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(Received 1974 February 8) Summary In many theoretical rcsearches and operations in geodesy the geonietrical and physical quantities under consideration must be framed in an appropriate system of co-ordinates with simple metric. An attempt is here made to establish a one-to-one correspondence between points of the actual gravity field of the Earth and the normal ellipsoidal one both referred to geographic co-ordinates and to potential, which permits the univocal definition of anomalies and deflections, and the use of the simple metric tensor of the normal field for their analytical manipulation. The fundamental equation of physical geodesy, as well as the integrability conditions for deflections and anomalies are derived. 1. One of the main problems in both theoretical and operational geodesy is that of framing the geometrical or physical quantities observable at or near the surface of the Earth, in a system of co-ordinates of known and possibly simple metric. Traditionally, the problem is solved in two dimensions by a series of reductions of the observed values to the normal ellipsoid referred to geographic co-ordinates, the simpte metric of which is thereafter used for all subsequent planimetric computations; whereas the third co-ordinate, height, is treated separately and is referred in various ways to the geoid as the reference level surface. An extension of the foregoing concept to three dimensions has been indicated by Hirvonen (1960, 1961) who introduced the notion of the ‘ telluroid ’ as the image of the Earth’s surface in the ‘normal’ ellipsoidal gravity field; extensive use of this concept has been made in Molodenski’s theory for the determination of the actual shape of the Earth and its gravity field from observations of gravity at its physical surface. In the following an attempt is made to establish with a systematic procedure a one-to-one correspondence between points P of the ‘actual’ gravity field of the Earth, and points Q of the ‘ normal ’ ellipsoidal field, in such a way that thereafter the simple metric of this field can be used for analytical computations and practical uses. Such a procedure permits the simultaneous treatment of planimetric and altimetric co-ordinates, thus avoiding ill-defined reductions and approximations, and permits the unequivocal definition of ‘ deflections ’ and ‘ anomalies ’, for which the integrability conditions which they must satisfy are easily found.

Way of Seeming, gave a picture of the world in which &TXr6 y--qq and a'& tecopcx ^were rationalised as a system of concentric rings or stephanai. In effect, this system was a sort of scientific version of the traditional poetic picture of the world in which a round, flat earth was surmounted by the dome of heaven wreathed with "signs" and had beneath it first Hades, then Tartarus, with gates, a bronze threshold at the top, a bronze fence round it, about its "neck" a triple coil of night, and a floor which was as far beneath earth as earth was beneath heaven., The Parmenidean rationalisation of poetry is in strong contrast to the more strictly mnaterialistic picture of the Ionian Greeks, according to which the world was either like a log of wood floating on water (Thales) 2, or like a cylindrical drum three times as broad as it was deep (Anaximander) 3, or like a lid floating upon air (Anaximenes Anaxagoras, Democritus) 4, and in which the poetical concept of Hades and Tartarus, so necessary to those preoccupied with rewards and punishments in an after life but quite unnecessary to those who were not, had been totally eliminated. I considered the Parmenidean picture in connexion with the general account of t& i'7ra yiq and the image of whorls given in the myth of Er in the Republic. It seems clear that in the Republic Plato had in mind the traditional poetic picture of Hades below a flat earth. It is true that he does not actually tell us where the gaxtotvtoq T63noq is, the judgment place of souls whence they proceed either upward through the X&al?oc -ou oi pavou or downward through a corresponding Xca[La into the equivalent of Tartarus, but since the souls on their return to the meadow from either journey

Исходя иэ определений гармоник геопотенциала по воэмущениям орбит спутников, дается вывод строгих формул полой кривиэны, направляющих косинусов внешней нормали и других характеристик формы соответствующей уровенной поверхности. Сопоставляя эти величины с соответствующими величинами для поверхности, эа относимости, эа которую принимается трехосный эллипеоид, получены выражения для аномалий кривизны и составляющих уклонений отвеса в длоскости меридиана и первиго вертикала. Под условием минимума суммыквадратов этих величин определимы фундаментальные параметры формы эемли, т. е. полярное и экваториальное сжатия и ориентировка осей экваториального эллипса грехосного еллипсоида, наиболее подходящего к исследуемой поверхности.

