angles equations errors triangles base triangulation values measured line stations
SURVEYING is the art of determining the relative positions of prominent points and other objects on the surface of the ground and making a graphical delineation of the included area. The general principles on which it is conducted are in all instances the same: certain measures are made on the ground and corresponding measures are protracted on paper, on a scale which is fixed at whatever fraction of the natural scale may be most appropriate in each instance. The method of operation varies with the magnitude and importance of the survey, which may embrace a vast empire or be restricted to a small plot of land. All surveys rest primarily on linear measures for direct determinations of distance ; but these are usually largely supplemented by angular measures, to enable distances to be deduced by the principles of geometry which cannot be conveniently measured over the surface of the ground where it is hilly or broken. The nature of a survey depends on the proportion which the linear and the angular measures bear to each other ; it may be purely linear or even purely angular, but is generally a combination of both methods. Thus in India there are numerous instances of large tracts having been surveyed by the purely linear method, in the course of the revenue surveys which were initiated by the native Governments. The operations were conducted by men who had no knowledge of geometry or of any other measuring instrument than the rod or chain, and whose principal object was the determination of fairly accurate areas ; their methods sufficed for this purpose and were accepted and perpetuated for many years by the European officers to whom the revenue assessments became entrusted after the subversion of the native rule. In India, too, there are extensive tracts of country which have been surveyed by the purely angular method, either because the ground did not permit of the chain being employed with advantage, as in the Himalayan mountains and hill tracts generally, or because the chain was considered politically objectionable, as in native states where it would have been regarded with suspicion.
Surveys of any great extent of country were formerly constructed on a basis of points whose positions were fixed astronomically, and in some countries this method of operation is still of necessity adopted. But points whose relative positions have been fixed by a triangulation of moderate accuracy present a more satisfactory and reliable basis ; for astronomical observations are liable, not only to the well-known intrinsic errors which are caused by uncertainties in the catalogued places of the moon and stars, but to external errors arising from deflexions of the plumb line under the influence of local attractions, and these of themselves materially exceed the errors which would be generated in a fairly executed triangulation of a not excessive length, say not exceeding 500 miles. The French Jesuits who made a survey of China for the emperor about 1730 appear to have been the first deliberately to discard the astronomical and adopt the trigonometrical basis. In India the change was made in 1800, when what is known as the Great Trigonometrical Survey was initiated by Major Lambton - with the support of Colonel Wellesley, afterwards Duke of Wellington - as a means of connecting the several surveys of routes and districts which had already been made in various parts of the country, and as a basis for future topography. This necessitated the inception of the survey as an undertaking calculated to satisfy the requirements of geodesy as well as geography, because the latitudes and longitudes of the points of the triangulation had to be determined for future reference, - as in the case of the discarded astronomical stations, though in a different manner, - by processes of calculation combining the results of the triangulation with the elements of the earth's figure. The latter were not then known with much accuracy, for so far geodetic operations had been mainly carried on in Europe, and additional operations nearer the equator were much wanted ; the survey was conducted with a view to supply this want. Thus a high order of accuracy was aimed at from the very first. In course of time the operations were extended over the entire length and breadth of Hindustan and beyond, to the farthest limits of British sway ; they cover a larger area than any other national survey as yet completed, and are very elaborate and precise. Thus, as triangulation constitutes the most appropriate basis for survey operations generally, a short account will be given of (1) the methods of the Great Trigonometrical Survey of India. This will be followed by accounts of (2) traversing as a basis for survey, (3) levelling, (4) survey of interior detail, (5) representation of ground, (6) geographical reconnaissance, (7) nautical surveying, (8) mapping, (9) map printing, (10) instruments.
I. GREAT TRIGONOMETRICAL SURVEY OF INDIA.
meridional arc - the "great arc " - was eventually deemed inadequate for geodetic requirements. A superior instrumental equipment was introduced, with an improved modus operandi, under the direction of Colonel Everest in 1832. The network system of triangulation was superseded by meridional and longitudinal chains taking the form of gridirons, and resting on base-lines at the angles of the gridirons, as represented in fig. 1. For convenience of reduction and nomenclature the triangulation west of meridian 92° E. has been divided into five sections, - the lowest a trigon, the other four quadrilaterals distinguished by cardinal points which have reference to an observatory in Central India, the adopted origin of latitudes. In the north-east quadrilateral, which was first measured, the meridional chains are about one degree apart ; this distance was latterly much increased, and eventually certain chains - as on the Malabar coast and on meridian 84° in the south-east quadrilateral - were dispensed with, because good secondary triangulation for topography had been accomplished before they could be commenced.
In all base-line measurements the weak point is the determination of the temperature of the bars when that of the atmosphere is rapidly rising or falling ; the thermometers acquire and lose heat more rapidly than the bar if their bulbs are outside, and more slowly if inside the bar. Thus there is always more or less lagging, and its effects are only eliminated when the rises and falls are of equal amount and duration ; but as a rule the rise generally predominates greatly during the usual hours of work, and whenever this happens lagging may cause more error in a base-line measured with simple bars than all other sources of error combined. In India the probable average lagging of the standard-bar thermometer was estimated as not less than 0°•3 Fahr., corresponding to an error of - 2 millionths in the length of a base-line measured with iron bars. With compound bars lagging would be much the same for both components and its influence would consequently be eliminated. Thus the most perfect base-line apparatus would seem to be one of compensation bars with thermometers attached to each component ; then the comparisons with the standard need only be taken at the times when the temperature is constant, and there is no lagging.
Expan- 3. Factor of Expansion of Standard Bar. - This was Plan of 4. Plan of Triangulation. - This was broadly a system length was governed by the height to which towers could be conveniently raised to surmount the curvature, under the well-known condition, height in feet = g x square of the distance in miles; thus 24 feet of height was needed at each end of a side to overtop the curvature in 12 miles, and to this had to be added whatever was required to surmount obstacles on the ground. In Indian plains refraction is more frequently negative than positive during sunshine ; no reduction could therefore be made for it.
