G. H. DARWIN

THE RIGIDITY OF THE EARTH

Recent investigations prove without any room for doubt that, quite apart from earthquakes, the solid earth is never at rest, but is heaving up and down almost as though it were breathing, and I wish explain the chain of argument by which we have become convinced of the truth of the existence of this rhythmic motion. The lines of reasoning by which this conclusion is established are of various kinds, but they all lead to results closely concordant with one another.

The first suggestion on the subject, which carried much weight, was by that great physicist Lord KELVIN. His recent death, at a ripe old age, has been deplored by men of science of every civilised country in the world. In Rome, where he was an honoured member of the Accademia dei Lincei, his loss is felt no less than elsewhere.

Lord KELVIN attacked the difficult problem of determining the deformation of an elastic sphere, with the object of considering the earth as placed under strain by the attraction of the moon. LAME had solved the problem previously, although I do not think Lord KELVIN was aware of the fact; but LAME's solution has comparatively little interest in connection with our present subject, because he drew no conclusions as to the rigidity of the earth.

Lord KELVIN found that if the earth were throughout its mass only as stiff as glass, it would rise and fall under the tidal forces due to the moon's attraction more than half as much as if it were liquid throughout. Even if it were made of steel the tidal movements would be one third as great.

Now we are only able to scratch the surface of the earth, and thus we know but little about the inside; yet it is difficult to suppose that the earth is constructed of a material possessing a degree of stiffness incomparably greater than that of any of the materials with which we are acquainted. Accordingly Lord KELVIN concluded that the solid earth must be rising and falling with the tide, like the waters of the ocean, but of course to a less extent.

It then remained to consider the means whereby we ought to be able to perceive this tidal motion of the solid earth, and he concluded that it would be by its effect on the movements of the waters of the ocean. If the solid earth rose and fell exactly

as much as if it were liquid, it is clear that the waters of the sea would not move at all relatively to the land, for the whole would move up and down together. Thus the existence of oceanic tides shows that the solid earth moves to a less extent than if it were liquid. This line of argument may be carried further, and we may conclude that if the solid earth moves by half as much as if it were liquid, the oceanic tides will be reduced to half as much as if the earth were liquid. Or again, since a globe of steel of the size of the earth would pulsate tidally one third as much as one of water, Lord KELVIN concluded that the oceanic tides would be two thirds as great as if the solid earth were at rest.

He then stated conjecturally that our tides are certainly more than half as high as they would be on an unyielding earth, although perhaps not more than twothirds as great. Hence he pronounced the earth to be stiffer than glass but perhaps not so stiff as steel. This conclusion is, as we now know, very nearly correct, but it must be considered more as an intuition than as a proof, although nearly 40 years ago his conclusion seemed justifiable.

If the earth were to rotate very slowly on its axis whilst the moon moved in the heavens with corresponding slowness, the waters of the ocean would have time to assume a definite position of rest at each moment of time. This position of rest or of equilibrium can be calculated with all the accuracy desired, and it enables us to determine what would be the height of tide according to this hypothesis, which is called the Equilibrium Theory.

In his celebrated investigation of the tides, LAPLACE, following NEWTON, showed that the equilibrium theory was useless as a means of tidal prediction and that the actual motion must be incomparably more complex. He proved in fact that it is impossible to predict the height of the tides of oceans broken by continents, as on the earth. And since the time when Lord KELVIN wrote, the researches of Mr. HOUGH have shown that the semidiurnal oscillations of the sea, on a planet completely covered with water, are of even greater complexity than was supposed by LAPLACE. Indeed Mr. HOUGH has proved that the semidiurnal oceanic tides may in some places have an amplitude ten times as great as that deduced from the equilibrium theory. In such places a reduction of the tide to even a half of its amplitude on a rigid earth might easily remain unnoticed.

Within a few years after Lord KELVIN had written, our knowledge of the tidest had begun to increase very rapidly. This was largely due to the inauguration of the harmonic analysis of tidal observations, which by the way was also due to Lord KELvin. Amongst the tidal oscillations which were then actually measured for the first time were those oscillations which were called oscillations of the first species by LAPLACE and are now more commonly described as tides of long period. These oscillations consist of a slow rise and fall of the mean level of the sea, continually disturbed as it is by the ordinary semidiurnal tide. The most important of these oscillations has a period of fourteen days, and another has a period of a month. They are called in English the fortnightly and monthly tides. These tidal oscillations are so minute that they escaped detection in all the older tidal observations, but they have now been measured with some degree of accuracy at a large number of places.

LAPLACE adduced considerations which seemed to prove that these tides of long period would, in consequence of fluid friction, obey the equilibrium law pretty closely that is to say the ocean would, as regards this group of forces, have time enough to assume the position of equilibrium. Thus he held that it was possible to calculate the amplitudes of oscillation of these tides on the hypothesis that the solid earth was absolutely unyielding.

