λ= 9° 53' 44" = 35624", λ= 2° 32′ 43" 9163",

=

. . — (a - b) = 32208, 1 ( a + b) = 20883305,

2

a=20915513, b=20851097, € =

1

325

1 -0.08

300

It will be seen in these successive examples that the ellipticity is nearer and nearer to that deduced from the fluid theory; when the arcs compared are near each other the resulting ellipticity differs much from that value; but when they are more distant from each other, as in the fifth example, the result is far more accordant. This agrees with what was deduced from the formulæ in the last Article. If there were no errors in the data, viz. in the observed amplitudes and measured arcs, the results ought to come out in complete accordance with each other, if the figure of the Earth be truly spheroidal; for the formulæ are sufficiently exact for this purpose.

PROP. To explain the cause of the ellipses, determined from the several pairs of arcs, differing from each other.

126. We have assumed, (1) that the meridian arc is an ellipse, that being the form which it would have were the Earth fluid: (2) that the plumb-line at all stations of the meridian is a normal to this ellipse. These suggest in what direction we are to look for an explanation of the discrepancies in the results.

First. It is obvious that the form of equilibrium no longer actually exists, as all the variety of hill and dale, mountain and table-land and ocean-surface, sufficiently testifies. Geology teaches the same more generally and philosophically. Extensive portions now dry land were once at the bottom of the ocean, receiving the fossil deposits and burying them in the detritus of rocks, which time wore down, to become, as they are now, the records of their own history. Changes of level must therefore have taken place on a large scale. Landmarks in Scandinavia, the temple of Serapis at Puzzuoli, the ancient and recent coral-reefs in the Pacific, as pointed out

WHY DIFFERENT ARCS GIVE DIFFERENT RESULTS. 125

by Mr Darwin, all testify that these changes of level are still slowly going on. It has been suggested, with great probability, that it is caused by the expansion and contraction of vast portions of rock in the interior of the Earth arising from variations in temperature produced by chemical changes. Whatever the cause, the fact is certain. The Earth's form can no longer be a form of fluid-equilibrium, although the average form may be so.

Secondly. The plumb-line may not in all cases be perpendicular to the mean ellipse. Local attraction is sufficient to produce material errors in the vertical, and therefore in the amplitudes determined by meridian zenith distances of stars. For instance (Art. 55, Ex. 2), an error as great as 5" was discovered at Takal K'hera in Central India by Colonel Everest, arising from the attraction of a distant table-land. Sir Henry James has shown that a deflection of about the same amount occurs at Arthur's Seat, Edinburgh (Phil. Trans. 1857). We have mentioned that the attraction of the Himmalaya Mountains produces a deflection amounting to as much as 28" at the northern extremity of the Great Indian Arc (Art. 62, Ex. 1). We have calculated elsewhere (see Art. 62, Ex. 2, and Phil. Trans. for 1859) that the deficiency of matter in the vast ocean south of India causes such deflections as 6", 9", 10" 5, 19"7 at various stations: and (Art. 64) we have shown that it is not improbable that extensive but slight variations of density prevail in the interior of the Earth, the causes of which are not visible to us as mountain masses and vast oceans are, sufficient to produce errors in the plumb-line quite as great as and even greater than most of those already enumerated. These seem abundantly to account for the variety in the calculated semi-axes and ellipticities in the last Article, derived as they are from uncorrected observations.

127. Mr Airy has entered very thoroughly into a comparison (see Figure of the Earth, Encyc. Metrop.) of the various arcs measured in different parts of the world. He has used them according to their importance and value, as determined by the circumstances under which they were measured and observed.

128. The late M. Bessel devised a method by which the

results of all the surveys in different parts of the world might be brought to bear simultaneously upon the problem. This method is followed by Captain A. Clarke, R.E. in his Chapter on the figure of the earth at the end of the British Ordnance Survey Volume. The arcs which he uses in his calculation for determining the mean figure of the earth are eight in number; viz. the Anglo-Gallic, Russian, Indian II (or Great Arc), Indian I, Prussian, Peruvian, Hanoverian, and Danish arcs. These consist of fifty-eight subordinate divisions, the lengths of which have been measured and the latitudes of their extremities described. The method which Bessel invented was this: corrections, expressed in algebraical terms, are applied to the latitudes of the several stations dividing the arcs into their subordinate parts, such as to make their measured lengths exactly fit an ellipse. The values of the axes of this ellipse are then so determined as to make the sum of the squares of these corrections a minimum: that is the ellipse which most nearly represents the observations and measures, and is therefore taken to be the mean ellipse.

PROP. To obtain a formula for correcting the amplitude of an arc, so as to make its measured length accord with a given ellipse.

129. The length of an arc is, by Art. 122,

Suppose now that xx' are small corrections which must be applied to the observed latitudes to make the measured arc fit the ellipse of which a and b are the semi-axes; then λ and m, being obtained from observation, will not, when substituted in the above formula, give the measured value of 8; but

must be substituted instead of them. Hence omitting very small quantities,

Why DIFFERENT ARCS GIVE DIFFERENT RESULTS. 127

1

= (a + b) ( x + x' — x) — 23 ( a − b) sin (1 + x' — x) cos 2m

S=

2

= 1 (a + b) λ — 232 (a - b) sin λ cos 2m

+ 1 (x' — x) { (a + b) x − 3 (a — b) cos λ cos 2m} ;

28 − (a+b) λ + 3 (a - b) sin λ cos 2m

(a+b)λ-3 (a - b) cos λ cos 2m

Now the mean radius of the earth is known not to differ

much from 20890000 feet, and the ellipticity from therefore convenient to put a and b under the form

1

300

It is

where the squares of u and v may be neglected. When these are substituted in the formula it may be put in the following form,

x' = m +au+ßv+x,

where m, a, ẞ are functions of the observed latitudes and the measured length and other numerical quantities only.

The values of m, a, ẞ have been calculated in the Ordnance Survey Volume for the 85 divisions of the 8 arcs mentioned in Art. 128.

130. In pursuing the process described in Art. 128, the ten quantities u, v, x1, x,... x, are all considered as variables, to be determined so as to make the sum of the squares of the corrections a minimum. The result is, that u=-0.3856,

v=1.0620; and these make the semi-axes and ellipticity of the mean ellipse as follows:

But this process, we think, is not correct. Although ... are unknown quantities, yet they are not variables independent of u and v. This we shall show in the following Proposition.

PROP. To determine the correction of the latitude of the reference-station of an arc, in terms of the axes of the variable ellipse and the deflection of the plumb-line at the station arising from local attraction.

131. In the accompanying diagram the plane of the paper