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THE THEORY OF ISOSTASY, ITS DEVELOPMENT AND RESULTS1
A. Prey.
Introduction.
When we demand of an uneducated person that he believe in the spherical form of the earth, we are, of course, thereby demanding great trust in our “learnedness.” Coastal inhabitants and dwellers on plains may perhaps still believe this; but for inhabitants of mountains it is indeed very difficult to imagine the earth in the form of a sphere. In fact, a sphere is a round, smooth body, whereas mountain dwellers see around them only the summits of cliffs reaching into the clouds and assuming the most fantastic shapes. Vertical distances seem to us far greater than horizontal distances; to climb 100 m is much harder for us than to walk the same distance across a plain. This is why the inhabitants of mountainous countries cannot free themselves from the notion that mountains are great and the earth small, whereas in reality precisely the opposite is the case. The irregularities of the earth’s surface, so striking to the eye, are very insignificant in comparison with its dimensions, but we can learn this only by means of measurement. One must marvel at the boldness of the ancient scholars who, relying on their conviction in the simplicity of the laws of nature, first arrived at the conclusion that the earth is spherical, and only afterward undertook measurements. The circumstance that Eratosthenes apparently was not too surprised when the volume of the earth found by him was expressed by too large a number may be explained by two reasons: first, he could have had as poor an idea of the actual magnitude of the earth as we do now, and, second, information about the dimensions of the “world” known at that time was so imprecise that people then did not know what part of the whole terrestrial globe this “world” represented.
The measurements that we can make are divided into two kinds: purely geometrical and physical. We shall first take up the first group of measurements.
Triangulation.
As has already been mentioned, the first measurements proceeded from the assumption that the visible irregularities in the distribution of masses on the earth’s surface are of no great significance; thus, the task was set of determining the radius of the terrestrial sphere. In doing so, the surface of the sea was regarded as the theoretical surface of the earth, since this part of the earth has the greatest similarity to the geometrical form of a sphere. The measurements were carried out on the continent; however, at first no importance was attached to this circumstance. Subsequently these measurements were reduced to the surface of the sea. When the consequences following from Newton’s law began to be derived, it turned out that the earth must be flattened. This supposition was also confirmed by measurements. From that time on the earth began to be regarded as an ellipsoid of revolution, and thus it was necessary to determine two further quantities: the major axis of the earth and its flattening. Under the influence of the desire to obtain these quantities as accurately as possible, the art of degree measurements developed. But it soon became clear that different degree measurements gave different values for one and the same quantities. These differences went far beyond the limits of the errors of measurement; thus it was seen that the nonuniformity in the distribution of masses on the earth’s surface could not be completely neglected. In attempts to free the results of computations from the influence of this nonuniformity in the distribution of masses, facts of such exceptional interest were discovered that a question which previously had aroused no interest—the question of insignificant corrections—imperceptibly became the chief subject of research. In this way the measurement of the earth came into contact with another science, which from the very beginning had studied the distribution of masses in the earth’s crust, namely geology. The joint work of these two sciences proved very fruitful.
After it had been found that the theoretical surface of the earth can be neither a sphere nor an ellipsoid of revolution, the necessity arose for a more exact determination of this surface. For this purpose use was made of the concept, known in hydrostatics, of a level surface, i.e. a surface assumed by the free envelope of a liquid, whose principal property consists in the fact that at each of its points it is normal to the force acting upon it. Such a level surface, for example, is the sea surface, if one leaves out of account all the irregularities caused by gusts of wind and by air pressure, by tides and ebbs, and similar phenomena. The form of this surface is determined by the earth’s gravitational field and by the centrifugal force due to the earth’s rotation. This level surface, and its continuation beneath the continents, constitutes the theoretical surface of the earth, which bears the name geoid. Since the attraction of masses—
depends on their magnitude and their internal structure, then small and minute irregularities in the masses are reflected on the geoid in the form of slight bends, and moreover in a very “flattened” form. Owing to the complexity of the earth’s surface, we cannot find any analytical expression for it and must construct this surface point by point.
This construction of the geoid proceeds in the following way. The angles measured in triangulation are horizontal angles, i.e., they lie in a plane perpendicular to the direction of gravity, and therefore they may be regarded as lying on the surface of the geoid. Thus, in order to reduce the network it is necessary also to know the curvature at various points of the geoid. This curvature, however, is unknown, and therefore we are forced to replace it by the simpler curvature of an ellipsoid of revolution. This substitution amounts to the fact that the network with the angles that were measured on the geoid is represented as being laid down on an ellipsoid of revolution. In order for this to be possible, it is necessary to fix the initial point and the direction of one side. For this purpose the geographical position of one of the vertices of the triangle and the azimuth of one of the sides beginning at this initial point are determined astronomically; then, on the ellipsoid of revolution, one finds the point that has the same geographical coordinates, takes it as the initial point, and draws the side whose azimuth would correspond to the measured value. In this way the position of the network on the ellipsoid of revolution proves to be fully determined. The mutual position of the ellipsoid of revolution and the geoid is determined by the fact that these two surfaces at the chosen initial point have a common tangent plane. As a result of the reduction of the whole network, the geographical coordinates of all vertices of the triangles and the azimuths of their sides are obtained as they would be if the network had been laid down on the ellipsoid of revolution. Having determined the geographical coordinates and azimuths of the other vertices of the triangle by astronomical means, one can compare the results of the astronomical and geodetic measurements. The difference between them is called the deflection of the plumb line, since this latter may also be represented as the difference between the zeniths computed by geographical means and determined astronomically.
These deflections of the plumb line have the following character: at the initial point the deflection of the plumb line is equal to zero, since at this point we made the ellipsoid of revolution coincide with the geoid. Starting from this point, the deflections of the plumb line increase irregularly. This increase in the deflection is explained by the fact that the ellipsoid of revolution has a different mean curvature than the geoid, and therefore, starting from the initial point, the two surfaces will diverge more and more (Fig. 1).
This systematic part of the deflections of the plumb line can be made to disappear by choosing an ellipsoid of revolution with another curvature, i.e., with anoth—
of another major axis and another flattening. It is still more convenient to construct the ellipsoid of rotation in such a way that the deviation of the plumb line vanishes not at the initial point, but on the average over the whole region subject to investigation, i.e. so that the positive and negative values of the deviations balance one another.
The residual which remains after this is distinguished by the irregularity of its distribution; it is this that is regarded as the actual disturbance of the plumb line. It can arise only from the fact that the geoid has a very irregular curvature as a result of those disturbances which the plumb line experiences because of the uneven distribution of masses on the earth.
Fig. 1.
Thus the idea was reached of taking into account the attraction of the disturbing masses, since these masses are known, and thereby freeing the measurements from the influence of these disturbances. As a result the irregularities in the disturbances of the plumb line were to disappear, and a much greater agreement with the ellipsoid of rotation was to be obtained. The first attempts of this kind were made in India.
As early as 1847, during the measurement of the great Indian arc passing through Everest, it became apparent that, if the curvature of the whole arc is taken to be equal to its mean curvature, then at the Kaliana station (Kaliana, \(\varphi = 29^\circ 31'\)) a disturbance of the plumb line of \(5''.236\) to the north is obtained, while at the southern station Damargida (Damargida, \(\varphi = 18^\circ 3'\)) this disturbance is equal to \(3''.791\) and is likewise directed to the north1. Sir Andrew Waugh, then superintendent of the geodetic works in India, asked the archdeacon of Calcutta, John Henry Pratt, to take up the question whether these disturbances could be explained by the existence of great mountains to the north. Pratt’s computations gave a striking result: the attraction of the mountain masses proved to be enormous; one had to wonder that this attraction did not affect the observations much more strongly. For the three latitude stations which then existed, the following values were obtained2:
Kaliana . . . . . . . . . \(27''.853\) to the north — \(16''.942\) to the east.
Kalianpur . . . . . . . \(11''.968\) „ „ — \(4''.763\) „ „
Damargida . . . . . . . \(6''.909\) „ „ — \(2''.723\) „ „
Pratt had far less information about the height and extent of the Himalayas than we do3, but, generally speaking, he rather underestimated all
heights than overestimated them, and thus the figures written above are more likely too small than too large.
Soon Airy1 explained this fact by suggesting that beneath mountains there is a large defect of mass, which counteracts the attraction of the visible part of the mountains. He proceeded from the idea that blocks of the earth’s crust float upon deeper-lying, denser layers, just as blocks of ice float in water. Since these blocks are immersed in a denser mass, this mass is displaced by them and, in this way, a compensating defect of mass is obtained. Although Pratt2 at first opposed this view and sought an explanation of the phenomena described above in a stronger curvature of the Indian arc as compared with the mean curvature of the earth’s surface, nevertheless in 1857 he agreed with this view and, developing it further, attempted to explain this defect by supposing that mountain ranges arose as a consequence of an expansion of the earth’s crust, with which there must be associated a small decrease in the density of the masses. The following figures show what result may be obtained by making various assumptions about the depth to which the defect extends.
| English miles | Kaliana | Kalianpur | Damargida |
|---|---|---|---|
| 100 | 26″.440 south | 12″.111 north | 6″.855 south |
| 300 | 21″.106 ” | 11″.678 ” | 6″.866 ” |
| 500 | 17″.106 ” | 9″.622 ” | 6″.670 ” |
| 1000 | 11″.199 ” | 7″.386 ” | 5″.220 ” |
Thus it actually proved possible that subterranean defects compensate a significant part of the attraction, but still it was not possible to explain exhaustively the peculiarities of the Indian stations. On this matter, however, there also existed another view. The ocean, with its enormous mass of water, whose density is equal to 1, whereas the density of the earth’s crust is equal to 2.7, must represent a very large defect of mass, the action of which is similar to a repulsion of the plumb line in the direction toward the continent. Pratt obtained,3 it is true, on the basis—
corrected according to the results obtained in measuring the height of the Himalayas by the brothers Strachey. See Pratt, J. H.: On the deflection of the plumbline in India caused by the attraction of the Himalayan Mountains and of the elevated regions beyond and its modifications by the compensating effect of the deficiency of matter below the mountain mass. Phil. Trans. of London, 149.
obtaining very inaccurate and erroneous information about the depth of the ocean, the following data:
| Cape Comorin | 19″.71 to the north | 2″.19 to the east. |
| Damargida | 10.44 ” | —11.80 ” |
| Kalianpur | 9.00 ” | — 0.48 ” |
| Kalian | 6.18 ” | — 0.09 ” |
These figures, although they do not claim to be exact, may nevertheless serve as reference material.
If these disturbances are added to the disturbances caused by the Himalayas (p. 34), it may be observed that, although the quantities thus obtained will be larger, they exhibit a more uniform course, and therefore their differences from the numbers for the initial station Kalianpur are smaller; nevertheless it proved impossible to obtain any satisfactory results from this. At this stage the investigation of the question was suspended for a time. The question of why the attraction of the Himalayas and the repulsion of the ocean are not perceptible remained unresolved. Pratt attempted to explain the peculiarities of the Indian latitude by the existence of an underground accumulation of masses south of Kalianpur, in the same way as the disturbances of the plumb line in the environs of Moscow had been explained.
