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214 MR. S. S. HOUGH OX THE APPLICATION OP HAKMONIC 
If we suppose that the free surface of the ocean is a spheroid of revolution about 
the axis Oz, it will be convenient to refer to a system of spheroidal coordinates /x, (f), v 
1 elated to x, y, z by the equations 
a: = c ^{1 + v^(l — cos cf), 
y = c y(l y(l — y?') sin (f), 
z = cuy. 
The line-element ds for this system of coordinates is given by 
7 0 d(v~ + y^^) 7 o , 9 /-. I 0\ /o 9,\ 7 lO | ^ (v” + /X") ,9 
ds~ = ^ _ dy- + c® (1 +v^){l — y^) dcfj- + dv^ ; 
w 
hence, if we identify y, (f), v with a, /3, y respectively, we have 
^ _ C \/(r" -t y~) 
h, ~ v^(I - y^ 
T = C v/(i + v") x/{l — y^), 
'^2 
jL _ c + y~) ^ 
Ag \/(l + v~) 
and, supposing that is the equation to the free surface, the equation (II) 
becomes 
^ ^ 1 J //I _ 21 7 TTI - 1 1 [ hY 1 
0/ c\/(y" -f- v) 0/x p ' ^ ^ J c^(vd + 1) d(f) iv/(l — y~) i 
We have already neglected on the right smah terms of the order h compared with 
those retained; we now propose to make the further hypothesis that the spheroidal 
surface of tlie ocean is of small ellipticity e. In this case c will be small and Vq 
large, in such a manner however that cvq is finite and equal to the polar radius a ; 
further Ipo” approximately equal to 2e. Hence we find 
07 
ri 
1_0/X 
P(1 -P)AU -f 
h V" 
— I 
dcf> lv/(l -P)JJ 
where the terms omitted on the right are of order h and of order e compared with 
those retained. 
^ 4. 'Transformation of the Dynaniical Equations. 
Let 6 denote the Inclination to the axis of 2 of the normal to the surface v — const, 
through any point ; then the direction-cosines of this normal will be 
sin 6 cos <^, 
sin 6 sin <h, cos 6 ; 
