and leaving out the constant term, we get for the last part of

our integral

mass

[ { αqQq + αg Qz+&c.}.

a

(C)

And finally, adding (A), (B), and (C) together, observing hat the coefficients of a++ &c.} in (B) and in (C)

together make up the whole mass of the earth (or Ampa3)

divided by a, equating to zero, and transposing, we get

whence we may successively determine a, a, &c., and obtain the amount of disturbance,

r—a=a{a2Q2+azQs+&c.}.

is so nearly equal to 1,

S

In the physical problem before us that we must take many hundreds of terms before would sen

доп

sn

sibly differ from 1; and in the meanwhile, Q being never greater than 1 and converging slowly*, while A, diminishes pretty rapidly, the terms would have become exceedingly small. We may therefore write 1 for, and a forr; and then we get for the general term,

aan

{

P

1

3 2n+1

I have had calculated for me the first fifty values of A, for a cap of ice extending to N.P.D. 30°, and the values of Q and of the product A, Qn corresponding to the disturbance at N.P.D. 35°, or about that of the north of England. For the divisor 1 I have then taken p=6, which I presume to be under

P

3 2n+1

the probable value, so as to make the calculation of the disturbance if anything too large; and, with the same object, I have used

* See Murphy, as above.

the ultimate limit, or 2, for this divisor as soon as I reached the first set of negative terms in the series. With these data I give the approximate results for the disturbance: 1st, at the north pole, where Q=1; 2ndly, at the south pole, where Q±1 alternately; 3rdly, at N.P.D.35°; and 4thly, at the corresponding southern latitude, at which the values of Q are also the same in magnitude as in the north, but with the alternate signs changed. 1. At the north pole, summing together each set of positive and negative terms, we have the elevation

B

2

{.13132+00623+00176+00101+00036},

or B× 07034. If we suppose such a mass as two miles thickness of ice, or say 10,000 feet of the density of water, this would give an elevation over the mean level of some 700 feet. But it is to be borne in mind that, to obtain such a mass, we must rob the existing sea of the greater part of it. In our ideal case of an ocean covering the whole surface, each mile of ice taken from it would lower the mean level by about 13ths, or 332 feet.

2. At the south pole we have also a relative elevation, but the amount is only

B

2

{03128+00128+00045+00042+00035},

or B01689, less than one-fourth of that at the north pole. 3. At north polar distance 35°, as might be expected, the series is much less regular. Grouping positive and negative terms separately, it gives

В

03719-01485+00112-00559+00003-00456

+0000100079+00008-00035+00019-00012 +00024-00004+00027, or B ×·00642.

It will be seen that the positive terms by themselves descend and then rise again, and the negative terms rise and then fall by themselves; but it is certain that the maxima in each group will grow less and less; and though a hundred terms would have been more satisfactory than fifty, we may feel very confident that the elevation over the mean level for two miles of ice will not exceed 80 feet, which is really a considerable depression from the existing level.

group of terms

B

2

{03046-00673+00145-00333},

after which the terms become so small and so nearly balanced as not to be worth setting down. The result for fifty terms is Bx01127, or near upon as much again as the quantity obtained for northern elevation.

The general result, then, of the disturbance of the earth's surface is not such as was anticipated by those who first started this speculation, though it cannot surprise those who are familiar with the theory of the tides. The solid nucleus of our ideal earth remaining central, as we have seen, there is an accumulation of water towards the two opposite regions where the difference between the extraneous attraction on the surface and on the centre, which is the true disturbing force, is greatest. In the tidal theory, owing to the great distance of the luminary, these differences follow the same law, sensibly, under and opposite to the disturbing body; but in our case the changes are very rapid under and near the ice, which lies close to the surface, while they are gradual in the opposite hemisphere. Hence results an egg-shaped form of sea surface, with a comparatively sharp little end at the north, balanced by a big-end or broadspread mass at the south, both ends combining to rob the equatorial region.

As I said at the beginning of this paper, the failure to account for the special facts of the glacial epoch does not deprive this theory of a certain positive value. If it cannot explain elevations or depressions of 1000 feet, it does teach us there is an agency at work in nature, which had perhaps been overlooked, which must be borne in mind in all speculations where tens of feet are material. The existing world, with its irregular continents and scooped-out seas, cannot have its water-surfaces level (regularly spherical or elliptical I mean); and with secular changes in the disposition of the solid parts, there must be local changes in the fluids, manifesting themselves in submersions or "raised beaches," which, when arising from this cause, must be in long and regular lines.

