Page:EB1911 - Volume 08.djvu/835

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808

EARTH, FIGURE OF THE

Stations.	Astronomical Latitudes.			Distance of Parallels.
	°	′	″	Ft.
Formentera	38	39	53·17	· ·
Mountjouy	41	21	44·96	982671·04
Barcelona	41	22	47·90	988701·92
Carcassonne⁠	43	12	54·30	1657287·93
Pantheon	48	50	47·98	3710827·13
Dunkirk	51	2	8·41	4509790·84

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.

	°	′	″	Ft.
Formentera⁠	· ·			· ·
Greenwich	51	28	38·30	4671198·3
Arbury	52	13	26·59	4943837·6
Clifton	53	27	29·50	5394063·4
Kellie Law	56	14	53·60	6413221·7
Stirling	57	27	49·12	6857323·3
Saxavord	60	49	37·21	8086820·7

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:—

	°	′	″	Ft.
Punnea	8	9	31·132	· ·
Putchapolliam⁠	10	59	42·276	1029174·9
Dodagunta	12	59	52·165	1756562·0
Namthabad	15	5	53·562	2518376·3
Daumergida	18	3	15·292	3591788·4
Takalkhera	21	5	51·532	4697329·5
Kalianpur	24	7	11·262	5794695·7
Kaliana	29	30	48·322	7755835·9

The data of the Russian arc (long. 26° 40′) taken from Struve’s work are as below:—

	°	′	″	Ft.
Staro Nekrasovsk	45	20	2·94	· ·
Vodu-Luy	47	1	24·98	616529·81
Suprunkovzy	48	45	3·04	1246762·17
Kremenets	50	5	49·95	1737551·48
Byelin	52	2	42·16	2448745·17
Nemesh	54	39	4·16	3400312·63
Jacobstadt	56	30	4·97	4076412·28
Dorpat	58	22	47·56	4762421·43
Hogland	60	5	9·84	5386135·39
Kilpi-maki	62	38	5·25	6317905·67
Torneå	65	49	44·57	7486789·97
Stuor-oivi	68	40	58·40	8530517·90
Fuglenaes	70	40	11·23	9257921·06

From the are measured in Cape Colony by Sir Thomas Maclear in long. 18° 30′, we have

	°	′	″	Ft.
North End	29	44	17·66	· ·
Heerenlogement Berg	31	58	9·11	811507·7
Royal Observatory	33	56	3·20	1526386·8
Zwart Kop	34	13	32·13	1632583·3
Cape Point	34	21	6·26	1678375·7

And, finally, for the Peruvian arc, in long. 281° 0′,

	°	′	″	Ft.
Tarqui	3	4	32·068	· ·
Cotchesqui	0	2	31·387	1131036·3

Having now stated the data of the problem, we may seek that oblate ellipsoid (spheroid) which best represents the observations. Whatever the real figure may be, it is certain that if we suppose it an ellipsoid with three unequal axes, the arithmetical process will bring out an ellipsoid, which will agree better with all the observed latitudes than any spheroid would, therefore we do not prove that it is an ellipsoid; to prove this, arcs of longitude would be required. The result for the spheroid may be expressed thus:—

a = 20926062 ft. = 6378206·4 metres.

b = 20855121 ft. = 6356583·8 metres.

b : a = 293·98 : 294·98.

As might be expected, the sum of the squares of the 40 latitude corrections, viz. 153·99, is greater in this figure than in that of three axes, where it amounts to 138·30. For this case, in the Indian arc the largest corrections are at Dodagunta, + 3·87″, and at Kalianpur, - 3·68″. In the Russian arc the largest corrections are + 3·76″, at Torneå, and - 3·31″, at Staro Nekrasovsk. Of the whole 40 corrections, 16 are under 1·0″, 10 between 1·0″ and 2·0″, 10 between 2·0″ and 3·0″, and 4 over 3·0″. The probable error of an observed latitude is ± 1·42″; for the spheroidal it would be very slightly larger. This quantity may be taken therefore as approximately the probable amount of local deflection.

