/g? — Re 
sin {m (a—&)+ my (mct/g?— he }dé d&dm 
Dc3 gle? men me? (z+6)/9 
ral, |). 8 camera 
cos {m (Ex) }e my G— mel? dEd&dm, (4) 
where tan e = —c?m?/gn. 
It may be verified directly that each term in (4) is a potential function and 
201 r 