Selection of Sites for Stations. - This, a very simple Sites for matter in hills and open country, is often very difficult in stations. plains and close country. In the early operations, when the great arc was being carried across the wide plains of the Gangetic valley, which are covered with villages and trees and other obstacles to distant vision, masts 35 feet high were carried about for the support of the small reconnoitring theodolites, with a sufficiency of poles and bamboos to form a scaffolding of the same height for the observer. Other masts 70 feet high, with arrangements for displaying blue lights by night at 90 feet, were erected at the spots where station sites were wanted. But the cost of transport was great ; the rate of progress was slow ; and the results were unsatisfactory. Eventually a method of touch rather than sight was adopted, feeling the ground to search for the obstacles to be avoided, rather than attempting to look over them ; the "rays" were traced either by a minor triangulation, or by a traverse with theodolite and perambulator, or by a simple alignment of flags. The first method gives the direction of the new station most accurately ; the second searches the ground most closely ; the third is best suited for tracts of uninhabited forest in which there is no choice of either line or site, and the required station may be built at the intersection of the two trial rays leading up to it. As a rule it has been found most economical and expeditious to raise the towers only to the height necessary for surmounting the curvature, and to remove the trees and other obstacles on the lines.
Structure of the Principal Stations. - Each has a cen- Principal tral masonry pillar, circular and 3 to 4 feet in diameter, stations. for the support of a large theodolite, and around it a platform 14 to 16 feet square for the observatory tent, observer, and signallers. The pillar is carefully isolated from the platform, and when solid carries the station mark - a dot surrounded by a circle - engraved on a stone at its surface, and on additional stones or the rock in situ, in the normal of the upper mark ; but, if the height is considerable and there is a liability to deflexion, the pillar is constructed with a central vertical shaft to enable the theodolite to be plumbed over the ground-level mark, to which access is obtained through a passage in the basement. In early years this precaution against deflexion was neglected and the pillars were built solid throughout, whatever their height ; the surrounding platforms, being usually constructed of sun-dried bricks or stones and earth, were liable to fall and press against the pillars, some of which thus became deflected during the rainy seasons that intervened between the periods during which operations were arrested or the commencement and close of the successive circuits of triangles. In some instances displacements of mark occurred of which the magnitudes were not ascertainable, but were estimated as equivalent to p.e.'s of about ± 9 inches in the length and ± 2"•4 in the azimuth of the side between any two deflected towers ; and, as these theoretical errors are identical with what may be expected at the end of a chain of 36 equilateral triangles in which all the angles have been measured with a p.e. = ± 0"-5, the old triangulation over solid towers had evidently suffered much more i from the deflexions of the towers than from errors in the measurements of the angles.
Theo. The principal theodolites are somewhat similar to the dolites. astronomer's alt-azimuth instrument, but with larger azimuthal and smaller vertical circles, also with a greater base to give the firmness and stability which are required in measuring horizontal angles. The azimuthal circles have mostly diameters of either 36 or 24 inches, the vertical circles having a diameter of 18 inches. In all the theodolites the base is a tribrach resting on three levelling foot-screws, and the circles are read by microscopes ; but in different instruments the fixed and the rotatory parts of the body vary. In some the vertical axis is fixed on the tribrach and projects upwards ; in others it revolves in the tribrach and projects downwards. .In the former the azimuthal circle is fixed to the tribrach, while the telescope pillars, the microscopes, the clamps, and the tangent screws are attached to a drum revolving round the vertical axis ; in the latter the microscopes, clamps, and tangent screws are fixed to the tribrach, while the telescope pillars and the azimuthal circle are attached to a plate fixed at the head of the rotatory vertical axis. The former system - called that of flying microscopes - permits the vertical axis to be readily opened out and cleaned, and presents the same clamp and tangent screw for employment during a round of angles; the latter - the system of fixed microscopes - necessitates the removal and replacement of all the microscopes, clamps, and tangent screws whenever the axis is cleaned, which is very troublesome, and it presents three sets of clamps and tangent screws for successive employment during a round of angles, which is a departure from true differeutiality. The vertical axis is perforated for centring over the station mark with the aid of a "look-down telescope" instead of a plummet. The azimuthal circle is invariably read by an odd number of microscopes, either three or five, at equal intervals apart. The .telescope rests with its pivots in Y's at the head of two pillars of a sufficient height to enable it to be completely turned round in altitude. The vertical circle is fixed to the transit axis of the telescope, and is read by two microscopes 180° apart, at the extremities of arms projecting from one of the pillars. The stand is a well-braced tripod, carrying an iron ring on which the theodolite rests and may be turned round bodily whenever desired, as for shifting the position of the zero of the azimuthal circle relatively to the points under observation. The ring is 3 inches broad and of the same diameter as the circle of the foot-screws of the theodolite. In some instruments the foot-screws rest directly on the ring ; but the instrument can be raised off the ring and turned round with the aid of an apparatus in the centre of the stand. In others they rest in grooves at the angles of an iron triangle which sits on the ring and can be shifted in position by hand ; thus with the stand well levelled in the first instance the circle may be set within 1' of any required reading. The centring over the station mark is performed by pushing screws placed either in the drum of the stand or at the angles of the triangle.
For travelling the theodolites were packed in two cases, the larger containing the body of the instrument, the smaller the telescope and the vertical circle ; the stand constituted a third package. Each was carried on men's shoulders as the safest method of transport ; the weights, of the heaviest 36-inch and of the lightest 24-inch instruments, as packed with ropes and bamboos, were, respect. ively, as follows : - body, 649 lb ; telescope, 130 ; stand, 232; total, 1011 lb ; and 300, 135, and 185, total G20 lb.
Signals. - Cairns of stones, poles, or other opaque Signals. signals were primarily employed, the angles being measured by day only ; eventually it was found that the atmosphere was often more favourable for observing by night than by day, and that distant points were raised well into view by refraction by night which might be invisible or only seen with difficulty by day. Lamps were then introduced of the simple form of a cup, 6 inches in diameter, filled with cotton seeds steeped in oil and resin, to burn under an inverted earthen jar, 30 inches in diameter, with an aperture in the side towards the observer. Subsequently this contrivance gave place to the Argand lamp with parabolic reflector ; the opaque day signals were discarded for heliotropes reflecting the sun's rays to the observer. The introduction of luminous signals not only rendered the night as well as the day available for the observations but changed the character of the operations, enabling work to be done during the dry and healthy season of the year, when the atmosphere is generally hazy and dust-laden, instead of being restricted as formerly to the rainy and unhealthy seasons, when distant opaque objects are best seen. A higher degree of accuracy was also secured, for the luminous signals were invariably displayed through diaphragms of appropriate aperture, truly centred over the station mark ; and, looking like stars, they could be observed with greater precision, whereas opaque signals are always dim in comparison and are liable to be seen excentrically when the light falls on one side.
A signalling party of three men was usually found sufficient to manipulate a pair of heliotropes - one for single, two for double reflexion, according to the sun's position - and a lamp, throughout the night and day. Heliotropers were also employed at the observing stations to flash instructions to the signallers.
