The object of this communication was to shew the application of Professor Thomson's method of images to the solution of certain problems in hydrodynamics. Suppose that there exists in an infinite mass of incompressible fluid a point from which, or to which the fluid is flowing with a velocity alike in all directions. Conceive now two such points, of intensities equal in magnitude and opposite in sign, to coexist in the fluid; and then suppose these points to approach, and ultimately coalesce, their intensities varying inversely as the distance between them. Let the resulting point be called a singular point of the second order. The motion of a fluid about a solid, oscillating sphere is the same as if the solid sphere were replaced by fluid, in the centre of which existed such a point. It is easy to shew that the motion of the fluid due to a point of this kind, when the fluid is interrupted by a sphere having its centre in the axis of the singular point, is the same as if the sphere's place were occupied by fluid containing one singular point of the second order. By the application of this principle may be found the resistance experienced by a sphere oscillating in presence of a fixed sphere or plane, or within a spherical envelope, the oscillation taking place in the line joining the centres, or perpendicular to the plane.