Page images
PDF
EPUB
[ocr errors]

arc of France by P. F. A. Méchain and J. B. J. 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 (see EARTH, FIGURE of the).

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; the determination of altitude for all stations.

For the computation, the points of the actual surface of the earth are imagined as projected along their plumb lines on the mathematical figure, which is given by the stationary sea-level, and the extension of the sea through the continents by a system of imaginary canals. For many purposes the mathematical surface is assumed to be a plane; in other cases a sphere of radius 6371 kilometres (20,900,000 ft.). In the case of extensive operations the surface must be considered as a compressed ellipsoid of rotation, whose minor axis coincides with the earth's axis, and whose compression, flattening, or ellipticity is about 1/208.

Measurement of Base Lines.

10 determine by actual measurement on the ground the length of a side of one of the triangles ("base line "), 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.

In order to avoid the difficulty in exactly determining the temperature of a bar by the mercury thermometer, F. W. Bessel introduced in 1834 near Königsberg a compound bar which constituted a metallic thermometer. A zinc bar is laid on an iron bar two toises long, both bars being perfectly planed and in free contact, the zinc bar being slightly shorter and the two bars rigidly united at one end. As the temperature varies, the difference of the lengths of the bars, as perceived by the other end, also varies, and affords a quantitative correction for temperature variations, which is applied to reduce the length to standard temperature. During the measurement of the base line the bars were not allowed to come into contact, the interval being measured by the insertion of glass wedges. The results of the comparisons of four measuring rods with one another and with the standards were elaborately computed by the method of least-squares. The probable error of the measured length of 935 toises (about 6000 ft.) has been estimated as 1/863500 or 1-2 (u denoting a millionth). With this apparatus fourteen base lines were measured in Prussia and some neighbouring states; in these cases a somewhat higher degree of accuracy was obtained.

The principal triangulation of Great Britain and Ireland has seven base lines: five have been measured by steel chains, and two, more exactly, by the compensation bars of General T. F. Colby, an apparatus introduced in 1827-1828 at Lough Foyle in Ireland. Ten base lines were measured in India in 1831-1869 by the same apparatus. This is a system of six compound-bars self-correcting for temperature. The bars may be thus described: Two bars, one of brass and the other of iron, are laid in parallelism side by side, firmly united at their centres, from which they may freely expand or contract; at the standard temperature they are of the same length. Let AB be one bar, A'B' the other; draw lines through the corresponding extremities AA' (to P) and BB' (to Q), and make A'P B'Q, AA' being equal to BB'. If the ratio A'P/AP equals the ratio of the coefficients of expansion of the bars A'B' and AB, then, obviously, the distance PQ is constant (or nearly so). In the actual instrument

1 An arrangement acting similarly had been previously introduced by Borda.

P and Q are finely engraved dots 10 ft. apart. In practice the bars, when aligned, are not in contact, an interval of 6 in. being allowed measured by an ingenious micrometrical arrangement constructed between each bar and its neighbour. This distance is accurately on exactly the same principle as the bars themselves.

The last base line measured in India had a length of 8913 ft. In consequence of some suspicion as to the accuracy of the compensation being conducted so as to determine the actual values of the probable apparatus, the measurement was repeated four times, the operations errors of the apparatus. The direction of the line (which is at Cape Comorin) is north and south. In two of the measurements the brass component was to the west, in the others to the east; the differences between the individual measurements and the mean of the four were +0.0017, 0·0049, 0·0015, +0.0045 ft. These differences are very small; an elaborate investigation of all sources of error shows that the probable error of a base line in India is on the average 2.8 μ. These compensation bars were also used by Sir Thomas Lacaille's arc at the Cape. The account of this operation will be Maclear in the measurement of the base line in his extension of found in a volume entitled Verification and Extension of Lacaille's Arc of Meridian at the Cape of Good Hope, by Sir Thomas Maclear, published in 1866. A rediscussion has been given by Sir David Gill in his Report on the Geodetic Survey of South Africa, &c., 1896. A very simple base apparatus was employed by W. Struve in his triangulations in Russia from 1817 to 1855. This consisted of four wrought-iron bars, each two toises (rather more than 13 ft.) 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 means 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 ft. from the stake in a direction perpendicular to the base. Struve investigated 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. He found that 0.8 μ was the mean of the probable errors of the seven bases measured by him. The Austro-Hungarian apparatus is similar; the distance of the rods is measured by a slider, which rests on one of the ends of each rod. Twenty-two base lines were measured in 1840-1899.

