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97. COR. If x be small, not exceeding 12°, we may put sin = 1 in this formula; then
3 cos 2m. λ 2
PROP. To obtain formulæ 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.
98. Let sam, 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. In the denominators of the resulting expressions the quantity
cos 2m - cos 2m'
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.
99. Ex. 1. Compare the two parts of the Indian Arc from Kaliana (lat. 29° 30'. 48") to Kalianpur (24'. 7. 11"), the length being 1961157 feet, and that between Kalianpur and Damargida (18°. 3'. 15"), the length being 2202926 feet. 1= 5o. 23'.37" = 19417", X'=6o.3'.56" = 21836",
2m= 53o. 37'. 59", 2m' = 42o. 10'. 26",
:-} (a – 1) = 54456, 3 (a+b) = 20929789 feet,
1 a= 20984245, b=20875333, E=
Ex. 2. Compare the two parts of the English Arc; viz. from Saxaford (60°. 49'. 39'') to Clifton (53o. 27'. 30"), measuring 2692754 feet, and from Clifton to Southampton (50°. 54'. 47''), measuring 928774 feet.
X = 70.22'.9" =26529", X'= 29.32'. 43" = 9163",
2m 114°. 17'.9", 2m' = 104°. 22'.17",
1 a=20923049, b=20804211, E=
176 Ex. 3. Compare the arc between Kalianpur and Damargida with that between Clifton and Southampton. 1 = 6'63'.56", l'=2o. 32'. 43",
2m = 42°. 10'. 26", 2m' = 104o. 22'. 17",
:: } (ab) = 33094, 3(a
1 a=20915864, b=20849676, E=
316 It will be seen in these examples that when the arcs compared are near each other the resulting ellipticity differs much from that deduced by the fluid theory: but when they are more distant from each other, as in the third example, the result is far more accordant. 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.
100. 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 by Mr Darwin, all testify that these changes of level are still slowly going on. It has been suggested, with great probability, to be 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 is so. Secondly. The plumb-line may not in all cases be
perpendicular even 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. 56), 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. 61). We have calculated elsewhere (see Art. 62 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. 63), 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.
101. 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. His result satisfactorily shows that the ellipticity of the mean spheroid is about zoo. The conditions, therefore, required for supposing the Earth to have received its present average form from having been once in a fluid state, are altogether satisfactorily fulfilled.
The same result has been obtained by another process, first used by the late M. Bessel and adopted by Captain A. Clarke, R. E. in the Volume of the British Ordnance Survey. This method we shall now explain, first introducing one or two propositions which we shall require for its application. Let the form of the meridian line be such, that
p= A + 2B cos 21+ 2C cos 41
is the radius of curvature at a point of which the latitude is 7.
Prop. To prove that if the meridian be an ellipse,
102. Let p'a'y be the radius of curvature and co-ordinates to a point in latitude 1, in an ellipse,
=1, =1–, ¢=1_o 72
Expanding this last, neglecting e®....
15 p’=a (1 – e") (1 + e* sin’l+
e sin 1)
Comparing this with p= A +2B cos 21 + 2C cos 41,