Canon Moseley's objections to the received theory be admitted. as valid, they will be found to apply with equal force to his own. theory. Observation abundantly proves that the resistance offered by the bed of great glaciers to the sliding of the lower surface is very much less than even the smallest amount of shearing-force derived from his several discordant observations. If the mass of the glacier were dragged onwards, as he contends, by the alternate expansion and contraction of the superficial strata, it is certain that the lower surface would slide on its bed at the same rate as the upper part. It is equally easy to show that the differential motion found in going from the central part to either bank would be an impossibility. The whole would move forward as one mass, like the sheet of lead upon the roof, and only where some small lateral portion of the ice encountered a fixed obstacle would the resistance lead to fracture-just as in Canon Moseley's cylinder experiments. The larger mass would move on, the smaller fragment would be left behind.

If I might presume to estimate the net results of this renewed discussion of the causes of glacier-motion, I should say that they are not considerable, but yet are far from worthless. Canon Moseley's experiments have added something to our knowledge, and especially those on the tenacity of ice, which have some bearing on the origin of crevasses. Of far greater importance are the observations on ice-planks made by Mr. William Mathews. The first of these, published in the Alpine Journal,' gave prominence to a fact which had long been familiar to myself, and probably to many others. I have often found that long icicles placed in an inclined position, and supported only at the upper end, will gradually resume the vertical direction, and I had, perhaps too lightly, assumed that this was a particular instance of the process by which ice changes its form through fracture and regelation. In Mr. Mathews's first experiment, conducted during a thaw, a thick plank of ice supported at each end was deflected at the middle through a space of 7 inches in as many hours. Although none but very minute fissures were observed, the facts did not seem to me altogether inconsistent with that explanation. In the second series of observations, made during the severe frost of February last, Mr. Mathews found that at temperatures notably below the freezing-point a plank of ice, supported as before, subsides slowly between the points of support under the sole influence of its own weight. The deflection under these circumstances was about 1 inch in twenty-four hours. Taking this observation in connexion with a multitude of facts recently brought to light, and especially the researches of M. Tresca, we are led to admit that ice, in common with very many apparently rigid bodies, does possess a certain degree of plasticity which is

exhibited by changes of form effected very slowly under the action of forces of moderate amount, rather than by the rapid action of more powerful agencies.

The admission of this conclusion may slightly modify, but will not materially alter, the views now generally held as to the causes of glacier-motion, which are mainly derived from the remarkable researches of Professor Tyndall. Whatever may be the final judgment of men of science, I feel quite sure that it will not confirm the opinion expressed by Canon Moseley in his latest publication: that" the phenomena of glacier-motion belong rather to mechanical philosophy than to physics." Every real advance that has been made towards the explanation of those phenomena has been due to the application of increased knowledge of the physical properties of glacier-ice; and if any thing be wanting to complete the explanation now generally accepted, it must be derived from such additional acquaintance with those properties as may be derived from continued observation and experiment.

II. Reply to M. Delaunay's objection to the late Mr. Hopkins's Method of determining the Thickness of the Earth's Crust by the Precession and Nutation of the Earth's Axis. By Archdeacon PRATT, M.A., F.R.S.

To the Editors of the Philosophical Magazine and Journal. GENTLEMEN,

IT

T is only two days ago that I saw for the first time M. Delaunay's strictures* upon the late Mr. Hopkins's method of ascertaining the least thickness of the Earth's Crust by means of the phenomena of Precession and Nutation, although I had previously seen a notice that such strictures had been laid before the French Academy. Having now that gentleman's paper before me, I write to endeavour to convince your readers that the point of Mr. Hopkins's reasoning has been altogether missed, and that his method stands altogether unimpaired by these strictures.

2. I am ready to allow, and so would Mr. Hopkins have allowed, that if the crust of the earth revolved round a steady axis, always parallel to itself in space, and if at some particular epoch a difference existed between the rate of movement of the crust and of the fluid within it, the resulting friction would gradually destroy this difference and bring about a conformity in the motion of both parts. I will even go further, and allow that the effect of the internal friction and viscidity of the fluid may be such that the resulting rotary motion may be the same as that which the whole mass would have had at the epoch if it * Translated in the Geological Magazine, November 1868, p. 507.

had suddenly become one solid body and thereby suddenly retarded the rotation. This, before proceeding, I will illustrate by an example for the use of your mathematical readers.

3. Suppose a spherical shell or crust of mass C to have within it a solid spherical nucleus, of radius b and mass N, fitting it exactly; and the crust to receive an angular velocity of rotation around an axis fixed in the crust, the nucleus at that moment having no angular velocity; but suppose that a slight force of friction between the surfaces gradually generates a rotary motion in the nucleus; and suppose this force to vary as the difference between the angular velocities of the crust and nucleusthat is, of the surfaces in contact. Let w and w' be the angular velocities at the time 't, k and k' the radii of gyration of the two bodies, F(w-o') the force at the equator of the nucleus which represents the friction between it and the crust. Then the equa tions of motion are

W

Suppose also that would have been the angular velocity, when the primitive impulse was given, on the hypothesis of the crust and nucleus being rigidly connected so as to be one mass. Then

B(Ck2+Nk12)=aCk2.