Pratt had at his disposal only three stations. In the investigations of 1901 there were 159 such stations. The system of plumb-line disturbances obtained at these stations, when referred to Kalianpur as the initial point, shows that, beginning from Calcutta, there extends in the WNW direction as far as the Indus a belt, almost without exceptions, of positive disturbances, i.e. disturbances directed toward the south; to the north and south of it lies a region of negative disturbances, while the far south of India again represents a region of positive disturbances. Attempts were made to investigate to what extent these facts may be explained by various assumptions concerning compensation. The attraction of the mountains was again computed on the basis of contemporary data; the influence of the ocean was also taken into account, and in both calculations the assumption was made that no compensation whatever exists. The following results were obtained. (See table, p. 37).
Thus it is seen that the attraction of the mountains and the repulsion of the ocean complement each other,—where the attraction of the Himalayas becomes smaller, there the influence of the ocean increases, so that the figures in the column in which the sums of these two deviations are given show comparatively small differences. If we take Kalianpur as the initial station and subtract 37.6 from all the figures, we obtain the figures standing in the seventh column. The last column contains the residuals between the computed values and the observational data. The difference between them, especially near the mountains, has already become quite small. Thus the investigations for different geographical latitudes argue against compensation of the Himalayas.
THE THEORY OF ISOSTASY, ITS DEVELOPMENT AND RESULTS
| Station | Himalayas | Ocean | India | Rest of Asia | Total | Reduced to Kalianpur | Observed deflection of the plumb line | Observed minus computed |
|---|---|---|---|---|---|---|---|---|
| Mussure | −64.″9 | −10.″3 | +.″5 | −3.″3 | −73.″5 | −35.″9 | −37.″08 | −1.″2 |
| Dehra Dun¹) | −72.2 | −10.3 | +12.6 | −3.3 | −73.2 | −35.6 | −37.15 | −1.6 |
| Kaliana | −36.2 | −11.0 | +1.9 | −2.0 | −47.3 | −9.7 | −7.00 | +2.7 |
| Kalianpur | −18.4 | −19.4 | +3.1 | −2.9 | −37.6 | 0.0 | 0.0 | 0.0 |
| Damargida | −10.4 | −26.2 | +0.1 | −2.0 | −33.1 | −0.5 | −2.73 | −2.2 |
| Punnae²) | −3.4 | −37.6 | −8.7 | −0.6 | −50.3 | −12.7 | +1.89 | +14.6 |
| Bombay³) | −7.9 | −29.6 | −1.3 | −2.2 | −41.0 | −3.4 | −10.56 | −7.2 |
| Mangalore⁴) | −4.9 | −28.6 | −6.3 | −2.0 | −41.8 | −4.2 | +2.76 | +7.0 |
| Madras⁵) | −6.8 | −28.0 | −3.6 | −1.1 | −39.5 | −1.9 | +3.6 | +5.5 |
| Waltair⁶) | −11.0 | −33.0 | −10.9 | −0.7 | −55.6 | −18.0 | −9.18 | +8.8 |
| Calcutta | −23.3 | −19.9 | −0.4 | −1.0 | −44.6 | −7.0 | +0.67 | +7.7 |
If, however, we examine the data for the first vertical, obtained partly from the azimuth data and partly from the longitude data, then we obtain:
| Station | Computed from the attraction of masses | Reduced to Kalianpur | Observed | Observed minus computed |
|---|---|---|---|---|
| Mussure | −41.″1 | −32.″6 | −26.″0 | +6.″6 |
| −38.6 | −30.1 | −22.1 | +8.0 | |
| Kaliana | −20.3 | −11.8 | −4.4 | +7.4 |
| Kalianpur | −8.5 | 0.0 | 0.0 | 0.0 |
| Damargida | −3.8 | +4.7 | −9.8 | −14.5 |
| Punnae | +0.7 | +9.2 | −1.8 | −11.0 |
| Bombay | −20.3 | −11.8 | +6.4 | +18.2 |
| Mangalore | −22.2 | −13.7 | +1.9 | +15.6 |
| Madras | +21.0 | +29.5 | −7.0 | −36.5 |
| Waltair | +17.5 | +26.0 | −3.1 | −29.1 |
| Calcutta | +0.8 | +9.3 | −10.1 | −19.4 |
With the exception of the first three mountain stations, the computed values deviate in one direction and the other from the values obtained
¹) Dehra Dun, ²) Punnae, ³) Bombay, ⁴) Mangalore, ⁵) Madras, ⁶) Waltair.
⁷) Burrard S. S.: l. c.
upon observation. Thus the deflections of the plumb line in the first vertical speak for the compensation of the Himalayas.
Finally, three more longitudes must be taken into account:
| Stations | Computed from attracting masses | Observed | Observed minus computed |
|---|---|---|---|
| Amristar–Multan¹) | −20″.0 | +8″.22 | +28″.0 |
| Valgar–Bombay | +39.6 | −10.06 | −49.7 |
| Madras–Mangalore | +44.7 | −9.27 | −5.40 |
These data also speak in favor of compensation. Thus these assumptions lead us to an obvious contradiction, and therefore there must exist some other cause influencing the deflection of the plumb line. It was found in the assumption that there exists an underground chain which begins at Balasore, passes along the mouth of the Hooghly in the direction toward Jodhpur in Rajputana, and then extends parallel to the Himalayas. The Himalayas are regarded as uncompensated, the Tibetan plateau as compensated by two-thirds, while the ocean is regarded as fully compensated, and therefore is not taken into account. Under these assumptions concerning compensation we obtain the following picture.
| Station | Latitude | Himalayas | Tibet | Underground chain | Total | Observed deflection of the plumb line | Observed minus computed |
|---|---|---|---|---|---|---|---|
| Dehra Dun | 30°19 | −19″ | −18″ | +1″ | −36″ | −35″ | −5 |
| Kaliana | 29 31 | −3 | −11 | +2 | −12 | −11 | +2 |
| Noh²) | 27 51 | −2 | −9 | +7 | −4 | −3 | +1 |
| Daiatghari³) | 24 38 | −1 | −6 | +4 | −3 | −2 | +1 |
| Kaliyanpur | 24 7 | −6 | +3 | −3 | −2 | 0 | |
| Loidi⁴) | 23 8 | −5 | −3 | −8 | −7 | 0 | |
| Badgaon⁵) | 20 44 | −4 | −7 | −11 | −10 | 0 | |
| Damargida | 18 3 | −3 | −2 | −5 | −4 | −1 | |
| Nathabad⁶) | 15 6 | −2 | −1 | −3 | −2 | −1 | |
| Punnal | 8 9 | −1 | 0 | 0 | 0 | 0 |
¹) Amristar — Mooltan.
²) Noh, ³) Daiathgari, ⁴) Loidi, ⁵) Badgaon, ⁶) Nathabad.
Since Kaliānpur lies in immediate proximity to an underground chain, Dehra Dun is taken as the initial point. Thus this investigation, which in fact began with the rejection of the assumption concerning the compensation of the Himalayas, showed, however, that at least the entire mass of the plateau of Central Asia is for the most part compensated.
The idea of compensation of external irregularities in the distribution of masses was, meanwhile, also confirmed by Helmert’s extensive investigations into the distribution of gravity. Thus the fact of the existence of compensation became indisputable, and it proved necessary in all geodetic operations to take this fact into account. The study of this question was broadly undertaken during the American works. The entire extensive system of deflections of the vertical, obtained on the basis of all the triangulations in the United States, was used.¹ All points were reduced to the United States standard, i.e., to the geographical position and astronomically measured azimuth at Meades Ranch
\[ (\varphi = 39^\circ 13',\ \lambda = 98^\circ 32'), \]
which was adopted as the initial point; the position of the ellipsoid of revolution was chosen so that at this point the geodetic and astronomical zeniths coincided with one another. The whole system comprised 381 latitude stations, 131 longitude stations, and 253 stations for determining azimuths. Among these stations there were 32 so-called “Laplace points,” at which both longitude and azimuth were determined simultaneously, and which served for control.
First of all, for all these points the influence of the visible irregularity in the distribution of masses was computed for an area with a radius of 2564 English miles (4125 km). The computation used was the same as that ordinarily employed in calculating similar mass effects: the entire region is divided into separate parts by means of circles whose center lies at the given station and radii drawn in such a way that the attraction of the separate sectors can be computed with sufficient accuracy from the mean height. Thus the sizes of these parts are determined chiefly by the distance of the individual points from the observing stations; moreover, these parts are chosen so that the computation of their attraction should be as simple as possible. Assuming that the disturbing masses are at the same height as the station, one may apply the formula
\[ D = 12''{,}44\,\frac{\delta}{\Delta}\,h(\sin a' - \sin a)\,\log \operatorname{nat}\frac{r'}{r}. \]
¹ Hayford John F.: The figure of the earth and isostasy from measurements in the U. S. Coast and geodetic survey. Washington, 1909. — The same author: Supplementary investigation in 1909 of the figure of the earth and isostasy. Coast and geodetic survey. Washington, 1910.
Here \(r\) and \(r'\) denote the inner and outer radii of the region under consideration; \(a\) and \(a'\) are the azimuths of the radii separating one part of the region from another; \(h\) denotes the mean height, and finally, \(\delta\) and \(\Delta\) the surface and mean density of the earth.
For greater simplicity, the difference of the sines is set equal to some simple fraction, for example, 0.25. In that case, for attraction directed from north to south, the western and eastern sectors will be larger than the northern and southern ones, which is not essential, since they give a very small component along the meridian. For computing the attraction in the east–west direction, everything is rotated by \(90^\circ\). For \(\delta\) the value 2.67 is taken, and for \(\Delta\)—5.576 (after Harkness); thus,
\[ \frac{\delta}{\Delta}=\frac{1}{2.09}. \]
Since on American maps heights are expressed in feet (1 mile \(=\) 5280 feet), we obtain the following formula
\[ D=12''.44\,\frac{1}{2.09}\,\frac{h\ \text{in feet}}{5280}\,0.25\,\log \operatorname{nat}\frac{r'}{r}. \tag{1} \]
If the region is divided into parts so that
\[ \frac{r'}{r}=1.426, \]
then it will turn out that
\[ D=0''.0001\ (h\ \text{in feet}). \tag{2} \]
Thus, in order to find the attraction of an individual sector, it is sufficient to read its height in feet from the maps; the attraction of this sector will be expressed by the same number \(0''.0001\) as the height is expressed in feet; all further calculations drop out. For very large distances it is necessary to take the curvature of the earth into account. Accordingly, the six outer radii are chosen somewhat larger, in order that the simplicity just mentioned may be preserved. For regions that lie entirely on the surface of the ocean, \(h\) must be taken as negative, since a portion of the earth’s crust is lacking here; instead, however, the water mass is added, whose density is equal to 1.027. If this value is inserted instead of 2.67, then the coefficient in our formula is reduced in the ratio
\[ \frac{1.027}{2.27}=0.385; \]
thus we have:
\[ D'=-0''.0001\,h\,(1-0.385)=-0''.0001\,h\,0.615. \]
But since on American maps sea depth is given in fathoms, the left-hand side in this equality must be multiplied by 6, and thus we obtain:
\[ D'=-0''.000369\times(\text{depth in fathoms}). \tag{3} \]
Thus the computations are, indeed, reduced to a very simple form. With very large differences in height between the individual points and the initial point, it is necessary to introduce still another small correction for the inclination of the attracting force. This correction is taken from a small table and is introduced, of course, only for the narrowest circles.
