303 T. H. Havelock 
in terms of the co-ordinates (21), with ¢ = ¢ = 1/e on the spheroid. Taking from 
(26) the terms which contribute to the resultant tangential force, we have 
ae 
al (AWOKE, + BOK(a%e? — 12) Gy} dk, (34) 
with 
= my WI, (Ky, @) I, {Kp(h? + k*)*} exp [—«k,(f—z)]sinne sinn(O— Qt), (35) 
and a similar aa for G',, derived from (28). If (34) is expanded in the form 
= BS (Atcossu + Bysin sw) Pru) PHS), (36) 
Tr=1 s=0 
we must add a similar expression with Q3(¢) in place of P3(€) so as to maintain the 
boundary condition at the surface of the spheroid; for this part of ¢ this is 0g/0¢ = 0 
for € = ¢). Hence on the spheroid we have 
SE = S&S CH(Apcossu + Bpsin se) PAG) Pew), 
with Cy = 1— Ps'(Eo) Or(Eo)/Pr(So) Or (So)- (37) 
We now expand (34) in the form (36), noting that for the value of R we only require 
the term in P9(z). For this purpose we have 
J, (Kn @) Xp [K,(f—2z)] sin n(8 — Qt) 
_ = exp[—2k,f +2] |" exp [2k,(xsin u—y cos uw) + 7k, h cos u]sinnudu, 
(38) 
with the origin now at the centre of the spherqid. Substituting from (21), we multiply 
by “dudw and integrate over the surface of the spheroid. It can be shown that 
1 2a 
| ndye| exp [7x,,{ay sin u — b(1 — 7)# (cos u cosw +7 sin w)] dw 
—1 0 
= 47i(7/2k,,ae°)tsin-*uJ;(k,aesinwu). (39) 
Using (39) in (38), we have, so far as this typical term is concerned, the integral 
cs 
4ni(77/2k,, act) | exp [ik, h cos u] J3(k,, ae sin u) sin wu sin nudu. (40) 
Su 
It is of interest to find that (40) can be put in the form (47?/a?e3)i"L,,, in the notation 
of (31). Collecting these results and including the factor C? from (37), we obtain for 
the wave resistance 
2 
R=— ae A Sn nL, (24 D,+B 7 0M) exp [— 2n?0?f/g]. eD) 
We could obtain the couple @ by similar calculations; or, using (32), we have 
2h O6 z 
¢ eS nM, (2AL, + BO n?M,) exp[—2n20%f/g]. (42) 
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