GEODESY(γη, the earth, δαιω, to divide) is the science of surveying extended to large tracts of country, having in view not only the production of a system of maps of very great accuracy, but the determination of the curvature of the surface of the earth, and eventually of the figure and dimensions of the earth. This last, indeed, may be the sole object in view, as was the case in the operations conducted in Peru and in Lapland by the celebrated French astronomers Bouguer, La Condamine, Maupertuis, Clairaut, and others; and the measurement of the meridian arc of France by Mechain and Delambre had for its end the determination of the true length of the “metre” which was to be the legal standard of length of France.
The basis of every extensive survey is an accurate triangulation, and the operations of geodesy consist in—the measurement, by theodolites, of the angles of the triangles; the measurement of one or more sides of these triangles on the ground; the determination by astronomical observations of the azimuth of the whole network of triangles; the determination of the actual position of the same on the surface of the earth by observations, first for latitude at some of the stations, and secondly for longitude.
To determine by actual measurement on the ground the length of a side of one of the triangles, wherefrom to infer the lengths of all the other sides in the triangulation, is not the least difficult operation of a trigonometrical survey. When the problem is stated thus—To determine the number of times that a certain standard or unit of length is contained between two finely marked points on the surface of the earth at a distance of some miles asunder, so that the error of the result may be pronounced to lie between certain very narrow limits,—then the question demands very serious consideration. The representation of the unit of length by means of the distance between two fine lines on the surface of a bar of metal at a certain temperature is never itself free from uncertainty and probable error, owing to the difficulty of knowing at any moment the precise temperature of the bar; and the transference of this unit, or a multiple of it, to a measuring bar, will be affected not only with errors of observation, but with errors arising from uncertainty of temperature of both bars. If the measuring bar be not self-compensating for temperature, its expansion must be determined by very careful experiments. The thermometers required for this purpose must be very carefully studied, and their errors of division and index error determined.
The base apparatus of Bessel and that of Colby have been described in Figure of the Earth (vol. vii. p. 598). The average probable error of a single measurement of a base line by the Colby apparatus is, according to the very elaborate investigations of Colonel Walker, C.B., R.E., the Surveyor-General of India, ± l ’ 5∕x,(μ, meaning “one millionth”). W. Struve gives ± 0'8μ, as the probable error of a base line measured with his apparatus, being the mean of the probable errors of seven bases measured by him in Russia; but this estimate is probably too small. Struve’s apparatus is simple: there are four wrought iron bars, each two toises (rather more than 13 feet) long; one end of each bar is terminated in a small steel cylinder presenting a slightly convex surface for contact, the other end carries a contact lever rigidly connected with the bar. The shorter arm of the lever terminates below in a polished hemisphere, the upper and longer arm traversing a vertical divided arc. In measuring, the plane end of one bar is brought into contact with the short arm of the contact lever (pushed forward by a weak spring) of the next bar. Each bar has two thermometers, and a level for determining the inclination of the bar in measuring. The manner of transferring the end of a bar to the ground is simply this: under the end of the bar a stake is driven very firmly into the ground, carrying on its upper surface a disk, capable of movement in the direction of the measured line by mean3 of slow-motion screws. A fine mark on this disk is brought vertically under the end of the bar by means of a theodolite which is planted at a distance of 25 feet from the stake in a direction perpendicular to the base. Struve investigates for each base the probable errors of the ,measurement arising from each of these seven causes:— alignment, inclination, comparisons with standards, readings of index, personal errors, uncertainties of temperature, and the probable errors of adopted rates of expansion.
The apparatus used in the United States Coast Survey consists of two measuring bars, each 6 metres in length, supported on two massive tripod stands placed at one quarter length from each end, and provided, as in Colby’s apparatus, with the necessary mechanism for longitudinal, transverse, and vertical adjustment. Each measuring rod is a compensating combination of an iron and a brass bar, supported parallel to one another and firmly connected at one end, the medium of connexion between the free ends being a lever of corhpensation so adjusted as to indicate a constant length independent of temperature or changes of temperature. The bars are protected from external influences by double tubes of tinned sheet iron, within which they are movable on rollers by a screw movement which allows of contacts being made within 10 ¾ 00 of an inch. The abutting piece acts upon the contact lever which is attached to the fixed end of the compound bar, and carries a very sensitive level, the horizontal position of which defines the length of the bar. It is impossible here to give a full description of this complicated apparatus, and we must refer for details to the account given in full in the United States Coast Survey Report for 1854. This apparatus is doubtless a very perfect one, and the manipulation of it must offer great facilities, for it appears to be possible, under favourable circumstances, to measure a mile in one day, 1Ό6 mile having been measured on one occasion in eight and a half hours. In order to test to the utmost the apparatus, the base at Atalanta, Georgia, was measured twice in winter and once in summer 1872-73, at temperatures 51°, 45°, 90° F.; the difference of the first and second measurements was +0 ’ 30 in., of the second and third + 0 ’ 34 in.,—the actual length and computed probableerror expressed in metres being 9338 ’ 4763 ± 0Ό166. It is to be noted that in the account of a base recently measured in the United States Lake Survey, some doubt is expressed as to the perfection of the particular apparatus of this description there used, on account of a liability to permanent changes of length.
The last base line measured in India with Colby’s compensation apparatus had a length of 8912 feet only, and in consequence of some doubts which had arisen as to the accuracy of this compensation apparatus, the measurement was repeated four times, the operations being conducted in such a manner as to indicate as far as possible the actual magnitudes of the probable errors to which such measures are liable. The direction of the line (which is at Cape Comorin) is north and south, and in two of the measurements the brass component was to the west, in the other two it was to the east. The differences between the individual measurements and the mean of the four are + Ό017, - , 0049, - ∙0015, + Ό045 in feet. The measurements occupied from seven to ten days each,—the average rate of such work in India being about a mile in five days.