The intenal structures of the moon, Mars, Venus, and Mercury are examined in the light of what is known about the constitution of the earth. A review of the seismic determination of the elastic constitution of the earth's mantie, using results on the stability of silicates at high pressures, leads to the following interpretation: the rapid increase of velocity beginning at 200 km depends on the olivine-spinel transition and the breakdown of silicates to oxides. Preliminary calculations of the stability field of periclase (MgO) and stishovite (high-density SiO/sub 2/) relative to olivine (MgSiO/sub 4/) indicate that the oxides are stable at pressures greater than 1.0 x 10/sup 5/ 1.5 x 10/sup 5/ bars. The oxide transition produces a change in volume of about 20 per cent. The gravitational figure of the earth, as obtained from satellite orbits, is used to estimate the possible deviations from hydrostatic equilibrium on other planets. The near coincidence of the present rate of heat production of a chondritic earth and the present surface heat flow is discussed as a limiting condition of the internal thermal structure of the earth. Observations of the orbital and rotational motion of the moon give its gravitational figure. Calculations ofmore » its thermal structure show that a model moon with uniform radioactivity and chondritic composition is inconsistent with the calculated gravitational figure. The inferred strength of the moon requires either that the radioactivity be substantially less than that of chondrites or that the heat sources be concentrated in the outer layers. The problem of differentiation without melting is noted. The average lunar material has a radioactivity perhaps one- half, or less, that of chondritic materials. The astronomical data on the mean density and gravitational figure of Mars are examined. If the mean radius of Mars is taken to be 3310 km, the planet must have a surface density of about 3.8 to 3.9 and must be nearly homogeneous. If Mars were homogeneous and had a radioactivity equal to that of chondrites, the interior would be molten and large- scale gravitational differentiation would be expected. Since this differentiation is not apparent in the gravitational data, it is concluded that the radioactive composition of Mars differs from that of chondrites. The internal structures of Mercury and Venus are examined in terms of their inferred rotational history. In conclusion, the inner planets differ both in the abundances of the heavy elements and in the abundances of potassium, uranium, and thorium. Chondrites may provide a satisfactory chemical model for the earth but not for the other inner planets; Venus is a possible exception, only on the grounds of ignorance of its intenal constitution. (auth)« less

During the past year a number of seismological questions have presented themselves. I will discuss three of them very briefly here. The first has to do with the problem of regional variation in the travel-time of seismic waves. An unusual example of this situation has been found namely, the shock of November 30, 1935, in the Caribbean Sea near Panama (see Fig. 1). The epicenter in this case was adopted as 10.°1 north latitude and 79.°5 west longitude. The distance from this point to each of the recording stations was then computed, using a formula for great-circle distances without correction for the figure of the Earth. These distances were then used to take out the (P-0) values from three different sets of well-known seismological tables. The (P-0) values were then subtracted from the arrival-time of the first wave at each station. The results in seconds may be seen in Figure 1. Each station is here indicated by an approximate direction-line from the epicenter. At the outer end of each direction-line is found the name of the recording station and the mean value of origin in seconds as determined from the three sets of tables. Note that these mean times for the group of stations from Buffalo to Harvard are about four seconds later than any of the other mean times of origin. This indicates that the travel-time of the preliminary earthquake-waves to the stations in this sector was slower by four seconds.

The professional surveyor, in the course of his training, is required to study certain problems of geodesy. He learns the significance of spherical excess; convergence of the meridian; the computation of the azimuth and distance between two stations, given their geographical latitudes and longitudes; and the relevant factors for the adopted Figure of the Earth.

The principal formulae of the geophysical geodesy are based on the famous explicit expression of Stokes (1849). Up to now, there has been no method for a computation of the corresponding explicit expression of a (non-spherical) surface with masses outside the geoid. In this paper there is a solution of this problem. Another paper on this subject was presented to “Nordiska geodetmotet” in Copenhagen May 1959 (in Swedish).