For stations that lie close to one another, the outer circles for the most part overlap, and therefore the difference in the effects of mountain masses at such stations is very small. Thus, if the computations have been carried out for three stations, then for a fourth station lying between them the attractions of the outer circles may be found by simple interpolation. Such a method of computation results in an enormous saving of labor1.
Consideration of the disturbances found in this way, caused by external masses, shows that in this case as well their values are much greater than the disturbances of gravity which they were supposed to explain. Since, however, the masses causing the disturbances undoubtedly exist and attract the plumb line, there must unquestionably be another cause which exerts the opposite effect on the plumb line, and we must assume that the external accumulations of mass correspond to subterranean defects of mass. If the compensation were complete, then the influence of the visible masses would have to disappear almost entirely, and the residuals should not depend on the external form of the earth. In the United States, however, this is by no means observed. One can enumerate a whole series of cases where, more or less distinctly, a connection may be noticed between external forms and disturbances of the plumb line. Yet these disturbances are very small in comparison with the large external irregularities in the distribution of mass, and therefore it must be supposed that the latter are compensated to a considerable degree in the deeply lying layers.
In what follows we shall proceed from the assumption of the existence of complete compensation.
Everything indicates that the masses located inside the earth are in a state of a certain plasticity, so that all differences of pressure within the earth disappear with time and hydrostatic equilibrium is established there. Surfaces of equal density here coincide with the so-called equipotential surfaces. But it is obvious that the upper layers of the earth do not obey this law. In these layers we observe a chaotic mixture of masses of different density and form. Thus, if we pass from the interior of the earth to its surface, we must undoubtedly encounter an equipotential surface which will be the last surface,
still corresponding to hydrostatic equilibrium. It must have the property that a unit of its area must everywhere be under the same pressure. The hypothesis which assumes complete compensation of the external masses consists in this: that the pressure on each unit of area of the above-mentioned equipotential surface depends only on the masses lying normal to it; and thus the weight of all these columns, resting on such unit areas and normal to them, must be the same over the whole earth. This state of equilibrium of masses is called isostasy, and the equipotential surface on which, according to our assumption, the unevenly distributed masses are piled up is called the isostatic layer.
For convenience of computation it is assumed that the compensating masses between various depths down to the isostatic layer are distributed uniformly. We shall see later that this assumption has its physical interpretation.
In order that the basic assumptions be fulfilled, it is necessary that every mass rising above sea level be matched by a deficiency of mass within the earth. If we denote the height above ocean level by \(h\), the surface density of the earth by \(\delta\), and the depth of occurrence of the isostatic layer by \(T\), then the defective density \(\delta_1\) is determined from the equality
\[ \delta h = \delta_1 T, \tag{4} \]
where this density, being defective, must be negative. By this amount \(\delta_1\) the density of the whole layer must be less than the normal density of the earth’s crust. Thus over the entire surface of the earth we have
\[ \frac{\delta_1}{h}=\mathrm{const}=\frac{\delta}{T}. \]
For points located on the surface of the oceans, this equality takes a somewhat different form. Since in the whole depression occupied by the ocean there is a mass whose density is not \(\delta=2.67\), but \(\delta_0=1.03\), the ocean (from outside) represents a large defect of mass, with depth \(h\) and density defect
\[ \delta-\delta_0 = 2.67 - 1.03 = 1.64. \]
Thus here we have
\[ (\delta-\delta_0)h=\delta_1 T;\quad \text{or}\quad \frac{\delta(\delta-\delta_0)}{\delta}\,h=\delta\cdot 0.615\,h=\delta_1 T \tag{5}. \]
Thus, instead of the entire depth of the ocean, we take into account only \(0.615\) of this depth. The density \(\delta_1\) will in this case be positive po-
that an excess of mass beneath it must correspond to the defect of the ocean. The density is always in a very simple relation to \(h\); consequently, the effect of the compensating masses can easily be found by multiplying the results found for the individual quadrilaterals by formulas (2) or (3) by a factor \(F\), having the following form:
\[ F=1-\frac{\log \dfrac{r+\sqrt{r^{2}+T^{2}}}{r'+\sqrt{r'^{2}+T^{2}}}}{\log 1.426} \]
Thus, having made a definite assumption concerning the values of \(T\), one can find the value of \(F\) for each ring. These values decrease rapidly when \(r\) increases, so that the outer rings already exert a very small influence. Concerning the depth at which the isostatic layer lies, the following assumptions are made. Assumption A: the depth at which the isostatic layer lies is equal to 0, i.e. the compensating masses, exactly like the external masses, are at the same elevation as the station. In this case both these and the other masses lie above the level of the ocean. This assumption amounts to paying no attention at all to the position of the masses. Assumption B: the depth at which the isostatic layer lies is infinitely great; in this case \(\delta_i=0\), i.e. the equilibrating masses have no density, and, despite their infinite extent, have absolutely no effect on the plumb line. This assumption amounts to regarding the external masses as entirely uncompensated. Assumption E: the depth \(T\), at which the isostatic layer lies, is equal to \(162.2\) km. Assumption H: \(T=120.9\) km. Assumption G: \(T=113.7\) km. These values of \(T\) were chosen on the basis of trials made during the computations. Proceeding from each of these assumptions, corrections to the deflections of the plumb line were computed, and then this entire system was subjected to painstaking smoothing. In doing so, the corrections to the geographic latitude and longitude of the initial station Meades Ranch, the corrections to its azimuth, and also the semi-major axis and flattening of the Earth were taken as unknowns. The measure of the suitability of one solution or another was furnished by the sums of the squares of the remaining deflections of the plumb line, which can no longer be explained by the extension of the external masses and by internal compensation. For these sums the following values were obtained:
| Assumption | Sum |
|---|---|
| B | 107385 |
| E | 10297 |
| H | 10063 |
| D | 10077 |
| A | 18889 |
Assumption B, under which the masses are considered uncompensated, proves to be completely unsuitable. Thus, the assumption that the external masses are piled up on the surface of the earth, within which there are no anomalies, in any case does not withstand criticism. Assumption A, under which the masses are entirely disregarded, already proves incomparably better, since the sum of the squares of the differences in this case is already six times smaller. The best result, apparently, is obtained under the assumption that a certain isostatic position exists, and the table cited above shows that solution $H$ is better than all the others; the investigations of 1909 led to replacing this solution by solution $G$. It is clear that there is no great difference between these two solutions, and therefore it is quite immaterial which of them is adopted as final. All the other criteria—such as the mean differences, irrespective of their sign, the percentage of deviations greater than $2''$ or $5''$, the maximum of deviations over the whole region or in its separate parts—all lead to the same conclusion. Thus the following result is obtained: the external inequality in the distribution of masses is compensated in the United States by a defect which is uniformly distributed among layers of various depth down to 122 km. The investigations of 1909 gave, for the depth to which the defect extends, 113.7 km, while the most recent observations gave 102 km.
Since differences still continue to remain, it is clear that complete compensation does not take place. These differences amount on average to 0.1 of the disturbance; thus it may be said that the masses are compensated to $^9/_{10}$ of their magnitude.
The very good agreement of the values obtained from the assumption of compensation with those values which were obtained by Helmert1 in his investigations of the force of gravity, and the circumstance that the study of earthquakes in its turn points to the existence of an isostatic layer,2 have led to the conviction of the real existence of this layer. Without adhering to any definite hypothesis, one may, however, assert that at a depth of 120 km the stratification of the earth changes in a manner characteristic of its structure, and the phenomena observed on its surface cease here.
During the American investigations other assumptions concerning isostasy were also made. Assuming that the deficient density reaches its maximum at the surface and, descending into the depths, decreases linearly, we shall find that the isostatic layer lies at a depth of 189 km. If all the compensating masses are concentrated in a layer 16 km thick, then the isostatic layer proves to be at a depth of 65 km.
If, however, one accepts the hypothesis of Chamberlin, according to which the defective density first increases and then, decreasing, disappears in deeper layers, then for the depth sought one obtains 310 km.
The success of Hayford’s investigations prompted the same method to be applied, and, proceeding from the same depth of occurrence of the isostatic layer, also to the Indian measurements. However, this work did not yield the expected success1. The discrepancies remained large (in mountainous localities on average 16″), and therefore one must conclude that in India, at a depth of 113.7 km, no level surface exists. True, Hayford in his computations proceeds only from this depth, but Bowie2 did not succeed in obtaining better results under other assumptions concerning the depth of occurrence of the isostatic layer. Apparently, in India the conditions are very complex, especially since India is very small in comparison with the United States, while here unusually high mountains and a deep ocean approach one another very closely; moreover, the quite young Himalayan mountains may have a special anomaly; finally, it is possible that an underground chain, which may be regarded as an accumulation of uncompensated masses, actually exists3, and that this chain, owing to the small size of the whole region, plays a large role in it.
Small accumulations of uncompensated masses and defects are also found in America, but there is nowhere there so complex a formation that it could be taken for an underground mountain chain.
II. Measurement of the Force of Gravity.
We now come to the second group of measurements. The first observations of the change in the force of gravity when moving over the earth’s surface belong to Richer. In 1671 Richer went to Cayenne to determine the solar parallax from observations of Mars. In Cayenne he found that, in order for the clock to keep correct time, its pendulum had to be shortened by 5/4 line. At first several incorrect explanations were given for this phenomenon, but it was soon pointed out that it is caused by the flattening of the earth. To determine the flattening of the earth on the basis of observations of gravity became possible only after Clairaut proved a remarkable theorem, according to which between the force of gravity on the earth, its flattening, and the speed of its rotation there exists a relation that does not depend
from the internal stratification of the earth, assuming that these layers are surfaces of revolution. This theorem is a consequence of Green’s much more general theorem.
To determine the shape of the earth on the basis of measurements of gravity, it was necessary to have a very large number of stations. Bouguer, during the Peruvian degree measurements, was already making observations with a pendulum; other observers soon followed him, and Laplace1, at the turn of the eighteenth and nineteenth centuries, was already able to attempt to determine the flattening of the earth. He had at his disposal 15 stations. But the result he obtained, \(1:336\), did not meet expectations; the observations proved unsatisfactory because of their extreme difficulty, the measurement of the length of the pendulum being especially difficult. A significant step forward was made in 1822 by Kater, who introduced the now familiar reversible pendulum. Measurements of gravity were thereby substantially facilitated; however, they continued to remain one of the most tedious and difficult tasks because of the numerous corrections that had to be introduced into the direct observations: corrections for temperature, air pressure, air friction, for the mass of air carried along by the pendulum, for the sliding and rotation of the pendulum prism on its support, and, finally, for the forced oscillations of the rod and support. The observations take place in dark cellars, continue for several hours in succession, and require concentrated attention. It is therefore easy to imagine that it was impossible to obtain the large quantity of material required for the problem posed.
The measurement of gravity in the eighties of the last century advanced greatly thanks to the introduction of relative measurements by means of Sterneck’s invariable pendulum.2 The fundamental principle, known in all sciences in which measurements are involved, is that differences can be measured with greater accuracy than absolute quantities. Proceeding from this principle, it was decided to measure only the differences between the force of gravity at a reference station and at the place where the observations are conducted. If a pendulum is arranged in such a way that it can remain invariable—this is best achieved by simplifying its construction—then, when differences are measured, precisely those corrections disappear whose determination constitutes the greatest difficulty, such as, for example, corrections depending on the shape of the knife-edge. In general, only the corrections for temperature and for air pressure remain; these can be made very easily. The only remaining difficulty is the forced oscillations of the support.—Thanks to the introduction
THE THEORY OF ISOSTASY, ITS DEVELOPMENT AND RESULTS
of these instruments, the number of observation stations over the course of several years increased very greatly; now there are 3000 such stations, whereas in 1882 Helmert had at his disposal only 122 of them.^1)
It soon became clear, on the basis of measurements of gravity, that in order to determine the compression of the earth it was necessary to take those values of this force which it has on a surface of the earth that is as smooth and undistorted as possible. They began to look for such observation stations as were known to be outside the action of disturbing forces. However, even at these stations corrections had to be introduced.
