Bae Ne de 1989 Havelock, Note on the sinkage of a ship at low speeds 203 
assume the submerged part of the ship to be ellipsoidal. The solution of the corresponding 
potential problem is well-known, and an exact expression can easily be found for the total 
defect of vertical pressure, and hence we obtain a certain equivalent sinkage. 
The problem is of some interest since Professor Horn’) has proposed to estimate the 
so-called form effect upon resistance by an approximate formula involving the sinkage at low 
speeds, The expressions obtained here for ellipsoidal forms are compared numerically with 
these results and with other experimental data, 
2. A solid, whose surface is given by 
ae y? 2? 
San yg GS CO SSOM ioe one aeenoneen DE 
is moving through an infinite liquid with velocity U parallel to the axis Ox. The velocity 
potential of the fluid motion is given by 
abe | adi 
One, U2 | (Gay eaye oe yee Neh 
i 
in which («, y, 2) are given in terms of orthogonal coordinates (A, 4, ) by 
»_(@ +4) (a? + w) (a +») 
(a? Se, b*) (a? ai Cc’) (3), 
with similar expressions for y and 2z. 
In these coordinates the, ellipsoid (1) is given by 2=0; and we have also 
C di 2abe 
a,—=abe | (2 As? (e+ Aji (c? Lye TE — 3?) GS cyt? (i= E) So, 6 9 8 (4). 
() 
F, E being elliptic integrals with parameters given by 
Sina—NOa—102))) (Qz—c2) csi (2 C2) 2) ee ee nn()e 
The fluid pressure is given by 
ee AI es =0pm 1 
P=P.+e < OU = — OU OW aa a poe o 6 (Oh 
If (l,m, ) are the direction-cosines of the normal at any point of the ellipsoid, the required 
total defect of resolved pressure Q is given by 
=0\\(uS24 ae )nds. SEL Nar. PEM Gy ent Sg Te (7) 
the integral being taken over the half surface of the ellipsoid lying on one side of the x y-plane. 
Using well-known properties of the coordinates 2, u,v (as given, for example, in Lamb’s. 
Hydrodynamics, p. 149), it can be shown that, on the ellipsoid 20, we have 
Op U J 2b* ce? (a? + mw) (a? +») 
0x 2—a, \% (a? — 8) (a? — ce?) wv J’ 
and 
0) zy +(¢ a =, 0c (a? + a) (a?-+ 7) 
oy (a? — b*) (a? —c?) wy 
(8). 
| Ay ) 7? { (6? + w) (c? + w) (6? + v) c? +) | @(e + we +v) 
=a ula? + Ww) (u—yr) ° v(a? +) (vy —p)J (a? — B?) (a? —e2) 
Further, we also have 
/2 
1 (4u— v) (vy — we) ; 
IS) ace Hea nG = wees CBee. al) 
1) Horn, Intern. Tagg. der Leiter der Schleppversuchsanstalten, 1937, S. 20. 
459 
