CHAPTER II.

THE FIGURE OF THE EARTH, DETERMINED BY GEODETIC OPERATIONS.

121. IN the previous Chapter we have shown that if the Earth be considered a fluid mass the form of the surface will be an oblate spheroid of small ellipticity, its axis coinciding with the axis of revolution, and the surface being everywhere at right angles to the direction of gravity; and further, that upon assuming that the density of the strata varies according to a certain probable law, the ellipticity is somewhat greater

We propose to submit this to the test of measurement, by enquiring whether an ellipse can be found with its axis coinciding with the axis of the Earth and cutting the plumb-line at stations along it at right angles; and whether the ellip

1

ticity of that ellipse is about 300

The method of doing this is as follows. A base-line, about 5 or 6 miles in length, is measured with extreme accuracy, near the meridian, the curvature of which we are to find. By a series of triangles this base is connected with a number of stations in succession lying near the meridian, the angles and sides of which are calculated or observed, as the case may Thus a connexion is established between the original base and a second base at the termination of the chain of triangles, and the length of this second base obtained by calculation. It is then measured, as the first was, and by a comparison of the calculated and measured results the correctness or not of

be.

the operations is tested. This having been satisfactorily performed, the projections of the sides of the triangles on the meridian are found, and their sum gives the length of the meridian arc between its two extremities. The latitudes of these extremities are then observed with great care, and from these data the form of the ellipse, of which the arc is a part, is found by the principles of conic sections, as we shall now

show.

It is obvious that the actual surface of the earth is of a very irregular form, being diversified by mountains and valleys. In our investigation, at any rate in the first instance, these are not taken account of; the whole is supposed to be levelled down. and all the measures which are taken are reduced down to the sea-level, the sea being supposed to have the spheroidal form, since it is a free surface. The sea-level at any place means the level at which sea-water would stand if let in from the sea by a canal.

§ 1. The determination of the Mean Figure of the Earth.

PROP. To find the length of an arc of meridian in terms of the amplitude, the semi-axis major, the ellipticity (the ellipticity being small), and the middle latitude.

122. Let l and l' be the latitudes of the extremities of the arc, m the mean of these or the middle latitude; λ the amplitude of the arc or the difference between the latitudes; a, b, and e the semi-axes and ellipticity; s the length of the arc, r the radius vector, and the angle r makes with the major axis. Then

3 4

.. s=a {( 1 − e) (1 − 1) — ¦ e (sin 27— sin 21')}

= (a + b) x − 23 (a − b) sin x cos 2m.

121

123. COR. 1. If λ be small, not exceeding 12o, we may put sin λ=λ in this formula; then

COR. 2. The value of λ in terms of s including the square of the ellipticity is given by the formula, which may easily be deduced from the last Article, viz.:

Then the

COR. 3. Let S be the length of an arc of longitude in latitude 7, L the longitudinal amplitude of the arc. radius of the circle of longitude

=r cos 0 = a cos / (1 + e sin2 7) = cos l {a + (a−b) sin2 7} ; .. S= Lcos l {a + (a - b) sin3 l}.

PROP. To obtain formula for finding the semi-axes and ellipticity, when the lengths, amplitudes, and middle latitudes of two small arcs are known; and to ascertain what arcs are adapted to give the best results.

124. Let sλm, s'a'm' be the lengths, amplitudes, and middle latitudes;

by which a and b and therefore e are found.

The effect on the axes of any error in the amplitudes will be found by differentiating the above formulæ. În the denominators of the resulting expressions the quantity

will appear. The errors in the axes consequent on errors in the observed amplitudes will, therefore, be least when this quantity is a maximum. Suppose one arc is chosen in the southern half of the quadrant, cos 2m is positive; then

2m' = 180° or m' = 90°

will give the best result. Suppose one arc is in the northern half, cos 2m is negative; then 2m'=0 will give the best result. Hence the nearer one arc is to the pole and the other to the equator, the less will errors in the data affect the calculated form of the ellipse. This will be illustrated in the following examples. The data are taken from the Volume of the British Ordnance Survey, pp. 743, 757.

125. Ex. 1. Compare the two parts of the English Arc, from Saxaford (60° 49′ 39") to Clifton (53° 27' 30"), measuring 2692754 feet, and from Clifton to Southampton (50° 54′ 47′′), measuring 928774 feet.

λ = 7° 22′ 9′′ = 26529′′, λ' = 2° 32′ 43′′ = 9163′′,

•·. 1 (a - b) = 59419, 1 (a + b) = 20863630.

2

a = 20923049, b=20804211, e=

1 1+0.70

176

300

SEMI-AXES AND ELLIPTICITY.

123

Ex. 2. Compare the two parts of the Indian Arc from Kaliana (lat. 29° 30′ 48′′) to Kalianpur (24° 7' 11"), the length being 1961138 feet, and that between Kalianpur and Damargida (18° 3′ 15′′), the length being 2202905 feet.

λ = 5° 23' 37" = 19417", λ= 6° 3′ 56" 21836",

=

Ex. 3. Compare the arc between Kalianpur and Damargida with that between Damargida and Punnæ (8° 9′ 31′′) the length being 3591784 feet.

1

2

.. — (a − b) = 26194, 1⁄2 (a + b) = 20867130 feet,

1 1-0.25

a=20893324, b=20840936, €=

399

300

Ex. 4. Compare the arcs between Kaliana and Kalianpur and between Damargida and Punnæ.

λ = 5° 23′ 37′′ 19417", X'= 9° 53′ 44′′ = 35624",

=

Ex. 5. Compare the arcs between Damargida and Punnæ

and between Clifton and Southampton.