George Davidson 1825-1911 Clantitr0n Series GEODESY CLARKE Hontron MACMILLAN AND CO. PUBLISHERS TO THE UNIVERSITY OF xforfc Clartniroti GEODESY BY COLONEL A. R. CLARKE, C.B. EOTAL ENGINEERS; F.B.S. ; HON. F.C.P.S. CORRESPONDING MEMBEB OF THE IMPERIAL ACADEMY OF SCIENCES OF ST. PETERSBURG xfotb AT THE CLAEENDON PKESS MDCCCLXXX [All rights reserved] ^ ct PREFACE THE Essay entitled ' Figure of the Earth/ by Sir G. B. Airy, in the Encyclopedia Metropolitana, is the only adequate treatise on Geodetic Surveys which has been published in the English language, and though now scarce, it will ever remain valuable both on account of the historic research it contains, and the simple and lucid exposition of the me- chanical theory there given. Since the date of its publication however have appeared many important volumes, scientific, descriptive, official, such as Bessel's Gradmessung in Ostpreussen ; Colonel Everest's Account (1847) of his Great Arc; Struve's two splendid volumes descriptive of the trigono- metrical chain connecting the Black Sea with the North Cape ; the Account of the Triangulation of the British Isles ; the Publications of the Inter- national Geodetic Association ; recent volumes of the Memorial du Depot General de la Guerre ; the Yearly Reports of the United States Coast and Geodetic Survey; the current volumes by General Ibanez, descriptive of the Spanish Triangulation, so remarkable for precision ; and last, though not least, the five volumes recently published by General Walker, containing the details of Indian Geodesy. Vi PREFACE. The subject has thus of late years become a very large one, and although the present work does not go much into details, it is hoped it will to some extent fill a blank in our scientific literature. The Astro- nomical aspect of the science is but lightly touched on for in this matter books are not wanting we have for instance the works of Briinnow and Chau- venet, the last of which contains almost everything that can be required. The once generally accepted ratio 298 : 299 of the earth's axes may be said to have disappeared finally on the publication (in 1858) of the investigation of the Figure of the Earth in the Account of the Tri- angulation of the British Isles, when it was replaced by 293 :294. At the same time that this ratio is, in the present volume, still further altered in the same direction, the formerly received value of the ratio as deduced from pendulum observations is now altered from something like 288 : 289 up to the same figures as now represent meridian measurements, namely, about 292:293. Thus, the disagreeable hiatus long supposed to exist between the result of actual meridian measure- ments and that deduced by Clairaut's Theorem from the actually observed variations of gravity on the surface of the earth, has now disappeared thanks to the energetic labours of General Walker and his efficient staff of Officers. A. B. CLARKE. CONTENTS. CHAPTER I. Geodetical Operations. Early Geodetic observers, Snellius, Picard, Cassini, Godin, Bouguer, and de la Condamine. Maupertuis' Swedish measurements. Labours of Bouguer and de la Condamine in Peru. The Toise of Peru. Connection of France and England by triangulation Ramsden's theodolite. Measure of base line on Hounslow Heath. The repeating circle. The French Meridian Chain by Delambre and Mechain. Borda's Rods. Determination of the length of the legal metre. Bessel's degree-measure in East Prussia, Colonel Everest's Indian Arc. Attraction of the range of ^mountains situated at the middle point of the arc. Indian chain of triangnlation continued by General Walker. British triangulation. Struve's Russian Arc of 25 20'. Sir Thomas Maclear's revision of Lacaille's Arc at the Cape of Good Hope CHAPTER II. Spherical Trigonometry. Fundamental equations of plane and spherical trigonometry. Corre- sponding variations of sides and angles. Right-angled triangles. Legendre's Theorem, and its errors. Spherical co-ordinates. Expan- sion of (i + 2cos0 + n a )~ 8 ......... 37 CHAPTER HI. Least Squares. Method of least squares. Law of facility of errors. Mean and pro- bable errors. Solution of systems of equations, when the number of equations is greater than the number of unknown quantities. Probable errors of the results. Numerical examples. Treatment of conditioned observations. Probable error of a function of the corrected angles of a triangle .......... 52 Vlll CONTENTS. CHAPTER IV. Theory of the Figure of the Earth. PA GK The potentials of confocal ellipsoids at an external point are as their masses. Expressions for the potential of a nearly spherical ellipsoid at an internal and at an external point. Potential of a spheroid formed of homogeneous spheroidal shells, whose density and ellipticity vary with their distance from the centre. Jacobi's theorem on the equilibrium of a rotating fluid ellipsoid. Equilibrium of a homogeneous rotating fluid spheroid. Clairaut's differential equation connecting p and e. Clair- aut's Theorem. Law of ellipticity of the strata resulting from Laplace's law of density. Comparison of resulting ellipticity of surface with facts. Irregularities of the earth's surface. Disturbance of sea-level caused by irregular masses and mountain chains. Theories of Sir G. B. Airy and Archdeacon Pratt as to the attraction of mountains. Effect of altitude on an observed latitude 66 CHAPTER V. Distances, Azimuths, and Triangles on a Spheroid. Principal radii of curvature on a spheroid. Mutual azimuths and zenith distances of two points on a spheroid. Elements of a spheroidal triangle. Of the various curves that may be taken as the sides of a spheroidal triangle. The curve of alignment. Curvature of surface. Theory of spheroidal triangles. Errors resulting from treating them as spherical . 102 CHAPTER VI. Geodetic Lines. Characteristic of a Geodetic line. Bessel's auxiliary spherical triangle. Path of the Geodetic line between two points referred to a plane curve joining those points. Azimuths of the Geodetic at its extremities. Length of the Geodetic. Numerical examples. Geodetic joining Bor- deaux and Nicolaeff. Geodetic distance of Strasburg and Dunkirk . 124 CHAPTER VII. Measurement of Base-Lines. Standards of length of different countries. Thermometers and their errors. Flexure. Coefficients of expansion. Personal errors in observ- ing. Values of various Geodetic standards, expressed in terms of the CONTENTS. IX PAGE Standard Yard of England. Length of the Metre. Struve's base ap- paratus. Bessel's. Colby's compensation apparatus. The United States base-apparatus. Apparatus of M. Porro. Base of Madridejos. Verifica- tion of the base by triangulation 1 46 CHAPTER VIII. Instruments and Observing. Ramsden's theodolites. Twenty -four inch theodolites of Troughton and Siniins. Theodolites used in India, in Russia, and in Spain. Mode of observing. Scaffoldings. Airy's zenith sector. The zenith telescope. Portable transit -instrument ; flexure ; reduction of observations. Transit in the vertical of Polaris. Transit in the prime vertical. Observations for latitude. For azimuth. Determination of differences of longitude by the American method. Personal equation. Transatlantic longitudes. Indian longitudes. Longitude of Algiers 1 74 CHAPTER IX. Calculation of Triangulation. Calculation of a polygon formed by several triangles with a common vertex. Numerical example, with probable errors of results. Calcula- tion of a chain of triangles, with one or two bases. Probable error of an observed angle. Geometrical equations of condition of a network of triangles. Bessel's method of reduction by least squares. Adaptation of this method to the British triangulation. Probable errors of results. Bases. Numerical example of calculation of a small network. Junction of triangulations of England and France. Junction of triangulations of Spain and Algiers. Method of treatment of crossing or closing chains as in the Indian Survey 217 CHAPTER X. Calculation of Latitudes and Longitudes. Having given the latitude of A, the azimuth there of B, with the dis- tance AB, to determine the latitude and longitude of B and the direction of the meridian there. Numerical examples. Distance of parallels of A and B. Differentiation of such results with reference to the elements of the spheroid supposed variable. Length of the parallel of 52 between the meridians of Valencia in Ireland and Mount Kemmel, in Belgium, on an indeterminate spheroid . .267 X CONTENTS. CHAPTER XL Heights of Stations. Terrestrial refraction. Coefficient of refraction Value obtained from observations in this country. Difference of height obtained from observed zenith distances. Eeduction of involved observations by least squares. Distance of the sea horizon. Condition of the mutual visibility of two stations . 28 CHAPTER XII. Connection of Geodetic and Astronomical Operations. Effect of irregularities of the earth's surface on an observed latitude, an observed longitude, and an observed azimuth The effect of the same on the observed angles of triangles. Determination of an ellipsoid which satisfies most nearly the astronomical observations made in a triangulation. Ellipsoid representing the surface of Great Britain and Ireland, and the residual errors. Apparent errors to be expected from the form of the ground around an astronomical station. Attraction caused by mountain ranges and table-lands. Effect of mountain attrac- tion on the operations of spirit -levelling. Mean density of the earth . 287 CHAPTER XIII. Figure of the Earth. Determination of the figure of the earth by Airy. By Bessel. By Clarke. Determination of an ellipsoidal figure, and position of the axes. Influence of the different arcs in the determinations of the semi-axes of the spheroid most nearly representing the earth. Introduction of the Indian longitudes into the problem. Final equations for the determina- tion of the spheroid. Values of the semi -axes and residual corrections to latitudes. Comparison of the figures of the individual arcs with that of the spheroid. Length of a degree of latitude and of longitude. True length of the ideal metre . . 302 CHAPTER XIV. Pendulums. Of the earlier pendulum observers, Picard, Richer, Bouguer. Bouguer's reduction of his observations. His formula for mountain attraction as influencing the movement of the pendulum. Maupertuis' observations in CONTEXTS. XI PACK Finland. Borda's apparatus. Kater's reversible or convertible pendu- lum for determining the absolute length of the seconds pendulum. His invariable pendulum Observation of coincidences. Reduction of re- sults to an infinitely small arc of vibration. Bessel's investigations. Repsold's form of pendulums. Observations of Sabine and Foster Rus- sian pendulum observations. Indian series by Basevi and Heaviside, with two invariable pendulums. Corrections for height of station, and the attraction of the underlying strata. Summary of observed vibration numbers. Determination of the ellipticity 323 NOTES AND ADDITIONS 351 ERRATA. Page 36, line 5, for du read der; c.1. T-> y itetv^ ?> r>o? j* j j ^ ** * 41, 12, cos| w cos |a; 225, in the figure, for Jc read fc ; 239, line 11 (from bottom), for ' corrections ' read ' errors of observation.' * %^"^ CHAPTER I. GEODETICAL OPERATIONS. OF the many discoveries made in modern times by men of science astronomers and travellers none have ever tended to shake the doctrine held and taught by the philosophers of ancient times that the earth is spherical. That the surface of the sea is convex anyone may assure himself by simply observ- ing, say with a telescope from the top of a cliff near the sea, the appearance of a ship on or near the horizon, and then repeating a few moments after at the foot of the cliff the same observation on the same ship. Assuming the earth to be a sphere, a single observation of a more precise nature taken at the top of the cliff would give a value of the radius of the sphere. The observation required is the dip or angle of depression of the horizon : this, combined with one linear measure, namely, the height of the cliff, will suffice for a rough approximation. This is an experiment that was made at Mount Edgecombe more than two centuries ago, and may have possibly been tried in other places. The depression of the sea horizon at the top of Ben Nevis is 1 4' 48"; this is the mean of eight observations taken with special precautions for the very purpose of this experimental calculation ; the height of the hill is 4406 feet. Now let x be the radius of the earth, h the height of the hill, the tangent drawn from the observer's eye to the horizon subtends at the centre of the earth an angle equal to the depression ; call this angle 8, then the length of the tangent is x tan 8. The square of this is equal to /i(2x + h), or with sufficient accuracy for our pur- pose to 2xh, hence x = 2^ cot 2 8. But this formula is not practically true, as the path of the ray of light passing from the horizon to the eye of the observer is not a straight line, 7/ 2 GEODETICAL OPEKATIONS. but a curved one. But the laws of terrestrial refraction have been carefully studied, and we know that the value just written down for x should be multiplied by a certain con- stant : that is to say, the true equation is # = 1'6866^ cot 2 8. This numerical co-efficient, obtained from a vast number of observations, is to be considered as representing- a phenomenon of variable and uncertain amount. On substituting the values of h and 8 we obtain for the radius expressed in miles x = 3960. Now this is really very near the truth ; but, except for the precaution of having made the observations at the proper hour of the day, the error might have been a hundred miles : in fact the method, though it serves for getting the size of the earth in round numbers, is totally inadequate for scien- tific purposes. Amongst the early attempts to determine the radius of the earth, that of Snellius in Holland is remarkable as being the first in which the principle of measurement by triangulation was adopted. The account of this degree measure was pub- lished at Leyden in 1617. Haifa century later, in France, Picard conceived the happy idea of adapting a telescope with cross wires in its focus to his angle measuring instruments. Armed with this greatly improved means of working, he executed a triangulation extending from Malvoisine, near Paris, to Amiens. From this arc, whose amplitude, deter- mined with a sector of 10 feet radius, was 1 22' 55", he deduced for the length of a degree 57060 toises. The accuracy of this result however was subsequently found to be due to a compensation of errors. One of the most important results of this measurement of Picard's was that it enabled Sir Isaac Newton to establish finally his doctrine of gravitation as published in the Prin- cipia (1687). In this work Newton proved that the earth must be an oblate spheroid, and, moreover, that gravity must be less at the equator than at the poles. Of this last pro- position actual evidence had been obtained (1672) by the French astronomer, Richer, in the Island of Cayenne in South America, where he had been sent to make astronomical observa- tions and to determine the length of the seconds' pendulum. Having observed that his clock there lost more than two GEODETICAL OPERATIONS. 3 minutes a day as compared with its rate at Paris, he fitted up a simple pendulum to vibrate seconds, and kept it under observation for ten months. On his return to Paris he found the length of this seconds' pendulum to be less than that of the seconds' pendulum of Paris by li line. This very im- portant fact was fully confirmed shortly after by observations made at other places by Dr. Halley, MM. Varin and Des Hayes, and others. Picard's triangulation was extended, between 1684 and 1718, by J. and D. Cassini, who carried it southwards as far as Collioure, and northwards to Dunkirk, measuring a base at either end. From the northern portion of the arc, which had an amplitude of 2 12', they obtained 56960 toises as the length of a degree, while the southern portion, 6 19' in extent, gave 57097 toises. The immediate inference drawn by Cassini from this measure was that the earth is a prolate spheroid. A subsequent measurement by Cassini de Thuri, and Lacaille, of this same arc, proved the foregoing results to have been erroneous, and that the degrees in fact increase, 'not decrease, in going northwards (Meridienne verif.ee en 1744). Nevertheless the statement, on so great an authority as that of Cassini, that the earth is a prolate, not an oblate, spheroid, as maintained by Newton, Huygens, and others, found at the time many adherents, and on the question of the figure of the earth the scientific world was divided into hostile camps. The French, however, still maintained the lead in geodetical science, and the Academy of Sciences resolved to submit the matter to a crucial test by the measurement of an arc at the equator and another at the polar circle. Accordingly, in May, 17 35, the French Academicians, MM. Godin, Bouguer, and de la Condamine, proceeded to Peru, where, assisted by two Spanish officers, after several years of laborious exertions, they succeeded in measuring an arc of 3 7', intersected by the equator. The second party consisted of Maupertuis, Clairaut, Camus, Le Monnier, the Abbe Outhier, and Celsius, Professor of Astronomy at Upsal : these were to measure an arc of the meridian in Lapland. It is not our intention to write a history of the geodetical operations which have been carried out at various times and B 2 GEODETIC AL OPERATIONS. Circle places ; we shall, however, give a somewhat detailed account of the measurement in Lapland, partly because it was the one which first proved the earth to be an oblate spheroid, and also because it will at the same time serve the purpose of present- ing a general outline of the method of conducting a geodetic survey. The party of Maupertuis landed at the town of Tornea, which is at the mouth of the river of the same name at the northern extremity of the gulf of Bothnia, in the beginning of July, 1736. Having explored the river and found that its course was nearly North and South, and that there were high mountains on every side, they determined to estab- lish their stations on these heights. The points selected are shown in the accompany- ing diagram, together with the course of the river Tornea. Taking the church of the town of Tornea as AA\ ^K ^ e sou thern extremity of \\lv<> the arc, the points were se- lected in the order Niwa, N ; Avasaxa, A ; Horrila- kero, H ; Kakama, K ; Cui- taperi, C ; Pullingi, P ; Kit- tis, Q ; Niemi, N ; the north end of the base JB ; and the south end of the base B. The signals they constructed 3 ^ on the hill tops which had Fig. i. first to be cleared of timber were hollow cones com- posed of many large trees stripped of their bark and thus GEODETICAL OPERATIONS. 5 left white so as to be visible at ten or twelve leagues' distance. They took the precaution to cut marks upon the rocks, or drive stakes into the ground, so as to indicate precisely the centres of their stations, which could thus be recovered in case of any accident to the signal. Accurate descriptions of the stations are given in Outhier's work, entitled, -Journal cTun Voyage au Nord en 173637. The arrangement of the stations in this triangulation, a heptagon in outline, having the base line at the middle of its length, is certainly very good, and they regarded it on its completion with pardonable satisfaction, remarking that it looked as if the placing of the mountains had been at their disposal. The angles were measured with a quadrant of two feet radius fitted with a micrometer. With respect to the accuracy of this instrument it is stated that they verified it a great many times round the horizon and always found that it gave the sum of the angles very nearly equal to 360. In making the actual observa- tions for the angles of the triangles they took care to place ,the instrument so that its centre corresponded with the centre of the station. Each observer made his own observation of the angles and wrote them down apart, they then took the means of these observations for each angle : the actual read- ings are not given, but the mean is. The three angles of every triangle were always observed, and, by way of check, several supernumerary angles sums or differences of the necessary angles at any station were also observed. The measurement of the angles was completed in sixty- three days, and on September the 9th they arrived at Kittis and commenced to prepare the station for astronomical work. Two observatories were built ; in one was a small transit instrument, having a telescope fifteen inches in length, placed precisely over the centre of the station, and a clock made by Graham. The second observatory, close by, contained the ze- nith sector, also made by Graham ; the zenith sector was thus not over the centre of the trigonometrical station, but measure- ments were taken whereby the observations could be reduced to the trigonometrical station. The clock was regulated every day by corresponding altitudes of the sun. The astronomical observations to be made included a determination of absolute 6 GEODETICAL OPERATIONS. azimuth, and this was effected by observing with the small telescope the times of transit of the sun over the vertical of Niemi in the south-east in the forenoon and over the ver- tical of Pullingi in the south-west in the afternoon. These observations were made on eight days, between September 30th and October 8th. The reduction of such observations requires the solution of a spherical triangle whose angular points correspond to the zenith, the pole, and the place of the sun ; then are given the colatitude, the sun's north polar distance, and the hour angle of the sun that is, the angle at the pole and the two adjacent sides are given, and from these is to be calculated the angle at the zenith, which is the required azimuth of the sun at the noted time of obser- vation. The zenith sector consisted of a brass telescope nine feet in length, forming the radius of an arc of 5 30', divided into spaces of 7' 30". The telescope, the centre to which the plumbline was hung, and the divided limb were all in one piece ; the whole being suspended by two cylindrical pivots, which allowed it to swing like a pendulum in the plane of the meridian. One of these pivots ending in a very small cylinder at the exact centre of the divided limb and in its plane formed the suspension axis of the plumbline. The divided limb had a sliding contact with a fixed arc below, and this arc carried a micrometer against the pivot of which the limb of the sector was kept pressed by the tension of a thread. This micrometer screw, by communicating to the telescope and limb a slow movement in the plane of the meridian, served to subdivide the spaces of 1' 30". The in- strument was not used to determine absolute zenith distances, but differences of zenith distance only. The observations of 8 Draconis, which passed close to the zenith, were commenced at Kittis on the 4th of October and concluded on the 10th. Leaving Kittis on the 23rd, they arrived at Tornea on the 28th, and commenced the observations of 8 Draconis on the 1st of November, finishing on the 5th. The observations of the star at both stations were made by daylight without artificially illuminating the wires of the telescope. The difference of the zenith distances, corrected for aberration, GEODETICAL OPERATIONS. 7 precession, and nutation, gave the amplitude of the arc 57'26".93. It remained now to measure the base line, and this had been purposely deferred till the winter. The extremities of the base had been selected so that the line lay upon the surface of the river Tornea, which, when frozen, presented a favourable surface for measurement. They had brought with them from France a standard toise (known afterwards as the Toise of the North), which had been adjusted together with a second toise, namely, that taken to Peru for the equatorial arc to the true length at the temperature of 14 Reaumur. By means of this they constructed, in a room heated artifi- cially to the temperature just mentioned, five wooden toises, the extremities of each rod being terminated in an iron stud, which they filed down until the precise length of the toise was attained. Having driven two stout nails into the walls of their rooms at a distance a trifle less than five toises apart the five toises, placed upon trestles, were ranged in horizontal line in mutual contact between these nails, which were then filed away until the five toises just fitted the space between them. Thus the distance between the prepared surfaces of the nails became a five toise standard. By means of this stan- dard they constructed for the actual measurement eight rods of fir, each five toises (about 32 feet) long, and terminated in metal studs for contact. Many experiments were made to determine the expansions of the rods by change of tempera- ture, but the result arrived at was that the amount was inappreciable. The measuring of the base was commenced on December 21st, a very remarkable day, as Maupertuis observes, for com- mencing such an enterprise. At that season the sun but just showed himself above the horizon towards noon ; but the long twilight, the whiteness of the snow, and the meteors that continually blazed in the sky furnished light enough for four or five hours' work every day. Dividing themselves into two parties, each party took four rods, and two independent measurements of the line were thus made. This occupied seven days : each party measured every day the same number of toises, and the final difference between the two measurements 8 GEODETIC AL OPEKATIONS. was four inches, on a distance of 8*9 miles. It is not stated how the rods were supported or levelled probably they were merely laid in contact on the surface of the snow. It was now an easy matter to get the length of the ter- restrial arc. Calculating the triangles as plane triangles they obtained the distance between the astronomical observatories at Kittis and Tornea, and also the distance of Tornea from the meridian of Kittis. The length of this last enabled them to reduce the direct distance to the distance of the parallels of their terminal stations. The calculation of the distance was checked in various ways by the use of the supernumerary angles. The distance of parallels adopted was 55023-5 toises, which gave them, in connection with the observed amplitude, the length of one degree at the polar circle. The absolute latitude of Tornea, as obtained from observa- tions, made with two different quadrants on Polaris, was 65 50' 50", a result which did not however pretend to much precision. The value they had obtained for the degree being much in excess of that at Paris showed decisively that the earth was an oblate and not a prolate spheroid. So great however was the difference of the two degrees that they resolved to submit the whole process to a most rigorous examination. It was concluded that the base line could not possibly be in error, considering the two independent measures : nor could the angles of the triangles, each of which had been observed so often and by so many persons, be conceived to be in error. They determined however to re-observe the astronomical amplitude, using another star, and also to observe the absolute azimuth at Tornea. The maker of the zenith sector, Graham, had pointed out that the arc of 5 30' was too small by 3"- 75 : this they de- termined to verify for themselves during the winter at Tornea. The sector being placed in a horizontal position, two marks were fixed on the ice, forming with the centre of the sector a right-angled triangle. The distances, very carefully measured, were such that the angle of the triangle at the centre of the instrument was precisely 5 29' 50" '0. The angle as observed with the instrument (and here there is a curious misprint in GEODETICAL OPERATIONS. 9 Maupertuis's book) was 5 2 9' 5 2"- 7: this was a satisfactory- check on Graham's 3"-75. The 15' spaces were all subse- quently measured with the micrometer, and also those two particular spaces of one degree each on which the amplitudes depend were compared. The star selected for the second de- termination of the amplitude was a Draconis which passed only one quarter of a degree south of Tornea. The observa- tions at Tornea were made on March 17th, 18th, 19th, and at Kittis on the 4th, 5th, 6th of April. The resulting amplitude was 57'30"-42. The azimuth at Tornea was obtained on May 24th by an observation of the horizontal angle between the setting sun, at a known moment of time, and the signal at Niwa. Again the following morning the sun was at that time of the year only about four hours between setting and rising the angle was observed, at a given moment, between the rising sun and the signal of Kakama. Thus, by an easy calculation, the azimuths of these two stations were obtained. The result differed about 34" from the azimuth as calculated from the observations that had been made at Kittis. This difference in the azimuth would not make any material difference in the calculated length of the arc ; and of the difference of 3"- 4 9 between the two determinations of ampli- tudes, one second was due to the difference of the two degrees of the sector used respectively with a and with 8 Draconis. Thus, the whole operations were concluded with the result that the length of the degree of the meridian which cuts the polar circle is 57437-9 toises. Notwithstanding the appearance of a considerable amount of accuracy in Maupertuis's arc-measurement, yet there is a notable discordance between his terrestrial and astronomical work, as if either his arc were 200 toises too long, or his amplitude twelve seconds or so too small. In order to clear up this point, an expedition was organized and despatched from Stockholm in 1801, and the arc was remeasured and extended in that and the two following years by Svanberg. The ac- count of this measurement was published in the work entitled Exposition des Operations faites en Lapponie, fyc. par J. Svan- berg, Stockholm, 1805. Svanberg succeeded fairly, though 10 GEODETICAL OPERATIONS. not perfectly, in refinding the stations of Maupertuis, and verifies his terrestrial measurement : but taking for his own terminal points two new stations not in Maupertuis's are, the amplitude obtained by the latter was not verified. The length of the degree which Svanberg obtained was about 220 toises less than that of Maupertuis. The valley in which Quito is situated is formed by the double chain of mountains into which the grand Cordillera of the Andes is there divided, and which extends in a nearly south direction to Cuenca, a distance of some three degrees. This was the ground selected by MM. Godin, Bouguer, and de la Condamiiie as the theatre of their operations. These moun- tains, which, from their excessive altitude, were a source of end- less fatigue and labour, offered however considerable facilities for the selection of trigonometrical stations which, taken alternately on the one side of the valley and on the other, regulated the lengths of the sides and enabled the observers to form unexceptionally well-shaped triangles. The chain of triangles was terminated at either end by a measured base line. The northern base near Quito had a length of 7-6 miles : the altitude of the northern end was 7850 feet above the level of the sea. This indeed is the lowest point in the work, seven of the signals being at elevations ex- ceeding 14,000 feet. The accompanying diagram shows the northern tri- angles of the arc, extending as far south as Cotopaxi. The southern base was about 1000 feet above the northern, and had a length of 6-4 miles : it occupied ten days (August, 1739) in the measurement, while the northern, on rougher ground, took five-and-twenty (October, 1736). The measuring Cochesqui Obs y . Pichincha GEODETIC AL OPERATIONS. 11 rods used in the base measurement were twenty feet in length terminated at either end in copper plates for contact. Each measurement was executed in duplicate : the whole party being divided into two companies, which measured the line in opposite directions. The rods were always laid horizontally, change of level being effected by a plummet suspended by a hair or fine thread of aloe. The rods were compared daily during the measurement with a toise marked on an iron bar and which was kept duly shaded in a tent. This working standard, so to call it, had been laid off from the standard toise which they had brought from Paris. De la Condamine thus refers to his standard, which, known as the Toise of Peru, subsequently became the legal standard of France : 'Nous avions emporte avec nous en 1735 une regie de fer poli de dix-sept lignes de largeur sur quatre lignes et demie d'epaisseur. M. Godin aide d'un artiste habile avoit mis toute son attention a ajuster la longueur de cette regie sur celle de la toise etalon, qui a ete fixee en 1668 au pied de 1'escalier du grand Chatelet de Paris. Je previs que cet ancien etalon, fait assez grossierement, et d'ailleurs expose aux chocs, aux injures de 1'air, a la rouille, au contact de toutes les mesures qui y sont presentees, et a la malignite de tout mal-intentionne, ne seroit guere propre a verifier dans la suite la toise qui alloit servir a la mesure de la terre, et devenir Foriginal auquel les autres devoient etre comparees. II me parut done tres necessaire, en emportant une toise bien verifiee d'en laisser a Paris une autre de meme matiere et de meme forme a laquelle on put avoir recoups s'il arrivoit quelqu'accident a la notre pendant un si long voyage. Je me chargeai d'office du soin d'en faire faire une toute pareille. Cette seconde toise fut construite par le meme ouvrier, et avec les memes precautions que la pre- miere. Les deux toises furent comparees ensemble dans une de nos assemblies, et 1'une des deux resta en depot a 1'Acade- mie : c'est la meme qui a ete depuis portee en Lapponie par M. de Maupertuis, et qui a ete employee a toutes les opera- tions des Acade'mieiens envoyes au cercle Polaire/ Both the bases were measured at a mean temperature very nearly 1 3 Reaumur : ' C'est precisement celui que le thermometre de M. de Reaumur marquoit a Paris en 1735, lorsque notre toise 12 GEODETICAL OPERATIONS. de fer fut etalonee sur celle du Chatelet par M. Godin.' (Mesure des trois premiers Degres du Meridien par M. de la Condamine, Paris, 1751, pp. 75, 85.) The difference between the two measures of the base in either case is said not to have exceeded three inches. The quadrants, of from two to three feet radius, with which the angles of the triangles were observed were very faulty, and much time was spent in determining their errors of division and eccentricity. M. de la Condamine obtained a system of corrections for every degree of his instrument, and in only four of the thirty- three triangles as observed by him does the error of the sum of the observed angles amount to 10"; that is, after being corrected for instrumental errors. All the three angles of every triangle were observed, and each angle by more than one observer. The azimuthal direction of the chain of triangles was de- termined from some twenty observations of the sun at various stations along the chain. The determination of the latitudes cost them some years of labour. Their sectors of twelve and eight feet radius were found very defective, and they were virtually reconstructed on the spot. A vast number of observations were rejected, and the amplitude was finally adopted from simultaneous observa- tions of e Orionis made by De la Condamine at Tarqui (the southern terminus) and Bouguer at Cotchesqui ; the observa- tions, extending from November 29th 1742, to January 15th 1743. By the simultaneous arrangement of the observations any unknown changes of place in the star were eliminated in the result. The zenith sector was used in a different manner from that of Maupertuis. In his case the plumb-line indicated the direction of the telescope, or the star, at the one station and at the other ; there was no attempt to ascertain the absolute zenith distance. In the observations in Peru the zenith sector was reversed in azimuth several times at each station, whereby the unknown reading of the zenith point was eliminated, and the double zenith distance of the star measured. The amplitude of the arc, as derived from e Ori- onis, they found to be 3 7' 1"-0. This was checked by GEODETICAL OPERATIONS. 13 observations on a Aquarii and Aquilse, which however they did not use, From this and the length of the arc, namely, 176945 toises (at the level of their lowest point, and taking the mean of the two lengths calculated by Bouguer and De la Condamine), the length of the degree was ascertained to be 56753 toises. Bouguer published his history of the expedition in a work entitled, La figure de la Terre, par M. Bouguer, Paris, 1749. The calculations of this arc were revised by Von Zaeh (Hon. Corresp. xxvi. p. 52), who finds the amplitude to be 3 7' 3"- 7 9 and the terrestrial arc 176874 toises, reduced to the level of the sea. Delambre, by a revision of the reduction of the ob- servations made with the zenith sector, obtained for the lati- tudes of Tarqui 3 4' 31".9 S and of Cotchesqui 2' 3l"-22 N, making the amplitude 3 7' 3"-12. In 1783, in consequence of a representation from Cassini de Thuri to the Eoyal Society of London on the advantages that would accrue to science from the geodetic connection of Paris and Greenwich, General Roy was with the King's ap- proval appointed by the Royal Society to conduct the opera- tions on the part of England, Count Cassini, Mechain, and Legendre being appointed on the French side. The details of this triangulation, as far as concerns the English observers, are fully given in the Account of the Trigonometrical Survey of England and JFales, Vol. I. The French observations are recorded in the work entitled, Expose des Operations faites en France en 1787 pour la jonction des Observatoires de Paris et Greenwich : par MM. Cassini, Mechain, et Legendre. A vast increase of precision was now introduced into geodesy. On the part of the French, the repeating circle was for the first time used ; and in England Ramsden's theodolite of three feet diameter was constructed and used for measuring the angles of the triangles and the azimuth by observations of the Pole Star. The lower part of this instrument consists of the feet or levelling screws, the long steel vertical axis, and the micrometer microscopes originally three in number whereby the graduated circle is read, these being all rigidly connected. The next part above consists of the horizontal circle, the hollow vertical axis fitting on to the steel axis 14 GEODETICAL OPERATIONS. before mentioned, and the transverse arms for carrying- the telescope, all strongly united. The circle has a diameter of thirty-six inches, it is divided by dots into spaces of 15', which by the microscopes are divided into single seconds. The vertical axis is about two feet in height above the circle. The telescope has a focal length of thirty-six inches and a transverse axis of two feet in length, terminated in cylindrical pivots, about which, when supported above the axis of the theodolite, it is free to move in a vertical plane. A second instrument almost identical in size and construc- tion was shortly afterwards added. Both of them have done much service on the Ordnance Survey, having been used at most of the principal stations. Notwithstanding all the travelling and usage they have been subjected to for so many years, they are both now, with perhaps the exception of some very trifling repairs, as good as when they came from Rams- den's workshop. Fortunately no accident has ever happened to either of them, which is remarkable when we consider how many mountains they have ascended. The measurement of a base on Hounslow Heath was the first step in the trigonometrical survey of Great Britain. The ground was selected from the extraordinary evenness of its surface and its great extent without any local obstructions to the measurement. The bases which had been measured previously to that time in other countries had generally been effected with deal rods. Accordingly, three such rods, twenty feet each in length and of the finest material, were obtained; they were ter- minated each in bell-metal tips, by the contact of which the measure was to be made ; but it does not appear that they were oiled or varnished. In the course of the work it became obvious that the rods were affected to such an extent by the variations of humidity in the atmosphere that the measure- ment was considered a failure. The base was then measured with glass tubes of twenty feet in length, of which the expansions were determined by actual experiment. The tem- perature of each tube was obtained during the measurement from the readings of two thermometers in contact with it. The length obtained from the glass tubes was 27404-0 feet GEODETICAL OPEKATIONS. 15 when reduced to the level of the sea and to the temperature of 62 Faht. With respect to the reduction of the base to the level of the sea, what is meant is this : when we speak of the earth being* a sphere or a spheroid we do not mean thereby that the external visible surface of the earth is such. What is intended is that the surface of the sea, produced in imagi- nation so as to percolate the continents, is a regular surface of revolution. As trigonometrical operations are necessarily conducted on the irregular surface of the ground, it is usual to reduce the observations or measurements to what would have been obtained at corresponding points on the surface of the sea. If S be any actual trigono- metrical station, s its projection on the surface of the sea, so that the line Ss = h is a normal to the water surface at s, then s is the point dealt with in all the calculations of tri- angulation. ,In this light a base line should be measured along the level of the sea as ab, but practically the section of a base line will be always some un- even line as AB. Generally, it will be measured in a succession of small horizontal portions as indicated in the diagram : we may suppose each hori- zontal portion to be a measuring rod. If I be the length of a rod and r the radius of the earth, then the length of the projection of I on a b by lines drawn to the centre of the earth is clearly i r ^ - 7 == ' ' ~ r + h r summing this from one end of the base to the other, we see that if i be the number of measuring rods in the base and il=L, then the length of the base as reduced to the level of the sea ab is Fig. 3- For the reduction of the base it is necessarv then that the 16 GEODETICAL OPEKATIONS. height of every portion of the base be known, in order to get the mean height of the line. To return to the measurement of the base at Hounslow. It was considered that the length obtained by the glass tubes ought to be verified, and it was decided to remeasure the line with a steel chain. For this purpose two chains of a hundred feet long were prepared by Ramsden. Each chain consisted of forty links, half an inch square in section, the handles were of brass, perfectly flat on the under side ; a transverse line on each handle indicated the length of the chain. One chain was used for measuring ; the other was reserved as a standard. At every hundred feet of the base was driven a post carry- ing on its upper surface a graduated slider, moveable in the direction of the base by a slow-motion screw; this post served to indicate, by a division on the scale or slider, the end of one chain and the initial point of the next. The chain, stretched by a weight of twenty-eight pounds, was laid out in a succession of five deal coffers carried on trestles, so that the handles of the chain rested upon two of the posts, or on the divided scales attached thereto. The final result exceeded by only some two inches that obtained from the glass tubes. The instrument introduced in these operations by the French for the measurement, not only of terrestrial angles, but for astronomical work, was one constructed on a principle pointed out by Tobias Mayer, professor in the University of Gottingen, in Commentarii Societatis Regiae /Scientiarum, Getting. 1752. The repeating circle, used then and for many years after to the exclusion of every other kind of instrument for geodetical purposes in France, soon attained an immense reputation, and was adopted in nearly every country of continental Europe, where precise results were desired. It was, however, never used in England. The aim of the principle of repetition was to eliminate errors of division, a class of errors which was certainly large at that time. But, as the art of dividing circles attained gradually to higher perfection, so the value of the repeating circle diminished. Besides it was found by pretty general experience that the instrument was liable to constant error, of which the origin was not explained satisfactorily. GEODETICAL OPERATIONS. 