MOMENT ON A SUBMERGED SOLID OF REVOLUTION 133 
The coefficient C1 is then given by 
4 1 2cr 
= Oh = | (l—p?)! dw i exp{«b(1—.2)#(cos w+-i sin w sin 6)+ 
“1 
++ ikap cos O}cos w dw. (15) 
The integrations can be reduced to known forms, and we obtain 
Cl = 3(7/2xa3e3)*b seci@ J(«ae cos A). (16) 
Hence, from (11), the corresponding term in the integral for A} is 
3(77/2xae)? sec!O J, («ae cos 8). (17) 
Using (7) and (17) in (1) we obtain the expression for G, which may be 
written in the form 
ae 47 co % 
Gh = IG p20 | k dk \ sec? dé | DAF —cosg)e-ts-ik se ds, 
—ae 0 0 
(18) 
where the real part is to be taken, and 
gq = xae cos 6, D = (+ K9sec?6)/i(e— Kg sec?4 iu sec A). 
After carrying out the integrations in « and k this leads to 
407 
1 2 
Gy = — 647pa’e?A B(c?/K,) | (“2 — eos) exo f sec’) sec @ dG, (19) 
Pp 
0 
with p = x aesecé. 
For computation it is convenient to express these results in terms of the 
so-called spherical Bessel functions, of which tables are available. If we 
iti 
ee S43(P) = (77/2p)'Jy 4a(p), 
T, = | S\(p)S4(p)e-21 =? seot9 a, 
0 
40 
1, = [ S3(p)e-27 #0"? e029 a, 
0 
Be tbewe R = 6479p, ate+A?],, 
Gy, = 64mgpatetA (ic, ael, —21,), 
Gy = —64ngpate4A BI,. (20) 
5. These results may be checked, to some extent, by taking the limiting 
case of a sphere. In the first place we may calculate directly the case for 
the sphere by the same method. For a sphere of radius a, we obtain 
+7 
G, = 4npc?a%3 | sec? e—2xof sec*d gp (21) 
579 
