1Aristotle regarded the Earth as having an upper inhabited half and a lower uninhabited one, and the air on the lower half as tending to flow upwards through the Earth. The obstruction of this passage brought about an accumulation of air within the Earth, and the increased pressure may occasion oscillations of the surface, which may be so intense as to cause earthquakes.
1Aristotle regarded the Earth as having an upper inhabited half and a lower uninhabited one, and the air on the lower half as tending to flow upwards through the Earth. The obstruction of this passage brought about an accumulation of air within the Earth, and the increased pressure may occasion oscillations of the surface, which may be so intense as to cause earthquakes.
EARTH, FIGURE OF THE.The determination of the figure of the earth is a problem of the highest importance in astronomy, inasmuch as the diameter of the earth is the unit to which all celestial distances must be referred.
Historical.
Reasoning from the uniform level appearance of the horizon, the variations in altitude of the circumpolar stars as one travels towards the north or south, the disappearance of a ship standing out to sea, and perhaps other phenomena, the earliest astronomers regarded the earth as a sphere, and they endeavoured to ascertain its dimensions. Aristotle relates that the mathematicians had found the circumference to be 400,000 stadia (about 46,000 miles). But Eratosthenes (c.250B.C.) appears to have been the first who entertained an accurate idea of the principles on which the determination of the figure of the earth really depends, and attempted to reduce them to practice. His results were very inaccurate, but his method is the same as that which is followed at the present day—depending, in fact, on the comparison of a line measured on the earth’s surface with the corresponding arc of the heavens. He observed that at Syene in Upper Egypt, on the day of the summer solstice, the sun was exactly vertical, whilst at Alexandria at the same season of the year its zenith distance was 7° 12′, or one-fiftieth of the circumference of a circle. He assumed that these places were on the same meridian; and, reckoning their distance apart as 5000 stadia, he inferred that the circumference of the earth was 250,000 stadia (about 29,000 miles). A similar attempt was made by Posidonius, who adopted a method which differed from that of Eratosthenes only in using a star instead of the sun. He obtained 240,000 stadia (about 27,600 miles) for the circumference. Ptolemy in hisGeographyassigns the length of the degree as 500 stadia.
The Arabs also investigated the question of the earth’s magnitude. The caliph Abdallah al Mamun (A.D.814), having fixed on a spot in the plains of Mesopotamia, despatched one company of astronomers northwards and another southwards, measuring the journey by rods, until each found the altitude of the pole to have changed one degree. But the result of this measurement does not appear to have been very satisfactory. From this time the subject seems to have attracted no attention until about 1500, when Jean Fernel (1497-1558), a Frenchman, measured a distance in the direction of the meridian near Paris by counting the number of revolutions of the wheel of a carriage. His astronomical observations were made with a triangle used as a quadrant, and his resulting length of a degree was very near the truth.
Willebrord Snell1substituted a chain of triangles for actual linear measurement. He measured his base line on the frozen surface of the meadows near Leiden, and measured the angles of his triangles, which lay between Alkmaar and Bergen-op-Zoom, with a quadrant and semicircles. He took the precaution ofcomparing his standard with that of the French, so that his result was expressed in toises (the length of the toise is about 6.39 English ft.). The work was recomputed and reobserved by P. von Musschenbroek in 1729. In 1637 an Englishman, Richard Norwood, published a determination of the figure of the earth in a volume entitledThe Seaman’s Practice, contayning a Fundamentall Probleme in Navigation experimentally verified, namely, touching the Compasse of the Earth and Sea and the quantity of a Degree in our English Measures. He observed on the 11th of June 1633 the sun’s meridian altitude in London as 62° 1′, and on the 6th of June 1635, his meridian altitude in York as 59° 33′. He measured the distance between these places partly with a chain and partly by pacing. By this means, through compensation of errors, he arrived at 367,176 ft. for the degree—a very fair result.
The application of the telescope to angular instruments was the next important step. Jean Picard was the first who in 1669, with the telescope, using such precautions as the nature of the operation requires, measured an arc of meridian. He measured with wooden rods a base line of 5663 toises, and a second or base of verification of 3902 toises; his triangulation extended from Malvoisine, near Paris, to Sourdon, near Amiens. The angles of the triangles were measured with a quadrant furnished with a telescope having cross-wires. The difference of latitude of the terminal stations was determined by observations made with a sector on a star in Cassiopeia, giving 1° 22′ 55″ for the amplitude. The terrestrial measurement gave 78,850 toises, whence he inferred for the length of the degree 57,060 toises.
Hitherto geodetic observations had been confined to the determination of the magnitude of the earth considered as a sphere, but a discovery made by Jean Richer (d. 1696) turned the attention of mathematicians to its deviation from a spherical form. This astronomer, having been sent by the Academy of Sciences of Paris to the island of Cayenne, in South America, for the purpose of investigating the amount of astronomical refraction and other astronomical objects, observed that his clock, which had been regulated at Paris to beat seconds, lost about two minutes and a half daily at Cayenne, and that in order to bring it to measure mean solar time it was necessary to shorten the pendulum by more than a line (about1⁄12th of an in.). This fact, which was scarcely credited till it had been confirmed by the subsequent observations of Varin and Deshayes on the coasts of Africa and America, was first explained in the third book of Newton’sPrincipia, who showed that it could only be referred to a diminution of gravity arising either from a protuberance of the equatorial parts of the earth and consequent increase of the distance from the centre, or from the counteracting effect of the centrifugal force. About the same time (1673) appeared Christian Huygens’De Horologio Oscillatorio, in which for the first time were found correct notions on the subject of centrifugal force. It does not, however, appear that they were applied to the theoretical investigation of the figure of the earth before the publication of Newton’sPrincipia. In 1690 Huygens published hisDe Causa Gravitatis, which contains an investigation of the figure of the earth on the supposition that the attraction of every particle is towards the centre.
Between 1684 and 1718 J. and D. Cassini, starting from Picard’s base, carried a triangulation northwards from Paris to Dunkirk and southwards from Paris to Collioure. They measured a base of 7246 toises near Perpignan, and a somewhat shorter base near Dunkirk; and from the northern portion of the arc, which had an amplitude of 2° 12′ 9″, obtained for the length of a degree 56,960 toises; while from the southern portion, of which the amplitude was 6° 18′ 57″, they obtained 57,097 toises. The immediate inference from this was that, the degree diminishing with increasing latitude, the earth must be a prolate spheroid. This conclusion was totally opposed to the theoretical investigations of Newton and Huygens, and accordingly the Academy of Sciences of Paris determined to apply a decisive test by the measurement of arcs at a great distance from each other—one in the neighbourhood of the equator, the other in a high latitude. Thus arose the celebrated expeditions of the French academicians. In May 1735 Louis Godin, Pierre Bouguer and Charles Marie de la Condamine, under the auspices of Louis XV., proceeded to Peru, where, assisted by two Spanish officers, after ten years of laborious exertion, they measured an arc of 3° 7′, the northern end near the equator. The second party consisted of Pierre Louis Moreau de Maupertuis, Alexis Claude Clairault, Charles Étienne Louis Camus, Pierre Charles Lemonnier, and Reginaud Outhier, who reached the Gulf of Bothnia in July 1736; they were in some respects more fortunate than the first party, inasmuch as they completed the measurement of an arc near the polar circle of 57′ amplitude and returned within sixteen months from the date of their departure.