17 The repeating circle has a tripod stand, with the usual levelling foot-screws, and a long vertical axis, at the base of which is a small azimuthal circle, which, however, is only a subordinate part of the instrument. At its upper extremity this vertical axis of rotation carries on a kind of fork a short horizontal axis, to which are united on opposite sides of it the repeating circle and its counterpoise ; the axis of rotation of the circle itself passing from the one to the other. By rotation round the horizontal axis the circle can be set at any inclination between the limits of horizontality and vertically; this, combined with azimuthal rotation round the long vertical axis, allows the circle to be brought into any plane whatever. The circle, which is divided on one surface only, is fitted with two telescopes ; the upper telescope carries with it four verniers for reading the angles ; the lower telescope carries no verniers, and is mounted eccentrically; the optical axis of each telescope is parallel to the plane of the circle. Moreover each telescope rotates round an axis coincident with that of the circle, and each may be inde- pendently clamped to the circle. The process of measuring an angle between two terrestrial objects is this ; let R and L designate respectively the right and left objects. The first thing is to bring the plane of the circle to pass through R and L. Suppose, to fix the ideas, that the divisions of the circle read from left to right (this was the French practice and is contrary to ours), (l) Having set and clamped the upper telescope at zero, the circle is turned in its own plane until R is bisected by the upper telescope, then the circle is clamped. (2) The circle and upper telescope remaining fixed, the lower telescope is brought to bisect L and then clamped to the circle ; this is the first part of the operation. (3) Without deranging the telescopes the circle is undamped and rotated in its own plane until the lower telescope comes to R and bisects it ; then the circle is clamped. Thus the upper telescope has been moved away from R in the opposite direction to L, and by an amount equal to the angle to be measured. (4) The upper telescope is now undamped and directed to L where it is clamped. If now the verniers be read it is clear that they indicate c 18 GEODETICAL OPERATIONS. double the angle between E and L. This compound opera- tion is repeated as many times as may be thought necessary, starting always from the point where the upper telescope has arrived at the close of the preceding double measure. It is hardly necessary to remark that the clamps are accompanied by the ordinary tangent screws. It is only necessary to read the circle at the commencement and at the end of the repetitions, keeping account of the number of total circumferences passed over. Then the result- ing angle, which may be many thousands of degrees, is divided by the number of repetitions ; thus the error of reading and of graduation is divided by so large a number that it is practically eliminated. There are, however, other sources of error at work ; the whole apparatus is not rigid as it is in theory supposed to be, and the play of the several axes doubtless affects the work with some constant error. Moreover it is a principle in observing generally, that to repeat the same observation over and over, under precisely the same circumstances, is a mere waste of time, the eye itself seems to take up under such circumstances a fixed habit of regarding the object observed, and that with an error which is for the time uniform. In some repeating circles a tendency has been found in the observed angle to continually increase or decrease as the number of repetitions was increased. W. Struve, in his account of his great arc in Russia, observes that if in measuring an angle the repetition be made first in the ordinary direction, and then again by reversing the direction of rotation of the circle, the two results differ systematically. Accordingly it became the practice to combine in measuring an angle rotations in both directions. Neverthe- less there was no certainty that even then the error was elimi- nated, and the method of repetition was soon abandoned. In March, 1791, the Constituent Assembly of France .received and sanctioned a project of certain distinguished members of the Academy of Sciences, Laplace and Lagrange being of the number, to the effect that a ten-millionth part of the earth's meridian quadrant should thereafter be adopted as the national standard of length, to be called the metre. GEODETIC AL OPERATIONS. 19 The length was to be determined by the immediate measure of an arc of the meridian from Dunkirk to Barcelona, com- prehending- 9 40' of latitude, of which 6 were to the north of the mean latitude of 45. This measurement was to in- clude the determination of the difference of latitude of Dun- kirk and Barcelona, and other astronomical observations that might appear necessary; also the verification by new observa- tions of the angles of the triangles which had been previously employed ; and to extend them to Barcelona. The length of the seconds' pendulum in latitude 45 was also to be deter- mined, and some other matters. Delambre was appointed to the northern portion of the arc, Mechain to the southern ; each was supplied with two repeat- ing circles made by Lenoir, and the work was commenced in June, 1792. The angles of all the triangles from Dunkirk to Barcelona were observed with repeating circles, and absolute azimuths were determined at Watten (a station adjacent to Dunkirk), Paris, Bourges, Carcassonne, and Montjouy. The sun was used in these determinations, in the evenings and mornings ; the angle between the sun and selected trigono- metrical stations being observed at recorded moments of time. The observations are numerous ; at Paris there are as many as 396, yet between that station and Bourges (120 miles south), where there were 180 observations, the discrep- ancy between the observed azimuths is as much as 39" -4. Delambre could not explain the discrepancies between his observed azimuths, but consoled himself with the reflection that a somewhat large error of azimuth did not materially influence the result he obtained for the distance between the parallels of Dunkirk and Barcelona. The latitudes were determined by zenith distances, prin- cipally of a and /3 UrsaB Minoris, at Dunkirk, Paris, Evaux, Carcassonne, Barcelona, and Montjouy. The length of the terrestrial arc was determined from two measured lines, one at Melun, near Paris, the other at Car- cassonne each about seven and a quarter miles long. The measuring rods were four in number, each composed of two strips of metal in contact, forming a metallic thermometer, carried on a stout beam of wood. The lower strip is of c 1 20 GEODETICAL OPERATIONS. platinum, two toises in length, half an inch in width, and a twelfth of an inch in thickness. Lying immediately on this is a strip of copper shorter than the platinum by some six inches. The copper strip is fixed to the platinum at one extremity by screws, but at the other end, and over its whole length, it is free to move as its relative expansion requires along the plati- num strip. A graduated scale at the free end of the copper, and a corresponding vernier on the platinum, indicate the varying relative lengths of the copper, whence it is possible to infer the temperature and the length of the platinum strip. At the free end of the latter, where it is not covered by the copper, there is a small slider fitted to move longitudinally in a groove, so forming a prolongation to the length of the platinum ; the object of this slider, which is graduated and read by help of a vernier, is to measure the interval between the extremity of its own platinum strip and that of the next following in the measurement. Both the verniers mentioned are read by microscopes. In the measurement each rod was supported on two iron tripods fitted with levelling screws, and the inclination of the rod was obtained by means of a graduated vertical arc of 10, with two feet radius, furnished with a level and applied in reversed positions. The whole apparatus was constructed by M. de Borda. The rod marked No. 1 was compared by Borda with the Toise of Peru, not directly, but by means of two toises which had been frequently compared with that standard; so that all the lengths in the French arc are expressed in terms of the Toise of Peru at the temperature of 16-25 Cent.= 13 Reaumur. The rod No. 1 was not after Delambre's time used in measuring bases, but was retained by the Bureau des Longitudes as a standard of reference. The Commission appointed to examine officially the work of Delambre and Mechain, and to deduce the length of the metre, after having verified all the calculations, determined the length of the meridian quadrant from the data of this new French arc combined with the arc in Peru. For the French arc they had obtained a length of 551584-7 as comprised between the parallels of Dunkirk and Montjouy, with an GEODETICAL OPERATIONS. 21 amplitude of 9 40' 25"; the latitude of the middle of the arc being 46 1 1' 58". For the arc of Peru they took (accord- ing to Delambre's statement) Bouguer's figures, namely, 176940-67, that is 176873 as the length reduced to the level of the sea, with an amplitude of 3 7' l", the latitude of the middle being 1 31' 0". It may be worth while here to go over, in an approximate manner, this historically interesting calculation. The latitude of a place on the surface of the earth, supposed an ellipsoid of revolution, is the angle the normal to the surface there makes with the plane of the equator. Let 2 A and 2 B be the sum and difference of the semiaxes of the elliptic meridian, which we suppose to be so nearly a circle that the square of the fraction B : A is to be neglected, then it is easy to show that the radius of curvature at a point whose latitude is < is E A 3^cos2, Multiply this by dcj) and integrate from to J TT ; this gives for the length of the quadrant Q = $ TT A. If we know the radii of curvature at two points whose latitudes are <$> and ' } then we have two equations such as the above, and eliminating between them, the result is putting 22, 2 A for the sum and difference of the radii, and o-, 6 for the sum and difference of the mean latitudes, A = 2 + A cot a- cot 5. If we divide the length of a short arc by its amplitude we get the radius of curvature at its centre : thus, from the numbers we have just given, the radii of curvature at the centres of the French and Peruvian arcs are respectively 3266978 and 3251285, thus 2 = 3259131 and A = 7846, log 7846 ... 3-89465, logcot(' p be the radii of curvature of the meridian at the middle points of the two sections p = 20803380, in latitude <#>'=: 22 36' 32"; p = 20813200, in latitude = 19 34' 34". Here we have an anomaly that has been met with in other places, namely, that the curvature of the meridian apparently increases towards the north. Such an effect might result from an error of latitude of the centre point of the arc, and Colonel Everest looked for the possible source of the error in the at- traction of a mass of mountains or table-land to the north of Takal Khera, called the Mahadeo P'har. This table-land ap- proaches in form to a rectangle of length ^^=120 miles, breadth B D = 6Q miles, Takal Khera T being distant 20 miles from and opposite to the middle point of CD. The mean height of the range above T is about 1600 feet or 0-3 mile. Colonel Everest, to obtain the de- flection of the direction of gravity at T, caused by the attraction of this mass, investigates a general expression for the attraction of a parallelepiped We may verify his result by a simple formula which will be found in chapter XII of this volume. The deflection at T depends on the angles tf, as marked in the diagram, and is expressed by the formula 5 = 1 2' r - 4 4 gh \og e (tan f tf cot % 0), where g is the ratio of the density of the hills to the mean density of the earth (^ = 0-6) and h the height of the plateau in miles (h 0-3). Thus, using common logarithms, 5 = 10"-31 log (tan \ 6' cot \ 0). Now 6'= 53 8' and = 18 26', and we have log tan 26 34'... 9-6990, log cot 9 13'... 0-7898, log tan i & cot \ B ... 0-4888, which multiplied by 10"- 3 gives 5"-0 as the required error of latitude. Colonel Everest then investigates the alteration at any external point. GEODETICAL OPERATIONS. 29 required to the latitude of Takal Khera in order that the two sections of the arc may conform to the (then) received value of the earth's ellipticity, namely, ^5. We may verify his result by an approximate calculation. A correction x to the latitude of Takal Khera makes the amplitudes a' x and and the radii of curvature become which are to be equated respectively to A3JBcos2'-, A 3cos2(j>, A and B being the half sum and half difference of the semi- axes of the earth, and A=6QQ. Then eliminating A, we have with close approximation a __ pp sin('+4>)... 9-82691, 000236 ... 6-37373, -005 ... 7-69897, 000178 ............... ...6-24947, 000414 =xia. Thus a being 10956", a =4"- 5. This agreement with the computed error, as caused by the attraction of the Mahadeo mountains, is very satisfactory. The accident to the great theodolite had the effect of turn- ing Colonel Everest's attention to the necessity of measuring every angle on different parts of the circle, the zero being shifted systematically through equal spaces a practice very rigidly adhered to on the Survey ever after. Nevertheless he was not satisfied with his arc between Damargida and Kalianpur : the errors in the sums of the angles of the tri- angles frequently amounting to 4" and 5". Accordingly, a few years after, the old theodolite was entirely re-made, a new one of the same size obtained, Ramsden's zenith-sector was replaced by two vertical circles of 36 inches diameter, and for base-line measures, Colby's Compensation-apparatus was ob- tained. Thus armed with the finest instruments, he revised entirely the arc in question and extended it northward to 30 GEODETIC AL OPERATIONS. Banog in latitude 30 29'. Here however the influence of the Himalayas on the latitude and also on the azimuth are very perceptible, and Kaliana, in latitude 29 30' 49", was adopted as the northern terminus of the arc. Base lines were measured at Damargida, Kalianpur, and Dehra Dun, near the northern extremity. The comparison of the measured lengths of the terminal bases with their lengths, as computed from the base at Kalianpur, stands thus DEHRA DUN. DAMARGIDA. Measured length in feet 39183-87. 41578-54. Computed 39183-27. 41578-18. Great improvements were also effected by Colonel Everest in the determination of azimuth by the increased number and systematic arrangement of the observations of circumpolar stars. Take for instance the following results of his own observations for azimuth of the c referring lamp ' at Kalianpur in 1836. By 130 observations of b Urs. Min. ... 179 59' 53"-120, 115 4 Urs. Min. Bode ... 53-565, 128 51 Cephei ... 53-420. But for the details concerning this arc, reference must be made to the work entitled An Account of the Measurement of two Sections of the Meridional Arc of India . . . , by Lieut. -Colonel Everest, F.R.S., etc. (1847). The subsequent history of the Great Trigonometrical Survey of India is to be found in the volumes now being published by Major-General Walker, C.B., F.R.S. Vol. i describes the measurements of the ten base lines ; vol. ii treats of the reduction of the triangulation by least squares. At page 137, vol. ii, is a comparison of the observed azimuth at Kalianpur with the observed azimuths at sixty-three different stations in India, exclusive of those under the influence of the Himalaya and Sulimani mountains. At thirty-four stations the discrepancy of azimuth is under 3", the largest discrepancy being one of 1 0". The conclusion on the evidence of all these meridional determinations is that the ob- served azimuth at Kalianpur requires a correction of 1"-10. The position of Kalianpur is at o in the adjoining diagram, which indicates by simple lines the various chains forming the GEODETICAL OPERATIONS. 31 Principal Triangulation of India. Some of these are chains of single triangles, others are double chains or strings of quadri- laterals and polygons. The letters alcdefgkij indicate the positions of the base lines. Fig 7. Sir A. Waugh, who succeeded Sir George Everest, relax- ing none of the precision introduced by his predecessor, extended (1843-61) the triangulation by about 7900 miles of chain, mostly double, with determinations of azimuth at 97 stations. Major-General Walker succeeding, added (1861-73) some 5500 miles of triangle chains, mostly double, with CKEODETICAL OPERATIONS. determinations of azimuth at 55 stations and of latitude at 89. This work includes the entire re-measurement of Col. Lambton's arc from Cape Comorin to Damargida. The data of the Indian arc as hitherto used for the problem of the figure of the earth are superseded by these revised and ex- tended triangulations. The following table contains the latitudes of certain points and the distances of their parallels expressed in terms of the standard foot of England. The points marked with an asterisk are on the meridian of 75, the others are in the line of the original arc l . STATIONS. LATITUDES. DISTANCE OF PARALLELS. Shahpur* Khimnana* Kaliana o / // 32 I 34-06 30 22 11-78 2Q ^O 48-32 8653153-1 8051100-7 773Qo6 sin c sinf sm ^ 7 but sin c x sin .F = sin # sin CJ_ , sin c 2 sin .F = sin a sin (7 2 , sin./ sin c sin .F = sin a sin sin C ; and by the substitution of these it follows that sin (7, sin (7 2 . . cot/ = cot ^ + cot- - (9) sm (7 sm (7 It is frequently necessary to determine the difference of two nearly equal angles from the given difference of their cosines, thus in the equation cos a cos/3 = 2 \ sn a cos a ,., . 0.10-.)- . 00) For instance, if in the spherical quadrilateral PQpq the angles P, Q are right angles, and the sides Pp, Qq are equal, say each = , then producing these sides to meet in a point A, cos a cos A = 2 sin 2 - tan 2 b, where A = PQ and a=pq. Then the formula (10) leads to this .*,* ^ sin b sin \lf sm ^1 = or, tanj(^( sm(C 17) and we get 2 sin 2 cose sin C sin(j5-0) + 2sin 2 tan Fig. ii, 1 + sec 2 e sin (B-0)' 2 sin 2 i sin B either of these expressions is convenient for the calculation of tan 77. The case applies to the determination of the azimuth of a circumpolar star, B being the pole of the heavens, C the zenith, and A the place of the star S when at its greatest azimuth. When is small, then approximately _ 2 sin 2 J cos 2 c tan C r] ~ l-cotJ?sin0 J of which the error is, very nearly, sin 4 \ sin 2 2 c tan C. 7. Consider next the case of a spherical triangle all of whose sides are small with respect to the radius of the sphere. Let A', B', G' be the angles, and A' the area of a plane triangle whose sides are a, b> c, the same as those of the spherical triangle ; then omitting small quantities of the sixth order, (?) becomes (sin o- sin o-, sin cr sin o-o)^ A = v 3/ cos a cos b cos (24) SPHERICAL TRIGONOMETRY. 47 Now we have sm ^ cos .. ., / or (T-, \ i /sin (r 9 sin a, <)b be the errors of the sides so computed. They will depend on the actually adopted value of e, which may be computed in more than one way : we shall therefore first express the errors in terms of an arbitrary e, thus l . sin (A - J e) . , / . sin A\ ^a = c . \~ r f sm" 1 (sine- -) sin ( C ^ e ) ^ sin U / By the verification of the following steps cot -cot A = a ,~ G n ( 1 + a ~ ~ C ) ' ad sin C ^ 12 in((7-Je)~ C 6 . / . sin^f\ sinyi a . _, / ^ 2 7c 2 \ sm- 1 ( sin c - ^) = c - -+ (a 2 c 2 ) ( 1 --- 1 -- )> smC'/ smC 6 v 20 60 / in the second of which e 2 is replaced by i a 2 b 2 sin 2 (7, we find a r 2 a\/ f 2e 3^ 2 7c 2 v J) = -( 2 c 2 ) (-T-^ TV 1 H -- ) 6 V ^absmC 60 / If therefore we calculate e by the formula 2 e = a b sin (7, the errors of the resulting sides will be But if we compute e by the formula e sin i # sin i b sin (7 sm - = - - - f - , 2 cos J c or, which amounts to the same thing, by the formula 3c-a-> e = i ao sm c; (^1 H -- - J ? fl= (_) (a^-Si'+c*), (27) the errors are Suppose, for example, to take a numerical case, that the sides of the triangle are a 220, b = 180, c=60 miles, 1 See 4ccozm< of the Principal Triangulation, page 245. SPHERICAL TRIGONOMETRY. 49 then by the first method of calculating the spherical excess, the errors of the resulting sides in feet would be ^a = +0-068, l>b = + 0-026. By the second method the errors would be da = -0-031, Zb = -0-030. We infer that the errors resulting from the use of Le- gendre's Theorem are of a minute order, and that they cannot prejudice any applications that can be made of it to actual use. In the case in which 0, b, and C are given to find A, , and c, if c = J a b sin C, we have c sin i (A-B) = ( b) cos J (C J e), (28) ccos i (A-B) = (a + b) sin i ( o', 50 SPHEKICAL TRIGONOMETRY. where e' is the spherical excess of P l p^ jt? 2 q 2 , we have Further, put then by Legendre's Theorem, in the triangle P l *t"l o-i - : - > I/ 1 = o-i - ; - COS if! COSje! Practically, we may neglect the divisor cos J e 15 and take x l = s 1 cos(a 1 ^ 1 ) J y^ = s 1 8m(a 1 $c l ). (29) The errors of these expressions, if e x is calculated by the formula 1 \ s sin a x cos a lf are s 5 *by l = - (22 sin 2 % 3) sin Oj cos 2 a, 360 s 5 ^^ = (1 13 sin 2 aj) sin 2 a a cos a a ; 90 which may always be neglected : that is, for a distance s of 3 they amount at a maximum to 0"'0007 and 0"-0027 respectively. So also in the triangle P 2 P x q 2 , < = ? 2 cos (a 2 - e 2 - i e'), y 2 ' = * 2 sin ( a 2- 3 2) 5 where c' = a? 2 ' y x ; and finally, ^2 = ^i +y 2 7 , * a = ! + < + i ^ e 7 ; (30) the last following from (11). 9. The expansion of (1 +2n cosO + ri*)- 8 is one of importance in geodetical as in other calculations : it is proposed to expand this *in a series proceeding by cosines of multiples of 6. Let z = e V , then z + - = 2cos0. and z r + = 2cosr0; z z thus we have which multiplied out becomes SPHERICAL TRIGONOMETRY. 51 I) 2 , J n*+... The term in n* is retained, though however it will not be actually required. We are more immediately interested in the cases in which s = i and s = f : they stand thus, (1 +2* cos + *)-* = l+-tt 2 + 4 + ... (31) (32) ^. + cos 26 _ q2 e ^-^ 3 + 3.5.7 We have also for the logarithmic expansion, = log (l+nz) + log (l + -) = 2 If jrccostf cos 2^+ cos 3^ ...[; (33) (23 ) where M is the modulus of the common system of logar- ithms : log M = 9-6377843. E 2, CHAPTER III. LEAST SQUARES. THE method of least squares, foreshadowed by Simpson and D. Bernoulli, was first published by Legendre in 1806. It had however been previously applied by Gauss, who, in his Theoria Motus, &c., 1809, first published the now well-known law of facility of errors, basing the method of least squares on the theory of probabilities. The subject is very thoroughly dealt with by Laplace in his Theorie analytique des probabi- lites : it is full of mathematical difficulties, and we can here give but the briefest outline. 1. The results of a geodetic survey, whether distances between points, or azimuths, or latitudes, are affected by errors which are certain linear functions of errors of observation ; thus the precision of the results depends first on the precision of the angular and linear measurements ; and secondly, on the manner in which those measurements enter into the results. Consider first the observations of a single angle. In order to avoid constant errors that would arise, for instance, from errors of graduation, and from any peculiarity of light falling on the two signals observed, the observations are repeated on different parts of the circle, and at different hours of the day, and on different days. The expert observer bears in mind that the probable existence of unrecognized sources of constant error renders it useless to repeat the same measurement a large number of times in succession under precisely the same cir- cumstances. With measurements thus carefully made, and in large numbers, it is to be assumed that the arithmetic mean is, if not the true, at any rate the most probable value of LEAST SQUARES. 53 the angle, and the differences between the individual observa- tions and the mean are the apparent errors of observation. Of course the sum of these errors is zero, and positive and^ negative signs are equally probable ; and it is a matter of observation, or fact, that if such errors be arranged in order of magnitude, the smaller errors are more numerous than the larger, and mistakes excluded beyond a certain (not weTT defined) limit, large errors do not occur. This leads to the conception of a possible law of distribution of errors. Suppose the number of observations indefinitely great, the errors being capable of indefinitely small gradations, then it is conceivable that the number, y, of errors lying between the magnitudes x and x + dx may be expressed by a law such as y =< (x 2 ) dx ; a function which is the same for positive and negative values of #, and which must rapidly diminish for increasing values of x. Here y also expresses the probability of any chance error falling between x and x-\-dx, provided the in- tegral of^JSMbetween the limits 00 be made = 1. The nature of the function < has been investigated from various points of view, each investigation presenting some difficult or questionable points, but all ending in one and the same result. "We shall here give the method proposed by Sir John Herschel, though its validity has been questioned. Let a stone be dropped with the intention that it shall strike a mark on the ground ; through this mark suppose two straight lines drawn at right angles. Taking these lines as axes of co- ordinates x,y, the chance of the stone falling between the distances x>x + dx, from the axis of y is $ (x 2 )dx } and the chance of its falling between the distances y, y + dy from the axis of x is (^ 2 ) dx dy, or generally <$>(x 2 )<$) (i/ 2 } da- when da- is an element of area about xy. But this chance is not de- pendent on the particular direction of the axes; if then x' y' be other coordinates having the same origin, ifa? 2 -f^ 2 =#' 2 + ^' 2 ; an equation of which the complete solution is <(# 2 ) = Ce cx *. Since however $(x 2 } diminishes as x in- creases, c must be negative ; write therefore 1 : c 2 instead 54 LEAST SQUARES. of c. Then, since the integral of $ (x 2 ] dx between + oo is to be unity, and since /00 __2 / e ~*dx = CX/TT, (l) ,/_QO it follows that CcVv = 1. Thus the probability of an error between x and x + das is and this expresses also the number of errors between x and x + dx, the whole number being supposed to be unity. 2. Let /UJL be the mean value of all the errors without regard to sign, fa the mean value of their squares, then o --, (3) \/7T 2 From the intimate relation thus shown between c and the average magnitudes of the errors, it has been called the modulus l of the system ; it is large or small according as the observations are of a coarse or a fine kind. It follows from this that the number of errors whose absolute magnitudes are between and tc is N=-^~ Te-^dt. (4) V-ffJo The values of this important and well-known integral have been tabulated for all values of t. For instance, the number of errors less than J c, c, 20, respectively are N^c - -520, N c = -843, N 2c = -995. Thus only five errors in a thousand exceed 2c. There is a certain value of c, call it c'=pc, to which corresponds the value of N= i, so that half the errors are greater and half are less than c. This c' is called the ' probable error,' since the probabilities of an error exceeding it or falling short of it are equal ; and the value of p found from the tabulated values of 1 Airy, Theory of Errors of Observation, page 15. LEAST SQUARES. 55 the integral is -477. The integral (4) gives also the number of errors between and ic' 9 if t = ip, thus for the successive values of i=l, 1, 2, 3, 4, 5, the values of N are 264, -500, -823, -957, -993, -999. So that in a thousand errors, seven, for instance, exceed four times the probable error. From (3) it follows that c f 0-674 pj. 3. If a X be a multiple of an observed quantity X, in the observations of which the modulus is c, then the modulus in the corresponding system of errors of a X is clearly ac. The probable error of the sum of two quantities affected by independent errors is the square root of the sum of the squares of their separate probable errors. Thus if X+ T= Z, and the moduli of the system of errors in X and Y be a and b respect- ively, then the law of facility of errors in Z is the same func- tion < as before, but with a modulus = Va 2 + b 2 . f For the error z in Z being the sum of an error x in the first system, and an error y = z x in the second, the chance of the concurrence of errors between x and x + dx, and between z x and zx + dz is .. JT 2 (Z-X)* 7 Z? a 2 + &2 , ZO? \2 - i * dx . e *~ dz = i e~*+ e~~"V~ <*+**' dx. irao Trao To include all combinations this expression must be inte- grated, considering z constant, from x = oo to x + oo . Thus, bearing in mind the integral (l), the probability of an error between z and z + dz is so reproducing, in a. remarkable manner, the function (7) t-/fc 3 3 -f ... = 0, 3 c)\ 2 +(cc)\s+ &c. Put V for the determinant formed by the coefficients of this equation, [ad] for the minor of (a a), &c., then V A! = [], VA 2 = [*], VA 3 =[)y+(ac)z +...(am) = 0, (8) (ab)x-\-(bb)y + (bc)z +... (bm) = 0, (ac}x + (bc)y + (cc)z -{-...(cm) = 0, &c. and these equations are in fact what we should have arrived at if we had set out with the intention of determining x, y, z, .. . , so that the sum of the squares of the errors 2 (e 2 ), or ^(aos + by + cz ...+ mf should be a minimum. Exactly in the same manner, if we had retained the separate values of w l9 w 2 , ..., we should have found that an, y> #, . . . are to be determined so as to make ?,w(ax + by + cz+...+m) 2 (9) or 2 (we 2 ), a minimum. This case practically therefore re- duces to the former, if we first multiply each equation by the square root of the corresponding weight. 5. Returning to the case of equal weights, let us determine the probable errors of any linear function, &8jf&+gy+%z... of the obtained values of #, y, z, .... Let the solution of the equations (8) be written thus = a+(aa)(am)+(ap)(bm)+(ay)(cm) ... , (10) =jr + (a0)(a = t+(ay)(am) + &c. Then, if ., (11) &c., it follows that, =f+(aa)A+(ab)3 + (ac)C..., (12) =f+ = h + &c.; LEAST SQUARES. 59 and thus Let S be the sum of the squares of these coefficients of % , m 2 , ..., then S = which by (11) and (12) gives finally . (13) when therefore we require the probable error of a function of xy z ... , it is necessary in solving the equations (8) to leave the absolute terms symbolical. Thus we have the required numerical quantities (a a), (a/3), .... The probable error of/a? 4-^4-^... might be taken as e v/ where e is the probable error of one of the equally well ob- served quantities m. The value of e is generally only to be determined by consideration of the residual errors of the equations : let o- be the sum of the squares of these residual errors, i the number of the equations, j that of the quantities x -> y> Z J ... > then the probable error ofjO'-f gy + hz ... is 0.674 4^)*. i) For the necessity of dividing by i j rather than i we must refer to treatises on least squares, for instance, Gauss, Tkeoria Combinations, 38, or Chauvenet's Spherical and Practical Astronomy, Vol. II, pages 519-521. A check on the calculated sum o- is afforded by the easily verified equation a- = Suppose the case of only two unknown quantities, then (10) becomes o-- i (**)()-(*)(*) > *-> )-(*)' 60 LEAST SQUARES. and the probable error of fx+gy is, (14) if /+# = 1, the probable error is a minimum, when 6. The following numerical examples will serve to elucidate the preceding theory. The annexed table contains in the first, third, fifth, and seventh columns, forty independent micro- meter measurements of equal weight, made for the purpose of determining the error of position of a certain division line on a standard scale. X ERROR. X ERROR. X ERROR. X ERROR. 3-68 -025 2-81 1. 12 5-48 + i-55 3-28 - 0-65 3-u - 0-82 4-65 + 0-72 376 -0-17 3-78 -0-15 4-76 + 0-83 3-27 -0-66 4-59 + 0-66 3-22 - 0-71 2-75 - 1.18 408 + 0-15 2-64 1-29 3-9 8 + 0-05 4-i5 + 0-22 4-51 + 0-58 2-98 - 0-95 39i O-O2 5-o8 + I-I5 4-43 + 0-50 4.21 + 0-28 5-21 + 1-28 2-95 - 0-98 3-43 - 0-50 5-23 + 1*30 4-43 + 0-50 6-35 + 2-42 3-26 067 4-45 + 0-52 2-28 -I-6 5 3-78 - 0-15 2-48 -i-45 3-95 + O-O2 4-10 + O-I7 4.49 + 0-56 4.84 + 0-91 2-66 - L27 4.18 + 0-25 The arithmetic mean of the measured quantities gives as = 3-93 ; and in the alternate columns are placed the errors, or differences between the individual measures and their The sum of the squares of these errors is 32-635, mean. hence the probable error of a single determination is Now if we arrange the errors in order of magnitude, we find that those two which occupy the centre position and so represent the probable error are -f 0-66 and 0.66. Again, as we have seen at page 55, the number of errors out of 40 LEAST SQUARES. 61 which should be less than half the probable error is 11, the actual number is 12. The number which according to theory should be under twice the probable error is 33, the actual number is 32. Finally, two errors should exceed three times the probable error, the actual number is 1 . The probable error of the determined value of x is 0-62 : \/40 =0-097. The unit of length in these measures is the millionth of a yard. Take now a case of two unknown quantities. The observed differences of length between the platinum metre of the Royal Society and a steel metre of the Ordnance Survey at certain temperatures are given in the accompanying table (Com- parisons of Standards, page 171): DlFF. TEMP. DlFF. TEMP. DlFF. TEMP. DlFF. TEMP. o 576 65.16 647 63.76 4!-59 36-12 36-57 35-22 669 65-20 7- 2 3 63-93 38-53 36-06 3854 37-33 5-00 65-45 4.17 64-21 41-08 36-08 39-47 37-49 5-9 65-51 6-32 63.90 39- i 3 36-23 41-10 37-60 6-39 64-57 7-31 64-08 41-80 35-57 40-10 37-79 577 64.77 38-07 38-36 38-65 35-48 330 64.88 39- 2 9 33-35 41-26 35-94 Let x be the excess of length of the platinum metre at 62, and y its excess at 32, above the steel metre at the same temperatures, then at the temperature t the excess is -32 62 t and this is to be equated to the observed difference. The first three equations for instance are M05# 0-105^ 5-76 = 0, M070 0-107^ 6-69 = o, M15# 0-115y 5-00 = 0, and so on. In forming the sums of squares and products (), (at), (it), (am), (6m), 62 LEAST SQUARES. it is to be remarked that in this case, since in each equation a -f b = 1 , we have (a) + (a4) = () and (ai) + () = () ; thus the final equations corresponding" to (8) are found to be 14-5497240?+ 0-684874^-163-0507 = 0, 0-684874#+ 10-080524^462-4393 = 0. The solution, leaving the absolute terms symbolical, gives # + 068950 (am) 004685 (bm) = 0, y 004685 (am) + - 099522 (bm) = 0. Restoring the numerical values of (am), (bm), we have x = 9-08, y = 45-26, and on substituting these in the 26 equations, the residual errors are 0-49 0-41 0-48 1-69 0-85 + 0-29 1-47 0-03 + 2-24 1-30 + 2-41 0-83 - 0-08 + 2-31 + 0-47 + 1-83 - 0-75 - 2-59 1-05 + 0-49 0-74 0-74 + 4'8i 1-82 0-48 + 1-03 the sum of the squares of which is 66-03, so that the probable error of a single comparison is 0-674 (H^)*=M2; and the probable error of x and y are x ... M2(-0690)* = 0-29, y ... M2(-0995)* = 0-35. We may calculate by (14) the probable error of the differ- ence of length corresponding to any temperature r, and it is easy to prove that this is a minimum when r is the mean of all the observed temperatures. 7. A case of frequent occurrence is that in which we have the observed values of a number of quantities t^, ft a , ..., u^ which, though independently observed, have yet necessary relations amongst themselves expressed by j ( < i) linear equa- tions. Let U 1 , U 2 , ... , U i be the observed values with weights \ 5 z 9 > if #1 , # 2 , . . . , # f are ^ ne errors of U lt U 2 , ... U i) they are connected by j equations of the form = + !! -M a a? a + 8 a? 3 .... (15) LEAST SQUAKES. 63 Now the probability of the concurrence of this system of errors may be thus expressed according to (5), p _ Q e -v>\ dz> it appears that ux vy = wx. Therefore x = 1 1 1 - -f - + U V W V U V W W * = 1 + 1 + 1 % # w 64 LEAST SQUARES. Thus, A, B, C being the true angles, and e lt e 2 , e$ the actual errors of the observed angles, the adopted values are % = A J^ u w *, 1 C = 1 1 v V W e<> 4- o 1 * 1 1 1 w U V W The actual error of any function as a& + /3B-{-y& of the adopted angles is thus where JL + J. + JL * = JL + A + z. ? w t; w ^ v w; Now the squares of the moduli for the law of facility of error in the observed angles, that is for e l9 e 2 , 3 , are the re- ciprocals of u, v, w, so that the square of the modulus for errors in a ( E + 3i3 + ( is \ (a-w'K? + I (p which may be put in the form and the probable error is the square root of this multiplied by p = -477. If, for instance, the side c be given, the probable error of the calculated side a is C u cot 2 C+ v (cot A + cot C) 2 + w cot 2 A H + pa < - ^ - '- - > ( uv + vw + wu ) If the three angles be equally well observed, each having the probable error e, that of a is t()*(cot 2 ^4cotJ(cot(7+cot 2 (7)^. (17) In the application of. such a formula as this, it is necessary to bear in mind that it is based distinctly on the hypothesis of the errors following the established law of facility of error LEAST SQUARES. 65 in (2). It could not be derived otherwise if, for instance, the observations were liable to unknown constant error. It is safer therefore, if there be any doubt on this point, to obtain e from the differences between the individual observations and 8, 33, dO dr, and f 2 " f n r i r*sm0dd0dr abc ~~ JQ Jo JQ R The variation of the function F-r- abc in passing from the ellipsoid abc to the adjacent confocal ellipsoid is 1 See also Todhunter, Attractions, &c., ii. 243 ; and the Quarterly Journal of Mathematics, vol. ii. pp. 333-7. 68 THEORY OF THE FIGURE OF THE EARTH, Now bx = r cos 08a = - 8#, and so on ; thus CL so that 1 or, putting N for the quantity in brackets, the integration extending throughout the ellipsoid. Consider now the shell which is bounded by the ellipsoidal surface whose semiaxes are ra, rl, re, and that whose semiaxes are (r + dr)a, (r + dr)6, (r + dr)c. Let e be the thickness of this shell, then A, ju, v being the direction cosines of the normal at so yz on the inner shell, o o o ^ 4- . + f!. - r* 2 7i2 2 '""" ' but \ f r + TT + ~r) 2 = 9 > and so for u, v ; hence V# 4 ^* c 4y a 2 ^ = Xrdr, - = Let ^/S be an element of one of the surfaces of the shell, then the element of volume is edS, by which we may replace a&cr 2 sin d$> dO in the triple integral (l). Thus we have N 1 dS *N ' 1 costidS where (^ is the angle at xyz between the normal drawn THEORY OF THE FIGURE OF THE EARTH. 69 wards there and the straight line drawn from thence to fg k. Consequently the integration with respect to S extending over the whole surface of the ellipsoid whose semiaxes are ra, rb, re. Therefore, by the Lemma at the commencement of this in- vestigation, if the point fgh he outside the ellipsoid, V : ale is constant ; that is, if H be the mass of the ellipsoid V : M is constant; that is, it is independent of the lengths of a be, depending only on the eccentricity of the ellipsoid. In other words, the potentials of confocal ellipsoids at an external point are as their masses. 2. An expression for the potential of an ellipsoid at an in- ternal point may be derived from the last-written equation. When fg h is internal, we have by virtue of the Lemma v N 2w8* r , -/-)= i-l rdr > abc j abc J where the integration is to be taken for all the shells outside the particle, viz. from r = / to r = 1, where so that " abc \a* ~~ b 2 c 2 ' The right hand member of this equation is the increment of the function V : abc in passing from the ellipsoid abc to the confocal ellipsoid a + ba, b + bb, c + ftc, where aba = bbb cbc = Ibt. We may now use the ordinary d instead of 6, and putting 2 = a 2 -^, 2 = /3 2 + ^ c 2 = y 2 + t, we have d 7 C / 2 70 THEORY OF THE FIGURE OF THE EARTH. V Integrate from t = to t = oo , since -^ vanishes when t is infinite, then we have -- <> where Q s= {(a 2 + 1) (/3 2 + 1) (y 2 + *)}*. This expresses the potential of the ellipsoid whose semiaxes are a/3y, at an internal point fgh. 3. We may adapt the result just arrived at, to the case of a very nearly spherical ellipsoid in the following manner. Let the semiaxes squared be k 2 -f- e l5 k* + e a , F + e 3 , where e 1} e 2 , e 3 are very small quantities whose squares are to be neglected, and fj + e 2 + e 3 = 0. Put k 2 + 1 = u, and then Q = u% - } and Jf being the mass of the ellipsoid V**a|Jf. Thus, at an internal point /^^, the integral being taken from ^ = k 2 to w = oo : thus r= i jr. f * h - ^ + !i/!M!^ ) ^fc wf ( w * 2 3 where r 2 '= f 2 -\rg* +ti 2 . The result of the integration is which is the potential required. From this we may obtain the potential of an ellipsoid at an external, point. When the internal point fgh arrives at, or is on the surface, / 2 ff 2 w ,-. ^ + e, + k^ + ^ + t? + e 3 = whence r 2 _ z.2 , fi f - & ~ THEORY OF THE FIGURE OF THE EARTH.. 7T 3 -. 1 r* ! ' Thus, for a point on the surface, Let the ellipsoid to whieh this refers be called the ellipsoid E, and let the point fgh on its surface be called E. Let there be another ellipsoid E' confocal with E, and interior to it, its mass M' 9 and squared semiaxes k^ + c lf k^ + e 2 , k-f + e 3 . Let V be the potential at R of E', then by Laplace's Theorem V':M' = r-.M. Therefore, in the preceding equation we may substitute V and M' for V and M, or in other words, that equation ex- presses the potential of an ellipsoid at an external point. If abc be the semiaxes of any ellipsoid, and ^^a-c 2 , e * = c 2 -a*, e 2 ,, 2 /.,, 2 / 2\ i P" ., 2 ,, 2 /. 2 x, 2> 1-^6 ^2 C 3 V 2 ^3 /"""-^e ^3 ^1 V^3 ^1 ; 42 /T and so on ; where P^, 1%', I?" are what Legendre's coefficient of the order i becomes when the direction cosines of the line r are substituted severally for the variable involved in the expression for P i , and Pisa symmetrical function of the direction cosines. The values of P 15 P 2 , P 4 , P 6 are P a 2 - , ^ ~ 2 * 2' 5.7 t 3.5 rt 1.3 P *=^~2A 2 + 2A 2.4,6 2.4.6 2.4.6 2.4,6 1 Philosophical Magazine, December, 1877. 72 THEORY OF THE FIGURE OF THE EARTH. We see from the law of this series for V that for an external point,, however near it may be to the ellipsoid, if the ellipsoid be very nearly a sphere, so that small quantities of the second order ( 2 _ q , . C 2 e 3 6 1> Z e l 6 2> e l e z ~^ 3> also p' ._ 3 f 1 2 ~ "2" ~& *J 2 " 8 " i substituting these values in the second term of the series, we get again the expression (4). 4. We shall now take the case of an oblate spheroid 1 , and obtain an expression for the potential of a spheroidal shell at an external and at an internal point. Let the semiaxes of an ellipse be c (1 -f- i e) and c (l f = C 2, e 1 = |^ 2 = e 2 , 3 = -i ec 2. thus we get for an internal point the potential n 1 = 2^(^- and for an external point Next take the shell, whose interior surface is a spheroid, whose elements are c, e, and its exterior surface the spheroid whose elements are c + dc and e + cle. Let e be a function of c, so that j de j de = - clc\ dc then the potentials of the shell at an internal and an external point are respectively dU, . , dU - dc. and - dc, do dc the density being unity ; if the density of the shell be p } the potentials are, at an external point, p ~~ dc ; dc at an internal point, p r dc. 5. Now consider a spheroid in which the density is not uni- form, but varies in such a manner that the surfaces of constant density are concentric and coaxal spheroids, the external surface being one of them. The ellipticity of the surfaces as well as the density is a function of the distance from the centre. Suppose that r = c {1 +e(% jut 2 )} is the equation of the generating curve of the surface of density p, then c being the independent variable, e and p are functions of c. Thus we may in other words suppose the spheroid formed of homo- geneous spheroidal shells of which the ellipticity and density 74 THEORY OF THE FIGURE OF THE EARTH. are functions of c. The external surface we shall particularise by the accented letters c', dx dy dz so that in this case Xdx-\- Ydy + Zdz is a complete differen- tial. And in the case of a fluid rotating, say round the axis of z, with uniform velocity, the corresponding part of Xdw+Tdy+Zdz is easily seen to be a complete differential. Therefore, for the forces with which we are concerned, namely, attraction of gravitation and the so-called centrifugal force where is some function of xyz, and it is necessary for equilibrium that dp = p d& be a complete differential, that is, p must be a function of 0, so also p becomes a function of 0, so that d = is the differential equation of surfaces of equal pressure and equal density. Although, since the earth is revolving about its axis, all problems relating to the relative equilibrium of the earth itself and the bodies on its surface are really dynamical problems, yet they may be treated statically by intro- ducing in addition to the attraction the fictitious force called centrifugal force. Let the earth be referred to rectan- gular coordinates, the axis of z being that of revolution. 