Wave Resistance of a Spherord. 281 
Taking the integration with respect to y first, we obtain, after integration 
by parts, 
(b2e’2 — 2?)t 2 
(ko sin 8 sec? 6) | y (b?e'? — y? — 2*)-V?2 sin (ky sin 8 sec? 6) dy 
—(b2e’2 —2?)1/? 
oe (b?e'2 Lee z2)il2 =a =: ‘ 
~ a ein O eae J, (ky V be? — 22 sin 6 sec? 8). (29) 
The integration with respect to z now becomes 
be’ 
| (b2e’2 — 22)U2 ere sec*O (ic, 4/b%e'2 — 2? sin 8 sec? 8) dz, (30) 
—be’ 
and this is equivalent to evaluating 
2 (ie cosh (« cos ¢) J, (8 sin ¢) sin? ¢ dd, (31) 
0 
where o = Kbe’ sec? 0, 8 = Kybe’ sin 0 sec? 0. 
The integral (31) may be evaluated as a special case of Sonine’s integral, or 
by expanding cosh(« cos ¢) in powers of cos ¢, integrating term by term, and 
summing the resulting series. The latter expression for (31)-is found to be 
« 2n 9n—1/2 
5 CRE OS T (m+ 3) jy 
n=o 2m! aS n+s/2 (B). (32) 
Noting that in the present problem, « <> 8, the value of (32), or of the integral 
(31), is 
1/2 
2 () = To/2 {(a? — B?)"?}, (33) 
where the Bessel function is given by 
__ f Be sinh x 
hn) = (2) (cosh « — smhe) (34) 
Collecting these results, we obtain 
(P + 1Q)/Bu e~*f °°? —2(77353e'3 /2icq3 sec? 8)"2 Taio (Kegbe’ sec 9). (35) 
Finally, from (19) we find 
77/2 
R=382 r'ox )b3e’SB2u2 | e~ *xof sect? fT. (gde’ sec 8)}2 sec? 8d0, (36) 
0 
or in the same form as (27), 
R=32 r4gob%e’SB2e-” | eP# {Toi (Kgde’ V1 + 22)}2 dt, (37) 
0 
where B is given in (11). 
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