The measurement of Bouguer and De la Condamine was executed with great care, and on account of the locality, as well as the manner in which all the details were conducted, it has always been regarded as a most valuable determination. The southern limit was at Tarqui, the northern at Cotchesqui. A base of 6272 toises was measured in the vicinity of Quito, near the northern extremity of the arc, and a second base of 5260 toises near the southern extremity. The mountainous nature of the country made the work very laborious, in some cases the difference of heights of two neighbouring stations exceeding 1 mile; and they had much trouble with their instruments, those with which they were to determine the latitudes proving untrustworthy. But they succeeded by simultaneous observations of the same star at the two extremities of the arc in obtaining very fair results. The whole length of the arc amounted to 176,945 toises, while the difference of latitudes was 3° 7′ 3″. In consequence of a misunderstanding that arose between De la Condamine and Bouguer, their operations were conducted separately, and each wrote a full account of the expedition. Bouguer’s book was published in 1749; that of De la Condamine in 1751. The toise used in this measure was afterwards regarded as the standard toise, and is always referred to as theToise of Peru.
The party of Maupertuis, though their work was quickly despatched, had also to contend with great difficulties. Not being able to make use of the small islands in the Gulf of Bothnia for the trigonometrical stations, they were forced to penetrate into the forests of Lapland, commencing operations at Torneå, a city situated on the mainland near the extremity of the gulf. From this, the southern extremity of their arc, they carried a chain of triangles northward to the mountain Kittis, which they selected as the northern terminus. The latitudes were determined by observations with a sector (made by George Graham) of the zenith distance of α and δ Draconis. The base line was measured on the frozen surface of the river Torneå about the middle of the arc; two parties measured it separately, and they differed by about 4 in. The result of the whole was that the difference of latitudes of the terminal stations was 57′ 29″ .6, and the length of the arc 55,023 toises. In this expedition, as well as in that to Peru, observations were made with a pendulum to determine the force of gravity; and these observations coincided with the geodetic results in proving that the earth was an oblate and not prolate spheroid.
In 1740 was published in the ParisMémoiresan account, by Cassini de Thury, of a remeasurement by himself and Nicolas Louis de Lacaille of the meridian of Paris. With a view to determine more accurately the variation of the degree along the meridian, they divided the distance from Dunkirk to Collioure into four partial arcs of about two degrees each, by observing the latitude at five stations. The results previously obtained by J. and D. Cassini were not confirmed, but, on the contrary, the length of the degree derived from these partial arcs showed on the whole an increase with an increasing latitude. Cassini and Lacaille also measured an arc of parallel across the mouth of the Rhone. The difference of time of the extremities was determined by the observers at either end noting the instant of a signal given by flashing gunpowder at a point near the middle of the arc.
While at the Cape of Good Hope in 1752, engaged in various astronomical observations, Lacaille measured an arc of meridian of 1° 13′ 17″, which gave him for the length of the degree 57,037toises—an unexpected result, which has led to the remeasurement of the arc by Sir Thomas Maclear (seeGeodesy).
Passing over the measurements made between Rome and Rimini and on the plains of Piedmont by the Jesuits Ruggiero Giuseppe Boscovich and Giovanni Battista Beccaria, and also the arc measured with deal rods in North America by Charles Mason and Jeremiah Dixon, we come to the commencement of the English triangulation. In 1783, in consequence of a representation from Cassini de Thury on the advantages that would accrue from the geodetic connexion of Paris and Greenwich, General William Roy was, with the king’s approval, appointed by the Royal Society to conduct the operations on the part of England, Count Cassini, Méchain and Delambre being appointed on the French side. A precision previously unknown was attained by the use of Ramsden’s theodolite, which was the first to make the spherical excess of triangles measurable. The wooden rods with which the first base was measured were replaced by glass rods, which were afterwards rejected for the steel chain of Ramsden. (For further details seeAccount of the Trigonometrical Survey of England and Wales.)
Shortly after this, the National Convention of France, having agreed to remodel their system of weights and measures, chose for their unit of length the ten-millionth part of the meridian quadrant. In order to obtain this length precisely, the remeasurement of the French meridian was resolved on, and deputed to J.B.J. Delambre and Pierre François André Méchain. The details of this operation will be found in theBase du système métrique décimale. The arc was subsequently extended by Jean Baptiste Biot and Dominique François Jean Arago to the island of Iviza. Operations for the connexion of England with the continent of Europe were resumed in 1821 to 1823 by Henry Kater and Thomas Frederick Colby on the English side, and F.J.D. Arago and Claude Louis Mathieu on the French.
The publication in 1838 of Friedrich Wilhelm Bessel’sGradmessung in Ostpreussenmarks an era in the science of geodesy. Here we find the method of least squares applied to the calculation of a network of triangles and the reduction of the observations generally. The systematic manner in which all the observations were taken with the view of securing final results of extreme accuracy is admirable. The triangulation, which was a small one, extended about a degree and a half along the shores of the Baltic in a N.N.E. direction. The angles were observed with theodolites of 12 and 15 in. diameter, and the latitudes determined by means of the transit instrument in the prime vertical—a method much used in Germany. (The base apparatus is described in the articleGeodesy.)
The principal triangulation of Great Britain and Ireland, which was commenced in 1783 under General Roy, for the more immediate purpose of connecting the observatories of Greenwich and Paris, had been gradually extended, under the successive direction of Colonel E. Williams, General W. Mudge, General T.F. Colby, Colonel L.A. Hall, and Colonel Sir Henry James; it was finished in 1851. The number of stations is about 250. At 32 of these the latitudes were determined with Ramsden’s and Airy’s zenith sectors. The theodolites used for this work were, in addition to the two great theodolites of Ramsden which were used by General Roy and Captain Kater, a smaller theodolite of 18 in. diameter by the same mechanician, and another of 24 in. diameter by Messrs Troughton and Simms. Observations for determination of absolute azimuth were made with those instruments at a large number of stations; the stars α, δ, and λ Ursae Minoris and 51 Cephei being those observed always at the greatest azimuths. At six of these stations the probable error of the result is under 0.4″, at twelve under 0.5″, at thirty-four under 0.7″: so that the absolute azimuth of the whole network is determined with extreme accuracy. Of the seven base lines which have been measured, five were by means of steel chains and two with Colby’s compensation bars (seeGeodesy). The triangulation was computed by least squares. The total number of equations of condition for the triangulation is 920; if therefore the whole had been reduced in one mass, as it should have been, the solution of an equation of 920 unknown quantities would have occurred as a part of the work. To avoid this an approximation was resorted to; the triangulation was divided into twenty-one parts or figures; four of these, not adjacent, were first adjusted by the method explained, and the corrections thus determined in these figures carried into the equations of condition of the adjacent figures. The average number of equations in a figure is 44; the largest equation is one of 77 unknown quantities. The vertical limb of Airy’s zenith sector is read by four microscopes, and in the complete observation of a star there are 10 micrometer readings and 12 level readings. The instrument is portable; and a complete determination of latitude, affected with the mean of the declination errors of two stars, is effected by two micrometer readings and four level readings. The observation consists in measuring with the telescope micrometer the difference of zenith distances of two stars which cross the meridian, one to the north and the other to the south of the observer at zenith distances which differ by not much more than 10′ or 15′, the interval of the times of transit being not less than one nor more than twenty minutes. The advantages are that, with simplicity in the construction of the instrument and facility in the manipulation, refraction is eliminated (or nearly so, as the stars are generally selected within 25° of the zenith), and there is no large divided circle. The telescope, which is counterpoised on one side of the vertical axis, has a small circle for finding, and there is also a small horizontal circle. This instrument is universally used in American geodesy.
