SPHEROIDS, AND ON THE OCEAN TIDES UPON A YIELDING NUCLEUS. 15 
The two latter terms give rise to two tides, in one of which (according to previous 
notation) 
3 012/ 
S cos(r£ + 77) = ^ - sin 3 6 cos 2<f> cos 2crt, 
and in the second of which 
S cos (vt-\-7])= — sin 3 6 sin 2 <j> cos( 2cot-\-- 
diurnal tide is where tan e= —= —— 
4w cr gciw 
Now e, which depends on the frequency of the tide generating potential, will clearly 
be the same for both these tides ; and therefore they will each be equal to the corres¬ 
ponding tides of a fluid spheroid, reduced by the same amount and subject to the 
same retardation. They may therefore be recompounded into a single tide; and 
since v will here be equal to 2oj, it follows that the retardation of the bodily semi- 
Also the height of the tide is less than the 
corresponding equilibrium tide of a fluid spheroid in the proportion of cos e to unity. 
Similarly by section (6) the height of the ocean tide on the yielding nucleus is given 
by the corresponding tide on a rigid nucleus multiplied by sin e, and there is an accelera- 
77" 6 
tion of relative high water equal to —-—. 
The case of the fortnightly tide is somewhat simpler. 
If 11 be the moons orbital angular velocity, and I the inclination of the plane of the 
orbit to the earth’s equator, then the part of the tide generating potential, on which 
the fortnightly tide depends, is—- 
Q uni 
g ^ wr % sin 3 1 — cos 3 6) cos 2 £lt 
and we see at once by sections (5) and (6) that tan e = ——. The bodily tide is the 
tide of a fluid spheroid multiplied by cos e ; the reduction of ocean tide is given by sin e ; 
and there is a time-acceleration of relative high water of or i — — of a 
& 4f2 2rt 2 7T 
week. 
In order to make the meaning of the previous analytical results clearer, I have 
formed the following numerical tables, to show the effects of this hypothesis on the 
semidiurnal and fortnightly tides. The coefficient of viscosity is usually expressed in 
gravitation units of force so that the formula for e becomes, tan e = ^ VC °- . In the 
VJCL 
tables v is expressed in the centimetre-gramme-second system, and in gravitation units 
of force ; a is taken as 6‘37 X 10 8 , and w as 5’5, and the angular velocity co of the moon 
relatively to the earth as ’00007025 radians per second. 
With these data I find v — LO 13 X 2’625 tan e. As a standard of comparison with 
the coefficients of viscosity given in the tables, I may mention that, according to some 