For a continent composed of matter between two and three times the density of water, and rising above its level, represents a sea over the same area with the addition of an attracting mass of more than the average depth and more than the density of this sea; and a hollow scooped out of a supposed level sea-bottom in like manner represents an abstraction of previously existing attractive force there. And these disturbances in the force will produce corresponding elevations or depressions of mean level

near the borders of the continent or hollow, and also at the antipodes. In such an irregular system as our globe actually is, these causes of disturbance are, no doubt, to a great extent antagonistic one to another; and he must have a wonderful sagacity, or be very rash, who should attempt to conjecture the total effect, at any place, of the existing configuration of land and sea. Still the fact remains, that the surface of the sea cannot be regular, and that the irregularities must shift and vary as the disposition of the land changes.

Kitlands, Dorking.

XXXII. A Speculation concerning the relation between the Axial Rotation of the Earth, and the Resistance, Elasticity, and Weight of Solar Ether. By Professor FREDERICK GUTHRIE*.

THE

HE line of argument in the following paragraphs is this:

If solar æther resists the translation of matter, and if solar space be filled with æther of uniform resisting power, then the effect of such an æther would be to tend to cause the earth to revolve on its axis in a direction opposite to that in which it actually revolves.

But if the æther has weight and elasticity, and if its resistance increases with its density, then solar æther might convert a portion of the orbital motion of the earth into axial rotation in the direction in which it actually revolves.

And hence it is not impossible that the axial rotation of the earth may be due to the conversion of a part of its orbital motion by the resistance of the solar æther, the solar æther having weight.

1. Gravitation explains both the translation of the earth and its retention in a circumsolar orbit. But the daily or axial rotation of the earth cannot be referred to the same force.

2. When a body moves in any path without rotation, that is, in such a manner that any fixed line in it retains its original direction, or any two positions of the same line are parallel to one another, then every point of the body traverses a path of the same length and shape.

3. If a circle moves without axial rotation in a circular orbit, all points on or in the circle describe circular orbits whose radii are all equal to the distance between the centre of the moving circle and the centre of the orbit.

4. If, on the moving circle, a point, which at starting is at a certain distance from the centre of the orbit, moves uniformly

in a circle round the centre of the moving circle in the same direction as the moving circle moves in its orbit, and in such a manner as to move round the centre of the moving circle in the same time as the circle performs one orbital revolution, that point remains at the same distance from the centre of the orbit.

5. Inversely, when a circle moves without axial rotation in a circular orbit round a point, the points of greatest and least distance between the circle and the orbital centre travel round the circumference of the moving circle in such a manner that the line joining them, when produced, passes through the orbital centre. This line completes one revolution as the moving circle completes one revolution.

6. Hence the successive series of points of maximum distance* traverses an orbital path longer than the path described by the successive series of points of minimum distance by the circumference of a circle whose radius is equal to the diameter of the moving circle.

7. The same is true, mutatis mutandis, for any points in any body moving in any closed orbit.

8. Suppose that the earth at some time had orbital (by the action of gravity or otherwise) but no axial rotation.

9. If the space through which the earth travels in its motion. round the sun be filled with a uniform medium of any degree of tenuity, which has any inertia or which in any way offers resistance to the passage of the earth through it, either frictional resistance to the earth's surface, or penetrative resistance towards its mass, then orbital motion must necessarily give rise to axial rotation.

10. Because the succession of points furthest from the sun will be continually subjected to the greatest resistance, and the points nearest to the sun to the least resistance during the earth's orbital motion; and because all other points of the earth's surface (if the resistance be only superficial) or of its mass (if it be penetrative) meet with greater or less opposition, according as they are further from or nearer to the sun.

11. Hence the resulting effect would be equivalent to a force upon the part of the earth furthest from the sun, tending to make the earth revolve on an axis perpendicular to its orbitplane and in a direction opposite to the orbital rotation of the earth.

12. Thus if a sphere were to move (at first without rotation)

*The successive (with regard to the circumference of the wheel) series of points in which a rolling wheel touches the ground is the point (with regard to position referred to a vertical line through the centre of the wheel) where the wheel touches the ground.