Mr Sharpe, Liquid Motion from a Single Source , etc. 223 
Liquid Motion from a Single Source inside a Hollow Un- 
limited Boundary. By H. J. Sharpe, M.A., St John’s College. 
[Read 20 May 1901.] 
Part I. 
1. The problem expressed by the title of this Paper suggested 
itself to me as an analogue of the well-known and difficult one, 
that of a Paraboloidal Reflector of Sound, with a single source 
of sound in the focus, a problem which, as far as I know, has 
never been solved. It is proposed to consider the problem of 
Liquid Motion from a Single Source inside Hollow Material 
Boundaries, — surfaces limited in one direction, but unlimited in 
the opposite direction, surfaces (when the problem is considered 
in three dimensions) having a general resemblance either to a 
tube closed at one end, a hyperboloid, or paraboloid of revolution. 
It will be shewn that there are an infinite number of surfaces for 
which complete solutions can be found. I think the results may 
throw some light on the general phenomena of the reflection of 
liquid motion and perhaps of sound reflection at curved surfaces. 
The problem can also be solved in two dimensions, and with this 
case we will begin. 
2. The liquid motion which is supposed to be in the plane of 
the paper is referred to two axes Ox, Oy, and is supposed to be 
symmetrical with regard to Ox. 0 is a single source of liquid 
supply. DABA'D' is a rigid boundary whose possible shapes will 
presently be explained. Liquid issuing from 0 is reflected against 
the rigid boundary, and goes to infinity in the direction Ox. To 
find the motion. 
3. AC A' is a circle with any radius a. Different expressions 
will be assumed for the liquid velocities inside and outside this 
circle, but such that these velocities are continuous at every point 
of the arc AC A' inside and outside. Let u x , u y be the liquid 
velocities parallel to Ox and Oy, expressed in the polar coordinates 
r and 6 of any point. 
