MEAN FIGURE, TAKING ACCOUNT OF LOCAL ATTRACTION.

129

is the plane of the meridian in which the arc, of which AB is one section, has been geodetically measured. A is the reference-station of the several portions of the whole arc. AZ is the vertical at A in which the plumb-line hangs. The

two curves, of which A'B' and ab are portions, are a variable ellipse and the mean ellipse having the same centre O and their axes in the same lines, the mean ellipse being what the variable ellipse becomes when the values are substituted for u and v which make the sum of the squares of the errors a minimum: Z'AA'N' and zAaN are normals through A to these two ellipses; AD, A'm', am are perpendicular to OD.

Now, if the earth had its mean form, a plumb-line at A would hang in the normal zA to the mean ellipse; but it hangs actually in ZA. Hence ZAz is the deflection (northward in the diagram) which the plumb-line suffers from the local attraction arising from the derangement of the figure and mass of the earth from the mean. This angle is some constant but unknown quantity t, t being reckoned positive when the deflection is northward. This quantity t is part of the correction ZAZ', or x, added to the observed latitude of A before applying the principle of least squares. The other part is ZAZ', which I will now calculate: it is the angle between the two normals drawn through A to the variable and the mean ellipses. By the property of an ellipse of which the ellipticity is small,

ON=2e. Om, and ON'=2e'. Om'.

Also as Om, Om', OD differ only by quantities of the order of the ellipticities, they may be put equal to each other in small terms, because we neglect the square of the ellipticities.

.. ZAZ' = ¿NAN' = < AN'DAND

Suppose that v and V are the values of v for the variable and the mean ellipses. Then by the value of e in Art. 129

PROP. To obtain formula for calculating the Mean Figure of the Earth, taking into account local attraction.

132. If we adopt Bessel's method with the necessary correction pointed out above, the sum of the squares of errors, which is to be differentiated with respect to u and v to obtain a minimum, is

{n, (v − V) + t,}2 + {m ̧ + a‚u + ß ̧v + n ̧ (v − V ) + t1}2

+ {m',+ a' ̧u + ß' ̧v + n ̧ (v − V) +t,}2 + ... {n2 (v − V) + t2}2 + {m2+ a„u + ß ̧v + n2 (v − V ) + t,}2

+

2

+ {m'2+ a' ̧u + ß' ̧v + n2 (v − V ) + t2}2 + ...

= a minimum.

The letters accentuated 1, 2... 8 appertaining to the eight Arcs.

Let U and V be the values of u and v which belong to the mean ellipse. These values, then, must be put for u and v in the two equations produced by differentiating the above with respect to u and v. We have

1

1

1

a1 (m1+a ̧U+ß‚V + t1) + a', (m', + a' ̧U + B'1 V + t1) + ..... +a1⁄2 (m2+¤ ̧U+B2V + t2) + a'1⁄2 (m21⁄2 + a'„U + ß'¿V + t2) + ...

+

and

2

2

2

n ̧t ̧ + (ß ̧ + n) (m1 + a ̧U+ß‚V+t1)

2

2

2

= 0;

MEAN FIGURE, TAKING ACCOUNT OF LOCAL ATTRACTION. 131

n2t2 + (B2+n) (m2 + a2U+ß2V + t2)

+

2

Let (m) be a symbol representing the sum of all the mes appertaining to the divisions of the same arc; and let Σ(m) represent the sum of all these sums for all the arcs; and similarly for other quantities besides m. Then the above equations become

i being the number of stations on the representative arc.

The numerical quantities involved in the first two lines of these equations have been calculated in the article on the Figure of the Earth in the British Ordnance Survey Volume; and the remaining ones have been calculated by the author, and the whole gathered together in an article published in the Proceedings of the Royal Society, Vol. XII. No. 64, p. 253. The numerical quantities are there substituted, the equations are solved, and the following results obtained.

a = 20928627+1057·8t, +342·9t2+152·3t ̧ +27·3t+93·6t ̧

b

+8.8t+63.7ty +62·9tg,

= 20849309 — 3762·6t, — 334·3t, — 661.3t - 101·5-372·6t

-140t-249-3-249.1t..

