MR. W. M. HICKS OH THE MOTION OF TWO SPHERES IN A FLUID. 487 
the spheres can neither ever come to rest; if 
<h or >m. 2 +±m 2 
the small sphere can never come to rest. 
The effect of the fluid on vibratory motions. 
18. Suppose each of the two spheres attracted to a fixed centre of force where the 
force varies as the distance. Let x x , x. 2 be the distances of the spheres at any time 
from them respective centres of force measured in the same direction. Then 
2T=A 1 w 1 2 + AoWo 2 —2 Bm ] w 2 =C— m x ix x x x — 2 x. 2 2 
Also since we neglect squares of small quantities in finding the small vibrations, the 
equations of motion become 
A 1 jc 1 —Bx 2 = — m 1 [x 1 x 1 
— Baq+Ayr* = — m, ? jj. 9 x 2 
and we suppose the spheres so distant, and their motions so small, that we may neglect 
the small changes in A, B during the motion. The spheres must not be too close, for 
dA 
at contact —f, &c., are infinite, as was shown in S 14. 
dr 0 
Solving the above equations in the usual manner we find 
£C 1 =L 1 sin (Ky + a)+N 1 sin (K a £+/3) 
x 2 =eL x sin (Ky + a) + e / N 1 sin (KA+/3) 
where 
K 1 2 | A iViiniio + A 2 // 1 m 1 + yf {(Apiq/r ,—Aniiijfjbfl + 4m 1 m 2 /i 1 //. 2 B 2 } 
K,»r 2(A7VB j ) 
_ A 1 K 1 2 -w lA t 1 _ BKp / _ A 1 K 3 3 -??t lAtl _ BK 3 2 
BKj 2 “AgK^-mg/A, e— BKv — A 2 K 
From this we see that, to the first order of small quantities, the mean position of 
the spheres is not altered, or to that degree of approximation there is no mean 
attraction or repulsion. 
If we regard the spheres as two pendulums swinging in the fluid, in the same 
horizontal line, of lengths l x , 1. 2 , then the motion is given by the above equations if 
we write 
3 r 2 
