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For the determination of latitudes, the instrument used from the commencement of the survey till the year 1836 was a zenith sector constructed by Ramsden.* It consisted of an arc of 1530, divided to every 5', and having a micrometrical division by which quantities of about a tenth of a second could be estimated. The radius of the arc was eight feet, and the aperture of the telescope was four inches. The instrument was read off by means of a plumb-line. The apparatus for placing it truly vertical, for reversing its position, and for bringing the plumb-line over the centre of the instrument, were all very perfect; but notwithstanding the general excellence of the instrument, it was not well adapted to constant transport from station to station, by reason of its complex construction and the number of separate parts of which it was composed. The necessary reversal of the instrument on successive nights, and the consequent delay during bad weather, was also a source of considerable inconvenience.

"In consequence of these deficiencies, an application was made to the Astronomer Royal by Major-general Colby for a new zenith sector, by which the observations of a single night might give an accurate determination of the latitude. The design was furnished, and eventually the construction of the instrument superintended by Mr. Airy.

"The first principle in this instrument, now known as Airy's zenith sector, was the arrangement for making successive observations in two positions of the instrument, face east and face west, at the same transit. The second principle was the substitution of a level or system of levels for the usual plumb-line. The third principle was the casting in one piece, as far as practicable, of each of the different parts of the instrument, in order to avoid the great number of screws and fastenings with which most instruments are hampered, and to secure, if possible, perfect rigidity."

These remarks are followed by a description of the instrument, and of the method of making observations with it.

Section III. Reduction of Observations.—In this section an exposition is given of the method, founded on the theory of probabilities, for reducing the observed to the most probable mean bearings.

Section IV. Observations, Terrestrial and Astronomical, for the principal Triangulations.-The first part of this section contains extracts from the reduced theodolite observations of terrestrial objects; the second part contains the principal azimuthal observations in detail, and the results of those which are less important, together with their range and probable error; the third part contains the results of the observations made with Airy and Ramsden's zenith sectors for the latitudes of various points in the triangulation.

* This instrument was destroyed by the fire which happened in the Tower of London in 1841.

Section V. Measurement of Base Lines. - Some details relative to the subject of this section have been already given in the Monthly Notices for April 1858 (vol. xviii. p. 221).

Section VI. Principles of Calculation.-The theory of spheroidal triangles is investigated. The error resulting from the solution of such triangles by spherical trigonometry is demonstrated to be inconsiderable. It is further shown that no appreciable error can arise from the application of Legendre's theorem to the computation of spheroidal triangles. This theorem is accordingly employed in the calculations, in preference to the chord method which was exclusively used in the earlier part of the operations. Formulæ are then investigated for calculating the longitudes, latitudes, and azimuths. The section concludes with an exposition of the method employed for determining the most probable values of the corrections to the observed mean bearings of the triangulation.

Section VII. Reduction of the Triangulation.-In order to obtain the most probable values of the corrections to the observed bearings, Colonel Yolland directed that the whole triangulation should be reduced according to the method of least squares. As the number of equations of condition amounted to 920, their solution in one mass was impracticable. However, a mode of treatment was devised which was equally effective for the object in view, and was naturally suggested by the relations in which the corrections stood towards each other in the equations. This consisted in breaking up the triangulation into a number of separate parts or figures, each of which afforded a limited number of equations of condition, and then applying the theory of probabilities to the solution of each group of equations.

Section VIII. Triangles and Distances.-The definitive values of the absolute distances are made to depend on the Lough Foyle and Salisbury Bases. The distance between the parallels of Dunnose and Saxavord is found to be 3,729,334 07 feet. The distance between Easington and Saxavord was computed by two different series of triangles. By the one the distance was found to be =2,288,427.29 feet; by the other it came out 2,288,427.38 feet. Both Easington and Saxavord lie nearly on the same meridian. The distance of Saxavord from the meridian of Easington was found

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Section IX. Observed Zenith Distances and Altitudes.The theory of probabilities is applied to the determination of the mean observed zenith distances. The coefficient of refraction is investigated in two different ways: first, by direct observation from one point to another, the heights of both points

being exactly ascertained by levelling; secondly, by combining reciprocal observations of zenith distance. The results are interesting. With regard to 31 values of the coefficient deduced from levelled heights, it is found that in the cases wherein the rays cross the sea, the mean value of the coefficient amounts to o'0817; while for the rays which pass over the land, the mean value of the coefficient is 0.0772. Omitting the case of Ben Nevis, in which the atmosphere appears to have been in an abnormal state, the value of the coefficient is found to be 0.0756 for rays not crossing the sea. The following are the results deduced from pairs of reciprocal observations :-In 35 cases wherein the ray passes for a considerable portion of its length over the sea, the value of the coefficient k = 0.0801; in 13 cases wherein the ray passes also over the sea, but for a smaller portion of its length, the value of ko'0778; in the remaining cases, wherein the ray is inland, the value of k = 0.0744. The definitive result is,

Coefficient of refraction

=

=

00809 for rays crossing the sea.