76 THEOKY OF THE FIGURE OF THE EARTH. At the point xyz within the mass let V be the potential of the mass, then the components of the force there are T dV dV dT X = ^ + *> Y= ^ + ^> Z = ~Tz' to being the angular velocity of rotation. Then according to what precedes, at every surface of equal pressure and density ^dx + ^dy -f jdz-\-u> 2 (scdx -\-ydy} = ; and integrating = V -\ (x 2 4- y 2 } = 0, (7} P 2 where is constant for a particular surface, but varies from one surface to another. This equation then is that of a sur- face of equal pressure and density ; generally termed a level- surface. At every point of a level-surface the resultant force is perpendicular to the surface, and its amount is evidently -=- i where dn is the element of the normal. dn 7. Let us now enquire whether it is possible for a homo- geneous fluid mass of the form of an ellipsoid, rotating round one of its axes, to be in relative equilibrium. If the semi- axes be a be, audfg/i the coordinates of any particle of the mass, then the potential at this point is given by equation (2). If we substitute this value of V in the equation of a level- surface, and then divide by f M, where M is the mass of the ellipsoid, we get, supposing the axis c to be that of revolution, dt a r dt T 00 dt _-/ J J where Q 2 = (a 2 + 1) (b 2 + ) (c 2 + 1), and C' is a constant. This equation must hold at the external surface which is that of zero pressure : but at this surface /o o 7 O 2 .2 22 THEORY OF THE FIGURE OF THE EARTH. 77 and comparing coefficients of y 2 , ^ 2 , ^ 2 , we have 2o> 2 C dt M, ~~J Qa 2 +t~~^ 2co 2 which are equivalent to two equations ; and we have to ascer- tain whether the results to which they point are possible. Subtract the second equation from the first and we get then eliminating //, by means of the third equation, the result is 'IJ Q(a 2 + t)(b' 2 +t) "" J Q (c 2 + t) this condition may be satisfied either by a = b, in which case the ellipsoid is one of revolution round c ; or by making the quantity within the brackets vanish, that is f J o but there can be no negative elements in this integral unless ad * Imagine a triangle having two sides a, 6, including a right angle, then the perpendicular from the right angle to the hypothenuse is ab(a?-\-b 2 )~^. From this it appears that c must be less than either b or a if the last-written integral is to vanish. If, however, c be very small the integral becomes negative. Therefore there is some value of c which will satisfy the equation. For a discussion of this very interesting problem see a paper in the Proceedings of the Royal Society, No. 123, 1870, by Mr. Todhunter. That the value of o> is real will appear from the first and third equations, which give 2 0,2 ^ a 2 c 2 r tdt ~ZM: = ~^~J Q( which is essentially positive. 78 THEORY OF THE FIGURE OF THE EARTH. This remarkable fact, that a homogeneous fluid ellipsoid of three unequal axes, revolving about its smallest axis, can be in a state of relative equilibrium, was discovered by Jacobi in 1834. 8. In the case in which a = b and the ellipsoid becomes an f oblate spheroid, there is but one equation of condition, namely, I that which connects the velocity of rotation with the ratio of ^ the axes. Let the axes be c and c (1 + e 2 )^ ; then if p be the density of the fluid mass, the last equation written down becomes _ 3 Now transform this integral by putting c 2 + 1 = e 2 c 2 cot 2 0, then Q = 3 ; call the right-hand member of the last equation E; then as e increases from zero, E increases from zero until, for a certain value of e derived from the equation dE = 0, or E becomes a maximum. As increases from this value, which is about 2-5, to infinity, E gradually diminishes to zero ; there is therefore a maximum limit to o>, and when the angular velocity is less than this limit, there are always two spheroids which satisfy the conditions of equilibrium : in one c is greater, and in the other less than 2-5. This fact was first indicated by Thomas Simpson, and subsequently proved by D'Alembert. It is to be remarked, however, that the same mass of fluid cannot take indifferently one or other of these THEORY OF THE FIGURE OF THE EARTH. 79 without an alteration in its moment of momentum. If the moment of momentum and the mass be given, there is but one possible form of equilibrium. We may now shew that the earth cannot be or have been J a homogeneous fluid. If p be the mean density of the earth, its mass is ^ irpa 2 Cj where a is the radius of the equator, and this mass divided by ac may be taken as the mean amount of the attraction at the surface ; then, if m be the ratio of centri- fugal force at the equator to gravity, Let I be the length of the seconds pendulum, then the acceleration due to gravity is -jr 2 1: at the equator, I 39-017 inches ; at the pole, I = 39-2 17; the mean of these is the length of the seconds pendulum in the latitude of 45. Also, the acceleration due to centrifugal force is, if t be the number of mean solar seconds corresponding to one revolution of the earth, 4#7r 2 hence on substituting the values =86164 and a 20 926000 feet, /= 39-117 inches, 40 1 Wl = -r-: ; = 289-1 Now when * 2 is very small, as in the case we are con- sidering, 0,2 3 + c 2 , 3 4 = -3- ta nie--3=-e 2 ; and this we have seen to be equal to f m, hence l6 2 = i^; and the ratio of the axes being 1 : 1-J-Je 2 , is 231-3 : 232-3, which differs materially from what we know to be the actual ratio. 9. Let us now consider the case of a revolving fluid spheroid which is 1 not of uniform density. Without assuming any law 80 THEORY OF THE FIGURE OF THE EARTH. for the density, let it be so far limited as that the surfaces of equal density shall be spheroids concentric and coaxal with the surface, and then determine the conditions which make equilibrium possible. In this case the surfaces of equal density are also surfaces of equal pressure. The potential at any point of such a mass is given in (6), and this has to be substituted in (7), which may be conveniently put in the form 0= F+^r^l-V)^ F+ir 2 o> 2 + Jr 2 a> 2 (i- M 2 ). (8) To conform with previous notation, r is to be here replaced by c /5 since small quantities of the second order are excluded. The result of the substitution is, if 4 TT U+ \ c* to 2 = 12, p Now this is to be constant for the spheroidal surface de- fined by c / and e /t but in order that it maybe so, H must vanish : hence, restoring the value of U from (6), W (9) ,o , and e 2 o> 2 c 2 C c ' de -!r / J -r 5J c r dc This very important equation, expressing the condition of equilibrium, was first given by Clairaut 1 . He transforms it thus : omitting the subscripts which specified the particular surface at which the potential was taken, multiply (10) by c 3 ; differentiate and divide by c* ; differentiate again, and then multiplying by c 2 , the result may be written ffie re where $ (c) is written for / pc 2 dc. /o 1 See Thtorie de la figure de la Terre, &c., pp. 273, 276. THEORY OF THE FIGURE OF THE EARTH. 81 10. The differential equation just arrived at can always be integrated, at least by series, when p is given in terms of c ; and the two arbitrary constants will enable us to make the value of e satisfy the equation from which it is derived. When therefore p is given in terms of e it is always possible to find the ellipticity of every surface of equal density and pressure so as to satisfy the condition of equilibrium. Thus we have a possible constitution for the earth. Without how- ever assigning any particular law of density, Clairaut made a very important deduction from the preceding ; it may be put thus: the mass M of the spheroid is = 4 TT (c'), and the ratio of centrifugal force at the equator to gravity being m = 2 : M, In (10) make c f = (f ; the result is then (5) gives for the potential at any point external to the earth r=^ + ^(/-f) (*-*). (is) If we differentiate this with respect to r, the differential coefficient taken with a negative sign gives the attraction in the direction of the earth's centre, which may be taken for the component in the direction of the normal as we are neglecting small quantities of the second order. In order to get the whole force of gravity which includes centrifugal force we must add to this the vertical component of the latter. Or, more directly, the value of g is J0 dV 2/l *j?--jr-"*(i-rt Performing the differentiation, and putting for r its value at the surface, the result is 82 THEORY OF THE FIGURE OF THE EARTH. Let G be the value of gravity at the equator, where pt = ; then if $ be the latitude, ff = <7{l+(f^-V)sm 2 }. (14) Hence, the formula known as Clairaut's Theorem : viz. if G, G' be the values of gravity at the equator and at the pole respectively, then '- G s = %m-e. In his demonstration, Clairaut makes no assumption of original fluidity ; he supposes the strata to be concentric and coaxal spheroidal shells, the density varying from stratum to stratum in any manner whatever : it is assumed however that the superficial stratum has the same form as if it were fluid, and in relative equilibrium when rotating with uniform angular velocity. Professor Stokes in his demonstration of Clairaut's Theorem in two papers 1 published in 1849, showed that if the surface be a spheroid of equilibrium of small ellip- ticity, Clairaut's Theorem follows independently of the adop- tion of the hypothesis of original fluidity or even of that of any internal arrangement in nearly spherical strata of uniform density. On this point it is needful to bear in mind that without altering gravity at any point on the surface of the earth, the internal arrangement of density may be altered in an infinity of ways : for since the attraction of a solid homo- geneous sphere is at any external point equal to that of any concentric spherical shell of the same mass as the sphere being homogeneous and not inclosing the point referred to it is clear that one might leave a large cavity at any part of the earth's mass by distributing the matter in concentric shells outside it. The fact that the variations of gravity on the earth's surface, as indicated by the pendulum, are in accordance with the law shown in Clairaut's Theorem is therefore no evidence of the original fluidity of the earth. 11. In order to determine the law of ellipticity of the surfaces of equal density, it is necessary to assume some law con- 1 Cambridge and Dublin Mathematical Journal, Vol. IV, page 194. Cam- bridge Philosophical Transactions, Vol. VIII, page 672. THEORY OF THE FIGURE OF THE EARTH. 83 necting p with c. The law assumed by Laplace, and not since replaced by any better hypothesis is, that the com- pressibility of the matter of which the earth consists is such that the increase of the square of the density is proportional to the increase of pressure. This law involves at least no- thing at variance with our experimental knowledge of the compressibility of matter. Expressed symbolically, it is dp = kpdp, p Now, by (9), if we omit the small term in o> 2 and replace k by 4 IT k 2 an arbitrary constant, the equation becomes 1 C c C c> = - \ pc 2 de + I pcdc. C JQ J c Multiply this by c and differentiate twice, thus of which the integral is pc = h sin (- + g) Now, in order that the density at the centre may not be infinite, it is necessary that g : this gives for the law of density Ji . c p = - sin y c k We may obtain the mean density p of the earth thus : the mass of the spheroid is But ic Q = I ch sin T dc, J Q K (* i^ (* chk COS T + kk I cos-rdc, K JQ K = hk (k sin T c cos T ) ; V k k' hk(k . ' = np , and put A which is another remarkably simple result following from 1 Laplace's law of density, and enables us from the observed \ constant of precession to deduce a value of the earth's ellip- ticity. This method was first pointed out by d'Alembert in his work, Eecherches sur la Precession des Equinoxes. 14. In the following table we give the numerical results of the preceding theory on six different suppositions as to the mag- nitude of the ratio of the mean density to the superficial density of the earth. The second column gives the value of THEORY OF THE FIGURE OF THE EARTH. 87 the subsidiary angle in arc, which is expressed in degrees in the next column ; the fourth column gives the ellipticity of the surface ; the next that of the strata at the earth's centre ; the last column gives the computed constant of precession : I n e *.,&> IT e' to C A C 1-9 2-4083 I38-0 i 2~8^4 I 354 i 295 2-O 2-4605 I4I.0 i 2~9?2 i 363 i 300 2-1 2-5058 143-6 I 2^6T 9 i 37^ i 305 2-2 2-5454 145-8 i i 379 i 309 300-2 2-3 2.5804 147-8 i y&i i 386 i 3~T2 2-4 2-6115 149-6 306-1 i 393 I 3^6 The actual value of CA : C determined from astronomical observations (Annales de Vobservatoire Imperial de Paris, tome V. 1859, page 324) is between ^ T and ^J-g-, which corre- sponds with the value of n = . As we shall see in the sequel the value of n = \ is that which corresponds to the ellipticity -^-J-g- of the earth as derived from the measured arcs of meridian. The results of pendulum observations have been supposed to give, by means of Clairaut's Theorem, an ellipticity of about ^| , which corresponds with n = ^ . We shall see, in a subsequent chapter, the bearing of recent ob- servations in India on this point. 15. That the agreement indicated in the last paragraph between the results of the preceding theory and the results of observa- tion and measurements is not more exact need not surprise us. 88 THEORY OF THE FIGURE OF THE EARTH. (I The substance of the earth is not of the nature supposed in 1 1 the theory ; that it was at one time entirely fluid is almost certain, but at present the crust at least is solid to a depth of many miles, and the whole visible surface is most irregular, presenting oceans, continents, and mountains. The surface which has to be compared with theory is that of the ocean continued in imagination to percolate by canals the con- tinents : this surface, represented always by the mean height of the sea, is what we understand by the mathematical surface of the earth. The irregular and unsymmetrical forms of oceans and continents forbids us to suppose that the form of the sea is any regular surface of revolution, and this irregularity must produce a discordance between the fluid theory and the results of measurements. Every mountain mass we assume to produce some disturbance of the mathe- matical surface, and any variation of density in the underlying portions of the crust will do the same. Having seen that the general figure of the earth is very fairly in accordance with theory, we shall now examine into the irregularities of the sur- face caused by disturbing masses, and in so doing, we may sim- plify matters by neglecting the earth's ellipticity and rotation, and consider it a sphere, whose density, except near the surface, is a function of the distance from the centre. Suppose then, in the first instance, the earth to be such a symmetrical sphere covered with a thin film of sea, its radius = c-, let matter m be now added all over and throughout the crust, of varying positive or negative density a function of the latitude and longitude, in such a manner as to represent the actual state of the earth's superficial density and inequalities. The total amount of the disturbing matter m is to be zero. Now take any point P on the surface of the no longer spherical sea, let y + c be the distance of P from the earth's centre, and let the potential at P of the mass m be 7. Then the surface of the sea being an equipotential surface must be represented when the constant is properly determined by the equation V H = constant. c+y c where M is the mass of the earth. Since y is very small, we THEORY OF THE FIGURE OF THE EARTH. 89 shall omit its square, thus y =. V -f constant, (20) Here C is to be determined so that ffy sin Odd dfr taken over the whole spherical surface, may be zero. Let us now, since it is impossible to assign any general form to F" in the equation just deduced, suppose the case of the disturbing mass being restricted to a certain locality. We shall suppose it to be a mass of great density and of such compact form that its potential shall be the same, or very nearly the same, as if the whole were gathered into its centre ; which is sup- posed, moreover, to be below the surface of the ground. Let JJ.M be the mass, he the depth of its centre below the surface. Let be the angle be- tween the radius drawn through m, the centre of the disturbing mass, and that drawn to P a point on the disturbed surface. Let p be the projection of P on the spherical surface, then since Pp is very small, we may put V =. pM-r- mp, and will be the equation of the curve which by revolution round mO generates the disturbed sur- face. The volume contained by this surface will be equal to that of a sphere of radius c if we make / y sin B dO = 0, J(\ or, r. sin 6 dO + 2C=0; 90 THEORY OF THE FIGURE OF THE EARTH. hence C+pc = 0, and, ~ MC "" expresses the elevation of the disturbed surface at every point. It will be seen that if we draw FG bisecting mO at right angles, then y is positive for all those points which are on the same side of FG with the point m, and negative at all other places. The maximum value of y corresponds to = 0, showing that the greatest elevation takes place directly over the disturbing mass. This maximum elevation is, neglect- _ ~~ k ' In order to get some definite numerical ideas from the result at which we have just arrived, let the disturbing mass be a sphere of radius = n miles, its centre being at the same distance n below the surface. Let its density, being that by which it is in excess of the normal density in its vicinity be half the mean density of the earth, then " 2o Here Y is expressed in miles : to express it in feet we must multiply this by 5280, also put c 3960 ; thus in feet Y=%n\ If then the diameter of the sphere of disturbing matter be one mile, n i, and the value of T is two inches. This shows that a large disturbing mass may produce but a very small disturbance of the sea-level whether indeed the mass be situated above or below the surface. A displacement of the sea-level, such as has just been supposed, could not make itself directly perceptible in geodetic operations, but indi- rectly it can, viz. through the inclination of the disturbed surface to the spherical undisturbed surface, or which is much the same, by means of the altered curvature of the surface : for careful geodetic operations enable one to assign the local curvature of the surface with considerable precision. THEOBY OF THE TIGUKE OF THE EARTH. 91 16. Let us confine our attention to the surface in the vicinity of the disturbance, and thus disregard powers of higher than the square, then the equation of the generating curve is uc r = c -f - - u.c, of which the part IJLC may be dismissed from consideration / being a very small constant. The angle between the surfaces b-e*t termed local deflection of the plumb-line is dr u.0 also, since the greatest local deflection corresponds to -TI* that is, it is found at a point on the surface whose distance from the radius of the earth passing through the centre of the disturbing mass is to the depth of that centre as 1 :V%. The maximum deflection \}/ then has the value Taking the same disturbing mass as before, that is to say, with a radius of n miles, this becomes in arc or expressed in seconds _ nx 180.60.60 _ 200V3 ^ " ' 39607r.3\/3 UTT which is almost exactly 10^. Taking n as in the previous case = i, \js = 5"0 . Now this in geodetic measurements is a large quantity, that is to say, that with ordinary j3are, one can determine the latitude of a place to half j^econd, so that i K J i * /* 92 THEOKY OF THE FIGUEE OF THE EAETH. 5" would be a very measurable quantity. Then if we con- sider two points which lie on opposite sides at the distance + ch\/\ from that point of the surface which is vertically over the disturbing mass, the angle between the normals to the disturbed surface at those points will be larger than the angle between the corresponding normals of the spherical surface by 10". This leads us to consider the curvature of the surface in a plane section passing through the earth's centre and the disturbing mass. The radius of curvature E may be obtained from the known formula 2 (^ r ^\ dPr ~R = we may omit the square of -=- , and thus with sufficient pre- dd cision R r\ r 2<9 2 -/& 2 ) -> i) To determine the maximum and minimum values of the curvature, we must put the differential coefficient of R~ l , with respect to 6, equal to zero: that is, 6(26 2 3h 2 ) = 0, which is satisfied either by = 0, or 6 = k \/|, the former corre- sponds to the maximum curvature which is found vertically over the disturbing mass, and is expressed by while the minimum curvature found at the distance is expressed by i--i(i- Now whatever be the radius of the disturbing sphere, if the depth of its centre be equal to its radius, and the dis- turbing density be half the mean density of the earth, then THEORY OF THE FIGURE OF THE EARTH. 93 fj. = J^ 3 , and the radii of the surface become changed in the positions indicated into so that enormous variations of curvature result from even small disturbing masses below the surface. That effects of a similar character would follow in the case of compact dis- turbing masses above the surface, is easy to see. 17. To take the case of a supposed mountain range, of which the slope is much more precipitous on one side than on the other; let us enquire into the difference of level of the dis- turbed surface of the sea at the foot of the one slope as compared with that at the other. Strictly speaking, the level will be one and the same, but there will be a difference with reference to the undisturbed spherical surface. For sim- plicity, suppose the range to be of a uniform triangular section as in the accom- panying diagram : let . Let h be the height, supposed uniform, of the plateau, so that an element of mass at P is c 2 h sin QdfydQ, then h being taken as indefinitely small with respect to e, the potential at .Fis sin \y d z /"** = 4ch I ' sn a cos ( 1 sin 2 a sin 2 <)* [-]* sin" 1 sin a sin $ = Icha . So also we may find that the potential at E is luck sin Ja, while at the opposite point E' of the sphere it is litck(l cosja). Then if, as in (21), y Q , y, if be the elevations in feet of the sea at E, F, and E f respectively, the density of the attracting region being half the mean density of the earth y = 2 eh - + a (l- M *) = 0; also, we have seen that g being the force of gravity, Hence it follows, that if h be the height of the lake being a small quantity gh = a constant. We may imagine the surface of the lake continued so as to surround the earth, then the distance of this surface from the surface of the sea at any place is inversely proportionaLtcL gravity at that _place. The surface of a lake then is not exactly parallel to that of the sea, the inclination of these surfaces being- measured in the meridian plane. Let < be the latitude, / the angle between the surface of the lake and that of the (imaginary) sea below it, then dk 1 = = > cd<$> da dk + = 102 THEORY OF THE FIGURE OF THE EARTH. but g- {l + (|^-*')sin 2 4>}; It follows from this, that the latitude of a station whose height is Ji^ as determined by observation, requires the cor- rection /. But this is practically a very small quantity only amounting to a few tenths of a second for ordinary mountain heights. To express / in seconds, the right hand member of the last equation must be divided by sin l". Then csin. 1" being the length of one second on the earth^s surface, is approximately 100 feet: also approximately, %m~e =0-0052. If then the height expressed in thousands of feet be H, /= - CHAPTER V. DISTANCES, AZIMUTHS, AND TRIANGLES ON A SPHEROID. 1. ASSUMING that the figure of the earth is an ellipsoid of revolution generated by an ellipse whose semiaxes are a and c, so that 2 a is the diameter of the equator, and 2c the polar axis, then the equation of the meridian curve is where x and z are the distances of any point in the meridian from the axis of rotation and from the plane of the equator respectively. This equation is satisfied by the values x = a cos u t z = c sin u. (l) The latitude of a point on the surface of the earth is the angle made by the normal at that point with the plane of the equator. Let be the latitude of the point determined as above by u, and let s be the length of the elliptic curve of the meridian measured from the equator as far as the point whose latitude is <$>, then dx = a sin udu = sin $ds, (2) dz = c cos udu = cos Qds, whence the relation of and u, akanu = ctanc/>. (3) The angle u is termed the reduced latitude. Let e be the eccentricity of the meridian, so that ate 2 a z c 2 , and put A 2 = 1 e 2 sin 2 <, V 2 = 1 e 2 cos 2 u ; (4) 104 DISTANCES AND AZIMUTHS ON A SPHEROID. then we may readily verify the following relations : AV = A/T^?, (5) V sin = sin u> cos $ = A cos u, sin 20 sin (0 u) sin 2^ A i \f\ $* ^ Thus the coordinates x and z may be written x = acos ^ z = a _^jL_(i_ e *Y ( 6 ) A A If we differentiate (3), and eliminate du by (2), we get ds^_ 1e 2 ~d$ ~ A 3 ' and this is the radius of curvature of the meridian. Call it g, and let p be the radius of curvature of the section of the surface perpendicular to the meridian, this being also the normal terminated by the axis of revolution. Then 2. In the adjoining figure, let be the centre, and OP the polar semiaxis of the spheroid, Q the equator, A, B points in the meridians PAE, PBQ: a, I the projections of A^ B on the axis, AN the normal at A, BN the inter- section of the plane ANB with the meridian of B. Let K be the projection of B on the plane PAEO, and draw BH t HK perpendicular to AN. Let a be the azimuth of B at A, namely, the inclination of the plane NAB to the plane NAP, and if 90 + /ui be the zenith distance Fig. 20. of B at A, then BAN 90-/u. Take OE, OP as axes of no and z, that of y being at right angles to these, then if u, u' be the DISTANCES AND AZIMUTHS ON A SPHEROID. 105 reduced latitudes of A and .#, o> their difference of longitude or the inclination of the planes PAS, PJ3Q, we have for the coordinates of the two points x = a cos Uj of = a cos u' cos o>, y 0, y' a cos u' sin CD, z = c sin u, z' = c sin u. If be the length of the chord or straight line AB 3 and AH=ksmp, HK k cos \L cos a. If we project the broken line Aff+HK first on 0^, then on OP, we have / (sin fj, cos $ + cos \j* cos a sin ) = # (cos w cos #' cos eo), (8) k ( sin /z sin -f cos ju cos a cos $) = c (sin w' sin u), and i? JT is equal to either member of the equation k cos /x sin a = # cos w' sin a>. From (8) eliminate first cosjit, then sin/^i, and in the results replace terms in < by their equivalents in u. Substitute in the expression for k the values of xx'yy 'zsf, and put, for brevity, 2 for sin u sin u, then we get the four following equations : 1 -- 2 = s i n u s ^ n w/ + cos w cos u ' cos ^ H -- ^ 2 j (9) 2 6i ^. V - cos UL cos a = cos u sin u' sin & cos w x cos &> -f e 2 cos u2, a k cos a sin a = cos u sm co, ^ ^ V - sin a = 1 sin u sin u' cos ?< cos u cos co. c The first three of these equations correspond with the fundamental equations, page 40, of Spherical Trigonometry. Let v be the third side of the spherical triangle of which two sides are 90 % and 90 u', including an angle CD, and let \// be a subsidiary angle such that sin \|/- sin - = e sin \ (u' u) cos \ (u f -f u) ; further, let a 7 // A' represent in relation to the point B what 106 DISTANCES AND AZIMUTHS ON A SPHEROID. ap, A represent in relation to A, then we get from the pre- ceding, the following system of equations : k 2sin- cosx/r, (10) . v sm IJL = A sin - sec \fr, v sin fjf = A' sin sec \jsj a , . sm a cos ju = j cos u sm a>, . , , a sm a cos ju = y cos % sin co, A? which express the distance with the mutual azimuths and zenith distances of two points on a spheroid. 3. If we divide the second of equations (9) by the third, we have V cot a cos u' sin a> = cos u sinit' sin u cos n' cos a> -f ,b! sin (b A sin 0\ cota cot/3 =- V( - -^ - ), A cos< v sin a> , e 2 cos A sin d>\ cot a x + cot Q = -7 - -i- ( - ^ - 2- ) v A By a series of reductions which we shall not here trace out (Memoirs of the E. A. Soc., Vol. xx, page 131), the following result may be obtained from these equations : e AS a 4- a' = /3 + /3' -\ -- (- J sin a cos 2 a sin cos 3 c/>. Now the maximum value of the small term in e* is i ^ f^ 3 . 8*7 W ' if the distance corresponding to k be n degrees, this expressed DISTANCES AND AZIMUTHS ON A SPHEROID. 107 in seconds is 0"-0000015ft 3 , which for a distance of even several hundred miles is practically zero. Hence, the follow- ing important theorem : If $, ' be the latitudes of two points, CD their difference of longitude, a, a' their mutual azimuths, then )cot^ (12) sin Thus it follows that the ' spherical excess ' of a spheroidal triangle is equal to that of a spherical triangle whose angular points have the same latitudes and longitudes as the corre- sponding points of the spheroidal triangle. Let S be any point in the curve AB\ from S draw SG perpendicular to AN-, let SG = AG = so that f, f are the coordinates of S, then putting 6 sin _, h = = cos $ cos a, (13) Vle* the equation of the curve ASS is found to be this may be obtained in the following manner : if J be the projection of S on the meridian plane of A, the distance of / from the axis of revolution is x = (p )cos cos a sin $, while the length of SJ is y = ( sin a, and the distance of S from the plane of the equator is z = (p(le 2 ) ) sin $ + f cos a cos 0. These being connected by the relation give the equation (14). From (14) we can deduce the radius of curvature of the vertical section at A, for it is the limit of the ratio of f 2 : 2 f 108 DISTANCES AND AZIMUTHS ON A SPHEKOID. when those quantities vanish. If R be this radius of curvature If we put = r cos and f = r sin 0, the equation (14) may be written thus r + ?(/& cos /sin 0) 2 2R(l +%?) sin0 = 0; from this we may, putting = AT + Br 2 + Cr* ... , obtain in terms of r by the method of indeterminate coefficients. We shall simply give the result which is, putting / 7 / O 7 O 7?_ f k TT-f ~ k * . which is another form of the polar equation of AB. If s be the length of the curve from A to 8 .de 2 I .dd* and if we substitute in this the values of the differential co- efficients derived from the equation of the curve just given, the result after integration is this If we substitute in the equation just obtained k for r, we have the length of the curve joining AB. Unless, however, in a case in which for some special reason an extreme pre- cision is required, several terms of the series may be rejected. For instance, that involving e 2 k 5 can only amount to a hundredth of a foot in 300 miles, the term in e*k* is still smaller ; so that we may safely put DISTANCES AND AZIMUTHS OX A SPHEROID. 109 Let Rf be the same function of a'' that R is of a ; then the length of the curve joining AB, which is formed by the intersection of the surface, and the plane which contains the normal at B and passes through A is Now the difference of s and / is of the order e* 6 , and is to be entirely rejected : and if we take for s the mean of the last two series, it will be seen that in adding them together the terms in e 2 & so far cancel, that their sum becomes a term of a higher order which may be neglected. In fact either series may be represented by where - 2 (i _ *2 COS 2 u S1 - n 2 \ V R Q being a mean proportional between R and R', or rather very nearly so since e 2 cos 2 u sin 2 a differs inappreciably from e 2 cos 2 u' sin 2 a. 5. As it may be interesting, as occasion offers, to compare precise results with others obtained by means of approximate formulaB, we here give the results of the calculation by the formulae just investigated, of the angles and sides of a spheroidal triangle of which are given the latitudes and longitudes of the angular points. Assume for A, B, C these positions Lat. Long. A ... 5157'N. ...4 46' W., B ... 53 4 N. ... 4 4 W., C ... 50 37 N. ... 1 12 W.; and take for the elements of the spheroid a = 20926060, c : a = 294 : 295. 110 DISTANCES AND AZIMUTHS ON A SPHEROID. Also loga= 7-3206874662, log* 2 = 7-8304712628, log V !#= 9-9985253144. The reduced latitudes of A, B, C, and the corresponding functions A 15 A 2 , A 3 are first found to be A ... Ui = 51 51' 19"-92163, log A, = 9-9990867071, B. ..w 2 =52 58 23 -43810, log A 2 = 9-9990589251. C...^ 3 =50 31 16 -40080; log A 3 = 9-9991202240 ; for the subsidiary angles corresponding to the opposite sides log sin ^ = 8-4217198302, log sin ^ = 8-6156259752, 2 logsin^ = 8-3562766510, logsin^ 2 = 8-4221562901, logsin^ = 8-0187180976; Iogsin\/r 3 = 8-6709531435. Counting azimuths continuously from north round by east and south, the azimuths of the sides are found to be AS ... 2039'17"-2401, BA ... 201 12 r 36 // -8177, ...142 55 50 -2183, G5...325 11 7 -4013, CA. ..302 10 54 -6710, ^...119 23 54 -3366; whence the angles A= 9844'37"-0965, = 58 16 46 -5994, C- 23 12 -7303, A + B + C=180 1 36 -4262. The distances, chords and curve lines, come out thus, ^ = 1104249-327, a = 1104377-386, 2 = 950259-744, b = 950341-187, 3 = 436473-497, c= 436481-410. Again, take a triangle near the equator, and let the positions of the angular points be as follows : Lat. Long. A ... l^O' S. ... 0' E., B ... 20 N. ... o 30 E., C ... 1 30 N.... 3 E.; DISTANCES AND AZIMUTHS ON A SPHEBOID. Ill then the azimuths, angles, and sides, true to the last place of decimals, are these AS, 1521'24"-0371; A, 195 21' 5"-7090; BC, 65 6 46 -6939; CB, 245 9 10 -7078; CA, 225 12 16 -2131; AC, 45 12 16 -2131; A 2950'52".1760; BC = 1006266-448 feet; .5=130 14 19 -0151; CA = 1544212-630 feet; C= 19 56 54 -4947; AB = 689666-750 feet. 6. If the two points whose distance apart is required are on the same meridian, and have latitudes <, ', then f* (i-e* J+ (I^'si It is convenient to replace here e 1 by another symbol n, such that _ a c = the result is r# n)(l-n 2 ) (! + 2n The expansion of (l +2n cos$-f ri*)~% will be found at page 51 : if we effect the required integration the result is (3 n + 3 n 2 + V n*) sin (<'- 0) cos ( + ) + (y ^2 + y ^3) s i n 2 (^'_ ^,) C os 2 (0' + #) -|| fl 3 sin 3 (4> / -0) cos 3 (0' + ^>). The part of 5 which depends on n* may always be safely omitted; in fact, for the Russian arc of upwards of 25, it amounts to only an inch and a half. We may therefore take sn 0-( cos + V ^ 2 sin 2 (0" - 0) cos 2 ((/>' + ). (17) 112 DISTANCES AND AZIMUTHS ON A SPHEROID. This expresses the length of an arc of the meridian between the latitudes $ and $' , the ratio of the semiaxes being 1 ft: 1+ft, and the polar semiaxis = c. It is customary in geodetical calculations to convert a dis- tance measured along a meridian when that distance does not exceed a degree or so into difference of latitude by dividing the length by the radius of curvature corresponding to the middle point, or rather to the mean of the terminal latitudes. And vice versa, small differences of latitude are converted into meridian distance by multiplying the difference of latitude by the radius of curvature at the mean latitude. The amount of error involved in this procedure may be readily expressed by means of the above series ; it depends on ft, and the higher powers of n ; these last we may leave out of consideration, requiring only the principal term of the error. Let ^a, + i a be the extreme latitudes, then 8 = c(l +ft) a Sen sin a cos 2$, but the radius of curvature isc(l-fft) 3nc cos 2$, so that S = tt+i ftd 3 COS 2$. The error we are in quest of is therefore iefta 3 cos20. This vanishes in the latitude of 45, and in latitude 60, it is (since n = -^ nearly) about ^T^Q go?. For one degree nearly) about 1 on 3 "2"40~0 * ' log/> ... 7-320, 2400- 1 ...4-620, sin 3 ! ... 6-726, 046 ... 2-666; the error is half an inch. For 100 miles it would amount to nearly two inches. 7. We may here notice a source of error that exists in all theodolite observations of horizontal angles. If B be the projection on the spheroidal surface of a signal B' at a height h above B, then to an observer at A, B and B' are not in the TRIANGLES ON A SPHEROID. 113 same vertical plane, unless B happens to be in the same latitude as A. The angle between B and J? at A is in fact, as may be easily verified, sin 2 a cos 2 <. This is a very small quantity : in the latitude of Great Britain it can only amount to an eighteenth of a second for every thousand feet of height. If h be such that, neglecting the consideration of refraction, to the observer at A, B ap- pears at a zenith distance of 90, then h =/ 2 :20, and the error is e 2 k 2 z sin 2 a cos 2 d>. 4 a 2 The plane containing the normal at A and passing through B, and that containing the normal at B and passing through A, cut the surface in two distinct plane curves. Suppose to fix the ideas that A and B are in the northern hemisphere, B having the greater latitude of the two : then the curve APB made by the plane containing the normal at A lies to the south of the curve BQA corresponding to the plane con- taining the normal at B. There is thus a certain ambiguity as to what is to be considered the distance AB : but this ambiguity is more apparent than real, for the shortest or geodetic distance does not, as we shall see, differ sensibly from the length of either of the plane curves. The direction more- over of BQA is correct at B, and that of APB is right at A. Among the various curves that may be traced on the surface connecting A and B, there are two which have a special claim to attention, viz. one which we shall call the curve of alignment and the other the geodetic line. We shall refer the course of both these to the plane curves, and shall first consider the curve of alignment. Suppose that an observer between A and B provided with a transit theodolite wishes to place himself in line between these points. Shifting his position transversely to the line AB, he will consider himself in line when he finds that at 114 TRIANGLES ON A SPHEROID. the point L the vertical plane described by his telescope passes through both A and B. In the adjoining figure let CQLP be a meridian plane cutting the plane curves in Q, P, and the curve of alignment in L. Let u n u' be the reduced latitudes of A and B ; those of P, Q, L being respect- ively U f , U', U: also let Fig. 21. BCQ = u. Then if a be the azimuth of B or P at A, (1 1) gives V cos u sin co cot a = cos u, sin &' sin ^, cos u' cos co e 2 cos ^, (sin &' sin M f ), V cos U, sin co, cot a = cos u, sin Z7, sin u, cos C/", cos co, e 2 cos ^, (sin U, sin %,), where V 2 = 1 e 2 cos 2 u t . The elimination of cot a from these equations, gives equation (18), viz. : sin U.Ncos U. = e? -5 sin co, cos ?7, . ~ sin U, -f sin ( ' smcocos^ where .^ sin w' cos u, sin co, 4- sin u t cos &' sin co' cos u, cos ^' sin co Let us here introduce an auxiliary spherical triangle ABC, in which AC = 90^,, BC = 90 #', and the angle ACB = co, so that A and B correspond respectively to A and B. In the side AB take D, such that ACD = co,, BCD = co', so that D corresponds to PQ or L. Moreover, let CD = 90 u , AD = c,, BD = c?', and AB = c?, then by (8) and (9), pages 41,42, sin co tan U Q = sin co' tan u, + sin co, tan u' t sin c sin u = sin c' sin u, -f sin c?, sin uf, so that tan u = 2V. It is unnecessary in this investigation to retain terms in e* or higher powers, so that in terms multiplied by e 2 we may replace U, by # . Making this TRIANGLES ON A SPHEROID. 115 substitution in (18), and multiplying- through by cos%, we have for P, on replacing sin (/", u ) by U,u Qi . c' . c, 2sm-sin-' U f U Q = e 2 cosu sinu f - (19) cos Similarly for the point Q, . c' . c, 2 sm- sin-' cos I In like manner the condition that the vertical plane at L passes through both A and B gives for L . c' . c, 2 sin - sin - UUQ = e 2 cos u sin U Q 2 . cos I Taking the differences of these equations, and multiplying them by a, we have . c' . c, 2 sin -sin-' QP = ae 2 cos U Q - (sin u' sin #,), cos 2 sin sin LP = a e 2 cos # - (sin w sin u,\ cos I 2sin |sin| QZ = ae* cos w - (sin u' sin w ). cos I These quantities completely determine the position of L with respect to the plane curves. Since the ratio of LP: AP vanishes when AP = c, = 0, it is evident that the curve of alignment touches at A the plane curve APB, and its azimuth there is consequently the azimuth of B. So also the curve of alignment has at B the true azimuth of A. In tracing this curve two cases arise : first, i 2 116 TRIANGLES ON A SPHEROID. sin^ may between A and B have its values entirely inter- mediate between sin u, and sin u' ; in this case the curve lies entirely between APB and BQA. But if A and B, not sup- posed to be many degrees apart, are nearly in the same latitude, so that the reciprocal azimuths are both (measured from the north) less than a right angle, then the values of sin U Q will not all be between sin u' and sin u, . In such case, QL, as is easily proved, vanishes when c, . c tan -f tan - = 2 2 snw , sin u + sin u, and this value of e f determines the point, say F } when the curve of alignment crosses the plane curve BQA. Thus, from A to F t L is between the plane curves, and from F to B it lies on the north side of FB, the actual distance being of the order e 2 c 4 . If A and B have the same latitude, the curve of alignment lies wholly to the north of the plane curve between A and B. The angle at which the plane curves intersect, either at A or J5, is Q 1 = e z cos 2 u sin 2 a sin 2 - , supposing c to be small : and if we compare this with the expression, page 130, for the angle which the geodetic curve starting from A towards B makes at A with the vertical plane there, we see that, neg- lecting quantities of the order e 2 c s , the angle which the geodetic curve makes at A with the curve APB is one third of the angle /, and similarly at B. But, as we shall see, if we take into account the higher powers of c, the geodetic crosses BQA Fig. 22. under some circumstances ; lying like the curve of align- ment wholly to the north of the plane curves when A, B having the same latitude, these curves coincide. Q TBI ANGLES ON A SPHEROID. 117 9. In strict analogy with the method followed in plane curves, Gauss defined the curvature of a surface thus : if we have a portion of a surface bounded by any closed curve, and if we draw radii of a unit sphere parallel to the normals at every point of the bounding curve, the area of the corresponding portion of the sphere is the total curvature of the portion of surface under consideration. And if at any point of a surface we divide the total curvature of the element of surface con- taining the point by the area of that element, the quotient is called the measure of curvature at that point. Let the ele- ment of surface be the very small rectangle made by four lines of curvature. Let a, /3 be the sides of this rectangle, f, p the corresponding radii of curvature. The normals drawn through the points of the contour lie in four planes cutting each other two and two at right angles. The cor- responding radii of the unit sphere form on its surface a rectangle whose sides are a : g and /3 : p, and its area aft : gp ; this divided by the area of the rectangle gives 1 : p as the measure of curvature. Gauss has shown that, if an inex- tensible but flexible surface be bent or deformed in any way, then the measure of curvature at every point remains the same. Thus, taking a very small portion of a surface at the centre of which the principal radii of curvature are $ p. this portion may / be fitted to a sphere whose radius is j (%p)^. Without attempting a rigid proof, this may be seen as follows : FPj PQ are the principal sections of a surface through P their radii of curvature ?, p respectively. P' is a point indefinitely near P in FP ; ^,\ P'Q' a section of the surface by a plane through P" perpendicular to the plane FP. Let q, q' be the projections of Q, Q' on the plane FP, so that Pq, P*C[ intersect at the distance f from P. PQ = P'Q" being a very small quantity ( = s) compared 118 TRIANGLES ON A SPHEROID, with f Or p, then since = Hence, PP' being given, the law of width of the elementary strip of surface PQP'Q' is the same as if belonging to a sphere of radius (p)'. Hence, a very small portion of sur- face round P may be bent to fit a sphere of that radius. When a surface is so bent, lines drawn on it remain un- changed in length, and angles of intersection remain unchanged. Thus, a small spheroidal triangle whose sides are geodetic lines may be fitted on a spherical surface of radius ($p)^ these quantities corresponding to the centre of the triangle the geodetic lines retaining their character become arcs of great circles, and the angles of this spherical triangle are the same as those of the spheroidal triangle before deformation. IO. We shall now compare the angles of a spheroidal triangle (viz. the true angles as observed or formed by joining the angular points by curves of alignment) having given sides lying in given azimuths, with a spherical triangle having sides of the same length, and the radius of the sphere being (l)*j which we shall denote by N. The higher powers of e 2 are to be neglected, and it is premised that the differences of the angles in question are of the order ?(? If #, y be the coordinates of any point of a curve which passing through the origin touches the axis of as there, then s being the length of the curve measured from the origin, we have by Maclaurin's Theorem TRIANGLES ON A SPHEROID. 119 or if be the radius of curvature at the origin where (-f) is the value of that differential coefficient at the \ds' origin. These may be written / Applying these expressions to the curve of intersection of the spheroidal surface with the plane containing the normal at A and passing through B : drop from B a perpendicular on the normal at A, and let , 77 be the coordinates of B, c the length of the curve AB, and R the radius of curvature of the section at A, then rj c _ = 1 _ cos _ _ which may be written thus . c_ c 3 r l ~= U - [ "~" YI C s \ 1 \ (* {] i\ "\ Here -^- = - ( I sin 2 d>'+# cos 2 a cos 2 <') > H a ^ * where a is the azimuth of B at ^, <' the latitude of A, and $ the mean latitude of the triangle. Now it is unnecessary to retain in the expression for f any term of a higher order than e z c 3 , or in 77 any term of a higher order than e^c 2 , so 120 TKIANQLES ON A SPHEROID. that we may dispense with the term containing (-^) ; and also in the expression for It we may substitute for a the azimuth of AB at its middle point, call this y; and for $ put <. Thus, 11 e* cos 2 $ cos 2 y and it follows that c (20) c e 2 c 2 -cos) + ^ We do not require the smaller terms. Thus the position of is definitely expressed : and the coordinates ', rf of C are obtained by substituting in the above b and /3 for c and y. 11. Let A) B, C be the angles of the spherical triangle whose sides are a, b, c to the radius N, and let the angles of the spheroidal triangle be and let the azimuths a, (3, y of the sides a, b, c (at their middle points) be reckoned consecutively from to 360, and in the same direction as the lettering A } B, C. Then regarding ^V as the unit of length, But we have also e z BC 2 = 2 (1 cos a) a* cos 2 cos 2 a. 1 2 Substitute for '7777', and put T ^ e 2 cos 2 < = ^ f : then on equating these values of BC 2 , we get after a slight reduction, = 2 (cos a cos # cos c sin b sin c cos ^ x ) 2 cos2y.JT, TRIANGLES ON A SPHEROID. 121 where H= K= 4bccosA'3b*c 2 = -a 2 2a6cosC. Now put for cos ^4' its equivalent cos A d A sin A, then, if A be the area of the triangle, the above expression becomes 4A ^-dA = a 4 cos 2 a + b 2 cos 2 /3 ( 2 + 2 ac cos 5) cos 2 y (a 2 -\-2al cos ), which by means of (2), page 38, is reduced to 4&dA = 2iabcsinA {> c I a b ) ab ( dA dB <)B = < cos A T cos^ c l a b _ab . c Substituting the values we have obtained for dA, dB, %B = iab {2 sin (a -I- /3) + sin 2 y cos (a ^3)}. Again, if with the side c of a spheroidal triangle, and the adjacent spheroidal angles, we calculate the other two sides by the rules of spherical trigonometry, their errors will be, as we may easily verify, TRIANGLES ON A SPHEROID. 123 n2/3cos(a-y)} 5 (24) sin u sin C/ Thus the greatest error that can arise in the calculated side of a triangle on account of the spheroidal form of the surface, is less than e 2 2 abc 4 sin C ' where C is the angle opposite the base or given side. If in the case of our spheroidal triangle of reference we cal- culate from the given side c the sides , b, their errors are + 0-5 ft. and +0-7 ft., which are very small in respect to those large distance^, viz. a = 209 miles, b = 180 miles. It follows therefore that spheroidal triangles may be calcu- lated as spherical triangles, that is to say, they may be calculated by using Legendre's Theorem, and obtaining the spherical excess from the formula *;. ( 25 ) CHAPTER VI. GEODETIC LINES. THE geodetic line has always held a more important place in the science of geodesy amongst the mathematicians of the continent, than has been assigned to it in the operations of the English and Indian Triangulations. Here, indeed, it has been completely set aside, partly because the long arcs measured are in the direction of the meridian itself a geo- detic line and partly because the actual angles of a geodetic triangle cannot be observed, since, as we shall see, the azimuth of a geodetic, as it starts from a point A to a point J3, is different from the astronomical azimuth of B at A. But the difference of length between the plane curve distance AB and the geodetic distance is all but immeasurably small for any such distance as three or four degrees. It may also be proved that the calculation of spheroidal triangles as sphe- rical is correct only when the observed angles have been reduced to the geodetic angles, that is, the angles in which the geodetic lines joining the three vertices intersect. Still the difference is so very small that for such triangles as are formed by mutually visible points on the earth's surface it has been generally disregarded. We do not however conclude that geodetic lines have no necessary place in geodesy. Both the extreme precision now attained in the measures of base lines and angles, and the vast extents of country over which triangulations are being carried, make the consideration of even the smallest refinements not superfluous. GEODETIC LINES. 125 We shall now investigate briefly the nature of the geodetic line as the shortest line on the surface of an ellipsoid of revolution. Suppose the position of a point on the surface to be defined by its distance f measured from one of the poles along a meridian, and by its longitude o> measured from a fixed meridian, then, r being the distance of the point from the axis of revolution, the length of a curve traced on the surface is This length is to be a minimum between the given ex- tremities. We shall most readily arrive at the characteristic of the curve by giving a variation 8o>, a function of to co. Thus, /V2 // d. ,/ ds r 2 t ds J " ~ \ ds ' ' consequently, for the minimum, To fix our ideas, let longitude be measured positively from west to east, and azimuths from north through east round to north. Let a be the azimuth of the element ds of the curve, then ds cos o = d(, ds sin a = /6?oo, and the second of these substituted in the characteristic of minimum gives r sin a = C. If u be the reduced latitude, r a cos u, where a is the semiaxis major of the spheroid; and if u t a f be the initial values of u a at the point A, say, then cos u sin a = cos u, sin a, . (2) The relation here expressed is that which exists in a spherical triangle 833 C, whose sides are 8C = 90 u,\ and 33 C = 90 #, and the angles opposite to them 833 = a,, and 126 GEODETIC LINES. = 180 a. Let the third side of this triangle be be the latitude of a point on the geodetic sin u sin