1. Introduction 1.1 The prime object of a national geodetic survey is the determination of the coordinates of a fairly close network of inter-connected survey stations distributed over the whole area of the survey.

The principal object of the calculations contained in the following paper, is to investigate the figure which a fluid, consisting of portions varying in density according to any given law, would assume, when every particle is acted upon by the attraction of every other and by a centrifugal force arising from rotatory motion. To what extent this may have been the original condition of the earth, is a doubtful question; and although observation does not fully warrant this supposition of the regular arrangement of parts, it has necessarily been made the foundation of most of the mathematical calculations connected with the investigation. Before proceeding to this problem, it is necessary to calculate the attraction of a body of any given figure, and consisting of strata, varying in their densities according to any given law; and it is in this problem that the principal difficulty lies. The elegant method of solution discovered by Laplace is well known; and I have followed his steps as far as the point where the equation, known by his name, first appears. In order to illustrate the nature of the deviation which I have there made, it will be necessary to mention some of the principal steps of the two methods. By means of a theorem, which Laplace laid down as true of all spheroids that differ but little from spheres, and the properties of the integral of the equation referred to, he was enabled to substitute the easy rules of differentiation for the more complicated inverse processes, and thus to compute the attraction of that class of figures. It has, however, been since discovered by Mr. Ivory, that this theorem is true only of spheroids of a particular kind; and, consequently, to this kind the solution of the problem is restricted. This defect, and the indirectness of his analysis, led other mathematicians to consider the question; and, in 1811, Mr. Ivory published his method, which has the great advantage of being more direct, though equally limited.

1. In former communications to the Royal Society I have shown that Local Attraction, owing to the amount it in some places attains, is a more troublesome element to deal with in geodetical operations than had generally been supposed. The Mountains and the Ocean were shown to combine to make the deviation of the plumb-line as much as 22".71, 17".23, 21".05, 34".16 (or quantities not differing materially from them) in the four principal stations of the Great Arc of India between Cape Comorin and the Himmalayas—viz. at Punnœ (8° 9' 31"), Damargida (18° 3' 15'), Kalianpur (24° 7' 11"), Kaliana (29° 30' 48") ; and how much these might be increased or lessened by the effect of variations of density in the crust below t was difficult to say. Deviations amounting to at least such quantities as 7''.61 and 7".87 were shown to exist in the stations of the Indian Arc, arising from this last cause (see Phil. Trans. 1861, p. 593 (4) and (5)). M. Otto Struve has lately called attention to similarly important deflections caused by local attraction in Russia—and especially to a remarkable difference of deflection at two stations near Moscow, only about eighteen miles apart, amounting to as much as 18", which is attributed to an invisible unknown cause in the strata below (see Monthly Notices of the Royal Astronomical Society, April 1862).

About three hundred astronomical observatories throughout the world give us their position to tenths of a second in Latitude and hundredths of a second in Longitude. These are their astronomical positions but they are often quite ignorant of their exact geographical location. The reason for this is that the earth is irregular in shape and only by elaborate surveys, computations, and a reduction to a preselected earth shape can the coordinates of a station be secured. The earth is assumed to have different shapes by different countries and the reference ellipsoid can be either the Everest, the Bessel, one of the three Clarke’s, the Helmert, the Hayford or the Madrid 1924 (International) figure. The tie-in reference point for Canada, Mexico and the United States is at Meades Ranch, Kansas with a Latitude of 39”13’ 26.686” and a Longitude of 98”32’ 30.506”.

Summary The equations to be represented in the computer of an inertial navigation system for use near the surface of the earth are derived by elementary methods, and a block diagram of the computer is presented. No restrictions are placed on the nature of the earth's gravity field and only minor restrictions on the shape of the earth. The shape and gravity field of the earth are dis­ cussed and specialization of the equations and block diagram is made for the case where the earth and gravity field are assumed to have axial symmetry.