(2)

Subtracting the second of equations (1) from the first, putting

The first of these expressions shows that the angular velocity of the crust begins with a; and when ct becomes very large indeed, it is reduced to B. Hence the effect of the constant friction of the nucleus against the inner surface of the crust is at last to reduce the velocity of the crust to what it would have been at first if the crust and nucleus had been one solid mass.

We may conclude perhaps that the same effect would be produced, though in a much longer time, if the interior were not a solid sphere, but a fluid mass.

The above reasoning shows that if the disturbing force producing precession and nutation did not exist, and the interior of the earth were fluid (whatever the thickness of the crust), it may be fairly assumed that the motion of rotation of the crust would now, the earth having existed so many ages, be exactly what it would have been had the earth been one solid mass, all difference of motion having been long ago annihilated by the internal friction and viscidity.

:

4. But the disturbing force producing precession and nutation does exist. It consists of two parts, one constant and the other variable and periodical. The constant part is that which produces the steady precession of the axis (and which I will call for convenience the precessional force); the other produces the nutation. I will consider the precession first. Suppose now, for the sake of argument, that at the present moment, as M. Delaunay imagines, the crust and the fluid are revolving precisely as one mass, all previous differences of motion, even under the action of the disturbing force which produces precession and nutation, having been annihilated by friction and viscidity. I ask, What will be the action of the precessional force from this moment? It tends to draw the pole of the crust towards the pole of the ecliptic and this tendency, as mathematical physicists well understand, combined with the rotary motion of the crust, produces this singular result, viz. the pole does not move towards the pole of the ecliptic, but shifts in a direction at right angles to the line joining the poles towards the west; so that the inclination of the axis to the ecliptic remains constant, but the axis shifts towards the west. The space through which it shifts in an infinitesimal portion of time varies as the length of the time and the force directly, and as the inertia of the mass to be moved inversely. The inertia of the mass depends upon the thickness of the crust only; for the friction of the fluid against the inner surface of the crust (which might, as I have shown, in the course of years produce a sensible effect) cannot do so during the infinitesimal portion of time I am considering before the precession is actually produced. The precessional force has its full effect in producing the precession of the solid crust, the fluid not having time to diminish that effect before the axis has assumed a new position; and in this new position of the axis the precessional force is precisely the same in amount as before, to go on causing the precession as before. The precessional force is, in fact, ever alive and active, and shows this in incessantly producing the effect I have described; and the precession goes on

steadily, the amount of it depending upon the mass of the crust thus moved, which the fluid has not time to retard or lessen. M. Delaunay says that "the additional motion due to the abovementioned causes [the disturbing forces which give rise to precession and nutation] is of such slowness, that the fluid mass which constitutes the interior of the globe must follow along with the crust which confines it, exactly as if the whole formed one solid mass throughout." In reply to this, I say that it is not the slowness of the motion, but the want of solid connexion between the crust and the fluid in contact with it which affects the problem. The motion, whatever its amount, is incessantly being generated by the disturbing force; and, owing to this want of solid connexion, the friction of the fluid has not time during the successive moments during which the precession is generated, to stop or even sensibly to check it.

It will thus be seen that at every instant the precessional force proceeding from the action of the sun and moon on the protuberant part of the earth's mass will, if the earth be a solid mass, have to move the whole mass; and if the earth have a solid crust only with a fluid interior, the force will have to move only the crust, against the evanescent resistance of the fluid within during so short a space of time as it takes to produce precession. The resulting precessional motion will be different in the two cases; and therefore the actual amount of the precession which the earth's axis has (and which is a matter of observation) is a good test of the solidity or fluidity of the interior. This is Mr. Hopkins's method.

The force producing nutation is much smaller, even at its maximum, than the precessional force. Its effect, however, is precisely the same in this respect that it depends upon the mass of the solid crust, and in no respect upon the friction of the fluid within it, which has not time to influence the nutation before the nutation is actually produced.

5. I do not here undertake to go into Mr. Hopkins's numerical calculations; I simply vindicate his method. I do not here consider what modification the elasticity of the solid material of the earth may have upon his numerical results. I conceive that it would have no effect, if the disturbing force were constant and there were no nutation. For, under the dragging influence (if I may so call it) of the constant precessional force, the solid material would be under a steady strain, and would communicate the effect of the force, continuously acting, from particle to particle of the solid part as if it were really rigid; and the resulting precessional motion would be greater or less as the mass of the solid part may be smaller or larger-that is, the solid crust thinner or thicker. But as the disturbing force is not constant,