1 € = {1+0·0608, +0.0085, +0.0103t,+00016, +0.0059t 263.9

+0·0003 + 0·0039t + 0·001639t。}.

PROP. To show the degree of uncertainty local attraction, if not allowed for, introduces into the problem of the Figure of the Earth.

133. The formulæ deduced in the last Article for the semiaxes and ellipticity of the mean figure of the earth show

us, that the effect of local attraction upon the final numerical results may be very considerable: for example, a deflection of the plumb-line of only 5" at the standard station (St Agnes) of the Anglo-Gallic arc would introduce a correction of about one mile to the length of the semi-major-axis, and more than three miles to the semi-minor-axis. If the deflection at the standard station (Damargida) of the Indian Great Arc be what the mountains and ocean make it (without allowing any compensating effect from variations in density in the crust below, which no doubt exist, but which are altogether unknown), viz. about 17"-3 (Art. 62, Ex. 1 and 2), the semiaxes will be subject to a correction, arising from this cause alone, of half a mile and two miles. This is sufficient to show how great a degree of uncertainty local attraction, if not allowed for, introduces into the determination of the mean figure. As long as we have no means of ascertaining the amount of local attraction at the several standard stations of the arcs employed in the calculation, this uncertainty regarding the mean figure, as determined by geodesy, must remain.

PROP. To state the result of applying the same principles to each of the three long arcs, the Anglo-Gallic, Russian, and Indian, and to obtain a Mean Figure of the Earth from them.

134. The first three of the eight arcs which have been used in the above calculation, viz. the Anglo-Gallic, Russian, and Indian, are of considerable length. If the method of the last Article is applied to each of these separately we shall obtain three pairs of values of the semi-axes, involving the three unknown expressions for local attraction at the three standard stations. If local attraction be neglected, these pairs will differ slightly from each other, suggesting an idea, which we shall notice in Art. 141, that the equator is not a circle. But local attraction is more important in its effects than any other known cause of derangement, and must not be neglected; and the inference that the equator is not a circle cannot be drawn without better evidence being adduced against it. All calculations which have hitherto been made of the mean figure of the earth have gone on the hypothesis that it is an oblate spheroid, and the à priori argument in its favour from the fluid-theory is so overwhelming that it must not be

MEAN FIGURE, TAKING ACCOUNT OF LOCAL ATTRACTION.

133

abandonel without sufficient evidence. In what follows we shall show, that so far from there being any evidence that the equator is not a circle, the three elliptic meridians of the Anglo-Gallic, the Russian, and the Indian arcs can be made almost precisely the same by a very moderate allowance for local attraction.

In the previous calculation t has represented the angle which the plumb-line makes, in the plane of the meridian, with the normal to the mean ellipse of the earth. We shall now use T as the angle which the plumb-line makes, in the plane of the meridian, with the normal to the mean ellipse of the particular arc under consideration.

The calculation will be found in the Royal Society's Proceedings already referred to. The following are the results for the Three Arcs :

a1 = 20928190 +15777 T1, b1 = 20847200 - 5885.9 T1,

a2 = 20927234 +345·2 T2, b2 =20861620 +2832-4 T2,

318.9

20926529 +13862.8 T,, b = 20855535+5555.6 T ̧,

1

€3 291-g (1 +0·11707).

If these ellipses are made equal to each other, that is,

these formulæ give four equations of condition connecting the three quantities T, T2, T. The most likely solutions of these four equations are found, by the method of least squares, to be

T1 =

=-1"37, T=2"-22, T=0"·033. When these are substituted in the semiaxes, they give

a1 = 20926029, a,= 20926468, a,= 20926072.

аз

b1 =20855264, b, 20855332, b1 = 20855352.

=

These three results are remarkably near each other; they differ from their average, 20926189 and 20855316, in no case