12 G\\'YTU.EY<., Range of Progressive Waves in Deep Water. 



- 2\Jf,{sEi) + JiiS^i) + ^1(5^1)^2^5^2) 



+ '>^.(5^x)'>^i(5^3)} 

 6^6 = 2 ; J6(6^i) + ^2(6^2) + J^{6E,)J^{6E,) 



+ J,{6E,),M6E,) + J,{6E,)J,(6E,) + J,{6E,)J,{6E,)) 



- 2\Jr{6Ee) + M6E,) + J,{6h\)[J,{6Ei) + J,(6E,)J,{6E,)] 

 + J,{6E,)J,{6E,)\ (17) 



The coefficients Jl/^ and /V,. in (11) and (13) can best 

 be found in conjunction, thus 



~{M,+ K) = J :^/J,SOs[{r-m)f + r{E,s\n<p+ ...)\di>...{iS) 



and 



Trr C 



—{M,.-N,^= / 2://,„cos;(r + ;«)0 + r(£'isin^ + ...)j^0--(i9) 



and the values of each integral can readily be written 

 down by means of (15) and (16). 

 1 thus obtain 



JA + iVi= 2H,JXE,)JXE.^ 



^2HJ,{E,)[i-J,{E.;)\ 



+ 2B,{ME\) + J,{E.^], 

 M, - N, = 2H,\J,{E,) - J,{E,)J,{E,) - JXE,)J,{E.^] 



+ 2B,\J,{E\)J,{E.^ - J,{E,) - J,{E.^] , 



2{A/, 4- A.^ = - 2B, [J,{2E.^[J,{2E,) + J,{2E,) + J,{2 E,)] 



+ J,{2E,)JX2E.;} + J,{2E\)J,{2E,)} 



^2H.Jl2E^JX2E.^ 



,2llM2E,)\l-Jl2E.^\ 

 + 2H,\J.l2K,) + J,{2E.^\ 