The principal work containing the methods and results of these operations was published in 1858 with the title “Ordnance Trigonometrical Survey of Great Britain and Ireland. Account of the observations and calculations of the principal triangulation and of the figure, dimensions and mean specific gravity of the earth as derived therefrom. Drawn up by Captain Alexander Ross Clarke, R.E., F.R.A.S., under the direction of Lieut.-Colonel H. James, R.E., F.R.S., M.R.I.A., &c.” A supplement appeared in 1862: “Extension of the Triangulation of the Ordnance Survey into France and Belgium, with the measurement of an arc of parallel in 52° N. from Valentia in Ireland to Mount Kemmel in Belgium. Published by ... Col. Sir Henry James.”
The principal work containing the methods and results of these operations was published in 1858 with the title “Ordnance Trigonometrical Survey of Great Britain and Ireland. Account of the observations and calculations of the principal triangulation and of the figure, dimensions and mean specific gravity of the earth as derived therefrom. Drawn up by Captain Alexander Ross Clarke, R.E., F.R.A.S., under the direction of Lieut.-Colonel H. James, R.E., F.R.S., M.R.I.A., &c.” A supplement appeared in 1862: “Extension of the Triangulation of the Ordnance Survey into France and Belgium, with the measurement of an arc of parallel in 52° N. from Valentia in Ireland to Mount Kemmel in Belgium. Published by ... Col. Sir Henry James.”
Extensive operations for surveying India and determining the figure of the earth were commenced in 1800. Colonel W. Lambton started the great meridian arc at Punnae in latitude 8° 9′, and, following generally the methods of the English survey, he carried his triangulation as far north as 20° 30′. The work was continued by Sir George (then Captain) Everest, who carried it to the latitude of 29° 30′. Two admirable volumes by Sir George Everest, published in 1830 and in 1847, give the details of this undertaking. The survey was afterwards prosecuted by Colonel T.T. Walker, R.E., who made valuable contributions to geodesy. The working out of the Indian chains of triangle by the method of least squares presents peculiar difficulties, but, enormous in extent as the work was, it has been thoroughly carried out. The ten base lines on which the survey depends were measured with Colby’s compensation bars.
The survey is detailed in eighteen volumes, published at Dehra Dun, and entitledAccount of the Operations of the Great Trigonometrical Survey of India. Of these the first nine were published under the direction of Colonel Walker; and the remainder by Colonels Strahan and St G.C. Gore, Major S.G. Burrard and others. Vol. i., 1870, treats of the base lines; vol. ii., 1879, history and general descriptions of the principal triangulation and of its reduction; vol. v., 1879, pendulum operations (Captains T.P. Basevi and W.T. Heaviside); vols. xi., 1890, and xviii., 1906, latitudes; vols. ix., 1883, x., 1887, xv., 1893, longitudes; vol. xvii., 1901, the Indo-European longitude-arcs from Karachi to Greenwich. The other volumes contain the triangulations.
The survey is detailed in eighteen volumes, published at Dehra Dun, and entitledAccount of the Operations of the Great Trigonometrical Survey of India. Of these the first nine were published under the direction of Colonel Walker; and the remainder by Colonels Strahan and St G.C. Gore, Major S.G. Burrard and others. Vol. i., 1870, treats of the base lines; vol. ii., 1879, history and general descriptions of the principal triangulation and of its reduction; vol. v., 1879, pendulum operations (Captains T.P. Basevi and W.T. Heaviside); vols. xi., 1890, and xviii., 1906, latitudes; vols. ix., 1883, x., 1887, xv., 1893, longitudes; vol. xvii., 1901, the Indo-European longitude-arcs from Karachi to Greenwich. The other volumes contain the triangulations.
In 1860 Friedrich Georg Wilhelm Struve published hisArc du méridien de 25° 20′ entre le Danube et la Mer Glaciale mesuré depuis 1816 jusqu’en 1855. The latitudes of the thirteen astronomical stations of this arc were determined partly with vertical circles and partly by means of the transit instrument in the prime vertical. The triangulation, a great part of which, however, is a simple chain of triangles, is reduced by the method of least squares, and the probable errors of the resulting distances of parallels is given; the probable error of the whole arc in length is ± 6.2 toises. Ten base lines were measured. The sum of thelengths of the ten measured bases is 29,863 toises, so that the average length of a base line is 19,100 ft. The azimuths were observed at fourteen stations. In high latitudes the determination of the meridian is a matter of great difficulty; nevertheless the azimuths at all the northern stations were successfully determined,—the probable error of the result at Fuglenaes being ± 0″.53.
Before proceeding with the modern developments of geodetic measurements and their application to the figure of the earth, we must discuss the “mechanical theory,” which is indispensable for a full understanding of the subject.
Mechanical Theory.
Newton, by applying his theory of gravitation, combined with the so-called centrifugal force, to the earth, and assuming that an oblate ellipsoid of rotation is a form of equilibrium for a homogeneous fluid rotating with uniform angular velocity, obtained the ratio of the axes 229:230, and the law of variation of gravity on the surface. A few years later Huygens published an investigation of the figure of the earth, supposing the attraction of every particle to be towards the centre of the earth, obtaining as a result that the proportion of the axes should be 578 : 579. In 1740 Colin Maclaurin, in hisDe causa physica fluxus et refluxus maris, demonstrated that the oblate ellipsoid of revolution is a figure which satisfies the conditions of equilibrium in the case of a revolving homogeneous fluid mass, whose particles attract one another according to the law of the inverse square of the distance; he gave the equation connecting the ellipticity with the proportion of the centrifugal force at the equator to gravity, and determined the attraction on a particle situated anywhere on the surface of such a body. In 1743 Clairault published hisThéorie de la figure de la terre, which contains a remarkable theorem (“Clairault’s Theorem”), establishing a relation between the ellipticity of the earth and the variation of gravity from the equator to the poles. Assuming that the earth is composed of concentric ellipsoidal strata having a common axis of rotation, each stratum homogeneous in itself, but the ellipticities and densities of the successive strata varying according to any law, and that the superficial stratum has the same form as if it were fluid, he proved that
where g, g′ are the amounts of gravity at the equator and at the pole respectively, e the ellipticity of the meridian (or “flattening”), and m the ratio of the centrifugal force at the equator to g. He also proved that the increase of gravity in proceeding from the equator to the poles is as the square of the sine of the latitude. This, taken with the former theorem, gives the means of determining the earth’s ellipticity from observation of the relative force of gravity at any two places. P.S. Laplace, who devoted much attention to the subject, remarks on Clairault’s work that “the importance of all his results and the elegance with which they are presented place this work amongst the most beautiful of mathematical productions” (Isaac Todhunter’sHistory of the Mathematical Theories of Attraction and the Figure of the Earth, vol. i. p. 229).