(l-* 2 cos 2 ^)* dr = ^f sm0 = a sinudu, whence we obtain ds =a(le*cos z u^d a/ the longitudes of (r and _5, uu' their reduced latitudes. Let (5r be a point on the side &B of the auxiliary spherical triangle &23 ( corresponding to so that &ffi = 0-, &i3 = 0-' GEODETIC LINES. 127 = 90^, 93C = 90 &' ; then we have the three follow- ing equations : sin u = sin u, cos a- -f- cos u t sin ' + ,, w f9 o-, for o> 7 o>, w 7 w, o-' 0-, then sin w = sin o> H o- sin a, cos &, cos a>, e 2 sin OT / = sin a/ H o- r sin a, cos w, cos a>', e 2 sin w, = sin a>, H o-, sin a, cos w, cos a> r . 2 We have now to substitute these in the equation above which contains tan u. If a be the azimuth of the geodetic at B 3 we have sin , o-, cos a sin co Take now, as at page 114, an auxiliary spherical triangle ABC corresponding point for point with ABC and &BC, so that BC = 90-/, AC = 90-w,, ACB = ' ; on ^5 take , such that ACG = o>, GCB = then if CG = 90 , sin w' tan u = tan w 7 sin CD + tan u, sin &>, , (6) as at the page above referred to ; u is the same in both cases on the supposition of A B being the same points in the one case as in the other, the intermediate point being also the same. 128 GEODETIC LINES. Now since e* is to be neglected, we may put within the paren- thesis in (5) , cos u, sin o-, sin o). = sin o> cosu sino- , cos u sn o- sin to = sin CD cos u sin o- After making this substitution, taking the difference of (5) and (6), and putting tan u tan^ = (u U Q ) sec 2 ^, we get finally e 2 ,a sin o-. u u n = cos u ( : j- cos u. cos a, u 2 V sin a- smcr a-, sin a- sin (/ 7 )- (7) To compare this with the equations at page 115; the points AB we are now dealing with are to be considered the same as the points AB of that investigation, and our present point G is on the same meridian with P, L and Q y thus the c c'c, of those formulae correspond respectively with o-'o-,o- of (7), and we know fully the course of the geodetic compared with either the plane curves or the curve of alignment. From the last equation written down and (19) we get for the distance of G north of P, 1 e^ No- sino- o^sincr , , PG = a cos u { . / cos&, cos a. -- '- 7 cos u cos a 2 / sin cr sin a- cos- We may alter the form of this by eliminating a' through the equation cos u' cos a" = sin u, sin a cos u, cos a cos a/ , thus getting e z PG = a cosM(Hcosu,cosa, + .STsin^,), (8) 1 See Philosophical Magazine, May, 1870. 'On the course of geodetic lines on the earth's surface,' by Captain Clarke, R.E., from which much of this chapter is taken. GEODETIC LINES. 129 where j-. a- sin cr, cr, sin cr sine/ tano-' , (T . cr, sin-sm- K = cr. sin cr 4 7 o- cos- If we desire to trace the course of a geodetic line, not as passing through two given points, but as starting from a point A in a given azimuth, we may refer its different points to the corresponding points of the curve of inter- section of the vertical plane at A which touches the geodetic at that point. In order to do this we must put in (8) v o- = cr,, and in the result put o- / =0, making the point B move up to A along the geodetic. The result of this operation is PG = cos u sin a- < (- l) cos u, cos a, -( tan (12) 3. Let us now consider the case of geodetic lines starting from a point on a spheroid of small excentricity and diverging in all directions. First to confine our attention to a single line, it is well known, and may be inferred from the auxiliary spherical triangle, that a geodetic touches alter- nately two parallels equidistant from the poles the differ- ence of longitude between the successive points of contact being constant, and something less, than 180 depending on the angle at which it cuts the equator. Now suppose a line starting from a point on the equator with azimuth a, the osculating plane at that point cuts the equator again at the opposite point N. As a point P moves along the geodetic towards N 9 the angle r of the auxiliary spherical triangle increases from 0, and when it becomes IT then rr also becomes = TT, and P has reached the equator, its longitude being TT J e 2 7rsma. Since, in (9), o-cotcr 1 is negative for all values of a- from to TT, the geodetic lies wholly on the south side of the osculating plane at the initial point, if we suppose a<90, and its distance south when o- = TT and P is on the equator is \ ire 2 cos a. We infer from this that all geodetics proceeding from the same point on the equator have an approximately equal length i -ne 2 about 36' in the case of the earth intercepted between the meridian through N and the equator. Consequently, the ultimate intersections of the geodetics will form an envelope like the evolute of an ellipse or the hypocycloid 2. 7 2 w +y* = #3, N being the centre of the curve. If geodetic lines start- ing from a given point intersect so as to form an envelope, then each line is a shortest one only up to its point of contact with the envelope and no further (Jacobi: Vorlesungen ilber Dynamik, Berlin, 1866). If the lines diverge from a point GEODETIC LINES. 131 not on the equator, but in latitude u t the diameter of the envelope will vary as cos 2 u. 4. In the case of a geodetic joining two points which are a short distance apart, the line will generally lie between the plane curves, and neglecting quantities of the order e 2 o- 4 it is easy to see from what precedes that PG _ substitute for the = 8% tan a sec U, and we may suppose all the points of the plane curve to be referred in this manner to the geodetic. Now when CD is increased by 8o> ds f Tr ( is increased by the first term of which when integrated is, by reason of the character of the geodetic, zero. Hence the increment of GEODETIC LINES. 133 length in passing from the geodetic to the plane curve is, since ds cos a = d s 2 u cos 3 a and as we require merely the first or principal term in the value of 8*, we may put d = adu, and so bs = \ a cos 2 u cos 3 a / ( -, ) du. Let the whole length of the line be c, the distance of G from the initial point A being as before = o-. Now, omitting small quantities of the fourth order, the equation (8) gives g 2 PG = a o- (c o-) (c + o-) cos 2 u cos a, whence we have 8o> = ^-^(c 2 a 2 ) sinacos?*, / 7 and du = du ' ^ da- ' du ae* f c .-. ds cos 4 u sin 2 2 a / (c^3a 2 ) 2 d(r 288 /0 = c 5 cos 4 # sin 2 2 a. 360 This gives an approximation to the truth, only when the distance c is not very large. The coefficient -5^-5 ae* is only 2-66 feet : and if c were for instance 10, the maximum value of bs would be less than a hundredth of an inch. For the curve of alignment, the difference of length between it and the geodetic is obtained by putting bw = ^e 2 a ( 2 (T I (7 '/ cos- \tan- which gives 46-6 feet as the distance of the geodetic north of the plane curve. The differences of the azimuths of the geo- detic from those of the vertical planes are, at Calcutta 3"- 7 6, and at Kurrachee 2 "-04. As a second example, take the line joining the Cathedral of Bordeaux in latitude 44 50' 20" with the observatory at Nicolaeff in latitude 4658 / 20 // ; the spherical distance of these points being 22 35' 30". By spherical trigonometry we get at Bordeaux the azimuth of Nicolaeff =7 2 55' 7", and at Nicolaeff the azimuth of (i.e. the angle between the north meridian and) Bordeaux =8 3 23' 14". Take Bordeaux as the initial point, and let the whole distance be divided into ten equal parts, and through each point of division let a portion of a meridian line be drawn intersecting the plane curves P, Q, and the geodetic. The formerplane curve is that which is formed by the plane containing the vertical at Bordeaux, 136 GEODETIC LINES. and it lies entirely to the south of Q, that is, between the terminal points. We have also included in the calculation the curve of alignment, intersecting each meridian PQ in the point L as the geodetic intersects it in G : thus the relative course of these two lines will be seen. The first column of the following table contains the successive distances of the intermediate points from Bordeaux : e, LATITUDE OFP Q NORTH OFP L NORTH or P G NORTH OFP L NORTH OF G G NORTH OF Q O 1 II t II ft. ft. ft. ft. ft. 000 44 5 20 o-o 0-0 0-0 o-o o-o 2 15 33 45 2 7 4i 18-26 5-40 14.00 - 8-60 -4-26 4 31 6 45 59 5 8 32.19 17.66 26-53 - 8-87 -5-66 6 46 39 46 26 58 41-93 31-80 36-70 -4-90 - 5-23 9 2 12 4 6 4 8 33 4762 44-04 43-77 + 0-27 - 3-85 ii 17 45 47 4 32 49-36 51-71 47-20 + 4-51 2-16 13 33 18 47 H 5i 47-23 53-19 46-54 + 6.65 o 69 15 4 8 5 1 47 19 2 3 41-25 4788 4 J -57 + 6.31 + 0-32 18 4 24 47 18 8 31.42 36-17 32-13 + 4-04 + 0-71 20 19 57 47 ii 6 17.70 19.42 18-24 + 1-18 + 0-54 22 35 30 46 58 20 o-o o-o 0-0 o-o o-o Here both the geodetic and the curve of alignment cross to the north side of the curve Q : the geodetic departing but very slightly to the north. In fact the azimuth of the geo- detic at Nicolaeff differs from the true azimuth by only 0"-152 by the formula (10). For here the condition shown in equation (11) is very nearly fulfilled. We may determine the point of intersection of the geodetic with Q from the equation (13) ; for the whole distance being the longitude of B. In this case the auxiliary triangle is right-angled, the sides containing the right angle are 90 U and. o-, the third side is 90 ^, and the two other angles are CT and a. Hence sin u = sin Ucos from tzr, we must develope the second of equations (15). Expand- ing the radical, we get . 9. The results we have just obtained enable us to solve the GEODETIC LINES. 139 more general problem : a geodetic starts from a given point A, whose reduced latitude is u n in a given initial azimuth a,, to determine the latitude and longitude of a point B on the geodetic whose distance from A measured along that curve is ?. In the solution of this problem we shall omit the terms in e 6 as unnecessary for our purpose. Let PHK be the auxiliary triangle, P corresponding to the pole ; H, K, to A and B respectively, so that PH=90 u n PHK = a,. Drop the perpendi- cular PM on HK, produced if necessary, and let H "' Fig- 25. PKM=a. Then we have from the right-angled triangle PHM, sin u, = sin U cos 2, cos a, cos u, sin U sin 2, sin a, cos v., cos U ; whence U and 2 are obtained with a check. Now as s cor- responds to o-, so let S be the linear distance which corresponds to 2, then by (19) since KM 2 o-, 8 S-s .'. ^ n- + 7? CQS (9. y. tr) sir> rr a*/l- follows from (20) in the following manner: first, in the right hand member of this equation put 2 for a ; secondly, write 2 0- for a, subtract the second result from the first, then we 140 GEODETIC LINES. have the co r of our present problem expressed thus co w = cosU {A'v .B'cos^S cr) sincr}, (23) where the values of A' 9 B' t are as given in (21). 10. We shall now give the working in full of the following numerical example. Given the latitude of the centre of the Tower of Dunkirk 51 2' 8"-41 as determined by observations with Ramsden's Zenith Sector ; the latitude of the vane of the Munster Tower of Strasburg Cathedral 48 34'55"-94, and the difference of longitude of these points (Annales de r Observatoire Imperial de Paris, Tome VIII, pp. 256, 320, 356) 522'28"-440; to determine the shortest distance between them, and their mutual azimuths. We take the elements of the spheroid, a = 20926060, 0:0 = 294:295, whence logtf 2 = 7-8304712, loge? 2 cosec 2" = 2-8438663, logaVle 2 = 7-3192128. Using only seven place logarithms, we shall omit the terms in fc 6 . It is unnecessary to give the calculation for determining the reduced latitudes of the stations, they are found to be u, = 50 56' 25"-837, log cos u, = 9-7994281, / =48 29 8-406, log cos u' 9-8213873, also, co = 5 22 28 -440, log sin co =8-9715838. In the auxiliary triangle whose sides are 90 &, 90 u', the included angle is w of which an approximate value is co, the true difference of longitude and the third side or. The other two angles are the azimuths of the geodetic line at the terminal stations. If 90 U be a perpendicular dropped on the side o- from the opposite angle, and 2 the distance from the point u, to the foot of this perpendicular, sin cr cos U = cos u, cos u' sin -ar, cos 2 sin 7= sin^,; GEODETIC LINES. 141 put now . e z sin 2 U 2 + y * 4 )> 2 + 3 K 4 , then s being the shortest distance required, o-) sin o- +(7 cos (4 2 2o-)sin2-f-5co, where e* / 8-59240 5co = 27 // -33 1-43667 log cosec o- 1-12981 o-j CT O = 14"- 41 1-15888 142 GEODETIC LINES. whence ^ = 5 22' 55"-77 and ^ = 4 15' 25"-6. From these we get U and K with sufficient approximation, log cos u, cos u' 9-6208154, log sin -53-! cosec o-j 0-1016022, log cos U 9-7224176, log sin U 9-9291162, log e 2 sin 2 U 7-0866436, log (1 + e 2 cos 2 U) 0-0008179, logK 2 7-0874615. The value of 2 from the equation cos 2 sin U = sin u, is 2 = 23 54' 49"- 3 ; it is negative because the perpendicular U falls on the opposite side of Dunkirk to that in which Strasburg lies. Hence, logcos (22 o-,) = 9-7885193. Then to get r from (25) the work stands thus, putting for a moment N for e 2 cos n, cos u sin -BTJ cosec 2", \ogN 1-436877 1-4369, log sec* o-j 0-000400 cos (2 2 o-) 9-7885, -f27 // -3701 1-437277 \ K 2 6-7864, log (i e 2 ^ K 2 ) 7-03362 0103 8-0118, + 0"-0296 8-47090 .'. tsr = o> + 27"-389 = 5 22' 55"-829. We have now to compute the third side and remaining angles of the spherical triangle whose sides are complements of u, and u' t and w the included angle. The most correct way of determining the angles is by computing the tangents of half their sum and half their difference, thus we obtain the azimuths Strasburg at Dunkirk ... 123 7 20 // -41, Dunkirk at Strasburg ... 52 46 11 -46 ; the last being measured in the direction north towards west. To determine the third side o-, we might proceed in different ways, but owing to the uncertainties of the seventh place of the resulting logarithm there will remain an uncertainty of between two and three units in the third decimal of the seconds. The value as far as seven-place logarithms can give it, is 0-= 415 / 25 // -710. GEODETIC LINES. 143 There remains no difficulty in computing the length s from the formula (25) as we are in possession of all the necessary logarithms. We shall therefore merely give the values of the parts of the different terms depending on the different powers of K : TERMS IN a sin a sin 2ff ft. I 540^0-6 ft. ft. K 2 1895-27 n6vj;6 * 7-54 4-27 0-14 The sum of these gives s = 1552630-4 with an uncertainty of one or two units in the place of decimals. The true distance is 1552630*300. The astronomical azimuths (com- puted with precision) are Strasburg at Dunkirk ... 123 7' 20"-165, Dunkirk at Strasburg ... 52 46 11 -725. -If now by the formula (12), page 130, we compute the differences between the true and geodetic azimuths, we find that to reduce the former to the latter the corrections -f 0"-24 and 0"-26 are to be applied. Adding these we get again the geodetic azimuths as before. 11. In the particular manner in which we have deduced the equation of the geodetic line, there is this disadvantage that we have lost sight of one of its principal characteristics. Let p, q be adjacent points on a curved surface; through s the middle point of the chord pq imagine a plane drawn perpendicular to pq, and let S be any point in the intersection of this plane with the surface. Then jpS+ Sq is evidently least when sS is a minimum, that is, when sS is a normal to the surface ; hence it follows, that of all plane curves joining pq, when those points are indefinitely near to one another, that is, the shortest which is made by the normal plane. That is to say, 144 GEODETIC LINES. the osculating plane at any point of a geodetic line on a curved surface 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 I, m, n, of BC be I', m', n, then %, y, z being the coordinates of B, those of A and C will be respectively sccl, ycm, zcn } z + cn', and consequently, the coordinates of M, the middle point of AC are therefore the projections of BM on the coordinate planes are and the direction-cosines of BM are proportional to I' I, m'm, n'n. If the angle made by BC with AB be in- definitely small, then the direction-cosines of BM are pro- portional to 8, bm, bn. Now if AB, BC be considered two contiguous elements of a geodetic curve, then BM must be a normal to the surface, and since bl, bm, bn are in this case represented by das , dy dz .-r-> #.-r-> a. -7-1 as ds ds we have, if u = be the equation of the surface, _ du du ^ ' dx dy dz In the case of the ellipsoid the equations of the geodetic line are GEODETIC LINES. 145 For the spheroid, where a = b, d*x d*y _ y ~d~ X d? = This being integrated gives ydxxdy = Cds, which leads to the equation r 2 dca = Cds, as before. The two equations (26) are equivalent to one only, for of its three members any one can be deduced from the other two by means of the equations du .. du du , -j-dx + dy-\-- r -dz-=Q, dx dy y dz ds* r ds 2 y r ds 2 On the subject of this chapter, see an interesting paper : Sur les ecarts de la ligne geodeslque et des sections planes nor- males entre deux points rapproches (Tune surface courbe, par F. J. Van den Berg (Extrait des Archives Neerlandaises, t. xii). CHAPTER VII. MEASUREMENT OF BASE-LINES. THE Geodetic Standards of length of different countries vary in length, in form, and in the material of which they are composed. They are divided into two classes, standards 'a traits' and standards c a bouts'; in the first, the lines or dots defining the measure are engraved on small disks of silver, platinum, or gold let into the bar ; in the second, the bar generally has its extremities in the form of a small cylinder presenting a circular disk, either plane or convex, of hard polished metal, or sometimes of agate, for the contact measurements. The unit of length, in which by far the greater part of the geodetical measurements in Europe are expressed, is the Toise of Peru, a measure, ' a bouts,' of which fortunately there exist two copies (compared with the original and certified by Arago), one made for Struve in 1821, and a second for Bessel in 1823 ; it has moreover a third representative in Borda's Rod, No. 1. The Standards of Belgiunrand Prussia are copies of the toise of Bessel ; and the Russian Standard, which is two toises in length, is measured from the toise of Struve. The Standard of the Ordnance Survey is ten feet in length and in section a rectangle of an inch and a half in breadth by two and a half in depth, supported on rollers at J and f of its length. The ends of the bar are cut away to half its depth, so that the dots marking the measure of ten feet are in the neutral axis. The standard yard of this country and its copies are bars, an inch square in section, of iron, steel, brass, or copper ; the lines defining the yard being in the axis of the MEASUREMENT OF BASE-LINES. 147 bar. The Standard of the Spanish Geodetical Survey is a bar of four metres in length, constructed of two plates of iron ri vetted together in the form of a JL. The denning lines are on the upper edge of the vertical bar. Standards of length are generally provided with thermo- meters which either lie in contact with the metal or have their bulbs bent downwards so as to enter into cylindrical holes in the upper surface of the bar. It is necessary that the errors of these thermometers be known with considerable accuracy, for an error of a tenth of a degree of temperature corresponds to an error of nearly a millionth of the length of an iron bar and quite that amount in a bronze bar ; it is therefore neces- sary that the error be less than, say, 0-04. These thermo- meters are compared with standard thermometers from time to time. A standard thermometer for geodetic purposes must be the best workmanship of the best workman, and the residual errors of the division-lines have to be determined from special observations and measurements. These consist in, the deter- mination of the boiling point, that also of the freezing point, tne determination of the errors of calibration, and finally, the comparisons together of the standards after the application of the corrections which shall have resulted from the foregoing operations. As all thermometers have an index error which is liable to slow variations in the course of time, it is necessary frequently to redetermine the freezing point by placing the thermometer in broken ice. A convenient method of pre- paring the ice is to plane it from a block with a rough plane. In the comparisons of thermometers with one another it is essential that they be held in water. In the comparisons at Southampton the thermometers are carried on a small plat- form of perforated zinc in the middle of a rectangular vessel measuring 16 inches square by 36 inches in length, thus the thermometers are covered with about seven inches of water when under comparison. There is a piece of mechanism for agitating the water throughout its mass at intervals, so as to prevent any local cooling. The thermometers, lying hori- zontally, are read by a vertical micrometer microscope from above. It may be interesting to give here the results of comparisons L 2 148 MEASUREMENT OF BASE-LINES. of two important standard thermometers ; one is that on which depend all the comparisons of standards made at Southampton, the other is a standard used for similar purposes in India. An examination of the boiling and freezing points of these thermometers made at the time of the comparisons show that the former requires the correction -010 (32) 0-41 where t is the thermometer reading, and the latter 0'010 (32). The calibration corrections given in the second and fifth columns result from a large number of micro- meter measurements of the capacities of the different portions of the tubes. The first and fourth columns contain each the mean of five simultaneous readings of the thermometers in water ; the room in which the comparisons were made having been kept at a temperature not differing more than a couple of degrees from that of the water. ORDNANCE SURVEY STANDARD. INDIAN SURVEY STANDARD. Reading. Cal. Corr. App. Error. Reading. Cal. Corr. App. Error. o o o 97.84 - 0-032 + 0-005 97.46 - 0-073 - 0-005 92.58 0-062 0-000 92-11 - 0-005 0-000 87.61 - 0-039 + 0-010 87-I5 O-OI2 O-OIO 82-72 - 0-095 0-005 82-25 O-O2O + 0-005 77.92 0-116 o-ooo 77-43 - 0-044 0-000 72.56 0-106 0-015 72-03 + 0-046 + 0-015 67.70 - o 083 + O-OIO 67-17 4- 0-029 O-OIO 62.64 0-068 0-005 62-15 + 0-035 + 0-005 57-95 - 0-059 - 0-015 57-48 + 0-036 + 0-015 52-63 0-041 O-O2O 52.20 + 0-016 + O-O2O The 'apparent error' in the third and sixth columns is the difference between the individual readings after correction and their mean. The standards when boiled were kept in a horizontal position. If T be the reading of the thermometer, S that of the barometer (reduced to 32) at the same moment, then the error E of the boiling point is E= T2I2 + 1-680 (33 B), where 93 is Laplace's standard atmospheric pressure, namely in MEASUREMENT OF BASE-LINES. 149 latitude $, and at the height of Ji feet above the sea, 23 = 29-9215 in. + - 0785 cos 2< + - 0000018 h. This is 0760 of a metre in the latitude of 45 or 30-000 inches at the equator ; in either case at the level of the sea. A few minutes after boiling, the thermometers are placed in ice for the determination of the index error. But it is not sufficient to define a measure as the distance between two marks on the upper surface of a bar of metal at a given temperature, for the bar is not a rigid but an elastic body, which changes its form according to the manner in which it is supported. If an uniform elastic rod be supported at its centre in a horizontal position, the whole of the material above the ' neutral axis ' is in a state of tension, while the lower half is compressed ; and an exactly opposite state exists if the rod is supported at its ends. Suppose the bar to be of length a, and its section a rectangle of breadth h and depth k : let w be its weight, and a the small quantity by which the bar would be either lengthened or shortened by an extending or jcompressing force equal to w. Then, supposing the bar to be in its unconstrained state perfectly straight, if p be the radius of curvature of the axis at any point q when the bar is slightly bent in the plane of k, the sum of the moments of the elastic forces developed in the transverse section of the bar at that point may be shown to be equal to > and this must be equal to the sum of the moments of the external forces tending to bend the bar round q. 4 2 . 5 LetP,P-bethe Fig ^ 2 points of support of the rod at the distances b, V from C, the middle point of AB. Let the equation of the rod, or of its axis rather, be expressed in rectangular coordinates #, y, the axis of x passing through the points of support, and x corresponding to the point C. Now the external forces tending to bend qB round q are its own weight and the reaction of the support at P 7 , and the sum of the moments of these forces is bw , . w . . 150 MEASUKEMENT OF BASE-LINES. Hence, the condition of equilibrium is a& 1 2&y a b'-b It is easy to see from this that there can be no points of inflection unless b + b' >\a. That is, unless the supports are further apart than half the length, the whole bar will be convex upwards. This equation however does not refer to the portion PE of the bar ; for that portion we find - -- - ~ ~ 6a p 4 a Now since the flexure of the bar is really very small, we -, dy z , ... 1 d^y 6 a , , may omit -~^. > and putting - = -~ and //, = ^ > the above dx* & p dx* ak 2 equations become .i + ^i^-i 2 , (i) ?-? (2) from whence we obtain by two integrations the equation of the axis. The distance between two points on the upper surface of the bar at its extremities will be variable, not only from the curvature of the neutral axis shortening its horizontal pro- jection, but from the compression of extension of the upper surface. The change of length arising from the first is generally quite inappreciable, that from the second is large, a source of error unless guarded against. Imagine normals drawn at A, C, and B to the axis of the bar in its vertical plane, the angle between the normals at G and B sup- posed to converge upwards being 6, and let 0' be the angle between the normals at A and ./o + 4 o+O a' J b t \ 4 a ,, w ^ ~ 24' MEASUREMENT OF BASE-LINES. 151 with a similar expression for fl 7 ; thus the contraction of the upper surface is, if we put a a = &, For a bar supported at its centre & = b'= 0, and the surface is extended to the amount \ c. For a bar supported at its extremities the contraction is \ e, being double the amount of the extension in the previous case. If supported on rollers at one-fourth and three-fourths of its length, the upper surface is extended T V e. But if we place the supports so that the distance between the extreme marks on the upper surface will be the same as if the bar were straight ; a particular case of a more general theorem due to the Astronomer Royal (Memoirs of the Royal Astronomical Society, vol. xv). The variations of length to which the upper surface of a bar is thus liable have given rise to the practice of engraving the lines indicating the measure on surfaces (gold, silver, or platinum disks) in the neutral axis. At the Ordnance Survey Office, Southampton, is a building specially constructed for comparisons of standards. The inner room, measuring 20 feet by 11, with thick double walls, is half sunk below the level of the ground, and is roofed with 9 inches of concrete. An outer building entirely encloses and protects the room from external changes of temperature ; so that diurnal variations are not sensible in the interior. Along one wall of the room are three massive stone piers on deep foundations of brickwork ; the upper surfaces of these stones (which are 4j feet above the flooring on which the observer stands) carry the heavy cast-iron blocks which projecting some seven inches to the front over the stones hold in vertical positions the micrometer microscopes under which the bars are brought for comparison. Each micrometer micro- scope is furnished with an affixed level for making its axis vertical ; one division of the micrometer is somewhat less than the millionth of a yard. It is a most essential point in the construction that the 152 MEASUREMENT OF BASE-LINES. foundations which carry the stone piers the supports of the bars under observation and the flooring on which the ob- server stands, are separate ; thus, no movement made by the observer communicates any motion either to the bars or to the microscopes. The illumination of the disks (on the bar) which bear the lines or dots indicating the measure, is effected by the light of a candle placed some ten inches behind each microscope : the light of the candle passes through a large lens which forms an image of the flame on the disk, giving abundant illumination with a minimum of heat. When two bars are to be compared they are placed generally in the same box side by side and close together; each bar rests immediately on rollers to which a fine vertical movement can be communicated. The first adjustment is to level one of the bars and bring the microscopes over the terminal dots ; the microscopes are then made truly vertical, brought per- fectly to focus, with the collimation axis closely over the dots. It is usual to arrange a pair of bars at least twenty-four hours before any comparisons are made, so that a steady equality of temperature may have been obtained. The bars are visited for the purpose of comparison three or four times a day ; all adjustments are frequently put out and renewed, and the bars themselves are made to interchange places so as to avoid constant error, the possibility of which requires to be ever kept in mind. The observations made at one visit and con- stituting ' a comparison' are these : (l) The thermometers in the bars are read ; (2) the bar A being under the microscopes the lines or dots at either end are bisected and the micrometer read ; (3) the second bar is brought under the microscopes and read ; (4) B is thrown out of focus, brought back again, and read again ; (5) A is observed a second time after renewed focussing; (6) the thermometers are read again. As no artificial temperature is used, it is the practice to compare bars when the temperature is near 62, which is the standard temperature for standards of length in this country, and again when it is much lower, so as to eliminate the dif- ferences of expansion. The absolute rates of expansion have been determined for bub few standards, although an elaborate MEASUREMENT OF BASE-LINES. 153 apparatus exists in the comparison room just described for the determination of absolute expansion of ten feet bars. The great point in this apparatus is the possibility of maintaining a bar at a high temperature, such as 90 or 100, without per- ceptibly heating the room; then comparing, if for instance there be two bars A, IB ; A hot with B cold ; and again, B hot with A cold. Each bar lies closely between two long narrow tanks of copper ; the cold bar has either ice or cold water in its tanks, while those of the hot bar are continuously supplied with hot water by flexible feed pipes from a large cistern maintained steadily at the required temperature outside the building ; the hot water continually running away from the tanks and passing out of the room by flexible waste pipes. A special mechanism permits of the rapid interchange of the bars, each with its tanks, under the microscopes. The following coefficients of expansion were obtained for four ten feet bars from 6500 micrometer and thermometer readings : Indian Standard: Bronze ... 0-0000098277 -0000000057, Steel ... 0-0000063478 -0000000056, Ordnance Survey: Iron ... 0-0000064729 -0000000031, Iron ... 0-0000064773 -0000000033. In expressing micrometer measurements and their probable errors it is convenient to use as unit the millionth of a yard. With this unit the probable error of a single micrometer bisection of a good line is, for an expert observer, 0-25, but for a coarse or ill defined line, or a dot, it may be considerably more. The probable error of a single comparison of two bars depends on their length as well as on the quality of the lines : for a yard it varies from 0-35 to 0-66; for a bar of 10 feet it may be between -65 and 1-30. The micrometer observations in the comparisons of standards are affected to a small extent by c personal error ' : that is to say, what one observer may consider a 'bisection', another observer may think to be in error. What is technically termed a bisection is the placing of the spider lines of the micrometer centrally over the line to be observed ; or the adjusting the parallel micrometer lines so that the engraved 154 MEASUREMENT OF BASE-LINES. line on the bar may appear equally distant from them. Thus, if one observer make a bisection, and two others were to make a drawing of what they see, they might produce such results as a and b, fig. 27, where the fine lines fl 5 are the micrometer lines, and the thick line is the engraved line on the bar. There is however not altogether a constant difference between two observers ; it is only on some lines that there is any personal error, and then it is but a very small quantity. The difference of opinion seems to arise from some inequalities about the edges of the engraved line. Those lines Fig 27 which bring out the greatest amount of personal error are fine or faint lines ; the best lines for observing are those whose edges are clean and parallel. In the platinum metre of the Royal Society the lines are very fine and difficult to observe. When the observations to be made with a micrometer micro- scope are such that a large number of divisions have to be measured, involving it may be perhaps several revolutions of the micrometer, it is absolutely necessary to investigate the errors of the screw. As the measurement of any space on a scale is affected by the error of focal adjustment, it is necessary in measuring spaces for the determination of the values of the screws of micrometers that the focus be readjusted at every measure. In the principal micrometer microscopes of the comparison apparatus at Southampton the value of a division of the micrometer is for the one microscope 0-79566 -00008, and for the other 0-79867 -00009; these values are not sen- sibly affected by temperature. The probable errors in the measurement of n thousand divisions are for the respective microscopes and \/- the larger quantity arising from that microscope having less perfect definition. In the case of standards, 'a bouts/ the surfaces of the circular terminating disks should be slightly convex ; but the radius MEASUKEMENT OF BASE-LINES. 155 of curvature of these surfaces is a disposable constant, which may be turned to account in the following manner. The true length, or, which is the same, the maximum length of the bar, is the distance of the centres C, C' of the two disks, these, as well as the corresponding centres of curvature being in the axis of the bar. If the measure be made from any point P on the surface of one disk to a point P' on the other disk, the distance PP', if taken as the length of the bar, will be in error. Now we may take the radius of curvature p of either disk such that the chance error consequent on measuring between any other than the centre points of the disks may be a minimum. Let 2 a be the length of the bar, 2c the diameter of the small disks. Take the centre point of the axis of the bar as the origin of coordinates, the axis itself being that of z, if r, r be the distances of P, P' from the axis, we may put for the coordinates of those points r 2 x = r cos 0, y = r sin 0, z = a -- , 2p One of these angles, as 0', we may put = 0. Then (2a- > 2p .-. 20 PP' = (2 - (r 2 + r' 2 )r 2 / 2 + 2 40 v p which is the error of the measurement. The sum of the squares of these errors for all pairs of points is -2a v which is a minimum when p = 2 a ; that is, the centre of curv- ature for either disk must be at the other end of the bar. In order that the triangulation of the continental countries of Europe might be put in connection with the triangulation of England, the Government of this country, at the suggestion of General Sir Henry James, then Director of the Ordnance 156 MEASUKEMENT OF BASE-LINES. Survey, invited the Governments of Russia, Prussia, Belgium, Spain, Austria, and also the United States of America to send their standards to Southampton to be compared. The in- vitation in each case was complied with, and an account of the comparisons, which are of the highest importance to Geodesy, will be found in two papers in the Philosophical Transactions for 1866 and 1873: fuller details are given in the work entitled Comparisons of the Standards of Length of England, France, &c., by Col. Clarke, R.E. The following are some of the principal results of these comparisons, the Old English capitals representing the true lengths of the Yard, Toise, Metre, and Klafter : NAME OF STANDARD. STAND. TEMP. ACCREDITED LENGTH \ LENGTH IN ENGLISH YARDS. Belgian Toise Prussian Toise Russian Double Toise o 61-25 n 1. { O-OOIOO {JD 0-00099 2 C 0-00560 2-13150851 If 2-13150911 4-26300798 mm. Spanish 4 Metre Bar. Platinum Metre, Koy 1 . Soc*. n 32-0 4iH+ 0-40710 $fl 0-01759 1. 1-09360478 Pulkowa copy of Klafter ... Milan copy 2 Kj 3 61-25 ii It 0-00029 2-t 0-00580 2-07403658 2-07401462 " " K ' ii 3& o-ooooo 2-07402990 The first three lines in this table afford, from many thou- sands of observations, three entirely independent values of the toise. The greatest divergence of any one of the three values from their mean is but half a millionth of a toise. Then the toise being known, the length of the metre follows by means of the definition 443296 & = 864000 fE. A further check on this value of the metre is afforded by the Spanish bar, of which the length, as taken from Borda's rod No. 1, is 4-0004071 M. 1 The ' line ' represents the 864th part of the Toise, or of the Klafter. 2 The Milan copy of the Klafter of Vienna has two measures of the Klafter laid off on it, one on its upper surface defined by dots i. 3, the other on its under surface by dots marked by Koman numerals I. II. MEASUBEMENT OF BASE-LINES. 