Page:EB1911 - Volume 08.djvu/834

 of the disturbing mass below the surface, $$\mu$$ the ratio of the disturbing mass to the mass of the earth, and $$a\theta$$ the distance of any point on the surface from that point, say $$\mathrm{Q}$$, which is vertically over the disturbing mass. The maximum value of $$y$$ is at $$\mathrm{Q}$$, where it is $$y = a\mu u (1 - u)$$. The deflection at the distance $$a\theta$$ is $$\Lambda = \mu u \sin \theta (1 - 2u \cos \theta + u^2)^{-\frac{3}{2}}$$, or since $$\theta$$ is small, putting $$h + u = 1$$, we have $$\Lambda = \mu \theta (h^2 + \theta^2)^{-\frac{3}{2}}$$. The maximum deflection takes place at a point whose distance from $$\mathrm{Q}$$ is to the depth of the mass as $$1 : \sqrt{2}$$, and its amount is $$2\mu /3 \sqrt{3h^2}$$. If, for instance, the disturbing mass were a sphere a mile in diameter, the excess of its density above that of the surrounding country being equal to half the density of the earth, and the depth of its centre half a mile, the greatest deflection would be 5″, and the greatest value of $$y$$ only two inches. Thus a large disturbance of gravity may arise from an irregularity in the mathematical surface whose actual magnitude, as regards height at least, is extremely small.

The effect of the disturbing mass $$\mu$$ on the vibrations of a pendulum would be a maximum at $$\mathrm{Q}$$; if $$v$$ be the number of seconds of time gained per diem by the pendulum at $$\mathrm{Q}$$, and $$\sigma$$ the number of seconds of angle in the maximum deflection, then it may be shown that $$v/\sigma = \pi \sqrt{3}/10$$.

The great Indian survey, and the attendant measurements of the degree of latitude, gave occasion to elaborate investigations of the deflection of the plumb-line in the neighbourhood of the high plateaus and mountain chains of Central Asia. Archdeacon Pratt (Phil. Trans., 1855 and 1857), in instituting these investigations, took into consideration the influence of the apparent diminution of the mass of the earth’s crust occasioned by the neighbouring ocean-basins; he concluded that the accumulated masses of mountain chains, &c., corresponded to subterranean mass diminutions, so that over any level surface in a fixed depth (perhaps 100 miles or more) the masses of prisms of equal section are equal. This is supported by the gravity measurements at Moré in the Himalayas at a height of 4696 metres, which showed no deflection due to the mountain chain (Phil. Trans., 1871); more recently, H. A. Faye (Compt. rend., 1880) arrived at the same conclusion for the entire continent.

This compensation, however, must only be regarded as a general principle; in certain cases, the compensating masses show marked horizontal displacements. Further investigations, especially of gravity measurements, will undoubtedly establish other important facts. Colonel S. G. Burrard has recently recalculated, with the aid of more exact data, certain Indian deviations of the plumb-line, and has established that in the region south of the Himalayas (lat. 24°) there is a subterranean perturbing mass. The extent of the compensation of the high mountain chains is difficult to recognize from the latitude observations, since the same effect may result from different causes; on the other hand, observations of geographical longitude have established a strong compensation.

The astronomical stations for the measurement of the degree of latitude will generally lie not exactly on the same meridian; and it is therefore necessary to calculate the arcs of meridian $$\mathrm{M}$$ which lie between the latitude of neighbouring stations. If $$\mathrm{S}$$ be the geodetic line calculated from the triangulation with the astronomically determined azimuths $$\alpha_{1}$$ and $$\alpha_{2}$$, then

in which $$2\alpha = \alpha_{1} + \alpha_{2} - 180^{\circ}, \Delta \alpha = \alpha_{2} - \alpha_{1} - 180^{\circ}$$.

The length of the arc of meridian between the latitudes $$\phi_{1}$$ and $$\phi_{2}$$ is

where $$a^2e^2 = a^2 - b^2$$; instead of using the eccentricity $$e$$, put the ratio of the axes $$b : a = 1 - n : 1 + n$$, then

This, after integration, gives

where

The part of $$\mathrm{M}$$ which depends on $$n^3$$ is very small; in fact, if we calculate it for one of the longest arcs measured, the Russian arc, it amounts to only an inch and a half, therefore we omit this term, and put for $$\mathrm{M}/b$$ the value

Now, if we suppose the observed latitudes to be affected with errors, and that the true latitudes are $$\phi_{1} + x_{1}, \phi_{2} + x_{2}$$; and if further we suppose that $$n_{1} + dn$$ is the true value of $$a - b : a + b$$, and that $$n_{1}$$ itself is merely a very approximate numerical value, we get, on making these substitutions and neglecting the influence of the corrections $$x$$ on the position of the arc in latitude, i.e. on $$\phi_{1} + \phi_{2}$$,

here $$d\alpha_{0} = x_{2} - x_{1}$$; and as $$b$$ is only known approximately, put $$b = b_{1}(1 + u)$$; then we get, after dividing through by the coefficient of $$d\alpha_{0}$$, which is $$= 1 + n_{1} - 3n_{1} \cos (\phi_{2} - \phi_{1}) \cos (\phi_{2} + \phi_{1})$$, an equation of the form $$x_{2} = x_{1} + h + fu + gv$$, where for convenience we put $$v$$ for $$dn$$.

Now in every measured arc there are not only the extreme stations determined in latitude, but also a number of intermediate stations so that if there be $$i + 1$$ stations there will be $$i$$ equations

In combining a number of different arcs of meridian, with the view of determining the figure of the earth, each arc will supply a number of equations in $$u$$ and $$v$$ and the corrections to its observed latitudes. Then, according to the method of least squares, those values of $$u$$ and $$v$$ are the most probable which render the sum of the squares of all the errors $$x$$ a minimum. The corrections $$x$$ which are here applied arise not from errors of observation only. The mere uncertainty of a latitude, as determined with modern instruments, does not exceed a very small fraction of a second as far as errors of observation go, but no accuracy in observing will remove the error that may arise from local attraction. This, as we have seen, may amount to some seconds, so that the corrections $$x$$ to the observed latitudes are attributable to local attraction. Archdeacon Pratt objected to this mode of applying least squares first used by Bessel; but Bessel was right, and the objection is groundless. Bessel found, in 1841, from ten meridian arcs with a total amplitude of 50°·6:

The probable error in the length of the earth’s quadrant is ± 336 m.

We now give a series of some meridian-arcs measurements, which were utilized in 1866 by A. R. Clarke in the Comparisons of the Standards of Length, pp. 280-287; details of the calculations are given by the same author in his Geodesy (1880), pp. 311 et seq.

The data of the French arc from Formentera to Dunkirk are—