_Measuring Horizontal Angles. - The theodolites were Ireasurinvariably set up under tents for protection against sun, fig horiwind, and rain, and centred, levelled, and adjusted for the Z„iIt:l s. runs of the microscopes. Then the signals were observed in regular rotation round the horizon, alternately from right to left and vice versa ; after the prescribed minimum number of rounds, either two or three, had been thus measured, the telescope was turned through 180°, both in altitude and azimuth, changing the position of the face of the vertical circle relatively to the observer, and further rounds were measured ; additional measures of single angles were taken if the prescribed observations were not sufficiently accordant. As the microscopes were invariably equidistant and their number was always odd, either three or five, the readings taken on the azimuthal circle during the telescope pointings to any object in the two positions of the vertical circle, "face right" and "face left," were made on twice as many equidistant graduations as the number of microscopes. The theodolite was then shifted bodily in azimuth, by being turned on the ring on the head of the stand, -which brought new graduations under the microscopes at the telescope pointings ; then further rounds were measured in the new positions, face right and face left. This process was repeated as often as had been previously prescribed, the successive angular shifts of position being made by equal arcs bringing equidistant graduations under the microscopes during the successive telescope pointings to one and the same object. By these arrangements all •periodic errors of graduation were eliminated, the numerous graduations that were read tended to cancel accidental errors of division, and the numerous rounds of measures to minimize the errors of observation arising from atmospheric and personal causes.
The following table (I.) gives details of the procedure at different times ; in the headings M stands for the number of microscopes over the azimuthal circle of the theodolite, Z for the number of the zero settings of the circle, N for the number of graduations brought under the microscopes, A=360°-N, the arc between the graduations, R the prescribed number of rounds of measures, and P=R x Z, the minimum number of telescope pointings to any station, excluding repetitions for discrepant observations : - Under this system of procedure the instrumental and ordinary errors are practically cancelled and any remaining error is most probably due to lateral refraction, more especially when the rays of light graze the surface of the ground. The three angles of, every triangle were always measured.
Vertical 10. Vertical Angles. Refraction. - The apparent altiangles. tude of a distant point is liable to considerable variations during the twenty-four hours, under the influence of changes in the density of the lower strata of the atmosphere. Terrestrial refraction is very capricious, more particularly when the rays of light graze the surface of the ground, passing through a medium which is liable to extremes of rarefaction and condensation, under the alternate influence of the sun's heat radiated from the surface of the ground and of chilled atmospheric vapour. When the back and forward verticals at a pair of stations are equally refracted, their difference gives an exact measure of the difference of height. But the atmospheric conditions are not always identical at the same moment everywhere on long rays which graze the surface of the ground, and the ray between two reciprocating stations is liable to be differently refracted at its extremities, each end being influenced in a greater degree by the conditions prevailing around it than by those at a distance ; thus instances are on record of a station A being invisible from another B, while B was visible from A.
When the great arc entered the plains of the Gangetic valley, simultaneous reciprocal verticals were at first adopted with the hope of eliminating refraction ; but it was soon found that they did not do so sufficiently to justify the expense of the additional instruments and observers. Afterwards the back and forward verticals were observed as the stations were visited in succession, the back angles at as nearly as possible the same time of the day as the forward angles, and always during the so-called "time of minimum refraction," which ordinarily commences about an hour after apparent noon and lasts from two to three hours. The apparent zenith distance is always greatest then, but the refraction is a minimura only at stations which are well elevated above the :lurface of the ground ; at stations on plains the refraction is liable to. pass through zero and attain a considc-.able negative magnitude during the heat of the day, for the lower strata of the atmosphere are then less dense than the strata immediately above and the rays are refracted downwards. On plains the greatest positive refractions are also obtained, - maximum values, both positive and negative, usually occurring, the former by night, the latter by clay, when the sky is most free from clouds. The values actually met with v: erefound to range from + 1.21 down to - 0-09 parts of the contained arc on plains ; the normal "coefficient of refraction " for free rays between hill stations below 6000 feet was about •07, which diminished to •04 above 18,000 feet, broadly varying inversely as the temperature and directly as the pressure, but much influenced also by local climatic conditions.
In measuring the vertical angles with the great theodolites, graduation errors were regarded as insignificant compared with errors arising from uncertain refraction ; thus no arrangement was made for effecting changes of zero in the circle settings. The observations were always taken in pairs, face right and left, to eliminate index errors, only a few daily, but some on as many days as possible, for the variations from day to day were found to he greater than the diurnal variations during the hours of minimum refraction.
Results deduced from Observations of Horizontal Angles ; Weights. - In the Ordnance and other surveys the bearings of the surrounding stations are deduced from the actual observations, but from the "included angles" in the Indian Survey. The observations of every angle are tabulated vertically in as many columns as the number of circle settings face left and face right, and the mean for each .setting is taken. For several years the general mean Weights. of these was adopted as the final ..sult; but subsequently a "concluded angle" was obtained by combining the single means with weights inversely proportional to g2+ o2÷n, - g beinc, a value of the e.m.s.1 of graduation derived empirically from the differences between the general mean and the mean for each setting, o the e.m.s. of observation deduced from the differences between the individual measures and their respective means, and n the number of measures at each setting. Thus, putting w1, w2, ... for the weights of the single means, w for the weight of the concluded angle, 211 for the general mean, C for the concluded angle, and c/5, d2, ... for the differences between If and the single means, we have 2C d + + C= M-+ -"- 7-2 (1) 2C2+ 20 = IC24and . (2).
C - if vanishes when is is constant ; it is inappreciable when g is much larger than o ; it is significant only when the graduation errors are more minute than the errors of observation; but it was always small, not exceeding 0"•14 with the system of two rounds of measures and 0"•05 with the system of three rounds.
The weights of the concluded angles thus obtained were employed in the primary reductions of the angles of single triangles and polygons which were made to satisfy the geometrical conditions of each figure, because they were strictly relative for all angles measured with the same instrument and under similar circumstances and conditions, as was almost always the case for each single figure. But in the final reductions, when numerous chains of triangles composed of figures executed with different instruments and under different circumstances came to be adjusted simultaneously, it was necessary to modify the original weights, on such evidence of the precision of the angles as might be obtained from other and more reliable sources than the actual measures of the angles. This treatment will now be described.