The first of these corrections concerns the influence of height above sea level. All points of the earth’s surface are situated above the level of the ocean; their distances from the center of the earth are greater than the distances of points of the sea surface, and therefore the correction can be computed with sufficient accuracy from the formula \(\frac{2h}{R}\), in which \(h\) is the height of the place above ocean level, and \(R\) is the mean radius of the earth. Soon, however, another correction was also introduced, namely: for the attraction of the layer whose thickness is equal to the height of the place above ocean level; in doing so they proceeded from the perfectly correct idea that such a mass exists, and therefore must undoubtedly exert some action on gravity. The extent of this layer may be considered infinite, since the masses lying to the sides at a great distance produce no influence. Thus a very simple formula was obtained for the disturbance of gravity \(\Delta g\). It has the following form:
\[ \Delta g=-\frac{3}{2}\frac{\delta}{\Delta}\cdot\frac{h}{R}\,g \]
where \(\delta\) and \(\Delta\) represent the surface and mean density of the earth. This correction is usually called the Young–Bouguer correction (Joung-Bouguer). But it was soon noticed that the attraction of this layer exceeds the value that could have been expected on the basis of observations of the variation of gravity. Therefore Faye proposed completely omitting this reduction and making only the correction for height. Only in those cases when the terrain is very steep or precipitous is it necessary to introduce a topographic correction. However, in most cases this latter is equal to zero, and therefore Faye’s method is practically identical with reduction to open air.
Attention was again drawn to these facts when Helmert, in the cited book (“Theory, etc.”), undertook calculations for deriving the normal formula of gravity, for which at his disposal—
^1) Helmert, R.: Die mathematischen und physikalischen Theorien der höheren Geodesie. II Teil, Leipzig 1884.
At that time there were 122 observing stations for measuring gravity, located at various latitudes; however, it was necessary not to take into account observing stations situated on islands. In order to obtain a normal formula, one must imagine that all observing stations lie at ocean level; thus they have to be imagined as transferred into the interior of the earth’s crust. If, as Helmert did, one starts from an expansion in series in negative powers of the distances of individual points from a certain center, then difficulties arise connected with the convergence of these series for such points lying inside the earth’s crust. Helmert tried to eliminate these difficulties by replacing the actual distribution of masses by a certain ideal one, assuming that the external masses are condensed on a surface lying 21 km below the physical surface of the earth. Since this surface lies deep inside the earth, all points on the surface of the ocean are, in relation to the masses, external points, and thus the convergence of the series is assured. It turned out that the values obtained in this way agree very well with the deviations from the normal formula for gravity. In laying masses inside the earth, something seems to occur that is in the closest connection with the very nature of things; it appears as though these masses have entered where they, properly speaking, should have been, or from where they may have originated.
Finally, the fact of the existence of underground mass defects appears most clearly in the measurements of Sterneck1, carried out in the Alps,—chiefly because the number of observations was very large, and they formed a connected series in which, when crossing the mountains, it was not difficult to notice their influence, now increasing and now weakening. For all 508 stations that were in Austria-Hungary at that time—stations whose elevation reached up to 1500 m—it was established that the connection with the height of the mountains is very weak. The results of the observations made at these stations can be expressed by means of the formula:
\[ g=\gamma_0+0.017-0.0003141\,h, \]
where \(g\) denotes the observed values of gravity, \(\gamma_0\) its normal value, and \(h\) the height in meters. After reduction to free air, one obtains:
\[ g=\gamma_0-0.0003086\,h. \]
This equality is justified only on the average; at individual points there are deviations which, evidently, are connected with the structure of the mountains. Thus, in the Tyrolean Alps gravity is everywhere too
was small, and thus it must be assumed that the visible masses situated on the surface of the earth correspond to an underground defect.
This latter was first computed in the form of the so-called ideal layer at sea level. The attraction of this layer is equal to the attraction of an infinitely thin mass-bearing surface, which is located above ocean level and is equivalent to a layer of some finite thickness with a defective density of −2.7 (absolute density 0). The thickness of this defect reaches up to 1200 m. Although in this way the existence of the defect was established, the introduction of the concept of an ideal layer located at sea level and causing a disturbance was not very satisfactory. In the first place, no physical conceptions could be associated with it, since the layer cannot both lie at ocean level and at the same time attain a thickness of 1200 km; and, in the second place, this assumption of a density equal to 0 led to the conception of the existence of cavities. True, such cavities near the surface of the earth are quite possible, but their existence at great depth, under the enormous pressure that prevails there, is hardly probable. Thus the task consists in finding for this defect the best possible physical representation, and, moreover, in localizing it in some way. According to potential theory, this problem remains indeterminate, since one can find an infinite number of mass configurations which from the outside will exert exactly the same disturbing influence on the force of gravity. However, by making certain plausible assumptions, one can obtain a whole series of conclusions. As regards the magnitude of the defect, apparently the assumption of complete compensation is closer to reality than the assumption of its complete absence. It therefore proved convenient to assume that complete compensation always exists, and to regard it as a normal phenomenon, while deviations from it are to be considered as a disturbance.
The determinations of the force of gravity on the ocean, which were carried out by Hecker (O. Hecker)¹ with the aid of the “boiling of thermometers”² during his many—
¹ Hecker O.: Bestimmung der Schwerkraft auf dem Atlantischen Ozean sowie in Rio de Janeiro, Lissabon und Madrid; Veröffentlich. d. preuss. geod. Institutes N. F. No. 11. Berlin 1903.—Ero же: Bestimmung der Schwerkraft auf dem Indischen und Grossen Ozean und deren Küsten, sowie erdmagnetische Messungen; Veröffentlich. d. Zentralbureaus der intern. Erdmessung N. F. No. 16. Berlin 1908.—Ero же: Bestimmung der Schwerkraft auf dem Schwarzen Meere und an dessen Küsten, sowie neue Ausgleichung der Schwerkraftmessungen auf dem Atlantischen, Indischen und Grossen Ozean; Veröffentlich. d. Zentralbureaus der intern. Erdmessung N. F. No. 20. Berlin 1910.—Wolf, H.: Die Schwerkraft auf dem Meere und die Hypothese von Prott; Zeitschrift f. Vermess. 45.—Helmert, R.: Die Erfahrungsgrundlagen der Lehre von allgemeinen Gleichgewichtszustande der Massen der Erdkruste. Berliner-Sitzungsbericht. 1912. 20.
² The idea of this method is extremely simple. The pressure of the atmosphere is determined, on the one hand, directly by a mercury barometer, and on the other hand—from tab—
A. Prey
…numerous voyages over all the great seas served as the concluding link in this construction: these measurements showed that gravity on the ocean may be regarded as normal. Since the density of the oceanic water mass is 1.03, i.e., in comparison with the earth’s crust, with a density of 2.7, represents a large mass defect, gravity at the surface of the ocean can be normal only if beneath the ocean floor there is a corresponding excess of density. Thus the seas, apparently, are completely compensated.
It is very difficult to make observations by means of “boiling thermometers,” and therefore it is not easy to increase the number of such observations. The successful observations which, on commission from the Dutch commission on degree measurements, were made by Vening-Meinesz¹) over a benchmark in a submarine, permit one to hope that the situation here will soon take a favorable turn.
Thus, on the basis of disturbances of the plumb line one can compute the depth of occurrence of the isostatic layer. The first attempt at such a computation was made by Helmert²). He proceeded from those positive disturbances in gravity which increased when approaching the steep coasts of the continents, and to whose connection with subterranean compensations Schioetz³) had already drawn attention. Helmert found that the isostatic layer, if Pratt’s hypothesis is adopted, lies at a depth of 118 km, in striking agreement with the value which had been obtained in the American investigations from the disturbance of the plumb line.
Further proofs of the correctness of this value were obtained by the Americans themselves⁴), who reduced all the stations for measuring gravity (124) to the isostatic layer, assuming that it lies at the depth found from measurements of the deflection of the plumb line; in this way there is obtained a system of values for gravity which already shows no influence at all of external masses.
…persons—by the boiling temperature of pure water. In the latter case, however, the pressure is measured by the weight of a mercury column reduced to latitude 45°. Hence, comparing the two values obtained for the pressure, one can find \(\dfrac{g}{g_{45}}\). Ed.
¹) Meinesz, Vening: Observation de pendule sur la mer pendant un voyage en sous-marin de Holland à Java. Publication de la Commission Géodésique Neerlandaise.
²) Helmert, R.: Die Tiefe der Ausgleichsfläche bei der Prattschen Hypothese für das Gleichgewicht der Erdkruste und der Verlauf der Schwerestörung von Inneren der Kontinente und Ozeane nach den Küsten. Sitzungsber. preuss. Akad. d. Wiss. 1909 № 48.
³) Schioetz, O. E.: Die Schwerkraft auf dem Meere längs dem Abfall der Kontinente gegen die Tiefe. Christiania 1907.—Helmert, R.: Unvollkommenheiten im Gleichgewichtszustande der Erde. Sitzungsber. d. preuss. Akad. d. Wiss. 1908. № 44.
⁴) Hayford, I. E. and Bowie, W.: The effect of topography and isostatic compensation upon the intensity of gravity. Coast and geodetic survey, spec. publ.—Bowie, W.: The effect of topographic etc. 2-nd paper, spec. publ. 12.
Bowie¹) also attempted to determine the depth of occurrence of the isostatic layer from observations of gravity; he did this with the aid of observation points in the high mountains of America. Bowie was guided by the same basic idea as Helmert, namely, he believed that only in those localities which deviate strongly from normal stratification can conclusions be drawn concerning the vertical distribution of masses. Indeed, for horizontal layers of great extent, the disturbance of gravity does not depend on the height of the attracted point above the layer. The depth at which the layers causing the disturbance are located has absolutely no effect on the magnitude of the disturbance itself. Therefore the depth of occurrence of the isostatic layer cannot be determined if the observation points are situated in a locality in which the influence of the disturbing forces is manifested only weakly. Bowie found that the isostatic layer lies at a depth of 95 km; in a later reduction for this depth the earlier value was obtained, i.e. 113 km.
In such a determination of the depth of the isostatic layer, a certain caution must always be borne in mind—as is shown by the investigations of Kohlschütter²) in German East Africa. Stations located on the edge of a plateau and in grabens are, in their situation, similar to stations on the steep margins of continents. The determination of the depth of the isostatic layer, in good agreement with other investigations, gave \(120 \pm 20\) km. However, further, more detailed investigations showed that compensation is fulfilled only on the average; in reality the plateau, since it rises 1290 m above the mean plane, represents an excess, while the graben represents a mass deficiency. But in such a case the determination of the depth of the isostatic layer is illusory.
For coastal stations, values ranging from 100 to 140 km were also found here for the depth of occurrence of the isostatic layer.
