SPHEROIDS, AND ON THE OCEAN TIDES UPON A YIELDING NUCLEUS. 23 
circumstances, a dynamical investigation of the effects of a tidal yielding of the earth 
on a tide of short period, according to the canal theory, is likely to be interesting. 
The following investigation will be applicable either to the case of the earth’s 
mass yielding through elasticity, plasticity, or viscosity; it thus embraces Sir W. 
Thomson’s hypothesis of elasticity, as well as mine of viscosity and elastico-viscosity. 
11. Semidiurnal tide in an equatorial canal on a yielding nucleus. 
I shall only consider the simple case of the moon moving uniformly in the equator, 
and raising tide waves in a narrow shallow equatorial canal of depth h. 
The potential of the tide-generating force, as far as concerns the present inquiry, is, 
--''It" sin 2 6 cos 2(</>— cot), where r=~ . This force will raise 
a) A ' Ac 6 
a bodily tide in the earth, whether it be elastic, plastic, or viscous. Suppose, then, 
that the greatest range of the bodily tide at the equator is 2E, and that it is retarded 
after the passage of the moon over the meridian by an angle Then the equation to 
the bounding surface of the solid earth, at the time t, is r—a- j-E sin 2 6 cos [2(<£— 
or with former notation cr=E sin 2 6 cos [2(<f> — cot)-\-e\. 
The whole potential V, at a point outside the nucleus, is the sum of the potential 
of the earth’s attraction, and of the potential of the tide-generating force. Therefore 
Esin 2 6>cos[2(^-^) + e]+^ sin 2 6 cos 2 (<f>-cot) 
=p^+{F cos [2(<£—&k) + e]-f G sin [2(<£—w£)-be]}^ sin 2 0 
3 T 
where F = ppE-f- cos e, G=- sin e. 
Sir George Airy shows, in his article on “ Tides and Waves” in the ‘ Encyclopaedia 
Metropolitana,’ that the motion of the tide-wave in a canal running round the earth is 
the same as though the canal were straight, and the earth at rest, whilst the disturb¬ 
ing body rotates round it. This simplification will be applicable here also. 
As before stated, the canal is supposed to be equatorial, and of depth h. 
After the canal has been developed, take the origin of rectangular coordinates in 
the undisturbed surface of the water, and measure x along the canal in the direction 
of the moon’s motion, and y vertically downwards. 
We have now to transform the potential V, and the equation to the surface of 
the solid earth, so as to make them applicable to the supposed development. If v be 
the velocity of the tide-wave, then coa=v ; also the wave length is half the circum- 
2 
ference of the earth’s equator, or na ; and let m =-. Then we have the following 
transformations :— 
with the old notation, 
