1891.] 



Fluid Motions in two dimensions. 



181 



pressions for the lengths of the straight boundaries, which will 

 determine the unknown constants a, and we shall find for z a 

 complex expression when u has values corresponding to points 

 on a free stream-line; i.e. the coordinates x and y of a point 

 on a free stream-line are expressed as functions of a real para- 

 meter u. 



All the problems solved in the first part of Mr Michell's paper 

 can be treated in the manner here explained. I propose now 

 to consider his first problem — that of the escape of a jet from a 

 vessel,, and then to give solutions applicable to other cases. 



3. Problem (i). Escape of a jet from a vessel. 



Suppose we have a polygonal vessel of such shape that the 

 fluid coming from oo must move parallel to the negative direction 

 of the axis of y in the z plane, and the jet is formed by fluid 

 escaping from a side parallel to the axis of x. 



a \ 



The boundary in the z plane will be as above, and as we are 

 going to transform this boundary into a straight line u real in 

 the plane of a new variable u it is convenient to denote points 

 on the boundary by the values of u at the corresponding points. 

 We shall assume then that the points u = l, u = — l correspond 

 to the edges of the hole, and u = go to the point at go in the 

 z plane in the direction of the jet, and we shall take u — c, 

 a t , a 2 , ... unknown constants for the points corresponding to the 

 point at co in the z plane from which the fluid comes and the 

 corners of the polygonal boundary. 



The boundary in the iv plane consists of two infinite straight 

 lines corresponding to the bounding stream-lines. If these be 

 taken to be ty = 0, ^r — tt, and the velocity along the free stream- 

 lines be taken as unity, the ultimate breadth of the jet will be ir. 



15—2 