If $\rho$ be the radius of curvature of the meridian in latitude $\phi ,\rho '\!$ that perpendicular to the meridian, $\mathrm {D}$ the length of a degree of the meridian, $\mathrm {D} '\!$ the length of a degree of longitude, $r$ the radius drawn from the centre of the earth, $\mathrm {V}$ the angle of the vertical with the radius-vector, then

${\begin{array}{rllll}&&\qquad {\text{Ft.}}\\&\rho &=20890606\!\cdot \!6&-106411\!\cdot \!5\;\;\cos 2\phi &+225\!\cdot \!8\cos 4\phi \\&\rho '&=20961607\!\cdot \!3&-\;\;35590\!\cdot \!9\;\;\cos 2\phi &+45\!\cdot \!2\;\;\cos 4\phi \\&D&=\quad 364609\!\cdot \!87&-\quad 1857\!\cdot \!14\cos 2\phi &+\;\;3\!\cdot \!94\cos 4\phi \\&D'&=\quad 365538\!\cdot \!48\cos \phi &-\quad \;\;310\!\cdot \!17\cos 3\phi &+\;\;0\!\cdot \!39\cos 5\phi \\{\text{Log}}&r/a&=9\!\cdot \!9992645&+\;\;\!\cdot \!0007374\cos 2\phi &-\!\cdot \!0000019\cos 4\phi \\&V&=700\!\cdot \!44''\sin 2\phi &-1\!\cdot \!19''\sin 4\phi .\end{array}}$

A. R. Clarke has recalculated the elements of the ellipsoid of the earth; his values, derived in 1880, in which he utilized the measurements of parallel arcs in India, are particularly in practice. These values are:—

${\begin{array}{c}a=20926202{\text{ ft. }}=6378249{\text{ metres,}}\\b=20854895{\text{ ft. }}=6356515{\text{ metres,}}\\b:a=292\!\cdot \!465:293\!\cdot \!465.\end{array}}\!$

The calculation of the elements of the ellipsoid of rotation from measurements of the curvature of arcs in any given azimuth by means of geographical longitudes, latitudes and azimuths is indicated in the article Geodesy; reference may be made to Principal Triangulation, Helmert’s Geodasie, and the publications of the Kgl. Preuss. Geod. Inst.:—Lotabweichungen (1886), and Die europ. Längengradmessung in 52° Br. (1893). For the calculation of an ellipsoid with three unequal axes see Comparison of Standards, preface; and for non-elliptical meridians, Principal Triangulation, p. 733.

Gravitation-Measurements.

According to Clairault’s theorem (see above) the ellipticity $e$ of the mathematical surface of the earth is equal to the difference ${\tfrac {5}{2}}m-\beta$ , where $m$ is the ratio of the centrifugal force at the equator to gravity at the equator, and $\beta$ is derived from the formula $\mathrm {G} =g(1+\beta \sin ^{2}\phi )$ . Since the beginning of the 19th century many efforts have been made to determine the constants of this formula, and numerous expeditions undertaken to investigate the intensity of gravity in different latitudes. If $m$ be known, it is only necessary to determine $\beta$ for the evaluation of e; consequently it is unnecessary to determine $\mathrm {G}$ absolutely, for the relative values of $\mathrm {G}$ at two known latitudes suffice. Such relative measurements are easier and more exact than absolute ones. In some cases the ordinary thread pendulum, i.e. a spherical bob suspended by a wire, has been employed; but more often a rigid metal rod, bearing a weight and a knife-edge on which it may oscillate, has been adopted. The main point is the constancy of the pendulum. From the formula for the time of oscillation of the mathematically ideal pendulum, $t=2\pi {\sqrt {l/\mathrm {G} }}$ , $l$ being the length, it follows that for two points $\mathrm {G} _{1}/\mathrm {G} _{2}=t_{2}^{2}/t_{1}^{2}$ .

In 1808 J. B. Biot commenced his pendulum observations at several stations in western Europe; and in 1817–1825 Captain Louis de Freycinet and L. I. Duperrey prosecuted similar observations far into the southern hemisphere. Captain Henry Kater confined himself to British stations (1818–1819); Captain E. Sabine, from 1819 to 1829, observed similarly, with Kater’s pendulum, at seventeen stations ranging from the West Indies