-10 MR. S. S. HOUGH ON THE APPLICATION OF HARMONIC 
But by detiiiition of xp we have 
pjp = const + V' + ^ oi® (a;- + 'if) —xp 
= const + V'o + v' + a + 1 .(7), 
where V'q denotes the potential at x, y, z in the steady motion, v' the potential due 
to the attraction of the layer of fluid contained between the actual free surface and 
its mean position, and v the disturbing potential, which may be regarded as due to 
some external attracting system. 
Since in (6) 'dpfii is already associated with the small factor in calculating cpjoii 
we may omit all small quantities of the order and thus replace this expression by 
its value in the steady motion. But from (7) we have in this case 
whence 
pip — const + Vh + ^ 0 )' [x^ + y~), 
1 
P 
C2J 0 
0/f d/I 
> (v; -b 
1 2 
{x^ + f) 
} 
s- 
Now, since the free surface of the ocean must be an equipotential surface, the 
resultant of gravitation, including centrifugal force, must be perpendicular to this 
suriace. Denoting by <j its value, we have 
and therefore 
U — 
~ pp + f) 
L It 
■) 
1 dp _ 
p dll' 
( 8 ). 
Introducing the values ofp, from (7), (8) into the expression (6), and equating 
0 Ih 
the latter to a constant, we And 
[ ^' u “1" H" ^ “h y") ~ ~ — const, 
or, on equating periodic parts to zero, 
i/; = v' — g 'C-\- V 
(9), 
vhere the bars are used to denote surface-values at the undisturbed free surface. 
