In the Reader for January 13, 1866, I advanced an objection to the submergence theory on the grounds that the lowering of the ocean-level by the evaporation of the water to form the ice-cap, would exceed the submergence resulting from the displacement of the earth’s centre of gravity. But, after my letter had gone to press, I found that I had overlooked some important considerations which seem to prove that the objection had no real foundation. For during a glacial period, say on the northern hemisphere, the entire mass of ice which presently exists on the southern hemisphere would be transferred to the northern, leaving the quantity of liquid water to a great extent unchanged.
Note on the preceding by Sir William Thomson, F.R.S.
“Mr. Croll’s estimate of the influence of a cap of ice on the sea-level is very remarkable in its relation to Laplace’s celebrated analysis, as being founded on that law of thickness which leads to expressions involving only the first term of the series of ‘Laplace’s functions,’ or ‘spherical harmonics.’ The equation of the level surface, as altered by any given transference of solid matter, is expressed by equating the altered potential function to a constant. This function, when expanded in the series of spherical harmonics, has for its first term the potential due to the whole mass supposed collected at its altered centre of gravity. Hence a spherical surface round the altered centre of gravity is the first approximation in Laplace’s method of solution for the altered level surface. Mr. Croll has with admirable tact chosen, of all the arbitrary suppositions that may be made foundations for rough estimates of the change of sea-level due to variations in the polar ice-crusts, the one which reduces to zero all terms after the first in the harmonic series, and renders that first approximation (which always expresses the essence of the result) the whole solution, undisturbed by terms irrelevant to the great physical question.
“Mr. Croll, in the preceding paper, has alluded with remarkable clearness to the effect of the change in the distribution of the water in increasing, by its own attraction, the deviation of the level surface above that which is due to the given change in the distribution of solid matter. The remark he makes, that it is round the centre of gravity of the altered solid and altered liquid that the altering liquid surface adjusts itself, expresses the essence of Laplace’s celebrated demonstration of the stability of the ocean, and suggests the proper elementary solution of the problem to find the true alteration of sea-level produced by a given alteration of the solid. As an assumption leading to a simple calculation, let us suppose the solid earth to rise out of the water in a vast number of small flat-topped islands, each bounded by a perpendicular cliff, and let the proportion of water area to the whole be equal in all quarters. Let all of these islands in one hemisphere be covered with ice, of thickness according to the law assumed by Mr. Croll—that is, varying in simple proportion of the sine of the latitude. Let this ice be removed from the first hemisphere and similarly distributed over the islands of the second. By working out according to Mr. Croll’s directions, it is easily found that the change of sea-level which this will produce will consist in a sinking in the first hemisphere and rising in the second, through heights varying according to the same law (that is, simple proportionality to sines of latitudes), and amounting at each pole to
(1 - ω)it/1 - ωw,
where t denotes the thickness of the ice-crust at the pole; i the ratio of the density of ice, and w that of sea-water to the earth’s mean density; and ω the ratio of the area of ocean to the whole surface.
“Thus, for instance, if we suppose ω = ⅔, and t = 6,000 feet, and take ⅙ and 1/(5½) as the densities of ice and water respectively, we find for the rise of sea-level at one pole, and depression at the other,
⅓ × ⅙ × 6000/1 − 2/3 × 1/5½ ,
or approximately 380 feet.
“I shall now proceed to consider roughly what is the probable extent of submergence which, during the glacial epoch, may have resulted from the displacement of the earth’s centre of gravity by means of the transferrence of the polar ice from the one hemisphere to the other.”