ERROR IN COMPARING TWO DIVISIONS OF AN ARC. 139

142. A similar calculation was afterwards made by Capt. Clarke, with Bessel's method (see Memoirs Roy. Ast. Soc. 1859-60, p. 25); but he neglects local attraction, as General de Schubert has done, although it is a disturbing cause of much more importance than any which the method of least squares is used to eliminate. In a subsequent paper indeed the General points out that local attraction may greatly modify, if not altogether destroy, the discrepancies between the different meridians (see Monthly Notices of Roy. Ast. Soc. No. 6, April 13, 1860, p. 264), a result which our calculation based upon the modification of Bessel's method fully confirms.

The following calculation shows the same in a simpler way. PROP. In comparing two divisions of an arc of meridian, to find the effect of a small deflection of the plumb-line at the middle station on the resulting axes.

143. Let +λ' be the astronomical amplitude of the whole arc, and λ and λ' the amplitudes of its two divisions. Then for determining the form of the meridian, we have, by Art. 124,

Suppose the latitude of the middle station is wrong, owing to unknown local attraction, by the quantity : then as λ+λ' is, by hypothesis, correctly determined, X' = — dλ ;

(by Art. 124), neglecting the ellipticity;

=

Ex. 1. In the Russian arc λ= 13° 1' 46860" from StaroNekrassowka to Dorpat, λ = 12° 17' = 44220" from Dorpat to Fuglenes; and twice the middle latitudes are 103° 43' and 129° 3'.

Ex. 2. In the Indian arc, divided at Damargida,

λ=11° 27' 33" = 41253"

from Kaliana to Damargida,

λ=9° 53' 44" = 35624"

from Damargida to Punno:

also 2m 47° 34', 2m' 26° 13'. If dλ=1",

=

=

DEVIATIONS FROM THE MEAN FORM.

141

These are large quantities; and if they are so large for only 1" of local attraction, they may be in fact much larger than this, without our having any means of knowing it. We have already shown (Art. 62) that there may be much larger deflections than 1" without any visible cause to produce them. The calculations referred to in the last Article regarding the elliptical form of the equator are, therefore, not to be considered as trustworthy.

§ 2. The form of separate parts of the surface,

144. What has gone before leads to the determination of only the Mean Figure of the Earth. Our knowledge, however, of the surface-diversified as it is with mountains, plains, and oceans-is sufficient to show that particular parts of the surface depart from this mean figure.

We have shown already that the large effects of the Himmalayas and the Ocean in India are very nearly compensated for by variations in density in the crust. The residual deflections, however, are not to be overlooked. It is to the consideration of these that we now call the attention of the student. In the course of our remarks some things will be explained which probably have not been so thoroughly understood in what has gone before as they will be now.

PROP. To explain what is meant by the Sea-level, and to point out its use.

145. In the diagram suppose A is the station from which we commence: and suppose the dark line AB to be the curve

N

in which still water would lie, if a canal were cut from the sea along the meridian through A northwards, and the sea were allowed to flow into it. This curve is called the SEA-LEVEL. Where the level changes owing to the ebb and flow of the tide the mean is taken.

The plumb-line at every place along this curve hangs at right angles to the curve at that place; because it is one condition of fluid equilibrium, that the resultant force at any point of the fluid surface acts in the normal at that point (Art. 75). This level-curve will partake, therefore, of all the irregularities of the plumb-line caused by local attraction. It indicates the general form of the surface, altered as it has become, since the earth ceased to be a fluid mass, by the upheavings and sinkings which geology teaches us have most certainly taken place.

It is this curve which is meant when we speak of the Arc of Meridian, and it is the work of the Trigonometrical Survey to determine its form, and to measure the elevations and depressions of places on the meridian with reference to it. A and B are in fact points in which verticals through these stations cut this level-curve, and are not necessarily the places themselves, which may be some feet above or below them. The exact contour of the earth's visible surface is obtained by finding the form of the level-curve or arc of meridian, and also the elevations or depressions of places, above or below this curve. The level-curve is not necessarily an ellipse: indeed most likely it is not: but as it evidently does not differ much from a circle short portions of it may be represented very well by an elliptic arc of small ellipticity.

PROP. To explain what is meant by Astronomical and Mean Amplitudes.

146. Let ONE represent an elliptic quadrant of the earth's mean figure, O being the centre of the earth. It does not necessarily follow that the local arc AB should lie on this quadrant; owing to the local departure from the mean figure AB may lie above it (as in the diagram) or below it. Let the dotted quadrant O'N'E' be exactly equal to the quadrant ONE of the mean ellipse, with its axes parallel to those of

SEA-LEVEL AND LEVEL-CURVE.

143

that ellipse, and its centre O' so situated that the circumference of the ellipse passes through both A and B. If the curvature of the arc AB is not that of the mean arç, but if some other line represent it, as the continuous line AB, then the plumb-line will hang in the normals aA, bB to this line, and the angle they include is the observed or astronomical amplitude of the arc, because it is measured by the corresponding arc in the heavens, defined by the points in which the plumb-line at its extremities intersects the celestial vault.

The dotted line AB represents the mean arc, and the inclination of the dotted normals a'A, b'B measures the mean amplitude of the arc, and can be calculated by the formula of Art. 123, when we know the length of the mean arc, as well as of the mean axes.

The astronomical amplitude can, therefore, always be obtained by observation. The mean amplitude could not be thus obtained, unless we happened to know that the actual arc coincided with the mean. We proceed to show how in all cases the mean amplitude may be determined by means of the geodetic arc measured by the Survey and the mean axes found as already described.

PROP. To prove that the lengths of the mean arc and of the geodetic arc of meridian between two places, as much as twelve degrees and a half apart, differ by an insensible quantity: and to show how the mean amplitude can be obtained by this theorem.

147. Lets be the length of the elliptic arc between the stations, land l' the observed latitudes of the extremities, λ and m the amplitude and middle latitude. Let c be the chord, r and 0, r' and ' the polar co-ordinates from the centre of the ellipse to the extremities of the arc, a and b the semiaxes;

.. c2=r2+r'2—2rr' cos (0—0')=2rr′ {1 — cos (0 – 0') } + (r−r')2,

Also

r=a (1— e sin2l), r′ = a (1 — e sin2l').

tan 0 = (1-2e) tan 7, 0=7-e sin 27;

... 00'λ - 2e sir. A ccs 2m;