General Carlos Ibañez employed in 1858-1879, for the measurement of nine base lines in Spain, two apparatus similar to the apparatus previously employed by Porro in Italy; one is complicated, the other simplified. The first, an apparatus of the brothers Brunner of Paris, was 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 and 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.

Quite similar apparatus (among others) has been employed by the French and Germans. Since, however, it only permitted a distance of about 300 m. to be measured daily, Ibañez introduced a simplifi cation; the measuring rod being made simply of steel, and provided with inlaid mercury thermometers. This apparatus was used in Switzerland for the measurement of three base lines. The accuracy is shown by the estimated probable errors. 0.2 μ to ±0.8 μ. The distance measured daily amounts at least to 800 m.

A greater daily distance can be measured with the same accuracy by means of Bessel's apparatus; this permits the ready measurement of 2000 m. daily. For this, however, it is important to notice that a large staff and favourable ground are necessary. An important improvement was introduced by Edward Jäderín of Stockholm, who measures with stretched wires of about 24 metres long; these wires are about 1.65 mm, in diameter, and when in use are stretched by an accurate spring balance with a tension of 10 kg. The nature of the ground has a very trifling effect on this method. The difficulty of temperature determinations is removed by employ ing wires made of invar, an alloy of steel (64%) and nickel (36%) which has practically no linear expansion for small thermal changes

Geodetic Survey of South Africa, vol. iii. (1995), p. viü; Les Nouveaux Appareils pour la mesure rapide des bases géod., par J. René Benoît et Ch. Ed. Guillaume (1906).

at ordinary temperatures: this alloy was discovered in 1896 by Benoit and Guillaume of the International Bureau of Weights and Measures at Breteuil. Apparently the future of base-line measurements rests with the invar wires of the Jäderin apparatus; next comes Porro's apparatus with invar bars 4 to 5 metres long. Results have been obtained in the United States, of great importance in view of their accuracy, rapidity of determination and economy. For the measurement of the arc of meridian in longitude 98 E., in 1900, nine base lines of a total length of 69.2 km. were measured in six months. The total cost of one base was $1231. At the beginning and at the end of the field-season a distance of exactly 100 m. was measured with R. S. Woodward's "5-m. icebar" (invented in 1891); by means of the remeasurement of this length the standardization of the apparatus was done under the same conditions as existed in the case of the base measurements. For the measurements there were employed two steel tapes of 100 m. long, provided with supports at distances of 25 m., two of 50 m., and the duplex apparatus of Eimbeck, consisting of four 5-m. rods. Each base was divided into sections of about 1000 m.; one of these, the "test kilometre," was measured with all the five apparatus, the others only, with two apparatus, mostly tapes. The probable error was about 0.8 μ, and the day's work a distance of about 2000 m. Each of the four rods of the duplex apparatus consists of two bars of brass and steel. Mercury thermometers are inserted in both bars; these serve for the measurement of the length of the base lines by each of the bars, as they are brought into their consecutive positions, the contact being made by an elastic-sliding contact. The length of the base lines may be calculated for each bar only, and also by the supposition that both bars have the same temperature. The apparatus thus affords three sets of results, which mutually control themselves, and the contact adjustments permit rapid work. The same device has been applied to the older bimetallic-compensating apparatus of Bache-Würdemann (six bases, 1847-1857) and of Schott. There was also employed a single rod bimetallic apparatus on F. Porro's principle, constructed by the brothers Repsold for some base lines. Excellent results have been more recently obtained with invar tapes.

The following results show the lengths of the same German base lines as measured by different apparatus:

Base at Berlin

"

Base at Strehlen

1854

[ocr errors]

Old base at Bonn

1879 1847

[ocr errors]

"

1892 1892 1892

[ocr errors]

"

New base at Bonn

[ocr errors]

metres.

1864 Apparatus of Bessel 2336-3920
1880
Brunner
·3924
Bessel 2762.5824
Brunner -5852
Bessel
2133-9095
.9097
2512.9612
Brunner .9696

It is 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 may be reduced to what it would have been had the measurement been made on the surface of the sea, produced in imagination. Thus if I be the length of a measuring bar, h its height at any given position in the measurement, the radius of the earth, then the length radially projected on to the level of the sea is (1-h/r). In the Salisbury Plain base line the reduction to the level of the sea is -0-6294 ft.

|

the measures. Leaving the base line, the sides increase up to 10, 30 or 50 miles occasionally, but seldom reaching 100 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 ft. high, is surmounted by a spire of 90 ft. 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 ft. above the ground, having its bearing upon timbers passing through the spire at that height. Thus the instrument, at a height of 178 ft. 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 anglesor 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 may be 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.