The problem of the figure of the earth treated as a question of mechanics or hydrostatics is one of great difficulty, and it would be quite impracticable but for the circumstance that the surface differs but little from a sphere. In order to express the forces at any point of the body arising from the attraction of its particles, the form of the surface is required, but this form is the very one which it is the object of the investigation to discover; hence the complexity of the subject, and even with all the present resources of mathematicians only a partial and imperfect solution can be obtained.
We may here briefly indicate the line of reasoning by which some of the most important results may be obtained. If X, Y, Z be the components parallel to three rectangular axes of the forces acting on a particle of a fluid mass at the point x, y, z, then, p being the pressure there, and ρ the density,dp = ρ(Xdx + Ydy + Zdz);and for equilibrium the necessary conditions are, that ρ(Xdx + Ydy + Zdz) be a complete differential, and at the free surface Xdx + Ydy + Zdz = 0. This equation implies that the resultant of the forces is normal to the surface at every point, and in a homogeneous fluid it is obviously the differential equation of all surfaces of equal pressure. If the fluid be heterogeneous then it is to be remarked that for forces of attraction according to the ordinary law of gravitation, if X, Y, Z be the components of the attraction of a mass whose potential is V, thenXdx + Ydy + Zdz =dVdx +dVdy +dVdz,dxdydzwhich is a complete differential. And in the case of a fluid rotating with uniform velocity, in which the so-called centrifugal force enters as a force acting on each particle proportional to its distance from the axis of rotation, the corresponding part of Xdx + Ydy + Zdz is obviously a complete differential. Therefore for the forces with which we are now concerned Xdx + Ydy + Zdz = dU, where U is some function of x, y, z, and it is necessary for equilibrium that dp = ρdU be a complete differential; that is, ρ must be a function of U or a function of p, and so also p a function of U. So that dU = 0 is the differential equation of surfaces of equal pressure and density.We may now show that a homogeneous fluid mass in the form of an oblate ellipsoid of revolution having a uniform velocity of rotation can be in equilibrium. It may be proved that the attraction of the ellipsoid x² + y² + z²(1 + ε²) = c²(1 + ε²); upon a particle P of its mass at x, y, z has for componentsX = − Ax, Y = − Ay, Z = − Cz,whereA = 2πk²ρ(1 + ε²tan−1ε −1),ε³ε²C = 4πk²ρ(1 + ε²−1 + ε²tan−1ε),ε²ε³and k² the constant of attraction. Besides the attraction of the mass of the ellipsoid, the centrifugal force at P has for components + xω², + yω², 0; then the condition of fluid equilibrium is(A − ω²)xdx + (A − ω²)ydy + Czdz = 0,which by integration gives(A − ω²)(x² + y²) + Cz² = constant.This is the equation of an ellipsoid of rotation, and therefore the equilibrium is possible. The equation coincides with that of the surface of the fluid mass if we makeA − ω² = C / (1 + ε²),which givesω²=3 + ε²tan−1ε −3.2πk²ρε³ε²In the case of the earth, which is nearly spherical, we obtain by expanding the expression for ω² in powers of ε², rejecting the higher powers, and remarking that the ellipticity e = ½ε²,ω² / 2πk²ρ = 4ε² / 15 = 8e / 15.Now if m be the ratio of the centrifugal force to the intensity of gravity at the equator, and a = c(1 + e), thenm = aω² /4⁄3πk²ρa, ∴ ω² / 2πk²ρ =2⁄3m.In the case of the earth it is a matter of observation that m = 1/289, hence the ellipticitye = 5m / 4 = 1/231,so that the ratio of the axes on the supposition of a homogeneous fluid earth is 230 : 231, as stated by Newton.Now, to come to the case of a heterogeneous fluid, we shall assume that its surfaces of equal density are spheroids, concentric and having a common axis of rotation, and that the ellipticity of these surfaces varies from the centre to the outer surface, the density also varying. In other words, the body is composed of homogeneous spheroidal shells of variable density and ellipticity. On this supposition we shall express the attraction of the mass upon a particle in its interior, and then, taking into account the centrifugal force, form the equation expressing the condition of fluid equilibrium. The attraction of the homogeneous spheroid x² + y² + z²(1 + 2e) = c²(1 + 2e), where e is the ellipticity (of which the square is neglected), on an internal particle, whose co-ordinates are x = f, y = 0, z = h, has for its x and z componentsX′ = −4⁄3πk²ρf(1 −2⁄5e), Z′ = −4⁄3πk²ρh(1 +4⁄5e),the Y component being of course zero. Hence we infer that the attraction of a shell whose inner surface has an ellipticity e, and its outer surface an ellipticity e + de, the density being ρ, is expressed bydX′ =4⁄3·2⁄5πk²ρf de, dZ′ = −4⁄3·4⁄5πk²ρh de.To apply this to our heterogeneous spheroid; if we put c1for the semiaxis of that surface of equal density on which is situated the attracted point P, and c0for the semiaxis of the outer surface, the attraction of that portion of the body which is exterior to P, namely, of all the shells which enclose P, has for componentsX0=8⁄15πk²f∫c0c1ρdedc, Z0=16⁄15πk²h∫c0c1ρdedc,dcdcboth e and ρ being functions of c. Again the attraction of a homogeneous spheroid of density ρ on anexternalpoint f, h has the componentsX″ = −4⁄3πk²ρfr−3{c³(1 + 2e) − λec5},Z″ = −4⁄3πk²ρhr−3{c³(1 + 2e) − λ′ec5},where λ =3⁄5(4h² − f²) / r4, λ′ =3⁄5(2h² − 3f²) / r4, and r² = f² + h².Now e being considered a function of c, we can at once express the attraction of a shell (density ρ) contained between the surface defined by c + dc, e + de and that defined by c, e upon an external point; the differentials with respect to c, viz. dX″ dZ″, must then be integrated with ρ under the integral sign as being a function of c. The integration will extend from c = 0 to c = c1. Thus the components of the attraction of the heterogeneous spheroid upon a particle within its mass, whose co-ordinates are f, 0, h, areX = −4⁄3πk²f[1∫c10ρ d{c³(1 + 2e)} −λ∫c10ρ d(ec5) +2⁄5∫c0c1ρ de],r3r3Z = −4⁄3πk²h[1∫c10ρ d{c³(1 + 2e)} −λ′∫c10ρ d(ec5) +4⁄5∫c0c1ρ de].r3r3We take into account the rotation