Summary Small changes in the orbits of satellites near to the earth have revealed new information about the earth's gravitational field and atmosphere. This article explains the main properties of satellite orbits and describes the discoveries which have been made with their help in the following branches of geophysics: the figure of the earth and the gravitational field; the density of the upper atmosphere and its variations, which have proved to be linked with solar activity; and, finally, other properties of the atmosphere, such as temperature and winds.

The propagation of radio waves of very low frequency to great distances is conveniently treated by regarding the space between the earth and the ionosphere as a wave-guide. Several authors have found that the least attenuated modes are profoundly affected by the earth’s curvature. This effect is investigated for several models of the ionosphere. It is found, in particular, that for frequencies greater than about 30 kc/s some modes are possible for which the energy is concentrated in a region near the base of the ionosphere, and the field strength near the ground is small. It is useful to think of such modes as being composed of waves repeatedly reflected at the inside spherical surface of the ionosphere, the rays being chords of this sphere. By analogy with sound waves these modes are called ‘whispering gallery modes’. The theory uses wave admittance and reflexion coefficient variables because these satisfy differential equations which are convenient for integration using a digital computer. The curvature of the earth is allowed for by using the method of the modified refractive index, but the earth’s magnetic field is neglected. Formulae for the m ode condition and the excitation of the various modes by a transmitter are given and discussed. A new way of dealing with an ionosphere having a continuous electron density profile is presented. The results of some numerical calculations are given both for a sharply bounded homogeneous ionosphere and for an exponential profile of electron density.

Summary The representation of the Earth's gravity field as the potential of a simple layer distributed over the surface of the Earth is determined by combining satellite observations and gravity anomalies. Density values of the simple layer for 192 surface elements are computed and converted into harmonic coefficients up to the 15th degree and order. These co-efficients are used to determine surface density values referred to a reference ellipsoid with the flattening of an earth in hydrostatic equilibrium. The geophysical implications of these values are outlined.

Horizon profile of Earth calculated from solutions of plane-parallel Rayleigh scattering atmosphere

(1818). XIX. Observations relating to the operations undertaken to determine the figure of the earth. The Philosophical Magazine: Vol. 52, No. 244, pp. 119-131.

1. In a paper published in 1902, “On the Vibrations and Stability of a Gravitating Planet,” I attempted to examine whether, owing to gravitational instability, an arrangement in concentric spherical shells might not be unstable for a planet of the size of our earth. It was not found possible to get exact results; progress only seemed possible by supposing that the problem would in its essentials be similar to a highly artificial problem in which gravitation was exactly annulled in the symmetrical configuration by the superposition of a countervailing field, so that in this state the matter was of uniform density and unstressed. The only possible justification for such an assumption was that I found myself unable to make any progress, either physically or mathematically, without it. Subject to this assumption, I found that, for a mass of the structure of our earth in its present state, the symmetrical configuration would undoubtedly be stable. It seemed likely, however, that in a past age in which the rigidity of the earth was considerably, although not enormously, less than it now is, the symmetrical configuration might have been unstable; the stable configuration would have been an unsymmetrical one, in which the surfaces of equal density were spherical but not concentric, so that the point of maximum density coincided neither with the centre of gravity nor the centre of figure of the earth’s surface. It was suggested that traces of this unsymmetrical configuration might still be found in the arrangement of oceans and plateaux on the earth’s surface. In 1906, Lord Rayleigh, in a paper “On the Dilatational Stability of the Earth,” drew attention to the wide departure from actuality implied in my assumptions, and suggested a set of assumptions which had more justification in nature. Lord Rayleigh’s suggestion was, in brief, that we might regard the symmetrical configuration as one in which the gravitational forces were balanced by hydrostatic pressures only. When disturbed, there is an additional stress superposed on to the hydrostatic pressure, and Lord Rayleigh suggested that this might be supposed connected with the additional strain— i. e ., the strain above and beyond that in the initial configuration by the ordinary elastic solid relations.

