MR. S. S. HOUGH ON THE APPLICATION OF HARMONIC 
;iG 
/XU + 2oN cos e = h, " 
C/U. 
/XV - 2aj (U cos - W sm /o I-.(13). 
/XW — '2a)Y sin 0 = ho 
The equations in tlie form we have just written them will hold good whatever be 
the de]:)th of the ocean or the ellipticity of its surface. We now proceed to introduce 
a})proximations similar to those of the last section. 
_ 
In the first j^lace we have, as in § 2, W = , and this by (12) we see is of the order 
h/a compared with IT or V. Hence, omitting terms of the order kja, and of order e, 
compared with those retained, the equations (13) take at the surface the approximate 
foi’ms 
iXU + 2wV cos 6 = 
/XV — 2ajU cos 9 = 
- 2coY sin ^ = XV 
\/ (1 — vb dy 
a dfj, ’ 
1 Oxfr q 
«v/(l -gb , 
d-yfr 
■ ■ (lU 
where, in conformity with the notation of ^ 2, we have denoted by cxjjjdn’ the rate of 
variation of xp in the direction of the normal to the surface of the ocean. 
From the equations (14) it appears that dxfj/cu is a quantity of the same order of 
— 3 
magnitude as xjj/a ; also if we apply the operator to each of the equations (13), 
we shall obtain equations which enable us to express dU/dn', dYjcn, oW/3n' in terms 
of U, V, W, and the surface-values of \jj and its differential coefficients. A little 
consideration will show that in general 0U/0»', 0V/0?f, 0XX"/0/f must be of the 
order U/«, Y/a. 
Now the approximations introduced in the last section will hold good provided 
that we may neglect 
(ro - Ti) 
j_a7 
and (yo - Ti) 
in comparison with [U/Ao/v]’ 5 io our present notation, that 
neglect 
h ^ {cv (e + V'' (1 - f) ej aiui h A 
cYv^ + V 
\/ (1 “ V ) \/ (1 + 
we may 
