132 T. H. HAVELOCK 
Treating the integration in « as before, and carrying out the integrations 
in x and k, this leads, after some reduction, to 
G, = 647pa?e?A?(c?/K9) | sec a=? — cos] x 
P 
0 
x (psinp +2 cosp— 2 SE |e tess db (9) 
Pp 
with p = x,aesec 0. 
We now determine the next approximation to ¢ so as to satisfy the 
condition at the surface of the spheroid, namely that the normal component 
of velocity from ¢,+-¢, must be zero over the spheroid. 
We use coordinates p,f,w given by 
x=aenl, y= ae(l—p*)?(C?—1)!sinw, z= ae(1—p?)'(f2—1)? cosw, 
(10) 
the spheroid being given by ¢ = ¢, = l/e. 
If, in the neighbourhood of the spheroid, ¢, is expressed in the form 
$s; = > > (AScossw-+ Bisin sw) P%(u) PX), (11) 
r=1 s=0 
then the required next approximation is given by 
= ie Q"( nn yharses soc Brains) ry )Qs(Z). (12) 
By considering the behaviour of the terms in ¢, as ¢— oo, we see that 
the only one which contributes to the moment JM referred to in (7) is the 
term in P}(.)Q}(¢)cos w; this latter quantity approximates to — 2a?e*z/3r3 
as > oo. Alternatively, we may get the same result from the expression 
of this term as a line distribution of doublets parallel to Oz along the axis 
of the spheroid between the two foci. Hence, putting in the value of the 
factor P}’())/Q1’ (fo), we have 
M = 2a°e? BA}, (13) 
with B- = flog{(1+e)/(1—e)}+-e(2e2— 1)/(1—e?). 
To. determine A} we take from the expression for ¢3 in (1) the term in the 
integrand involving the coordinates, namely 
Tr=1s= 
exp «(z-++72 cos 6+ iy sin 6), 
and expand the value of this on the spheroid in the form 
r 
S > (C8cos sw +Dssin sw) Psu). (14) 
s=1 
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