157 According to the observations at Southampton the Spanish bar is 4-0004052 H, a difference of only half a millionth of the length. The final results are these : H= l-09362311g, &= 2-07403483g. The lengths adopted for measured bases have varied accord- ing to the circumstances of each case. That of Bessel in East Prussia, as we have seen, was but little more than a mile in length whereas the base line of Ensisheim in France measured by Col. Henry was 11-8 miles. Between these limits they may be found of all lengths. In India, with the exception of the line at Cape Comorin of 1-7 miles, the remaining nine bases are between 6-4 and 7-8 miles. In the Spanish triangu- lation are several short bases of about a mile and a half; the principal base, near Madrid, is 9*1 miles long, and there is one of just a mile in length in the Island of Ivica. In selecting ground for a base measurement, the conditions to be secured are that it be fairly even, and free from obstacles, and that the extremities should not only be mutually visible, but command views of more distant stations of the triangu- lation, so that the sides of the triangles, commencing with Fig. 28. the base, may gradually increase. The first of the annexed diagrams shows the connection of the base measured at 158 MEASUREMENT OF BASE-LINES. Epping, Maine, United States, with the adjoining trigono- metrical stations. The second shows the connection of the base measured near Ostend by General Nerenburger in 1853. The knowledge of the length of a metal bar at any moment involves three distinct matters : the length at some specified temperature, the coefficient of expansion, and the temperature of the bar at the moment in question. The first is known by repeated comparisons with the Standard ; the second can be obtained only from special experiments ; the exact tem- perature of a bar at any moment can only be inferred from the indications of thermometers in contact with it, involving the assumption that the temperature of the bar is the same as that of the mercury in the thermometers. But experiments have shown that we may be deceived in this. To evade the temperature difficulty two different forms of construction have been adopted, one that of Borda where the measuring bar is composed of two rods of quite different rates of expansion, forming a metallic thermometer ; the other that of Colby, where by a simple mechanical arrangement two rods of different expansions are made to present two points at an invariable distance. Whatever be the apparatus used, it is essential that the measure be confined strictly to the vertical plane containing the extremities of the base ; and that the deviations, in the vertical plane, of the line actually measured or traced by the individual bars from a straight line, be precisely measured. The first part of this sentence requires however to be qualified it is sometimes necessary that two or more segments of a base be not absolutely in the same straight line ; this is no disadvantage when the angles the different parts make with one another are known. But in each segment the measure must be in one vertical plane. As a preliminary operation to the measurement of a base it is usual after getting an accurate section of the line by spirit levelling, to measure the distance in an approximate manner. In making this measure one or two or more points are selected in positions convenient for dividing the base into segments. The selected points are subsequently adjusted into the line of the base with the utmost precision by means of a theodolite MEASUREMENT OF BASE-LINES. 159 or transit instrument erected at either or both extremities ; or if they be not absolutely in the line, angles measured at them indicate their real position. These intermediate points are sometimes preserved in the same permanent manner as the terminal points of the base, namely, by a fine mark on a massive block of stone set in brickwork. The mark itself may be a microscopic cross drawn on the surface of a piece of brass cemented into the stone, or it may be a dot on the end of a piece of platinum wire set vertically in lead run into a hole in the stone. In some cases the ends of a base have been indicated by small vertical facets. A more detailed aligning of the base follows. By means of the theodolite or transit instrument over the ends and inter- mediate points, pickets are driven into the ground at regular intervals ; each picket carries a fine mark indicating exactly the line of measurement. As the errors resulting from faulty alignment do not tend to cancel, being always of the same sign, this operation always receives the last degree of care. ... In attempting to give a description of the apparatus and processes for measuring base lines it would be quite beyond the purpose of this work to enter into the details which are very complex. Abundant information can be obtained from such works as the following : Compte rendu des operations . . . a la mesure des bases geodesiques Beiges, Bruxelles, 1855 ; Triangu- lation du Royaume de Belgique, Bruxelles, 1867; Experiences faites avec I'appareil a mesurer les bases, Paris, 1860; Base Centrale de la triangulation geodesique d'espagne, Madrid, 1865 ; the Account of the measurement of the Lough Foyle base, London, 1847; and other works. In the apparatus used by the Russian Astronomer F. W. Struve in the measurement of base-lines, there were four bars, each two toises in length, of wrought iron. One end of each bar terminates in a small steel cylinder coaxal with the bar, its terminal surface being slightly convex and highly polished, the other end carries a contact lever of steel connected directly with the bar. The lower arm of this lever terminates in a polished hemisphere, the upper arm traverses a graduated arc also rigidly connected with the bar. When an index line at the end of the longer arm points to a certain central division on 160 MEASUREMENT OF BASE-LINES. the graduated arc, the bar is at its normal length, but its length is also known corresponding to any reading of the arc. The annexed figure shows the contact lever. In measuring, the bars are brought into contact, which is main- tained by a spring acting on the lever. Each bar held at two points is protected within a box from which its extremities project ; it is further protected from variations of tem- "Jj ~~ perature by being wrapped in many folds of cloth and raw cotton. Two thermometers, whose bulbs are let into the body of the bar, indicate its temperature. The end of a day's work is marked by driving into the ground under the advanced end of the front bar a very large iron picket to the depth of two feet. This picket carries an arm with a groove, in which slides, and can be fixed, a metallic cube having a fine mark on its upper surface. The projection of the end of the bar over this mark is effected by means of a theodolite established as a transit instrument at a distance of 25 feet in a direction perpendicular to the base. Struve investigates very carefully for his several bases the probable errors arising from the following causes. 1 . Errors of alignment. 2. Errors in the determination of the in- clinations of bars. 3. Error in the adopted length of the working standard. 4. Error in the adopted lengths of the measuring bars. 5. Error in reading the lever index and of the graduation. 6. Personal errors of the observers. 7. The uncertainty of temperature. This last was subdivided into four headings, (1) uncertainty in the expansion of the standard, (2) uncertainty in the expansion of the measuring bars, (3) uncertainty of temperature during the comparisons of the bars and standard, (4) uncertainty of the mean temperature at which the base was measured. The probable errors of the seven bases measured with these bars range from 0-73/x to 0-91ju, where jx is a millionth part of the length measured. The remainder of the Russian bases, three in number, were measured by the apparatus of M. de Tenner. In this system ~^> r C S i 1 1 /" N S o v y 1 1 H } kf _JS n MEASUREMENT OF BASE-LINES. 161 the measuring bar is of iron, and the intervals between bars in the line is measured by a fine sliding scale. The accuracy of these bases is not so great, the probable errors are about 3-ljut. Borda's measuring rods have been already described in connection with the work of Delambre. In Bessel's system the platinum and copper of Borda are replaced by iron and zinc, and the intervals are measured with a glass wedge. The annexed figure shows the small interval forming the metallic thermo- meter. The upper or zinc rod termi- nates at either end in a horizontal knife edge. The small piece affixed to the upper sur- face of the iron rod has two ver- ' tical knife edges, Fig. 30- one forming the end of the measuring rod, while the other, or inner knife edge, forms with horizontal edge of the zinc the small interval which constitutes the thermometer. The rods are supported on seven pairs of rollers carried by a bar of iron, the whole being protected in a case from which the contact ends of the rod project. A small longitudinal movement of the rods by rolling on the supporting rollers is communicated to them by means of a slow motion screw of which the milled head projects from the box. The glass wedge has a length of about four inches, being 0-07 of an inch thick at the smaller end and 0-17 at the larger end; it has engraved on its face 120 division lines 0-03 of an inch apart. Denote by r the standard temperature to which the measure- ments and comparisons are reduced. Let the lengths of the zinc and iron rods at this temperature be I, I', then at any other temperature they will be thus expressed, M 162 MEASUREMENT OF BASE-LINES. Let the difference between them, as measured by the wedge, be i at the temperature t, then i = I' l+(e e) (t r), and eliminating t T, W-l'e e'i that is, the length of the rod is expressed in the form A where A and B are constants to be determined for each compound rod. The lengths of the rods may be written thus L 2 = L 3 = The small differences of the bars represented by the quanti- ties a?, the sum of which is zero, and the values of the thermo- metric coefficients y+ are determined by comparisons of the rods inter se. From these comparisons a system of eight equations is deduced by the method of least squares from which the a?'s and y's are obtained. Finally, the comparison of one of the rods with the standard gives I. The length of the base line is finally expressed in the form 31 = n 1 + a^ + fi% 2 + y % 3 4- 5^ 4 + ay^ + tfy 2 + y> 3 + 6> 4 . The probable error of Bessel's base was found to be 2-2^. Bessel's apparatus was used in the Belgian bases near Beverloo and Ostend, measured (1852, 55) with every imagin- able precaution by General Nerenburger. The mode of terminating a day's work by the use of the plummet, a weak point in Bessel's base, was replaced by the following procedure. The exact end of a day's work being decided in advance, a mass of brickwork was built from a depth of a couple of feet up to the surface of the ground ; in this was built a cast-iron frame presenting a surface flush with the brickwork. Another frame of iron two feet high, and which could be screwed to the former or removed at pleasure, carried on its upper surface a groove in which a small measuring rule 14 inches long was free to slide in the line of the base and to be clamped where required. This rule terminated in a vertical knife edge at one the advanced edge, and in a horizontal knife edge at the MEASUREMENT OF BASE-LINES. 163 following end. When the vertical knife edge of the last bar, at the end of the day's work, arrived near this apparatus, the small rule was set and clamped so as to leave between it and the measuring bar the usual small interval for measurement with the glass wedge. On the following morning the work was resumed by starting from the other end of the rule which thus formed a part of the base measure. The mean error of the base near Beverloo, 2300 metres in length, was 0-59ju; that of the Ostend base, 2488 metres, was 0-45 n ; these at least are the quantities as computed. In Colby's Compensation Apparatus the component bars are of iron and brass, 1 feet in length, firmly connected at their centres by a couple of transverse cylinders. At either extremity is a metal tongue about six inches long, pivoted to both bars in such a manner as to be perfectly firm and im- moveable, while yet not impeding the expansion of the bars. A silver pin let into the end of each tongue carries a micro- scopic dot, marked f a triangulation is dependent on the precision with which the observing theodolites are centred over the station marks. Whatever be the form of the signal o erected over a trigonometrical station, it is essential that it be symmetrical with respect to the vertical line through the centre mark, so that the observation of the signal shall be equivalent to an observation of a plumb-line suspended over the mark. For very distant stations a heliostat is used, which centred over the station observed, reflects the rays of the sun to the -observing theodolite. On the Ordnance Survey the heliostat is a circular looking-glass provided with a vertical and a horizontal axis of rotation, kept constantly directed by an attendant. It is essential that the theodolite be supported on a very solid foundation. The mode of effecting this must depend on the nature of the ground : generally it is sufficient to drive strong stakes as far as possible into the earth, then to cut them off level with the surface, and so form an immediate support for the stand of the instrument. In all cases the theodolite is sheltered by an observatory, the floor of which has no contact with the instrument stand. In order to command distant points it is sometimes neces- sary to raise the instrument by scaffolding 40, 60, or as much as 80 feet above the ground ; in such cases an inner scaffold carries the instrument, a second or outer scaffold supporting the observatory, as shown on the next page l . The method of observing is this : let A, B, C ... II, K be the 1 Fig. 38 is the drawing of a scaffold, seventy feet high, built by Sergeant Beaton, R.E. INSTRUMENTS AND OBSERVING. 181 points to be observed, taken in order of azimuth ; then, the instrument being in adjustment and level, A is bisected and the microscopes read, then B is similarly observed, then in succession the other stations C...H, K-, after K the movement of the telescope is continued in the same direction round to A, which is observed a se- cond time. This constitutes what is termed on the Trigonometrical Survey of Great Britain * an arc ' ( French , m ise ; Ger- man, satz). A more ordinary pro- cedure is to observe the points as be- fore in the order A, B, C...H, K, then reversing the direction of motion of the telescope, to reobserve them in the inverted or- Fig. 38. der^, H...C, B,A. Thus each point in the arc is observed twice. In order to eliminate errors of graduation it is the practice to repeat arcs in different positions of the horizontal circle ; some observers shift the zero of the circle after each arc, others take a certain number of arcs in each position of the zero. Supposing the circle to remain really fixed during the taking 182 INSTRUMENTS AND OBSERVING. of an arc (which is executed in as short a period of time as possible), the probable error of an observed angle will depend on the errors of bisection of the objects observed, on the errors of reading the circle, and on errors of graduation. If a be the probable error of a bisection, (3 the probable error of the mean of the readings of the microscopes, y the error of the angle due to faults in the division lines actually used, then the error of the angle as measured by n arcs in the same position of the circle is But taking the angle from n measures in each of m positions of the circle, the probable error is * I mn where y, having reference only to accidental errors of division, is a constant peculiar to each instrument. With a first-rate instrument in favourable circumstances the probable error of a bisection, including that of reading the circle is, 0"-20. The probable error of an observed angle depends on the instrument, on the observer, and on the numbers n, m. In the best portions of the Indian triangu- lation it is 0"-28; in Struve's observations in the Baltic Provinces it was 0"-38. Ramsden's Zenith Sector,- which had a telescope of 8 feet in length, was destroyed in the fire at the Tower of London, and was replaced by Airy's Zenith Sector. This instrument, represented in the next page, fig. 39, is in three parts; the outer framework, the revolving frame, and the telescope frame. The framework is cast in four pieces ; the lower part, an inverted rectangular tray with levelling footscrews; two uprights with broad bearing pieces screwed to the inverted tray ; and a cross bar uniting the tops of these uprights. Through the centre of this bar passes down- wards a screw with a conical point, which, together with the vertex of a cone rising from the centre of the inverted rect- angular tray, determine the axis of revolution and form the bearings of the revolving frame. INSTRUMENTS AND OBSERVING. 183 The revolving frame is of gun metal cast in one piece. It is also in the form of a tray strongly ribbed at the back, Fig- 39- having four lappets or ears acting as stops in the revolution. In the centre of the front of this frame is a raised ring of about nine inches diameter, forming the bearing plate of the telescope frame. Concentric with this ring at each end of the frame are the divided limbs, which have a radius of 20-5 inches, and are divided on silver to every five minutes. There is also at each end a raised clamping-limb roughly divided, to which the clamp for securing the telescope at the required zenith distance is attached. On the reverse side of the re- volving frame are mounted three levels. The telescope frame revolves in a vertical plane by a hori- zontal axis passing through the revolving frame. Cast in one piece with the telescope frame are, the ring for holding the object-glass cell of the telescope, the four micrometer micro- 184 INSTRUMENTS AND OBSERVING . scopes, which are afterwards bored through the metal, and the eye-piece. The micrometers are of the usual construction, the threads intersect in an acute angle, and have a range of about 10 minutes on the divided limb. In the eye-piece of the telescope are five meridional threads, carried by a fixed plate, and a single thread at right angles to them, moved by a micrometer screw. The tube of the telescope is merely a protection from dust, and carries no essential part of the instrument except a simple apparatus for regulating the amount of light illuminating the threads, which by the turning of a screw, increases or diminishes the orifice through which the light enters. The focal length of the telescope is 46 inches, the diameter of the object-glass 3-75 inches, and the magnifying power usually employed about 70. The deviation of the plane of the instrument from the meridian, which is generally very small, being carefully ascertained by observations of the transits of north and south stars, and the axis being as nearly as possible vertical, the observer sets the telescope to the approximate zenith distance of the star to be observed, clamps it, and before the star enters the field reads the four micrometer microscopes and the levels. The star on its appearance is bisected by the eye-piece micro- meter on one of the threads, the name of the thread being re- corded with the reading of the micrometer. The telescope is then undamped and the revolving frame reversed by turning it through 180 on its vertical axis, so that the face which before was east is now west. The telescope is quickly re-set to the approximate zenith distance and clamped, and the star again bisected by the telescope micrometer on one of the threads, generally the same one on which it was previously observed. The five micrometers are then read and the levels on the reverse side. This completes the double ob- servation. The amount of the azimuthal deviation is ascertained by comparing the differences of the observed times of transit of north and south stars with their differences of right ascension. If A be the excess of the difference of right ascension of two stars over the observed difference of times of their transits, 5 b' their declinations, and $ the latitude of the instrument. INSTRUMENTS AND OBSERVING. 185 then expressing A in seconds of time, a, the azimuthal deviation, is in seconds of space, 1 5 A cos 8 cos 6' ~ cos$ sin (8 8')' The correction to the zenith distance z t on account of this deviation, is cos 6 sin 2" The correction for the distance i from the meridian, of the thread on which the star is observed is i 2 tan 8 sin 1", the upper sign applying to south stars, the lower sign to north stars. The latitudes of 26 stations of the principal triangulation of Great Britain and Ireland have been observed with this instrument. The latitudes of a still larger number have been determined with the Zenith Telescope. This instrument, which is of very simple construction and very portable, is represented on the next page. The telescope, 30 inches in length, is fixed at one end of a short horizontal axis, and is counterpoised at the other ; thus the optical axis describes a vertical plane, that of the meridian when in use. The lower part consists of a tripod with levelling screws connected with a steel axis about 15 inches high, and an azimuthal setting circle. On the steel axis fits a hollow axis which carries at its upper extremity the horizontal axis of the telescope. The latter has a setting circle and a very sensitive level. The reticule consists of the ordinary five transit threads and a transverse thread moved by a micrometer screw of long range, by which an angle of 30' may be measured in zenith distance. In the plate forming the horizontal circle are four circular holes, by means of one of these, the telescope being pointed to the nadir, the col- limation is corrected by means of a Bohnenberger eye-piece and a basin of mersury. For observing, the first thing is to ascertain the reading of the meridian on the setting circle; this is done by a few transits observed. The observer is provided with a list of 186 INSTRUMENTS AND OBSERVING. stars in pairs ; each pair is subject to the condition that the interval of their right ascensions is between 2 m and 1 O m and the difference of their zenith dis- tances not greater than 15'; one star passes north of the zenith, the other south. Now let the telescope be set to the mean of the zenith distances and directed to the south, say, supposing the first star to pass south of the zenith. The star as it passes is bisected by the micrometer thread on the centre thread. The instrument is then rotated through 180 of azimuth, not disturbing the telescope ; the se- cond star will then at the proper time 1 ass through the field, and is in like manner bisected on the centre thread. Knowing the value ^W^ of a division of the micrometer we have at once, by the dif- ference of the mi- Fig 40. crometer readings, the difference of zenith distance of the INSTRUMENTS AND OBSERVING. 187 stars which leads immediately to the knowledge of the latitude. Each star observation is accompanied by readings of the level. In this method, refraction is virtually eliminated, since it is only the difference of the refractions at the two nearly equal zenith distances which has to be applied. The value of the micrometer-screw is determined by observing- on the micrometer thread, transits of Polaris, while its movement is vertical or nearly so, that is, from 20 m before to 20 m after its time of greatest azimuth. Thus, in connection with observations of the level, an accurate knowledge of the screw over its whole range is obtained. This instrument is the invention of Captain Talcott, U. S. Engineers, and is exclusively used for latitudes in the U. S. Coast Survey. Its weak point is that in the selection of the pairs of stars, it may be necessary to use some stars whose places are but indifferently known. In this latitude, however, no great difficulty is found in obtaining pairs of stars whose places are given either in the Greenwich or Oxford Cata- logues. As made by Wurdemann it is an instrument of extreme precision and most pleasant to observe with. We have had a case for instance at Findlay Seat in Elginshire, where thirty-one pairs observed successively in one night presented a range not exceeding 2"-00. The drawing in Fig. 41 represents a very excellent portable transit instrument used on the Ordnance Survey in connection with the Zenith Telescope. The uprights are of mahogany, built of pieces screwed together ; it has a reversing apparatus by which the telescope can be reversed in 15 s . The focal length is 21 inches and its aperture 1*67 inches. 2. A telescope mounted on a transverse axis, as that of an altazimuth or transit instrument, as it rotates round that axis, experiences alterations of force which, since the material of which both telescope and axis are composed is not rigid but rather flexible, tend to change its form. Suppose, in the first place, that the instrument is perfectly rigid, perfectly 188 INSTRUMENTS AND OBSERVING. collimated, and perfectly level, its centre thread tracing out, eay, the meridian plane; then if flexure be introduced, at every Fig. 41. zenith distance there will be a deflection of the telescope out of the meridian. It has been shown by Sir George Airy (Monthly Notices of the E. A. S., January 1865) from me- chanical considerations, that z being the zenith distance of the point to which the telescope is directed, this deflection is of the form A sin z + B cos z, from which it follows that the path traced by the centre thread is still a great circle. The pole of this great circle, instead of being at the east point of the INSTRUMENTS AND OBSERVING. 189 horizon, will have azimuth 90 + , and zenith distance 90-{-#, where a and b are minute angles. Now when the transit is reversed in its Y's, the pole of the great circle described in this case is in azimuth 2 70 + a, and its zenith distance is, as before, 90-j-3. That is, the instrument, though it collimates on a horizontal point, will not be in collimation at the zenith ; there will appear an error of the nature of a level error, changing sign with change of pivots, combining in fact with the error of inequality of pivots. The diagonal form of transit instrument, in which the rays of light instead of passing straight from the object-glass to the eye-piece are bent at right angles by a prism in the central cube and so pass out at one of the pivots, is not so well known in this country as in Germany and Russia. The advantages of this construction are, that the observer without altering his position can observe stars of any declination, that the uprights are short, and that the level can remain on the pivots as the telescope sweeps the meridian, nor need it be taken off on reversing the telescope. There is a special apparatus for reversal ; from very numerous observations made with one of these transits the disturbance due to reversal of the telescope was found to be 0"-19 in azimuth, and 0"-13 in level. But the effect of flexure in this instrument is very obvious. The weight of the telescope, the central cube, and the counterpoise, cause the prism to be displaced vertically downwards by a nearly constant quantity ; so that the image of a star in the field is always vertically below its proper place at a distance, say f. Thus every micrometer reading of an object in the field requires a correction -fcosz; the magnitude of /"can be obtained by comparing the reading of the collimation centre, as determined on a horizontal mark, with the same as determined on a collimator in the zenith or at any zenith distance not near 90. To determine f in the case of a Russian Transit of this description employed for a time on the Ordnance Survey, a collimator was arranged so as to be capable of being set at any zenith distance whatever ; the result from 172 observations was/= 3"-16 0".04. In the reduction of the observations, this quantity is added to difference of pivots, which in the same instrument was 190 INSTRUMENTS AND OBSERVING. 0"-65 0"-02 (see a Paper on this subject in the Mem. R.A. Soc., Vol. xxxvii). 3. Let $ be the latitude of the place of observation, z, a being the zenith distance and the azimuth of an observed star S whose declination is 8, its hour angle being h. We shall suppose that h is zero at the upper culmination, increasing from to 360 ; and that the azimuth is zero when the star is north, and increases from to 360 in the direction from north to east. Then in the spherical triangle ZPS, Z being the zenith and P the pole, ZP = 90- 0, PS = 90-8, PZS=a, jgPS=360-; and the following equations express z and a in terms of <, 8, /t, cosz = sin 5 sin $-f-cos8 cos$ cos^, (l) cos a sin z = sin 8 cos (/> cos 8 sin < cos ^, sin a sin z = cos 8 sin h. If T be the reading of the clock, r its correction, A the right ascension of the star, then the hour angle is given by h= 15(T+r-A). The equations (14) of spherical trigonometry express the influence upon z and a of variations in 8, , and h j thus, S being the parallactic angle, sin z da = sin ^8 + cos 8 cos SdA + cosz sinad(/>, dz = cos Sdb cos 8 sin Sdh cos a dcfr. When the place of a star is required with great precision it is necessary to take into account the effect of diurnal aberration, whereby the star is displaced towards the east point, e, of the horizon by the amount 0"311 cos sin Se, increasing thus the azimuth and zenith distance by quantities 8 a and bz, given by the equations, sin z8a = 0"-31 1 cos cos a, bz = 0"'31 1 cos , c c c= -8) 4. When the transit instrument is in the meridian a is near 90. In equation (5) for a write 90 + a, and suppose a to be so small that cos a may be put = 1 ; then also h will be 1 Gradmessung in Ostpreussen, page 312. O 194 INSTRUMENTS AND OBSERVING. very nearly or 180, and we may put cosh = 1. Thus asin (0 b) + bcos((j) 8) + c ^cos8 = 0, for an upper transit, where the coefficient cos z of I has been replaced by cos(c/> 5). For brevity let a, 6, c now stand for the azimuth, level, and collimation errors, divided each by 1 5 to reduce them to the unit of seconds of time, then the correction to the clock time is, since -^ h = T+r A, T = A T+ sin(0 6) sec 6 + ^ cos( 8) sec8 + . The last of these is specially convenient for the reduction of transits of stars near the zenith. On reversing the in- strument, which is done at least once or twice in each evening's work, the sign of c is changed, being positive for circle east and negative for circle west. The sign of b is positive when the east end of the axis is high. In these formulae, for lower culmination (sub polo) 180 8 must be written for 8, and 12 h + ^ for A\ also A must be increased by s - 02 cos $ sec 8 for daily aberration when great precision is aimed at. The method of least squares is generally adopted for the deter- mination of the azimuth, the error at a stated moment, and the rate of the clock ; every transit giving one equation. In commencing to observe with a portable transit at a new station, the first matter is to secure a very firm foundation, and to remove or reduce to a minimum the collimation error ; then having placed the instrument as nearly in the meridian as can be done by any ready means of estimation, to level the transverse axis. If the clock error be known the observer has merely to take a quick moving star of large zenith distance approaching the meridian, and follow it up to the moment of transit with the middle thread of the telescope. Suppose, however, the clock error to be unknown : in this case let two stars differing considerably in declination be observed, let the first give an apparent clock correction r 13 and the INSTRUMENTS AND OBSERVING. 195 second an apparent clock correction r 2 , then the formula , tan 1 . . will give very approximately the real correction of the clock, which will serve for placing 1 the instrument nearly in the meridian. The formulae (7) or (8) show that stars near the zenith are best suited for the determination of the time when there is uncertainty of azimuth. For determining the azi- muth it is desirable to include in an evening's observations one or more transits of close circumpolar stars, even if ob- served only on one thread. In order to secure this the portable transit is sometimes used out of the meridian, namely, in the vertical plane passing through a circumpolar star. The method of time determination by a transit instrument set in the vertical of Polaris is very generally adopted in continental Europe, having the advantage of securing the knowledge of the azimuthal position of the instrument with- out any uncertainty, the transit of each time star being immediately accompanied by an observation of Polaris. The instrument, instead of being placed in the meridian, is placed with its centre thread slightly in advance of the position of Polaris, and accurately levelled. For this method of observing, the instrument must have a micrometer carrying a vertical thread across the field ; also it must have an arrangement such as the screw shown in the transit instrument fig. 41, page 188, for altering the position of the instrument by definite small quantities. Let m be the micrometer reading of the colli- mation centre of the instrument, m the reading of the star when bisected, if ju, be the value of one division of the screw, p(m m ) will be the distance of the star, call it c\ from the great circle described by the collimation centre. It is sup- posed that micrometer readings increase as the thread moves towards the circle-end of the axis. The method of arranging the observations would then be somewhat as follows, de- pending of course on circumstances of weather, &c. : Circle East Transit of a time star and two bisections of Polaris. Circle West Polaris, two time stars, and Polaris. Circle East Transit of a time star and bisections of Polaris. o 2 196 INSTRUMENTS AND OBSERVING. These observations, supposed to constitute one complete time determination are to be accompanied by readings of the level. The azimuth and zenith distance of Polaris are to be com- puted and tabulated for every five minutes of time during the period the star is under observation. Let T' be the reading of clock corresponding to the observation of Polaris, T an approximate value, as near as can be obtained, of the clock correction, r-f Ar the real correction. Let the computed azimuth of the star corresponding to the time T' + r be a' , then if /3 be the change of azimuth for one second of time, a' + /3.Ar will be the true azimuth of the star at the moment of observation. In the equation (3) replace a by 90 + a, when it becomes c + l cos 2 + sin 2 sin (a a) 0. Thus, for the position circle east, we get for the pole star and the time star respectively, Pole star ; a = a' + 3 AT H -- : T + -. 7 sin / sm / $ cos z c Time star : a = 180 + a -\ -- : -- \- -- sm z sin z It is supposed that the level error 6 does not change between the observation of the time star and of Polaris ; also that c is the collimation error, either of the ' mean thread J or of the centre thread, according to the manner in which the transits have been reduced : generally the reductions are made to the centre thread. The hour angle h of the time star and the azimuth are connected by the relation cos 8 sin h = sin z sin a ; .*. cos 8 sin^ = sin 2 sin + ( cos 2 + c) cos a. (10) If we put a = a' + c cosec /, and cos 8 sin h Q = sin z sin a Q1 (11) the difference of (10) and (11) gives cos 8 cos h (hh^ sin z cos a (aa^) + (# cos z + c) cos a ', 7 , 7 sin (,& + /) c 8 AT sin z - .. Q - , -- - - > cos 8 sin z cos 8 cos 8 where the factor cos a : cos h has been replaced by unity. If INSTRUMENTS AND OBSERVING. 197 T be the observed time of transit of the star, and A its right ascension, T+r + Ar A = T \^. Hence, finally, y.Ar = A T r + J- / + sec $ + c sec 5, (12) where y = 1 ^j (3 sin z sec 5, and b, c are expressed in seconds of time. The zenith distance of the time star is given by z = $ 8+i 2 sin($ 8) cos sec 8 . sin I" ', (13) where the azimuth a is expressed in seconds. The formula (6) for the reduction of the time of transit over a side thread at the distance c from the centre thread gives in this case T 1 T c{sec(8-f n) sec (8 #)}^, where n = acos(f). The subject is fully treated in an essay entitled Die Zelt- fiestimmung vermittelst des tragfiaren Durchgangsinstmments im Vertlcale des Polarsterns, von W. Dollen, St. Petersburg. The method of reduction of the observations given above is virtually that adopted in the operations of determining the difference of longitude of Poulkowa (St. Petersburg), Stock- holm, and intermediate stations. Mem. Acad. Imp. Sci. St. Petersburg, Tome XVII, Nos. 1 and 10. 5. If the vertical plane described by a transit instrument freed from level and collimation error be intersected once by the diurnal path of a star, it will be intersected a second time. Let h fi Ji f be the hour angles corresponding to the two times of transit, then by (5) cos a cos sin 8 + cos a sin < cos 8 cos h f -f sin a cos 8 sin h t = 0, cos a cos $ sin 8 + cos a sin cos 8 cos h f -f sin a cos 5 sin Ji ; and from these we have tan 8 cos J (h, + h') = tan $ cos J (&,&) , = sin tan ' If the times of transit of a star be observed, giving k f and 198 INSTRUMENTS AND OBSERVING. Ji ', the first equation gives a value of the latitude, and the second the azimuth of the plane. When the instrument is in the prime vertical, a = 0, and h^ + h' = ; if therefore h be the hour angle of the star at either transit, tan < = tan 8 sec h. The method of determining latitudes by observations of the transits of stars over the prime vertical was originated by Bessel. A great advantage is the facility it offers for the elimination of instrumental errors by the reversal of the telescope either between the observations of two stars, or even in the middle of the transit of a star, or by using the instrument circle north one night and circle south the next. But the disadvantage is that the method demands a very precise knowledge of the time, and it is better suited to high latitudes than to low ones. The error d(f> of latitude, as de- pending on errors of 8 and k, is given by the equation 2d(j> , .. 2^8 -: = tan hdh + > sin 2$ sm28 which shows that the hour angle, or the zenith distance of the star when observed, should be as small as possible. The equation (5), if we make a a very small angle, applies to the case of transits in the prime vertical. Here b is posi- tive when the northern end of the axis is high. Putting cos a == 1 , and cos 8 sin h = sin z (when the azimuth of the star is very nearly 90), our equation becomes asinz + b cos z -f c + cos $ sin 8 sin (f) cos 8 cos^ = 0. (15) Let (/>' be determined from the equation tan (/>' = tan 8 sec //, then if fy' < = e cos sin 8 sin < cos 8 cos k = e (sin $ sin 8 -f cos $ cos 8 cos h) but the quantity within the last parenthesis is cos z ; hence $ = (/>'+# tan z + b + c sec z, (16) an equation which can be verified geometrically : z must be taken negatively for western transits. When the observed star is near the zenith there is time to reverse the instrument in the middle of the transit. Thus a star may be observed at its eastern transit on the north INSTRUMENTS AND OBSERVING. 199 side of the prime vertical upon those threads which are to the south of the collimation centre ; then, after reversing the instrument, the star may be observed again on the same threads. Leaving the telescope in the last position until the star comes to the western transit, it is observed again on the same threads to the south of the prime vertical, and then reversing the telescope the star again crosses the same threads on the north side. Thus each thread gives a latitude deter- mination freed from instrumental errors. Let I be the angle corresponding to the interval of time between two transits over one thread on the north side, I' that corresponding to the observations on the same thread on the south side, H cor- responding to the difference between the star's right ascension and the mean of the four times of transit, then either by (15) or geometrically, we get cot (<-) = cot 5 cos J (/+/') cos J (/'-/) sec H. (17) But practically this requires rather too many reversals of the instrument. It is probably best to select a number of stars for which c/> b does not exceed 2, such that they can be observed first on the east side of the zenith, circle N. say, and again on the western side, circle S. Then on the next night in the positions E. circle S. and W. circle N. For this case of very small values of $ 8, if we put e for J-(0 8) 3 cosec 1", and calculate , which will be less than l"-5, from an approximately known value of $, then the equation (15) may for the two observations of the star E. and W. be written thus for each thread : 2 h $ 5 c = -77 sin fy cos b sin 2 -f a sin z -f b cos z +e, sin 1 i (f) b + c=. -77 sin cos 8 sin 2 + a sin /-f b' cos / + e, sin 1 2 where the unit is 1". From the mean of these c is eliminated, and since / differs but little from z, a enters with a small coefficient. The value of a for the evening's work may be obtained thus : suppose the two equations just written down to appertain to the centre thread, then a and c remaining symbolical, the difference of the equations will give one of the form c + a sin z = g. Each star will give one such equation. 200 INSTRUMENTS AND OBSERVING. 6. The determination of latitudes for geodetic purposes is effected by one or other of the following methods : (l) deter- minations of the meridian zenith distances of stars ; (2) by zenith distances of Polaris, a method which has the advantage that the observations may be made at any part of the star's apparent orbit, and by day as well as by night; (3) by transits in the prime vertical ; (4) by the zenith telescope. In the first method it is desirable that stars be observed equally on both sides of the zenith, so that in the end the mean of the zenith distances may be nearly zero. When the star is observed at a small hour angle from the meridian which should not be done in the case of stars near the zenith if / be the meridian distance, z the observed zenith dis- tance, k the small hour angle from culmination, then , 2 cos cos 8 sin 2 \h . . Z = # + : i / / . \ > (18) ~~ sin if*-*-~\ v ' the upper sign applying to upper, the lower sign to lower culminations. This formula includes the term in ^ 4 ; on the right side of the equation sf is to be obtained from the ap- proximately known latitude. In the hands of an expert observer it is certain that very excellent results for latitude can be obtained from small circles. The latitudes of the greater part of the stations in the Russian arc were determined with circles of 1 1 inches and 1 4 inches diameter. We have described the instrument used in the Spanish geodetic operations. The following results for latitude at three different stations by three different methods are interesting : METHOD. CONJUROS, 36 44'- DIEGO GOMEZ, 40 55'- LLATIAS, 43 29'. No. of Days. Seconds of Latitude. No. of Days. Seconds of Latitude. No. of Days. Seconds of Latitude. Polaris 5 9 6 22-41+. 10 21-99 + -io 22.43 + - 12 5 5 6 39-H + -IO 38-26 + -07 38-42 + . 12 5 5 4 // 28.78 + - 10 29-02 + -io 2 9-45 - T 3 Other stars ... Prime Vertical INSTRUMENTS AND OBSERVING. 201 In the case of the second station there is a difference between the results given by the first and second methods amounting to 0"'85. This, however, is not much greater than the difference between the latitudes of Balta as obtained from the zenith sectors of Bamsden and Airy. In the zenith telescope let us suppose the micrometer readings to increase as the zenith distance decreases. The instrument being set, approximately, to the mean zenith distance of a pair of stars about to be observed, and the level indication being zero, let Z Q be the angle made with the vertical by a line joining the optical centre of the object glass with a point in the centre of the field whose micrometer reading is m . It is presumed that during the short period of time required to observe a pair of stars the relation of the level and telescope remain unchanged ; hence, if when one of the stars as the north star is observed, the north end of the level has the reading n, while the southern has the read- ing ?, then the zenith distance of the point m is Z Q + i (s n\ the level readings being converted into angular measure. Let 5' S, be the declinations of N. star and of S. star. ' m' m f micrometer readings of N. star and of S. star. Ef Rj refractions for N. star and for S. star. n' s' level readings for N. star. , *, S. star. [JL A angular values of one division of micrometer and level. Then the apparent zenith distances of the stars are 5 ,. . 8. ..-*,++-&, = gi-- and eliminating Z Q and m Qi ,). (20) Here it is supposed that the observation is made on the centre thread, and that the instrument is in the plane of the meridian. If the micrometer bisection is made when the star is on a side thread at a distance c from the centre thread, the correction + J c 2 tan 6 is required to the zenith distance ; the 20.2 INSTRUMENTS AND OBSERVING. uppor sign for south stars, the under sign for north stars. Thus the above expression for