Thus, all observations led to the isostatic layer, and subsequently it was necessary to recognize the existence of an isostatic stratification of masses. It therefore proved necessary to subject all observations to the corresponding reduction. American investigations showed that, in the treatment of observations of gravity³), it is necessary to take into account the masses of the whole earth, down to the mass of the antipodes, since, by neglecting distant masses, we commit a very large error. Thus, for example, if station 49 (Salt Lake City) is taken, the total attraction of the 9 inner zones proves to be equal to 0.1230
¹) Bowie, W.: Investigations of gravity and isostasy. U. S. Coast geodetic survey, special publ. 40.
²) Kohlschütter E.: Über den Bau der Erdkruste in Deutsch-Ostafrika. Göttingen, Nachrichten. 1911.
³) Bowie, W., l. c.
cm/sec², but in this case the attraction of the zones decreases with distance from the center; hence one might have thought that the remote zones exert no influence at all. In reality, however, these values change sign and then again increase in absolute magnitude. If one goes as far as the 17th zone, the magnitude of the attraction proves to be equal to —0.062 cm/sec²; the final value of this quantity, obtained if all masses are taken into account, is —0.0414 cm/sec². As auxiliary means for such a complete reduction one may mention the tables of the above-mentioned American bulletins and the tables of Meissner1 and Niethammer2. The author3 hopes to bring his expansions in spherical functions up to the 16th order of the relations between elevations and depths on the earth into such a form that for every point of the earth the corresponding correction may be obtained. Unfortunately, all these numerous computations have not yet been published. It would be highly desirable to carry out similar isostatic reductions for all observing stations measuring the force of gravity. Since the attraction of very remote masses over large regions of the earth’s surface may be regarded as constant, a reduction performed by simple methods also gives good results. These results are especially suitable when it is necessary to compare small regions with one another. In the latter case the perturbation due to the action of remote masses enters, as a component part, into the normal force of gravity or into the disturbing influence of the continents.
III.
Thus, on the basis of observations, we have definitely established that isostasy takes place, i.e. that there exists a compensation between the visible and invisible irregularities in the distribution of masses. Our next task is to connect this fact with physical conceptions and to indicate how isostasy is connected with facts known from geology. Usually two different conceptions are associated with the idea of isostasy. One of them belongs to Pratt, and it has hitherto been used in carrying out the majority of computations; the other was given by Airy.
a) Pratt’s Hypothesis.
The fundamental idea of Pratt is as follows: equilibrium above the isostatic layer is due to the fact that the rise of masses above the level—
the ocean was accompanied by loosening of the underlying rocks and the associated decrease in their density. This explanation agrees with the views of modern geologists on certain types of mountain-building. Geological studies have shown that magma, highly compressed within the earth, when the pressure decreases, becomes not only more mobile and fluid, but that at the same time there also occurs an extraordinary increase in volume and the development of large quantities of heat. This decrease in density is, naturally, associated with this increase in volume. Therefore, when blocks are raised upward under the action of certain forces, the cavities arising in the process are filled by magma that has been relieved of its load and has expanded. In this way a mountain chain can rise, and this will not be accompanied by an increase in mass; the force of gravity will change insignificantly, and the mountains will be compensated from the very beginning. Such a case apparently occurs on the Colorado Plateau, which, without doubt, once rose and which now displays full compensation.
In order to facilitate the mathematical treatment of the material, it is assumed that the compensating masses lie directly beneath the disturbing masses and are distributed uniformly down to a certain depth, where the layer that we have called isostatic is already located. This hypothesis1 was used by Hayford and other scholars, and the investigations set out above showed that everywhere one obtains one and the same depth of the isostatic layer. What is involved here, of course, is a certain mean value, and it is possible that in some mountainous countries complete equalization of the masses takes place at a higher level, while in other places it occurs somewhat deeper.
The relation between the height of the external masses and the density of the corresponding defect is obtained from equality (4) or (5) on p. 42.
$$ \delta h = \delta_1 T \quad \text{and} \quad \delta h \cdot 0.615 = \delta_1 T $$
Since the rising masses, obviously, themselves lose density, it would be more correct to reckon the mass deficit on land not from sea level, but from the physical surface of the earth, and in the oceans to begin the reckoning from the bottom. Then, instead of formulas (4) and (5), we would obtain the formula
$$ \delta h = \delta_1 (T + h) \quad \text{and} \quad \delta h \cdot 0.615 = \delta_1 (T + h); $$
in the second formula \(h\) represents the depth of the sea, and therefore it must be taken with a negative sign. Since, in comparison with \(T\), \(h\)
represents a very small quantity (on average it amounts to from \(1/30\,T\) to \(1/40\,T\)), then this difference may for the most part be regarded as a quantity of the second order.
In accordance with this, Washington1 found that the mean density decreases with increasing height of the mountain.
Hübner2 determines the density of the defect from the equality between the visible masses and the masses lacking beneath the land, taking into account that the radii diverge outward. It is clear that this no longer corresponds to the concept of isostasy, since the latter requires determination of the defective density directly from equality (4). On the basis of the known fundamental equations of hydrodynamics it follows that
\[ dW=\frac{1}{\delta}\,dp \]
where \(\delta\) denotes density, \(p\) pressure, and \(W\) potential. If we put \(-\dfrac{dW}{dh}=g\), then we obtain \(dp=-\delta gdh\). Thus the pressure is proportional only to height and is wholly independent of the fact that, owing to the divergence of the radii, there is a greater mass in the upper layers than in the lower ones. Hübner’s construction is quite in accord with the idea of an isostatic distribution of masses, but it does not satisfy the condition of constancy of pressure on the isostatic layer. The contradiction here consists simply in the fact that Pratt’s assumption is, from the hydrostatic point of view, wholly unrealizable.
Fig. 2.
According to Pratt’s hypothesis, masses of equal weight must rest upon every arbitrarily small element of area of the isostatic layer. This assumption undoubtedly has its weak points. Indeed, let us consider Fig. 2a. Let \(ABEF\), \(BCGH\), etc., represent columns of masses of equal weight. Thus equal loads are imposed on the areas of the isostatic layer \(AB\), \(BC\), etc. In order that Pratt’s assumption could be fulfilled, it is necessary that the columns \(ABCF\), \(BCGH\), etc., be perfectly unyielding and entirely independent
THEORY OF ISOSTASY, ITS DEVELOPMENT AND RESULTS
from one another, as, for example, solid prisms standing side by side.
According to Bowie1, the picture should have the following form: let us imagine prisms of identical cross-section and identical weight, but made of different material; the lengths of such prisms must be different: if one imagines that these prisms are immersed side by side in some liquid, for example in mercury, then all of them will sink into this liquid to the same depth, and therefore will have a common base, which corresponds to the isostatic layer. Equality of cross-sections and weight is, obviously, even superfluous; it is sufficient that the length of the prism be inversely proportional to its specific weight.
The earth’s crust, however, cannot be imagined in the form of separate solid and mutually independent prisms extending all the way to the isostatic layer. If, for example, as a result of erosion part of the mass \(GHFF'\) were transferred to the surface \(EF\), then the prism \(ABEF\) would become heavier and would sink more deeply: owing to this the plane \(ABCD\) would lose the character of an isostatic layer, and the latter would have had to be chosen deeper from the very beginning. In general, changes so strong are always possible in the earth’s crust that they also affect the isostatic layer. In this sense the position of this layer depends on time.
In accordance with the fundamental ideas of Pratt’s theory, it is necessary to choose for the isostatic layer such a depth of occurrence that all phenomena on the earth’s crust, generally speaking, do not affect it; however, under the assumption to which Fig. 2a corresponds, this is impossible.
Let us now make the opposite assumption: let us suppose that the masses are mobile, as if they were in a liquid or in a plastic body. If we now consider the level \(FEF'\), then the pressure on \(FF'\) will prove greater than the pressure on \(EF\), although the density of the right-hand column is less than the density of the left-hand one. But above \(FF'\) there is a column of mass, whereas above \(EF\) there is no such column. Thus, in the right-hand column we in general have a greater pressure than in the left-hand one, and only at the isostatic layer does this difference disappear; the same will occur when we pass to purely density relations (Fig. 2b), replacing the step \(GFE\) by a continuous slope. On the right we shall constantly have an excess of pressure, and the moving masses will follow this excess; at the same time leveling will take place, the mountain will sink deeper, and throughout the whole mass hydrostatic equilibrium will be established of itself. But this, obviously, again—
this is not feasible, since the visible mountain chains are extraordinarily hard and, under the influence of the pressure that prevails at the surface, exhibit no properties of plasticity.
Thus there must exist some intermediate stage, characterized by the fact that the surface consists of solid blocks, but downward it becomes, under the influence of increasing pressure and temperature, more and more plastic. Let, in Fig. 2c, \(CDEF\) represent a very hard surface block; then part of the weight is transmitted to the left side, and the block will withstand the corresponding differences of pressure. At great depth the masses will be displaced until equilibrium is established. From this, however, it follows that adjustment between surface formations and deep ones cannot take place over arbitrarily small areas: such adjustment is possible only over large surfaces, and thus the question arises as to the magnitude of the minimum surface on which such adjustment takes place.
Thus we must establish at what depth and under what kind of load the difference of stresses becomes so great that it overcomes the resistance of the materials and causes the masses to move.
Darwin (Darwin)\(^{1}\) proceeded from the assumption that the earth is a homogeneous elastic sphere, and calculated that a mountainous region whose elevations and valleys deviate from the mean level by \(\pm 2000\) m and extend over a distance of 315 English miles between parallel ridges, with an average density of 2.8, must already sink into the earth if the material lying at a depth of 50 miles possesses no greater hardness than tin or lead. This corresponds to a difference of stresses of \(0.4\ t\) per \(\mathrm{cm}^2\).
Jeffreys (Jeffreys)\(^{2}\) calculated that, for a thickness of the solid earth’s crust amounting to more than half the distance between mountain ridges, with a difference of heights of \(\pm 1500\) m and a density of the mountain rocks of 2.7, the difference of stresses will have a maximum value of \(0.6\ t\) per \(\mathrm{cm}^2\), whereas basalt under ordinary conditions can withstand, without breaking, a stress of \(2\ t\). Jeffreys made calculations for mountains of smaller dimensions and lower density than Darwin, and nevertheless his stresses turned out larger than Darwin’s. This occurred because Jeffreys took into account only the earth’s crust, which accordingly has a lower density,
\(^{1}\) Darwin, G. H.: On the stresses caused in the inferior of the earth by the weight of continents and mountains. Phil. Trans. of the R. S. 1882, p. 173; Scientific Paper 2, 9.
\(^{2}\) Jeffreys, H.: The earth, its origin, history and physical constitution. Cambridge 1924.
between them, whereas Darwin proceeded from the density of the entire terrestrial globe. Therefore Jeffreys’s deformations turn out to be much larger, and the stresses also turn out to be larger, just as, under the same load, a stronger beam bends less than a weaker one, in which the limiting stress is reached much sooner.
If a range is relatively broad and heavy, the crust will bend and thereby force the underlying plastic substance to yield, owing to which an isostatic equilibrium of the masses will be established.
According to Jeffreys’s calculations, the Alps are still too small to be compensated. They could not bend the earth’s crust, 50 km thick; the latter would bear their weight without any deformation. For such bending it is necessary that the distance from one mountain range to another be equal to 1200 km. The Caucasus likewise should still be uncompensated, while the Himalayas should lie at the boundary of compensation; the continents, however, should be fully compensated.
