MOMENT ON A SUBMERGED SOLID OF REVOLUTION 13] 
The integration in « may be transformed in the usual manner by treating 
« as a complex variable and integrating round a suitable quadrant accord- 
ing as ~—k > or < 0. Finally we obtain 
in 
R = 327°gpa%e? A | sec? J3(Ky ae sec A)e-2xof sec*d 1g), (4) 
0 
which is the known expression for the wave resistance. 
The vertical force, Y, apart from that due to buoyancy, can be obtained 
similarly from a 
Y = 4p | Acx(éf4/éz) dev. (5) 
—ae 
This involves the real part of the contour integrals in « referred to above, 
and leads to double integrals; the expression for Y can easily be written 
down, but it is not very suitable for numerical computation. 
3. The moment of the forces about Oy requires more consideration, and 
we shall take it in two parts. 
We calculate first the moment on the initial source distribution arising 
from the vertical component of the velocity derived from the term ¢3 
in (1); we denote this part by G,. Thus 
ae 
G, = 4mp | Acx?(6b3/0z) dv, (6) 
—ae 
taken along the axis. 
But we have to proceed to a further approximation to the velocity 
potential, because the uniform stream produces on this second approxi- 
mation a contribution to the moment of the same order as G,; we denote 
this second part by G,. Let 4, be the term to be added to (1) for the next 
approximation. This term represents some distribution of sources and sinks 
within the spheroid; if M is the total moment of this distribution resolved 
parallel to Oz, then we have 
CG, = —4npcM, (7) 
and the total moment on the solid to this stage is G+ Gs. 
4. From (6) and (1), we have 
too) 
ae ae 7 
G, = spate [ atde | kak [ «| K (ict Hey 800%) 
K— 
—ae =u 0 
Kg sec?0-+-1p sec 0 
—ae 
sen 2kf+ix(e—hcos8 de, (8) 
ae 
