110 On Stationary Waves in Water. 



The stream-bed i/r = — has two singularities, viz. at 



<f)=-\ and a depression will occur under the highest 



— jjua : r ° 



part of the wave. 



If «i</3 a similar 'motion will ensue, but the stream-bed 

 will have only one singularity and will be elevated under 

 the highest point of the wave. 



Case (b). If 2a 2 > 1, i.e. c 2 >2cjh, the motion with a free 

 surface of this form is impossible : the discussion follows 

 the same lines as in the previous example. 



IV. Progressive waves in deep water. 



It is not without interest to obtain the expressions given 

 by Stokes (Lamb, 'Hydrodynamics,' art. 230). 



Interchange the variables w, z by writing ~F(w) = H(z), 

 (1) becomes 



^={l-2iK'(z)}i{c* + 2gh-2 9 K(z)}i, . (11) 



and will include cases of flow with a froe surface. 



There will be no singularities in the finite part of the 

 ?#-plane if 



l-2iR'(z)=\{c* + 2 9 h-2 9 K(z)}, 



i.e. it H = C + Be-^, 



Le.it dw =^(l~2iK'{z)), 



dz v\ 



'ci 



i. e. (f) — a / j- ai + fte 1 ^ sin hx 



^ = a / f y + (3e kl J cos he 



(12) 



here g\ = Jc. 



V. Conclusion. 



The above analysis shows that although the problem of the 

 flow of liquid under gravity with a free surface, and given 

 form of rigid boundary, is very difficult of solution, yet the 

 main characteristics can be determined by noting the form 

 of the free surface in such cases and using the methods of 

 this paper. 



My thanks are due to Prof. A. R. Forsyth and Prof. A. 

 N. Whitehead for their assistance. 



