MR. W. M. HICKS ON THE MOTION OF TWO SPHERES IN A FLUID. 461 
case where B is moving in the line joining the centres, the image becomes simplified 
and reduces to a single doublet. _ For let us find the image of a doublet whose axis 
passes through the centre of a sphere. 
The doublet is formed by allowing an equal source and sink P, P' to indefinitely 
approach one another, their magnitudes increasing indefinitely, yet so that p.PP' is 
finite. Now let P, P' lie on the line through the centre of the sphere, and let Q, Q' 
be their inverse points ; moreover, let the limit of /x.PP ' = k. Then the image of P, P' 
consists of a source ~p at Q, a sink at Q', and a line source (supposing P outside 
P', and therefore Q' outside Q) along Q Q' with line density j also the quantity 
-■QQ', together with the sink at Q', is equal and opposite to the source at Q, and we 
may suppose it added to the sink at Q', when they become equal. Now as P, P' 
approach to coincidence so do Q, Qj and the image of the doublet k at P becomes the 
doublet at Q, whose magnitude is the limit of 
zucv_z. ^ QQ_ 
Op-w * — ^oP’pp 7- 
OP 3 ’ 
i.e., one of opposite sign and magnitude (^ () pj X that at P. The same result can 
easily be shown to follow from the analytical formula in § 2. 
The case where the doublet has its axis perpendicular to the line joining the centres 
has more analogy with the case of a source. The image here consists of a doublet of 
the same sign at the inverse point, with a trail of doublets of opposite sign extending 
to the centre. 
Fig. 2. 
Let, as before, P, P' be equal source and sink, Q, Q' their inverse points with respect 
to the circle. 
Then at Q, Q' we have a source and sink of magnitude and in the limit we have 
a doublet 
fiaQQ'_ h f_a_Y 
op L v°p/ 
Also, if P, R' be corresponding points on 0 Q, O Q', we have a line density at 
3 o 
MDCCCLXXX. 