The underlying theory pertaining to the shape of the earth is briefly reviewed. The operations and results that have led to the theory of isostasy are given. Hopfner's recent inquiry into the problem is cited in which he has shown that by means of strict potential theory isostatic reductions are highly questionable, and, furthermore, that there is no need to postulate an isostatic state in order to account for gravity data. High points of geologic and seismologic data are also cited. The conclusion is that the theory of isostasy. rests upon a much more insecure foundation than has heretofore been supposed.

A new type of mathematical model has been proposed and computed for the advantages of natural and artificial test areas to be combined.

A modal solution for the em propagation in a spherical stratified troposphere is derived, via the Marcuvitz-Schwinger formalism. The approximations due to the small curvature of the earth are illustrated. The representation of the sources via the complete set of modal solutions is presented, as well as the excitation factors, computed for the vertical electric or magnetic dipole. The possible use of digital and analog computers is highlighted. Future plans for the work are the computations of a great deal of digital and analog solutions, and also the study of the influence of the random variability of the tropospheric refraction index. In the Appendix, a preliminary case [f(z)=az] is presented.

The ellipticity of the earth, deduced by Captain Sabine from a series of pendulum experiments the most extensive, and apparently the most deserving of confidence, that has ever been made, differs considerably from that which, as is generally believed, is indicated by geodetic measures. The difference can only be explained by errors of observation, by peculiarities of local circumstances, or by some defect in the theory which connects the figure of the earth with the variation of gravity on its surface: under the last head may be placed defects in the mathematical part of the theory, and errors in the assumptions of the original constitution and present state of the earth. It was with a view to ascertain the sufficiency of the mathematical theory, that I undertook the investigations contained in this paper. The celebrated proposition called Clairaut's theorem, by which the earth's ellipticity is inferred from the variation of gravity on its surface, is obtained only by the rejection of the squares and higher powers of the ellipticity. It is by the same rejection that the figure of the earth, supposed a heterogeneous fluid, is proved to be an elliptic spheroid. It appeared therefore probable, that a more accurate theory might introduce some modification into Clairaut's theorem, and might also show he figure of the earth to differ from an ellipsoid ; and there was no reason to think that the first approximation to that figure was more accurate, than the first approximation to the motion of the moon’s perigee. The result of my investigation does not at all serve to reconcile the pendulum observations made by Captain Sabine with the measures of degrees : and with respect to one object, which I hoped to obtain, I am therefore completely unsuccessful. The theory shows, however, that the earth’s figure, on the usual suppositions as to its constitution, is not an elliptic spheroid; and the formulæ which I have obtained will give the means of determining very exactly the figure of the earth, when the experiments on the variation of gravity, or the measures of arcs on the earth’s surface, shall be thought sufficiently accurate. As the subject is one whose interest is not confined to the present time, I have ventured to offer my investigations to the Royal Society. The first part of the following sheets contains the theory of the heterogeneous earth, pushed so far as to include all the terms of the second order: it is succeeded by a comparison of this theory with Captain Sabine’s results, and with the best arcs of the meridian that have been measured and in the conclusion, I have offered some suggestions on the propriety of repeating some of these measures.

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A radio antenna on a satellite within the ionosphere can receive extraterrestrial radiation incident only within a definite cone of directions. A new mathematical technique is applied to ray tracing in a spherically stratified isotropic ionosphere, and results are presented which show how the cone size varies with receiver frequency and satellite height. The contributions of the different directions of incidence to the total electric field measured in the ionosphere are calculated.

Gravimetric methods of determining the figure of the Earth can be divided into two groups. The first employs an auxiliary surface—the geoid. The second employs the so-called quasigeoid as an auxiliary surface. Its main advantage is in that it only uses surface measurements and that it does not require knowledge of the structure of the Earth's crust. This method was treated in [1]. The results display an accuracy of the order of the Earth's flattening. The purpose of this paper is to show that this method can be applied theoretically to reach an arbitrary accuracy.

