.. 1

cos (0—0') = 1 − cos λ - 2e sin3λ cos 2m

= 2 sin2 = {1 − 2e (1 + cos λ) cos 2m} ;

.•. c2 = 4a2 sin2 — \ {1 — 2e (1 + cos λ) cos 2m — e (sin37+sin°7′')}

1

--

= 4a2 sin2 — \ [1 − e {1 + (2 cos λ) cos 2m}];

λ

=

с

с

2a

[

1+

+ 1/2 € {1 + (2 + cos λ) cos 2m} 2a

2a

1

с

√ 4a2 - c2

1

+ e {1 + (2 + cos λ) cos 2m} tanλ.

2

= (a+b) sin1 2 + (a−b) { 1 + 1 (1 − cos λ) cos 2n

2a

{

2

Taking the variations of s with respect to a and b, c being constant, as also λ and m because they occur in small terms, we have the difference in length of two arcs joining the stations and belonging to different ellipses, only having their axes parallel.

= (da + db) (√

145

=

λ-tan

1 1

— tan 12 › ) + (da — db) | tan | › (1–cosλ) cos 2m

(Sa + db) P+ (da — db) Q cos 2m, suppose,

= (P + Q cos 2m) da + (P − Q cos 2m) 8b;

=

Sa and Sb are two arbitrary increments of a and b. We will find the least values of these which will produce a given increase ds to the arc: that is, the values which make da2 + b2 a minimum.

.. {(P– Q cos 2m)2 + (P+ Q cos 2m)2} da

Let one of the ellipses be equal to the ellipse of the earth's mean figure, a and b being the semi-axes; then da and b will be the excess (or defect, if negative) of the semi-axes of the other ellipse: this latter ellipse is the ellipse which most nearly coincides with the actual arc s of the level curve and therefore represents it. The first ellipse is not necessarily the mean ellipse itself, but is only equal to it in dimensions and parallel to it in position; for the actual arc may lie above or below the mean ellipse. The result of this is, that the arc of the mean ellipse which corresponds with s of the actual measured arc will not necessarily have precisely the same middle latitude, although the chord c is of the same length. But as the middle latitude will differ only by a quantity of the order of the ellipticity this difference will not appear in the result because we neglect the square of the ellipticity.

We will put ds = arc 1" = 0.0193 mile, 1° being 69.5 miles: and will find the value of λ which will make da~ db as large as the whole compression of the earth's pole, viz. 13 miles. This gives

P2 + Q2 ÷ Q = 0.0193 ÷ 13 = 0·0015,

A slight inspection of this equation shows that λ must be small. Expand in powers of λ; then

( + 1 ) ( )*°

3

= 0.0015, or

3

= 0.00135;

.. λ = 0·22 (in arc) = 0·22 × 57°.3 (in degrees) = 12o·6.

This shows, that in an arc of meridian as much as twelve degrees and a half in length it would require a departure from the mean ellipse equal to the whole actual compression of the pole of the earth in order to produce so slight a difference in the length as 1". Hence we may conclude that the difference in length between the mean arc and the actual

RELATIVE AMOUNT OF LOCAL ATTRACTION.

147

arc is in fact an insensible quantity, since an extravagant hypothesis regarding the departure of the form of the arc in question from the mean form will not produce a difference of length of more than 1".

148. The property here proved shows us at once how the mean amplitude of the arc may be found. By the formula in Art. 122 the mean amplitude may be calculated from the mean axes and the length of the mean arc when it is found. But the property now proved shows that this length is sensibly the same as the length of the geodetic arc, that is, the arc actually measured in the Survey, even though it may be altered in position by geological changes. This latter, then, may be used in the formula instead of the length of the mean arc, which but for this property would be unknown.

149. From what goes before it appears, that the difference between the astronomical amplitude and the mean amplitude thus found measures exactly the difference of meridian deflection caused by local attraction at the two extremities of the arc. The following PROP. will illustrate this.

PROP. To estimate the relative amount of local attraction in the plane of the meridian at stations on the Indian Arc.

150. The stations we shall take are Kaliana (29° 30′ 48′′), Kalianpur (24° 7' 11"), Damargida (18° 3′ 15′′), and Punno (8° 9' 31"). The lengths of the arcs connecting these stations (see Volume of the British Ordnance Survey, p. 757, where the data are all brought together) are 1961138, 2202905, and 3591784 feet respectively. By Art. 123, Cor. 1, we have the following formula:

λ=

or, since we neglect the square of the ellipticity,

a, b, and e are 20926180, 20855316,

(Art. 135).

295.3

Each of these differences measures the difference of local attraction in the meridian at the two stations at the extremities of the arc. They do not lead to the absolute amount of local attraction, but only to the difference of its amount in passing from one station to the next (see Art. 149).

151. The quantities above deduced are independent of any theory regarding the structure of the earth's mass. We may, however, endeavour to trace these resulting effects to their causes. In a former part of this treatise (Art. 62) it has been explained that two visible causes exist producing deflection, viz. the mountain mass on the north of India and the vast ocean on the south. It has also been shown (Art. 64) that a hidden cause of deflection may lie below, in the variation of the density of the earth's crust. The effect of the two visible causes has been estimated approximately by the author as follows (Phil. Trans. 1859):

* Colonel Everest brings out the first and second of these quantities -5" 24 and +3'79; but he works with different mean axes. We have used those of Art. 135.