of the earth by adding the centrifugal force fω² = F to X. Now, the surface of constant density upon which the point f, 0, h is situated gives (1 − 2e) fdf + hdh = 0; and the condition of equilibrium is that (X + F)df + Zdh = 0. Therefore,(X + F) h = Zf (1 − 2e),which, neglecting small quantities of the order e² and putting ω²t² = 4π²k², gives2e∫c10ρd{c³(1 + 2e)} −6∫c10ρd(ec5) −6∫c10ρde =3π.r³5r55t²Here we must now put c for c1, c for r; and 1 + 2e under the first integral sign may be replaced by unity, since small quantities of the second order are neglected. Two differentiations lead us to the following very important differential equation (Clairault):d²e+2ρc²·de+(2ρc−6)e = 0.dc²∫ρc² dcdc∫ρc² dcc²When ρ is expressed in terms of c, this equation can be integrated. We infer then that a rotating spheroid of very small ellipticity, composed of fluid homogeneous strata such as we have specified, will be in equilibrium; and when the law of the density is expressed, the law of the corresponding ellipticities will follow.If we put M for the mass of the spheroid, thenM =4π∫c0ρd{c³(1 + 2e)}; and m =c³·4π²,3Mt²and putting c = c0in the equation expressing the condition of equilibrium, we findM(2e − m) =4π ·6∫c0ρ d(ec5).35c²Making these substitutions in the expressions for the forces at the surface, and putting r/c = 1 + e − e(h/c)², we getG cos φ =Mk²{1 − e −3m +(5m − 2e)h²}fac22c²cG sin φ =Mk²{1 + e −3m +(5m − 2e)h²}h.ac22c²cHere G is gravity in the latitude φ, and a the radius of the equator. Sincesec φ = (c/f){1 + e + (eh²/c²)},G =Mk²{1 −3m +(5m − e)sin² φ},ac22an expression which contains the theorems we have referred to as discovered by Clairault.The theory of the figure of the earth as a rotating ellipsoid has been especially investigated by Laplace in hisMécanique celeste. The principal English works are:—Sir George Airy,Mathematical Tracts, a lucid treatment without the use of Laplace’s coefficients; Archdeacon Pratt’sAttractions and Figure of the Earth; and O’Brien’sMathematical Tracts; in the last two Laplace’s coefficients are used.
We may here briefly indicate the line of reasoning by which some of the most important results may be obtained. If X, Y, Z be the components parallel to three rectangular axes of the forces acting on a particle of a fluid mass at the point x, y, z, then, p being the pressure there, and ρ the density,
dp = ρ(Xdx + Ydy + Zdz);
and for equilibrium the necessary conditions are, that ρ(Xdx + Ydy + Zdz) be a complete differential, and at the free surface Xdx + Ydy + Zdz = 0. This equation implies that the resultant of the forces is normal to the surface at every point, and in a homogeneous fluid it is obviously the differential equation of all surfaces of equal pressure. If the fluid be heterogeneous then it is to be remarked that for forces of attraction according to the ordinary law of gravitation, if X, Y, Z be the components of the attraction of a mass whose potential is V, then
which is a complete differential. And in the case of a fluid rotating with uniform velocity, in which the so-called centrifugal force enters as a force acting on each particle proportional to its distance from the axis of rotation, the corresponding part of Xdx + Ydy + Zdz is obviously a complete differential. Therefore for the forces with which we are now concerned Xdx + Ydy + Zdz = dU, where U is some function of x, y, z, and it is necessary for equilibrium that dp = ρdU be a complete differential; that is, ρ must be a function of U or a function of p, and so also p a function of U. So that dU = 0 is the differential equation of surfaces of equal pressure and density.
We may now show that a homogeneous fluid mass in the form of an oblate ellipsoid of revolution having a uniform velocity of rotation can be in equilibrium. It may be proved that the attraction of the ellipsoid x² + y² + z²(1 + ε²) = c²(1 + ε²); upon a particle P of its mass at x, y, z has for components
X = − Ax, Y = − Ay, Z = − Cz,
where
and k² the constant of attraction. Besides the attraction of the mass of the ellipsoid, the centrifugal force at P has for components + xω², + yω², 0; then the condition of fluid equilibrium is
(A − ω²)xdx + (A − ω²)ydy + Czdz = 0,
which by integration gives
(A − ω²)(x² + y²) + Cz² = constant.
This is the equation of an ellipsoid of rotation, and therefore the equilibrium is possible. The equation coincides with that of the surface of the fluid mass if we make
A − ω² = C / (1 + ε²),
which gives
In the case of the earth, which is nearly spherical, we obtain by expanding the expression for ω² in powers of ε², rejecting the higher powers, and remarking that the ellipticity e = ½ε²,
ω² / 2πk²ρ = 4ε² / 15 = 8e / 15.
Now if m be the ratio of the centrifugal force to the intensity of gravity at the equator, and a = c(1 + e), then
m = aω² /4⁄3πk²ρa, ∴ ω² / 2πk²ρ =2⁄3m.
In the case of the earth it is a matter of observation that m = 1/289, hence the ellipticity
e = 5m / 4 = 1/231,
so that the ratio of the axes on the supposition of a homogeneous fluid earth is 230 : 231, as stated by Newton.
Now, to come to the case of a heterogeneous fluid, we shall assume that its surfaces of equal density are spheroids, concentric and having a common axis of rotation, and that the ellipticity of these surfaces varies from the centre to the outer surface, the density also varying. In other words, the body is composed of homogeneous spheroidal shells of variable density and ellipticity. On this supposition we shall express the attraction of the mass upon a particle in its interior, and then, taking into account the centrifugal force, form the equation expressing the condition of fluid equilibrium. The attraction of the homogeneous spheroid x² + y² + z²(1 + 2e) = c²(1 + 2e), where e is the ellipticity (of which the square is neglected), on an internal particle, whose co-ordinates are x = f, y = 0, z = h, has for its x and z components
X′ = −4⁄3πk²ρf(1 −2⁄5e), Z′ = −4⁄3πk²ρh(1 +4⁄5e),
the Y component being of course zero. Hence we infer that the attraction of a shell whose inner surface has an ellipticity e, and its outer surface an ellipticity e + de, the density being ρ, is expressed by
dX′ =4⁄3·2⁄5πk²ρf de, dZ′ = −4⁄3·4⁄5πk²ρh de.