In the past the Eotvos torsion balance has been used only in commercial geophysical explorations and not for geodetic purposes. Even in geophysics, after approximately 30 yrs of use, it was superseded by the faster, and handier gravimeters. The lack of interest in the geodetic applications can be explained partly by the fact that geodesists of the past did not need very detailed information about the shape of the earth. The torsion balance offers several possibilities in geodetic applications, of which the most important are: The interpolation of deflections of the vertical and of geoid undulations between reference stations, where gravity material is not sufficient; the determination of the geopotential values in a given area for the purpose of leveling reductions; the determination of the radius of curvature of the plumb line for reducing astronomical coordinates to the geoid; and the investigation of curvature properties of the equipotential surface for more accurate baseline reductions to the ellipsoid of reference.

The importance of close confrontation of Prof. Sommerfeld's attenuation formula with field-strength measurements is stressed. The fallacy of Prof. Zenneck's formula in most practical cases is pointed out, and a short summary given of Prof. Sommerfeld's conclusions, with approximate expressions for the "numerical distance" and the angle determining various shapes of theoretical attenuation curves. At very great distances, the signals over flat ground are shown always to decrease as the inverse square of distance, while at short distances short waves die away approximately as the inverse square root of distance. At intermediate distances, curious phenomena are shown to occur; in some cases, as illustrated by examples, the field vanishes altogether at a finite distance, to reappear farther away. An abac is included and instructions given for its use to obtain the inductivity and conductivity of the ground over which a series of field-strength measurements have been made. Results of such computations in Sweden and England are given. By using a graph reproduced in the paper an easy method of predicting field strengths for all wavelengths over soil, whose electrical constants are known, is devised. A simple semi-empirical formula to account for the curvature of the Earth at moderate distances is presented, and lines for better antenna construction touched upon, minimizing jamming from down-coming rays, and saving power as well.

The refraction of internal acoustic-gravity waves by the earth's spherically symmetric gravitational field is investigated by using the approximation of geometrical acoustics in an isothermal atmosphere. Instead of following straight lines the ray paths of internal acoustic waves bend away from the earth, whereas the ray paths of internal gravity waves bend toward the earth so as to nearly match the earth's curvature. Consequently, the use of a flat-earth approximation for describing long-distance propagation around a spherical earth is well justified for internal gravity waves, although not for internal acoustic waves.

The fundamental study of the propagation of radio waves has formed an important part of the programme of the Radio Research Board, which was established under the Department of Scientific and Industrial Research in 1920. The object of the present paper is to survey the progress made in this field during the decade ending in 1946, and to indicate the manner in which the investigations concerned have been developed to meet the requirements of those responsible for applying radio waves to various practical uses. The researches described cover the radio-frequency spectrum from about 10 kc/s up to 50000 Mc/s or more. They deal with the transmission of radio waves along the surface of the earth, and illustrate the manner in which this transmission is influenced by the electrical properties of the ground and the curvature of the earth and by the corresponding properties and slopes of natural and man-made protuberances on the ground. The characteristics of the ionosphere have been studied intensively for over twenty years, and during the past decade considerable advance has been made in our knowledge of this region of the atmosphere and of the manner in which radio waves are propagated through it. With the advance of radio technique and developments to shorter and shorter wavelengths, it has become clear that wave propagation in the lower atmosphere is greatly affected by the vertical gradient of the temperature and humidity of the air; the associated phenomena have been studied directly, and, in addition, the electrical properties of the constituents of the atmosphere, solid, liquid and gaseous, have been determined at the very high radio frequencies relevant to this portion of the investigation. Finally, the important part played by radio noise of either atmospheric or extraterrestrial origin has been studied with a view to understanding more clearly the manner and circumstances in which such noise can either influence the successful transmission of intelligence by radio waves, or provide a means of increasing our knowledge of such phenomena as those associated with thunderstorms and activity on the surface of the sun. This survey attempts to show what an important part the study of wave propagation plays in numerous applications of radio technique, and demonstrates that there is still much further work to be done before the full requirements of those using the technique can be met in a satisfactory manner.

(1862). LV. Tests of the truth of the fluid theory of the figure of the earth. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science: Vol. 24, No. 163, pp. 409-417.

