14 Gw'YTHER, Rang-e e>/ Progressive Waves in Deep Water. 



6( J/e + .V,) = 2 H^\Ji6E,)[J,{6E,) - J,(6^J] 

 + U6E„)[JleE,) + J,{6E:)-\ 



- J,[6E,) - J,{6E,) - J„_{6E,)J,{6E,)} 

 + 2 H,{J,\6E,) + ^,(6^,) - ^4(6^4) + -^(6 E,)J,{6 E.^) 



-J,{6E.;)J,{6E,)] 



+ 2lQJ,{6E,)J,{6E.) - J.16E,) - JA6E.:)] 

 + 2I/,{J,{uE,)-J,{6E;)} 

 -2H,J,{6E,) 



M,-]S\ = . (20). 



The Numerical Values of the Coefficients in 

 THE Case of the Free Surface. 



From the conditions (7) which we have found for the 

 free surface we may find, from (8), the corresponding 

 values of the E's (which I shall write in small type when 

 b = 0) in terms of /r^. I shall also omit the suffix from hi, 

 since this is now the only one of the set retained. 



These values are found to be 



e^= 2h-¥ 2//+ lo/f, 



^6 = i^-^^« . . (21) 



Also, we shall have 



J^rey) = I - rVr -Ur^- -V^^ -(iir-r' + ^h\ 

 JXre^ =1 -r'h'- ^/z«, 