where t, the instrument is set to the zenith distance of the pole star at its greatest azimuth, and directed to the star half-an-hour or so before the time of greatest azimuth. The micrometer screw is set at successive single revolutions in advance of the star, and the corresponding times of vertical transit observed ; the level is also read at each transit. Let / be the zenith distance of the star at one of these observations, f being that at the time of greatest azi- muth. If E be the refraction corresponding to f we may put also in the first of equations (19) put z Q -\-R= C^j and write / for 6'$, then expressing /fin seconds, Each observed transit gives an equation of condition of this form. The solution by least squares is simplified by sub- stituting M -f y for the unknown /ut, where M is an ap- proximate value, and^ the required correction : / f is easily calculated from the recorded time of observation. It is supposed in these formulae that the instrument is used with the micrometer screw below, as represented in the drawing. It may, however, be used in the other position, in which case, the sign of /u being changed, the formula still apply. In the two instruments used on the Ordnance Survey the values of one revolution of the micrometers are 62"-3560".003 and 63".3250"-006, derived in each case from the combined observations made at six stations. The lists of stars prepared for these instruments comprised from thirty to fifty pairs for each night, and of these a con- siderable proportion were found in the Greenwich and Oxford Catalogues, though some stars were dependent on the British INSTRUMENTS AND OBSERVING. 203 Association Catalogue places. The following table contains the final results for latitude at the station, a summit of the Grampians, where the smallest number of stars was observed : JULY i, 1868. JULY 4, 1868. JULY 6, 1868. PAIRS. 56 58' 56 58' 56' 58' I. 40-22 40-28 39^4 II. 41.46 40-26 III. 40-57 40-06 39-9 2 IV. 39 47 40-28 40-29 V. 39-34 ... 40-54 VI. ... 40-60 IX. 41-63 41-40 X. 41-18 40-55 3986 XI. 38-91 39 6 3 39-70 XII. 3964 40-01 XIII. 39-94 .. . 3901 XIV. 38-76 39-o 2 XV. 39.21 40-87 XVI. 40-37 XVII. 4-37 ... 4071 XVIII. 40-46 ... ... XIX. ... 40-33 Daily ) Means ) 40-01 4038 40-16 Latitude 56 58' + o"-o8. The simplicity of construction of the zenith telescope ex- empts it from several of the recognised sources of instrumental error, while its portability and ease of manipulation eminently fit it for geodetic purposes. It is exclusively adopted for lati- tudes in the United States, and it is probable that no one who has used it would return to graduated circles for latitude. The form of the expression for the latitude as determined by the zenith telescope shows that the error of a single result is affected by the errors in the assumed decimations of two stars, by the errors of two bisections of these stars, and by errors in the assumed values of the micrometer and level divisions. The last two sources of error can be made very small. The discussion of a large number of observations 204 INSTRUMENTS AND OBSERVING. shows that the probable error of observation only in a single determination of latitude from a pair of stars is between 0"-S5 and 0"-65, according to the skill of the observer and the sensitiveness of the level of the instrument. Hence, if e n e' be the errors of the declinations of two stars, and they be observed n times, the error of the resulting latitude may be expressed by If , , e' be the probable errors of the declinations, then the probable error of latitude resulting from n observations of this pair is Hence, in combining the results given by pairs of stars, the weight to be given to each result may be taken as The probable error of a declination will depend on the catalogue from which it is taken ; from the Nautical Almanac or Greenwich Catalogues e may be about 0"-5, but from the British Association Catalogue it would be probably double that amount. In the official Report * on the North American Boundary, the subject is very fully discussed. In those operations the probable error of a single determination is in several cases less than 0"-3. 7. The method that has been generally followed for the deter- mination of absolute azimuth in this country is the measure- ment of the horizontal angle between a terrestrial mark and a close circumpolar star, when at or near its position of greatest azimuth. The practice in other countries, as in Russia, in 1 Reports upon the Survey of the Boundary between the Territory of the United States and the Possessions of Great Britain. Washington, 1878; pp. 95-169- INSTRUMENTS AND OBSERVING. 205 Spain, and in America differs from this only in that the observations are not always confined to the position of greatest azimuth of the star. The most frequently used star is Polaris, then 8, e, and A Urs. Minoris, 5 1 Cephei, and others. The formula (4) shows that the level and collimation errors enter with large factors, large at least in high latitudes ; therefore it is necessary to determine the collimation before and after the star observations, and the level must be read in reversed positions during the observations. The error of level must be scrupulously kept as small as possible, and the value of one division of the level should be known at all temperatures. The difference of pivots must be accurately known, but no instrument with irregular pivots is fit for this work. The terrestrial mark not less than a mile off is generally for night work a lamp behind a vertical slit : the opening is sometimes covered with oiled paper. There are slight differences of detail in the modes of con- ducting the observations, but the following may be taken as virtually the ordinary procedure. The level being on the axis, and the instrument, say circle west : (l) the mark is ob- served ; (2) the star is observed ; (3) the level is read and reversed ; (4) the star is observed a second time ; (5) the level is read; (6) the mark is observed. The telescope is then reversed, and with circle east the operations just specified are repeated. The double operations complete one determination of the angle. The chronometer times of observation of the star are noted for the calculation of its azimuth. As in terrestrial observations the errors of graduation are eliminated by shifting the zero of the horizontal circle. The probable error of a determination of azimuth increases with the latitude : it may be expressed by the formula 6 = In the azimuth determinations by Struve in connection with his great arc of meridian, the probable error of a single deter- mination (as just defined) increased from 0"-75 in latitude 45 to l"-98 in Finmark. The determinations of azimuth in the recent geodetic operations in Spain, effected with theodolites of Repsold, are excellent. In the triangulation of Great Britain azimuths were determined at sixty stations; 206 INSTRUMENTS AND OBSERVING. at twelve of these the probable error of the final result is under 0"-50, and at thirty-four, under 0"-70. Generally speaking, in these observations the observer has had only an approximate knowledge of the time, and hence at each greatest azimuth of a star only a single determination was effected : each observation since the year 1844 has been cor- rected for level and collimation errors. At fifty-seven stations out of the sixty the observations were made by N. C. Officers of Royal Engineers. In Colonel Everest's work in India it was the rule to take four measures circle east and four circle west, at each zero, on each side of the pole : the number of zeros was four, making in all sixty-four measures as sufficient. But this number was often exceeded. Another method of determining absolute azimuths is by erecting a mark either to the east or west of north or one to the east and another to the west in such positions that the pole star shall cross the vertical circle of the mark a little before and a little after its greatest azimuth. The observations are made with a transit instrument furnished with a moveable vertical thread for micrometer measurements. The instrument is set with the centre thread nearly on the mark, then the telescope being elevated to the star at the proper time, the star will move slowly across the field. Read- ings of the micrometer thread on the mark, the star, the star, the mark are taken, and combined with level readings in reversed positions. This operation is repeated in the alternate positions of the instrument circle east, circle west. The observations should be so arranged that the star is taken as much on one side of the field as on the other ; thus the final result will be nearly independent of the assumed value of a division of the micrometer. Let /u be the micrometer reading of the collimation centre, jut that of the star, [/ that of the mark : suppose these readings to increase as the thread moves towards the circle end of the axis : also let d be the angular value of a micrometer division, then in accordance with equation (3) we have (n ju ) d+ 1 cos z H- sin z cos (a a) = 0, (// Po) d + b cos / -|- sin / cos (a'- a] ; INSTRUMENTS AND OBSERVING. 207 and hence, since a a is only a few minutes, CL ^^ OL ~~ sins z*^_ a sin (*'-*). (21) sin / sin z sin / v ' The following tahle contains the results for azimuth at a station in Elginshire, in latitude 5 7 3 5'; the observations were made with the Russian transit instrument previously alluded to. Each figure is a complete single determination, including both positions of the instrument, in the manner described : 1868. NORTH WEST MARK. Az. 177 45'... Oct. 14. Oct. 16. Oct. 17. Oct. 18. Oct. 20. Oct. 25. Oct. 26. // ft // ii // 37-n 38-46 3988 38-79 365 37-14 37-98 3650 37-21 3890 35-13 35-85 39' 1 3 37.21 3843 38-08 37-65 36-98 NORTH EAST MARK. Az. 182 if... Oct. 1 6. Oct. 17. Oct. 20. Oct. 21. Oct. 23. Oct. 25. // // // ii ii // I5-I3 15-04 I4-OI 16-31 1640 14-89 1575 16-13 IS-S^ 1566 14.83 15-40 15.29 14.50 1499 1564 1467 1697 15 oo Hence we have the azimuths reckoned from the south, North West Mark ... 1 77 45' 37"-61 0"-1 9. North East Mark ... 182 17' 15"-37 0".ll. In the case of the first mark the probable error of a com- plete single determination is 0"-820, and for the second it is i 0"'489. The difference in the precision of the results is due to the circumstance that the former mark was observed in the morning twilight, sometimes with a lamp, and with difficulty ; the latter mark was observed in good daylight in the afternoon. The observations were made (in stormy 208 INSTRUMENTS AND OBSERVING. weather at a height of 1100 feet) by Quarter-Master Steel and Serjeant Buckle, R. E., and indicate both expertness in the observers and perfection in the instrument. If in connection with the observation of the star its re- flection in an artificial horizon be observed, then the level may be dispensed with, unless indeed the zenith distance of the mark differ materially from 90. As the spherical co- ordinates of the star are a and 2, so those of its reflected image are a and 180 z\ and if ju, /u, be the readings of the star and of its reflection, (m ju ) d -f b cos z -f sin z cos (a a) = 0, (ft p^dbcosz + sinz cos (a a) o, neglecting the small change of zenith distance between the two observations. From the mean of these b disappears as far as the star is concerned, and = a j } sm sin z The azimuth of a circumpolar star at any point of its path may be obtained from the formula tan a sin H sin h 1 cos -5" cos /^ (22) where If is the hour angle corresponding to the maximum azimuth A. Or if the observations of the star are confined to times within an hour of the greatest azimuth, the formula (23), page 46, is sufficiently accurate and even in this, if the star be within 20 minutes or so of its greatest azimuth (this depends on the latitude of the observer) the denominator of the right side of the equation may be replaced by unity. The azimuth obtained from observations of the pole star requires the correction 0"-311 on account of diurnal aber- ration. 8. For the determination of the difference of longitudes of two stations for geodetic purposes the lunar methods are not sufficiently precise. The requirements of the case are in one INSTRUMENTS AND OBSERVING. 209 sense simple : the correct keeping of the time at A, the same at B, and some means of comparing simultaneous readings of the clocks at A and B. The extended system of electric telegraphs, now in use in all countries, affords the most pre- cise mode of comparing local times : the details of the method will be found in the U. S. C. Survey Reports for 1857, 1867, 1874; in the Reports of the Survey or- General of India ; in the Annales de Vobservatoire Imperial de Paris, vol. viii ; the Me- morial du Depot general de la Guerre, vol. xi ; the Publication des Konigl. Preussischen Geoddtischen Institute, Berlin, 1876; and other works. The method of recording time and astronomical observa- tions on a revolving cylinder originated in the U. S. C. Survey in the first attempt to determine longitude by electro-mag- netic signals. Bond's chronographic register is a cylinder of about twelve inches long by six inches diameter : it revolves once per minute, a uniformity of velocity being secured by a centrifugal fly-regulator in connection with a pendulum. As the cylinder revolves it is drawn uniformly along a screw- formed axis : its surface is covered with paper, removeable at pleasure, and a pen held in contact with the paper under the influence of an electro- magnet draws on the paper a con- tinuous spiral. The galvanic circuit passing through the clock is broken every second by the clock : this break, the duration of which is regulated to about one twentieth of a second, demagnetises the electro- magnet, and the pen under the influence of a spring draws a small offset at right angles to the continuous spiral : thus the beats of the clock are trans- formed from audible to visible intervals or signals. The man- ner in which the clock breaks the circuit will be understood from the adjoining figure, in which PP is the pendulum- rod, B a brass plate carried by the back of the clock-case from which projects a brass arm carrying an ivory bracket /. To this is affixed and adjusted a very small tilt-hammer of p Fig- 43 210 INSTRUMENTS AND OBSERVING. platinum, of which the left hand end rests on a metal disk plated with platinum, connected with B and the line wire w. A fine pin projecting from the pendulum rod strikes the obtuse angle of a small bend in the tilt-hammer, and for an instant as the pendulum passes its lowest position tilts up the left end of the hammer, so breaking the circuit. A signal key under the hand of the observer enables him also at pleasure to break the circuit in the same manner as does the clock ; thus, the instant of a star passing a thread of the transit instrument is recorded on the chronograph by an offset. The offset of the observer is readily distinguished from that of the clock by a difference of form. A small portion of the register has something of this appearance Fig. 44. It is the breaking of circuit whether by clock or observer that constitutes a signal, hence, in reading off the chrono- graphic record it is the right hand or first edge of the offset that is used in subdividing the seconds. In order to facilitate the reading of the time, one second, viz. that numbered 60 is omitted in the chronograph every minute, and also two seconds are omitted every five minutes. This omission is effected by means of a very ingenious arrangement whereby the clock itself completes the circuit at those instants. To determine the difference of longitude of two stations, A and jft, there must be at each an astronomical clock, a chrono- graph and a transit instrument. The transit instruments used with the most excellent results in Paris and Algiers, for the recent determination of the difference of longitude of those places have telescopes of 31 inches focal length and 2-5 inches aperture ; they are in fact meridian circles, the dia- meter of the circle being 1 6 inches. Those used in India are much larger, viz. 5 feet focal length of telescope with 5 inches aperture, but it is certain that the precision of results does not keep pace with increase of dimensions. From a discussion of a very large number of observed transits in the U. S. C. Survey it was ascertained that the probable error of INSTRUMENTS AND OBSERVING. 211 an observed transit (chronographic registry) over a single thread, the star's declination being 8, was expressed by e = + ((0-063) 2 + (0-036) 2 tan 2 )* or e = ((0-080) 2 + (0-063)* tan 2 &)', the former applying to instruments of about 47 inches focal length, the latter to a focal length of 26 inches. A much more formidable source of error is ' personal equation. 5 Every observer has his own peculiarity of habit in observing and recording transits which takes the form of a ' personal error.' In the eye and ear method a certain small error exists in associating the position of the star in the field with the audible beats of the clock, the eye and ear not acting in simultaneous accord : moreover, this may be com- bined with an erroneous habit of subdividing seconds. It is probably due in part to the same species of error of vision which will cause one observer with a microscope to bisect a line on a standard measure differently from the bisection of another observer, a difference which is tolerably persistent. In the chronographic method of recording observations, per- sonal error also exists, referrible to peculiarity of vision and manner of touching the signal key. Personal error may be affected by the brightness of a star and its velocity, it cer- tainly is affected by its direction of movement, north stars and south stars giving for some observers different personal errors. In those instruments in which the rays of light are turned through a right angle by a central prism, the personal error has been found to be different in the two positions, east and west, of the eye-piece. The investigation of personal error is therefore one of the most important elements in the question of longitudes. The ordinarily practised method of ascertaining relative personal error of two observers, A and , is this : they observe transits of the same star in the same instrument. A observes the first star over all the threads before it arrives at the centre, E then observes the same star over all the remaining threads. For the next star, B observes in the first half of the field, A in the second half, and so on alternately. The observation in 212 INSTRUMENTS AND OBSERVING. this manner of a large number of stars, in which those north of the zenith are to be separated from those south of the zenith gives the difference of the personal equations of A and B. Unfortunately personal equation is not altogether con- stant, depending on the nervous condition or state of health of the observer. The threads in transit instruments used with chronograph registry are generally numerous for instance, they are often arranged in five groups of five, the members of a group being at the equatorial interval of 2 s - 5 : in fine weather the three centre groups are found sufficient for observing. Supposing the apparatus and instruments to be in perfect adjustment, the observations for longitude each evening are preceded by observations of transits for the determination of instrumental errors and clock error : say six or eight zenith stars with one or two circumpolars and two reversals of the instrument. The clock at the eastern station A is then put in connection with the circuit and graduates the chronograph at A and the chronograph at B. The observer at A on the arrival of the first star on the list of signal stars made out for the longitude work, observes its transit, his signal key mark- ing it both on the chronograph at A and on that at B. On reaching the meridian of B the same star is observed in the transit instrument there and is recorded by the observer on the chronograph at B and on that at A : and so for the other stars. When half the evening's work is done the clock A is disconnected from the circuit and replaced by that at B. Each star gives thus a difference of longitude on each chronograph, a result independent of the star's place. The difference of longitude given by the western chronograph will be too small by the interval of time occupied in transmission of the signals, that at the eastern will be too great by the same amount. Hence, in taking the mean of the two chrono- graphic results this small interval is eliminated, provided the strength of the batteries has been kept constant. But the result is still affected with personal error of the observers. This may be eliminated by the interchange of observers when the station is half completed. When the stations are very far apart this method becomes INSTRUMENTS AND OBSERVING. 213 impracticable, and the following is that adopted. After the necessary observations have been made for the determination of clock error at each station, the eastern clock is first put into connection with the circuit, so as to graduate both chronographs : then that at the western is put into circuit and also graduates both chronographs. The western chrono- graph will give the longitude too small by transmission time, the eastern gives it too large by the same amount. The observers, each with his own instrument and apparatus, are collected (either after or before, or both) into one spot, and determine the difference of longitude of their respective instruments. In a very interesting account by J. E. Hilgard, Esq. of the transatlantic longitude work in 1872, we find the following statement of the results of three determinations of the longi- tude of Harvard College Observatory, Cambridge, U.S., west of Greenwich : By Anglo- American cables in By French cable to Duxbury in By French cable to St. Pierre in A very extensive series of longitude determinations has been carried out recently in India with most admirable pre- cision under the direction of M.-General Walker, C.B., F.R.S., Surveyor-General of India. In his yearly report for 1877-78 are found the results of eleven differences of longi- tude by electro-telegraphy with the corresponding geodetic differences. They are between Bombay (B) and Mangalore (N) on the west coast, Vizagapatam (F) and Madras (M) on east coast, and Hydrabad (H\ Bangalore (R), and Bellary (L) in the interior. The eleven lines observed are drawn in the diagram. In every triangle it will be noted there is a check on the accuracy of 1866 1870 1872 h. ... 4 ... j, m. 44 55 S. 30-99 30-98 + 30-98 8. 0-10. 0-06. 0-04. II 214 INSTRUMENTS AND OBSERVING. the work : thus, referring- to difference of longitude, the triangle LRM gives LM= LR + RM. 'When the opera- tions were commenced,' says General Walker, ; I determined that they should be carried on with great caution, and in such a manner as to be self-verificatory, in order that some more satisfactory estimate might be formed of the magnitudes of the errors to which they are liable than would be afforded by the theoretical probable errors of the observations . . . the simplest arrangement appeared to be to select three trigono- metrical stations A^ J3, (?, at nearly equal distances apart on a telegraphic line forming a circuit, and after having measured the longitudinal arcs corresponding to AB and BC to measure AC independently as a check on the other two.' The follow- ing table contains the observed differences of longitude : YEAB. ABC. OBSERVED DlFF. OF LONGITUDE. COBEECTIONS. 1872-73 Madras Bangalore O / /i 2 39 45-63 X t = + 2-OIO ,, Bangalore Mangalore ... 2 44 11.54 x z = + 1-690 1875-76 Hydrabad Bombay 5 4 2 I2> 74 x s =-0-452 Bellary Bombay 4 6 44-39 aj 4 =_ 0-393 Hydrabad Bellary i 35 28-25 x & = + 0-040 Madras Hydrabad i 43 40-38 x 6 =-0-412 Madras Bellary 3 19 8-45 x 7 = 0-192 ,, Bangalore Bellary o 39 20-46 x s = + 0-160 1876-77 Vizagapatam Madras ... 3 2 26-78 x a = + 0-401 Yizagapatam Bellary ... 6 21 35-84 x l6 -= o 401 Mangalore Bombay 2 I 50-54 * n = + o8 4 5 The first two determinations were the earliest made, and are affected with some fault in one of the transit instruments not fully known at the time, hence these have less weight than the others. With respect to the corrections in the last column, these arise in the following manner : it will be seen that the figure presents four triangles and a quadrilateral, each of these five presents a condition to* be fulfilled by the observed longitudes. Suppose that in consequence of errors in the concluded results INSTRUMENTS AND OBSERVING. 215 they require corrections a l9 x. 2 . . . # n in the order in which they are written down. Take for instance the triangle SLR, the sum of the fourth and fifth observed differences of longitude should be equal to the third, that is, 5 42' 12"-64 + # 4 + # 5 = 542'12"-74+# 3 ; hence a linear relation between # 3 , # 4 , and # 5 . The following equations can be thus verified : #1 # 7 + # 8 -2-36 = 0, tf 7 +# 9 -# 10 -0-61 = 0, # 5 + # 6 -# 7 +0-18 = 0, (23) # 3 + # 4 + # 5 0-10 = 0, # 2 + # 4 + # 8 # u + 2-77 = 0. Now the #'s cannot be determined from these equations. But the theory of probabilities shows that the values which are the most probable are those which, in addition to the con- ditions above, make the function a minimum. Here the symbols a lt a. 2 are the weights of the first two determinations, those of the remaining nine being taken as unity. We cannot assign precise values to a lt a. 2 , we shall assume them to be each = i. Proceeding by the ordinary method of the differential calculus, multiply the equations (23) severally by indeter- minate multipliers u 1} u 2 ...M 5 , then we get a?! = 2u lt x 2 = 2?/ 5 , # 3 = w 4 , and so on. Substitute the equivalents of ^ ... # n so expressed in terms of u^ ... % in the equations (23), they are thus trans- formed to 4 Wl _ u 2 + u., ...... + 6 - 2-36 = 0, f/j + Stta u 3 0-61 = 0, ! u z +3u z + u +0-18 = 0, (24) tt 3 +3z/ 4 + U 5 0-10 = 0, ! ............ + w 4 +5% + 2-77 = 0. Solving these equations we get numerical values of % . . . u 5 , whence immediately follow those of x 1 ... # n . These are written down in the last column of the table. The smallness of the corrections is abundant proof of the 216 INSTRUMENTS AND OBSERVING. remarkable precision attained in these observed differences of longitude. In volume xi. of the Mem. du Dep. gen. de la Guerre will be found a valuable account, in full detail, by M. le Commandant Perrier of the operations for determining the difference of longitude of Paris and Algiers by means of the submarine cable connecting Algiers and Marseilles : the daily results collected at page 167 stand thus : m. s. s. m. s. s. Nov. 2, 2 50-3720-049; Nov. 17, 2 50-3550-021 ; 3, 2 50-2840-050; 23, 2 50-3180- 25; 6, 2 50-298 + 0-047; 24, 2 50-2950-025 ; 7, 2 50-3380-046; Mean, 2 50-3260-010. This result requires the correction of 9 -093 for personal errors of the observers. Hence the difference of longitude is 2 m 50 8 -233. A check upon this is afforded by the inde- pendently observed differences of Paris Marseilles 12 m 13 8 .4350 8 .011, and Marseilles Algiers 9 m 23 s .2190 s -011 } of which the difference is 2 m 50 S -216 s - 01 6, differing only 8 -017 from the direct result. CHAPTER IX. CALCULATION OF TETANGULATION. IF the observed angles of a triangulation were exempt from error, the calculation of the distances between pairs of points would present no difficulty. But the errors with which every observed angle is burdened lead to conflicting results, and it becomes necessary to find a systematic method of treating these errors. In the case of a single triangle, if the three angles were equally well observed, and if the sum of those angles exhibited an error of, for instance, -f 3", we should naturally and rightly apply to each observed angle the correction 1": or if they are observed with unequal precision then we know by the method explained in the chapter on least squares how to divide the error among the angles. Still this applies generally only to isolated triangles, and it will be necessary to consider other combinations of points and angles. 1. Consider first a polygon of i sides represented in the an- nexed figure. Suppose that each angle in each of the i triangles is observed with an equal degree of pre- cision. In each triangle the sum of the observed angles will show a certain amount of error, also in adding up the angles at the central point P the sum will differ slightly from 360. Moreover, if Fig. 46. 218 CALCULATION OF TRIANGULATION. we start with the side PP[, and calculate in succession the sides PP 2 , PP 3 , ... PP., finally returning to PP A we shall find a difference between the length of PP t so calculated and that with which the calculation was commenced. This numerical difference is a function of the errors of observations : we have in fact i -f 2 numerical values of as many functions of the 3i observed angles. We shall adopt the following notation A 1} 19 C 1 the true angles of the first triangle ; AI, BI, C{ the observed angles of the same ; e i> fit 9\ the corresponding errors of observation ; it y\ > z \ * ne corrections to be computed ; &!, 33 1, Ci the finally adopted angles. For the n ih triangle the subscript unity is replaced by n. Thus A n ' = A n + e n , <& n = A n + e n + x n , Cn' = O n +ff n , C w = C n + ff n + Z n . We propose to investigate the most probable values of the corrections which should be applied to the observed angles. Put a lt p lt y l for the cotangents of A 19 B lt C lt and further let so that a 1 + & 1 + c l = 0. In the # tb triangle let the sum of the observed angles exceed the true sum by e n , also let the sum of the observed angles at P be 360 4- e . Then we have in all i+l equations. We may express each g in terms of the corresponding e and /*, and substituting in the last equa- tion it becomes Then we have the further geometrical condition PP 2 PP i _ sin BI sin^ 2 PP 2 *PP 8 'PP 1 " each side of this equation being unity (in spherical triangles we have merely to write on the left side sin PP l instead of CALCULATION OF TRIANGULATION. 219 PP lt &c.). The observed angles however will not fulfil this condition. Suppose the calculation made with the angles A', B' ... , and that the result is sin B-f sin B ... sin B? _ i _ _ L ]! _|_ sin A-[ sin A^ ... sin Af ~ where is a very small quantity. Then since sn it will follow that Now since the adopted angles 21, B, C are to fulfil all the requirements of the case, it follows from (2) and (3) that and also from (l) We have now to find such values of x lt y 19 x^ y^, &c. as being subject to the necessary conditions (4) shall further render the function a minimum. Differentiating this equation, the condition of minimum is dil _ ss a^ Ak + ^*i-taf^t5i*5fc+- = (2 ^ + jfj + ej ^^ + (^ + 2ft + Differentiate the equations (4), and having multiplied them by multipliers 3Q and 3P respectively, let them be added to the equation just written down, and we have + ( Now according to the principles of the differential calculus the coefficients of dx^ dy lt ... dx^ dy i must be severally equal 220 CALCULATION OP TRIANGULATION. to zero. Hence we are led to these equations : a} 1 =$c 1 + a l P Q, #2= Je 2 -M 2 P Q, (5) ft^-K + ^P- Q, ft=-t*2+*2P- Q, and so for the other triangles. Now by substituting the values of x l y l , # 2 y 2 , & c - so expressed in the equations (4), we get two others from which P and Q can be eliminated. If we put then and consequently, if further we put ^iJch 2 = U 3 UP = - hM+2iN, (6) UQ = +2kM- hN, which fully determine P and Q: and the values of x^y^z^ ^2^2 ^2> x iyi z i follow at once from equations (5). This com- pletely solves the question : the finally adopted values of the angles are, for instance in triangle 1 : - Q, (7) 2. We shall now ascertain how these adopted angles are individually affected by the actual errors of the observed angles. If we substitute in (5) the values of M and N> after replacing 6 e 1 6 2 ...e i by their equivalents in terms of the actual errors of observation from (l) and (3) we get CALCULATION, OF TBIANGULATION. 221 where = h 2ia n , 3F n ' = Among these quantities we shall require for the investiga- tion of probable errors the following relations which are easily verified : 2^2 + F-2 + &=iU The substitution of the last expressions for P and Q in the equations (7) leads to the following : where 2 fl ~ 3" 1 1 --^K#i-^i')> *2 = Jj( a i E 2 ^O'"' fl= -^ H --l^-^-), 1 1 i 1 l=-gH " 77 ^ x "" x *' * /_ /3 /nr /\ fc = -jf(i -&; -j 1 i , , j fl =~3 H - -jj- (O^ -j /^i ) > U c/ 2 1 1 *- 3 H i/ t/= lr ft^-^)...> 1 K-i^ (?!_ ---< h^M+s^ , 1 f 2 =-yw* + ^r)- 1 1 i f 1 = ~~ 3 ~ f-^ftA+2^/), ^ =-^ft^ 2 +2^/)... 2 // *-/y- .^x^iAV 8l = -f7 Put now A^, $', iS' 7 for the sum of the squares of the above 222 CALCULATION OF TRIANGULATION. coefficients of the actual errors in the expression for the adopted angles, then after a little reduction, we get, on put- ting e for the probable error of an observed angle, the follow- ing probable errors of the adopted angles for For the probable errors of the corrections x^y-^z^ to the observed angles we should obtain the following for i i 3. As a numerical example of the application of these formulae we shall take a very large polygon which embraces the greater part of Ireland. The central point is Keeper (P) in the county of Tipperary ; then in succession Baurtregaum (PJ near Tralee ; Bencorr (P 2 ) in Connemara ; Nephin (P 3 ) in Mayo ; Cuilcagh (P 4 ) near Enniskillen ; Kippure (P 5 ) near Dublin ; and Knockanaffrin (P 6 ) in Waterford. In taking this piece of work as an example it is necessary to remark that the conditions are not such as are supposed in our pre- ceding investigations : the angles are not observed inde- pendently, and they are not of equal weight. The angles were observed as explained at page 181, consequently the sum of the angles at P is necessarily 360: hence e = 0. Nevertheless, with this proviso the polygon will serve our purpose of illustration. The following table contains the CALCULATION OF TBIANGULATION. 223 data for the solution, and from these we have to calculate the 1 8 corrections to the observed angles : A OBSERVED ANGLES. ERROR OF A a, 6, c, cc. a' + M^ ^ o / // 4i' = 58 46 5-46 BI = 52 16 22-32 d' = 68 58 8-90 Sum 36-68 ,= -1-68 a, = -f- 1-986 6, =- 2-154 d =4 0-168 .(',=_ O*282 8-612 ^ Sph. 6x0688 = 38-36 A,' = 102 34 5-26 /= 54 38 2777 C a ' = 22 47 43-81 Sum = 16-84 .,=-,,. a 2 = + 0-264 6 2 =- 1-197 c 2 = + 0-933 2373 PP.P. Sph. excess = 20-63 A! - 74 5 54-27 B t ' = 68 22 0-37 PP>P* PP e P: u a ti // H A' Xi = + 0-32 + I-2I -0-73 0-06 - 3.81 + 0-41 B' y l= + 0-78 + i-37 - 0-50 + 0-33 - 2-18 + i-37 ' 2l = + 0-58 + 1-18 - 0-58 + 025 - 2-17 + o-73 If now we apply these corrections to the observed angles, each triangle will close correctly, and the reproduction of the side PP l by calculation through the angles of the polygon will stand thus : log sin 13. 9-8981414,6 9-9114487 9-9682783 9-9709969 9-9952856,6 9-6411438,7 sum = 9-3852949 log sin &. 9-9320053,4 9-9894661,7 9-9830544 9-8960580,8 9-6947090,6 9-8900019 9-3852949,5 = sum. CALCULATION OF TRIANGULATION. 225 The probable errors of the adopted angles $, 33, ( are for the first triangle 0"-766e, 0"-762e, 0"-746e, and similarly for the others. Here e is the probable error of an observed angle expressed in seconds. 3. Let the adjoining figure represent a chain of triangles, F-L, F 2 , ...F i being the points in which perpen- diculars from the trigo- nometrical stations P lt P 2) ... meet the meridian through P: let the length of the side %+!=*.. and the angle the di- rection P n P n +i makes with the north meridian K n . Suppose in the first place that each angle of each triangle is equally well observed, the probable error of an observed angle being e. Then the last side of the chain is ^ = sin BI sin B. 2 . . . sin B i sin A^ sin A. 2 . . . sin A i If this be calculated by using the observed angles A^ 3 B, ... the result k- will be , , _ , sin BI sin B% ... sin B- * sin AI sin A%' . . . sin A? " 226 CALCULATION OF TEIANGULATION. Using the same notation as before, and putting k f i lc i k { e, we have But if we correct each observed angle in each triangle by applying to it with a negative sign the third part of the excess of the sum of the observed angles above the truth, then the corrected angles are in the first triangle, and so for the others. If we calculate k t with these corrected angles and put still k'k = e we have which expresses the error of the resulting length of in terms of the actual errors of the observed angles. The probable error of Jc[ is thus (11) The reciprocal of t 2 (a 2 + a/3 -f /3 2 ) for any triangle or the reciprocal of the mean value of e 2 2 (a 2 4- a/3 -f ft 2 ) for a chain of triangles is called by Struve the * weight of continuation ' of the triangle or of the series. It is greatest when A and IB are nearly right angles, but in this case the angle C is very small and the triangle makes little ' progress ' in the chain. Hence the weight of a triangle is proportional to the progress it makes multiplied by the weight of continuation. Consider now the error of the calculated direction K{ of the last side ^ of the chain. Between K n , C n+l , and J n+1 there exists the relation whence it follows that A.J ^ TT c/j K. , K 3 = TT the law of which is obvious. Hence the azimuth of the last CALCULATION OF TRIANGULATION. 227 side as obtained from the corrected angles is or w -C 1 + 2 -43 + ...-, according as i is even or odd. Thus the error of the calcu- lated direction of k i is, disregarding its sign, and considering K as free from error, and the corresponding probable error 'A/T- Using the accent still to denote calculated quantities, the calculated value of PF i+l is k Q cos K+ k{ cos KI -f / cos K% + 3 ' cos K B ' + . . . k{ cos JT/, and in order to determine the probable error of this result it is necessary to express each term as a linear function of e \f\9\"> e -2/29-2> -" an( l then to ascertain the sum of the squares of the coefficients of those symbols. In thus estimating the errors of the calculated length of the chain and of the azimuth of the last side, we have treated the triangles as plane triangles, a simplification which can lead to no incorrect result. 4. Suppose that '_ both the sides k Q and k i are measured lines, free from error, and that it is required to correct the observed angles of the intervening chain so as to bring them into harmony with these lengths. Then when we use the thus corrected angles to calculate k i from k we arrive at a true result: thus sin B^ sin B 2 . . . sin JS t _ sin Bj sin B 3 . . . sin B^ sin A l sin A^ . .. sin A i ~ sin g x sin <&% ... sin ^ or Suppose that first ^ is calculated by means of the observed angles A^ /, A, B, ..., and Tf i being the result, let ^ k = &, then 228 CALCULATION OF TRIANGULATION. also using the same notation as before n +fn + &n = *, # +# n + Z n = - * n , and = cii^ + fay^a^ + P^ ...-a^ + /3^. (12) Further, let us take the more general case in which the angles are not equally well observed, and let the weights of A n ', J9 M ' C n ' be the reciprocals of w n , iv^ w n ". Then the most probable values of a^i^i, # 2 ^2 Z 2 are those, which, sub- ject to the condition (12), render a minimum, ) 2 4. "** . 9? -- r > To the differential of li add the differential of the right hand member of (12) multiplied by 3Q; then make the co- efficients of dx^ dy lt dx 2) dy%, ... severally zero. The first two give whence we have - ", (is) . aag H Wi the sum a?j +y x + z l making up 1 . Now if we substitute in (12) the values thus obtained of the %'s and ^'s the re- sulting equation will give Q in terms of known quantities, and thus all the required corrections to the angles follow from (13) and similar equations for each triangle. This completely solves the problem. Q being expressed in terms of e, e 1} 6 2 , ... these may be replaced by their equivalent expressions in terms of the 3* actual errors of observed angles, and thus, finally, the adopted angles and the length and CALCULATION OF TKIANGULATION. 229 direction of any side may be expressed in terms of the 3i errors. To avoid complexity, suppose the weights of all the ob- served angles equal, thus i=i i + i& a? 2 = -ie 2 -f 2 Q..., y\ = i *i + &iQ> #2 = i * 2 + ^<2 *i = -i e i + C \Q> Z 2 = - J * 2 + ^26 ; substituting in (12) and putting 2 (# 2 -f & 2 + c*) = 6/fc as at page 220, we have Thus we find for the actual errors of the adopted angles in the first triangle the expressions e f 2 - * lgl ) +^ (- 1 - glgl ) + ^ ( 1 - ai l ) T , i " " ~ " and by taking the sum of the squares of the coefficients of e ififfn & c ' we 8" e ^ ^ or ^ ne probable errors of the adopted angles : Probable error of 3 M . . . t ^/ (| - ||) The error of the direction of the side Tc i (that of the first side being free from error) depends on C x C 2 + C 3 &c. : the expression for the error is 230 CALCULATION OF TRIANGULATION. and if we add the sum of the squares of the coefficients as before, we find for the probable error of the direction in question, where 5. The probable error t of an observed angle must depend not only on the excellence of the instrument employed, the ex- pertness of the observer, and the number of observations taken ; but also on the care and skill with which the opera- tions generally are conducted. In as far as it depends on errors of bisection, of reading the circle, and of graduation, a value of e may be obtained from the internal evidence of the observations themselves at any station by comparing in- dividual measures of an angle with their mean. But the observed angles are affected with other errors which are only brought out in combining the observations made at different stations. For instance, if there be any residual error of centering the instrument over the station-mark, or if a signal observed be not truly centred on the station-mark, a constant error will result. In some instances local configuration of the surface may give rise to a lateral refraction, doubtless very small in amount, but persistent. Again a signal in the direction of east or west is liable to be differently illuminated on the north side and on the south so as to present a phase : any signal which is habitually seen in some peculiar light may present a phase. A more trustworthy method therefore than the evidence of the observations themselves is presented by the errors in the sums of the observed angles of triangles. In the 107 triangles between Dunkirk and Formentera the mean square of error of a triangle is 4-161, hence the mean square of error of an observed angle is one third of this or 1-387 : the proba- ble error of an observed angle therefore in this work is -6745\Xl-387 = 0"-794. CALCULATION OF TEIANGULATION. 231 In Colonel Everest's chain of triangles between the Dehra Dun and Seronj bases the 86 triangle errors give t = 0"'517 ; and for his 72 triangles between the Beder base and the Seronj base e = 0"-370. In the Russian arc, the probable error of an observed angle in the Baltic provinces is t = 0"-387 ; in Bessarabia e = 0"-573; in Finland e=0"-589; in Lapland e=0"-843, while in the 12 extreme northern triangles t = l"-466. 6. The chain of triangles joining Dunkirk and Formentera is a simple chain such as we have been considering. The same may be said of the greater part of the Russian chain, though in the portion of the work north of Tornea as far as Fuglenaes the work is strengthened by the ob- servation of a greater number of points at each station. The ac- companying diagram shows a por- tion of the meridional chain of Madrid starting from the base of Madridejos. It is clear that such chains cannot be dealt with in the same manner as the simple chain of triangles we have been considering, and this remark ap- plies still more forcibly to triangu- lations like that of Great Britain and Ireland where the lines of observation are interlaced in every possible manner. The equations of condition of a triangulation are those which Fig. 48. exist between the supernumerary observed quantities and their calculated values : that is to say, after there are just sufficient observations to fix all the points, then any angle subsequently observed can be compared with 232 CALCULATION OF TRIANGULATION. its calculated value. If a triangulation consist of n + 2 points, two of which are the extremities of a base line, then the re- maining n points will require 2n observed angles for their fixation, so that if m be the number of observed angles there will be m2n equations of condition. The manner of ob- taining these is as follows. Suppose a number of points A, B, C, ... already fixed, and that a new point P is observed from and observes back m of these points, then there will be formed m 1 triangles, in each of which the sum of the observed angles must be equal to 180 plus the spherical excess; this gives at once m 1 equations of condition. The m 2 dis- tances will each afford an equation of the form called a side equation,, viz. : PC PB PA_ ~PB' PA PC~ not however limited to three factors. Should P observe and be observed from only two points then there will be but one equation of condition ; when m is not less than 2 every other bearing not reciprocal whether from P to the fixed points or from the fixed points to P will give a side equation. When independent angles are observed another species of equation enters arising from the consideration that at every point where all the angles round the horizon have been observed, their sum must = 360. In what follows this case is not supposed. If then there be M observed bearings at N stations there will be MN angles for fixing N2 points, which require only 2N4: angles, so that the number of equations of con- dition is M 37V+4. If further there be P points at which there are no angles observed the number of equations of condition will be M 3N 2P-f 4, where N is the number of observing stations To these must be added equations which may arise from there being more than one measured base ; thus n bases would give rise to n I side equations. The side equations take the form 1 = sn + yi sn +y 2 sn 3 . . . . sin (AS + arj) sin (A + a? 2 ) sin (A + x 3 ) . . . ' where x^y^ % z y 2 > are corrections to the observed angles CALCULATION OF TBIANGULATION. 233 i'Bi, A 2 'B 2 ', .... This may be written sin AS sin AS... * Let #! ^ , . . . be expressed in seconds, and take the logarithm of (l 4) ; then if we put mod. sin l" cot A{ = a^ , mod. sin l" cot B = b L , and so on, we shall have = 2 (log sin J? logsin A') a^x^ -\- b^y-^a.