Determination of Theoretical Absolute Errors of TheoretiObserved Angles. - Values of theoretical error for groups cal errors of angles measured with the same instrument and under of angles.
similar conditions may be obtained in three ways, - (i.) from the squares of the reciprocals of the weight w deduced as above from the measures of such angle, (ii.) from the magnitudes of the excess of the sum of the angles of each triangle above 180° + the spherical excess, and (iii.) from the magnitudes of the corrections which it is necessary to apply to the angles of polygonal figures and networks to satisfy the several geometrical conditions (indicated in the next section). Let e1, e2, and e3 be the values of the e.m.s. thus obtained ; then, putting 22, for the number of angles grouped together, we have cI [w] and c ;,2 = [squares of triangular errors] also, putting TV for the mean of the weights of the t angles Refraction.
of a polygonal figure having In geometrical equations of condition, and x for the most probable value of the error of any observed angle, we have Puke] 1 a e,2= W .as m for a single figure, -[a] for a group of figures, [m] the brackets  in each case denoting the sum of all the quantities involved. e3 usually gives the best value of the theoretical error, then e2. As a rule the value by e, is too small; but to this there are notable exceptions, in which it was found to be much too great. The instrument with which the angles were measured in these instances gave very discrepant results at different settings of the circle ; but this was caused by large periodic errors of graduation which did not affect the " concluded angles," because they were eliminated by the systematic changes of setting, so the results were really more precise than was apparent.
When weights were determined for the final simultaneous reduction of triangulations executed by different instruments, it became necessary to find a factor p to be applied as a modulus to each group of angles measured with the same instrument and under similar conditions, to convert the as yet relative weights into absolute measures of precision. p was made = et+ e, whenever data were available, if not to e,± e2; then the absolute weight of an observed angle in any group was taken as avp2 and the e.m.s. of the angle as 1 p NilV. The average values of the e.m.s. thus determined for large groups of angles, measured with the 36-inch and the 24-inch theodolites, ranged from ± 0"•24 to + 0".67, the smaller values being usually obtained at hill stations, where the atmospheric conditions were most favourable.
Harmon- 13. IIarnzonizing Angles of Trigonometrical Figures. - izing Every figure, whether a single triangle or a polygonal netangles. work, was made consistent by the application of corrections to the observed angles to satisfy its geometrical conditions. The three angles of every triangle having been observed, their sum had to be made = 180° + the spherical excess ; in networks it was also necessary that the sum of the angles measured round the horizon at any station should be exactly = 360°, that the sum of the parts of an angle measured at different times should equal the whole, and that the ratio of any two sides should be identical, whatever the route through which it was computed. These are called the triangular, central, toto-partial, and side conditions ; they present n geometrical equations, which contain t unknown quantities, the errors of the observed angles, t being always >n. When these equations are satisfied and the deduced values of errors are applied as corrections to the observed angles, the figure becomes consistent. Primarily the equations were treated by a method of successive approximations; but afterwards they were all solved simultaneously by the so-called method of minimum squares, which leads to the most probable of any system of corrections ; it is demonstrated under EARTH, FIGURE OF THE (vol. vii. p. 599). The following is a general outline of the process :- Let x be the most probable value of the error and u the reciprocal of the weight of any observed angle X, and let a, b, . . . n be the coefficients of x in successive geometrical equations of condition whose absolute terms are ea, ep, ... e,,; then we have the following group of n equations containing t unknown quantities to be satisfied, the significant coefficients of x being 1 in the triangular, totopartial, and central, and ± cot X in the side equations + a2x2 + . . + aext.e„ +b2x2+ . ..+ btx,=c1, (3).
next+ nex2 + . . . + ntx, = e„ whose values are obtained by the solution of the following equations :- [aa.u]X„ + [ab.u]X,, + . . . + [an. u]X„ = e“ [ab.u]X„+[bb.u]X„,+ . . . +[bn.u]X„=e, (4), • • ..... • • [a n.u]X„ + [b9t.u]X,, + . . . + [nn.u]X„ = e„ the brackets indicating summations of t terms as to left of (3). Then the value of any, the pth, x is = it2,21a.„X„+ b.„Xb + . . . + npX„1, (5).
The minimum or H is = [eX] (6).
In the application to a single triangle we have xl+x2+x3=e, X= c÷ (ui + 742+ up) ; xi=u,X ; x,=u2X ; x3= u,X.
In the application to a simple polygon, by changing symbols and putting X and Y for the exterior and Z for the central angles, with errors x, y, and z and weight reciprocals u, v, and w, a for cot X and b for cot Y, e for any triangular error, e, and e, for the central and side errors, X. and X, for the factors for the central and side equations, and IV for u + v + w, the equations for obtaining the factors become rw(ciu - bv)ix _e rue [5U- w-f-p21X, L [w(au - bv)1, . Ea, ,„ au - by], _ [ (au- We J (7), v A, + -1- 0 11 - w A, - c, w - v le- (b W + - bv)X,-Xw„) (S) z= IV - (au - bv)X,+ (u + v)X,) Calculation of Sides of Triangles. - The angles Sides of having been made geometrically consistent inter se in each triangles. figure, the side-lengths are computed from the base-line onwards by Legendre's theorem, each angle being diminished by one-third of the spherical excess of the triangle to which it appertains. The theorem is applicable without sensible error to triangles of a much larger size than any that are ever measured.
Problem.-Assuming the earth to be spheroidal, let A and B be two stations on its surface, and let the latitude and longitude of A be known, also the azimuth of B at A, and the distance between A and B at the mean sea-level ; we have to find the latitude and longitude of B and the azimuth of A at B.
The following symbols are employed :-a the major and b the minor semi-axis ; e the excentricity, _ as bs ; p the radius of - curvature to the meridian in latitude X, = a(1 - es) (1- essinsX) ; v the normal to the meridian in latitude X, = a „ X and L the given (1 - essinsX) latitude and longitude of A; X + AX and L + AL the required latitude and longitude of B ; A the azimuth of B at A ; B the azimuth of A at B; AA =B- (r+,4); c the distance between A and B. Then, all azimuths being measured from the south, we have (9), - -c cos A cosec 1" - c2 sinsA tan X cosec 1" z.an .-,. 3 c2 es +1 sin2A cos A(1 +3 taus X) cosec 1" ( 6 p.p2 w cos cosec 1" + 1 - e2 sin 2A tan X cosec 1" " 2 v2 cos X '"= 1 cs (1 + 3 tan2 X) sin 2A cosA(10), cased 1" - sin A tan X cosee 1" "cos2 AA' or 1+-1 cv-22 + 2 tan2 1- X + sin 2A cosec 1" X B + A)= cs v ( 5 - -3-6+t an2 N) tan X sin °A cos A cosec 1" + ? c3sin2A tan X (1 + 2 tan"' X) cosec 1" 6 Each A is the sum of four terms symbolized by 81, S., 83, and 8, ; the calculations are so arranged as to produce these terms 'n the order 8N, 8L, and 5A, each term entering as a factor in calculating the following term. The arrangement is shown below in equations in which the symbols P, Q,. . . Z represent the factors which depend on the adopted geodetic constants, and vary with the latitude ; the logarithms of their numerical values are tabulated in the Auxiliary Tables to Facilitate the Calculations of the Indian Survey. 53X - P.cos A.6 SiL = + diX.Q.sec X.tan A OIA = + L.sin X 3,N= + A.6 52L= - 52X.S. cotA 82./1=+52.L.7' (12).
