476 MR. W. M. HICKS ON THE MOTION OF TWO SPHERES IN A FLUID. 
d Q C IQ' 
14. We may easily find — at contact of the spheres. If Q /t denote the rf h 
term of Q 1? then it may be shown that, x denoting 
a +b 
dQn __ n(n + l)(n — 1 + 3rd) 3 
dr a(n+xY 
dQ' n _ n 2 — l + 3a?(l— x) 2 
cv’dr mr’ 
both of which are of the order -. Hence the values of —f 2 , -y at contact 
~ dr dr dr 
are 
= —co. But though this is the case, the value of p.(Qi —yQO at contact is finite. 
The n th term is 
n s (n + 1)(?7 — 4) + 3(1 — x)(n + a;) 4 + (?r — 1 )(6w 2 + 4 nx + x^x 3 
ar?(n + xf 
which is of the order —, and therefore the whole sum is finite. Also when nz 2 the 
n~ 
n th term is positive, when 7^=1 the sign depends on the value of x. But by consider¬ 
ing the values of Q, &c., in terms of r, expanding them in ascending inverse powers of 
r, it can be shown that ^Q — ^Q') is positive always. Further, at contact Q — ~ is 
a negative quantity, whilst at an infinite distance it is zero. Hence, on the whole, it 
must increase with r, and if this takes place continuously, y-/Q — yQ j would always 
dr\ ~ a° 
be positive. Though I have convinced myself that such is the case, I have not been 
able to prove it in general. When the spheres are at a great distance the values of 
Q and Q' depend only on their first terms, and Q — ~Q' only on the term of Q', which 
& 3 
is of the order -. Hence here also the differential coefficient is positive. I have cal¬ 
culated and laid down curves representing the magnitudes of the Q and Q'-functions 
in the case of equal spheres, and when the radius of one sphere is twice that of the 
other, and in both cases the value for ~ (^Q — yQ'j comes out positive for all distances. 
In what follows we shall suppose that this quantity is always positive, but it must be 
understood throughout as only proved for the case of equal spheres and the case in 
which the radius of one sphere is double that of the other. 
15. Although the rapidly increasing complexity of the successive images when the 
spheres move perpendicularly to their line of centres would lead us to regard the 
