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. is a single source of liquid 

 supply. DABA'D' is a rigid boundary whose possible shapes will 

 presently be explained. Liquid issuing from 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 AG 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. 