The method of Μ. Porro, adopted in Spain, and by the French in Algiers, is essentially different from those just described. The measuring rod, for there is only one, is a thermometric combination of two bars, one of platinum and one of brass, in length 4 metres, furnished with three levels and four thermometers. Suppose A, B, C three micrometer microscopes very firmly supported at intervals of 4 metres with their axes vertical, and aligned in the plane of the base line by means of a transit instrument, their micrometer screws being in the line of measurement. The measuring bar is brought under say A and B, and those micrometers read; the bar is then shifted aud brought under B and C. By repetition of this process, the reading of a micrometer indicating the end of each position of the bar, the measurement is made. The probable error of the central base of Madridejos, which has a length of 14664 ’ 500 metres, is estimated at ÷ 0 ’ 17^. This is the longest base line in Spain; there are seven others, six of which are under 2500 metres in length; of these one is in Majorca, another in Minorca, aud a third in Iviça. The last base just measured in the province of Barcelona has a length of 2483 ’ 5381 metres according to the first measurement, and 2483 ’ 5383 according to the second.
The total number of base lines measured in Europe up to the present time is about eighty, fifteen of which do not exceed in length 2500 metres, or about a mile aud a half, and two—one in France, the other in Bavaria—exceed 19,000 metres. The question has been frequently discussed whether or not the advantage of a long base is sufficiently great to warrant the expenditure of time that it requires, or whether as much precision is not obtainable in the end by careful triangulation from a short base. But the answer cannot be given generally; it must depend on the circumstances of each particular case.
Itis necessary that the altitude above the level of the sea of every part of a base line be ascertained by spirit levelling, in order that the measured length maybe reduced to what it would have been had the measurement been made on the surface of the sea, produced in imagination. Thus if l be the length of a measuring bar, h its height at any given position in the measurement, r the radius of the earth, then the length radially projected on to the level of the sea is l - l. In the Salisbury Plain base line the reduction to the level of the sea is - 0 - 6294 feet.
In working away from a base line ab, stations c, d, e, f are carefully selected so as to obtain from well-shaped triangles gradually increasing sides. Before, however, finally leaving the base line it is usual to verify it by triangulation thus: during the measurement two or more points, as p, q (fig. 1), are marked in the base in positions such that the lengths of the different segments of the line are known; then, taking suitable external stations, as h, k, the angles of the triangles blip, plιq, hqk, kqa are measured. From these angles can be computed the ratios of the segments, which must agree, if all operations are correctly performed, with the ratios resulting from the measures. Leaving the base line, the sides increase up to ten, thirty, or fifty miles, occasionally, but seldom, reaching a hundred miles. The triangulation points may either be natural objects presenting themselves in suitable positions, such as church towers; or they may be objects specially constructed in stone or wood on mountain tops or other prominent ground. In every case it is necessary that the precise centre of the station be marked by some permanent mark. In India no expense is spared in making permanent the principal trigonometrical stations—costly towers in masonry being erected. It is essential that every trigonometrical station shall present a fine object for observation from surrounding stations.
Horizontal Angles.
In placing the theodolite over a station to be observed from, the first point to be attended to is that it shall rest upon a perfectly solid foundation. The method of obtaining this desideratum must depend entirely on the nature of the ground; the instrument must if possible be supported on rock, or if that be impossible a solid foundation must be obtained by digging. When the theodolite is required to be raised above the surface of the ground in order to command particular points, it is necessary to build two scaffolds,—the outer one to carry the observatory, the inner one to carry the instrument,—and these two edifices must have no point of contact. Many cases of high scaffolding have occurred on the English Ordnance Survey, as for instance at Thaxted Church, where the tower, 80 feet high, is surmounted by a spire of 90 feet. The scaffold for the observatory was carried from the base to the top of the spire; that for the instrument was raised from a point of the spire 140 feet above the ground, having its bearing upon timbers passing through the spire at that height. Thus the instrument, at a height of 178 feet above the ground, was insulated, and not affected by the action of the wind on the observatory.
At every station it is necessary to examine and correct the adjustments of the theodolite, which are these :—the line of collimation of the telescope must be perpendicular to its axis of rotation; this axis perpendicular to the vertical axis of the instrument; and the latter perpendicular to the plane of the horizon. The micrometer microscopes must also measure correct quantities on the divided circle or circles. The method of observing is this. Let A, B, C.... be the stations to be observed taken in order of azimuth; the telescope is first directed to A and the cross-hairs of the telescope made to bisect the object presented by A, then the microscopes or verniers of the horizontal circle (also of the vertical circle if necessary) are read and recorded. The telescope is then turned to B, which is observed in the same manner; then C and the other stations. Coming round by continuous motion to A, it is again observed, and the agreement of this second reading with the first is some test of the stability of the instrument. In taking this round of angles—or “arc,” as it is called on the Ordnance Survey— it is desirable that the interval of time between the first and second observations of A should be as small as maybe consistent with due care. Before taking the next arc the horizontal circle is moved through 20° or 30°; thus a different set of divisions of the circle is used in each arc, which' tends to eliminate the errors of division.
It is very desirable that all arcs at a station should contain one point in common, to which all angular measurements are thus referred,—the observations on each arc commencing and ending with this point, which is on the Ordnance Survey called the “referring object.” It is usual for this purpose to select, from among the points which have to be observed, that one which affords the best object for precise observation. For mountain tops a “referring object” is constructed of two rectangular plates of metal in the same vertical plane, their edges parallel and placed at such a distance apart that the light of the sky seen through appears as a vertical line about 10" in width. The best distance for this object is from one to two miles.
It is clear that no correction is required to the angles measured by a theodolite on account of its height above the sea-level; for its axis of rotation coincides with the normal to the surface of the earth, and the angles measured between distant points are those contained between the vertical planes passing through the axis of the instrument and those points.
The theodolites used in geodesy vary in pattern and in size—the horizontal circles ranging from 10 inches to 36 inches in diameter. In Ramsden’s 36-inch theodolite the telescope has a focal length of 36 inches and an aperture of 2 ’ 5 inches, the ordinarily used magnifying power being 54; this last, however, can of course be changed at the requirements of the observer or of the weather. The probable error of a single observation of a fine object with this theodolite is about 0" ’ 2.
