1SG WAVES IN LIQUIDS. [CHAP. VII. 



sions x, y ; and to be such as may have been generated from rest, 

 so that there exists a single -valued velocity -potential &amp;lt;f&amp;gt;. We 

 retain, for a first approximation, the assumption that the squares 

 and products of the velocities and relative displacements may be 

 neglected. Our equations then are 



with the boundary conditions 



% + ff + ff 



at ax dx dy dy 

 at the free surface [(10) Art. 10 J, and 



(21) 

 (21) 



(22) &amp;gt; 



when y=h. 



Now (21) is satisfied by the sum of any number of terms of 

 the form 



each multiplied by an arbitrary function of t, provided f + & 2 = 0. 

 Now j must in our case be wholly imaginary, for otherwise we 



should have -~- infinite for either a; = + oo , or x oo . Hence 

 doc 



we must have k real and j = ik. We write therefore 



&amp;lt;/&amp;gt; = 2 {e** (A cos kx + B sin lex) + e~ k * (A cos kx + B sin kx)} t 



the coefficients being functions of t as yet undetermined. The 

 condition (24) gives 



Ae-* h = A e kh , Be~ kh = B e kh , 

 so that the assumed value of &amp;lt; takes the form 



= 2 [e^y + + e-*&amp;lt;* +J }(P cos kx+Q sin kx) ...... (25). 



So far our work is rigorous. If we now neglect squares and pro 

 ducts of small quantities, (23) becomes on substitution from (22) 