53X = - 52,4. V.cotA 63L=+ 8,X. U. sinAx 53A= + 83L.JV 84A=- 53A.MtanA 84L= + 54x- Y. tanA 84A =+ 64L-Z By this artifice the calculations are rendered less laborious and made susceptible of being readily performed by any persons who are acquainted with the use of logarithm tables, Limits of 16. Limits within which Geodetic Porntilm may be emgeodetic ployed without Sensible Error. - Each A is expressed as a f"'"12. series of ascending differentials in which all terms above the third order are neglected ; for the side length c in no case exceeded 70 miles, nor was the latitude ever higher than 36°, and for these extreme values the maximum magnitudes of the fourth differential are only 0"•002 in latitude and 0"-004 in longitude and azimuth.
Far greater error may arise from uncertainties regarding the elements of the earth's figure, which was assumed to be spheroidal, with semi-axes a = 20,922,932 feet and b = 20,853,375 feet. The changes in M, AL, and AA which would arise from errors da and db in a and b are indicated by the following formulm : - d.AX = - AX.- Sea.e 6 (1 x( 2de ) 28 X.dv p P 3 (1 e2)c d.AL = - AL. - - 52E.-- (83L+54L)2dv - v The adopted values of the semi-axes were determined by Colonel Everest in an investigation of the figure of the earth from such data as were available in 1826. Forty years afterwards an investigation was made by Captain (now Colonel) A. B. Clarke with additional data, which gave new values, both exceeding the former.' Accepting these as exact, the errors of the first values are da= – 3130 feet and db= – 1746 feet, the former being 150, the latter 84 millionth parts of the semi-axis. The corresponding changes in arcs of 1° of latitude and longitude, expressed in seconds of arc and in millionth parts (is) of arc-length, are as follows : - In lat. 5° d.AX= -"-069 or 19 and d.AL= - ".540 or 150 At ; „ 15° „ -"'113„ 31„ „ „ ".554 „ 154 „ ; „ 25° „ -".195 „ 54„ „ „ -".581 „ 161 „ ; „ 35° „ -"'303„ 84„ „ „ -417 „ 171 „.
I See Account of the Principal Triangulation of the Ordnance Survey, 1858, and Comparisons of Standards of Length, 1866.
any points for which both may be available : they indicate the extent to which differences may be attributable to errors in the adopted geodetic constants, as distinct from errors in the trigonometrical or the astronomical operations.
the several trigonometrical figures consistent inter se, and principal r to give preliminary values of the lengths and azimuths of lotion. the sides and the latitudes and longitudes of the stations.
The results are amply sufficient for the requirements of the topographer and land surveyor, and they are published in preliminary charts, which give full numerical details of latitude, longitude, azimuth, and side-length, and of height also, for each portion of the triangulation - secondary as well as principal - as executed year by year. But on the completion of the several chains of triangles further reductions became necessary, to make the triangulation everywhere consistent inter se and with the verificatory baselines, so that the lengths and azimuths of common sides and the latitudes and longitudes of common stations should be identical at the junctions of chains, and that the measured and computed lengths of the base-lines should also be identical.
How this was done will now be set forth. But first it must be noted that the triangulation might at the same time have been made consistent with any values of latitude, longitude, or azimuth which had been determined by astronomical observations at either of the trigonometrical stations. This, however, was undesirable, because such observations are liable to errors from deflexion of the plumb-line from the true normal under the influence of local attraction, and these errors are of a much greater magnitude than those that would be generated in triangulating between astronomical stations which are not a great distance apart. The trigonometrical elements could not be forced into accordance with the astronomical without altering the angles by amounts much larger than their probable errors, and the results would be useless for investigations of the figure of the earth. The only independent facts of observation which could be legitimately combined with the angular adjustments were the base-lines, and all these were employed, while the several astronomical determinations - of latitude, differential longitude, and azimuth - were held in reserve for future geodetic investigations.
As an illustration of the problem for treatment, suppose a corn- Specific bination of three meridional and two longitudinal chains comprising illustraseventy-two single triangles, with a base-line at each corner, as shown lion sistent. Let A be the ori- A AVAST Awl measured and made congin, with its latitude and 115 longitude given, and also 11 the length and azimuth of 111" the adjoining base-line.
With these data processes 11,141 through the triangulation ■Av to obtain the lengths and azimuths of the sides ands A the latitudes and longitudes of the stations, say in the following order : - from A through B to E, through F to E, through F to D, through F and E to C, and through F and D to C. Then there are two values of side, azimuth, latitude, and longitude at E, - one from the right-hand chains via. B, the other from the left-hand chains via F ; similarly there are two sets of values at C; and each of the baselines at B, C, and D has a calculated as well as a measured value. Thus eleven absolute errors are presented for dispersion over the triangulation by the application of the most appropriate correction to each angle, and, as a preliminary to the determination of these corrections, equations must be constructed between each of the absolute errors and the unknown errors of the angles from which d.AA = AA.r/r, ;2A v dv dp\ t v P‘ P 2 tan2X + - (83A+ 84,4)2 - v d-2= - •000,000, 0478 (da - 2db - 3(da - db) sine X) they originated. For this purpose assume X to be the angle opposite the flank side of any triangle, and Y and Z the angles opposite the sides of continuation ; also let x, y, and a be the most probable values of the errors of the angles which will satisfy the given equations of condition. Then each equation may be expressed in the form [ax + by + cz]=E, the brackets indicating a summation for all the triangles involved. We have first to ascertain the values of the coefficients a, 5, and e of the unknown quantities. They are readily found for the side equations on the circuits and between the base-lines, for x does not enter them, but only p and z, with coefficients which are the cotangents of Y and Z, so that these equations are simply [cot Y.y - cot Z.z] =E. But three out of four of the circuit equations are geodetic, corresponding to the closing errors in latitude, longitude, and azimuth, and in them the coefficients are very complicated. They are obtained as follows. The first term of each of the three expressions for AX, AL, and B is differentiated in terms of c and A, giving d.AL = AL fc-c-ic + dA cot A sin 1" j- (15), in the course of the triangulation up to it from the base-line and the azicarried directly along one of the flanks, traverse, which may either run from vertex to vertex of the successive triangles, zigzagging between the flanks as in fig. 3 (2). For the general sym111111111stitute At,,, the side between stations a. and (1) AL„, AA„, c,,, A,,, and B„ : se -F 1 of the traverse ; and let Sc„ and Fig. 3. ( ) SA„ be the errors generated between the sides c„_1 and e„; then , _.1.=4; -=I + - de Se de, ac 8c2 dc,, .[Sc - p • • • - = - -1; el ci e2 c1 c2 c.„ 1 c dAi=3Ai; dA2=dBi+8,12; . . . (121,,=dB,,-1-1-aAn. Performing the necessary substitutions and summations, we get ("[AA]+IA11]8-ci .