Fig. 2 represents an altazimuth theodolite of an improved pattern now used on the Ordnance Survey. The horizontal circle of 14 inches diameter is read by three micrometer microscopes; the vertical circle has a diameter of 12 inches, and is read by two microscopes.
In the Great Trigonometrical Survey of India the theodolites used in the more important parts of the work have been of 2 and 3 feet· diameter,—the circle read by five equidistant microscopes. Every angle is measured twice in each position of the zero of the horizontal circle, of which there are generally ten; the entire number of measures of an angle is never less than 20. An examination of 1407 angles showed that the probable error of an observed angle is on the average ÷ 0 ’ 28.
For the observations of very distant stations it is usual to employ a heliostat In its simplest form this is a plane mirror 4, 6, or 8 inches in diameter, capable of rotation round a horizontal and a vertical axis. This mirror is placed at the station to be observed, and in fine weather it is kept so directed that the rays of the sun reflected by it strike the distant observing telescope. To the observer the heliostat presents the appearance of a star of the first or second magnitude, and is generally a pleasant object for observing.
Astronomical Observations.
The direction of the meridian is determined either by a theodolite or a portable transit instrument. In the former case the operation consists in observing the angle between a terrestrial object—generally a mark specially erected and capable of illumination at night—and a close circumpolar star at its greatest eastern or western azimuth, or, at any rate, when very near that position. If the observation be made t minutes of time before or after the time of greatest azimuth, the azimuth then will differ from its maximum value by
(4δ00 2 8 inΓ⅛¾ . ; . .
sin z in secouds of angle, omitting smaller terms. Here the symbol δ is the star’s declination, ε its zenith distance. The collimation and level errors are very carefully determined before and after these observations, and it is usual to arrange the observations by the reversal of the telescope so that collimation error shall disappear. If b, c be the level and collimation errors, the correction to the circle reading is b cot z ÷ c cosec z, b being positive when the west end of the axis is high. It is clear that any uncertainty as to the real state of the level will produce a corresponding uncertainty iu the resulting value of the azimuth,—an uncertainty which increases with the latitude, and is very large in high latitudes. This may be partly remedied by observing in connexion with the star its reflexion in mercury. In determining the value of “one division "of a level tube, it is necessary to bear in mind that in some the value varies considerably with the temperature. By experiments on the level of Ramsden’s 3~foot theodolite, it was found that though at the ordinary temperature of 66° the value of a division was about one second, yet at 32° it was about five seconds.
The portable transit in its ordinary form hardly needs description. In a very excellent instrument of this kind used on the Ordnance Survey, the uprights carrying the telescope are constructed of mahogany, each upright being built of several pieces glued and screwed together; the base, which is a solid and heavy plate of iron, carries a reversing apparatus for lifting the telescope out of its bearings, reversing it, and letting it down again. Thus is avoided the change of temperature which the telescope would incur by being lifted by the hands of the observer. Another form of transit Î3 the German diagonal form, in which the rays of light after passing through the object glass are turned by a total reflexion prism through one of the transverse arms of the telescope, at the extremity of which arm is the eye-piece. The unused half of the ordinary telescope being cut away is replaced by a counterpoise. In this instrument there is the advantage that the observer without moving the position of his eye commands the whole meridian, and that the level may remain on the pivots whatever be the elevation of the telescope. But there is the disadvantage that the flexure of the transverse axis causes a variable collimation error depending on the zenith distance of the star to which it is directed; and moreover it has been found that in some cases the personal error of an observer is not the same in the two positions of the telescope.
To determine the direction of the meridian, it is well to erect two marks at nearly equal angular distances on either side of the north meridian line, so that the pole star crosses the vertical of each mark a short time before and after attaining its greatest eastern and western azimuths.
If now the instrument, perfectly levelled, is adjusted to have its centre wire on one of the marks, then when elevated to the star, the star will traverse the wire, and its exact position in the field at any moment can be measured by the micrometer wire. Alternate observations of the star and the terrestrial mark, combined with careful level readings and reversals of the instrument, will enable one, even with only one mark, to determine the direction of the meridian in the course of an hour with a probable error of less than a second. The second mark enables one to complete the station more rapidly, and gives a check upon the work. As an instance, at Findlay Seat, in latitude 57° 35', the resulting azimuths of the two marks were 177° 45' 37"∙29 ÷ 0' ’ ∙20 and 182° 17' 15"∙61 ÷ 0"∙13, while the angle between the two marks directly measured by a theodolite was found to be 4° 3Γ 37"∙43 ÷ 0" ’ 23.
We now come to the consideration of the determination of time with the transit instrument. Let fig. 3 represent the sphere stereographi-cally projected on the plane of the horizon,— ns being the meridian, we the prime vertical, Z, P the zenith and the pole. Let p be the point in which the production of the axis of the instrument meets the celestial sphere, S the position of a star when observed on a wire whose distance from the collimation centre is c. Let a be the azimuthal deviation, namely, the angle uι7ιp, b the level error so that Zp = 90° - b. Let also the hour angle corresponding to ρ be 90° - n, and the declination of the same = m, the star’s declination being δ, and the latitude <∕>. Then to find the hour angle ZPS = τ of the star when observed, in the triangles ∕>PS, jt>PZ we have, since· pPS = 90 + τ - n,
-Sin c=sin m sin δ + cos7∕ι cos δ sin (π-τ), Sin m≈ sin b sin φ - cos b cos φ sin α, Cos m sin 7i=sin b cos φ + cos b sin φ sin a.
And these equations solve the problem, however large be the errors of the instrument. Supposing, as usual, a, b, rn, n to be small, we have at once τ = n + c sec δ + m tan δ, which is the correction to the observed time of transit. Or, eliminating m and n by means of the second and third equations, and putting z for the zenith distance of the star, t for the observed time of transit, the corrected time is
α sin z + & cos z+c cos δ
Another very convenient form for stars near the zenith is this—
τ = δ sec φ + c sec δ+n (tan δ-tan φ).