To apply this to our heterogeneous spheroid; if we put c1for the semiaxis of that surface of equal density on which is situated the attracted point P, and c0for the semiaxis of the outer surface, the attraction of that portion of the body which is exterior to P, namely, of all the shells which enclose P, has for components
both e and ρ being functions of c. Again the attraction of a homogeneous spheroid of density ρ on anexternalpoint f, h has the components
X″ = −4⁄3πk²ρfr−3{c³(1 + 2e) − λec5},
Z″ = −4⁄3πk²ρhr−3{c³(1 + 2e) − λ′ec5},
where λ =3⁄5(4h² − f²) / r4, λ′ =3⁄5(2h² − 3f²) / r4, and r² = f² + h².
Now e being considered a function of c, we can at once express the attraction of a shell (density ρ) contained between the surface defined by c + dc, e + de and that defined by c, e upon an external point; the differentials with respect to c, viz. dX″ dZ″, must then be integrated with ρ under the integral sign as being a function of c. The integration will extend from c = 0 to c = c1. Thus the components of the attraction of the heterogeneous spheroid upon a particle within its mass, whose co-ordinates are f, 0, h, are
We take into account the rotation of the earth by adding the centrifugal force fω² = F to X. Now, the surface of constant density upon which the point f, 0, h is situated gives (1 − 2e) fdf + hdh = 0; and the condition of equilibrium is that (X + F)df + Zdh = 0. Therefore,
(X + F) h = Zf (1 − 2e),
which, neglecting small quantities of the order e² and putting ω²t² = 4π²k², gives
Here we must now put c for c1, c for r; and 1 + 2e under the first integral sign may be replaced by unity, since small quantities of the second order are neglected. Two differentiations lead us to the following very important differential equation (Clairault):
When ρ is expressed in terms of c, this equation can be integrated. We infer then that a rotating spheroid of very small ellipticity, composed of fluid homogeneous strata such as we have specified, will be in equilibrium; and when the law of the density is expressed, the law of the corresponding ellipticities will follow.
If we put M for the mass of the spheroid, then
and putting c = c0in the equation expressing the condition of equilibrium, we find
Making these substitutions in the expressions for the forces at the surface, and putting r/c = 1 + e − e(h/c)², we get
Here G is gravity in the latitude φ, and a the radius of the equator. Since
sec φ = (c/f){1 + e + (eh²/c²)},
an expression which contains the theorems we have referred to as discovered by Clairault.
The theory of the figure of the earth as a rotating ellipsoid has been especially investigated by Laplace in hisMécanique celeste. The principal English works are:—Sir George Airy,Mathematical Tracts, a lucid treatment without the use of Laplace’s coefficients; Archdeacon Pratt’sAttractions and Figure of the Earth; and O’Brien’sMathematical Tracts; in the last two Laplace’s coefficients are used.
In 1845 Sir G.G. Stokes (Camb. Trans.viii.; see alsoCamb. Dub. Math. Journ., 1849, iv.) proved that if the external form of the sea—imagined to percolate the land by canals—be a spheroid with small ellipticity, then the law of gravity is that which we have shown above; his proof required no assumption as to the ellipticity of the internal strata, or as to the past or present fluidity of the earth. This investigation admits of being regarded conversely, viz. as determining the elliptical form of the earth from measurements of gravity; if G, the observed value of gravity in latitude φ, be expressed in the form G = g(1 + β sin² φ), where g is the value at the equator and β a coefficient. In this investigation, the square and higher powers of the ellipticity are neglected; the solution was completed by F.R. Helmert with regard to the square of the ellipticity, who showed that a term with sin² 2φ appeared (see Helmert,Geodäsie, ii. 83). For the coefficient of this term, the gravity measurements give a small but not sufficiently certain value; we therefore assume a value which agrees best with the hypothesis of the fluid state of the entire earth; this assumption is well supported, since even at a depth of only 50 km. the pressure of the superincumbent crust is so great that rocks become plastic, and behave approximately as fluids, and consequently the crust of the earth floats, to some extent, on the interior (even though this may not be fluid in the usual sense of the word). This is the geological theory of “Isostasis” (cf.Geology); it agrees with the results of measurements of gravity (vide infra), and was brought forward in the middle of the 19th century by J.H. Pratt, who deduced it from observations made in India.
The sin² 2φ term in the expression for G, and the corresponding deviation of the meridian from an ellipse, have been analytically established by Sir G.H. Darwin and E. Wiechert; earlier and less complete investigations were made by Sir G.B. Airy and O. Callandreau. In consequence of the sin² 2φ term, two parameters of the level surfaces in the interior of the earth are to be determined; for this purpose, Darwin develops two differential equations in the place of the one by Clairault. By assuming Roche’s law for the variation of the density in the interior of the Earth, viz. ρ = ρ1− k(c/c1)², k being a coefficient, it is shown that in latitude 45°, the meridian is depressed about 3¼ metres from the ellipse, and the coefficient of the term sin²φ cos²φ (= ¼ sin²2φ) is −0.0000295. According to Wiechert the earth is composed of a kernel and a shell, the kernel being composed of material, chiefly metallic iron, of density near 8.2, and the shell, about 900 miles thick, of silicates, &c., of density about 3.2. On this assumption the depression in latitude 45° is 2¾ metres, and the coefficient of sin²φ cos²φ is, in round numbers, −0.0000280.2To this additional term in the formula for G, there corresponds an extension of Clairault’s formula for the calculation of the flattening from β with terms of the higher orders; this was first accomplished by Helmert.
For a long time the assumption of an ellipsoid with three unequal axes has been held possible for the figure of the earth, in consequence of an important theorem due to K.G. Jacobi, who proved that for a homogeneous fluid in rotation a spheroid is not the only form of equilibrium; an ellipsoid rotating round its least axis may with certain proportions of the axes and a certain time of revolution be a form of equilibrium.3It has been objected to the figure of three unequal axes that it does not satisfy, in the proportions of the axes, the conditions brought out in Jacobi’s theorem (c : a < 1/√2). Admitting this, it has to be noted, on the other hand, that Jacobi’s theorem contemplates a homogeneous fluid, and this is certainly far from the actual condition of our globe; indeed the irregular distribution of continents and oceans suggests the possibility of a sensible divergence from a perfect surface of revolution. We may, however, assume the ellipsoid with three unequal axes to be an interpolation form. More plausible forms are little adapted for computation.4Consequently we now generally take the ellipsoid of rotation as a basis, especially so because measurements of gravity have shown that the deviation from it is but trifling.
Local Attraction.
