Stationary Waves in Water. 101 



Equation (1) with G(V) = B - cos olio gives the exact 

 solution of a case of flow illustrated in fig. 3, drawn for 

 the values B = 2, a='9. 



The singularities of — are given by 



dw 

 B = COS aii\ 



sin aiv= + 



dz 



dw 



1 



B — cos aw] 



P 



■A { 1 — a 2 sin 2 uiv} 2 + iot sin olw 



if a<l and B>1 yjr=0 gives a free surface. Now 

 surface 6 depends only on a, and not on B, so that an 

 tion in B will not alter the concavity or convexity 

 free surface. 



= -«* 



on the 

 altera- 

 of the 



The shape of the stream-bed, however, alters as indicated 

 in fig. 4, an elevation of the surface taking place over a 

 depression of the stream-bed. 



