DR. HAROLD JEFFREYS ON TIDAL FRICTION IN SHALLOW SEAS. 
263 
straits has been systematically ignored in this paper, except where it has been auto¬ 
matically taken into account in the determination of the excess of the entering over 
the issuing energy. The chief reason for this is the utter impossibility of finding 
it. The fjords on the west coasts of Norway, Greenland, and North and South 
America are innumerable, and in many of them, perhaps in all, there is a strong tidal 
current, so that the dissipation per unit area in these places must very much exceed 
that in any of the areas here treated. On the other hand, the total area must be less, 
and it is uncertain whether the increase in velocity is enough to counterbalance the 
decrease in area and make the total dissipation in these places comparable with that 
here found for the larger shallow seas. The same is true of shoals ; though the 
agreement between the results given by the two methods of finding the dissipation 
in shoaly waters, as in the Yellow Sea and the Strait of Malacca, indicates that the 
shoals at any rate do not contribute to the dissipation an amount overwhelmingly 
greater than the normal places, for one method necessarily includes the effect of the 
shoals and the other systematically omits it. Along the open shore again there 
must be some dissipation ; the currents there do not usually extend many miles out to 
sea, but they exist along a very long stretch of coast, and the aggregate dissipation 
in them may be appreciable. 
The hypothesis that the secular acceleration of the moon is due to dissipation of 
energy in the tides in shallow coastal regions therefore seems capable of satisfying all 
the quantitative demands on it, and it is also free from objections that have been 
urged against other attempted explanations.* It therefore occupies a strong position. 
Appendix. 
The Secular Change in the Obliquity of the Ecliptic. 
In consequence of the dissipation of energy in the diurnal tides there must be-a 
couple always acting on the earth so as to tend to resist its angular motion about an 
axis in the plane of the orbit of the moon or the sun, as the case may be. If 8 be the 
declination of the moon, the angular velocity of the earth about the diameter that 
points to the moon is O sin 8, and the angular momentum about it is CD sin 8, where C 
is the earth’s moment of inertia. Let L be the couple about this diameter. Then the 
rate of dissipation of energy in the diurnal tide is LD sin 8. Also the rate of change 
in the inclination is given by 
Now L must contain fl sin 8 as a factor, since it depends for its existence on the 
existence of the diurnal tide, whose coefficient is proportional to sin 8, and whose speed 
* Cf. ‘ Monthly Notices of R.A.S.,’ vol. lxxx., 1920, pp. 309-317. 
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