130 T. H. HAVELOCK 
The first two terms in (1) satisfy the condition at the surface of the solid. 
The third term, which we shall denote by 9s, is the first approximation to 
the wave motion; its form is determined so as to ensure that the three 
terms together satisfy the condition at the free surface of the water (2). 
The second term in (1), which is the velocity potential for the axial 
motion of a prolate spheroid, is usually given (3) as 2AcaeP(u)Q,(¢) in 
terms of coordinates specified by « = aeul, y = ae(1—p?)?(€?—1)? cosa, 
2 = ae(1—p,7)}(€2—1)'sinw; it can readily be verified that the two forms 
are equivalent. This equivalence is a particular case of a general relation 
which does not seem to have been stated explicitly, and the opportunity 
is taken of recording it here in view of its use in problems dealing with the 
motion of a spheroid. The relation expresses prolate spheroidal harmonics 
in terms of axial distributions of poles or multi-poles. Using the appropriate 
form of the known general expansion of reciprocal distance (4), it follows 
at once that ne 
ba! P. (k/ae) dk 
EOO=— | Se 
“ 
—ae 
For the general case, forming the corresponding expansion for the potential 
of a multi-pole, it can be shown that 
s SA comb iC AOUNG ( (ae? ~ k*)*SP S(k/ae) dk. 
P*(u)Q_ (Oe =+(5;+ taal | (y2 4 22 +(e@—k yt 
We use the theorem that the forces on the solid can be obtained as the 
resultant of forces on the internal sources, the force on a typical source m 
being —4zpmq, where q is the fluid velocity at the point other than that 
due to the source itself; in fact, we may omit the part of the velocity due 
to all the other internal sources and sinks. Thus for the horizontal force, 
or wave resistance R, we have 
ib 
[ Acx(éd3/x) dx, ( 
—ae 
R = —4zp 
bo 
~— 
taken along the axis y = z = 0. 
Taking 4, from (1) and omitting terms which, on account of the integra- 
tions in x and k, give no contribution to the final result, this reduces to 
ae ae $7 
a /19 1 ale 36 d6 r e—2f +ix(x—k)cos 8 
R= —16px?c?A* | x dx | f | see (_—= = Kk, 
0 0 
(3) 
a 
—ae —ae 
where the imaginary part is to be taken. 
576 
