MR. W. M. HICKS ON THE MOTION OE TWO SPHERES IN A FLUID. 485 
If a— 2b, x=l~, y—\, and we find from Legendre’s tables of the Eulerian integrals 
D 3 lcg 10 r(l+a!) = --485 
D 3 log 10 r(l+2/)=“--275 
and the ratio is 
1 p + 1174 
8 p + ’4b42 
which when the densities of the spheres and fluid are equal becomes 
-^-= ‘0954 
We And the velocities of the 
between 
and 
whence 
spheres relatively to the fluid by eliminating u 
Uy=u -\|-ar 
pu-\-lr—cl 
and 
d . A, —B 
P 
P 
U -2 
cl A, —B • 
-i- r 
p p 
Suppose now the same spheres projected with the same initial circumstances except 
that now the spheres have changed places, and let u\, u\ be the corresponding velo¬ 
cities at the same distances. Then 
d A x —B • 
P 
since cl and r do not depend on the question which of the two is foremost. 
Now if a > h we see at once from the expressions given for A ls A 2 in terms of the 
distances that A 1 >A 2 , and hence that the foremost will be most accelerated when it 
is the smallest. 
If now u x , u . 2 denote the velocities at any moment which we may regard as the 
velocity of projection 
/, ^ {JA ~ BK + (A 2 -BK} 2 
2T A 1 u l i + A i u^-2Bu 1 u i 
Writing £ for the ratio J1 the equation to find £ in order that h may have a given 
Vj ^ 
value, is 
3 R 
MDCCCLXXX. 