Suppose that in commencing to observe at a station the error of the chronometer is not known; then having secured for the instrument a very solid foundation, removed as far as possible level and collimation errors, and placed it by estimation nearly in the meridiaD, let two star3 differing considerably in declination be observed—the instrument not being reversed between them. From these two star3, neither of which should be a close circumpolar star, a good approximation to the chronometer error can be obtained; thus let ∈ 1 , ε 2 be the apparent clock errors given by these stars, if δ 1 , δ 2 be their declinations the real error is tan φ - tan δ 1 e = e 1 + (e 1 -⅛)tan8 1 -tanδ∕
Of course this is still only approximative, but it will enable the observer (who by the help of a table of natural tangents can compute ε in a few minutes) to find the meridian by placing at the proper time, which he now knows approximately, the centre wire of his instrument on the first star that passes—not near the zenith.
The transit instrument is always reversed at least once in the course of an evening’s observing, the level being frequently read and recorded. It is necessary in most instruments to add a correction for the difference in size of the pivots.
The transit instrument is also used in the prime vertical for the determination of latitudes. In the preceding figure let q be the point in which the northern extremity of the axis of the instrument produced meets the celestial sphere. Let nZq be the azimuthal deviation = a, and b being the level error, Zq = 90° - b; let also 7⅛P⅛ = τ and P# = ψ. Let S' be the position of a star when observed on a wire whose distance from the collimation centre is c, positive when to the south, and let h be the observed hour angle of the star, viz., ZPS'. Then the triangles gPS', ⅛PZ give
— Sin c = sin δ cos ψ - cos δ sin ψ cos (h+τ), Cos ψ = sin b sin φ + cos b cos φ cos a,
Sin ψ sin τ = cos b sin a.
Now when a and b are very small, we see from the last two equations that ψ = φ-b, a≈τ sin ψ, and if we calculate <∕>' by the formula cot ψ' = cot δ cos h, the first equation leads us to this result—
, a sin z + b cos z + c φ — φ + f r ψ CGS Z the correction for instrumental error being very similar to that applied to the observed time of transit in the case of meridian observations. When a is not very small and z is small, the formulae required are more complicated.
The method of determining latitude by transits in the prime vertical has the disadvantage of being a somewhat slow process, and of requiring a very precise knowledge of the time, a disadvantage from which the zenith telescope is free. In principle this instrument is based on the proposition that when the meridian zenith distances of two stars at their upper culminations—one being to the north and the other to the south of the zenith—are equal, the latitude is the mean of their declinations; or, if the zenith distance of a star culminating to the south of the zenith be Z, its declination being δ, and that of another culminating to the north with zenith distance Z' and declination δ', then clearly the latitude is ^(δ + δ') + ^(Z-Z'). Now the zenith telescope does away with the divided circle, and substitutes the measurement micrometrically of the quantity Z'-Z.
The instrument (fig. 4) is supported on a strong tripod, fitted with levelling screws; to this tripod is fixed the azimuth circle and a long vertical steel axis. Fitting on this axis is a hollow axis which carries on its upper end a short transverse horizontal axis. This latter carries the telescope, which, supported at the centre of its length, is free to rotate in a vertical plane. The telescope is thus mounted excentrically with respect to the vertical axis around which it revolves. An extremely sensitive level is attached to the telescope, which latter carries a micrometer in its eyepiece, with a screw of long range for measuring differences of zenith distance. For this instrument stars are selected in pairs, passing north and south of the zenith, culminating within a few minutes of time and within about twenty minutes (angular) of zenith distance of each other. When a pair of stars is to be observed, the telescope is set to the mean of the zenith distances* and in the plane of the meridian. The first star on passing the central meridional wire is bisected by the micrometer; then the telescope is rotated very carefully through 180° round the vertical axis, and the second star on passing through the field is bisected by the micrometer on the centre wire. The micrometer has thus measured the difference of the zenith distances, and the calculation to get the latitude is most simple. Of course it is necessary to read the level, and the observations are not necessarily confined to the centre wire. In fact if n, s be the north and south readings of the level for the south star, n', s' the same for the north star, l the value of one division of the level, m the value of one division of the micrometer, r, √ the refraction corrections, μ, μ' the micrometer readings of the south and north star, the micrometer being supposed to read from the zenith, then, supposing the observation made on the centre wire,—· Φ- ⅛(δ + δ ’ ) + ⅜(μ - μ ’ )rn +i(n+n , -s-d)l+i(rr , ). .
It is of course of the highest importance that the value m of the screw be well determined. This is done most effectually by observing the vertical movement of a close circumpolar star when at its greatest azimuth.
In a single night with this instrument a very accurate result, say with a probable error of about 0" - 3 or 0" , 4, could be obtained for latitude from, say, twenty pair of stars; but when the latitude is required to be obtained with the highest possible precision, four or five fine nights are necessary. The weak point of the zenith telescope lies in the circumstance that its requirements prevent the selection of stars whose positions are well fixed; very frequently it is necessary to have the declinations of the stars selected for this instrument specially observed at fixed observatories. The zenith telescope is made in various sizes from 30 to 54 inches in focal length; a 30-inch telescope is sufficient for the highest purposes, and is very portable. The zenith telescope is a particularly pleasant instrument to work with, and an observer has been known (a sergeant of Royal Engineers, on one occasion) to take every star in his list during eleven hours on a stretch, namely, from 6 o’clock p.M. until 5 a. μ., and this on a very cold November night on one of the highest points of the Grampians. Observers accustomed to geodetic operations attain considerable powers of endurance. Shortly after the commencement of the observations on one of the hills in the Isle of Skye a storm carried away the wooden houses of the men and left the observatory roofless. Three observatory roofs were subsequently demolished, and for some time the observatory was used without a roof, being filled with snow every night and emptied every morning. Quite different, however, was the experience of the same party when on the top of Ben Nevis, 4406 feet high. For about a fortnight the state of the atmosphere was unusually calm, so much so, that a lighted candle could often be carried between the tents of the men and the observatory, whilst at the foot of the hill the weather was wild and stormy Calculation of Triangulation.
