and η measured westwards. Now the great circle joining Z′ with the pole of the heavens P makes there an angle with the meridian PZ = η cosec PZ′ = η sec φ, where φ is the latitude of the station. Also this great circle meets the horizon in a point whose distance from the great circle PZ is η sec φ sin φ = η tan φ. That is, a meridian
mark, fixed by observations of the pole star, will be placed that
amount to the east of north. Hence the observed latitude requires
the correction ξ; the observed longitude a correction η sec φ; and
any observed azimuth a correction η tan φ. Here it is supposed
that azimuths are measured from north by east, and longitudes
eastwards. The horizontal angles are also influenced by the deflections of the plumb-line, in fact, just as if the direction of the vertical axis of the theodolite varied by the same amount. This influence,
however, is slight, so long as the sights point almost horizontally at the objects, which is always the case in the observation of distant points.
The expression given for N enables one to form an approximate estimate of the effect of a compact mountain in raising the sea-level. Take, for instance, Ben Nevis, which contains about a couple of cubic miles; a simple calculation shows that the elevation produced would only amount to about 3 in. In the case of a mountain mass like the Himalayas, stretching over some 1500 miles of country with a breadth of 300 and an average height of 3 miles, although it is difficult or impossible to find an expression for V, yet we may ascertain that an elevation amounting to several hundred feet may exist near their base. The geodetical operations, however, rather negative this idea, for it was shown by Colonel Clarke (Phil. Mag., 1878) that the form of the sea-level along the Indian arc departs but slightly from that of the mean figure of the earth. If this be so, the action of the Himalayas must be counteracted by subterranean tenuity.
Suppose now that A, B, C, ... are the stations of a network of triangulation projected on or lying on a spheroid of semiaxis major and eccentricity a, e, this spheroid having its axis parallel to the axis of rotation of the earth, and its surface coinciding with the mathematical surface of the earth at A. Then basing the calculations on the observed elements at A, the calculated latitudes, longitudes and directions of the meridian at the other points will be the true latitudes, &c., of the points as projected on the spheroid. On comparing these geodetic elements with the corresponding astronomical determinations, there will appear a system of differences which represent the inclinations, at the various points, of the actual irregular surface to the surface of the spheroid of reference. These differences will suggest two things,—first, that we may improve the agreement of the two surfaces, by not restricting the spheroid of reference by the condition of making its surface coincide with the mathematical surface of the earth at A; and secondly, by altering the form and dimensions of the spheroid. With respect to the first circumstance, we may allow the spheroid two degrees of freedom, that is, the normals of the surfaces at A may be allowed to separate a small quantity, compounded of a meridional difference and a difference perpendicular to the same. Let the spheroid be so placed that its normal at A lies to the north of the normal to the earth’s surface by the small quantity ξ and to the east by the quantity η. Then in starting the calculation of geodetic latitudes, longitudes and azimuths from A, we must take, not the observed elements φ, α, but for φ, φ + ξ, and for α, α + η tan φ, and zero longitude must be replaced by η sec φ. At the same time suppose the elements of the spheroid to be altered from a, e to a + da, e + de. Confining our attention at first to the two points A, B, let (φ′), (α′), (ω) be the numerical elements at B as obtained in the first calculation, viz. before the shifting and alteration of the spheroid; they will now take the form
| (φ′) + f ξ + gη + hda + kde, |
where the coefficients f, g, ... &c. can be numerically calculated. Now these elements, corresponding to the projection of B on the spheroid of reference, must be equal severally to the astronomically determined elements at B, corrected for the inclination of the surfaces there. If ξ′, η′ be the components of the inclination at that point, then we have
| ξ′ | = (φ′) − φ′ + f ξ + gη + hda + kde, |
| η′ tan φ′ | = (α′) − α′ + f ′ξ + g′η + h′da + k′de, |
| η′ sec φ′ | = (ω) − ω + f ″ξ + g″η + h″da + k″de, |
where φ′, α′, ω are the observed elements at B. Here it appears that the observation of longitude gives no additional information, but is available as a check upon the azimuthal observations.
If now there be a number of astronomical stations in the triangulation, and we form equations such as the above for each point, then we can from them determine those values of ξ, η, da, de, which make the quantity ξ2 + η2 + ξ′2 + η′2 + ... a minimum. Thus we obtain that spheroid which best represents the surface covered by the triangulation.
In the Account of the Principal Triangulation of Great Britain and Ireland will be found the determination, from 75 equations, of the spheroid best representing the surface of the British Isles. Its elements are a = 20927005 ± 295 ft., b : a − b = 280 ± 8; and it is so placed that at Greenwich Observatory ξ = 1″.864, η = −0″.546.