^ + b 2 y 2 .... (15) Here a^b^ ... are the differences of the logarithmic sines for one second. Thus we see how to obtain in any given triangulation all the necessary conditions that exist among the observed angles and to express them in the form of linear equations among the corrections to be applied to these angles. The correc- tions are as yet indeterminate ; but according to the theory of probabilities the most probable values are those that render a minimum the sum of the squares of all the errors -of observation. 7. Consider first the angles observed at one station only. Selecting one signal, R say, from amongst those observed, as that to which the direction of all the others are to be referred ; let the directions of the other signals taken in azimuthal order make with the direction of R the angles A, B, C, ... j these being the most probable values to be determined. Let the first arc give the readings %, %', %", %'", ..., % cor- responding to the arbitrary reading of 7?, of which let ^ be the true or most probable value : then the first, second, and third arcs will give the equations 0^- a?! = 0, m^XiA = 0, m^'x^B = 0, m 2 x 2 = 0, m%x 2 A = 0, m^'x 2 B = 0, m% # 3 = 0, m^ # 3 A = 0, wi^' a/ 3 B = 0, and so on. These equations at least would hold good were the observations free from error; as it is, the left hand members are the errors of observation, the sum of the squares of which must be made a minimum in order to obtain the 234 CALCULATION OF TRIANGULATION. V most probable values of A, , C, ... and of the arbitrary dis- tances #!, # 2 , #? 3 , ... of the zero of the circle from the signal of reference. In order to make the result general, multiply these equations by multipliers each of these to be unity when there is an ^observation, or zero when the corresponding observation is wanting. Then the sum of the squares of all the errors of observation at this station is Pi fa- &c. The diflPerential coefficients of this sum with respect to a?!, # 2 , # 3 ... A, , C, ... being severally equated to zero, we have these equations : &c.; &c.; substitute in the second set the values of &L , # 2 , ... given by the first, and the result will be a series of equations which may be written thus (aa)A + (ati)B + (ac)C+... = (an), (16) (ab)A+(bb)B+(bc)C+... = (bn\ (ac)A + (be)B+(ce)C+... = (en), &c. These equations determine A,B,C, ____ That is to say, they determine certain values which are the most probable with reference to the observations at that station only. But the condition to be satisfied is that the sum of the CALCULATION OF TRIANGULATION. 235 squares of the errors of observation at all the stations is to be a minimum. Let this sum be expressed by 211 = fco., which is to include all the stations. Suppose there are i equations of condition in the triangulation : these will be expressed thus tf+..., (18) fee. Multiply these equations by multipliers 7 1} I 2 , I 3 ,..I { of which the values are to be determined. Then the condition of minimum requires the following : AA dQ, JB dC ' " &c. To abbreviate, use this notation (19) &c., a symbol of the form \n\ corresponding to each observed angle. With this substitution the preceding equations become 236 CALCULATION OF TBIANGULATION. [3] +A" &c. Substitute now the values of the no's from the first equa- tions in the second equations, and the result will be ., (20) &c., corresponding with equations (16): (a a), (ab), (an), ... being the same in both. But the-A,lt, C, ... in (16) are not the same as the A, B } C, ... in (20) : the former are only ap- proximate values, let them be denoted by A lt B 19 C IJ ... ) and let and so on. Substitute these values in (20) and we have ., (21) &c. Each station will present a group of equations of this form in number less by unity than the number of signals there observed. These equations are to be solved and brought into the form (1) = (a) [l]+(a/3) [2]+(ay)[S] + ... , (2) = (a/3) [l]+(/3/8) [2]+(|8y) [3] + ... , (22) (3) = (ay) [l]+(/3y) [2]+(yy) [3] + ... , &c., this notation for the coefficients being adopted for the sake of symmetry. Substitute in these last the equivalents of [l], [2], [3], ... from (19), and we get (l), (2), (3), ... expressed in terms of the multipliers I 19 7 2 , 7 3 , ... . The equations of condition are to be formed with the angles CALCULATION OF TBIANGULATION. 237 A lt B l} ?!,... as resulting from the observations at the separate stations : this brings them into the form =ft+a 1 (l)+ft(2)+y 1 (3)+... , 0=^+a 2 (l)+ft(2)+y 2 (3) + ..., (23) =^+ 3 ( 1 )+/ 3 3(2)+y 3 (3)+..., &c., which are i in number. Next substitute in these the ex- pressions for (l), (2), ... in terms of /j ... I i} and the result is a system of equations from which the i multipliers are to be obtained by elimination. When this elimination, which is the most serious part of the operation, is effected, the numerical values of (1), (2), (3), ... easily follow. In this manner Bessel calculated his triangulation in East Prussia, which contained 31 equations of condition and re- quired the solution of a final system of equations of 31 unknown quantities. The method has since been extensively used notwithstanding the very heavy calculations demanded. It is now being carried out in the reduction of the Spanish triangulation. 8. In the principal triangulation of Great Britain and Ireland there are 218 stations, at sixteen of which there are no ob- servations, the number of observed bearings is 1554, and the number of equations of condition 920. The reduction of so large a number of observations in the manner we have been describing would have been quite impracticable, and it was necessary to have recourse to methods of approximation. In the first place, the final results of the observations at each station were obtained by an approximate solution of the equations (16), the nature of this approximation will be understood from the following table. The first division of this table contains the observations to six signals on six different arcs ; a constant quantity having been applied in each arc to the several circle readings in that arc so as to make R read the same (approximately the true azimuthal reading) on all. The means of the vertical columns are then taken, and the second part of the table contains the 238 CALCULATION OF TBIANGtJLATION. differences between the individual results in the vertical columns and their mean. In a column to the right of this E 4 21' A n / B 37 34' G 97 54' D 220 3 ' E 271 43' MEAN. 29-21 20-21 36-04 fvQI // 14-07 47.84 ii 19-00 18-18 39-17 38-22 29-21 29-21 29-2I 34-21 32-4 1 n-86 10-71 II-QI 46-05 48-30 16-30 H-I7 39-4 2 29-21 i8-*o 41-04 29-2I 34-64 I2-I 4 47-40 I7-25 39-46 O.QO o-oo o-oo 0-00 + 1-40 + 1-27 -0-43 2-23 + I-93 0.28 1-43 + 0-44 j.ac + i-75 + 0-93 O.QK 0-29 -1-24 -0-04 + 0-87 + 0-24 0-19 V>-IQ o-oo 0-23 + O-QO -3-08 *i 0-60 o-oo + 1-34 + 1-58 + 0-07 28-34 2897 35-!7 35-67 13-20 4697 18-13 17-04 38-30 37-98 29.40 34-40 12-05 39-61 30-40 29-81 28-24 33-6o 11-90 12-51 47-24 48-90 17.49 14-77 17-62 40-07 29-19 34-7i 12-42 47-70 17-19 38-99 portion of the table are placed the means of these small quantities in the corresponding horizontal lines. In the third part of the table these last small quantities, one cor- responding to each arc, are applied with contrary signs to the corresponding readings in the first part of the table. The means of the vertical columns as they now stand are taken as the true or most probable bearings. The weights of these values of the bearings are formed by taking the differences between the individual results in each vertical column and their mean and summing the squares of these differences : thus, see page 56, w = CALCULATION OP TRIANGULATION. 239 n being- the number of observations of the bearing in question. The final weights so obtained would have been greatly increased if it had been allowable to reject discordant observa- tions, but this has never been done unless the observer has made a remark that such an observation ought to be rejected. Observations taken under favourable circumstances are doubt- less more valuable than observations under less favourable circumstances ; but how to assign their relative numerical value is a question admitting of no general solution. ' It appears that the longer time one is compelled to bestow upon observations under less favourable circumstances, in a great measure compensates external disadvantage, and that causes of error of observation of which the observer himself has not been conscious, often influence him no less than those which obtrude themselves upon him ' (Bessel, Gradmessung in Ost- preussen}. It has indeed been often noticed that an observa- tion to which the observer has attached a remark to the effect that the bisection was unsatisfactory, or that the light was bad, or any other expression of doubt, has been found to agree with singular precision with the general mean. Thus then are obtained at each station the bearings of all the other stations with their respective weights. The problem then takes this shape : to determine a system of corrections to these bearings such that the sum of their squares, each multiplied by the corresponding weight, shall be a minimum : this is different from Bessel's solution, where the actual sum of the squares of all the corrections at all the stations was made a minimum. The modified problem, without sacrificing much, is very much more practicable. Let the observed bearings be numbered consecutively, and let x n be the correction to the th bearing, of which let the weight be w n . Then we have .. , (24) &c.; 212 = wx--\-wx 240 CALCULATION OF TBIANGULATION. being the equations of condition of the system, and a function 2H, of the corrections which is to be made a minimum. Differentiate the equations of condition after multiplying them by 7 15 7 2 , 7 3 , . . ., and adding to cll, make the coefficients of dx^ dx^ dx^ ... in the sum severally equal to zero. Thus w 1 se l = a i r i + d l I 2 + c 1 I 3 ... t (25) &C. Substitute these in the equations of condition, and the result is (aa\ ., /Qb\ -r /ac\ _ , x -H+C-H+C-H-. (26) &C. Here we have a system of numerical equations equal in number to the equations of condition, and by their solution are obtained numerical values of l lt I 2 , 1 3 , . . . . These substituted in (25) give directly the required values of a^ # 2 a? 3 , . . . . After the application of these corrections to the observed bearings, all the geometrical requirements will be fulfilled, and that with the least possible alteration, in the aggregate, of the original observations. The different steps of the process are then as follows : First : the obtaining of the geometrical equations of condition sup- plied by the connection of the triangulation. Second: the substitution in these equations of the observed bearings, each with its unknown correction appended. Third : the equations of condition being written out in their algebraic form (24), and unknown multipliers assumed, the equations (25) are formed. Fourth : from these equations the corrections must be obtained in terms of l lt 7 2 , J 3 , . . . and substituted in the equations of condition. Fifth : these equations must now be solved and numerical values will result for 7 t , 7 2 , 7 3 , ... . Sixth : the substitution of the values of these multipliers in the equa- tions (25) whereby the corrections x-^ # 2 #3* become known. CALCULATION OF TRIANGULATION. 241 Seventh : the verification of the work by the substitution of the corrections in the equations of condition, and by the work- ing out of the whole triangulation. The following test may also be applied. Supposing x^ x z x 3 , ... to be the corrections to the bearings at any one station, w l w 2 w%, . . . the correspond- ing weights, then it is easy to show that w^ x l + w 2 # 2 + w. d r 3 . . . = . If e n be the actual error of the n ih bearing, the error of the adopted value of that bearing is e n + # M . Now we may express the ay's in terms of all the e's : for by inversion of the equa- tions (26) the Z's may be expressed in terms of the e's, and these last are connected with the actual errors by the equa- tions 3 = c : e 1 + c. 2 e. 2 -f c 3 e 3 . . . , &c. Let the multipliers 1 1 I Z I^ ... by means of (26) be expressed in terms of e x 2 e 3 , ... thus ) 2 + (ay) 9 ..., (27) &c.; and make use of symbols A-jAgAg, ... such that A 1 = (aa)fl 1 + (a^)d 1 + (ay)c 1 ..., (28) A = a &c. Then ... = (A 1 1 + A 2 ^i + A 3 q ...)e it -f A 3 e? 2 ...)e? 2 , &c. Similarly x.^y^ ... may be expressed in terms of e 1 e. 2 e^ ... Consider now the probable error of e l + x l . If we write then w l (x l Suppose that t is the probable error of an observed bearing to R 242 CALCULATION OF TBIANGULATION. which appertains the weight unity, then the probable error corresponding to a weight w will be t:\fw. Therefore the probable error of w l (x l -f ^) is (29) l 2 3 Now which we shall arrange thus w \ 1 1 w \ 1 2 w i l 3 w \ d-i b-t u-i b-i u-i c-t -j- AiAn -|- AoAn -|- An Ac -}- .. 1 2 Wl 2 2 w, 2 3 Wl &c. These we have to add to similar expressions in thus &C. But the mutual relations of the equations (26) (27) give the following transformation of (28), &C. Thus by addition of the vertical columns of (30) CALCULATION OF TRIANGULATION. 243 which substituted in (29) that expression becomes r(%4A)*j and restoring the value of _p lt we have for the probable error of the corrected bearing, corresponding to x v (31) The probable error of the distance between any two stations in the triangulation, or of the angle subtended at any station by any two other stations, may also be expressed ; but for this we must refer to Gauss : Supplementum theoriae combina- tionis observationum erroribus minimis obnoxia, Gottingen, 1826; or to the investigations of General Walker, in the second volume of the Account of the Great Trigonometrical Survey of India, where the subject is very elaborately worked out. The minimum value of 211 = w^x^ -f w^xf -f w 3 # 3 2 ... is easily shown to be 2fl = 7 1 6 1 + I 2 e 2 + / 3 e 3 +.... In order to avoid the solution of the equations containing #20 unknown quantities in the triangulation of Great Britain and Ireland, the network covering the kingdom was divided into a number of blocks, each presenting a not unmanageable number of equations of condition. One of these being corrected or computed independently of the others, the corrections so obtained were substituted (as far as they entered) in the equa- tions of condition of the next block, and the sum of the squares of the remaining equations in that figure made a minimum. The corrections thus obtained for the second block were substituted in the third and so on. Four of the blocks are independent commencements, having no corrections from adjacent figures carried into them. The number of blocks is 2 1 : in 9 of them the number of equations of condition is not less than 50 : and in one case the number is 77. These calculations all in duplicate were completed in two years and a half an average of eight computers being employed 1 . The equations of condition that would have been required 1 In connection with so great a work successfully accomplished, it is but right to remark how much it was facilitated by the energy and talents of the chief computer Mr. James O'Farrell. E, 1 244 CALCULATION OF TRIANGULATION. to make the t Hang-illation conform to the measured lengths of the base lines were not introduced, as they would have very greatly increased the labour, already sufficiently serious. 9. When once the corrections to the several observed bearings have been found as described above, the calculation of the distances by Legendre's Theorem is sufficiently simple and straightforward. But if the equations of condition binding the triangulation to an exact reproduction of the lengths of the measured base lines have been omitted, we have still to consider what shall be taken as the absolute length of any one side in the triangulation. Let x be the required length of any one side ; and let ^ #,-& # 5 3 # be the lengths of the base lines as inferred from the ratios ^ 2 f 3 ... given by the triangulation of the specified side to those base lines. Then if B 1 B 2 B Z ... be the measured lengths of the base lines, w l w 2 w 3 ... the corresponding weights, x must be taken so as to render a minimum the expression x - that is to say In this kingdom six base lines have been measured, the earlier ones with steel chains, the two most recent with Colby's com- pensation apparatus. The absolute length of any side in the triangulation is made to depend entirely on the two last. The following table contains the measured lengths of the bases, and their lengths in the corrected triangulation DATE. PLACE. MEASURED. IN TRIAN- GULATION. DIFFER- ENCE. COUNTY. ft. ft. ft. 1791 Hounslow Heath 27406-19 27406-36 + 0-17 Middlesex. 1794 Salisbury Plain 3 6 576-83 36577-66 + 0-83 Wilts. 1801 Misterton Carr 26344-06 26343-87 0-19 Lincoln. 1806 Khud.llan Marsh 24516-00 24517-60 + i -60 Flint. 1817 Belhelvie 26517-53 26517.77 + 0-24 Aberdeen. 1827 Lough Foyle ... 41640-89 41641-10 + 0-21 Londonderry. 1849 Salisbury Plain 36577.86 36577-66 O-20 Wilts. CALCULATION OF TRIANGULATION. 245 The only serious difference here shown is in the case of the base at Rhuddlan, and this is owing in great measure to the bad connection of the line with the adjacent triangulation. 10. We shall now give a simple example of the calculation of corrections to observed bearings in a small piece of the triangu- lation of this country. The points are South Berule, B, in the Isle of Man; Merrick, M, in Kircudbrightshire ; Slieve Donard D, in the County Down, D> Ireland ; Snowdon, S t in the North of Wales ; and Sea Fell, F, in Cumberland. The line DF in the diagram being broken to- wards F t intimates that Sea Fell did not observe Slieve Donard. Now in the triangle BMD, the three angles being observed, we have first = 180 Fig. 49. where l is the spherical excess of the triangle. Secondly, the triangle BDS gives similarly Thirdly, the triangle BSF gives BSF+SFB+FBS= 18 Now the points MBF being fixed the observation of the angles BMF 9 BFM, which are known, brings in two equations of condition ; one is = and the second, the ' side equation ' sin BSF . sin BDS . sin BMP . sin BFM sin BF8 . sin BSD . sin BDM . sin BMF Finally, the observation DF brings in the side equation sin FMD . sin FBM . sin FDB sin FDM . sin FMB . sin FBD 246 CALCULATION OF TKIANGULATION. In order to express these equations in numerical form we give in the following table the results of the observations at the different stations on which the calculation is to be based. The last column but one gives the reciprocal of the weight of each observed bearing : and the last column the number of observations in each case. For te lt # 2 , # 3 , ... } we write, as is usual (1), (2), (3), OBSERVING STATIONS. STATIONS OBSERVED. OBSERVED BEARING WITH SYMBOLICAL CORRECTION. i w No. OF OBSER- VATIONS. Merrick South Berule 647' 45-72 + (i) O-II 35 Slieve Donard 41 52 49-08 + (2) 0-19 24 Sea Fell ... 312 48 54-59 + (3) 0-13 32 Slieve Donard Merrick 220 4 I 42-50 + (4) 0-08 8 Sea Fell ... 259 5 10-02 + (5) 0-46 13 South Berule 271 53 16-88 + (6) o-37 19 Snowdon 314 38 39-43 + (7) O-22 21 South Berule Slieve Donard 92 54 35-32 + (8) 0-61 2 4 Merrick 1 86 38 21-37 + (9) 0-82 20 Sea Fell ... 249 44 48-27 + (10) 0-50 17 Snowdon 341 41 50-13 + (ii) 0-62 2O Sea Fell ... Snowdon 20 38 31-85 + (12) ^93 3 South Berule 70 55 36-88 + (13) 0-98 6 Merrick 133 50 41-88 + (14) 1-88 7 Snowdon Slieve Donard 136 7 5I-72 + (15) 2-17 4 South Berule 162 10 9.13 + (16) IO-IO 4 Sea Fell ... 199 56 40-20 + (17) i-35 3 The formation of the four angle equations will then stand as follows, South Berule ... 9343 / 46' / 05-(8)4-(9) Slieve Donard... 51 11 34-38 (4) + (6) Merrick ... 35 5 3-36 -(l) + (2) 180 023-79 l = 22-922 /. = + 0.868-(l)4(2)-(4)-t(6) CALCULATION OF TRIANGULATIOX. 247 South Berule SlieveDonard Snowdon Ill 12 45-19 + (8) (11) 42 45 22-55 (6) + (7) 26 2 17-41 (15) + (16) 180 025-15 2 = 24-433 =+ 0-717- South Berule Snowdon Sea Fell 9157 / 1^86 (10) + (11) 37 46 31-07 (16) + (17) 5017 5-03 (12) + (13) 180 037-96 e= 32-258 = + 5-702-(10) + (ll)- South Berule Sea Fell Merrick 63 e2690 6255 5-00 (13) + (14) 53 58 51-13 + (l) (3) 180 023-03 e= 25-230 = - 2-200 + (l) (3) In the calculation of the side equations, one third of the spherical excess has been subtracted from the observed angles in the different triangles : this is not necessary, but it is generally convenient to do so. Using eight figures of logarithms, and understanding that (l), (2) ... are expressed in seconds the calculation stands thus, log sin BSF ... 9-7871239,0+27,17 { (16) + (17)}, BDS ... 9-8317751,2 + 22,77 {- (6) + (7)}, BMD ... 9-7594791,7 + 29,98 { (l) + (2) }, BFM... 9-9495547,9 + 10,77 { (13) + (14)}, logcosec BF S ... 0-1139629,9 17,49 { -(12) + (13)}, BSD ... 0-3576004,5-43,10 {_(15) + (16}}, BDH... 0-1083304,916,93 { (4) + (6) }, BMF... 0-0921606,2-15,31 {+ (l) - (3) }, Sum 124,7 + &c.; 248 CALCULATION OF TEIANGULATION. log sin FMD ... 9-9999417,3+ 0,34 { + (2) (3) }, FBM ... 9-9502860,1 + 10,68 {_(9) + (10)|, FDB ... 9-3455070,3 + 92,67 {-(5)+ (6)}, logcosec FD M ... 0-2069267,5-26,58 {-(4)+ (5) }, FMJB ... 0-0921606,2-15,31 {+(l)_(3)} 5 FED... 0-4052080,5-49,21 { + (8)-(lO)j, Sum 301,9 + &c. Thus the fifth and sixth equations are = 124-7 45-29 (l) +29-98 (2) +15-31 (3) + 1693 (4) -39-70 (6) +22-77 (7) + 17-49 (l2)-28-26 (13) + 10-77 (14) + 43-10 (15) 70-27 (16) + 27-1 7 (17). 0= +301-9 -15-31(1)+ 0-34 (2)+ 14-97 (3) + 26-58 (4)- 119-25 (5) + 92-67 (6) -49-21(8)- 10-68(9) + 59-89(10). Now multiply these six equations by 7 15 7 2 ... 7 6 , and form the equations (25) page 240 ; they will stand thus, 01 W =-/i + /4-45-29J 6 - 1.5-31 7 6> an d using the method of page 39, dS will involve generally six of the symbols a?, y with coefficients not unity. Then having written down the expres^ons for the successive quantities dv, dS, the formula (3) gives the total increments d(X) and CALCULATION OF TRIANGULATION. 257 d(Y) to be added to the values obtained for (Z), (7). Thus the equations (C) and (D) are Then we have to make a minimum the sum or, which is the same, 2 {# 2 +# 2 4-(#+y) 2 } is to be a mini- mum subject to four conditional equations : =a = = = a Proceeding as in previous cases, this resolves itself into the determination of four multipliers, by which finally the a?'s and ys are obtained. Had another side, as HK, been a measured base this cir- cumstance would have introduced an additional equation of condition. The most elaborate calculations that have ever been under- taken for the reduction of triangulation by the method of least squares are those of the Indian Survey. The principal triangulation of India is formed of chains of triangles dis- posed as shown in the diagram, page 31. The axis of the system is the great arc of Colonel Everest, running from Cape Comorin to the Dehra Dun base in the Himalayas : the principal chains divide the triangulation into five geo- graphical sections, four of which may be roughly described as quadrilaterals, the fifth in the south being trilateral. At the corners of the quadrilaterals are the base lines which, with one exception, were measured with Colby's apparatus. As the extent of the operations quite precluded the idea of reducing the whole in one mass, as required by theoretical considerations, General Walker decided to treat separately the five sections specified above, reducing the figures in succession, upholding and maintaining the results determined for the one first reduced in the contiguous figures when they in turn were to be undertaken. This arrangement made it necessary to commence with that section of the work which was in all 258 CALCULATION OF TBIANGULAT10N. its parts of the highest accuracy: this is the north-west quadrilateral 1 alfi. In this section of the work there are 128 single triangles and 110 polygons including in that term quadrilaterals and complex figures comprising a total number of 2418 observed angles. These polygons present in the aggregate 955 equations of condition, without consider- ing the closings of circuits. This being as a whole still unmanageable, it became neces- sary to obtain corrections to each separate figure, whether a simple triangle or a polygon, with regard only to the con- ditions presented by that figure itself. Thus, in the first instance, all the angles received corrections without regard to the closings of circuits. For the purpose of the final adjustments of the circuits, the complex chains composed as they are of single triangles and polygons were replaced by simple chains composed of these single triangles and a selection of continuous triangles from the polygons : the polygons having been made consistent, it was so far immaterial how these triangles were selected. Then the already partially corrected angles J, i?, C of any triangle receive symbolical corrections #, y> and (x+y), the sum of the squares of which multiplied by the respective weights is to be a minimum. The corrections #, y, . . . are con- nected by equations which ensure the closing of the chains and the correct reproduction of the measured base line lengths. An inspection of the diagram shows that in the north-west quadrilateral there are five circuits to close, and, according to what we have seen in the preceding pages, this requires 4 x 5 = 20 equations of condition. Besides these the four base lines give 4 1 = 3 additional equations. With respect to the reproduction, by the corrected triangu- lation, of the measured length of the bases, it is necessary to remark that since in this calculation the exact weight of every observed angle has been strictly considered and brought into play so as to influence duly the final results, so also should the probable errors or weights of the measured bases, which are not errorless any more than the observed angles. 1 Account of the Operations of the Great Trigonometrical Survey of India, by Coi, J. T. Walker, C.B., E.E., F.R.S. ; vol. ii, pages 30-32. CALCULATION OF TRIANGULATION. 259 But General Walker has shown (vol. ii, page 265) that it is but reasonable, after considering all the facts of the case, to treat the base lines as errorless in comparison with the triangulation. Let the adjoining diagram represent a succession of sides in a closing chain of triangles. Each side and each angle has a definite nu- ^\\^ merical value as given by the already partially corrected triangles, and to this numerical value is to be attached in each case a symbolical increment which can be expressed in terms of #'s and y's. Now suppose, that starting from the point a with a definite latitude and a definite azimuth of one line there, the latitude and longitude of b is calculated and also the back azimuth of a at b. Let Fio ._ .. this process be repeated from b to c, and so on, until p is reached. The result will be that we get the latitude and longitude of p and the azimuth of the line p q. Then again, if proceeding by b', c', ... we make corresponding calculations, we get a second set of values of latitude, longi- tude, and azimuth at p ; and the two sets must be equated respectively. Thus arise three equations l : Left route. Right route. Latitude, <' + 2 (a af + b' y') = 4> + 2 (a x + b y\\ \ / / \ j i ' Longitude, & + 2 (a^x + by) = H + 2 (a^ x + l^y] ; Azimuth, A' + 2 (a^ a' + // ) = A + 2 ( 2 a? + b. 2 y] . The fourth equation, an ordinary side equation, establishes the equality in length of pq with p' tf . 4>' 4>, H'l, and A x A are the circuit errors in latitude, longitude, and azi- muth, as shown by the partially corrected triangles. The manner in which the equations were formed is this. Take first the linear equations which secure the repro- duction of base line lengths (A, 13, C, D are the Sironj, 1 Comprised in Col. Walker's formulas (121), vol. ii, page 176. S 2, 260 CALCULATION OF TRIANGULATION. Dehra Dun, Chach, and Karachi bases) and also identity of side lengths at the junction of chains. The discrepancies, re- ferred to the seventh place of decimals in the logarithms, between the measured and com- puted base lengths, or between the lengths of sides at the junctions as computed by two Fig. 54- AB + + + 44,0; 68,2; 71,9; 79, 6; different routes, stand thus : 1. Dehra and Sironj bases 2. Triangle side at a ..... BaAd, da 3. Dehra and Chach bases . . BC 4. Sironj and Karachi bases . AD 5. Karachi and Chach bases .DC ....... -f 163, 8 ; 6. Side at b .......... da, ab de, eb . 124, 6; 7. Side at c .......... Tib, bc-hf, fc . + 150, 9 ; 8. Side at C ......... fc, cC-fg, gC . - 5, 3. Next follow the discrepancies of latitude, longitude, and azimuth at a, b, g, c, C : A-A' + 0-17 + 0-21 + 0-29 0-29 0-29 + 5-91 ; + 1-55; 3-25; 4-23; 3-00. AB, BaAd, da ... +0-39 da y abde, eb ...... 0-39 eh, ligeD, Dg ... +0-39 hb, bchf,fc ...... +0-04 fc,cC-fg, 9 C ...-0-00 These numbers supply the absolute terms of the 23 equa- tions of condition. After the formation of these equations the next step is to form the equations for the 23 multipliers, and solve them numerically. The values of the corrections, viz. the #'s and fs resulting from this voluminous calculation, are remarkably small. The total number is 1650 ; of which 1511 are less than a tenth of a second, 116 are between 0"-1 and 0"-2, 20 are between 0"-2 and 0' r '3, and 2 between 0"-3 and 0"-4 ; 1 only amounts to O r/ -46. This is the merest sketch of the elaborate system of calcu- lation followed in the reduction of the Indian triangulation ; CALCULATION OF TEIANGULATIOX. 261 its brevity might convey the impression that the matter is simple ; a reference however to General "Walker's second volume will dispel such impression. We have referred to the reduction of the north-west quadrilateral only, but similar methods are followed in the other sections of the work. In the reduction by least squares of the triangulation of Spain 1 , it has been found necessary in order to solve the equations of condition which are more than seven hundred in number to divide the whole network into ten groups : the number of equations of condition in these groups being from 60 to 83. Each group is reduced independently of the adjacent groups, and finally certain equations of condition are introduced in order to reconcile discrepancies that must other- wise appear at the common lines of junction. The number of base lines is four. 14. The French have recently (1859-1869) executed in Algiers, with modern instruments and methods, a network of triangu- lation extending from the frontiers of Morocco to those of Tunis, embracing 10 of longitude. M. le Commandant Perrier who conducted the western half of this chain was one of the officers who, in 1861-62, acted in co-operation with the English in the connection of the triangulation of England with that of France which connected in fact the Shetland with the Balearic Isles. In the course of a reconnaissance of the mountains near Oran in August, 1868, M. Perrier satis- fied himself that it was possible to connect geodetically Algiers with the peaks of the Sierra Nevada, some sixty leagues distant in Andalusia. He observed in fact two of these peaks from several of his stations, determining their approximate distances and heights, and proving that the path of the ray of vision did not in any case come within 300 metres of the surface of the sea. The only remaining diffi- culty was to obtain a visible signal suitable for the purpose of strict geodetic observations. 1 Informs sobre la compensation, por trozos, de los error es anyulares de la Red geodesica de Espana. Madrid, 1878. 262 CALCULATION OF TRIANGULATION. By the co-operation of the French and Spanish officers, and the liberality of their respective governments, the junction has just (in the autumn of the present year, 1879) been com- pleted in the most perfect manner by means of the electric light. Twenty miles south-east of Granada is the highest peak in Spain Mulhacen 11420 feet in height; distant fifty miles E. N.E. from this is Tetica (6820 ft.); the line joining these points forms one side of a quadrilateral of which the opposite side is in Algiers. The terminal points of the Algerian side, which is 66 miles in length, are Filhaoussen (3730 ft.) and M'Sabiha (1920 ft.), each of which is about 1 70 miles from Mulhacen. The other two sides and the diagonals of the quadrilateral span the Mediterranean. Each station observes the other three, so forming four triangles whose spherical excesses are 43"-50, 60"-07, 70"-73, 54"-16. At each station the signal light was produced by a steam engine of six horse-power working a Gramme's magneto- electric machine in connection with the apparatus of M. Serrin. The labour of transporting to such altitudes this machinery, with the requisite water and fuel in addition to the ordinary geodetic instruments and equipment and the maintenance of the whole in action for two months, necessi- tated at each station the formation of a military post. After incredible difficulties, the whole was ready on August 20th, the Spanish stations occupied by Colonel Barraquer and Major Lopez, the Algerian by M. le Commandant Perrier and Major Bassot. It was not however until the 9th of September that the electric light of Tetica was seen in Algiers a red round star-like disk visible at times to the naked eye; on the following day Mulhacen was seen, and the observations were thenceforth prosecuted until the 1 8th of October. The errors of the sums of the observed angles in the four triangles were + 0"-18, 0"-54, + l"-84, +1-"12, leaving nothing to be desired on the score of precision. Thus -^-^^^ /S T CALCULATION OF TRTANGULATION. 263 a continuous triangulation now extends from Shetland into Africa. Not content with this brilliant achievement, the deter- mination of the difference of longitudes of Tetica and M'Sabiha was resolved upon and carried out by M. Perrier and his Spanish associates. The method that should be adopted for this purpose had been previously made the subject of elaborate investigation and study at Paris with the apparatus actually used. The signals adopted were the eclipsing of the light every alternate second, the interval between the eclipse and the reappearance being one second. The signals numbering 640 each evening, divided into six- teen series, were issued alternately from Tetica and M'Sabiha from the 5th of October to the 16th of November; the obser- vations being registered chronograph ically. The difference of the personal errors of the observers, M.M. Perrier and Merino, had been thoroughly investigated at Paris, so that nothing was wanting to render the results absolutely satisfactory. We shall conclude this chapter by giving the formulae for the solution of a simple quadrilateral with diagonals as in Fig. 55- the annexed figure. Let the spherical excesses of the four 264 CALCULATION OF TEIANGULATION. triangles with common vertex be f 1} 2 , f 3 , f 4 , then denoting by accents the observed values of the angles, the three angle- equations may be written thus where e Q , e^ e^ result from the errors of observation. From these we have to form the quantities The eight unknown corrections to the eight observed angles (which we suppose to have been independently observed) may by means of the three angle-equations be reduced to five, and expressed thus : and it remains that these must fulfil the condition sin &! sin & 2 sin ^ 3 sin & 4 _ sin iSi sin iS 2 sin B 3 sin |g 4 If we put sin (A{ + eQ sin (A 2 ' + c a ) sin (^ + g 3 ) sin (^/ = 1+tjsml", sinljtf/ +e 1 )sm(j^ 9 +e ) sm(^r -----" - 2 + s fij + f^ & + ii or the spherical excesses of the four large triangles, are 43 // -50, GO 7 '- 07, 70 7/ -73, 54"-16; and the errors of the sums of the observed angles of the same triangles are respectively _ 1".04, + 1 77 -G8, +2 7/ -80, -fO"-08, and we have ?{(c 2 ) + i Co 2 }- 1 8-6711; 266 CALCULATION OF TEIANGULATION. so that t] {(c 2 ) + i c 2 }~ 1 = + 0"-0469. The values of a? , follow at once : // % =0-010, ^ = +0-040, x. 2 = +0-263, # 3 = +0-036, a? 4 = + 0-257; and the resulting corrected angles are o / ff o ' /' ! = 50 25 41-650, ^ = 88 53 30-990, & 2 =2410 1-927, iS 2 = 16 31 28-933, & 3 =79 134-054, 58 3 = 60 17 55-156, & 4 =1757 9-613, 38 4 = 22 44 31-907. CHAPTER X. CALCULATION OF LATITUDES AND LONGITUDES. THE problem : Given the latitude of A, and the distance and azimuth of B from A, to determine the latitude and longitude of B and the azimuth of A at B, would be very simple if the earth were a sphere, requiring merely the solution of a single spherical triangle. But the calculation is not quite so simple on a spheroid. In the accompanying figure AN is the normal (=p) at A, NAB is the vertical plane at A passing through B, and the inclination of this plane to the meridian of A is the azimuth ( = a) of B. The inclination of the plane ANB to the meridian of B call it a, is not however the azimuth of A at B. This azimuth, a, is the inclination of the plane BMA to the meridian of B, BM being the normal at B. Let 0, <', Q> be the lati- tudes and difference of longitudes of A and B, and let ANB = 0, BNO = ^. The difference of the angles a, and a' is a very small Fig. 56. 268 CALCULATION OF LATITUDES AND LONGITUDES. quantity : it may be thus investigated. In the spherical triangle formed by the directions BA> BM, BN sin a, sin ABM NB sin ABM NB cos ^ sin a' ~" sin^.ZW~ NA smBAN~ NA cos/x where /u, // have the same signification as at page 104. Now the equations (10), page 106, give A' sin p A sin //, and thence we get sin ^x = 1-* sn - sn - = 1 + i e 2 3 cos a sin cos d> ; cos/x then, as we shall see in equation (2) following, the ap- proximate value of NB : NA is On substituting these in the expression for sin a, : sin a', there results, after some little reduction, aa / = --- ; 2 cos 2 $ sin 2 a -f- : - sin 2 $ sin a If I/Z' be the number of degrees in the differences of longitude and of latitude between A and B, and if the ratio of the semiaxes be 294 : 295, a'-a, =-0 / '-21407.Z;i/cos 3 i(4/ + , log sec J (90 + 0) 0-03429587, log cosec 0-41767856, logcosi(90 $ 0) 9-98273156, log sin 9-44167235, log cot i a 0-06093582, log cot 0-06093582, logteni(a, + a>) 0-07796325, log ten 9-92028673; J(a, + co) = 50 6 55-3757, a, = 89 53 11-1759, 4(a,-a>) = 39 46 15-8002, w = 10 20 39-5755; the small correction a' a, amounts in this case to 0"-2319. Thus we have a'. The third side of the same spherical triangle, namely that opposite to Greenwich, is easily found to be 90 i/r = 38 4' 42"-2145. 272 CALCULATION OF LATITUDES AND LONGITUDES. Then by (5), since \l/ 4> is only a few minutes, p(^f $) is the distance of the parallels of Greenwich and Feaghmain ; and this distance divided by the radius of curvature for the ap- proximate mean of the latitudes of the two stations gives their actual difference of latitude. 4. Returning to the general question, let us take first the case in which the value of a is a right angle, and s not greatly exceeding, say, a degree. The azimuth a in this case will not differ from a, by any perceptible quantity ; put of = 90 - v ; here v is called the ' convergence of meridians.' Take a point B' on the meridian of A in the same latitude as B, so that the angle B' NO = BNO, and let B"NA = 77, then by (21), page 45, s 2 n = i (-) tan 0, and $ being the radius of curvature of the meridian at A, Thus we have, referring again to the equations (21), *2 <-<' = tan <'. This we may do by at once computing s 2 ' = 2j~ tan $ = !], (7) for the error thus introduced into the last two equations of (6) is of the order e 2 s 3 , and it is easy to convince oneself by a short calculation that for the distances we are contemplating CALCULATION OF LATITUDES AND LONGITUDES. 273 this may be neglected. The errors of these values of o> and v are in fact From the results just arrived at we can at once solve the more general problem when a has any value whatever. From B draw a perpendicular to the meridian of A meeting it in P, and let the spherical excess of the triangle ABP be c, then AP = s cos (a J e), JBP = s sin (a J e). Let the latitude of P be <,, the radius of curvature for the latitude \ (0 + 0,) being f : let also the difference of latitude , <' = 77, then * 2 c = - sin a cos a, r/ = sin 2 a tan < (8) $ = cos (a ) ?/, O) = v = (o sin (^ 4- 1 77) e. Here and p correspond to P, that is to latitude , . It is to be remembered that in this calculation 6 is negative when cos a is negative. With respect to the last equation expressing the convergence, the angle at B between A and P is equal to 90 a + e, and the azimuth of P at B is 90 to sin ((/>, J??); thus the azimuth of A at B being the sum of these is 5. We shall now give a numerical example of the application of these formulae, and for this purpose shall select the shorter 274 CALCULATION OF LATITUDES AND LONGITUDES. side of the spheroidal triangle, of which we have at page 110 the exact elements. We require the following table : a = 20926060, a : o = 295 : 294. i I p sm i" & smi" " b 2p sini" o / 5 1 5 7.9928272927 7'9939559 I 5 2 0-371326 52 o 8231241 943494 3" 10 8189614 9309214 2 95 20 8148048 9184517 278 30 7.9928106545 79939060007 0-371262 4 8065106 8935689 245 50 8023732 8811567 228 53 o 7982424 8687645 212 10 79927941185 7-9938563927 037H95 The data for the calculation are 5 = 436481-4, a= 20 39' 17"-240, = 51 57 -000. It is necessary first to calculate an approximate value of , by means of the formula s (j) f (/> = cos a, the result is , = 53 4' B". Then putting E~ l for the calculation will stand thus : sin l log* 2 11-27994 log sin a 9-54745 log cos a 9.97115 logE 0-37120 t = i 4 "-782 1-16974 log tan a 9-57630 log tan (fr 0-12397 *7 = 7"-4 J 3 0-87001 log s 5-6399657 log sin (a -Je) 9-5474 2 3o log (p sin i")- 1 7-9927965 log sec ($' + 77) 0-2212153 252O"-OOO 3.4014005 log sin ((/>' + 77) 9-9027368 a f a-| 20 39 12-313 20 39 7-386 log s 5-6399657 log cos (a-fe) 9-9711550 log (f sin i")' 1 7-9939053 |.n 3-6050260 = i = 53 6 59-998 3 59-998 ^ ai />' 3-30413^3 = 53 4 2-469 = o 42 o-ooo = 53 4 4-940 = o 33 19-579 CALCULATION OF LATITUDES AND LONGITUDES. 275 Thus we have the azimuth of A at B 158 47' 23"-181 ; it should be 158 47' 23"-182. And the errors of latitude, longi- tude, and azimuth are d<' = 0"-002, dco = 0"-000, da'= 0"-001. 6. In the case of distances exceeding a hundred miles it may be necessary to proceed in the following manner. The angle 6 can be obtained with any degree of accuracy that may be required from the series already investigated, then by a simple application of the rules of spherical trigonometry we have a, , \j/, and o>, with any accuracy that may be required a following from a,, as we have seen by a very small and easily calculated correction. With respect to <', the latitude of B, the only direct expression for it is obtained thus : join B with the centre of the spheroid and let A be the geocentric latitude of 5, thenif^V=r, tan X r sin \(/ ePpsincf) tan vjf ~ r sin \// but tan A = (1 e 2 ) tan $', therefore tan \lr r sin y and a very approximate value of p : r may be written down at once from (2). Still, the formula (9) is inconvenient for actual calculation, and it is practically easier to find the dis- tance of the parallels of the two stations and then to divide this distance by the radius of curvature at the mid-latitude. Let S be the distance of the parallels of A and B, then from the expression (5) for in terms of s it is easy to show that * = *=*. (10) s e To be very precise, + -^e 2 0* sin 2 2 a sin 2 (f> should be added on the right side of this equation, but it may be safely neglected as in all cases quite evanescent : therefore by equation (22), page 45, it follows that T 2 276 CALCULATION OF LATITUDES AND LONGITUDES. 7. In the calculations we have been exemplifying, the results depend on the elements assumed for the figure of the earth ; but it is often necessary to get results not so limited, and which can be readily modified to any change of the elements. Starting from the point A with a given latitude