The surface of Great Britain and Ireland is uniformly covered by triangulation, of which the sides are of various lengths from 10 to 111 miles. The largest triangle has one angle at Snowdon in Wales, another on Slieve Donard in Ireland, and a third at Scaw Fell in Cumberland; each side is over a hundred miles, and the spherical excess is 64".
The more ordinary method of triangulation is, however, that of chains of triangles, in the direction of the meridian and perpendicular thereto. The principal triangulations of France, Spain, Austria, and India are so arranged. Oblique chains of triangles are formed in Italy, Sweden, and Norway, also in Germany and Russia, and in the United States. Chains are composed sometimes merely of consecutive plain triangles; sometimes, and more frequently in India, of combinations of triangles forming consecutive polygonal figures. In this method of triangulating, the sides of the triangles are generally from 20 to 30 miles in length—seldom exceeding 40.
The inevitable errors of observation, which are inseparable from all angular as well as other measurements, introduce a great difficulty into the calculation of the sides of a triangulation. Starting from a given base in order to get a required distance, it may generally be obtained in several different ways—that is, by using different sets of triangles. The results will certainly differ one from another, and probably no two will agree. The experience of the computer will then come to his aid, and enable him to say which is the most trustworthy result; but no experience or ability will carry him through a large network of triangles with anything like assurance. The only way to obtain trustworthy results is to employ the method of least squares, an explanation of which will be found in Figure of the Earth (vol. vii. p. 605). We cannot here give any illustration of this method as applied to general triangulation, for it is most laborious, even for the simplest cases. We may, however, take the case of a simple chain—commencing with the consideration of a single triangle in which all three angles have been observed.
Suppose that the sum of the observed angles exceeds the proper amount by a small quantity e: it is required to assign proper corrections to the angles, so as to cause this error to disappear. To do this we must be guide*, by the weight of the determinations of each angle. When a series of direct and independent observations is made, under similar circumstances, of any measurable magnitude —as an angle—the weight of the result is equal to half the square of the number of observations divided by the sum of the squares of the differences of the individual measures from the mean of all. Now let A, k, l be the weights of the three measured angles, and let x, y, z be the corrections which should be applied to them. We know that x + y + c + e = 0; and the theory of probabilities teaches us that the most probable values are those which make hx 2 + ky i + lz- a minimum. Here we arrive at a simple definite problem, the result of which is hx=ky≈lz, showing that c has to be divided into three parts which shall be proportional to the reciprocals of the weights of the corresponding angles. In what follows we shall, for simplicity, suppose the weights of the observed angles to be all equal.
Suppose now that A, B, C are the three angles of a triangle, and that the observed values are A + e 1 , B + e 2 , C+e a ; then, although e 1 , c 2> c 3> the θ> rors of observation, are unknown, yet by adding up the observed angles and finding that the sum is in excess of the truth by a small quantity e, we get e- l + e i +c 3 =c. Now, according to the last proposition, if we suppose the angles to be equally well observed, we have to subtract ⅛β from each of the observed values, which thus become A +1 e 1 - ⅜e 2 - ⅛e 3 , B - ⅛e 1 + ∣e 2 - ⅜¾, C - ⅜e 1 - ⅜e 2 +1⅛ Then to obtain a and b by calculation from the known side c s we have
α sin (C - ½ - ½ + f e 3 ) ≈ c sin (a + fy l - ⅛β 2 - ⅛e 3 ), with a similar expression for the relation between b and c. Put a, β, γ for the cotangents of A, B, C, then the errors of the computed values of a and b are expressed thus—
5α = Jα{e 1 (2u + y) + ef-a + f) + e 3 (- a -2 7 )} b 7 > β i⅛{e l ( -∕J + γ) + e 2 ( 2∕5 + γ) +e 3 ( - β - 2γ)}
Now these actual errors must remain unknown; but we here make use of the following theorem, proved in the doctrine of probabilities. The probable error of a quantity which is a function of several independently observed elements is equal to the square root of the sum of the squares of the probable errors that would arise from each of the observed elements taken singly. Now suppose that each angle in a triangle has a probable error c, then we replace e 1 , e 2 , e. 3 by e, and adding up the squares of the coefficients find for the probable error of a, ÷ jα∈ χ ∕6 χ ∕(α 2 + ay + γ 2 ), and for that of b, ± ⅜δe v ∕6 √ z (β 2 +βγ÷γ 2 ). Suppose the triangle equilateral, each side eight miles, and the probable error of an observed angle 0" ’ 3; then the probable error of either of the computed sides will be found to be 0 ’ 60 inches.
Take a chain of triangles as indicated in the diagram (fig. 5); suppose all ths angles measured, and that the sides MN, H J are me xsured bases; it is required to investigate the necessary corrections to the observed angles in order not only that the sum of the three angles of each triangle fulfil the necessary condition, but that the length of HJ, calculated from that of MN, shall agree with the measured length.
Let X 1 , Y 1 , Z∣, &c., be the angles as observed, x 1 , y 1 , z 1 , &c., the required corrections; then each triangle on adding up the angles gives an equation x l + y l + z l + ∈ 1 = 0. Let the corrected angles be X 1 = X + z, Y 1 ≈Y + y, &c., then ë
HJ sin X} sin X* sin X* sin X} MN j sin Y} sin YJ sin YJ sin YJ sin Xi sin X 2 sin X 3 sin X, ι r . .
- sin Y 1 sin Y s sin Y 3 sin Y 4 ' M *
Let α 1 , 0 1 , 7 1 , . . . be the cotangents of the angles, so that sin X 1 = sin X(l + ax), then u in this last equation is easily seen to be the right hand member of the equation f≈a 1 x 1 -β 1 y 1 + a 3 x 2 -β i y 2 + . . . .
Here/is known numerically, for the ratio of the measured bases is known, and the product of the ratios of the sines of the observed angles is known by computation. The most probable values of x 1 , y 1 , z 1 are those which make the sum 2(∙z ,2 + y 2 + z 2 } a mini mum, or, as we may write it, φ - 2(« 2 + y t + (e + x + y) 2 ) a minimum. This, and the previous equation in f determine all the corrections. Differentiate both and multiply the former by a multiplier P, then
2x,∙ + yi+e i +Pαi=0, 2y,+a⅞ + i i -P0, = O, 3a⅛∙= - P(2α, + β,∙) - e,∙, 3y ’ = P(αi+20,∙)-c i .
