106 Prof. A. P. Richardson on 



Part II. 



In this part a different form is given to equation (I). As 

 in Part I., the solutions obtained are exact, and suffer from 

 the same disadvantage, viz. that the rigid boundaries obtained 

 are not straight. 



I. Discussion of the method. 

 Evidently if both Y'(w) and { }i are real over y = yjr 



££ = / 1 -WHw)V 



dw \c 2 + 2gh-2<jF(w)-?(w) r W J 



+ iY(w)=-±e ie (1) 



will give flow, under gravity, with a free surface over 

 which y =y + F (w) , and pressure P(fctf). 



dz 

 Exclude all the singularities of -j- in the w-plane by 



small circles, and divide up the ^-plane into strips so that 

 these points lie on the boundaries of the strips. Then, so 



long as i/r is confined to a single strip, ~=^ is uniform. 



Different problems will be solved according to the par- 

 ticular strip or strips over which ifr is allowed to range. 



For example, in the problem of the surface-wave due to a 

 submerged body i/r would range over two adjacent strips*. 

 The next example illustrates the sort of considerations which 

 arise. 



II. Flow over a corrugated stream-bed. 



In (1) put P(w)=0, F (w)=\ cos fiw, and /3' 2 = c 2 + 2gh. 



dz f 1 "1 * 



.*. y— = < -== ?r^— \ 2 a 2 s'm 2 aw > — iXll sin uiv. 



dw LP 2a\ COS /jLW r~ r~ j i~ 



■ ■ ■ (2) 



The singularities in the finite part of the z^-plane are the 



r °° tS0f ^-2 9 \cosfjLW = (3) 



and l = \V(-l-X 2 )(/3 2 -2#A,X), ... (4) 



where X = cosyu,?/;. 



There will be no real roots of (3) if /3 2 >2g\. 



* I have not succeeded in constructing" a case of this kitid. — A. R. R. 



