481 T. H. Havelock. 
This is a particular case of Sonine’s integtal,* and we obtain finally 
ies GRRE nn lor (5) 
A similar integral which we require is 
B= \| (1 = = = )" e*” cos ax daxdy, (6) 
taken over the ellipse (2). 
This may be evaluated in the same manner. To avoid possible ambiguity 
we distinguish between various cases according to the relative magnitude of 
ma and nf. We find 
B = 2"2n32ann Jap {( (mPa? — n2Q?)"?\ 
(mPa? — n282)3/4 ” ma > 1B; 
1, Layo {(n2B2 — mPa)? 
SO 1/ 23/2; 3/2 F 
= 227 32yy, ee ea? ma <nB 5 
= 3nmn, ma=npB ; (7) 
where I denotes the usual modified Bessel function. 
3. Consider a solid bounded by the ellipsoid 
7 22 
Sew Jk = 7] 8 
a a b a Ce ; (8) 
moving with uniform speed w in the direction of Ox, the axis Oz being hori- 
zontal and at a depth f below the surface of the water, while the axis Oy is 
vertical. 
We shall consider first the case a > b> c. 
The image of a uniform stream in the ellipsoid is a distribution of doublets 
over the plane area bounded by the elliptic focal conic 
eee 5= 1, 2=0; (9) 
BN 5 eon BODO EA) Seite el Me aa, i 
T (2 — ap) (a2 — c?)¥? (6? — c?)V2 ( 2 
where 
2 du 
tial cor CCE 00) 
For numerical calculation «) may be expressed in terms of elliptic integrals. 
*G.N. Watson, “Bessel Functions,” p. 376 (1st edn., 1922). 
324 