Born (A. Born)¹ raises the question of how thick a layer of earth must be in order for it, by its weight, to break through the underlying earth’s crust; in doing so he uses the strength data given in Hirschwald’s handbook and obtained on the basis of experimental measurements of building stones; naturally, these data can be used only with great caution. Here, too, it turns out—the discussion concerns only sedimentary rocks—that to break through a block having a small diameter a very large load is necessary; for example, to break through a circle 10 km in diameter a load of 21 km of sediments is necessary, with a crustal thickness of 120 km. However, in all probability the crust first bends strongly and only then breaks through.
A breakthrough, just like bending, displaces plastic subterranean masses, owing to which isostatic equilibrium is established.
All theoretical investigations, apparently, agree that compensation can exist only for very large formations, while observations point to the existence of the opposite phenomenon²). Not only con—
¹ Born, A.: Isostasie und Schweremessung, ihre Bedeutung für geologische Vorgänge. Berlin, 1923.
² Love (Love: Some problems of Geodynamics, Cambridge, 1911), proceeding from the assumption of the existence of complete isostasy and making the same assumptions as Darwin, obtained smaller values for the differences of stresses (0.26 t per 1 cm² against Darwin’s 0.41 t). For continents these differences of stresses are completely negligible, which is entirely understandable: after the leveling of the masses has occurred, the differences of stresses must disappear completely. Small differences remain still more—
continents, but also all large mountains, such as the Alps, the Caucasus, the Himalayas, rocky mountains, etc. According to Niethammer¹), compensation may already be expected within an area of 64 km², while the Americans used for their calculations circles with a radius of 18.8 km, i.e. an area of 1000 km² (approximately). This disagreement between observations and theoretical investigations arose because, in the theoretical investigations, it was apparently assumed that the substance within the earth behaves exactly as it does on its surface. However, if one takes into account the high temperature that prevails inside the earth, this proves to be unlikely; making a calculation with the aid of the usual geothermal gradient (3° per 100 m), we obtain at a depth of 50 km a temperature of 1500°, and at a depth of 80 km (50 miles)—2400°, i.e. a temperature at which the yielding capacity of rocks must increase considerably.
True, the geothermal gradient decreases as one approaches the center of the earth; otherwise, near the center there would be an impossible temperature. However, this change of the gradient within the upper layers is not especially great.
Owing to the absence of reliable conceptions about these matters, it is best of all to change the very course of the investigations: conclusions about the state of the earth’s crust and isostatic deductions should be made not on the shaky ground of theoretical constructions, but, on the contrary, starting from observations, which show that isostatic leveling already exists in small regions,—to draw conclusions about the state of the earth’s crust and thence infer the properties of the substance situated inside the earth or judge the temperature occurring there. Then it will turn out that the earth’s crust at a depth of 50 km can by no means be solid, and still less is it solid at a depth of 120 km. It may be that the plastic properties of matter appear already at a depth of 20–30 km, especially since a very long time is available for the manifestation of these properties.
With the supposition of a uniform distribution of defects within the earth one may associate a physical representation. Let the density of the external masses be \(\delta_0\), and their height before deformation be \(h\); in that case the pressure per unit surface is proportional to \(\delta_0 h\). Owing to this pressure the earth’s crust will bend and displace the plastic mass situated within the earth; if the density of this pla—
—because isostatic equilibrium is not identical with hydrostatic equilibrium. The results obtained by him show that, after isostatic equilibrium has already been established, it is sufficient to apply a very insignificant force in order to bend masses covering a large area. But the question of what forces must be overcome in order to establish isostatic equilibrium remains unresolved in Love’s work.
¹) Niethammer, Th.: Die Schwerbestimmungen der schweizerischen geodet. Kommission und ihre Ergebnisse. Verhandl. der schweiz. naturf. Ges. Schaffhausen, 1921.
of the mass is equal to $\delta_1$, then the thickness of the displaced layer, according to the basic law of isostasy, must be equal to $\dfrac{\delta_o}{\delta_1} k$, and if we do not take compressibility into account, this will hold throughout the entire bent layer. Thus the mass which, without deformation, ought to have been at the depth $x - \dfrac{\delta_o}{\delta_1} k$, is now found at the depth $x$. This descended mass carries with it the density $\delta - \dfrac{d\delta}{dx}\cdot \dfrac{\delta_o}{\delta}\cdot k$. Consequently, the density of each layer proves to be less than the normal density by $\dfrac{d\delta}{dx}\cdot \dfrac{\delta_o}{\delta}\cdot k$. If we assume that the density decreases linearly in approaching the center of the earth, then it follows that $\dfrac{d\delta}{dx}=\mathrm{const.}$, and therefore $\dfrac{d\delta}{dx}\cdot \dfrac{\delta_o}{\delta}\cdot k$ is the same throughout the whole defective layer. Thus the difference between the density of the defective and the normal layer is everywhere the same.
The quantity $\dfrac{d\delta}{dx}\cdot \dfrac{\delta_o}{\delta_1}\cdot k$ represents the defective density. If the mass defect, according to Pratt’s hypothesis, extends to the depth $T$, then the defective density by formula (4) is equal to $\dfrac{\delta_o h}{T}$, where $h$ is the height of the mountain after the bending of the earth’s crust. Thus we obtain $h = k - \dfrac{\delta_o}{\delta_1}\cdot k = k\cdot \dfrac{\delta_1-\delta_o}{\delta_1}$, and we arrive at the equality
\[ \frac{d\delta}{dx}\cdot \frac{\delta_o}{\delta_1}\cdot k = \frac{\delta_o}{T}\cdot k\cdot \frac{\delta_1-\delta_o}{\delta_1} \]
or
\[ \frac{d\delta}{dx}=\frac{\delta_1-\delta_o}{T}. \]
The same value of $\dfrac{d\delta}{dx}$ is also obtained under the assumption that $T$ represents the thickness of the earth’s crust, and $\delta_1$ its density at the lower boundary. Since $\delta_1$ previously represented the density of the mass displaced from within the earth, it follows from this that the density experiences no discontinuity in passing from the earth’s crust to the masses within the earth, which is quite in accord with our conception of the structure of the earth. Difficulties arise only insofar as it is scarcely possible to suppose that a crust 120 km thick is bent. Jeffreys found that even with a crustal thickness of 50 km the greatest mountains can be sustained by the earth’s crust without deformation. Therefore the depth of occurrence of the isostatic layer should be chosen much smaller.
If one imagines those varied phenomena which led to the formation of the earth’s crust, mountains, and oceans, it is impossible to believe that an isostatic state has prevailed continuously from the very beginning. It is far more probable that after all disturbances it is restored again, and we are convinced that this is still happening now, although after the solidification of the earth’s crust these phenomena must slow down more and more. A disturbance of isostatic equilibrium can be caused only by a horizontal displacement on the surface of, or within, the earth, and can be eliminated only in the same way. These movements may occur as a result of the accumulation or transfer of masses. One should not, of course, think that the restoration of isostatic equilibrium follows immediately after its disturbance.
The earth’s crust does not yield at once and can withstand a certain load not exceeding a definite limit. After this limit has been reached, the process of restoring isostatic equilibrium proceeds for some time at an accelerated rate and is sometimes accompanied by strong earthquakes; this process continues until, owing to internal resistance, i.e. owing to the friction and rigidity of the acting masses and of the material found within the earth, it again slows down. Such processes may be repeated many times.
The accumulation of masses may occur in various ways. Let us first consider the process of the “swelling” of magma, which has already been discussed on p. 531. So long as this phenomenon affects only that magma which lies directly beneath the hollow space that has formed, the isostatic state established from the very beginning does not change. But in this case one can speak only of the vertical uplift of blocks. When, however, as a result of lateral displacements, fissures are formed, the isostatic state changes, since the material that formerly lay in the place where the fissure formed departs, and no new material flows in to replace it. In this case a defect is formed. It may happen, however, that the space in the fissure that has formed is much greater than that which the underlying magma can fill as a result of the increase in its volume. Then new magma flows into the fissure from the side. As a result, subsidence of the ground occurs around the fissure, just as is in fact observed. In this process, however, the resulting bulging may reveal an excess of masses. This, for example, explains why young volcanic formations show an excess force of gravity. The excess, however, must disappear with time, and in fact—
THE THEORY OF ISOSTASY, ITS DEVELOPMENT AND RESULTS
therefore, on volcanic islands we observe a slow subsidence of the surface, which can be detected thanks to the growth of coral reefs.
The second cause producing accumulations of masses is shifts and mutual displacements of folds: here too isostatic displacements begin only after this load reaches a certain definite magnitude and the plastic masses are displaced to where, as a result of the shift, an excess of weight has formed at the surface. Here also, in young formations, complete isostasy is still evident.
Let us consider, finally, sedimentation as the last case of mass accumulation. The masses that are carried away in this process come from mountains and are deposited in foothills or, finally, in large river deltas. Lawson (Lawson1) pointed out an essential difference, connected with the fact that in one case the masses travel a short distance, while in the other they are transported over great distances. In the first case, for example, during erosion in mountains, in which a large part of the masses is deposited already in the foothills, the regions in which both positive and negative disturbances occur lie close to one another. In this case such regions can compensate one another. In the second case, for example in the formation of deltas, the masses are transported far from their source. The delta forms a region of disturbance, surrounded on all sides by regions in which no disturbance has occurred. The delta must become leveled with them, and therefore the disturbances will spread with diminishing intensity.
The height of the surface does not increase to the same extent as the thickness of the sediments. When the weight of the accumulating masses increases, it presses on the underlying layers, and part of the plastic masses must yield in response. However, since the density of these masses is greater than the density of the deposits (the density of the former is 2.9, and the density of the latter 2.2), the thickness of the displaced layer, after equilibrium is again established, must amount to only \(22/29\) of the thickness of the deposits. Thus, under the pressure of 1 km of deposits, the layer lying beneath them will be compressed to 0.75 km, whereas the upper level will rise only by 0.25 km. Thus it becomes clear that the assumption of isostatic equilibrium makes the accumulation of thick sediments much more comprehensible than the assumption that the lower layers do not yield.
Phenomena of loading correspond to phenomena of unloading, which are likewise capable of producing isostatic movement.
Here, first of all, one should point to the phenomena of erosion. Under the influence of water, enormous masses of mountains are transported over considerable
distances and settle somewhere else. The mountain range is unloaded, while the regions surrounding the mountains, which are in equilibrium or overloaded, press the plastic substance in under the mountain and thereby cause the mountains to rise. Such an uplift of mountains partly compensates for the lowering of their mean level caused by erosion. Since the density of the mass pressed in is approximately equal to 2.9, while the density of the mountain is 2.7, in order for equilibrium to be established, a layer whose thickness is equal to \(27/29\) of the eroded layer is sufficient. A kilometer of mass carried away by erosion corresponds to a subsoil layer \(0.93\) km thick—by this amount the mountain rises, whose mean height is thereby lowered only by 70 m. Thus erosion has a very weak influence on the height of mountains.
The second case, in which isostatic processes are caused by unloading, is observed in regions that underwent glaciation during the glacial period. The weight of the ice blocks forced the plastic masses lying beneath them into adjacent regions. The ice masses have now disappeared, and these regions reveal a defect of gravity. The displaced masses are again slowly returning to their former position, which can be detected from the uplift of the regions that were formerly covered by glaciers. This can be seen most clearly in the Fennoscandian massif and in Scotland; the smallest such mass deficit after the thawing of the glaciers proved to be in Labrador1.