= 0.0750 for rays not crossing the sea.

The heights of a considerable number of the stations were determined by spirit-levelling; the heights of others were deduced from the levelled heights, combined with the observed zenith distances. Thus from the levelled heights of Ben Nevis (4406.3 feet) and Ben Lomond (3192.2 feet), the heights of four other stations were determined. In this way the height of Ben Macdui was found to be 4295.6 feet. This result has recently received a remarkable confirmation. By levelling up the western side of the mountain, the height has been found to be 4295.70 feet; and by levelling down the eastern side, the result was 4295.76 feet.

servations.

Section X. Connexion of Geodetical and Astronomical Ob- In this section the subject of local attraction is generally considered, and formulæ are investigated for computing the amount of deflection upon various suppositions with respect to the disturbing mass. From observations made at Arthur's Seat, near Edinburgh, in 1855, the relative deflection at the north and south stations of the hill was found to amount to 4"07, whence the mean density of the earth was concluded to be 5.316 ±√ 3·725 e2 + 0029, being the probable error of the mean density of the hill, which is assumed by Colonel James to be 2.75, from the examination of a great many substances.

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Section XI. Determination of the Amount of Local Attraction at various Stations in the Triangulation. -An explanation of the method of computation is given, and is applied to several of the stations for which sufficient data are available. The attraction is also approximately ascertained for a few other stations which have indicated an unusually large amount of disturbance.

Section XII. Determination of the Spheroid most nearly representing the Surface of Great Britain and Ireland.Airy's elements of the earth's figure are assumed as the basis of investigation. The distance between Dunnose and Saxavord, when computed from these elements, is found to be 3,729,335.69; the triangulation of the survey gives 3,729,334 07 for the distance between the same stations. The difference, consequently, amounts to only 1.62 feet. However, a comparison of the observed and computed latitudes exhibits systematic indications of a small error in the elements. This leads to a thorough investigation, founded upon a comparison of the computed latitudes, longitudes, and azimuths, with the corresponding results of observation. The following are the elements of the spheroid which is found to agree most nearly with the surface of the British Isles:

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Section XIII. Of the Length of the Degree, &c.: Longitudes and Latitudes and Directions of the Meridian at the different Stations. The following is the expression obtained for the length of a degree of the meridian in the British Isles:

--

Length

=

364,5941-1953-8 cos 2 λ + 4'4 cos 4 a

the unit being a foot.

Section XIV. Figure of the Earth.

In the Philosophical Transactions for 1856 there is a paper containing a new determination of the Figure of the Earth, founded upon the arcs used by Bessel, but including the extensions of the English and Indian arcs (Monthly Notices, vol. xviii. p. 220). The investigation in the present volume includes also the Russo-Scandinavian arc, extending from Ismail, in lat. 45° 20', to Fuglenas, in lat. 70° 40', and having, consequently, an amplitude of 25° 20' (Monthly Notices, vol. xiii. p. 201). The data of this arc were furnished to the Ordnance by Professor Struve; the latitudes are not final, but the probable errors of the latitudes stated are less than half a second. The elements of the earth's figure are determined on two suppositions, the non-elliptic and the elliptic. On the first hypothesis the mean value of a correction to an observed latitude is 2064; on the second the mean value is ±z"·098. The figure deduced, independently of the elliptic assumption, does not differ sensibly from a spheroid of the same axes, the amount of the protuberance of a meridional section being represented by

dr = (177 feet. 5±70°9) sin2 à, à denoting the latitude.

But the deviation is found to be greater, when the absolute values of the axes are considered.

*The reciprocal of the ellipticity.

The following table affords a view of the definitive results derived from the two hypotheses:

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Combining Colonel Everest's Indian arc from Daumergida to Kaliana with the English arc from St. Agnes to Saxavord, the value of the mean degree is found to be 364,623°50±7·15 and a : b =292*39: 29139; probable error ±2.67.

=

Combining the same Indian arc with the Russian arc, the value of the mean degree is found to be 364,616.07 ± 6.59 and a : b = 29458293 58; probable error ± 2.27. The three arcs combined together give 364,619.19 ± 6.14 for the value of the mean degree, and a:b probable error ± 2.26.

294 44: 293 44;

The foregoing determinations conclude with this remark:"Until the exact latitudes of the thirteen stations in the Russian arc are known we cannot state the precise elements best representing all the geodetical operations. They cannot,

however, be far from the following:
:-

Mean Degree of Meridian

Ratio of Semi-Axes

364,616 feet of Ordnance Standard

293: 294."

The work concludes with some remarks upon the Indian arc.

Errata in the Art. Physical Astronomy (Encyclopædia Metropolitana) communicated by Captain Tennant.

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To be disposed of, an Iron Equatoreal Stand adapted for carrying a refractor of from 3 to 4 inches aperture. May be seen at the rooms of the Society.

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