i. e. 



Deep Water Waves. 387 



1 gX 



j,[.(„-,')-20]. . . . (3 ) 



6^77 drj' 2ir> 



2| If' 



We must, further, have it^O, i.e. rj = fj, when z/ = oo,or 

 £=—ico. This ^condition is satisfied if! we assume the 

 expansion 



97 = ? + iA 1 e-*i-iA 2 e- 2 * + 



as the form of t] *. Making the assumption, and substituting)- 

 in equation (3), we find, as the equation to determine the 

 coefficients A», 



= -^ + [C + lA„cos nf] [^1 + JB^A,, cos ^j + / J wAftgin wf \ "J 



The terms of this equation must be re-arranged in cosines 

 of multiples of f, and then the coefficient of each cosine is to 

 be equated to zero. The general form of the equations so 



* I find it difficult to persuade myself that this is the only form of F. 

 It is possible, if we do not make this assumption, to satisfy equation (3), 

 together with the condition of rest at the bottom of the liquid, in an 

 infinite number of ways, and in certain cases the exact expression for 

 the function F may be obtained. In fact if we put ■q = -q 1 -\-Lr) 2 , where 

 r)i and 772 are real when £ is real, equation (3) is satisfied provided that 



y 



and the only remaining condition is that 7 — £, when £ = — too . A function 

 satisfying this condition is that given by the equation 



n 2 e wc2 ^=Acos/-2& 



which presents some remarkable points of resemblance to Stokes's wave. 

 Such " waves ; ' are, however, not in general such that the crests of all 

 the stream-lines are vertically under the crests of the free surface. 

 Stokes's solution is the only one which satisfies this condition. 



The wave considered in the paper " On the Highest Wave in Deep 

 Water " (Phil. Mag. Dec. 1913, pp. 1053-8) is of the type considered in 

 this note. Another example, perhaps even more curious, is that of the 

 steady motion represented by the equations 



c 



in which the complete cycloid obtained by making 6 a real quantity is a 

 free surface, for which \^ = 0. The fluid is, moreover, at rest at the 

 bottom, where y=— 00, 6——ioo. In this case there is no need, as in 

 the paper referred to, to "' tit on " various distinct arcs of the complete 

 curve. 