In speaking of the figure of the earth, we mean the surface of the sea imagined to percolate the continents by canals. Thatthis surface should turn out, after precise measurements, to be exactly an ellipsoid of revolution isa prioriimprobable. Although it may be highly probable that originally the earth was a fluid mass, yet in the cooling whereby the present crust has resulted, the actual solid surface has been left most irregular in form. It is clear that these irregularities of the visible surface must be accompanied by irregularities in the mathematical figure of the earth, and when we consider the general surface of our globe, its irregular distribution of mountain masses, continents, with oceans and islands, we are prepared to admit that the earth may not be precisely any surface of revolution. Nevertheless, there must exist some spheroid which agrees very closely with the mathematical figure of the earth, and has the same axis of rotation. We must conceive this figure as exhibiting slight departures from the spheroid, the two surfaces cutting one another in various lines; thus a point of the surface is defined by its latitude, longitude, and its height above the “spheroid of reference.” Calling this height N, then of the actual magnitude of this quantity we can generally have no information, it only obtrudes itself on our notice by its variations. In the vicinity of mountains it may change sign in the space of a few miles; N being regarded as a function of the latitude and longitude, if its differential coefficient with respect to the former be zero at a certain point, the normals to the two surfaces then will lie in the prime vertical; if the differential coefficient of N with respect to the longitude be zero, the two normals will lie in the meridian; if both coefficients are zero, the normals will coincide. The comparisons of terrestrial measurements with the corresponding astronomical observations have always been accompanied with discrepancies. Suppose A and B to be two trigonometrical stations, and that at A there is a disturbing force drawing the vertical through an angle δ, then it is evident that the apparent zenith of A will be really that of some other place A′, whose distance from A is rδ, when r is the earth’s radius; and similarly if there be a disturbance at B of the amount δ′, the apparent zenith of B will be really that of some other place B′, whose distance from B is rδ′. Hence we have the discrepancy that, while the geodetic measurements deal with the points A and B, the astronomical observations belong to the points A′, B′. Should δ, δ′ be equal and parallel, the displacements AA′, BB′ will be equal and parallel, and no discrepancy will appear. The non-recognition of this circumstance often led to much perplexity in the early history of geodesy. Suppose that, through the unknown variations of N, the probable error of an observed latitude (that is, the angle between the normal to the mathematical surface of the earth at the given point and that of the corresponding point on the spheroid of reference) be ε, then if we compare two arcs of a degree each in mean latitudes, and near each other, say about five degrees of latitude apart, the probable error of the resulting value of the ellipticity will be approximately ±1⁄500ε, ε being expressed in seconds, so that if ε be so great as 2″ the probable error of the resulting ellipticity will be greater than the ellipticity itself.
It is necessary at times to calculate the attraction of a mountain, and the consequent disturbance of the astronomical zenith, at any point within its influence. The deflection of the plumb-line, caused by a local attraction whose amount is k²Aδ, is measured by the ratio of k²Aδ to the force of gravity at the station. Expressed in seconds, the deflection Λ is
Λ = 12″.447Aδ / ρ,
where ρ is the mean density of the earth, δ that of the attracting mass, and A = ƒs−3xdv, in which dv is a volume element of the attracting mass within the distance s from the point of deflection, and x the projection of s on the horizontal plane through this point, the linear unit in expressing A being a mile. Suppose, for instance, a table-land whose form is a rectangle of 12 miles by 8 miles, having a height of 500 ft. and density half that of the earth; let the observer be 2 miles distant from the middle point of the longer side. The deflection then is 1″.472; but at 1 mile it increases to 2″.20.
At sixteen astronomical stations in the English survey the disturbance of latitude due to the form of the ground has been computed, and the following will give an idea of the results. At six stations the deflection is under 2″, at six others it is between 2″ and 4″, and at four stations it exceeds 4″. There is one very exceptional station on the north coast of Banffshire, near the village of Portsoy, at which the deflection amounts to 10″, so that if that village were placed on a map in a position to correspond with its astronomical latitude, it would be 1000 ft. out of position! There is the sea to the north and an undulating country to the south, which, however, to a spectator at the station does not suggest any great disturbance of gravity. A somewhat rough estimate of the local attraction from external causes gives a maximum limit of 5″, therefore we have 5″ which must arise from unequal density in the underlying strata in the surrounding country. In order to throw light on this remarkable phenomenon, the latitudes of a number of stations between Nairn on the west, Fraserburgh on the east, and the Grampians on the south, were observed, and the local deflections determined. It is somewhat singular that the deflections diminish in all directions, notveryregularly certainly, and most slowly in a south-west direction, finally disappearing, and leaving the maximum at the original station at Portsoy.
The method employed by Dr C. Hutton for computing the attraction of masses of ground is so simple and effectual that it can hardly be improved on. Let a horizontal plane pass through the given station; let r, θ be the polar co-ordinates of any point in this plane, and r, θ, z, the co-ordinates of a particle of the attracting mass; and let it be required to find the attraction of a portion of the mass contained between the horizontal planes z = 0, z = h, the cylindrical surfaces r = r1, r = r2, and the vertical planes θ = θ1, θ = θ2. The component of the attraction at the station or origin along the line θ = 0 is
By taking r2− r1, sufficiently small, and supposing h also small compared with r1+ r2(as it usually is), the attraction is
k²δ (r2− r1) (sinθ2− sinθ1) h/r,
where r = ½ (r1+ r2). This form suggests the following procedure. Draw on the contoured map a series of equidistant circles, concentric with the station, intersected by radial lines so disposed that the sines of their azimuths are in arithmetical progression. Then, having estimated from the map the mean heights of the various compartments, the calculation is obvious.
In mountainous countries, as near the Alps and in the Caucasus, deflections have been observed to the amount of as much as 30″, while in the Himalayas deflections amounting to 60″ were observed. On the other hand, deflections have been observed in flat countries, such as that noted by Professor K.G. Schweizer, who has shown that, at certain stations in the vicinity of Moscow, within a distance of 16 miles the plumb-line varies 16″ in such a manner as to indicate a vast deficiency of matter in the underlying strata; deflections of 10″ were observed in the level regions of north Germany.
Since the attraction of a mountain mass is expressed as a numerical multiple of δ : ρ the ratio of the density of the mountain to that of the earth, if we have any independent means of ascertaining the amount of the deflection, we have at once the ratio ρ : δ, and thus we obtain the mean density of the earth, as, for instance, at Schiehallion, and afterwards at Arthur’s Seat. Experiments of this kind for determining the mean density of the earth have been made in greater numbers; but they are not free from objection (seeGravitation).
Let us now consider the perturbation attending a spherical subterranean mass. A compact mass of great density at a small distance under the surface of the earth will produce an elevation of the mathematical surface which is expressed by the formula
y = aμ {(1 − 2u cosθ + u²)−1/2− 1},
where a is the radius of the (spherical) earth, a (1 − u) the distanceof the disturbing mass below the surface, μ the ratio of the disturbing mass to the mass of the earth, and aθ the distance of any point on the surface from that point, say Q, which is vertically over the disturbing mass. The maximum value of y is at Q, where it is y = aμu (1 − u). The deflection at the distance aθ is Λ = μu sinθ (1 − 2u cosθ + u²)−3/2, or since θ is small, putting h + u = 1, we have Λ = μθ (h² + θ²)−3/2. The maximum deflection takes place at a point whose distance from Q is to the depth of the mass as 1 : √2, and its amount is 2μ/3 √3h². If, for instance, the disturbing mass were a sphere a mile in diameter, the excess of its density above that of the surrounding country being equal to half the density of the earth, and the depth of its centre half a mile, the greatest deflection would be 5″, and the greatest value of y only two inches. Thus a large disturbance of gravity may arise from an irregularity in the mathematical surface whose actual magnitude, as regards height at least, is extremely small.