If, indeed, isostasy took place in all parts of the terrestrial globe, then neither in the disturbance of the plumb line nor in the disturbance of gravity, after proper reduction, should there remain any irreducible residuals. However, such residuals are always obtained, and they may be explained in two ways: either there is in fact a disturbance of isostatic equilibrium, or an error was made in the reduction.
It is clear, however, that disturbances of isostatic equilibrium do occur. This follows already from the circumstance that isostatic shifts cannot take place at the same rate as the disturbances that cause them. An excess of pressure on one side or the other must first pass beyond its limiting value, and only then can it exert some action on the underlying layers. This latter, in all probability, occurs in jumps and with a certain constant delay. The masses within the earth are also not easily set in motion. The materials of which they are composed are considered, at the surface of the earth, to be very hard. Only under the influence of enormous pressure and high temperature do these masses become plastic. It is therefore necessary to take into account that young formations
must be compensated only in part, or not compensated at all; whereas the old formations have already reached full compensation.
The fact that the masses causing the disturbances are not always situated above the masses compensating them is one that cannot be denied, although American geophysicists assert that this cannot be concluded from their observations. However, the American observation stations are not sufficiently dense; even the 314 stations that were established during the 1916 survey1, for the expanse of the United States, constitute a very sparse network in comparison, for example, with the 231 stations available in Switzerland, which is 200 times smaller.
The necessity of such a relative displacement of the masses causing the disturbances and the compensating masses follows directly from the fact that isostatic equilibrium is established very slowly. One may say outright that the compensation of the mass defect of Scandinavia at the present time still lies in the vicinity of the region once covered by ice. The map of gravity anomalies in Switzerland2 indicates the existence of a mass defect in the Rhine valley, in the region of Chur, i.e. by no means where the large visible masses are located, as, for example, in the Bernese Oberland. Therefore the map of isostatic anomalies also indicates a strong disturbance of isostasy in this region.
In Tyrol the mass defect likewise lies not beneath the mountain chain, but is shifted somewhat to the north3.
When there is a large distance between the masses causing the disturbances and the masses compensating them, the connection between them is naturally difficult to notice.
It must be noted, however, that the fact that anomalies still remain in the disturbance of gravity is not yet proof of a violation of isostatic equilibrium. These anomalies may arise because the compensating masses are distributed in the inner layers of the earth not in the way we have assumed for simplicity of computation. In such a case the anomalies occur as a consequence of the imperfection of the reduction. American investigations have shown that the majority of observation stations lying on Precambrian formations have positive residuals, while stations lying on Cenozoic formations have negative ones. The former formations have a considerably greater
density than the latter. In the first case, despite complete compensation, a disturbance of equilibrium might occur because the denser masses lie above, in the immediate vicinity of the observation points, while below there are masses possessing lower density. In the Cenozoic stations this phenomenon might proceed precisely in the reverse order. However, on the basis of theoretical considerations it has been found that irregularities in the vertical distribution of layers of different density can influence the plumb line only in the case when these layers do not extend especially far in the horizontal direction. Thus, such an apparent disturbance of isostatic equilibrium is possible only where there are no formations occupying large spaces.
b) Airy’s Hypothesis.
We have now come to the second hypothesis on the structure of the earth, based on isostasy. According to this hypothesis, the earth’s crust consists of blocks of lighter material which float upon a more deeply lying, heavier and plastic mass. The depth of the isostatic surface is then determined by the lower surface of the thickest blocks. Considering the ratios of heights on the continents and depths of the sea floor, one can clearly perceive the existence of two selected levels. Namely, if one draws profiles in which the vertical dimensions are considerably exaggerated—for example, the profiles of Prof. Heiderich (Heiderich)—this becomes very clear. Indeed, the margins of the continents are generally not especially steep—their slope is on average equal to 5%—although there are regions of very great steepness. It must, however, be remembered that with the passage of time all slopes are smoothed out; thus it is possible that all the margins of the continents were once steeper, and that the continents then did indeed have the appearance of blocks.
Fig. 3.
Two conceptions are associated with this second hypothesis of isostasy. According to one of them, which was introduced by Heiskanen, the sea floor likewise represents the surface of a similar block; whereas, according to the other conception, the sea floor already lies at the boundary of the plastic layer. Under both the one and the other assumption, the upper and lower layers differ from one another in their composition, and this difference is expressed in a difference of densi-
THEORY OF ISOSTASY, ITS DEVELOPMENT AND RESULTS
of rocks: sial and sima of the geologists1. If we denote (Fig. 3) the thickness of the continental blocks by \(d\), and their density by \(\delta\), then these blocks will sink into the underlying layer, whose density is \(\delta_I\), to a certain depth \(x\), and will protrude from this layer by \(d - x\). If we suppose that the continental blocks are surrounded by water to a depth \(t\), then, on the basis of Archimedes’ law, we arrive at the following relation:
\[ d \cdot \delta = x\delta_I + t \cdot 1.03 \tag{6} \]
For a layer of thickness \(d'\), situated beneath the ocean, the weight of the water filling the ocean constitutes a load. Taking this into account, we arrive at the relation
\[ d' \cdot \delta + t \cdot 1.03 = x'\delta_I \tag{7} \]
It is further necessary to take into the calculation that the mean height of the continents is about 800 m above sea level, while the mean depth of the ocean is (approximately) 3700 m; and, thus, the continents rise above the ocean floor by 4.500 km. This leads us to the equality:
\[ (d - x) - (d' - x') = 4.5 \tag{8} \]
Let us put \(\delta = 2.7\), \(\delta_I = 2.8\), i.e. \(\delta_I - \delta = 0.1\), and for \(d\) take the series of values 100, 80, 60, 40, 20 km. From equality (6) we find for \(x\) the following series of values:
\[ 95.1,\quad 75.8,\quad 56.5,\quad 37.3,\quad 17.9\ \text{km} \]
these blocks therefore rise above the level of the layer in which they float by:
\[ 4.9,\quad 4.2,\quad 3.5,\quad 2.7,\quad 2.1\ \text{km}; \]
but, since the ocean floor lies more than 4.5 km below the surface of the continents, only the first result has any meaning. Under this first assumption, from equality (8) it follows that \(d' - x' = 0.4\), and from equality (7) we obtain \(d' = 49\) km.
For \(\delta = 2.7\) and \(\delta_I = 2.9\), i.e. \(\delta_I - \delta = 0.2\), we obtain the following mutually dependent values:
| \(d =\) | 100 | 80 | 60 | 40 | 20 km |
| \(x =\) | 91.8 | 73.2 | 54.5 | 35.9 | 17.3 ″ |
| \(d - x =\) | 8.2 | 6.8 | 5.5 | 4.1 | 2.7 ″ |
| \(d' - x' =\) | 3.7 | 2.3 | 1.0 | — | — |
| \(d' =\) | 72.5 | 52.5 | 33.5 | — | — |
in the second representation (Fig. 4) we must, in the equations given above, put \(d'\) and \(x'\) equal to zero and omit equation (7); thus these equations reduce to the following two equations:
\[ d \cdot \delta = x\delta' + t \cdot 103 \]
\[ d - x = 4.5. \]
Since now only two unknown quantities remain, the thickness of the layer is determined uniquely, for given \(\delta\) and \(\delta'\).
For \(\delta = 2.7\) and \(\delta_1 - \delta = 0.1\), one obtains \(d = 88\) km.
For \(\delta = 2.7\) and \(\delta_1 - \delta = 0.2\), one obtains \(d = 46.5\) km.
Of these two representations the second is undoubtedly the more plausible. Under the first representation we would have had to assume that, during the cooling of the earth, blocks of double thickness (and perhaps also of double density) were formed; for example, for \(\delta - \delta_1 = 0.1\) these blocks would have had to be 100 and 49 km thick. But nothing compels us to suppose that such a division existed. It should also be noted that, according to this hypothesis, the blocks would have had to collide with one another in such a way that no cracks remained anywhere; one must suppose that, in addition, there must also have existed a third level surface, namely the surface of that plastic layer upon which the blocks of the first and second kind floated. Or perhaps this was brought about by the greatest sea depths, which are found near the island of Tonga or near the Kuril Islands. According to the second representation, the very bottom of the ocean is the surface on which the stone blocks float, or, in the extreme case, this bottom was covered by a thin layer of sediments. For the Pacific Ocean this representation is apparently confirmed also by observations of the velocity of propagation of earthquakes. True, in the Atlantic Ocean and in the Arctic seas other relations obtain.
Fig. 4.
All isostatic phenomena, according to Airy’s hypothesis, take place exactly as in Pratt’s. Namely, continental blocks have an enormous extent for their density. Under a local load the continental block does not sink as a single whole, but bends and displaces the sima, i.e. these phenomena occur just as under Pratt’s hypothesis. If, however, the load proves too great, then the block breaks apart and then sinks as deeply as
THE THEORY OF ISOSTASY, ITS DEVELOPMENT AND RESULTS
This corresponds to Archimedes’ law. The edges of the fracture are then again washed away; however, they represent the place of least resistance, and therefore subsequent new displacements will occur more easily here.
A local reduction of the load has as its consequence the formation of a bulge, since at this place the crust offers less resistance to the subterranean pressure; beneath this bulge heavier masses will gather, replacing the masses of the upper layers carried away during unloading.
If the pressure of the subterranean layers on the crust is sufficiently great, or if the crust becomes sufficiently thin, then a break will occur, and this break will proceed upward from below until the block comes to rest in accordance with Archimedes’ law. In this way a state will in fact be attained that fully corresponds to Airy’s hypothesis, and the earth’s crust, in the place where it rises above the surface of the oceans, will be immersed more deeply into denser layers (Figs. 3 and 4).
Closely connected with the theory of blocks is Wegener’s well-known theory of continental displacements. Assuming that the continents float on the inner layers, in accordance with the fundamental laws of hydrostatics, we thereby ascribe to these layers certain properties of a liquid. At this point the question arises: to what extent are these inner layers fluid? Can the continents still move in these layers, or are they, to a certain degree, fastened to them? And what is the force capable of moving the continents?
Fig. 5.
Here, first of all, there comes to mind the so-called retreat of the pole. Let us imagine that the earth is an equilibrium figure of a rotating liquid, and let us replace a part of the surface mass by a floating continent. The center of gravity of the latter is located higher than the center of gravity of the liquid mass displaced by it. A higher position of the center of gravity corresponds to a decrease in the force of gravity and an increase in centrifugal force. In the equilibrium figure of a rotating liquid, the resultant of the forces of gravity and centrifugal force is perpendicular to the surface of the liquid. In this case the force of gravity gives a component directed northward, and the centrifugal force gives a southern component, and these two components balance one another (Fig. 5). If the force of gravity decreases, then the northern component also decreases, and the centrifugal force outweighs it; this excess is especially significant in the case when the centrifugal force itself increases. As a result, both of these causes produce a force directed southward. Thus it is quite possible that under this influence
continents are moving southward. True, one cannot say with certainty that the matter is so simple. It is possible that, under the influence of changes in the force of gravity and of the centrifugal force over an enormous extent of the globe, the latter oscillates somewhat, as a result of which the southern component disappears.
