T. H. Havelock 298 
We then integrate with respect to 7 term by term and obtain the limiting form of the 
resulting integrals in x as t->0o. This process readily gives the result 
K e-k(S—2) 
b= 7-7 +amsP |” xO] J, (kw) J,,(Kh) cos n(@ = Ot) dt 
Lemos. dn? J, (v?O2a/q) J, (n?Q?h/g) exp [ — n?07( f —z)/g] sinn(O— Ot), (5) 
1 
where P denotes the principal value of the integral. It may be verified directly that 
this solution satisfies the conditions for the quasi-steady state. 
3. If a sphere, of radius a, is moving uniformly in a circle we may, as a first 
approximation, take it as equivalent to a doublet of moment M equal to 4a*hQ. 
It is easily seen that the velocity potential for this doublet can be derived from (5) 
by taking 0¢/ot and replacing mhQ by M. Thus we obtain 
a Qt) Masin (6— Qt) 
re 
4M 2 © K mel 
plea uP EEOUE ko) J, (Kh) sin n(8 — Qt) dk 
ee 
Sm, (n?Q?a/9) J, (n?Q7h/g) exp [ — n?.Q?2(f —z)/g] cosn(O— Qt). (6) 
It may be noted that the second term in (6) is acai to 
2Mosin (0 — 4) | 4MQ? @ {= et 
ie gh 4 0 K— — 
We may deduce the wave resistance from the energy propagated outwards through 
a cylindrical surface, namely, 
foXe) 
-[ dz p $ sods (8) 
Taking the cylinder of large radius, we require the first terms in the expansion of 
(6), which are seen to be of order 7 ?. One such term comes from the integral in (7). 
Referring to (4), since we are concerned with large values of w, we may replace 
J,(K@) in the expansion by H®)(xaw) and take the real part. Thus we have to evaluate 
the real part of 
n3J,, (ko) J,(Kh) sin n(6— Qt)dk. (7) 
—k(f—2) 
| = = HH (ka) J,(kh)dk (o>h;z<0), (9) 
0 US, 
where we have put x, = n?Q?/g. Regarding « as a complex variable, we may change 
the path of integration to the positive half of the imaginary axis; taking account of 
the indentation at k = k,, we obtain for (9) 
) K,(@m) [, (hm) dm 
— TY (Ky @) In (Kh) exp[—K,(f—z2)]. (10) 
Collecting the results from (6), (7) and (10), and using the asymptotic expansions 
for J, and Y,, we obtain, for @ large, 
4MQ (= 
De a =) 3 AACS) SSO EN CO OES Ga ee (11) 
2 (°msin m(f—z)+«, cosm(f—z 
m+ Ky 
555 
