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 5; 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

Co

W.J.WELCHE

FIG. 2.-Altazimuth Theodolite.

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 bserved angle is on the average ± 0·28.

For the observations of very distant stations it is usual

mirror 4, 6, or 8 inches in diameter, capable of rotation rouud 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.

in seconds of angle, omitting smaller terms. Here the symbol 8 is the star's declination, z 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 zc cosec 2, 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 in the resulting value of the azimuth,—au 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 is 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 azimuthis.

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 meridiau 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′′-610"-13, while the angle between the two marks directly measured by a theodolite was found to be 4° 31′ 37′′·43 ± 0′′·23.

P

P

N

n

We now come to the consideration of the determination of time with the transit instrument. Let fig. 3 represent the sphere stereographically 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 w 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 wZp, b the level error so that Zp=90° - 6. Let also the hour angle corresponding to p be 90°-n, and the declination of the samem, the star's declination being 8, and the latitude 4. Then to find the hour angle ZPS = T of the star when observed, in the triangles PS. PZ we have. since PPS 90+T-N2

Fig. 3.

d+cos m cos d sin (n-T), - cos b cos sin a, + cos b sin o sin a.

-Sin c-sin m sin Sin m sin b sin Cas m sin nsin b cos And these equations solve the problem, however large be the errors of the instrument. Supposing, as usual, a, b, m, n to be small, we have at once rn+c sec 8+m tan 8, 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

a sin +b cos z+c t+ cos d

Of course this is still only approximative, but it will enable the observer (who by the help of a table of natural tangents can computee 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 nPq=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

the correction for instrumental error being very simila 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 formulæ 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 8, and that of another culminating to the north with zenith distance Z' and declination S', then Now the clearly the latitude is 1(8+8')+1(Z-Z'). 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 azi

Tb sec+c sec d+n (tan 8-tan 4). 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 meridian, let two stars differing considerably in declination be observed the instrument not being reversed between them. From these two stars, neither of which should be a close circumpolar star, a good approximation to the chronometer error can be obtained; thus let 1, 2 be the apparent clock errors given by these stars, if 8, 8, be their declinations the real error

is

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

W. WELCH.S

FIG. 4.-Zenith Telescope.

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, I the value of one division of the level, m the value of one division of the micrometer, r, r' the refraction corrections, 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,

=1(8+8′)+1(4 − 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"-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.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 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. mencing with the consideration of a single triangle in which We may, however, take the case of a simple chain--comall three angles have been observed.

Suppose that the sum of the observed angles exceeds the proper amount by a small quantity : it is required to assign proper corrections to the angles, so as to cause this error to disappear. To

do this we must be guided 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 h, 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+z+=0; and the theory of probabilities teaches us that the most probable values are those which make hx2+kya + lz2 a minimum. Here we arrive at a simple definite problem, the result of which is he-ky-lz, showing that 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+е1, B+е C+еg; then, although 1, eg, eg, the errors 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 ete+e-c. Now, according to the last proposition, if we suppose the angles to be equally well observed, we have to subtract je from each of the observed values, which thus become A+je - je, jeg, B-Je+je-jeg, C-je-je + fez. Then to obtain a and b by calculation from the known side c, we have

86-bb{e (−B+y) +eg( 28+ y) +ez( − B-2y)} 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 e, then we replace eees by e, and adding up the squares of the coefficients find for the probable error of a, ae √6 √(a2+ay + y2), and for that of b, ± }be √6 √(B2 + By +72). 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.

H

a minimum. This, and the previous equation in ƒ, determine all the corrections. Differentiate both and multiply the former by a multiplier P, then 2x+y+e;+ Pa; =0, 2y; + x;+e; - PB; =0, 3x=-P(2a + Bi) - li, 3y= P(a;+2Bi) — ci. Now, substitute these values in the ƒ equation, and P becomes known; then follow at once all the corrections from the two lastwritten 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 a, ß, 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 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 A, B, C 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 a, b, c the measures of curvature at the angular points, a=A+ (2x+b+c),

12

The geodesic line being the shortest that can be drawn on any surface between two given points, we may be con.. ducted to its most important characteristics by the follow ing considerations: let p, q be adjacent points on a curved surface; through s the middle point of the chord pq imagine in the intersection of this plane with the surface; then a plane drawn perpendicular to pq, and let S be any point PS+ Sq 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 normal to the surface at that point. Imagine now three points in space, A, B, C, such that AB = BC=c; 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-cly-cm : z-cn x+cl': y+cm': z+cn'.

Hence the coordinates of the middle point M of AC are x+c(l' − 1), y + c(m' — m), z + c(n'-n), and the direction

which, however, are equivalent to only one equation. the case of the spheroid this equation becomes

which integrated gives ydx-xdy-Cds. This again may be put in the form r sin a= = 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 plaues, 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 AB. If AS lie between this last value and AB, 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 ALA, BLA 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, B, y, these being measured from 0° to 360° continuously, and the angles of the triangle lettered in the same cyclical 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 A'-A- cos 2(b2 sin 28-c2 sin 2y), 12r2

When the angles of a triangulation have been adjusted by the method of least squares, the next process is to calcu late 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 formula 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 , 'be the latitudes of two stations A and B; a, a' their mutual azimuths counted from north by east continuously from 0° to 360°; w their difference of longitude measured from west to east; and s the distance AB.

8

Ρ

First compute a latitude, by means of the formula $1 = ¤ + -cos a, where p is the radius of curvature of the meridian at the latitude ; this will require but four places of logarithms. Then, in the first two of the following, five places are sufficient