MR. W. M. HICKS ON THE MOTION OF TWO SPHERES IN A FLUID. 483 
attract one another relatively according as — j l is positive or negative. This 
condition does not depend on their relative motion at any time, but only on their 
distance and the ratio of the constant energy to the constant momentum. The above 
. . d? 
condition may also be expressed, writing —=Jc 2 , as the sign of 
F|(A 1 A 2 -B*)-{(A 2 -B)^+(A 1 -B)^+2(A 1 -B)(A,-B)f} 
The last term is positive, for A l5 A. : , B all decrease as r increases. Now k must 
always be <p since r is always real. If we put Jd=p= A 1 +A 3 ~2B in the above, 
the criterion reduces to the sign of 
(A 1 A 3 -B 2 )|;{A 1 +A i ,-2B} 
i.e., since A 1 A : , —B 2 is always positive to the sign of 
|(A 1 -B)+|(A 4 -B) 
Now we are led to conclude from the argument in § 14 that —-(Aj — B) . . . are 
always positive. Hence when k has its greatest possible value the criterion is positive, 
much more then is it so for any other value of k. Hence we are led to conclude that 
whatever be the relation between the momentum and energy the spheres always move 
so that r tends to decrease, whilst in the case of equal spheres, or that in which the 
radius of one is twice that of the other, we know for certain that such is the case. 
We cannot prove from this that the spheres move with reference to a Jixecl point as if 
they repel one another, for it might happen that both the spheres might be accelerated, 
the extra energy of the motion of the spheres themselves being taken from the fluid 
motion; or that both are even retarded. We can easily show, however, that both 
cannot be accelerated if r is positive and both move in the same direction, for the dis¬ 
tance in this case increases, and therefore so do A x —B, A 2 —B, and hence because 
(A x — B)iq + (A. 3 — B)% is constant u x , u. 2 cannot both increase. Also if r is negative 
and u x , u 2 of the same sign the same result holds. 
In the case where the spheres are projected so that the momentum is zero 
2Tp 
' AAo-b 3 
and the relation between the velocities of projection that this may be the case is 
given by 