Now, substitute these values in the ∕ equation, and P becomes known; then follow at once all the corrections from the two last-written equations. These corrections being applied to the observed «angles, every side in the triangulation has a definite value, which is obtained by the ordinary method of calculation.
A spheroidal triangle differs from a spherical triangle, not only in that the curvatures of the sides are different one from another, but more especially in this that, while in the spherical triangle the normals to the surface at the angular points meet at the centre of the sphere, in the spheroidal triangle the normals at the angles A, B, C meet the axis of revolution of the spheroid in three different points, which we may designate α, β, y respectively. Now the angle A of the triangle as measured by a theodolite is the inclination of the planes BAα and CAα, and the angle at B is that contained by the planes AB/3 and CB∕3. But the planes ABα and AB/3 containing the line AB in common cut the surface in two distinct plane curves. In order, therefore, that a spheroidal triangle may be exactly defined, it is necessary that the nature of the lines joining the three vertices be stated. In a mathematical point of view the most natural definition is that the sides be geodesic or shortest lines. Gauss, in his most elegant treatise entitled Disquisitiones generales circa superficies curvas, has entered fully into the subject of geodesic triangles, and has investigated expressions for the angles of a geodesic triangle whose sides are given, not certainly finite expressions, but approximations inclusive of small quantities of the fourth order, the side of the triangle or its ratio to the radius of the nearly spherical surface being a small quantity of the first order. The terms of the fourth order, as given by Gauss for any surface in general, are very complicated even when the surface is a spheroid. If we retain small quantities of the second order only, and put 3β, (Γ for the angles of the geodesic triangle, while A, B, C are those of a plane triangle having sides equal respectively to those of the geodesic triangle, then, σ being the area of the triangle and π, h, t the measures of curvature at the angular points, ^=A + ^(2a + b + i ), S≡B+⅛a+2b +i ), ff = C+⅛ + b + 2c).
12
The geodesic line being the shortest that can be drawn on any surface between two given points, we may be conducted to its most important characteristics by the following considerations: let p, q be adjacent points on a curved surface; through s the middle point of the chord py imagine a plane drawn perpendicular to 2><7, and let S be any point in the intersection of this plane with the surface; then pS + S? is evidently least when sS is a minimum, which is when sS is a normal to the surface; hence it follows that of all plane curves on the surface joining p, q, when those points are indefinitely near to one another, that is the shortest which is made by the normal plane. That is to say, the osculating plane at any point of a geodesic line contains the normalto the surface at that point. Imagine nowthree points in space, A, B, C, such that AB = BC = cj let the direction cosines of AB be l,m,n, those of BC l',m',n', then x,y,z being the coordinates of B, those of A and C will be respectively— x-cl: y -cm ∙.z-cη x + cl ’ : y + crn' : z + cn'.
Hence the coordinates of the middle point M of AC are x + ⅛c(f -l},y + ic(m' - rn), z + ∣c(n' - n ∖ and the direction cosines of BM are therefore proportional to l' - l: m' - m: ri - n. If the angle made by BC with AB be indefinitely small, the direction cosines of BM are as δl∙. δrn , . δη. Now if AB, BC be two contiguous elements of a geodesic, then BM must be a normal to the surface, and since δ∕, δm, δn are in this case represented by δ y g , δ ^∙, we have dfκ <By dfz ds i ds i ds i du du, du ’ dx dy dz which, however, are equivalent to only one equation. In the case of the spheroid this equation becomes d i x d i y y ds i ~ x ds 1 ° ’ which integrated gives ydx - xdy = C,ds. This again may be put in the. form r sin α = C, where a is the azimuth of the geodesic at any point—the angle between its direction and that of the meridian—and r the distance of the point from the axis of revolution.
From this it may be shown that the azimuth at A of the geodesic joining AB is not the same as the astronomical azimuth at A of B or that determined by the vertical plane AaB. Generally speaking, the geodesic lies between the two plane section curves joining A and B which are formed by the two vertical planes, supposing these points not far apart. If, however, A and B are nearly in the same latitude, the geodesic may cross (between A and B) that plane curve which lies nearest the adjacent pole of the spheroid. The condition of crossing is this. Suppose that for a moment we drop the consideration of the earth’s non-sphericity, and draw a perpendicular from the pole C on AB, meeting it in S between A and B. Then A being that point which is nearest the pole, the geodesic will cross the plane curve if AS be between ∣AB and f AB. If AS lie between this last value and ⅜AB, the geodesic will lie wholly to the north of both plane curves, that is, supposing both points to be in the northern hemisphere.
The circumstance that the angles of the geodesic triangle do not coincide with the true angles as observed renders it inconvenient to regard the geodesic lines as sides of the triangle. A more convenient curve to regard as the side of the spheroidal triangle is this: let L be a point on the curve surface between A and B, λ the point in which the normal at L intersects the axis of revolution, then if L be subject to the condition that the planes ALλ, BLλ coincide, it traces out a curve which touches at A and B the two plane curves before specified. Joining A, B, C by three such lines, the angles of the triangle so formed coincide with the true angles.
Let the azimuths (at the middle point, say) of the sides BC, CA, AB of a spheroidal triangle be a, β , γ, these being measured from 0° to 360° continuously, and the angles of the triangle lettered in the same cvclical direction, and let a, b, c be the lengths of the sides. Let there be a sphere of radius r, such that r is a mean proportional between the principal radii of curvature at the mean latitude <∕> of the spheroidal triangle, and on this sphere a triangle having sides equal respectively to a, b, c. If A', B', C' be the angles of the spheroidal triangle, A, B, C those of the spherical triangle, then e*
A' - A = 12 - 2 cos ∙φ{b i sin 20 - c 2 sin 2γ), c 2
B' - B = cos 2 ≠(c 2 sin 2γ - α 2 sin 2a;,
C' - C -AL cos 2 ψ(α 2 sin 2α - δ 2 sin 20).