Taking Durham Observatory as the origin, and the tangent plane to the surface (determined by ξ = −0″.664, η = −4″.117) as the plane of x and y, the former measured northwards, and z measured vertically downwards, the equation to the surface is
.99524953 x2 + .99288005 y2 + .99763052 z2 − 0.00671003xz − 41655070z = 0.
The precise determination of the altitude of his station is a matter of secondary importance to the geodesist; nevertheless it is usual to observe the zenith distances of all trigonometrical points. Of great importance is a knowledge of the height of the base for its reduction to the sea-level. Again the height of a station does influence a little the observation of terrestrial angles, for a vertical line at B does not lie generally in the vertical plane of A (see above). The height above the sea-level also influences the geographical latitude, inasmuch as the centrifugal force is increased and the magnitude and direction of the attraction of the earth are altered, and the effect upon the latitude is a very small term expressed by the formula h(g′ − g) sin 2φ/ag, where g, g′ are the values of gravity at the equator and at the pole. This is h sin 2φ/5820 seconds, h being in metres, a quantity which may be neglected, since for ordinary mountain heights it amounts to only a few hundredths of a second. We can assume this amount as joined with the northern component of the plumb-line perturbations.
The uncertainties of terrestrial refraction render it impossible to determine accurately by vertical angles the heights of distant points. Generally speaking, refraction is greatest at about daybreak; from that time it diminishes, being at a minimum for a couple of hours before and after mid-day; later in the afternoon it again increases. This at least is the general march of the phenomenon, but it is by no means regular. The vertical angles measured at the station on Hart Fell showed on one occasion in the month of September a refraction of double the average amount, lasting from 1 p.m. to 5 p.m. The mean value of the coefficient of refraction k determined from a very large number of observations of terrestrial zenith distances in Great Britain is .0792 ± .0047; and if we separate those rays which for a considerable portion of their length cross the sea from those which do not, the former give k = .0813 and the latter k = .0753. These values are determined from high stations and long distances; when the distance is short, and the rays graze the ground, the amount of refraction is extremely uncertain and variable. A case is noted in the Indian survey where the zenith distance of a station 10.5 miles off varied from a depression of 4′ 52″.6 at 4.30 p.m. to an elevation of 2′ 24″.0 at 10.50 p.m.
If h, h′ be the heights above the level of the sea of two stations, 90° + δ, 90° + δ′ their mutual zenith distances (δ being that observed at h), s their distance apart, the earth being regarded as a sphere of radius = a, then, with sufficient precision,
| h′ − h = s tan ( s | 1 − 2k | − δ), h − h′ = s tan ( s | 1 − 2k | − δ′). |
| 2a | 2a |
If from a station whose height is h the horizon of the sea be observed to have a zenith distance 90° + δ, then the above formula gives for h the value
| h = | a | tan2 δ | ||
| 2 | 1 − 2k |
Suppose the depression δ to be n minutes, then h = 1.054n2 if the ray be for the greater part of its length crossing the sea; if otherwise, h = 1.040n2. To take an example: the mean of eight observations of the zenith distance of the sea horizon at the top of Ben Nevis is 91° 4′ 48″, or δ = 64.8; the ray is pretty equally disposed over land and water, and hence h = 1.047n2 = 4396 ft. The actual height of the hill by spirit-levelling is 4406 ft., so that the error of the height thus obtained is only 10 ft.
The determination of altitudes by means of spirit-levelling is undoubtedly the most exact method, particularly in its present development as precise-levelling, by which there have been determined in all civilized countries close-meshed nets of elevated points covering the entire land. (A. R. C; F. R. H.)
GEOFFREY, surnamed Martel (1006–1060), count of Anjou, son of the count Fulk Nerra (q.v.) and of the countess Hildegarde or Audegarde, was born on the 14th of October 1006. During his father’s lifetime he was recognized as suzerain by Fulk l’Oison (“the Gosling”), count of Vendôme, the son of his half-sister Adela. Fulk having revolted, he confiscated the countship, which he did not restore till 1050. On the 1st of January 1032 he married Agnes, widow of William the Great, duke of Aquitaine, and taking arms against William the Fat, eldest son and successor of William the Great, defeated him and took him prisoner at Mont-Couër near Saint-Jouin-de-Marnes on the 20th of September 1033. He then tried to win recognition as dukes of Aquitaine for the sons of his wife Agnes by William the Great, who were still minors, but Fulk Nerra promptly took up arms to defend his suzerain William the Fat, from whom he held the Loudunois and