The effect of the disturbing mass μ on the vibrations of a pendulum would be a maximum at Q; if v be the number of seconds of time gained per diem by the pendulum at Q, and σ the number of seconds of angle in the maximum deflection, then it may be shown that v/σ = π√3/10.
The great Indian survey, and the attendant measurements of the degree of latitude, gave occasion to elaborate investigations of the deflection of the plumb-line in the neighbourhood of the high plateaus and mountain chains of Central Asia. Archdeacon Pratt (Phil. Trans., 1855 and 1857), in instituting these investigations, took into consideration the influence of the apparent diminution of the mass of the earth’s crust occasioned by the neighbouring ocean-basins; he concluded that the accumulated masses of mountain chains, &c., corresponded to subterranean mass diminutions, so that over any level surface in a fixed depth (perhaps 100 miles or more) the masses of prisms of equal section are equal. This is supported by the gravity measurements at Moré in the Himalayas at a height of 4696 metres, which showed no deflection due to the mountain chain (Phil. Trans., 1871); more recently, H.A. Faye (Compt. rend., 1880) arrived at the same conclusion for the entire continent.
This compensation, however, must only be regarded as a general principle; in certain cases, the compensating masses show marked horizontal displacements. Further investigations, especially of gravity measurements, will undoubtedly establish other important facts. Colonel S.G. Burrard has recently recalculated, with the aid of more exact data, certain Indian deviations of the plumb-line, and has established that in the region south of the Himalayas (lat. 24°) there is a subterranean perturbing mass. The extent of the compensation of the high mountain chains is difficult to recognize from the latitude observations, since the same effect may result from different causes; on the other hand, observations of geographical longitude have established a strong compensation.5
Meridian Arcs.
The astronomical stations for the measurement of the degree of latitude will generally lie not exactly on the same meridian; and it is therefore necessary to calculate the arcs of meridian M which lie between the latitude of neighbouring stations. If S be the geodetic line calculated from the triangulation with the astronomically determined azimuths α1and α2, then
in which 2α = α1+ α2− 180°, Δα = α2− α1− 180°.
The length of the arc of meridian between the latitudes φ1and φ2is
where a²e² = a² − b²; instead of using the eccentricity e, put the ratio of the axes b : a = 1 − n : 1 + n, then
This, after integration, gives
where
α0= φ2− φ1α1= sin (φ2− φ1) cos (φ2+ φ1)α2= sin 2(φ2− φ1) cos 2(φ2+ φ1)α3= sin 3(φ2− φ1) cos 3(φ2+ φ1).
α0= φ2− φ1
α1= sin (φ2− φ1) cos (φ2+ φ1)
α2= sin 2(φ2− φ1) cos 2(φ2+ φ1)
α3= sin 3(φ2− φ1) cos 3(φ2+ φ1).
The part of M which depends on n³ is very small; in fact, if we calculate it for one of the longest arcs measured, the Russian arc, it amounts to only an inch and a half, therefore we omit this term, and put for M/b the value
Now, if we suppose the observed latitudes to be affected with errors, and that the true latitudes are φ1+ x1, φ2+ x2; and if further we suppose that n1+ dn is the true value of a − b : a + b, and that n1itself is merely a very approximate numerical value, we get, on making these substitutions and neglecting the influence of the corrections x on thepositionof the arc in latitude,i.e.on φ1+ φ2,
here dα0= x2− x1; and as b is only known approximately, put b = b1(1 + u); then we get, after dividing through by the coefficient of dα0, which is = 1 + n1− 3n1cos(φ2− φ1) cos(φ2+ φ1), an equation of the form x2= x1+ h + fu + gv, where for convenience we put v for dn.
Now in every measured arc there are not only the extreme stations determined in latitude, but also a number of intermediate stations so that if there be i + 1 stations there will be i equations
x2= x1+ f1u + g1v + h1x3= x1+ f2u + g2v + h2: : :: : :xi= x1+ fiu + giv + hi
x2= x1+ f1u + g1v + h1
x3= x1+ f2u + g2v + h2
: : :
: : :
xi= x1+ fiu + giv + hi
In combining a number of different arcs of meridian, with the view of determining the figure of the earth, each arc will supply a number of equations in u and v and the corrections to its observed latitudes. Then, according to the method of least squares, those values of u and v are the most probable which render the sum of the squares ofallthe errors x a minimum. The corrections x which are here applied arise not from errors of observation only. The mere uncertainty of a latitude, as determined with modern instruments, does not exceed a very small fraction of a second as far as errors of observation go, but no accuracy in observing will remove the error that may arise from local attraction. This, as we have seen, may amount to some seconds, so that the corrections x to the observed latitudes are attributable to local attraction. Archdeacon Pratt objected to this mode of applying least squares first used by Bessel; but Bessel was right, and the objection is groundless. Bessel found, in 1841, from ten meridian arcs with a total amplitude of 50°.6:
a = 3272077 toises = 6377397 metres.e (ellipticity) = (a − b) / a ≈ 1/299.15 (prob. error ± 3.2).
a = 3272077 toises = 6377397 metres.
e (ellipticity) = (a − b) / a ≈ 1/299.15 (prob. error ± 3.2).
The probable error in the length of the earth’s quadrant is ± 336 m.
We now give a series of some meridian-arcs measurements, which were utilized in 1866 by A.R. Clarke in theComparisons of the Standards of Length, pp. 280-287; details of the calculations are given by the same author in hisGeodesy(1880), pp. 311 et seq.
The data of the French arc from Formentera to Dunkirk are—
The distance of the parallels of Dunkirk and Greenwich, deduced from the extension of the triangulation of England into France, in 1862, is 161407.3 ft., which is 3.9 ft. greater than that obtained from Captain Kater’s triangulation, and 3.2 ft. less than the distance calculated by Delambre from General Roy’s triangulation. The following table shows the data of the English arc with the distances in standard feet from Formentera.
The latitude assigned in this table to Saxavord is not the directly observed latitude, which is 60° 49′ 38.58″, for there are here a cluster of three points, whose latitudes are astronomically determined; and if we transfer, by means of the geodesic connexion, the latitude of Gerth of Scaw to Saxavord, we get 60° 49′ 36.59″; and if we similarly transfer the latitude of Balta, we get 60° 49′ 36.46″. The mean of these three is that entered in the above table.
For the Indian arc in long. 77° 40′ we have the following data:—
The data of the Russian arc (long. 26° 40′) taken from Struve’s work are as below:—
From the are measured in Cape Colony by Sir Thomas Maclear in long. 18° 30′, we have