MR. W. M. HICKS OK THE MOTION OF TWO SPHERES IK A FLUID. 477 
problem of finding the energy in this case as almost hopeless, yet we can carry the 
approximation to any number of images with less labour than might at first sight 
appear. For suppose we wish to take into account 2 n images in A, due to A’s 
motion, that is on the whole 4 n reflections. We need only first calculate the distribu¬ 
tion of doublets for a general position of the original one, in the n th image in A, and 
find the amount of the first n images. We can then treat the second portion of the 
2 n images as the images resulting from the different parts of the n th image, and 
employ our first result to find the amount of the second portion by a single integra¬ 
tion. Suppose we proceed as if we were going on indefinitely: we suppose an original 
doublet in A at a distance p and calculate the density of the parts of the first image 
in A, say f(r) at a distance r, and thence its amount. We employ this result to find 
the density at any point of the second image, regarding it as made up of images of the 
different parts of the first, and this we do by using the expression found before, sub¬ 
stituting for the original doublet at p, an amount f(r)dr at a distance r, and integra¬ 
ting with respect to r over the first image. Thus we find the distribution for the 
second image and its amount, and therefore the amount for the first two images 
together. Starting now from this, and proceeding in the same way, we find the dis¬ 
tribution and amount of the first four images, then of the first eight, and so on. 
Thus to find the distribution of the 2Ah image we only require p-\- 1 operations, and to 
find its amount only p operations. Even with this method of proceeding the work 
would be exceedingly laborious. But for all practical purposes the first two images in 
A, i.e., the motion due to four reflections, will be sufficient—except when the spheres 
are in contact. We proceed then to find the values of the coefficient of v 3 and of v y v. 2 
to this degree of approximation. 
Suppose we have at P inside A a doublet k at a distance p x from A, whose axis is 
perpendicular to A B. 
i. First image in B.—Then we have at Q ls its inverse point in B, a doublet [flpj ^ 
Jc r 
and a line doublet thence to B, whose line density = — — 
ii. First image in A.—The image of this in A consists of two parts, that depending 4 
on the single doublet in B, and that depending on the line doublet. 
(a) Image of (fi.—A doublet at P, ( AP 3 
Aty 
a negative line doublet from P 3 to A wdiose line density = 
) whose magnitude is 
h I b \3 R 
ab 
AQ^BPj 
k, and 
Biy AQj 
(/3) Image of negative line doublet. —At a distance r from B we have a negative 
Iv V % • • C(? 
doublet — — dr. This has (1) a negative doublet at a distance from A=-— = R 
equal to — 
C—7 
3 A 
b BP 1 dr ' 
That is from P 3 to A we have a negative line doublet 
whose density at a distance R is 
3 Q 
MDCCCLXXX. 
