Waves at the Surface of Deep Water. 383 



to which yjr is subject, whatever may be the values of the 

 constants a, /3, &c. And, much as before, we shall find that 

 the surface condition can be satisfied to the order of a 7 

 inclusive : {3, 7, 8, e being respectively of orders a 4 , a , a 6 , a 7 . 

 We suppose that the free surface is the stream-line yfr — 0, 

 and the constancy of pressure there imposed requires the 

 constancy of U 2 — 2gy, where U, representing the resultant 

 velocity, is equal to */ {(d^'dx) 2 - r [dyjr/dy) 2 \ 9 and g is the 

 constant acceleration of gravity now to be determined. Thus 

 when ^r = 0, 



V 2 -2gy = 1 + 2(l-(f)y + a 2 e- 2 * + 2j3e~^ cos 2x 



_j_ 4^-% cos 3#+6Se~ 4 ^ cos 4# + See~ 5 ^ cos ox 



+ ±*fie-*y cos x + 6arye- 4 y cos 2# + 8aS<r^cos Sx . (2) 



correct to a 7 inclusive. On the right of (2) we have to 

 expand the exponentials and substitute for the various 

 powers of y expressions in terms of x. 



It may be well to reproduce the process as formerly given, 

 omitting S and e, and carrying (2) only to the order a 5 . We 

 have from (1) as successive approximations to y : — 



y = ae~ y cos .!'=:« cos x ; (3) 



y = a(l — y) COS x= — \ a 2 -f a cos X — \ a 2 cos 2x ; . (I) 

 y=a(l — y -+-t5_y 2 )cos X 



a? , /' , 9a!\ a 2 3 a' J 



= — - + a ( 1 + 77- ) cos # — —cos 2a- + — cos 3a?, (0) 



which is correct to a 3 inclusive, /3 being of order a 4 . In calcu- 

 lating (2) to the approximation now intended we omit the 

 term in ay. In association with a/3 and y we take e~ 3y =l ; 

 in association with ft, e~ 2lJ = \ — 2y ; while 



Thus on substitution for y 2 and y 3 from (5) 



«\>-^ = a 2 !l-2j/ + a 2 -^ 3 COS l i' + a 2 COs2.i'-Ja 3 COs3 t l'}. 



In like manner 



2f3e~ 2lJ cos 2x = 2/3 cos 2x — 2«/3(cos x + cos 3x). 



Since the terms in cos x are of the fifth order, we may 

 replace <x cos x by y, and we get 



U 2 -2^ = H-a 2 + a 4 + 2 /7 (l-(7-a 2 -2a 4 + /3) 



+ (a 4 + 2 y e)cos2.i' + (-ta ;i + 4 7 -2«/3)cos3.r. . (6) 



2 D 2 



