Newman 



-1/2 

 masked by the first-order potential cji - 0(R ). 



Extending these results to third-order involves straightfor- 

 ward but tedious analysis. The third-order free-surface condition 

 is analogous to (13), but involves more terms on the right-hand 

 side: 



- «('ix(<l^2zr+ ^'gxxz) (15) 



A particular solution, anadogous to (14), is given by the triple integral 

 ^t-rr ^Zw -Ztt k *ik x 



c^3= \ de, \ de^l dGjfO,) fOg) fOj) w(e, ,e2,e3)e '" '^^ (i6) 



Jq Jq -Jq 



where 



k = k(e ) + k(e ) + k(e ). 



-123 -^ r -^ 2 -^ z' 



The weight function W(0 ,9,0) is singular at points where its 

 denominator vanishes or where 



k,23 - (sec e, + sec d^ + sec Q^ = (17) 



and it is necessary, therefore, to study the roots of this equation. 

 It can be shown that the strongest singularities occur along the cusp 

 line; for example, at the point 



e^ = 9^= 63 - TT = 35°16'. 



The integral (16) is improper at these points and we, therefore, 

 conclude, as in many linear wave problems, that a steady- state 

 solution cannot be assumed a priori, but must be derived as the 

 appropriate limit of an initial value problem. 



An expedient initial value problem is obtained by regarding 

 the right-hand side of (15) as a pseudo-pressure distribution, im- 

 posed on the free surface from an initial state of rest, and then 

 looking for the steady-state limit which results. To avoid unneces- 

 sary algebra we rewrite (15) In the unsteady form 



536 



