488 MR W. M, HICKS OH THE MOTION OF TWO SPHERES IN A FLUID, 
Pi- 1 9 
.Ps-i.P 
b 1 D i H ' i 
Pi l i Pi H 
2tt 
where p 1 , p. 2 are the densities of the spheres compared to the fluid. 
If in the above we make m x = co we get the case of a forced vibration of period —j=^ 
V P'1 
In this case 
m 2 /4 3 
N 1= 0 K 2 2 = : 
A, 
aq = L sin (vW+ a ) 
L sin (' / ^+“)+ N sin (V ¥ ,+/3 
If the sphere (b) is set free when (a) is for the moment at rest, and the time be 
reckoned from this moment 
r 2 =el/cos y/^t —cos /\Jb£ht 
and the motion of (b) consists of two periodic terms whose amplitude is e times that 
of (a). 
Let now the strength of the centre of force on (b) diminish indefinitely. Then 
cc 2 =— L(cos vVi£ —1) 
Ag 
and (b) would oscillate in the same period as (a), without being attracted or repelled 
towards it except by forces depending on the square of the amplitude of (a). To find, 
then, whether the action of (a) on (b) is attractive or repulsive we must take account 
of quantities of the second order of small quantities. 
The full equation of motion of ( b ) is 
A 2 % - Bmj - {W-u l u. 2 ) C ^+^£(A l - 2B) u ^=0 
For a first approximation we have 
x. 2 = ~L(cos \/pF— 1) 
BL . 
'll. 2 — ^ V P-1 a/ Pi t 
aq=— L(cos \/^t — l)-\-z 
Write 