The total number of base lines measured in Europe up to the present time is about one hundred and ten, nineteen of which do not exceed in length 2500 metres, or about 1 miles, and three-points which have to be observed, that one which affords the best

one in France, the others in Bavariaexceed 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. With Jäderin's apparatus, provided with invar wires, bases of 20 to 30 km. long are obtained without difficulty.

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 pg (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 bhp, phq, hgk, kga are measured. From these angles can be computed the ratios of the segments, which must agree, if all erations are correctly performed, with the ratios resulting from

FIG. 1.

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 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 I to 2 miles.

This method seems at first sight very advantageous; but if, however, it be desired to attain the highest accuracy, it is better, as shown by General Schreiber of Berlin in 1878, to measure only single angles, and as many of these as possible between the directions to be determined. Division-errors are thus more perfectly eliminated, and errors due to the variation in the stability, &c., of the instruments are diminished. This method is rapidly gaining precedence.

The theodolites used in geodesy vary in pattern and in size-the horizontal circles ranging from 10 in. to 36 in. in diameter. In Ramsden's 36-in. theodolite the telescope has a focal length of 36 in. and an aperture of 2.5 in., 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 o"-2. Fig. 2 represents an altazimuth theodolite of an improved pattern used on the Ordnance Survey. The horizontal circle of 14-in. diameter is read by three micrometer microscopes; the vertical circle has a diameter of 12 in., 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 ft. diameter-the circle read by five equidistant microscopes. Every angle is measured twice in each position of the zero of the the entire horizontal circle, of which there are generally ten;

[graphic]

number of measures of an angle is never less than 20. An examina- | sponding uncertainty in the resulting value of the azimuth,-an tion of 1407 angles showed that the probable error of an observed uncertainty which increases with the latitude and is very large angle is on the average 0.28. miltis com elatog light in high latitudes. This may be partly remedied by observing in For the observations of very distant stations it is usual to employ connexion with the star its reflection in mercury. In determining a heliotrope (from the Gr. Aos, sun; rpóños, a turn), invented by the value of " one division" of a level tube, it is necessary to bear Gauss at Göttingen in 1821. In its simplest form this is a plane in mind that in some the value varies considerably with the temperamirror, 4, 6, or 8 in. in diameter, capable of rotation round a horizontal ture. By experiments on the level of Ramsden's 3-foot theodolite, and a vertical axis. This mirror is placed at the station to be ob- it was found that though at the ordinary temperature of 66° the served, and in fine weather it is kept so directed that the rays of the value of a division was about one second, yet at 32° it was about sun reflected by it strike the distant observing telescope. To the five seconds.dopo la 200 to a Finomia et Biom observer the heliotrope presents the appearance of a star of the In a very excellent portable transit used on the Ordnance Survey, first or second magnitude, and is generally a pleasant object for the uprights carrying the telescope are constructed of mahogany. observing. each upright being built of several pieces glued and screwed together; Observations at night, with the aid of light-signals, have been the base, which is a solid and heavy plate of iron, carries a reversing repeatedly made, and with good results, particularly in France apparatus for lifting the telescope out of its bearings, reversing it by General François Perrier, and more recently in the United and letting it down again. Thus is avoided the change of temperaStates by the Coast and Geodetic Survey; the signal employed ture which the telescope would incur by being lifted by the hands being an acetylene bicycle-lamp, with a lens 5 in. in diameter. of the observer. Another form of transit is the German diagonal Particularly noteworthy are the trigonometrical connexions of form, in which the rays of light after passing through the objectSpain and Algeria, which were carried out in 1879 by Generals glass are turned by a total reflection prism through one of the transIbañez and Perrier (over a distance of 270 km.), of Sicily and Malta verse arms of the telescope, at the extremity of which arm is the in 1900, and of the islands of Elba and Sardinia in 1902 by Dr eye-piece. The unused half of the ordinary telescope being cut away Guarducci (over distances up to 230 km.); in these cases artificial is replaced by a counterpoise. In this instrument there is the Sonceb0 dailyna od boroved aniblolas dyin to VM advantage that the observer without moving the position of his eye douda bo 3 sobateni sot eye commands the whole meridian, and that the level may remain on do ads sol blen 11 op lo siga yd bangome a ulgid the pivots whatever be the elevation of the telescope. But there is of sead of mon bonus ew volvis the disadvantage that the flexure of the transverse axis causes a hies ew Jasmani ady 10 variable collimation error depending on the zenith distance of the i goived buong sd star to which it is directed; and moreover it has been found that in gied ads 16 siga some cases the personal error of an observer is not the same in the bauerg ads ovoda two positions of the telescope.me loine ad no bniw ods ate love A dr to enement odo

[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors][subsumed][ocr errors][ocr errors][ocr errors]
[ocr errors][ocr errors][ocr errors][merged small][subsumed][ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]
[subsumed][ocr errors]
[ocr errors][ocr errors]
[ocr errors][subsumed][subsumed][ocr errors][subsumed][subsumed][subsumed]
[ocr errors]

let a, a, be the apparent clock errors given by these stars if 8, & the telescope, which latter carries a micrometer in its eye-piece, be their declinations the real error is

=+(-) (tan ø-tan &1)/(tan ¿1-tan ¿1⁄2).

