192 
PACIFIC SCIENCE, Vol. IX, April, 1955 
Elimination of y- Co-ordinate 
The equation to be solved is from (28) 
D] (d 4 
2tt 2 W 
h. I 4ctl\ _ IL ( i 4dtl\ „ CQS t ^1 ■ 2 / dr x dr A 
c 4 ^ dy 4 ) 8 \dx 2 “ 1 " dy 2 ) R dx^pwXdy dx ) K } 
where 
A 
4 
A i 
poo 
(36) 
Between the equator and 60° N or S, cos $ varies from 1 to 3^2- Since this is not a large 
variation, we may take the average value for cos 4> for this range, or 
cos <f) = 
■ r 
2i r/3 J 
iji 3V3 
cos 4>d<f) = — — 
2?r 
Thus, the third term in (35) now becomes approximately 
cos <j) d\f / 1 
3V3 djn 
2tt R dx 
R dx 
Substituting (30), (32), (34), and (37) in (35), we have 
1 
D) 
d“N,. 
L2x 2 UzjlYR 1 ' dV 
x 3 ' 
+ N m (\\ T]) — 
d i N, 
R 4 d<f>' 
1 
d 2 N, 
8 • dX 2 
v 3 x 
. ilf m (0) + N m (\; rj) — 
d 2 Mr 
+ AT m (X; ri) 
1 
d 2 M 7 
R 2 dtf 
3^ 3 
~2^r 
2 *R 2 
dNr, 
d \ 2 
i? 2 df>‘ 
— — E T 
where we have 
d 2 M m 
dtf 
X = 
= 0 
(2tt R/3)’ 
(37) 
(38) 
(39) 
so that the ocean is supposed to be bounded by the meridians of 0° and 120°, and the 
western and eastern boundaries are given by X = 0 and X = 1 respectively. 
Now it is possible to express ” f--- and m terms of M m ($), or 
and 
d 2 M 7 
* dy 2 
d 2 M ri 
dy 2 
d 4 M* 
dy - 
Substituting (40) in (38) we have 
dy 4 
= aTMi(4>) 
= Z PTMiti). 
(40) 
