282 T. H. Havelock. 
Prolate Spheroid at Right Angles to its Axis.—The distribution is given in 
(12), and in this case we use (21) instead of (20); apart from this, the caleu- 
lation follows the same course and we obtain finally 
a | 2. 
1, = 128rigpateAa’ | enol 8008 OL J 5/9 (egze sin 0 sec? Q)}2 cos 0 d0/sin? 6 
0 
= 128n’goa%e2A’e7? I, e {Tajo (ko aet V1 + t)}2 t-3 dt, (38) 
with A’ given in (13). 
Oblate Spheroid at Right Angles to its Axis.—The distribution given in (15) 
les in a plane parallel to the direction of motion, so we now use (20); the 
integrals are, however, of the same type as those already discussed and the 
analysis need not be given in detail. Using (15), (19) and (20), we obtain 
after some reduction 
R = 32n4gpb3e’3B’2e” [ e {Tajo (Kqbe't V1 + )}2 1-8 dt. (39) 
where B’ is given in (16). 
Sphere.—It may easily be verified that in the limit when e, or e’, becomes 
zero, all these expressions (27), (37), (38) and (39), reduce to the known result 
for a sphere, namely 
R = 4rgox,%b&e-” i (1 + ¢2)32 e- 2% dt 
0 
= / IL\ 
= rgpxctiPe! {Ky (bp) + (1+4) Kp}, (40) 
where K,, is the Bessel function defined by 
K, @) = \" e~7°osh4 cosh nu du. (41) 
0 
6. The resistances for prolate and oblate spheroids have been worked out 
independently in the preceding section. It is of interest to note that the 
results have the same analytical form and may, in fact, be deduced from each 
other by taking the eccentricity to be imaginary instead of real. For the 
prolate spheroid, e* = 1 — 6/a?; while for the oblate spheroid, e’2 = 1 — a/b. 
It may be verified that if in (27) we write e = ie’b/a, the expression transforms 
precisely into (37) ; and the same relation holds between (38) and (39). 
7. The integrals in the various expressions can be transformed into alter- 
native forms, or expressed in infinite series in several ways; but either the 
series do not converge rapidly enough for the values of the parameters which 
are of interest, or else the functions involved have not been tabulated. I¢ 
319 
