1 6 G\NYTll^R, Range of Progressive Waves zn Deep Water. 



^,=j/,=.y,=^« = j/6=iv;=o . (23). 



Indicating by the suffix that the symbols refer to 

 the free surface, we have now obtained these values 

 correct to the proposed degree of approximation : 



Jat^ = ^„ = ^„ - (3// 4- ih' + -jV/5) sin ^<, - {\h' - HM^) sin 2^, 



- gV/i^ sin 3/z„ - ^io//'' sin 4^„ . . (24). 



kx, = ix, - {h + 1// + ^th') sin ^, - (1 /^* - \yf) sin 2;., 



- fV/= sin 3^„ - ^V^/' sin 4^„ . . (25). 



ky^ = - (/^ + yf + ?lh') cos ^„ - (1/^* - U^') cos 2;x„ 



- ^Vi^ cos 3^,, - T^^/z'' cos 4^/„ . . (26). 

 where, from (8), 



^„ = k{ct + «)/( I + /r + 4/^' + '^^-h'), 



We also have the relation, from the surface-conditions 

 in (7), that 



and we can readily verify that the surface-condition, in the 

 form that 2gy^ — ai^ shall be constant, is satisfied to the 

 degree of our approximation. 



The noteworthy features which the results just 

 obtained present to us are their striking simplicity as 