'+ . + AA,,a-e= i ei 2 c2 C„ Thus we have the following expression for any geodetic error - & Sc„ Aq-j-11+ • • • + SA +• • • + ...(16), where is and q5 represent the respective summations which are the coefficients of Sc and SA in each instance but the first, in which 1 is added to the summation in forming the coefficient of SA.
The angular errors x, y, and z must now be introduced, in place of Sc and SA, into the general expression, which will then take different forms, according as the route adopted for the line of traverse was the zigzag or the direct. In the former, the number of stations on the traverse is ordinarily the same as the number of triangles, and, whether or no, a common numerical notation may be adopted for both the traverse stations and the collateral triangles ; thus the angular errors of every triangle enter the general expression in the form ±¢,x +cot Y.key - cot Z.g'z, in which if =k sin 1", and the upper sign of cp is taken if the triangle lies to the left, the lower if to the right, of the line of traverse. When the direct traverse is adopted, there are only half as many traverse stations as triangles, and therefore only half the number of /Zs and 95's to determine ; but it becomes necessary to adopt different numberings for the stations and the triangles, and the form of the coefficients of the angular errors alternates in successive triangles. Thus, if the pth triangle has no side on the line of the traverse but only an angle at the nth station, the form is + + cot l'",./.el.yp - cot Zp.p.'pzp.
If the qth triangle has a side between the nth and the (/ +1)th stations of the traverse, the form is cot X,(/21- (01+ lil,+, cot 172)y2 - (00.5 - fc cot Z9)z,.
As each circuit has a right-hand and a left-hand branch, the errors of the angles are finally arranged so as to present equations of the general form [ax + by + cz],.-[ax +by + cz]t= E.
The eleven circuit and base-line equations of condition having been duly constructed, the next step is to find values of the angular errors which will satisfy these equations, and be the most probable of any system of values that will do so, and at the same time will not disturb the existing harmony of the angles in each of the seventy-two triangles. Harmony is maintained by introducing the equation of condition x+y + = 0 for every triangle. The most probable results are obtained by the method of minimum squares, which may be applied in two ways.
A factor X may be obtained for each of the eighty-three equa,2 titans under the condition that [ - s2 + v -1 is made a minimum, at, v, and w being the reciprocals of the weights of the observed angles. This necessitates the simultaneous solution of eighty-three equations to obtain as many values of X. The resnIting values of the errors of the angles in any, the pth, triangle, are up[a.„X] ; y= vp[b,,X] ; = wp[cpX] (17).
One of the unknown quantities in every triangle, as x, may be eliminated from each of the eleven circuit and base-line equations by substituting its equivalent - (y+z) for it, a similar substitution being made in the minimum. Then the equations take the form [(b - a)y + (c - cz)z]= E, while the minimum becomes r(y+z)3±y2+0-1.
L t6 'V 212_1 Thus we have now to find only eleven values of X by a simultaneous solution of as many equations, instead of eighty-three values from eighty-three equations ; but we arrive at more complex expressions for the angular errors as follows ; - Yp =26, V Wp vP I (up + wp)[( bp ap) - top[(cp - ap)X] '1[AX]ali +"2[AXl2+ ... + AX„aedx.+1=( - ( i[AX tan A]SA, +:[AX tan 48-42+ .. .
+ AX,, tan A,,321,:) sin I."
l'ICALlaZ4 e,2 +:[AL]4e2+ • • • +AL'ae-L" c„ + i[AL cot A13,41+7AL cot A]3A2+ . . .
+ AL„ cot A „3A,,) sin 1" wP ((up+ vp)[(cp - ap)X] - vp[(bp - ap)Xrc up + vp + ;up The second method has invariably been adopted, originally be- Reduecause it was supposed that, the number of the factors X being re- tion of duced from the total number of equations to that of the circuit and principa base-line equations, a great saving of labour would be effected. But triaugnsubsequently it was ascertained that in this respect there is little latiou. to choose between the two methods ; for, when x is not eliminated, and as many factors are introduced as there are equations, the factors for the triangular equations may be readily eliminated at the outset. Then the really severe calculations will be restricted to the solution of the equations containing the factors for the circuit and base-line equations, as in the second method.
In the preceding illustration it is assumed that the base-lines are errorless as compared with the triangulation. Strictly speaking, however, as base-lines are fallible quantities, presumably of different weight, their errors should be introduced as unknown quantities of which the most probable values are to be determined in a simultaneous investigation of the errors of all the facts of observation, whether linear or angular. When they are connected together by so few triangles that their ratios may be deduced as accurately, or nearly so, from the triangulation as from the measured lengths, this ought to be done ; but, when the connecting triangles are so numerous that the direct ratios are of much greater weight than the trigonometrical, the errors of the base-lines may be neglected. In the reduction of the Indian triangulation it was decided, after examining the relative magnitudes of the probable errors of the linear and the angular measures and ratios, to assume the base-lines to be errorless (see § 19, p. 704 below).
The chains of triangles being largely composed of polygons or other networks, and not merely of single triangles, as has been assumed for simplicity in the illustration, the geometrical harmony to he maintained involved the introduction of a large number of " side," "central," and " toto-partial " equations of condition, as well as the triangular. Thus the problem for attack was the simultaneous solution of a number of equations of condition= that of all the geometrical conditions of every figure + four times the number of circuits formed by the chains of triangles+ the number of baselines -1, the number of unknown quantities contained in the equations being that of the whole of the observed angles ; the method of procedure, if rigorous, would be precisely similar to that already indicated for " harmonizing the angles of trigonometrical figures," of which it is merely an expansion from single figures to great groups.
