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= sin

1

1
+ € {1+ (2 + cos x) cos 2m} tan X.
2a

2

Now s= a

=a (

1

1

E a

3

ae sin cos 2m, by Art. 96, 2

= a (2-6) sin-1 + ae {1+(2+cos x) cos 2m} tan

2a

3

ae sin a cos 2m 2

= (a +6) sin". +(a – b) {1+2 (1 – cos ) cos 2m) tank x.

Taking the variations, c being constant,

a + 6

cda 88 = (a +86) sin

с

2a

a

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+ (8a – 8b) {1 + (1 –cos 1) cos 2m} tan a. The terms being small we may approximate ;

1
.. ds= (da +86) 5X-2 tan-x.da

)

+ (8a – 86) {1+; (1 – cos a) cos 2m} tanî = (8a+81) (fx-tan ») + (8a – 87) tanf(1–cos») cos 2m.

115. Secondly. The distance between the arcs. The equation to the local ellipse is

(oc - a)? (y - 3)2

B! a

= 1; 72

+

::. **+ya or pe=a? +202 + 2By 2(a* — c)

= a + 2aa cos 0 + 2aß sin 0 – 2a'e sin 0;

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Let R, C, C' be the values of r at the mid-latitude and at the extremities of the arc;

.: R=a + a cos m+ B sin m - (a - b) sin’m,

C=a +a cos l + B sin ? (a - b) sinol,
C"= at a cos l' + B sin l' (a - b) sino l'.

Multiply by 1, M, and N; add, and make the coefficients of a and B vanish;

1

... cos m +M cos + N cos l' = 0, sin m + M sin l+N sin l' =0);

sin (m 7)

sin (l' –1) R+MC+NC

.. M

seco a = N;

= a (1 + M+N) – (a - b) (sinm + M sin'l + N sin l')

=a (1+2M) – } (a –b) {1 – cos 2m +2M (1 – cos A cos 2m)} =} (a+b) (1 +2M) + (a 8) (1+2M cos 2) cos 2m =} (a + b) (1– sec ) +ă (a – 0) (1 – see a cos x) cos 2m.

1
2

Taking the variations, the distance required, or SR,

= } (Sa+80) (1 – secur)

+ (80 – 80) (1

80) (1 - sec a cosa)

cos 2m.

116. In order to apply these expressions for ds and SR to the arc in question, we must put

1=11'. 27'. 11", 2m= 47°. 34'.25";

.. ds = 0.0003397 da – 0·0010097 86,

SR= 0.0025536 da - 0.0075756 86.

Substituting the values of da and 86, Art. 113,
Arc I.

Arc II. Arc III.
Os - 0.0000936, 0.0009543, 0.0001977 miles,
SR=-0.0006492, 0.0071414, 0.0089798

These quantities are so small as to be practically insensible : the largest value of ds being 1 foot in an arc of 800 miles, and the

largest value of SR being less than 12 yards.

The result is, that the differences which are found to exist, between the observed amplitudes of arcs and the same'amplitudes calculated geodetically, can in no respect be accounted for by supposing the arcs to be curved differently from the mean ellipse; because, as the above calculation shows, the ellipses may differ considerably in form without producing a sensible effect upon the length of the arc. This conclusion differs from that come to in the first edition of this work. In that edition (Art. 104) the distance of one extremity of the arc from the centre of the ellipse was taken to be the same in the local and the mean ellipse; that is, those ellipses were supposed to be concentric, which they need not be, and are seen from the above investigation not to be. PROP. The differences between the astronomical and

geodetical amplitudes of an arc of meridian arise solely from local attraction, and are an accurate measure of the differences of local attraction at the extremities of the

117. The truth of this Proposition appears from the last Article. But we will establish it further by ascertaining how great a departure from the mean ellipse may exist without its producing even 1" of difference only in the amplitudes, as measured by the heavens and by the Earth.

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By Art. 116, as 1°= 69.5 miles,

0.00033978a-0.00100978b =1" = 0.0193056;

.. da= 56.8 +2.9786 miles.

The values of da and 86 which satisfy this equation and make the sum of their squares least, are da=5, 86=-17 miles. But this variation in b is greater even than the whole compression of the pole, which is only 13 miles, to say nothing of the value of da in addition. It may be safely concluded, therefore, that no hypothesis regarding the curvature of the Indian Arc will account for the defect of 3":791 and the excess of 5".236 in the geodetic measure, which Colonel Everest found in the two arcs between Kalianpur and Damargida and between Kaliana and Kalianpur, when compared with the astronomical latitudes.

There is no other possible cause, but local attraction affecting the plumb-line and level. These errors, therefore, become the accurate measure of the differences of the resultant local attraction, arising from causes visible and hidden, at the extremities of the arc. The effect of the two visible causes, the mountain-mass and the ocean, taken together is very well represented (as already explained) by 3":82 and 13".1i. To change these to – 35.79 and 5".24 (the values obtained by Colonel Everest from the comparison of the arc with the heavens) we must suppose some invisible cause, counteracting the effects of both the mountains and the ocean, and diminishing their combined effect in these two arcs respectively by 3":82 + 3"79 or 7":61 and 13":11 – 5"-24 or 7":87. These quantities are nearly equal, and point to some cause existing in the crust beneath near the middle of the arc, that is, in the neighbourhood of Kalianpur. That even a slight excess of density through a large space around Kaliana is capable of producing such an effect we have shown in Art. 65. An endless variety of other hypotheses may be conceived to produce this result, e. g. a deficiency of density beneath the mountains, accompanied by a corresponding deficiency south of Damargida towards Cape Comorin and the ocean. This double hypothesis is not, however, so simple as the single hypothesis above given. Whatever may be the facts of the case, this is certain, that the difference of the local attractions in the meridian at Damargida and Kalianpur is 3"79 south and the difference at Kalianpur and Kaliana is 5"-24 north.

PROP. To explain what effect local attraction will have upon the mapping of a country.

118. From what goes before it is clear that although the elliptic elements of the actual arc between two places may differ considerably from those of the mean ellipse, no sensible error will thence arise if we calculate the latitudes geodetically and with the elements of the mean ellipse.

If however the latitudes are laid down in a map from observations of the sun or stars they will be erroneous by the whole amount of deflection of the plumb-line by which the vertical and horizontal are determined. Thus in the case before us the deflections are as follows:

Kaliana. 27".98

At Damargida, Kalianpur, By mountains

6"•79 12.05 ocean

10 44 9.00 hidden cause

7 61

.00 Total deflections 24 84 21 05

6 -18
7 87

26 29

These angles converted into miles, at the rate of 1° to 69.5, or 51":8 to 1 mile, are 0.48, 0:41, 0.51 miles; by which quantities would the stations be wrongly placed on the map. "The relative error is largest in the upper division of the arc, and in that case is not more than 1-10th of a mile; but the positive error in each case is about half a mile.

If, then, the principal places are all marked down geodetically they will be correctly placed on the map, but if other places are filled in from observations of the sun or any other heavenly body they will be out of place by the whole of the error, viz. about half a mile.

PROP. Geodesy furnishes no evidence, in proof or disproof, of the upheaval or depression of the Earth's surface as suggested by geological phenomena.

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