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Mr. J. EL C. Searle on the 



where the bed is strictly uniform they have been placed at 

 the end of the work. 



§ 2. The figure below represents a vertical section of the 

 stream taken parallel to the banks, the motion being two- 

 dimensional. 



z is measured upwards from the bed of the lower stream. 



A A' is the free surface (/^i); 



BB' ,, stream-bed (^n); 



CC is any stream-line (yjr); 



Ji is the depth of the upper stream; 



d „ total drop of the stream-bed; 



U „ surface velocity at A; 



u „ velocity at C at a height z. 



By accenting these quantities we obtain their values on the 

 lower stream-bed, as shown. 



A, d, U, and u ( = function of z) are supposed given; 

 h', U', and u' are to be found. 



Let p be the pressure: then 



At A', Tr^X^-^'-iU' 2 . 

 Hence, TJ' 2 -\J 2 = 2 9 (h + d-h'). 

 Put L T '/U = i>, 2gh/TJ*= K , 1+J/A=a, K'/h=v, 



.'. l>"-l=S:*(8-tT) 



• (1) 



■ (2) 



Here tc and 8 are given, while v and vr are unknown. 

 Let now -\jr-\-dyfr be a stream-line in the neighbourhood of 

 ijr; and let 



s-frfs be its height in the upper stream; 

 z + <£?' be its height in the lower stream. 