The rigorous treatment would, however, have involved the simultaneous solution of about 4000 equations between 9230 unknown quantities, which was quite impracticable. The triangulation was therefore divided into sections for separate reduction, of which the most important were the five between the meridians of 67° and 92° (see fig. 1, p. 696), consisting of four quadrilateral figures and a trigon, each comprising several chains of triangles and some base-lines. This arrangement had the advantage of enabling the final reductions to be taken in hand as soon as convenient after the completion of any section, instead of being postponed until all were completed. It was subject, however, to the condition that the sections containing the best chains of triangles were to be first reduced ; for, as all chains bordering contiguous sections would necessarily be " fixed " as a part of the section first reduced, it was obvionsly desirable to run no risk of impairing the best chains by forcing them into adjustment with others of inferior quality. It happened that both the north-east and the south-west quadrilaterals contained several of the older chains ; their reduction was therefore made to follow that of the collateral sections containing the modern chains.
But the reduction of each of these great sections was in itself a very formidable undertaking, necessitating some departure from a purely rigorous treatment. For the chains were largely composed of polygonal networks and not of single triangles only as assumed in the illustration, and therefore cognizance had to be taken of a number of " side " and other geometrical equations of condition, which entered irregularly and caused great entanglement. Equations 17 and 18 of the illustration are of a simple form because they have a single geometrical- condition to maintain, the triangular, which is not only expressed by the simple and symmetrical equation +y + z. 0, but - what is of much greater importance - recurs in a regular order of sequence that materially facilitates the general solution. Thus, though the calculations must in all cases be very numerous and laborious, rules can be formulated under which they can be well controlled at every stage and eventually brought to a successful issue. The other geometrical conditions of networks arc expressed by equations which are not merely of a more complex form but have no regular order of sequence, for the networks present a variety of forms ; thus their introduction would cause much entanglement and complication, and greatly increase the labour of the calculations and the chances of failure. Wherever, therefore, any compound figure occurred, only so much of it as was required to form a chain of single triangles was employed. The figure having previously been made consistent, it was immaterial what part was employed, but the selection was usually made so as to introduce the fewest triangles. The triangulation for final simultaneous reduction was thus made to consist of chains of single triangles only ; but all the included angles were " fixed " simultaneously. The excluded angles of compound figures were subsequently harmonized with the fixed angles, which was readily done for each figure per se.
This departure from rigorous accuracy was not of material importance, for the angles of the compound figures excluded from the simultaneous reduction had already, in the course of the several independent figural adjustments, been made to exert their full influence on the included angles. The figural adjustments had, however, introduced new relations between the angles of different figures, . causing their weights to increase exteris paribus with the number of geometrical conditions satisfied in each instance. Thus, suppose it; to be the average weight of the t observed angles of any figure, and it the number of geometrical conditions presented for satisfaction ; then the average weight of the angles after adjustment may be taken as w.t- , the factor thus being 1.5 for a triangle, 1.8 for a hexagon, 2 for a quadrilateral, 2.5 for the network around the Sironj base-line, &c.
In framing the normal equations between the indeterminate factors X for the final simultaneous reduction, it would have greatly added to the labour of the subsequent calculations if a separate weight had been given to each angle, as was done in the primary figural reductions; this was obviously unnecessary, for theoretical requirements would now be amply satisfied by giving equal weights to all the angles of each independent figure. The mean weight that was finally adopted for the angles of each group was therefore taken as p being the modulus already indicated in section 12.
The second of the two processes for applying the method of minimum squares having been adopted, the values of the errors y and z of the angles appertaining to any, the pth, triangle were finally expressed by the following equations, which are derived from (18) by substituting it for the reciprocal final mean weight as above determined : - y, =11 [(21, - a, - cp)X] (19).
Zp= 2 -3P[(9.rp - a j,- b,)X] The most laborious part of the calculations was the construction and solution of the normal equations between the factors X. On this subject a few hints are desirable, because the labour involved is liable to be materially influenced by the order of sequence adopted in the construction. The normal equations invariably take the form of (4), the coefficients ou the diagonal containing summations of squares of the coefficients in the primary equations, while those above and below contain summations of products of the primary quantities, such that the coefficient of the pth X in the qth equation is the same as that of the qth X in the pth equation. In practice, as any single angular error only enters a few of the primary equations of condition, many of the coefficients vanish, both in the primary and in the normal equations ; and it is an object of great importance so to arrange the normal equations that most blanks shall occur above and fewest blanks between the significant values on each vertical line of coefficients ; in other words, the significant values above and below the diagonal should lie as closely as possible to the diagonal, every value on which is always significant. This advantage is secured when the primary equations are arranged in groups in which each contains a number of angular errors in common and as many as possible of those entering the group on each side. Thus the arrangement must follow the natural succession of the chains of triangles rather than the characteristics of the primary equations ; if, for example, all the side equations were grouped together, and all the latitude equations, and so on, great entanglement would arise in the solution of the normal equations, enormously increasing the labour and the chances of failure. The best arrangement was found to be to group the side and the' hree geodetic equations of each circuit together in the order of sequence of the meridional chains of triangles, and then to introduce the side equations connecting base-lines between the groups with which they had most in common.
The following table (II.) gives the number of equations of condition and unknown quantities - the angular errors - in the five great sections of the triangulation, which were respectively included in the simultaneous general reductions and relegated to the subsequent adjustments of each figure per se : - The magnitudes of t ie 2481 angular errors determined simu taneously in the first two sections were very small, 2240 being under 0'1, 205 between 0".1. and 0"•2, 33 between 0"•2 and 0"-3, 2 between 0"•3 and 0•4, and 1 between 0"•4 and 0"•5. In the third section, which contained a number of old chains, executed with instruments inferior to the 2 and 3 foot theodolites, they were larger: 780 were under 0"•1, 911 between 0".1 and 1"•0, 27 between 1"•0 and 2'0, and 1 between 2'0 and 2'1. Thus the corrections to the angles were generally very minute, rarely exceeding the theoretical probable errors of the angles, and therefore applicable without taking any liberties with the facts of observation.
by a purely algebraical process soon led to results of intolerable complexity, so that it was desirable to introduce numbers as soon as possible for every symbol except the absolute terms of the geometrical or primary equations of condition. But on continuing the algebraical process certain relations were found to exist between the coefficients of the indeterminate factors in the normal equations of the minimum square method and the coefficients of the unknown quantities in the primary equations of condition, which enormously simplified the process and led to a general algebraical expression of no great complexity ; it was also found that, the number of primary equations being n, the labour of calculation by the formula was reduced to an nth of that involved by resorting at once to numbers.