About 1881 I had the honour of helping Lord KELVIN to pass through the press the second edition of THOMSON and TAIT's Natural Philosophy, and it occurred to me that it might be possible to supplement Lord KELVIN'S interesting conjecture by a numerical evaluation from the data which were at that time available. Accordingly I took the observed results for 33 years of the fortnightly and monthly tides at various places, and compared them with the results as calculated by the equilibrium theory. I concluded that these tides had on the average an amplitude of two thirds as much as they would have had if the earth had been quite unyielding. The result exactly confirmed Lord KELVIN'S suggestion, and it seemed safe to assert that the earth was about as stiff as steel.

However within a year or two it occurred to me to doubt the justice of LAPLACE'S reasoning by which he believed that he had proved that the tides of long period should nearly obey the equilibrium law. On investigating the dynamical problem offered by these tides of long-period on an ocean-covered planet I found that they might differ in height considerably from their equilibrium value. This then seemed to throw doubt on the exactness of the conclusion that the earth was as stiff as steel, although it confirmed the view that the degree of rigidity of the earth's mass must be very great.

The matter rested in this state until 1903 when Lord RAYLEIGH (1) pointed out that the existence of the land barriers formed by our continents would have the effect of annulling those modes of motion in the ocean which are responsible for making these tidal oscillations on an ocean-covered planet so different from what they would be according to the equilibrium theory. He thus went far to reinstate LAPLACE's conclusion, as to the rigidity of the earth.

Since that time an immense amount of tidal data has been accumulating, and in 1907 Dr. WILHELM SCHWEYDAR of Potsdam reduced the observations of the tides of long period for 195 years at a number of ports all over the world (2). The result attained by him from this extremely laborious reduction is that these tides have a height between .62 and .60 of their equilibrium amount, my old result being .67. The weight to be attached to Dr. SCHWEYDAR's result, which points to a rigidity a little less than that of steel, is of course immensely greater than that which my conclusion possessed, based as it was on far more limited data. From this line of research then we conclude that the earth is a little less stiff, on the average, than steel, and we must now turn to another line of investigation.

If the earth were to yield like a perfect fluid and were to be a figure of equilibrium at every moment the moon's tide generating force would seem to vanish to (1) Phil. Mag., vol. 5 (1903), p. 136.

(2) Beiträge zur Geophysik, vol. 9 (1907), p. 41.

an observer capable of measuring it, who supposed the earth to be absolutely unyielding. The horizontal force due to the moon's attraction only amounts at greatest to one eleven-millionth part of gravity, and a pendulum will oscillate to and fro through an angle which is only 1/55th of a second of arc. It obviously then requires observations of extraordinary delicacy to measure so small a deflection of the plumb-line; indeed the measurement is quite impossible without some apparatus by which the deflections may be multiplied enormously. My brother and I made the attempt some 25 years ago, but found ourselves altogether defeated by those minute movements of the soil which form the subject of research of seismologists. Since our time however von REBEUR PASCHWITZ invented, or rather revived and perfected, an instrument of great delicacy, the horizontal pendulum. It must suffice to say here that it can be given almost any degree of sensitiveness to minute horizontal forces which disturb the apparent direction of the vertical. PASCHWITZ himself, and after his death EHLERT and KORTAZZI all undertook the measurement of the moon's horizontal force, and concluded that its apparent amount was from one half to two thirds of the amount on a rigid earth. Dr. SCHWEYDAR has also combined the observations of PASCHWITZ, EHLERT and KORTAZZI, and finds them to give a factor of oceanic tide of two thirds.

And now at length in the skilful hands of Dr. HECKER at Potsdam the problem has received a far more complete solution, the substantial correctness of which is not open to doubt (1). He erected in a recess in the side of a well at 25 m. below the surface at Potsdam two horizontal pendulums arranged in azimuths at right angles to one another, so that it is possible to measure the deflections of the vertical in the two directions N. E. and N. W. These two pendulums are arranged so as to furnish continuous automatic records, and have been under observation almost continuously from December 1902 to April 1905. The series still continues, but Dr. HECKER has published a first account of his researches.

The situation of these pendulums is incomparably superior to those used by any previous observer, and Dr. HECKER has known how to take full advantage of the superiority of his installation. The reduction of his observations shows that the oscillations of the pendulum are almost exactly two-thirds of the amount they would have on a rigid earth. The result derived from tidal observations is thus in substantial agreement with that from the horizontal pendulum. This mutual confirmation is important, and it goes far to remove the uncertainty which is inherent in the tidal method, in consequence of the doubt as to whether the oceanic tides of long period conform exactly to the equilibrium theory. It will perhaps not be generally recognised how remarkable is the accuracy of Dr. HECKER's observations, and I therefore propose to point out some of the most striking details. They appear to me moreover to be of the greatest interest in themselves, and to open the way to other researches as to the constitution of the earth.

The great difficulty which besets measurements of this kind resides in the fact that it is impossible to avoid the disturbances due to the sun's radiation. Whatever (1) Beobachtungen an Horizontalpendeln. k. Preus. Geodätischen Institutes. Neue Folge, N. 32

(1907).