530 EIGHTH PACIFIC SCIENCE CONGRESS 



In order that this relation should be always valid, the coefficients 

 of M^ {(f) must vanish. Thus we have 



81 Df dW„, _9 ^ dm^ 9^/5 dN^ 



32 W 'dX' 32^ "'^ dX^ "" 4^^ dX 



+ S( - -g-^'?' + ^ ^-:^)^' + T^rSak^i = (42) 

 where Tj is given by (34). 



It may be anticipated from (39) and (40) that the values of ^^ 

 are larger than that of a|.^for any corresponding set of values of i and 

 m. But the ratio 



R^ ~ po) R^ 

 is of the order of 10-' for the usual value of ^^ (=10^), so that we 

 may neglect the terms of /?» in the coefficients of N^ in (42). Then 

 (42) becomes 



81 D\ d'^N^ 9_ ^ dm^ 9Vr dN^ _^ Va^ A^ 



32 'R^~dX} 3277^ '^^ dX' ~ iir^ dX Si"""' 



+ _1 V a^Tj - 0. (43) 



i 



7. Determination of the Values of «' 



m 



The coefficients of M^^ (^) in the expansions of ~ were eval- 



d(^'^ 



uated as far as Mio(^), the higher terms being neglected. 

 For the ocean under consideration, we have 



M-{^) = - _±\ ^^A4>) + 8Af3(<^)- 6M,(,^) 

 11 1 + 4M,(<^) - 2M,(4>) 



9 1 - 3M,(c^) + 20Mo(</,) + ISM^icf) 



— 12M, (<^) + 6Ma (</)) 



9 I 



M-U\ - - ^J "^ ^'^^^^'^^ ~ ^^^^s(^) + 69M,(</,) 

 b(<^) -jX^ _^ 20M,((/.) — 10M9(</>) 



, _ _9^J - 7M, (<^) + 2IM3 (<^) - 35M, (4>) 



^h{4>) - 11"^ ^ 148M,(<^) + UM,{<}>) 



_ _ _9_J + 9M,(^) - 27M3(<^) + 45M,(</>) 



^^9(«^) - 11^ - 63M,(<^) + 257M9(<^) 



and 



3 I 7M,(cp) + 16M,{cj>) — 12Me(</,) 



M'/ (<^) = ; 



(44) 