The present configuration of the terrestrial parts does not make it possible to draw a conclusion as to the existence of such deviations of the pole. One may even say, with some justification, that the continents, with the exception of Africa, are moving away from the equator; between Asia and Australia the connection has already been severed, while in America this connection is reduced to a very narrow bridge in the tropics. True, if one assumes that the pole shifts in such a way that at different times the south lies in different directions, then the present equator will play no role. But, from the geophysical point of view, it is quite impossible to admit that the pole could shift so greatly, especially since, as we know, the continents are compensated. Thus it is difficult to imagine that the earth could be brought out of a state of equilibrium.
Secondly, one should consider the influence of tides and ebbs. Darwin calculated that the tides and ebbs caused by the moon are associated with a distortion of the earth’s surface. In this case, the closer a given point lies to the equator, the more strongly it is attracted westward; however, these displacements are so small that at present they cannot be taken into account. Darwin found that the displacement of longitudes during the last 46 million years amounts to \(19' \cos^2 \varphi\). This number may in reality be still larger; it was calculated on the condition that the earth represents a homogeneous mass, whereas the density of the earth’s surface is considerably less than its mean density. But even after multiplying by 4 or by 5, this number nevertheless remains very small. However, the possibility is not excluded that in times long past this influence of the moon was stronger, since the displacement of longitudes is inversely proportional to the 6th power of the distance of the earth from the moon and directly proportional to the relative angular velocity of the earth with respect to the moon. If we calculate the displacement of longitudes, relying on Darwin’s theory of the evolution of the earth–moon system and taking into account the corresponding distances of the earth and moon, as well as the periods of revolution of the earth and moon as they are obtained from the theory of tidal friction, we obtain the following table:
| Time of the earth’s rotation | Time of the moon’s revolution | Distance in earth radii | Change of meridians per year | |
|---|---|---|---|---|
| Now . . . | \(23^{h}56^{m}\) | \(27.32^{t}\) | 60.4 | \(5.10^{-10}\cos 2\varepsilon \cos^2 \varphi\) degrees |
| 46,300,000 years ago . . | 15 30 | 18.62 | 46.8 | \(4.10^{-8}\) |
| 56,600,000 ” ” ” . . | 9 55 | 8.17 | 27.0 | \(1.6.10^{-6}\) |
| 56,800,000 ” ” ” . . | 7 50 | 3.59 | 5.6 | \(5.10^{-5}\) |
| 56,810,000 ” ” ” . . | 6 45 | 1.58 | 9.0 | \(1.6.10^{-3}\) |
These values were calculated by Darwin on the assumption that, under present conditions, the influence of tidal friction on the inclination of the ecliptic is maximal. This corresponds to a relatively strong mobility of the masses. But such strong mobility does not exist at the present time. Therefore the intervals of time given earlier should be increased still more substantially. The values chosen by Darwin for the retardation of the tides and ebbs are obtained from \(\varepsilon = 17.5^\circ\) for semidiurnal tides and for the present time. If we take \(\varepsilon = 0\), then we obtain the maximum value of the change in longitude. It would occur if the surface of the earth were entirely fluid. To this correspond, therefore, for \(\varphi = 0\) (the equator), the numerical coefficients of the last column. The value of \(\varepsilon\) depends on the viscosity chosen. From Darwin’s formulas we find the following mutually related quantities:
| \(\eta t^3 = 10\) CGS | \(2\varepsilon = 0\) | \(\cos 2\varepsilon = 1.00\) |
|---|---|---|
| \(10^4\) | 0.13 | 1.00 |
| \(10^5\) | 2.10 | 1.00 |
| \(10^6\) | 20.10 | 0.94 |
| \(10^7\) | 75.1 | 0.26 |
| \(10^8\) | 88.28 | 0.03 |
Here \(v\) denotes the coefficient of rigidity, and \(t\) the relaxation time; thus \(vt\) represents the coefficient of viscosity. It follows from this that, up to a very considerable degree of rigidity, the earth behaves in the same way as a fluid. In this connection a fluidity of \(10^{18}\) CGS corresponds, for example, to a body possessing the rigidity of glass \((v = 2.44 \cdot 10^{11})\) with a relaxation time of 11 hours, or the rigidity of steel \((v = 7.8 \cdot 10^{11})\) with a relaxation time of 3–5 hours. On the basis of the figures in the last column of the preceding table, we find that in the course of 10,000 years there will already occur a displacement of longitude by \(16^\circ\), and in the course of 100,000 years by \(160^\circ\). However, for this motion we may take larger intervals of time; then, taking much larger values for the viscosity, we shall nevertheless obtain fairly large displacements of longitude. This effect was maximal when the time of the earth’s rotation about its axis was equal to 5 hours 50 minutes, and the time of the moon’s revolution about the earth was equal to 7 hours 10 minutes—it was then 56 million times greater than now. Thus it follows that the annual displacement at that time was equal to \(18^\circ\). It is therefore beyond doubt that in the early periods of the earth’s development there could have been strong shifts of continents, even if one assumes that the viscosity at that time was as great as it is now. These displacements will prove still more probable if one assumes that the earth’s crust was formerly much softer than now. Apparently Darwin himself takes account of the geological significance of this fact when he points to the form of the Asian shores of the Pacific Ocean and to the shores of the Atlantic Ocean
of Europe and America. But in any case the effect of these displacements may have been appreciable at least 50 million years ago.
One should not, however, imagine that the continents floated on the underlying layers like ships; in reality, these layers also took part in the motion: the continents were carried away by the flow of these subterranean layers just as floating ice floes are carried away by an ocean current. This, it seems to me, considerably facilitates the conception of the movements of the continents. One should not be surprised that the continents did not remain joined together, but tore away from one another, rotated, and so forth. For in all these movements the continents had to overcome various obstacles and resistances. Here it is impossible to indicate to what extent the fundamental principles of geology contradict the assumptions of this theory.
On the basis of the observations available, it is not yet possible to decide definitively which of the two isostatic hypotheses should be preferred: the hypothesis of Pratt or the hypothesis of Airy. Heiskanen found that for the Caucasus and for the United States the Airy hypothesis is somewhat better, but the difference in this case is very slight. Therefore it is almost immaterial which of these two hypotheses is correct. The chief difference is that in Pratt’s hypothesis the entire defect is distributed uniformly until it reaches the isostatic layer, while the defective density varies depending on the height of the visible masses. According to Pratt, the normal density increases in the earth’s crust with increasing depth into the earth. In Airy’s hypothesis the defective density is always equal to the difference between the sial and sima layers, and the compensation of the overlying layers depends entirely on the lower boundary. The density of the earth’s crust may in this case be regarded as constant. If the space occupied by blocks is not especially large in comparison with their thickness, then the results of reduction on the basis of these two methods may differ greatly from one another. In their investigations the Americans made reductions by Pratt’s method and did not obtain large discrepancies. On this basis the Americans reject Airy’s hypothesis. Heiskanen carried out the same reductions by Airy’s method and likewise did not obtain significant discrepancies. Apparently, those cases in which reductions by different methods give results differing from one another are very rare, and in these exceptional cases the divergence can easily be attributed to local disturbances. Thus, it is possible to decide which of these hypotheses is correct only on the basis of geological structures. The theory of blocks, at least with respect to large forms, proves to be more probable; however, in some cases the second hypothesis is also justified. Thus, for example, it cannot be denied that the second hypothesis explains very well the phenomena associated with the unloading of magma.
THE THEORY OF ISOSTASY, ITS DEVELOPMENT AND RESULTS
Movements are undoubtedly connected with isostatic phenomena, and these movements belong to the geological development of the earth’s crust. It would be wrong, however, to think that this development took place only thanks to isostatic phenomena. One cannot agree with Bowie when he completely excludes large lateral displacements of mountain formations and reduces everything to the expansion of the substance of which the earth’s crust consists, and from this comes to the conclusion that disturbances of isostatic equilibrium are generally impossible.
First of all, it must be noted in this regard that geologists by no means deny the theory of lateral displacements. On the contrary, in the modern theory of nappes this theory plays a very large role.
In Bowie’s theory the question is not of that unloading of magma of which we have already spoken, but of sediments which settle under the influence of their own weight and lead to mountain formation. Let us suppose that a thick layer of sediments has formed and that this layer, under the influence of its own weight, has subsided so strongly that its upper boundary has coincided with ocean level. According to Bowie’s theory, the lower parts thereby increase very greatly in their volume, lift the layers lying above them, and are the cause of the formation of mountains. Folds and dislocations occurred in the mountains as a result of the unevenness of these phenomena. Upon a more attentive study of Bowie’s theory, however, so many difficulties emerge that one has to doubt the possibility of such a development of the earth. Let us assume, for example, that the thickness of the layer at first is equal to 10 km, and that the new mountain rises by 2,000 m; in that case the thickness of the layer must increase by \(1/5\). Since the density of the mass of which the mountain consists is 2.7, before expansion this density must have been 3.4. This is still admissible. But, according to Bowie, the defect must be confined to the upper layer, 10 km thick, with a deficient density of 0.7, and this in any case is wholly inconsistent with Hayford’s assumption.
If one asks why these expansions occur, then first of all one thinks of explaining them by the action of heat. But, assuming that the mean temperature of the layer is \(150^\circ\)—at a depth of 10 km a higher temperature is hardly possible—we obtain, for an expansion of 20%, an incredibly large coefficient of expansion (almost 100 times greater than the coefficient of expansion of iron). If, however, one supposes that the sediments descended much deeper and that their temperature was therefore much higher, then, in order for them to reach the surface of the earth, their expansion must also be much greater. True, in this case Bowie has in mind not the influence of temperature, but chemical and physical changes; but the existence of such changes is still unknown.
At the same time, the uplift of the earth by 2,000 m would prove still too insignificant. Indeed, if the mountain region does not rise
for thousands of meters (we are not here taking into account the action of erosion), then where do such massive sedimentary deposits go? Let us suppose, finally, that a new mountain region has formed and, as a result of erosion, a layer of sediments of very great thickness has again arisen near it. Can these masses now undergo the former metamorphosis with an increase in volume?
It therefore seems to me that if such a formation of waters were, generally speaking, possible, it would have had to manifest itself at once.
Thus, the known forces of orogenesis arising as a result of the lateral displacement of mountains cannot be completely ignored. On the contrary, the influence of these forces is very great; near the earth’s surface colossal forces arise very quickly; these forces are the chief cause disturbing the state of isostatic equilibrium. The action of orogenetic forces occurs rapidly and suddenly and entails a disturbance of order; the work of isostasy, which strives to restore order and equilibrium, proceeds slowly and continuously.
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“Sial” is the name of a large group of rocks forming the basis of the upper layer of our continents. This name was proposed by Zoss and is composed of the initial syllables of the elements that play the chief role in them, Si Al—silicon—aluminium. Analogously, “Sima”—silicon—magnesium (Silicium-Magnesium)—is the name for the heavier deep-seated rocks (basalt). Ed. ↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩
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Astronomisch-geodätische Arbeiten in der Schweiz, 15, 16 — Niethammer, Th. Die Schwerebestimmungen der Schweiz, geodät. Kommission und ihre Ergebnisse. Verhandl. der Schweiz. naturf. gesellschaft. Schaffhausen 1921. ↩↩↩↩↩↩↩↩
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Prey, A.: Untersuchungen über die Isostasie in den Alpen auf Grund der Schweremessungen in Tirol. Sitzungsber. d. Akad. d. Wiss. in Wien I. Mitteilung 121; II. Mitteilung 123. ↩↩↩↩↩