482 
MR. W. M. HICKS ON THE MOTION OP TWO SPHERES IN A FLUID. 
and m l3 M' respectively denote the mass of the sphere (A), the mass of fluid 
displaced by it, and the mass of fluid in a unit sphere. 
It is to be remarked that A l5 A 2 , B are functions only of the distance between the 
d d 
spheres, and that therefore — + —= 0 . Since no forces are supposed to act on the 
(ic/_2 tii ICz) 
system, both the energy and momentum are constant. Hence 
2 T = constant 
~- + r~= constant =d 
OU x OUv 
feT 6 T 
The last equation also follows at once from Lagrange’s equation since =0, 
and may be written 
(A x —B) w 1 +(A 3 — H)it%=d 
We shall transform these equations by referring the motion to the velocity of an 
arbitrarily chosen point P between the spheres, and the distance between them. 
Let P divide the distance (r) in the constant ratio —— = 7 -. Then if x is the 
' 1 —« p 
distance of P from the origin, u its velocity 
and 
whence 
x 1 =x-)-otr, x 2 =x—/3r 
u 1 = u-j-ar, Uo=u—(3r 
(Ajd-Ao — 2B)?d-(- (A L a -+A 2I 8 2 +2a/3B)r 2 
+ 2 {a(A 1 — B) — f3( A 3 —B)} ur= 2 T 
(Aj + A 2 — 2 B) u -f {a(A 2 — B) — /3( A 3 —B) }r—d 
( 21 ) 
which we shall write 
whence 
or 
p id + qr 2 + 2 lur = 2 T 
pu-\-lr—d 
(l x l~ l 2 ) r2 — 2 T p—d? 
( 22 ) 
in which we are to take the positive or negative sign according as the spheres are 
separating or approaching one another. The spheres will move as if they repel or 