There are two outstanding problems relating to the propagation of wireless waves over the earth’s surface which at present remain unsolved, viz.— What is the agency which causes the waves to follow the curvature of the earth, thus rendering long-distance communication possible ? And what is the cause of the large and rapid variations of the intensity and apparent direction of the waves, very commonly observed at the receiving station and confined almost entirely to the hours of darkness ? Both phenomena can be explained to some extent by the well-known Heaviide-layer theory, with the modifications proposed by Eccles, but it is generally admitted that further experimental evidence of the existence of the layer is needed. If the theory is correct and is sufficient, it follows that, a receiver experiencing either of the above phenomena, part of the energy must be arriving in a downward direction ( i. e ., inclined to the horizontal); and that during the occurrence of directional variations, this downcoming wave must have at horizontally polarised component ( i. e ., with its electric force horizontal). I has, therefore, been generally recognised for some time that a conclusive experimental demonstration of the presence or absence of such waves, by suit able quantitative measurements in three dimensions, would aid considerably in proving or disproving the Heaviside-layer theory.

Human error concerning the geography of the earth has been recurrent throughout historic time. Some kinds of error arise because of the problems of dealing with phenomena in the three-dimensional zone forming earth-space. The broad curvature of the earth, the sheer diversity of earth-space, and the patterns of change through time all form pitfalls for the geographer who is, thereby, liable to error in his descriptive analyses. The problems of placing percepts (things perceived) against concepts (mental abstractions) make for difficulties in analyzing the three-dimensional spatial zone in which geographers work. Among the kinds of error committed there are avoidable varieties, such as the inexact use of descriptive and analytical language, the uncritical use of statistical data, the acceptance of unreliable data, and the confusion of the several kinds of spatial distributions. Less avoidable error derives from the formulation of hypotheses accounting for phenomena which, when critically examined a...

Geodesy is one of the oldest of the geophysical sciences. Originally it embraced only a limited field, consisting of the determination of the shape of the earth and its size and the means by which this was done. But now the subject of geodesy is very closely allied with other matters and there are few geophysical questions that do not, either directly or indirectly, concern the geodesist in the solution of his problems. Almost every country of the world that pretends to any standing has organized a service for the purpose of carrying on what are called geodetic surveys. As a matter of fact, any survey of the land in which the shape and size of the earth are taken into consideration can be called geodetic; thus, the hydrographic surveys along the coast made for a sailing chart are really geodetic surveys and, similarly, a topographic survey of a large area may be considered to be a geodetic survey; but what I have in mind in speaking of geodetic surveys, are triangulation, base measurements, precise leveling, etc.

Most upper mantle and crustal velocity determinations are made using conventional seismic refraction techniques under the assumption that the earth may be considered flat for distances up to about 1500 km. The flat earth assumptions do however lead to systematic errors which take the following forms: (i) the apparent velocity determinations are too large, and are a function of the refractor depth, (ii) apparent ’lateral variations’ in the upper mantle are produced, (iii) refractor depth calculations depart from correct values as the depth is increased beyond 60 km, (iv) misleading discrepancies occur when spherical ray theory is used with parameters determined from flat earth formulae. The measured apparent velocities v′ may be corrected for curvature to true velocities using v = [(R-H)/R] y , where R is the radius of the earth and H the depth of the refractor. The magnitude of the corrections shows that the curvature of the earth cannot be ignored in crustal and upper mantle refraction surveys when the depth of the refractor is more than 10 km. An apparent 8.50 km/sec layer at a depth of 124 km, for example, becomes an 8.33 km/sec layer at a depth of 130 km when curvature correction are applied. Since curvature effects have been neglected in many refraction surveys, other geophysical investigations (such as gravity, near earthquake etc.) which have made either direct or indirect use of flat earth data, may be significantly affected.

