Theory of Wave Resistance. 341 
with 
re = (« —hP+ y—kP +e +f? 
ta = (c —h)? + (y¥ — hk? + 2 —f)? 
@ = (x — h) cos 0 + (y — k) sin 0. 
It is assumed that the distribution is such that the various integrals are 
convergent. 
3. To calculate the wave resistance R we use the method of the previous 
paper to which reference has already been made. With the inclusion of the 
frictional term in the equations of fluid motion, energy is dissipated at a rate 
equal to 2u’ times the total kinetic energy of the liquid and this must be equal 
to the product Re. As yu’ is made to approach zero the quantity so calculated 
approaches a finite limiting value, and its physical interpretation in the limit 
when there is no fluid friction is the rate at which energy is propagated out- 
wards in the wave motion. 
The rate of dissipation of energy js given by 
— we | 6 fas, (7) 
taken over the boundaries of the liquid. As we require only the limiting value, 
we have the wave resistance given by 
R= Lim po i 
u>0 
—a 
[ b 2 de dy (8) 
taken over the free surface z = 0. 
Referring to (6), and putting the first two terms in the same integral form as 
the third, we obtain, at z = 0, 
7 (o) —xkf+ix 3 
a =| ds | dé | peli bustesiCiee Sint IL eg, (10) 
™ = 
- ? 
oz 0 K — Ky Sec? 6 + zy sec 0 
T 
ie) eT ita 
sec? 6 d0 | (9) 
Seely 
0 K — Ky sec? 9 + tu sec 0 
where the real parts are to be taken. 
After some reduction, we may write the real part of (9) in the form 
d= [. dé L {F, cos (kx cos @) cos (xy sin 0) 
—T 0 
++ F, sin («a cos 0) cos (xy sin 0) + Fy cos (kx cos 0) sin (ky sin 0) 
+ F, sin («x cos 6) sin (ky sin 0)} « dk, (11) 
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