Let F be any function whatever of the corrected angles (X, – .r,), (X2–x,), . of a trigonometrical figure ; let Jt= dx; 2= d2r2- - • ; also let up 2c2, . ., symbols hitherto employed to represent the relative reciprocal weights of the observed angles X,,X2 in future represent absolute measures of precision, the p.e.2 of the observed angles ; then the following formula expresses the p.e. of any function of the corrected angles rigorously : - p.e.2 of F + [fa .n] Lfa. + Lib . ttlAb + . + [fn. it]A„) =f21t – + rib. nil[fa.u]B.+Efb.u1Bb+ +[,(21••11.]B,) (20).
+ Lin ad ([fa. ic]Na + Vb .2t] + + Vii.21j1V4 The symbols a, b, . . . lb have the same signification as in (3) to (6) of section 13. A, B, . • .11r are coefficients which must be determined in the process of solving the normal equations as follows : - x,==dce.+ 244+ • • • + A„e„ Xb = B.e.+Bbeb+ . . . + B.e„ (21), X.= N.c.+21,764+. . . + where the coefficient represented by any two letters in one order is identical with that represented by the same letters in the reverse order; thus A.= N.. Hence to find thep.e. of any angle, as (X, – in a single triangle we have A=1, and A.= fact . tej= u1-1- 1(2+ 1.43; all the other factors vanish, and p.e.2 of (X, – x,)=2t, 2c, ++–p.c.2 of X,– p.c.2 of x1.
To find the p.c. of the ratio B of either side to the base, - if B= sin (X,– x,)÷sin (X2–x2), then f2 =B cot X, sin 1", f2=B cot X2 sin 1", /2=0, and p.e.2 of B = B2 sin.„1" 21, cot2X, + u.2eot2X2 OtIcot X1– u2cot X2)2 (22).
2c, + u, +us When the function of the corrected angles is the ratio of the terminal to the initial side of an equilateral triangle or a regular quadrilateral or polygon (either of two sides being taken if the figure has an odd number of exterior sides), then, assuming all the angles to be of equal weight, we have the following values of the p.e.'s and the relative weights of the ratios :- Figure. p.e. Weight. Figure. p.e. Weight.
Triangle +.821/csin 1" 1.49 Pentagon ±l'21 \A sin 1" 0.68 Quadrilateral 1.00 „ 1'00 Hexagon 1.29 „ 0.60 Trigon 1.05 „ 0'90 Heptagon 1.41 „ 0.50 Tetragon 1.15 „ 0.75 Octagon 1.57 „ 0.41 In ordinary ground seven single triangles will span about as much as two hexagons and the weights of the terminal sides would be as twenty-one by the-former to thirty by the latter. In a flat country two quadrilaterals would not span more than one hexagon, giving terminal side weights as five to six ; but in hills a quadrilateral may span as much as any polygon and give a more exact side of continuation. Thus in the Indian Survey polygons predominate in the plains and quadrilaterals in the hills.
The theoretical errors of the lengths and azimuths of the sides, and of the latitudes and longitudes of the stations, at the termini of the chains of triangles or at the circuit closings, might be calculated with the coefficients a, b, and c of a", y, and z in the circuit and base-line equations as the f's, and the known p.e.'s of X, Y, and Z and the other data of the figural reductions. Such calculations are, however, much too laborious to be ordinarily undertaken. Thus the exactitude of a triangulation is very generally estimated merely on the evidence of the magnitudes of the differences between the trigonometrical and the measured lengths of the base-lines ; for, though the combined influence of angular precision and geometrical configuration is what really governs the precision of the results, it is not readily ascertainable, and is therefore generally ignored. But, when questions as to the intrinsic value of a triangulation arise, the theory of errors should always be appealed to, and its intimations accepted rather than the evidence of base-line discrepancies, which if very small are certainly accidental, and if seemingly large may be no greater than what we should be prepared to expect. Good work has occasionally been redone unnecessarily, and inferior work upheld, because their merits were erroneously estimated. The following formulae will be found useful in acquiring a fairly approximate knowledge of the magnitude of the errors which theory would lead us to expect, not only in side, but in latitude, longitude, and azimuth also, at the close of any chain of triangles. They indicate rigorously the p.e.'s at the terminal end of a chain of equilateral triangles of which all the angles have been measured and corrected and are of equal weight ; the results may be made to serve for less symmetrical chains, including networks of varying weight, by the application of certain factors which can be estimated with fair precision in each instance.
Let e be the side length, c the p.c. of the angles, n the number of triangles, and the ratio (here = 1) of the terminal to the initial side, then p.e. of B=e shin? In p.c. of azimuth = 0/5 (23)1 p.e. of either coordinate =ee --6--v2n3+3n2+102t When the form of the triangles deviates much from the equiangular, the p.e. must be multiplied by a factor increasing up to 1.4 as the angles diminish from 60° to 30°, and a mean value of c must be adopted. When the chain is double throughout, the p.c. must be diminished by a factor taking cognizance of the greater weight of compound figures than of single triangles. Whets the chain is composed of groups of angles measured with different instruments, a separate value of e must be employed for each group, and the final result obtained from(INFT.,- The p.c. of B may be determined rigorously for any chain of single triangles, with angles of varying magnitudes and weights, by (22), with little labour of calculation.
the triangulation ; being fallible quantities, their errors triangumust be included among the unknown quantities to be in- lation. vestigated simultaneously, if their respective p.e.'s differ sensibly, or if the p.e.'s of their ratios are not materially smaller than those of the corresponding trigonometrical ratios. By (23) the p.e. of the ratio of any two sides of an equilateral triangle is e sin 1" s/2 3 ; but the p.e. of the ratio of two base-lines of equal length and weight is n where n is the p.e. of either base-line ; thus weight of trigonometrical ratio : weight of base-line ratio : 32/2 :E2 sins 1", or as 3 : 1 when e = ± 0'3 and n = + 1.5 millionth parts, which happens generally in the Indian triangulation. But the chains between base-lines were always composed of a large number of triangles, and the average weight of the base-line ratios was about eleven times greater by the direct linear measurements than by the triangulation, even when all the unascertainable constant or accidental errors - as from displacements of mark-stones - which might be latent in the latter were disregarded. Moreover, the baselines were practically all of the same precision ; they were therefore treated as errorless, and the triangulation was made accordant with them.
If a base-line AD be divided at /3 and C into three equal sections connected together by equilateral triangles, and every angle has been measured with a p.e. = e, the p.e. of any trigonometrical ratio may be put = IC.E sin 1", IC being a coefficient which has two values for each ratio, - the greater value when the triangulation has been carried along one flank of the line, the smaller when along both Tanks, as follows : - for ratio K = 1.41 and 1; for Ay) , 1.83 and 1.23; for A-1-3, 2.94 and 1.99; for -130- , 2-16 and