462 MR. W. M. HICKS ON THE MOTION OF TWO SPHERES IN A FLUID. 
II and -f - at Id. Consequently when P, P' approach indefinitely so do R, R', and 
cc 
we get a line doublet along 0 Q, whose line magnitude at any point R is the limit of 
—-.RR'= 
fi OR 
fl'OF 
PP = 
k OR 
a OP 
i.e., proportional to the distance from the centre. 
Fig. 3. 
5. Supposing that the positions of all the images of the doublets and their mag¬ 
nitudes are known when the sphere A is moving along the line B A, we proceed to 
find an expression for the kinetic energy. Let p„ be the distance of the n th image in 
A from A, and <t„ the distance of the n th image in B. Also let the magnitudes of the 
doublets there be v a respectively. Let <f> be the velocity potential of the motion, 
and the parts of cf> due to and v, r Then denoting the kinetic energy by T 
— 27 ra 2 u\ [</>] sin 9 cos 6cl9 
Jo 
where [<f> J is the value of <f> at any point (a. 9) on the sphere. Now and 
the part of T due to <£„ will be 
Now 
2T = 
— \L-na?u j' 
= — 27 TCl 2 fJb n U [ 
"/*„(« cos 6 + p d ) sin 6 cos 6 
{a 2 + 2p„a cos 6 + p 2 } 1 
d9 
+i 
(p u -\-ttpd)pdp, 
) ^ x {a 2 + p 2 ,^ 2p/ipuf 
f +1 (p + ap)pdp, cl f + 1 _ pudpu _ 
) - x {ct 2 + p 2 + 2 pap,}** clp) _ 1 ^/ « 2 + p 3 -f 2 pap. 
d 1 
/ • O 0 o 
dp op*Ci" 
i ( P+ a ) ( P '+« 2 —p«) — {p ''«) (a + a ' ■ J rP a ) 
i 
s 
When p= Rfl p„ <a and the above becomes 
