Waves at the Surface of Deep Water. 385 



The next step is the further development of the pressure 

 equation (2), so as to include terms of the order a 7 . "Where 

 /3, 7, &c. occur as factors, the expression for y to the third 

 order, as in (5), suffices ; but a more accurate value is 

 required in a 2 e~ 2y . Expanding the exponentials and re- 

 placing products of cosines by cosines of sums and differ- 

 ences, we find in the first place 



\j*-%jy^l-g-OL 2 )y + l + u 2 + **+^^--4tc?p 



+ COS X - — 4a-> + 2a/3 + r^- — —,— ' \ 



{ - b 2 J 



■f cos 2x | « 4 + 2(3 + ijp -2u2j3 | 



-r-eos3tf j ~ -«- -2a/3 + 47— -j-- + —r-^- + 3« 2 7 — 4aS j 



-t-cos4.r j -r^ h 4 2a 2 /3— 0*7+65 [ 



., ( 37a. 7 25a?B , 15a 2 7 „ , - _ 1 ,,„_ 



+ cob 5* j --go 12 + "^2 ~ 12aS + 8e r ^ 



From the terms in cos x we now eliminate cos x by- 

 means of 



a COS 07: 



/, 9« 2 \ , a 2 « 2 

 3^1- _j+ -+-cos2#, 



thus altering those terms of (12) which are constant, and 

 which contain y and cos 2x. Thus modified, (12) becomes 



\J 2 -2gy = l + «? + ** + ^ - 3« 2 /3 



+ 2^|l-^-a 2 -2 a 4 + /3-7 a «4- 24 4 J 



+ cos 2.i- { **+ 2/3+ ^ - * 2 /3 } 



+ cos 3# j — — - 2oiB + ±y- — + — -p + 3a 2 y ~ 4a$ } 



f 1 9a' 3 1 



+ co> 4;c -j p- + 2a 2 £ - G«7 + 63 [ 



+ cos o.i 1 1 — -^g- - -y^~- + - v - 12aS + 8e I . (13) 



The constant part has no significance for our purpose, and 

 the term in y can be made to vanish by a proper choice of g. 