An essential part of almost all air-defense computer simulation models is the visibility subroutine. This subroutine performs the necessary computations and provides the data required to determine whether enemy aircraft or missiles may be detected by stationary ground-based air-defense system sensors- that is, whether they are within sensor range and whether their visibility is obstructed by terrain irregularities and/or the curvature of the earth. This paper presents a new approach to the visibility subroutine, entitled MAVD (Minimum Altitude Visibi Zi ty Diagram). MAVD provides a visibility subroutine which could be incorporated into existing models relatively easily. The authors believe that it would result in a more realistic simulation and provide a significant reduction in computer time, in comparison with other simulation methods with which the authors are familiar.

1. Any estimate of the rigidity of the Earth must be based partly on some observations from which a deformation of the Earth’s surface can be inferred, and partly on some hypothesis as to the internal constitution of the Earth. The observations may be concerned with tides of long period, variations of the vertical, variations of latitude, and so on. The hypothesis must relate to the arrangement of the matter as regards density in different parts, and to the state of the parts in respect of solidity, compressibility, and so on. In the simplest hypothesis, the one on which Lord Kelvin’s well-known, estimate was based, the Earth is treated as absolutely incompressible and of uniform density and rigidity. This hypothesis was adopted to simplify the problem, not because it is a true one. No matter is absolutely incompressible, and, the Earth is not a body of uniform density. It cannot be held to be probable that it is a body of uniform rigidity. But when any part of the hypothesis, e. g ., the assumption of uniform density, is discarded, the estimate of rigidity is affected. Different estimates are obtained when different laws of density are assumed. Again, whatever hypothesis we adopt as regards the arrangement of the matter, so long as we consider the Earth to be absolutely incompressible and of uniform rigidity, different estimates of this rigidity are obtained by using observations of different phenomena. Variations of the vertical may give one value, variations of latitude a notably different value. It follows that “the rigidity of the Earth” is not a definite physical constant. But there are two determinate constant numbers related to the methods that have been used for obtaining estimates of the rigidity of the Earth. One of these numbers specifies the amount by which the surface of the Earth yields to forces of the type of the tide-generating attractions of the Sun and Moon. The other number specifies the amount by which the potential of the Earth is altered through the rearrangement of the matter within it when this matter is displaced by the deforming influence of the Sun and Moon. If we adopt the ordinarily-accepted theory of the Figure of the Earth, the so-called theory of “fluid equilibrium,” and if we make the very probable assumption that the physical constants of the matter within the Earth, such as the density or the incompressibility, are nearly uniform over any spherical surface having its centre at the Earth’s centre, we can determine both these numbers without introducing any additional hypothesis as to the law of density or the state of the matter. We shall find, in fact, that observations of variations of latitude lead to a determination of the number related to the inequality of potential, and that, when this number is known, observations of variations of the vertical lead to a determination of the number related to the inequality of figure. [ Note added , December 15, 1908.—This statement needs, perhaps, some additional qualification. It is assumed that, in calculating the two numbers from the two kinds of observations, we may adopt an equilibrium theory of the deformations produced in the Earth by the corresponding forces. If the constitution of the Earth is really such that an equilibrium theory of the effects produced in it by these forces is inadequate, we should expect a marked discordance of phase between the inequality of figure produced and the force producing it. Now Hecker’s observations, cited in § 6 below, show that, in the case of the semidiurnal term in the variation of the vertical due to the lunar deflexion of gravity, the agreement of phase is close. If, however, an equilibrium theory is adequate, as it appears to be, for the semidiurnal corporeal tide, a similar theory must be adequate for the corporeal tides of long period and for the variations of latitude.]

(1823). LXI. Proposal for a new method of determining a fixed unit of measure, by deducing the same from the curvature of the earth. The Philosophical Magazine: Vol. 61, No. 300, pp. 266-269.

1. The Secondary and Tertiary Triangulations of the six counties of Northern Ireland which were observed about 1900 were computed county by county each on its own meridian on a Cassini projection using Airy's figure of the earth. Although a number of points common to two or more counties were fixed no attempt was made to bring the separate counties into sympathy either with each other or even with the old Primary triangulation as adjusted by Clarke in 1856.