By adding these together, it appears that, to the order of terms here retained, the sum of the angles of the spheroidal triangle is equal to the sum of the angles of the spherical triangle. The spherical excess of a spheroidal triangle is therefore obtained by multiplying its area by —,, Gauss’s measure of curvature.
Further, let A 1 , β r 1 , C 1 be the angles of a plane triangle having still the same sides a, b, c, then it may be shown by spherical trigonometry that, r being the radius of the sphere as before,
Λ-∆ _ Δ ∕' 1 α 2 + 7δ 2 + 7c 2 ∖
A A1 3p(1 + -^20^-
Δ∕ 7α 2 + δ 2 + 7c 2 ∖
b - b i = 3≠ k 1 + '^120r 2 J’ p p Δ ∕ 1 7α 2 + 7δ 2 + c 2 ∖ C_C1 = 3K 1 + —2Ö?—λ
It is but seldom that the terms of the fourth order are required. Omitting them, we have Legendre’s theorem, viz., “If from each of the angles of a spherical triangle, the sides of which are small in comparison with the radius, one-third of the spherical excess be deducted, the sines of the angles thus diminished will be proportional to the length of the opposite sides, so that the triangle may be computed as a plane triangle.” By this means the spherical triangles which present themselves in geodesy are computed with very nearly the same ease as plane triangles. And from the expressions given above for the spheroidal angles A', B', C' it may be proved that no error of any consequence can arise from treating a spheroidal triangle as a spherical, the radius of the sphere being as stated above.
When the angles of a triangulation have been adjusted by the method of least squares, the next process is to calculate the latitudes and longitudes of all the stations starting from one given point. The calculated latitudes, longitudes, and azimuths, which are designated geodetic latitudes, longitudes, and azimuths, are not to be confounded with the observed latitudes, longitudes, and azimuths, for these last are subject to somewhat large errors. Supposing the latitudes of a number of stations in the triangulation to be observed, practically the mean of these determines the position in latitude of the network, taken as a whole. So the orientation or general azimuth of the whole is inferred from all the azimuth observations. The triangulation is then supposed to be projected on a spheroid of given elements, representing as nearly as one knows the real figure of the earth. Then, taking the latitude of one point and the direction of the meridian there as given—obtained, namely, from the astronomical observations there—one can compute the latitudes of all the other points with any degree of precision that may be considered desirable. It is necessary to employ for this purpose formulae which will give results true even for the longest distances to the second place of decimals of seconds, otherwise there will arise an accumulation of errors from imperfect calculation which should always be avoided. For very long distances, eight places of decimals should be employed in logarithmic calculations; if seven places only are available very great care will be required to keep the last place true. Now let φ, φ' be the latitudes of two stations A and B ; a, a their mutual azimuths counted from north by east continuously from 0° to 360°; ω their difference of longitude measured from west to east; and s the distance AB.
First compute a latitude φ l by means of the formula φ 1 -φ- ’ r - cos α, where p is the radius of curvature of the meridian at the latitude φ ; this will require but four places of logarithms. Then, in the first two of the following, five places are sufficient—
s’ . ε = sin α cos α ,
2pit s 2 η = sin 2 a tan φ i , 2pη φ -φ = - cos (a — ∣e) - 7j ,
P°
_ s sin (a - ⅛ε)
11 COS (≠'+ ⅛η) a' - a ∙= ω sill (φ'+⅜v)~e + lδθ°∙
Here n is the normal or radius of curvature perpendicular to the meridian; both n and p correspond to latitude φ 1 , and p 0 to latitude o(φ + ≠')∙ For calculations of latitude and longitude, tables of the logarithmic values of p sin 1", n sin 1", and 2npsin Γ' are necessary. The following table contains these logarithms for every ten minutes of latitude from 52° to 53° computed with the elements a = 20926060 and a:c = 295 :294 :— [table]
The logarithm in the last column is that required also for the calculation of spherical excesses, the spherical excess of a triangle being expressed by α ^. s i n .θ. . 6 h j 2 p n sin 1"
It is frequently necessary to obtain the coordinates of one point with reference to another point; that is, let a perpendicular arc be drawn from B to the meridian of A meeting it in P, then, α being the azimuth of B at A, the coordinates of B with reference to A are
AP=scos (a- ∣s), BP=s sin (a - ⅛e), where e is the spherical excess of APB, viz., s 2 sin α cos a multiplied by the quantity whose logarithm is in the fourth column of the above table.
Irregularities of the Earth's Surface.
In considering the effect of unequal distribution of matter in the earth’s crust on the form of the surface, we may simplify the matter by disregarding the considerations of rotation and excentricity. In the first place, supposing the earth a sphere covered with a film of water, let the density p be a function of the distance from the centre so that surfaces of equal density are concentric spheres. Let now a disturbance of the arrangement of matter take place, so that the density is no longer to be expressed by p, a function of r only, but is expressed by p + p', where p' is a function of three coordinates θ, φ, r. Then p' is the density of what may be designated disturbing matter; it is positive in some places and negative in others, and the whole quantity of matter whose density is p' is zero. The previously spherical surface of the sea of radius a now takes a new form. Let P be a point on the disturbed surface, P' the corresponding point vertically below it on the undisturbed surface, PP' = u. The knowledge of w over the whole surface gives us the form of the disturbed or actual surface of the sea; it is an equipotential surface, and if V be the potential at P of the disturbing matter p', M the mass of the earth,
M __ Μ M , τ α -^+V=C≡--- 3 u + V.
