204 Havelock, Note on the sinkage of a ship at low speeds Re Ne ns oeee 
Hence we obtain 
72 162 
ie abo U? | om iy slain 
0= gar=ayer ayn \an\ 3) ae \==cu 2) (G@a=ab3) (G2 =e) ya 
+( a, \ get ee (Cee te 
" \2—a,) 2 (a — 6°) (a? — cc?) \u(@ +) (u—») 
(u—) vw) 7 
@F DEL @ +) +7) 
Carrying out the integrations, and writing 
(8 +) (2? +) \ » 2) @O:. 
¥@L)0—M 
OQ srig oral O hi MIEN. PER RSN OSES ING See AAS Teen ean (i101) 
we obtain, for a>b>c, 
gh _ Ao J 2+a, — be? [ey 
> 2—a, " 220—a)@+b)(e@—e) * \2—a, HeF Ha) 
2 abe? b (a*® — c?)*? + a (b? — cc?) ° 
lo 
[eda 
~ Oa) @ ey — ot 8 oa ey Lc (Fey 
3. We require also the corresponding expression for an ellipsoid with a>c>6. This 
may be deduced directly from (12); or, alternatively, we may proceed as in the previous 
section but replacing ndS in (7) by mdS, given by 
(u — v) (v — pw) Ne 
(b? —c?) (8? —a@*) (24+ w(@+7) (F+W (e+ r)f 
After carrying out the integrations, we interchange 6 and ¢ so that we may express the 
result by means of (11). 
l 
maS= act dudy. .. . (13). 
We obtain, for a>c>b 
gh Ay a 2+ a, be ' a, ) a(a>+ab—c*) 
iy 2=a,  2CSa) C20) (Ge) "\2—ai) 2@+b(e@—c) 
; (14). 
2 abe? a {(a? — c?) (CP — 67)? 
9 =< ()° (a? ay Gz) uie(G2 ae Game ar ctan a b +¢? 
In this case, instead of (4) and (5), we have 
2abe 
OTe 2 2 2)1/2 vi E ) | 
8 (a? — c?) (a? — b?)!! ( ) (15). 
sin a={(a? —c*)|(a? — b?)H, siny = (a — oa 
4, The prolate spheroid may be considered separately, or may be deduced from the 
two previous cases. Taking limiting values, both (12) and (14) reduce to the expression for 
this case. 
For a>b; b=c, we obtain 
gh Ao 2+ ao b? | Ay a (a+ 2b) 
72 2=a, 2@=0, Gt’ Ba) 2@+0): (16) 
and in this we have 
MLA 1dbe : a, 
=> = 5 log, e); PS WGP oo 8 0 6 a 6 (Mb 
5. To apply these results to the problem under consideration we imagine a ship for 
which the immersed portion is ellipsoidal, the « y-plane being the water surface and the sides of 
the ship above water being vertical. Owing to the defect of buoyancy, which has been 
denoted by Q, the ship will sink in the water. This will, of course, alter the fluid motion; 
but for approximate comparison with experimental results, we define the equivalent sinkage 
h so that Q is equal to the weight of a volume of water of height h and of cross section 
equal to the section of the ship by the water surface; that is, h is defined by (11). 
460 
