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By these quantities the latitudes are diminished. Therefore the errors in the amplitudes are

— 13′′·11, — 3′′·82, +4′′·98.

These differ considerably from the differences of amplitude deduced from the arcs in the last Article. This shows us that there must be irregularities in the density of the crust below : their effect on the amplitudes is shown as follows:

Differences of amplitude

determined in last Article Effect of mountains and ocean Consequent effect of the

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hidden causes in the crust below + 8 91, +7 ·43, - 6 48. The hidden cause increases the amplitudes of the northern and middle of the three divisions of the Great Indian Arc, that is, makes the plumb-lines hang at a greater angle to each other; and diminishes the amplitude of the southern division, or makes the plumb-lines at its extremities hang less inclined to each other. An infinite variety of hypothetical arrangements of the materials of the crust may be conceived so as to produce this effect. The general result pointed to by this calculation is quite in accordance with the speculations of Art. 137; as the diminished attraction of the less dense parts of the crust below the mountains would, as it were, let the plumbline go at the northern end of the arc, and therefore increase its inclination to the plumb-line in the middle parts; and the increased density below the ocean would produce the opposite effect in the southern portion of the great arc.

PROP. To prove that the length of a mean arc of longitude is sensibly the same as the geodetically measured arc, if it do not exceed fifteen degrees in length.

This amount was not calculated in the Paper in the Philosophical Transactions alluded to above, as it was not there required. It has been since roughly obtained, in the same manner as the others, for the present purpose.

152. Let S be the length of the arc, 7 the latitude, L the longitudinal amplitude (i.e. the difference of the longitudes of its two extremities), c the chord. Then by Art. 123, Cor. 3, S= L cos l {a + (a − b) sin2 7},

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When a and b vary, c and 7 remain constant, but S and L vary. Hence

SS = SL cos l {a + (a − b) sin2 l} + L cos l {da + (da – db) sin3l},

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1

.. 88 = (L− 2 tan — L) cos 1 {da + (da — db) sin3l} ;

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Sa and Sb are arbitrary increments of a and b and produce the increment SS in the arc of longitude. We will find the least values of Sa and 8b, or those which make Sa2 +862 a minimum ;

.. sin* l da2+{(1 + sin2 1) da — n}2 = a minimum ;

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.. {sin*7+ (1 + sin2 7)2} dan (1 + sin l);

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=

L-2 tan L

Now put Sab 13 miles, SS arc 1" of a great circle

=

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This shows that I must be small. Expanding we have

L= 0·018, L= 0.262 (in arc) = 0·262 × 57°3 = 15o,

This shows that in an arc of longitude as much as fifteen degrees long (the length in miles depending, of course, on the latitude) it would require a departure from the mean ellipse equal to the whole actual compression of the pole of the earth to produce a difference in the length of the arc of only 00193 mile, or 102 feet. If it require so extravagant an hypothesis regarding the departure of the form of the arc from the mean form to produce so small a difference in the length, we may conclude that the actual difference in length of the actual arc and the mean arc of longitude is insensible, if the arc be no longer than fifteen degrees.

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

153. If the distances of places on the earth's surface referred to the mean spheroid were accurately known in miles, then by the use of the formulæ in Art. 123, and the mean axes the differences of latitude and longitude might be accurately determined, and the places laid down accordingly in a map would have their relative positions correctly assigned. But we have no direct means of ascertaining these distances. In the Propositions of Arts. 147, 152, however, it has been shown that the actual lengths of arcs measured by the Survey (that is, on the disturbed spheroid, so to speak,) differ from the lengths of the arcs on the mean spheroid by inappreciable quantities, if the

arcs are not chosen inordinately long, a thing which is never done. These measured arcs may therefore be used in this calculation instead of the mean arcs; and this convenient result is arrived at, that the relative position of places laid down on a map as determined by the Survey operations is not sensibly affected by any deviations of the form of the surface from the mean form, caused by those upheavings and depressions which geology shows us have undoubtedly taken place. The position of the map itself on the mean terrestrial spheroid would be fixed by ascertaining the absolute latitude and longitude of some one place in it. These would, of course, be affected by local attraction.

It thus appears that a map constructed wholly from geo-, detical measurements will be accurate in itself, that is, the relative position of places marked down in it will be correct. But the map itself will be as much out of its place on the terrestrial spheroid as the latitude and longitude of the station which fixes the map are erroneous in consequence of local attraction at that place. Also if any place is afterwards inserted in the map by observations made upon the heavens, the place will be out of its proper position by the difference in deflection of the plumb-line at that place and at the place the latitude and longitude of which fix the map.

PROP. To estimate the degree of departure of an arc of meridian between two stations from the curvature of the mean

arc.

154. Suppose an ellipse to be drawn through the extremities of the arc and so nearly coinciding with the arc so as to represent it. Let the origin of co-ordinates be very near the centre of this ellipse; r and 0, r' and ' polar co-ordinates to the extremities of the arc from the centre of the ellipse; a and ẞ rectangular co-ordinates to the centre of the ellipse, and therefore very small quantities. Hence the equation to this ellipse is

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•. x2 + y2 or r2 = a2 + 2ax+2ẞy — 2€ (a3 — x2)

=

a2+2aa cos 0+2aß sin 0-2a'e sin3 0;

CURVATURE OF ANY LOCAL ARC.

.. r = a + a cos 0 + ẞ sin - ae sin2 0.

153

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

.. R

= a + a cosm+ ẞ sin m

(a - b) sin3 m,

C = a + a cos l + B sin l - (a - b) sin2 1,

C': = a + a cos l' + ß sin l' — (a — b) sin2 l'.

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

.. cosm+M cos 1+ N cos l' = 0, sin m + M sin l + N sin l' = 0;

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= a (1 + M + N) − (a − b) (sin2m + M sin2 1 + N sin2 7′)

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Let Sa and 8b be the excess of the semi-axes of the actual arc above the axes of an ellipse equal to the mean ellipse and passing through the extremities of the arc, the axes of the two ellipses being parallel. Then taking the variations, the distance required, or SR,

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155. Ex. Let the arc be that between Kaliana (29° 30′ 48′′) and Damargida (18° 3′ 13′′): and let it be supposed to be part of the ellipse deduced in Art. 124, Ex. 2.

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