Of course this is still only approximate, but it will enable the observer (who by the help of a table of natural tangents can compute e 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=4, and b being the level error, Zq=90°-b; let also Pq=r and Pq=y. 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 qPS', qPZ give -Sin c=sin & cos-cos & sin cos (h+r), sin b sin +cos b cos cos a, cos b sin a.

Cos

Sin sin Now when a and b are very small, we see from the last two equations that y=4-b, ar sin y, and if we calculate o' by the formula cot '=cot & cos h, the first equation leads us to this result

='+(a sin z+b cos z+c)/cos 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 2 is small, the formulae required are more complicated.

[ocr errors]

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 me. ridian 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 8, and that of another culminating to the north with zenith distance Z' and declination &', then clearly the latitude is (8+)+ (Z-Z'). Now the zenith telescope does away with the divided circle, and substitutes the measurement micrometrically of the quantity Z-Z.

In fig. 4 is shown a zenith telescope by H. Wanschaff of Berlin, which is the type used (according to the Central Bureau at Potsdam) since about 1890 for the determination of the variations of latitude due to different, but as yet imperfectly understood, influences. The instrument 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 with a level. 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 eccentrically with respect to the vertical axis around which it revolves. Two extremely sensitive levels are attached to

FIG. 4.-Zenith Telescope constructed for the International Stations at Mizusawa, Carloforte, Gaithersburg and Ukiah, by Hermann Wanschaff, Berlin.

with a screw of long range for measuring differences of zenith dis tance. Two levels are employed for controlling and increasing the accuracy. 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 distelescope is set to the mean of the zenith distances and in the plane tance of each other. When a pair of stars is to be observed, the 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 of the zenith distances, and the calculation to get the latitude is the centre wire. The micrometer has thus measured the difference 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, the value of one division refraction corrections, μ, the micrometer readings of the south of the level, m the value of one division of the micrometer, r, r' the and north star, the micrometer being supposed to read from the zenith, then, supposing the observation made on the centre wire,— 6=1(8+8')+1(u−μ')m+}(n+n'~s-s′)l+}(r−r').

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"-2, 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, two nights at least 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 in. in focal length; a 30-in. telescope is sufficient for the highest purposes and is very portable. The net observation probable-error for one pair of stars is only #0".1.

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.M.. 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 ft. 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.

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 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 an electric circuit every second. In order to record the minutes as well as seconds, one second in each minute, namely that numbered o 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 corrections, which are recorded only at the station of obser vation, 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.

These errors can nevertheless be almost entirely avoided by using | For the sphere a=b=r, and making this simplification, we obtain the the impersonal micrometer of Dr Repsold (Hamburg, 1889). In theorem previously given by A. M. Legendre. With the terms of the this device there is a movable micrometer wire which is brought by fourth order, we have (after Andrae): hand into coincidence with the star and moved along with it; at fixed points there are electrical contacts, which replace the fixed wires. Experiments at the Geodetic Institute and Central Bureau at Potsdam in 1891 gave the following personal equations in the case of four observers:

[blocks in formation]

These results show that in the later method the personal equation is small and not so variable; and consequently the repetition of longitude determinations with exchanged observers and apparatus entirely eliminates the constant errors, the probable error of such determinations on ten nights being sca. cely 001.

Calculation of Triangulation.

[ocr errors]

(m2-a2

20

'm2 - b2.

20 k+).

'm2.
20

in which e=ok(1+(m2k/8)}, 3m2=a2+b2+c2, 3k=a+b+r. For the ellipsoid of rotation the measure of curvature is equal to 1/pm, p and being the radii of curvature of the meridian and perpendicular.

It is rarely that the terms of the fourth order are required. As a rule spheroidal triangles are calculated as spherical (after Legendre), i.e. like plane triangles with a decrease of each angle of about e/3: e must, however, be calculated for each triangle separately with its mean measure of curvature k.

