II. TIDE-GENERATING FORCES (cont.)

9. Correction to Equilibrium Theory for Continents

In the equilibrium theory as worked out by Newton and Bernoulli it is assumed that the figure of the ocean is at each instant one of equilibrium under the action of gravity and of the tide-generating forces. Sir W. Thomson has, however, reasserted [Footnote 358-2] a point which was known to Bernoulli, but has since been overlooked, namely, that this law of rise and fall of water cannot, when portions of the globe are continents, be satisfied by a constant volume of water in the ocean. The law would still hold if water were appropriately supplied to and exhausted from the ocean; and, if in any configuration of the tide-generating body we imagine water to be instantaneously so supplied or exhausted, the level will everywhere rise or fall by the same height. Now the amount of that rise or fall depends on the position of the tide-generating body with reference to the continents, and is different for each such position. Conversely, when the volume of the ocean remains constant, we have to correct Bernoulli’s simple equilibrium theory by an amount which is constant all over the globe at any instant, but which changes in time. Thomson’s solution of this problem has since been reduced to a form which is easier to grasp intelligently than in the shape in which he gave it, and the results have also been reduced to numbers.[Footnote 358-3] It appears that there are four points on the earth’s surface at which in the corrected theory the semi-diurnal tide is evanescent, and four others where it is doubled. A similar statement holds for the diurnal tide. As to the tides of long period, there are two parallels of latitude of evanescent and two of doubled tide.

Now in Bernoulli’s theory the semi-diurnal tide vanishes at the poles, the diurnal tide at the poles and the equator, and the tides of long period in latitudes 35° 16' north and south. The numerical solution of the corrected theory shows that the points and lines of doubling and evanescence in every case fall close to the points and lines where in the uncorrected theory there is evanescence. When in passing from the uncorrected to the corrected theory we speak of a doubled tide, the tide doubled may be itself nil, so that the result may still be nil. The conclusion, therefore, is that Thomson’s correction, although theoretically interesting, is practically so small that it may be left out of consideration.





Footnotes

358-2 Thomson and Tait, Nat. Phil., § 807.

358-3 Darwin and Turner, Proc. Roy. Soc., 1886.



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