301 T. H. Havelock 
M’g(a/f)° 
0-4 0:8 Ses 1-6 
hQ/V (Gf) 
FIGURE 1 
The motion of the spheroid is made up of a linear velocity hO parallel to Cx and 
a rotation Q about Cz. In terms of spheroidal co-ordinates given by 
a=aeul, y=ae(l—p*)i(C?—1)Feosw, z=ae(l—p?)i(€2—1)'sinw, (21) 
the known solution for this motion in an infinite liquid is (Lamb 1932) 
= 2AaehQAP,(1) Q,(6) — xBare?QP3(11) Q3(C) cos@, (22) 
with A-1 = 2e/(1 —e”) —log {(1 + e)/(1—e)}, a 
B= ee arene Mena) ee 
It is well known that the linear motion can be expressed in terms of a certain source 
distribution along the axis of the spheroid, and it can easily be shown that the angular 
motion can be ascribed to a doublet distribution along the axis. In fact, (32) is 
equivalent to 
ae kdk a k(a*e? — k?) dk 
= re —— 
? 410” acne A cscercees: 
We may now obtain the required solution by integration of the expression for a 
source given in (6). For the first term in (24) we have to replace a typical factor 
J,,(Kh) cos n(8— Qt) in (6) by J,,{K(h? + k?)*} cos n(@ — Qt—«), where tana = k/h; and, 
taking account of the integration in k, this may be replaced by 
J,,{k(h? + k?)*} sin nx sin n(8— Qt). 
Further, so far as the co-ordinates x, y, z are concerned, the second term in (24) may 
be derived by taking 0/éy of the first term; and when the expressions are put in terms 
of the fixed co-ordinates this is equivalent to operating by 0/ch. Also we have 
(24) 
= [J,,{k(h? + k?)*} sin na] 
= 4k[J,_,{k(h? + hk?) sin (n —1)a—d,,s{x(h? + k)#} sin (n+ 1) a]. (25) 
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