The geodetic 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. The surface of Great Britain and Ireland is uniformly covered by be adjacent points on a curved surface; through s the middle point triangulation, of which the sides are of various lengths from 10 to of the chord pq imagine a plane drawn perpendicular to pq, and let 111 miles. The largest triangle has one angle at Snowdon in Wales, S be any point in the intersection of this plane with the surface; another on Slieve Donard in Ireland, and a third at Scaw Fell in then pS+Sq is evidently least when sS is a minimum, which is Cumberland; each side is over a hundred miles and the spherical when sS is a normal to the surface; hence it follows that of all excess is 64". The more ordinary method of triangulation is, however, plane curves on the surface joining p, q, when those points are inthat of chains of triangles, in the direction of the meridian and definitely near to one another, that is the shortest which is made perpendicular thereto. The principal triangulations of France, by the normal plane. That is to say, the osculating plane at any Spain, Austria and India are so arranged. Oblique chains of tri-point of a geodetic line contains the normal to the surface at that angles are formed in Italy, Sweden and Norway, also in Germany point. Imagine now three points in space, A, B, C, such that AB = and Russia, and in the United States. Chains are composed someBC-c; let the direction cosines of AB be l, m, n, those of BC, times merely of consecutive plain triangles; sometimes, and more m', n', then x, y, z being the co-ordinates of B, those of A and C will frequently in India, of combinations of triangles forming consecutive be respectivelypolygonal figures. In this method of triangulating, the sides of the x-dy-cm-cn triangles are generally from 20 to 30 miles in length-seldom exceedx+c': y+cm': z+cn'. ing 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. We cannot here give any illustration of this method as applied to general triangulation, for it is most laborious, even for the simplest cases.

Three stations, projected on the surface of the sea, give a spherical or spheroidal triangle according to the adoption of the sphere or the ellipsoid as the form of the surface. 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 a, B, Y respectively. Now the angle A of the triangle as measured by a theodolite is the inclination of the planes BAa and CAa, and the angle at B is that contained by the planes ABB and CBB. But the planes ABa and ABB 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 geodetic or shortest lines. C. C. G. Andrae, of Copenhagen, has also shown that other lines give a less convenient computation.

K. F. Gauss, in his treatise, Disquisitiones generales circa superficies curvas, entered fully into the subject of geodetic (or geodesic) triangles, and investigated expressions for the angles of a geodetic 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 A, B, C for the angles of the geodetic triangle, while A, B, C are those of a plane triangle having sides equal respectively to those of the geodetic triangle, then, being the area of the plane triangle and a, b, r the measures of curvature at the angular points,

A=A+(2a+b+c)/2.
BB+o(a+b+c)/18,
α=C+o(a+b+20)/12.

Hence the co-ordinates of the middle point M of AC are x+c(l'-D).
+} c(m2−m), z+{c(n'-n), and the direction cosines of BM are
therefore proportional to -1: m'-m: n'-n. If the angle made
by BC with AB be indefinitely small, the direction cosines of BM
are as &: dm : dn. Now if AB, BC be two contiguous elements of
a geodetic, then BM must be a normal to the surface, and since &,
and if the equation of the surface be u=0, we have
om, in are in this case represented by 8(dx/ds), 8(dy/ds), 8(dz/ds),
dx du dy/du dz/du
ds2/dxds3/dyds/dz'

=

[merged small][ocr errors]

which integrated gives ydx-xdy=Cds. This again may be put in
the form sin a=C, where a is the azimuth of the geodetic at any
point-the angle between its direction and that of the meridian-
and the distance of the point from the axis of revolution.
From this it may be shown that the azimuth at A of the geodetic
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 geodetic 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 geodetic 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 considera-
the pole C on AB, meeting it in 5 between A and B. Then A being
tion of the earth's non-sphericity, and draw a perpendicular from
that point which is nearest the pole, the geodetic will cross the plane
curve if AS be between AB and AB. If AS lie between this last
value and AB, the geodetic will lie wholly to the north of both
plane curves, that is, supposing both points to be in the northern
hemisphere.

The difference of the azimuths of the vertical section AB and of the geodetic AB, i.e. the astronomical and geodetic azimuths, is very small for all observable distances, being approximately:

I

[ocr errors]

Geod. azimuth-Astr. azimuth 121 (cos2 + sin 2a+ sin 24 sin a), in which: e and a are the numerical eccentricity and semi-major axis respectively of the meridian ellipse, and a are the latitude and azimuth at A, s=AB, and and n are the radii of curvature of the meridian and perpendicular at A. For $100 kilometres, only the first term is of moment; its value is o°-028 cos sin 2a, and it lies well within the errors of observation. If we imagine the geodetic AB, it will generally trisect the angles between the vertical sections at A and B, so that the geodetic at A is near

« ՆախորդըՇարունակել »