As far as we know, u is always a very small quantity, and we have with sufficient approximation w = where δ is the mean density of the earth. Thus we have the disturbance in elevation of the sea-level expressed in terms of the potential of the disturbing matter. If at any point P the value of u remain constant when we pass to any adjacent point, then the actual surface is there parallel to the ideal spherical surface; as a rule, however, the normal at P is inclined to that at P', and astronomical observations have shown that this inclination, amounting ordinarily to one or two seconds, may in some cases exceed 10, or, as at the foot of the Himalayas, even 30 seconds. By the expression “mathematical figure of the earth” we mean the surface of the sea produced in imagination so as to percolate the continents. We see then that the effect of the uneven distribution of matter in the crust of the earth is to produce small elevations and depressions on the mathematical surface which would be otherwise spheroidal. No geodesist can proceed far in his work without encountering the irregularities of the mathematical surface, and it is necessary that he know how they affect his astronomical observations. The whole of this subject is dealt with in his usual elegant manner by Bessel in the Astronomische Nachrichten, Nos. 329, 330, 331, in a paper entitled “Ueber den Einfluss der Unregelmässigkeiten der Figur der Erde auf geodätische Arbeiten, &c.” But without entering into further details it is not difficult to see how local attraction at any station affects the determinations of latitude, longitude, and azimuth there.
Let there be at the station an attraction to the north-east throwing the zenith to the south-west, so that it takes in the celestial sphere a position Z', its undisturbed position being Z. Let the rectangular components of the displacement ZZ' be ξ measured southwards 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 = 77 cosec PZ' = ι∕ 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 <∕> sin φ — η tan <∕>. 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 expression given for u 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 inches. 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 is shown in a paper in the Philosophical Magazine for August 1878 by Colonel Clarke 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 excentricity 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 qarth’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, a, but for φ, φ + ξ, and for a, 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 (<∕>'), (a), (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
(≠') + ∕ξ+ gη+hdα +kcle, (α') +f'ξ +fη + h , dα + Bdc, (ω) +f"ζ + g'v + h! ’ dα + B ’ de, where the coefficients /,<7,... &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 r[ be the components of the inclination at that point, then we have
ξ' = (≠')-≠' + ∕ξ + gη+ hdα + kde, √ tan φ' = (a) - d+f'ξ + fη+ h'da + Bde, √ sec φ ’ ≈ (ω) - ω + ∕"ξ + (fη + h"da + k"de, where φ ’ , a, 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 ξ 2 + η 2 + 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 feet, 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 ’ 99524953z 2 + ∙99288005y 2 + ∙99763052z 2
- 0 ∙00671003≈z - 41655070z=0.
Altitudes.
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. The height of a station does indeed influence the observation of terrestrial angles, for a vertical line at B does not lie generally in the vertical plane of A, but the error (which is very easily investigated) involved in the neglect of this consideration is much smaller than the errors of observation. Again, in rising to the height h above the surface, the centrifugal force is increased and the magnitude and direction of the attraction of the earth are altered, and the effect upon the observation of latitude is a very small error expressed by the formula
· sin 2<∕>, where g, f are the values of gravity at the equator and at the pole. This is also a quantity which may be neglected, since for ordinary mountain heights it amounts to only a few hundreths of a second.
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 = Ό813 and the latter Zr=∙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"O 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≈3 tan h-h'=s tan fs-—δ' ) .
∖ 2α J
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 , α tan 2 δ a- 2∙1^∙
Suppose the depression δ to be n minutes, then h = 1 ∙054n 2 if the ray be for the greater part of its length crossing the sea; if otherwise, h = 1 ’ 040⅛ 2 . 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= 1O47u 2 = 4396 feet. The actual height of the hill by spirit-levelling is 4406 feet, so that the error of the height thus obtained is only 10 feet.
Longitude.
The determination of the difference of longitude between two stations A and B resolves itself into the determination of the local time at each of the stations, and the comparison by signals of the clocks at A and B. Whenever telegraphic lines are available these comparisons are made by electro-telegraphy. A small and delicately-made apparatus introduced into the mechanism of an astronomical clock or chronometer breaks or closes by the action of the clock a galvanic circuit every second. In order to record the minutes as well as seconds, one second in each minute, namely that numbered 0 or 60, is omitted. The seconds are recorded on a chronograph, which consists of a cylinder revolving uniformly at the rate of one revolution per minute covered with white paper, on which a pen having a slow movement in the direction of the axis of the cylinder describes a continuous spiral. This pen is deflected through the agency of an electromagnet every second, and thus the seconds of the clock are recorded on the chronograph by offsets from the spiral curve. An observer having his hand on a contact key in the same circuit can record in the same manner his observed times of transits of stars. The method of determination of difference of longitude is, therefore, virtually as follows. After the necessary observations for instrumental correctione, which are recorded only at the station of observation, the clock at A is put in connexion with the circuit so as to write on both chronographs, namely, that at A and that at B. Then the clock at B is made to write on both chronographs. It is clear that by this double operation one can eliminate the effect of the small interval of time consumed in the transmission of signals, for the difference of longitude obtained from the one chronograph will be in excess by as much as that obtained from the other will be in defect. The determination of the personal errors of the observers in this delicate operation is a matter of the greatest importance, as therein lies probably the chief source of residual error-
Since the article Figure of the Earth was written, considerable additions to the data for the determination of the semiaxes of the earth have been obtained from India, viz., a new meridian are of 20°, the southern point of which is at Mangalore, together with several arcs of longitude, the longest of which, between Bombay and Nizagapatam, extends over 10° 30'. The effect of the accession of these new measures is to alter the figure previously given to the following: the semiaxes of the spheroid best representing tire large arcs now available are
« = 20926202; c=20854895; c: a = 292∙465: 293*465.
This value of the major semiaxis exceeds that previously given by 140 feet, whereas the new polar semiaxis is less than the old by 226 feet. If we admit that the figure may possibly be an ellipsoid (not of revolution), then the investigation leads us, through the solution of 51 equations, to these values of the semiaxes—
α = 20926629, 5=20925105, c=20854407.
The greater axis of the equator lies in longitude 8" 15' west of Greenwich, a meridian which passing through Ireland and Portugal cuts off a portion of the north-west corner of Africa, and in the opposite hemisphere cuts off the north-east corner of Asia. The apparent ellipticity of the equator is much reduced by the addition of the new data, and it would not be right to put too much confidence in the ellipsoidal figure until many more arcs of longitude shall have furnished the means of testing the theory more decisively than can be done at present. (See Philosophical Magazine, August 1878.)
