£02 Prof. W. B. Morton on the Discontinuous 



A survey of the conditions of the problem may be ob- 

 tained as follows. Start with a single plane lamina set 

 at angle a. to the direction of the stream at infinity. The 

 stream divides at a point on the upstream side of the centre. 

 Now build out a second plane from the upstream edge of 

 the first, at angle j3 with the stream, until it reaches the 

 free surface of the liquid. This happens when the ratio of 

 the second breadth to the first is less than the Bobyleff value. 

 Up to this point, of course, the former state of motion 

 persists, the second face of the wedge lying entirely in the 

 region of dead water behind the first. But now a new kind 

 of motion sets in : the stream-. ine, which passes round the 

 angle of the wedge, is interrupted where it runs along the 

 second plane near its edge. Beginning at the corner, it 

 first sweeps round a pocket of dead water enclosed between 

 it and the second plane, it then becomes tangent to the 

 plane and runs along it to its edge and thence to infinity. 

 There is evidently a point of inflexion before the stream-line 

 joins the plane. As the second plane is extended, the point 

 on the first plane, where the stream divides, moves towards 

 the corner, until we reach Bobyleff' s case. Still further 

 extension gives the reverse change in the character of the 

 motion, the roles of the planes being interchanged until we 

 get to the one-plane case round the second plane, with the 

 first plane lying entirely in the dead water. 



It is proposed in the present note to discuss the general or 

 transitional case. The treatment is quite straightforward 

 on the well-known method of conformal representation, but 

 it derives some interest from the fact that there are two 

 different constant values of the velocity on the free portions 

 of the stream-line, one along the infinite branches and 

 another, smaller than this, round the pocket of dead water. 



The two critical breadth-ratios can be found from the two 

 known solutions, Kirchhoff's and BobylefFs. The expressions 

 obtained are, in general, complicated, but become manageable 



for the special case of a right-angled wedge 1/5= „--—«). 



Taking the breadth of the "first plane"' as unity, the breadth 

 of the second when it just touches the free surface is 



sin a ( sin a sin 0(1 — cos a cos 0) , sin^^+a)) 

 7rsina + 4\ (cos a — cos 0) 2 °sini(# — «) J ' 



. . . . (i) 



where a is the inclination of the first plane to the stream 

 and 6 is given by cos 6 — cos «/(2 -f-cos a). And the value 