= sm\j/ sin &> 80 sin0 cosa'8a sina'80, (13) b\js = cos a) 8 + sin sin aba cosa'80. In the first two of these equations \j/ may be replaced by 0', and in the third sin sin a' may be replaced by cos $ sin o>. In order to determine 80' we must revert to (9), putting it in the form e 2 x e 2 le 2 cos\^ v 1 tf 2 where is put for sin\//- sin 0. Differentiating this and omitting the last term, which is ver small, ^ - r (1 - e 2 ) cos 2 v/f 1 e 2 cos\l/ cos \//- (1 . or with sufficient approximation 2ebe cos\l/- 7 ^ Put for brevity 1 + e 2 cos 2 < = , then the third of equa- tions (13) gives If then 0', &>, a' are the same functions of + 80, a + 8 a, # + 8 a, -f be that (0'), (o>), (a'): are of 0, a, #, e, we have finally equations (14) which are of great importance: .. (r) = cos w5^> n sin '{a> (cu)| = sin ^>' sin a>30 sin cos a'5a sin a'89 cos'{a' (a')} = sina;5c/>+ cos sin 1" cos 52, CD being = 37239 " 5755 as we have seen, then log 37239"-5755 4-57100472, log p' sin I"cos52 1-79652713, log 230944-07 6-36753185. This value of P is however dependent on the particular elements assumed for the figure of the earth: increments ba, bb to the semiaxes will produce an increment of the form yba + y'bb to be added to the length of parallel just obtained. We have ^4 P = 2 and taking the logarithmic differential, we get without much trouble bp ba / p 2 \ bb , p 2 , \ 1 = ( 2 ~ 2 cos 4>) - -r (l - - cos 2 0), p a \ a 2 b \ a 2 which we shall write thus CALCULATION OF LATITUDES AND LONGITUDES. 279 Similarly, // corresponding to $ = 52, let P 8P 8co 8p' then = +-!L. .r to p Now the latitude of Greenwich in this case has itself to receive an increment 8$ = h,ba + kfib, so that putting Se = 6-?- and 8a = 0, P the second of equations (13) gives 8co sinco ^ -sin a (0 Thus we get sn a> , > , . , _ A - = 5 ( ^ tan \/A - - -f- ^ - sin a sec ^ + h ) J^ CO CO + bb (k t tan v/^ - - + k - sin a 7 sec \{/ + ') ' And finally ; the length of the arc of parallel in latitude 52 between Greenwich and Feaghmain, the semiaxes of the earth being 209263 48 + 80 and 20855233 + 8^ feet, is 2330944 ft -07 + 0-0062 8^ 0.0006 8. So also for Greenwich and Mount Kemmel in Belgium, the length of the arc of parallel in 52 is 634157 ft -39 + 0-0027 80-0-0006 8. CHAPTER XI. HEIGHTS OF STATIONS. THE direction in which a signal B is seen at an observing point A is determined by the direction of the tangent at A to the path of the ray of light passing between A and B. This direction differs from that of the straight line joining A and B on account of terrestrial refraction. The displacement, resulting from refraction, of the direction of B> takes place almost wholly in the vertical plane which we may with sufficient accuracy consider common to A and B. Lateral refraction sometimes exists, affecting to a very small extent horizontal angles, but we here are concerned only with the more ordinary phenomenon which affects the measurement of zenith distances. For the theory of this refraction we may refer to some able papers by Dr. Bauernfeind in the Astronom- ische Nackrichten for 1866. The amount of terrestrial re- fraction is very variable and not to be expressed by any simple law : the path of a ray of light, inasmuch as it depends on the refractive power of the atmosphere at every point through which it passes, is necessarily very irregular. This irregularity is very marked when the stations are low and the ray grazes the surface of the ground. In the plains of India it has been observed that the ground intervening between the observer and the distant signal, from being apparently convex in the early part of the day, changes gradually its appearance as the day advances, to a concavity so that at sunset the ground seems to slope up to the base of the signal tower which in the early morning was entirely below the HEIGHTS OF STATIONS. 281 horizon 1 . Under such conditions refraction is often negative: the coefficients ranging from 0-09 to -j-1-21. In Great Britain the refraction is greatest in the early mornings ; towards the middle of the day it decreases, again to increase in the evenings but this rule is not without remarkable exceptions. From a series of carefully conducted observations by Colonel Hossard at Angouleme 2 it appeared that refraction is greatest about daybreak ; from 5 or 6 A.M. until 8 A.M. it diminishes very rapidly; from 8 A.M. until 10 A.M. the diminution is slow; from this hour until 4 P.M. refraction remains nearly constant; after that it commences to increase. The average amount of refraction, by which is meant the difference between the true and the apparent directions, varies from about a twelfth to a sixteenth of the angle sub- tended by the stations at the centre of the earth. The larger values are found generally on the seaboard, the smaller values remote from the sea. The amount of refraction may be determined thus : let h, h' be the known heights of two stations A, B obtained for instance by spirit levelling : at A let Z be the true zenith distance of JB, and at B let Z' be the true zenith distance of A, C being the centre of the earth, which we may suppose a sphere of radius r, let the angle ACB v, then in the triangle ACB tan - tan \ (Z'-Z) = ., h ' ~~ ' ' 2 which determines Z and Z' '. If we substitute for tan \ v the first two terms of its expansion in series, the second equation is easily put in the form h'-h = .to 4 (Z'-Z) (! where s is the distance of the stations A, B measured on the sea-level. The assumption that the earth is a sphere may be practically remedied by using the measure of curvature of the surface in the vicinity of the stations for 1 : r 2 . 1 Account of the Great Trigonometrical Survey of India, vol. ii, page 77. 2 Mem. de Dep6t Gen. de la Guerre, vol. ix, p. 451. 282 HEIGHTS OF STATIONS. The coefficient of refraction is the ratio of the difference between the observed and real zenith distance at either station to the angle v : thus k being the coefficient of re- fraction, z 9 z' the observed zenith distances, Z-z Z'-z' 7c = or K = These two values however do not always agree. The fol- lowing table contains some determinations selected at random of the value of k obtained in this manner from observations made on the Ordnance Survey : STATION. HEIGHT. Z,Z' z,z' No. OBS. V k Ben Lomond Ben Nevis ... ft. 3192-2 4406-3 O 1 II 90 I 21-2 90 37 2-0 1 II 89 58 35-8 9 33 25-3 5 J 9 \ 2300.7 | -0719 -0942 Dunkerry . . . Precelly 17064 17579 90 31 51-0 90 32 45-0 90 26 56-1 90 27 44-4 4 5 j 38709 j 0762 -0777 High Wilhays Precelly 2039-6 17579 90 42 22-7 90 38 26-7 90 36 14-3 9 3 1 36-4 M 6 j 4843-0 | 0761 -0847 High Wilhays Hensbarrow 2039-6 1027-0 9 33 5 1 ' 8 90 i 540 90 31 32-8 89 59 20-1 12 17 j 2I42-9 j 0649 0718 Coringdon . . . Dunnose 655-6 771-9 90 12 54-0 9 J 7 154 90 10 30-4 90 14 57-1 16 12 j 1806-9 | 0?95 0765 Trevose Head Karnminnis 242-6 799-8 90 3 18-2 90 25 17-8 90 i 5-6 9 2 3 9-5 20 IO j I7I3-7 | -0774 0749 The most abnormal coefficient here shown is that at Ben Nevis, and it is worthy of notice that for a fortnight when the greater part of the observations were made the state of the atmosphere at the top of the hill was most unusually calm, so much so, that a lighted candle could often be carried from the tents of the men to the observatory, whilst at the foot of the hill the weather was wild and stormy. The value of the coefficient of refraction may also be ob- tained from the reciprocally observed zenith distances of A and Bj independently of the knowledge of the heights of those stations. For assuming the refraction to be the same at both HEIGHTS OF STATIONS. 283 stations (and for this end the observations should be simul- taneous) Z, ' = z + kv and Z r = sf + kv, therefore z + af+2kv = 180 -ft?; 24V 180 .-. l-2/ = - v From the mean of 144 values of k determined from the observations of the Ordnance Survey it appears that the mean coefficient of refraction is -0771. If we arrange the different values in order of magnitude, the extremes are -0320 and 1058, while the 72 which hold the central position are con- tained between -0733 and -0804, the mean of these 72, viz. 0768 differing but little from the general mean. Thus it would appear that the probable error of a single determination of k is about -0035. But it is necessary to discriminate between rays which cross the sea and those which do not. The result, having regard to the weights of the single determinations, is finally ttys for rays not crossing the sea, k = -0750 ; for rays crossing the sea, k = -0809 ; a result which is borne out by observations in other parts of the world ; for instance \ in the Survey of Massachusetts the value of k adopted for the % sea coast is -0784, while for the interior it is -0697. From the preceding equations we get J (Z' Z) = Let z, the observed zenith distance of B at A, be replaced by 90 + 6, so that 5 is the * depression ' of B, then (1) becomes /I 2k \ / h + h' $ 2 \ h -Ji = 5 tan (9 -- 8 ) (1 + + -- 5 ) : (2) V 2r ' V 2r r 12r 2 ' or if we have also the depression 6' of A as observed at B (3) The last factor in the last two equations may be safely omitted. 1 Professional Papers of the Corps of Engineers U.S.A., No. 12, page 143. 284 HEIGHTS OF STATIONS. If further we consider only the cases of distant stations when 8 seldom exceeds a degree, we may put 1'-* = ,!=*- ,8. (4) 2 T Put p - 7 > then taking the earth as a sphere of mean 1 - 1 K radius such that \ogr = 7-32020, for = -0750, log ju = 7-69181, k -0809, log/A = 7-69788, which suffice for ordinary purposes. When the height h' of .Z? has been determined from its ob- served zenith distance at A by the formula (4), the error of h f is r Suppose the distance s to be n miles, the probable error of the observed zenith distance 6 seconds, that of the co- efficient of refraction dt -004', then the probable error, expressed in feet, of h' will be approximately When the observations at A and B are mutual, we may either eliminate the coefficient of refraction by using the formula (3), or we may get two different results from a mean assumed value of k, and then combine these results by assigning them weights deduced from the consistency of the observations from which they are separately derived : the reciprocal of such weight according to the formula just given will be e 2 + -^ n 2 . When at each of three stations we have the observed zenith distances of the other two, the deduced differences of height will exhibit a discrepancy. Let the computed differences of height be C - B = Ji^ B A = h 2 , A C=h ?>) then we ought to have 7^ + ^ + ^3 = 0. This will not generally be the case, and we must apply corrections as lt a? a , # 3 to these quantities, such that 1 + a 2 + a? 3 + ^ 1 + ^2 + ^3 = 0, and if w ly w 2 , w^ be the weights of the determinations of k lt &j, h z the quantity HEIGHTS -OF STATIONS. 285 must be a minimum : a case analogous to that of getting the corrections to the angles of a single triangle. Or we may proceed thus : suppose there are four points, mutually ob- serving one another. Referring the heights to that of any one of the points, let them be 0, #, y^ z, then we nave six equations x + a = 0, '= 0, z-y+l'= 0, z+c = 0, x z+ c'=. 0, with assignable weights, from which #, y> z are to be obtained by least squares. The principal lines of spirit levelling covering England and Wales were reduced in this manner : the number of unknown quantities being ninety-one. In the reduction of the levelling of Scotland there were seventy-seven unknown quantities. .Suppose that at a station of height ^, the horizon of the sea is observed to have a depression A, if its distance be 2, equations (3) and (4) give ^ = 2taniA, & = !,*-; M and eliminating 2, k and A are thus connected, h = p tan 2 J A. (5) Let //, h" be the heights of two stations A, _6, whose distance apart is c, 8 the depression of B as observed at A. Let C be any other point in the ray joining A B : for instance, C may be a signal which appears to be exactly in line with B. Let h be the height of C, s + \c, s\c its distance from A and B respectively, then we have these three equations : 286 HEIGHTS OF STATIONS. of which the third, obtained by eliminating 8 from the first two, is the equation to the path of the ray of light joining AB. The point at which this ray approaches nearest the surface is determined by making the differential coefficient of Ji with respect to s zero ; thus we get p. n where s determines the place of the minimum height of the ray and h Q its amount. If we take for instance the case of Precelly in Pembrokeshire and High Wilhays in Devonshire, whose heights are given in the table, page 282, and which are 93 miles apart (\ogc = 5-69221) we find the minimum height of 677 ft. occurring at about 44 miles from the lower and 49 from the higher station. So for the stations Tetica and M'Sabiha, page 262, whose mutual distance is 225-6 kilometres, the nearest approach of the visual ray to the surface of the Mediterranean is 1077 ft. Two stations whose heights in feet are h\ h" will not under ordinary circumstances be mutually visible over the surface of the sea, if their distance in miles exceeds CHAPTER XII. CONNECTION OF GEODETIC AND ASTEONOMICAL OPERATIONS. 1. THEORETICAL considerations, as we have seen in chapter IV, combined with observation and measurement, have shown that the figure of the earth is very closely represented by an ellipsoid of revolution. It is not however exactly so ; the visible irregularities of the external surface and the variations of density of the material composing the crust, superinduce on the ellipsoidal form undulations which we cannot express by any formula. Extensive geodetical operations enable us however to determine a spheroid to which the mathematical surface of the earth can be very conveniently referred, and from which its deviations are probably very small. Designate this spheroid of reference E, and the actual mathematical surface of the earth S. Let A, B, C, ... be a series of points on S, A 19 B-x C 19 ... their projections on E, then, as far as our present knowledge extends, the linear magnitudes AA^ BB 1} CC^ ... are extremely small ; that is to say, that representing the normal distance of S above E by no observations yet made lead us to suppose that f is anywhere anything but very minute compared with the difference of the semiaxes of the spheroid. In the early period of geodetic science the irregu- larity of the earth's figure made itself apparent principally by the very discordant values that were obtained by different combinations of short arcs for the ellipticity of the surface. The discordances resulted from the fact with which we are now familiar, that the observed latitude of any station, 288 CONNECTION OF GEODETIC although from its surroundings it may be apparently quite free from any suspicion of local attraction, is yet liable to an error of one or two seconds. This amount indeed is often exceeded, and it is not very uncommon to find, as in the vicinity of Edinburgh, a deflection of gravity to the extent of 5", while in the counties of Banff and Elgin there are cases of still larger deflections, the maximum of 10" being found at the village of Portsoy. At the base of the Himalayas, where we should naturally expect a large attraction, it amounts to about 30", diminishing somewhat rapidly as the distance from the mountains increases. A very remarkable instance of such irregularities exists near Moscow 1 , brought to light through the large number of observed latitudes in that district. Drawing a line nearly east and west through the city, this line for a length of 50 or 60 miles, is the locus of the points at which the deflection of the direction of gravity northwards is a maximum, amounting nearly to 6" in the average, while along a parallel line eighteen miles to the south are found the points of maximum deflection southwards. Midway between these lines are found the points of no deflection. Thus there is plainly indicated the existence beneath the surface, if not of a cavity, yet of a vast extent of matter of very small density. Deflections much exceeding these in amount exist in the Caucasus and in the Crimea. 2. If we conceive the small quantity to be expressed in terms of the latitude and longitude, then the surface S is strictly defined. If we put f=^, df pd<^> p cos (f)d(*> then f, rj are the inclinations of the surface S at A to the surface E at A lt thus the latitude of A l is greater by than 1 Untersuckungen ueber die in der Naehe von Hoskau stattjlndende Local- Attraction, von G. Schweizer. Moskau, 1863. AND ASTRONOMICAL OPERATIONS. 289 the latitude of A, and the longitude of A L is greater than that of A by 77 sec <. With respect to the observed direction of the meridian, in the adjoining figure, let Z, Z l be the zeniths of A, A 1 : P the pole, Q the place of a terrestrial signal referred by its direction to the sphere of the heavens. Z^PH and Z^K being each equal to 90, as also ZPh and ZQ&-, the azimuth of Q as measured with a theodolite is a = hk, whilst HK = a t is the same azimuth as referred to the spheroid E. Produce HPZ l to and draw ZO perpendicular to Z^O, then Z 1 = ^ and ZO = 77. Then since PZ = 90-0, HPh is 77 sec <#>, and therefore the distance of h from PH is 77 tan 0. Again the angle . _ fsina 77 cos a Fig. 57. cos e where e is the angle of elevation of the signal, and the distance of k from KQ is consequently ( sin a 77 cos a) tan e. Thus we have c^ = a + T7tan + ( sin a 77 cos a) tan e. (l) Supposing then that, as is always the practice, the pole star is observed in connection with a mark which is at a zenith distance of very nearly 90, the term in tan e becomes insensible and may be omitted. Thus there exist the fol- lowing relations between A and A lt If , o> be the latitude and longitude of A } ^^ the latitude and longitude of A lt a the observed azimuth of a terrestrial signal, a A the same azimuth as referred to A-, , then (2) Since measured base lines are reduced to the surface of the sea, and since the angles measured by theodolites are the same as if measured also on that surface, it follows that u G> + 77 sec < tan <>. 290 CONNECTION OF GEODETIC trigonometrical operations may be considered as virtually conducted on the surface S. But the angles measured amongst the points A> B, C, ... are not identical with the angles measured amongst their projections A lt J5 13 C 1} ... It appears in fact from (l) that the horizontal angle mea- sured between two objects whose azimuths and elevations are a, a', e, and calculating it as indicated in chapters ix and x. 3. Suppose now that we have given the distance of two points, A, j, at each of which there are astronomical determinations of latitude, longitude, and azimuth: and let the following notation express the correspondence of the observed elements with the reduced quantities appertaining to A 15 JB 1 A A l B B l rjseccj) o> > + rj sec < a a + rj tan a a + r{ tan $'. If starting from A with the given distance, and a, $ we calculate the elements at j5, and get the numerical results (<') (a>) (a), then with the same distance, and a -f rj tan for a, + f for and a longitude rj sec $ for A^ , we should by equations (14) page 277, omitting dd, get for J3 l $>i = (0') + cos to f n sin o> sin (fry, (3) , . sin d>' sin o> . / sin 6 cos a . ,\ a)! = (a)) + - 7 f (- 7 tan sec 0)77, cos < V cos $ , , /N sin w sin cos co AND ASTRONOMICAL OPERATIONS. 291 these being equated to the elements at S^ as written above, we get f ' = $' (j/ -\- cos o> f n sin o> sin $ rj, / sin cos a + (~ cos.fr' sin o> , tan *'' = (a')-a' Hence it appears at once that the observation of the difference of longitude gives us no information that is not also given by the observation of azimuth ; it affords however a check upon the work. If instead of two points we have a network of triangles, then at every point where there are astronomical determina- tions we can express the ' and rf belonging to that point in terms of and 77. Now and rj are unknown, but we may suppose the spheroid E so placed with reference to & its axis parallel to the earth's axis of rotation or rather so placed with reference to the portion of S covered by the triangulation that the sum of the squares of all the 's and r/'s is a minimum. This condition determines and 17, and ', rf ... for all the other stations follow from the equations. It is further to be supposed with reference to the position of E that the mean value of f over the surface considered is evanescent. By the method just explained w ^ see how in a network of triangulation a system of deflections may be assigned having reference to a definite ellipsoid E in a definite position : for any other ellipsoid differing slightly from E a somewhat different system would have resulted, and in fact if we leave the semiaxes indeterminate that is to say express them in a symbolical form, then we can from the equations of con- dition which arise determine that particular ellipsoid, call it 4H, for which the sum of the squares of all the deflections is a minimum. Thus we obtain an ellipsoid which we may consider to represent better than any other that portion of S over which the triangulation extends. u 2 292 CONNECTION OF GEODETIC Let the semiaxes of (& be (a) + da and (c) + dc t where (a), (c) are approximate numerical values, da, dc small quantities to be determined. It is necessary to remark in the first place that the spherical excesses of the triangles having been calcu- lated from the elements (a), ( = f, da = 17 tan $, and for brevity write those equations thus : +C da+E de, +C'da QI ' = (a') +A" + "1+ where ' includes a term sec$ as in (3). Here (<'), (to), (a) are the astronomical elements at B l as calculated with the observed latitude and azimuth at A and with the (')' + -4 +B rj + C da + E de, secY = () tan <' rf = (a') - a' In the Account of the Principal Triangulation of Great Britain and Ireland will be found at pages 693 and 694 seventy-six equations of the kind just written down ; of these, thirty-five arise from observed latitudes and forty-one from observed azimuths and longitudes. The solution of these equations by the method of least squares determines the axes of that particular spheroid <& which most nearly represents the surface of this country, and also, by 77, the inclination of the surface of < at Greenwich Observatoiy to which 77 belong to the surface S there. From these follow the ' rf of every other point. The semiaxes of & are a = 20927005, c = 20852372. The azimuth and longitude equations are, from the nature of those observations, entitled to much less weight than the latitude equations : the azimuth equations in particular are directly affected by accumulation of errors of the observed angles of the triangulation. Hence the explanation of the fact that the average values of the resulting quantities 17 is somewhat larger than that of the f 's. It is interesting to compare the values of the quantities f, which we may take to be local deflections of gravity in the direction of the meridian, obtained as above, with the deflections calculated from the form of the ground around the stations for those stations at least where the means of making such calculation exist. In estimating from the form of the ground the deflection of gravity, an element of uncertainty exists in our not know- ing exactly to what distance from the station the calculation should be extended; for according to the views explained at page 97 it is very doubtful whether the influence of distant masses should be taken into account. Accordingly in the fourth column of the following table the influence of masses 294 CONNECTION OF GEODETIC at distances exceeding nine or ten miles is excluded in the fifth column the calculation is extended to these more distant masses : NAME OF STATION. SITUATION. DEFLECTIONS. t FROM THE GROUND. Dunnose Boniface Week Down Port Valley Near Vei* nor Isle of Wight Yorkshire Cork Kerry Wexford Mayo Londonderry ... Fifeshire Hebrides Sutherland Edinburgh Banffshire - 1-62 + o 80 + 0-58 + 1-61 -2-56 -3-54 + 2-92 - 0-88 + 0-26 -0-95 -4.48 + 1-82 -H 1-36 - 2-86 - 5-30 -9-55 1-02 + i-94 + 1-50 + 2-81 0-90 -3-03 + 3-85 - i-95 -0-17 - i-43 - 2-15 + 2-08 + 0-47 - 1-63 - 2-43 (-) -0-54 + 2-42 + 1-9 + 3- 2 9 - 4-55 + 5-4 + 1-13 - 2-30 - 4-02 2-OI - 3-57 (-5) Clifton Burleigh Moor Hungry Hill Forth Tawnaghmore Lough Foyle Kellie Law Monach Ben Hutig CaltonHill Cowhythe The quantities in the last two columns for Cowhythe are only roughly calculated. The triangulation being considered as projected on the ellipsoid fl* as finally determined, we can at each point where azimuth observations have been made obtain an apparent error of observed azimuth at that point. On forming these errors for sixty-one stations in this country it is found that twenty-three errors are under 3", ten between 3" and 4", and there is one error of 11". The probable error of azimuth of the triangulation of Great Britain as a whole is 0"-69. 5. The calculation of the disturbance of the direction of gravity at any station / due to the irregular distribution of masses of ground in the surrounding country presents no difficulty AND ASTRONOMICAL OBSERVATIONS. 295 if we possess a map of the district, showing by contours or otherwise the heights of the ground. Let there be drawn on this map a number of circles having / for a common centre, and also a series of radial lines through /: thus dividing the country into a series of four-sided compartments. Let a,, a be the azimuths of two consecutive lines, /, , / the radii of two consecutive circles, and let it be required to find the attraction at / or rather the component of the attraction acting in the direction north of the mass M of the compartment contained between those limits of azimuth and distance : the upper surface of M being supposed a plane. Put a, r for the azimuth and horizontal distance of any particle of this mass, its density, z its height above J, then its mass is grdadrdz, and the component of attraction required is { being sup- posed constant /v rr 1 rh ^ cos a d /a d r (lz A = * / / ~^ ' J+J^J* (r 2 + z 2 )? where h is the height of the upper surface of M above /. Hence = gh (sin a sm a.) log. " In exceptional cases only is it necessary to take into account h 2 , namely when the station is in the immediate vicinity of very steep ground, generally it may be neglected. In ordinary cases then the attraction due to M is $ Ji (sin a sin a,) log e - > r i or if the straight lines be so drawn that the sines of their azimuths are in arithmetic progression, having a common difference , and the radii of the circles in geometric pro- gression, the logarithm (Napierian) of the common ratio being /, then the whole attraction to the north is 296 CONNECTION OF GEODETIC where 2 (ft) is the sum of the heights of compartments north of the station, 2 (h') the sum of the heights of those south. Now if we regard the earth as a sphere of radius c and mass |H then the angular deflection of the direction of gravity D resulting from an attraction A is l>-*-A. m Taking c as 3960 miles, and putting for the mean den- sity of the earth, D expressed in seconds is D= 12"-44-, fo where it is supposed that the unit of length in the calculation of A is the mile. In the case we have just been considering = 12''-44-?-H {2 ()-:2 (/&')}. (4) o This method was first adopted for the calculation of the attraction of Shiehallion in the celebrated experiment of Dr. Maskelyne for the determination of . Here we have supposed that the calculation is not extended so far as to require any notice of the curvature of the earth's surface. If it is necessary to take this into consideration then it is easy to see that r being now angular distance, the component, in the direction of north, of the attraction of the mass standing in a compartment limited by azimuths a, a and distances r, / is 1 T f ' > \ f COS 2 * j A = jMfsma sin a.) / -. = dr, u J r . sin A r sinjr N /, tan i r \ = %h (sm a sin a,) (log e - ^ f- cos J r cos J r, J ^ belli 4 7*^ The attraction of an elevated table land whose upper surface is nearly a plane and its outline rectangular may be obtained thus. Taking the attracted point as origin of rectangular co-ordinates xy in the horizontal plane and z vertical, let the solid be bounded by the planes os = a, y z = x = a' y = b z = //, AND ASTRONOMICAL OBSERVATIONS. 297 h being very small in comparison with the other dimensions a' a, and b. The ^-component of the attraction is A = . f a> f f h xdxdydz lx dx dz Now if we neglect the higher powers of z* and put b = a, cot tj)f = a' cot <' this becomes The corresponding deflection is obtained by replacing f by 6"- 2 2, if the attracting mass be of a density equal to half the -mean density of the earth. If for example we suppose a table land extending twelve miles in length by eight miles in breadth, and having a height of 500 feet, then at an external point two miles per- pendicularly distant from the middle point of the longer side the deflection would be l"-47. In this case the term in A 3 is not perceptible. The attraction of a prism of indefinite length whose section is a trapezoid HSS'H', at a point in the plane of one of the faces SS', admits of a tolerably simple ex- pression from which we may obtain an approxima- Fig. 58. tion to the deflection that would be caused by a rectilinear range of mountains of some- what regular section. Let HOS=, H'08 =', OS=c, OH =6, 298 CONNECTION OF GEODETIC put further z 2 + (c + z cot cr) 2 = M 2 , z 2 + (c'-z cot a-') 2 = u' 2 . Then the limits being for y, and co ; for x, czcotv' and c + z cot o- ; for z, and ^, we have ,v / d , ,\ 2 , = f 'log C-)-^*^ log (-)<**. The integration presents no difficulty : the result is c'sin2a' ff csin2, V S*. (?).- (7) I + 2^ {^^ sin 2 a' c$ sin 2 o-} ; and if we replace by 6"-22, we get the deflection on the supposition that the moun- tains are of half the mean density of the earth. Suppose the section to be triangular as SVS'. From the formula last written down we can obtain the following expression for the deflection at any point as P on the slope SF; drawing PQ, PX parallel and perpendicular to the base, the deflection, call it v, is r T> Q" 2 T> C' 2 6"-22 \PX log (^) + * P q sin 2 x + 9-6, + 11-^ + I2-2/ ^-^9-5 + 97^- ^--7.4 Fig. 60. AND ASTRONOMICAL OBSERVATIONS. 299 The maximum attraction finds place here at about one- fourth of the height from the base. 6. It is interesting to enquire into the amount of error that would be introduced by such "attractions into the process of levelling over a chain of mountains. For instance, if spirit levelling is carried from S over V to S f and again from S through an imaginary tunnel in a straight line from S to &', what is the discrepancy produced by the attraction of the hill between the two results obtained for the height of S'? In the operation of levelling, the observer has his instrument midway between two levelling staves, one in advance, the other behind. The difference of level is always given by R,R', where R, is the reading of the back staff, K that of the front. But if there be a deflection of the amount v the attraction being counted positive in the direction in which the leveller is moving and if dx be the small hori- zontal distance between the staves in any one position of the instrument, then the measured difference of height R,Rf requires the correction vdx. Let v be the deflection at any point of SS', then if H be the height obtained at S' by the route SVS? and H' the height obtained by the direct route SS" so that the difference between the heights obtained is fvdxfv'dx. In the Astronomische Nachrichten, No. 1916, pp. 314-318, is an investigation of the integral fv dx : we may express the result thus, putting SF = *, FS'= /, SS f =a, the area of S7S f = A, and omitting for a moment the coefficient 6"- 2 2 COS ( (T 300 CONNECTION OF GEODETIC Again fv'dx is the excess of the potential of the mass at S ahove its potential at S' : this, see p. 94, is , , .(s, a /. #) / ^ \( 5/(T/ 5 -cos(cr+" 50209-2 sin 2 60-0 sin 40, where <" is the latitude expressed in seconds. This curve is not restricted to the elliptic form : it is depressed below an ellipse described on the same axes, but the maximum de- pression is only 59 feet, in the latitude of 45. In the Encyclopedia Metropolitana, under the heading * Figure of the Earth,' is the elaborate investigation of the Astronomer Royal. It is based on the discussion of fourteen meridian arcs and four arcs of parallel. The resulting semi- axes are a = 20923713, c = 20853810, with, a : c = 299-33 : 298-33. Bessel's investigation made a few years after, viz. in 1841, is to be found in the Astronomische Nachrichten, Nos. 333, 438. He obtained a = 20923600, c = 20853656, with, a:c = 299-15 : 298-15. The agreement of these results of Airy and Bessel, obtained by very different methods of calculation, is very striking ; but we now know that, owing to the defectiveness of the then existing data, both are considerably in error. During the sixteen years following, great additions were made to the data ; the Russian arc was extended from 8 to 25, the English from 3 to 11, and the Indian arc extended 5i. An investigation,, by Captain Clarke, R.E., based on these new arcs is to be found in the Account of the Principal Triangulation of Great Britain and Ireland. The data used in this investigation are : 1st, the combined French and English arcs, 22 9'; 2nd, the Russian arc 25 20' ; 3rd, the Indian arc 21 21' ; 4th, an earlier Indian arc of 1 35' ; 5th, Bessel's Prussian arc of 1 30'; 6th, the Peruvian arc 3 7'; 7th, the 304 FIGURE OP THE EARTH. Hanoverian arc 2l'; and 8th, the Danish arc 132 / . The small arcs however have very little influence in the result. In this investigation the figure of the meridian is not restricted to the elliptic form. The radius of curvature is supposed to be expressed by the formula e = ^4 + 2.5 cos 20+2(7 cos 40. (l) This represents an ellipse if 5_5 2 QAC 0. The coordi- nates x, y> of the point Q, whose latitude is 0, are x /gsin0^0 = (AB) cos0 + J(J5 C) cos 30 y = t cos -f ^ C sin 5 ; whence follow at once the semiaxes, a = --, c = Let OD ', if be the coordinates of a point P in latitude in an ellipse described on these same semiaxes : measure PS along the ellipse and SQ perpendicular to it, then PS = (as of) sin -f (yy) cos 0, /SQ = (x of) cos + (yy') sin $ ; expressing these by bs and br, it may be shown that this last expressing the protuberance of the curve (l) above an ellipse described on the same axes. The distance of two points on the curve (l) whose latitudes are J a and $ + J a is s = ^a+2i?.sinacos2 + C r sm 2 a cos 40. Suppose now that each of the observed latitudes in the different arcs has a symbolical correction x attached, such as to bring them into harmony with the curve (l). Then A, J3, C are determined so that 2 (x 2 ) is an absolute minimum. The resulting semiaxes are a = 20927197, c = 20855493, and a:c= 291-86 : 290-86. FIGURE OF THE EARTH. 305 The quantity 8r by which the curve is more protuberant than an ellipse on the same axes is (177 70) sin 2 2$. This is a satisfactory indication that the actual curve differs but very slightly from the ellipse. But on restricting the curve to an ellipse, the same data give a = 20026^48, c = 20^55^33, and a :c 294-26 : 293-26. But these conclusions were vitiated by the then existing uncertainty as to the unit of length on which the southern half of the Indian arc depended an uncertainty which is now removed by the recent remeasurement of the arc from Damargida to Punnse. 2. In the Memoirs of the E. A. Society, vol. xxix, is an in- vestigation of the figure of the earth regarded as a possible ellipsoid, suggested by General T. F. de Schubert's ' Essai Wune determination de la veritable Figure de la Terre.' In this enquiry one has first of all to define parallels and meridians ; the colatitude of a point, being still the angle made by the normal to the surface there with the axis of rotation. A meridian may be defined either as the locus of points, whose zeniths lie in a great circle of the heavens, whose poles are in the equator or at which the normals are perpendicular to a fixed line in the plane of the equator or we may define the meridian as a line whose direction is north and south. But these lines as we shall see are of different characters, and we shall call this last a north line. If a, 6 be the semiaxes of the equator of which we shall suppose a to be the greater, c the polar semiaxis, the equation of the surface is The direction cosines of the normal at #, y> z being pro- portional to L L ' ' 306 FIGURE OF THE EARTH. if (f) be the latitude of points on a parallel it is easy to see that r 2 */ 2 z z ~\ + I* - -2 cot2 * = 0- ( 3 ) a 4 b* c 2 Again, for a meridian, if the normals are to be perpendicular to the line whose direction cosines are proportional to sin (i) : cosco : 0, we must have for the equation of a meridian /** At _- sinco-f ^ cosco = 0; (4) (JU U the positive extremity of the semiaxis a being in longitude 0. From the three equations (2), (3), (4), if we put we get If ftf\O A.% (5) Let us now consider the north line. Suppose that a point on the surface of the ellipsoid moves always towards a given fixed point #'.//, and let it be required to determine the nature of the curve traced by this moving point. Two con- secutive points on the curve having coordinates xyz, x + dx, y + dy> z + dz give the condition 5<** + |^jr + rf*=0. (6) The equation of a plane passing through xyz and x'y'z' is A (*-*) + E (/-y) + C (/-*) = 0. This plane is to contain the normal at xyz and the point x + dx t y + dy, z + dz, which conditions give two other equa- tions in ABC: and eliminating these symbols we have the FIGURE OF THE EABTH. 307 differential equation of the required curve expressed by the determinant x'-x, y'-y, z'-z, dx, dy, dzj x y z = 0; the north line is a particular case of this general curve, viz. when a?' = 0, y = 0, / = oo : its equation is a?ydx-b z xdy=Q, (7) of which the integral is x a2 = Cy b *. It is therefore a line of 1 greatest slope ' with respect to the equator. Let S be any point on the surface of the ellipsoid, say in that octant where xyz are all positive. Let SN, SM, SP be indefinitely small portions of the north line, meridian line, and parallel passing through S: and let it be required to find the angles these lines make with one another. If from (6) and (7) we determine the ratios dx : dy : dz, we find them to be as and these are proportional to the direction cosines of SN. So also on differentiating the equation (4) of the meridian we find the ratios dx : dy : dz to be expressed by a 2 z cos to : b 2 z sin co : c 2 (x cos co-fy sin o>), and these are proportional to the direction cosines of SM. Again, differentiating the equation (3) of the parallel we find for the ratios dx : dy : dz \ cot 2 : ~ tan 0, x i 308 FIGURE OF THE EARTH. by the substitution of these in the foregoing ratios we obtain finally ,, OT) TT . . . ic 2 sin sin 2 to MSP = 2 sm d> sin 2 o> , . - > (8) 2 c 2 sm 2 + kr cos 2 TT ^c 2 sin sin 2 co = 2 ~ c 2 sir Thus for instance, if there be a difference of a mile between # and #, the angle between a meridian and parallel at a point in longitude 45, and latitude

o *r w j 2 sec2 4> RSN = 1 sm 2 a) sm < - ~ > 2i fC ^C which however does not hold good in high latitudes : for in the vicinity of the umbilics the lines of curvature are ap- proximately confocal conies having the umbilics as foci. The results of the calculation referred to in vol. xxix. of the M.R.A. S. as subsequently corrected for errors in comparisons of standards stand thus : Major semiaxis of Equator (long. 15 34' E.) a = 20926350 ; Minor semiaxis (long. 74 26' W.) b = 20919972 ; Polar semiaxis c = 20853429. But these are affected by the error in the southern half of the old Indian arc. A revision of this calculation based on the revision and extension of the Indian geodetic operations is to be found in the Philosophical Magazine for August 1878, resulting in the following numbers : Major semiaxis of Equator (long. 8 1 5 7 W.) a 20926629 ; Minor semiaxis (long. 81 45' E.) I = 20925105; Polar semiaxis c = 20854477. FIGUKE OF THE EARTH. 309 The meridian of the greater equatorial diameter thus passes through Ireland and Portugal, cutting off a small bit of the north-west corner of Africa : in the opposite hemisphere this meridian cuts off the north-east corner of Asia and passes through the southern island of New Zealand. The meridian containing the smaller diameter of the equator passes through Ceylon on the one side of the earth and bisects North America on the other. This position of the axes, brought out by a very lengthened calculation, certainly corresponds very re- markably with the physical features of the globe the dis- tribution of land and water on its surface. On the ellipsoidal theory of the earth's figure, small as is the difference between the two diameters of the equator, the Indian longitudes are much better represented than by a surface of revolution. But it is nevertheless necessary to guard against an impression that the figure of the equator is thus definitely fixed, for the available data are far too slender to warrant such a con- clusion. For the reduction of arcs of longitude in this enquiry we must form the expression for the length of an arc of parallel between given longitudes. By differentiating (5) we get dx k sin w f 1 c 2 \ 7 = --- - --- r ( - - r + 77 tan 2 (b ) 0o>, "i 1 i \li fc 2 sn the square root of the sum of the squares of these is, neg- lecting i 2 , ds = k(l + ?ft tan 2 <#>)"* { 1 - (i + J cos 2 0) cos 2, and o>' gives * _ ', 1^(1 4- i cos 2 <) (sin 2a>' sin 2a>,) But we return now to the ellipsoid of revolution. 310 FIGURE OF THE EARTH. 3. What we obtain by the linear measure of a few degrees of the meridian and the observation of the latitudes at the terminal points of the measure, is in reality nothing more than a value of the radius of curvature of the meridian at the middle point of the arc. If s be the length, the middle latitude, and a the difference of the extreme latitudes, then excluding quantities of the order n 2 in (17), page 111 \(a + c) f ( c) cos 2 < = - Or putting r for s : a, which is the radius of curvature, 1 3 cos 20 l + 3cos2cf> a 2- -+<- P - = r. If we have another arc, giving in latitude $' the radius of curvature /, 1 3 cos 2 (j/ I -f 3 cos 2 $' _ , 2 2 and from these two a and c can be determined. But to do so effectually, the coefficients of a and c in the two equations must not be nearly the same, that is the arcs must be situated in quite different latitudes. There are in particular two points in the meridian giving special results : at one of these the radius of curvature is #, viz. when cos 20= J or 0=54 45', and at the other where cos 2 = J or = 35 15', the radius of curvature is c. So that arcs measured in these latitudes give a and c directly and separately. The English arc is in the position to give a, while the French arc whose mid-latitude is 45 gives \a + \c. The individual influence of the existing arcs in the determinations of a and c may be seen using only round numbers from the following calcu- lation. Let the northern 10 of the Russian arc between 70 and 60 give a radius of curvature r l : let the English arc between 60 and 50 give r 2 ; the French arc 50 to 40 FIGURE OF THE EARTH. 311 giving r 3 , and the ten southern degrees of the Indian arc giving r, then we get these equations : | a \ cr^ 0, a r 2 =0, \ a+ i c r 3 = 0, the solution of which by least squares is c = + -0688 T-J + -1645 r 2 + -2602 r 3 + -5064 r 4 . We see here the great influence the southern part of the Indian arc has in determining ' $,, aj = \ sin 2 <$>' \ sin 2 <, , a 2 = J sin 4 $' J sin 4 $, . Suppose however that '; then if the left hand side of (9), that equation becomes dF dF dF dF , which, if we neglect the small terms n 2 b f } 8 <^ ( 1 -j- n 3 n cos 20 j dd> . Let the coefficients here of 8 <, , 8 $' be /u, , /x 7 , and write the equation thus Let the approximate values be those of a spheroid E^ c = 20855500, M = T J T , and let 7 = IO?00' 8^'Ot-sinl". (10) Moreover, if the corrections 6 0, , Sc/>' expressed in seconds be #,, #', and if we put 8 I -^T77 -- /- , . , sm l /x v a sm I' 7 8/sinl s " 1 0000 c/ sin I"' then FIGURE OP THE EARTH. 313 expresses the relation that must exist between corrections a?,, a?',' applied to the observed latitudes in an arc so that that arc shall belong to the ellipsoid of revolution E t whose elements are c= 20855500(1+^^), n = ^ + lOv sin l". (E) Here the case is supposed of merely the two terminal lati- tudes being observed. But if there be any number n of observed latitudes in the arc, then the correction to the southern terminal latitude being a?, each of the corrections to the remaining n I stations will be of the form Suppose then that we have several measured arcs of meri- dian, in each of which are several observed latitudes : then the sum of the squares of the corrections to all the latitudes necessary to bring them into harmony with the ellipsoid (E) is ' v 4 V &c.; u + b 2 v + c. 2 ar 2 ) 2 , V v + c 2 ' a? 2 ) 2 , ^ + &c. 5 and so on ; where a- l5 a? 2 are the corrections to the southern terminal observed latitudes of the first and second arcs. And it is assumed that that ellipsoid of revolution most nearly represents the figure of the earth which renders the sum U an absolute minimum. Thus u, v, ar x , a? 2 , ... are derived from the equations: dU dU and so on. This resolves itself into equating the several symbolical corrections to zero and solving the equations by the method of least squares. 314 FIGURE OF THE EARTH. 5. It is supposed in the preceding paragraph that the data consist in meridian arcs only but they are not now so limited. In the Indian longitudes we -have a most valuable addition to the measurements on which the calculation of the figure of the earth is to depend. Of the precision attained in these electro -telegraphic deter- minations we have a good means of forming an estimate by a consideration of the corrections calculated at pages 214, 215. On the whole it may be admitted that they are little inferior in weight to latitude determinations when we take into account the accidental effect of local attraction to which all are liable. We shall therefore form expressions for the easterly deflection at each of the longitude stations, and include in the U of the preceding paragraph the sum of the squares of these deflections. In the fifth column of the following table are given the differences of longitude as determined by electro-telegraphy and corrected as at page 214 for internal discrepancies with reference to Bellary as a central point. The second column contains the latitudes, the third and fourth the quantities ty, 0, used in the same sense as at page 267 : the last column contains the longitudes as calculated with the elements of Everest's spheroid E' in which n = -00166499, c = 20853284, from the triangulation connecting the several stations with Bellary. The latitude of Bellary is = 15 8' 33" : SECONDS STATION. 0' e OBSERVED LONGITUDE. OF GEOD. LONG. / // o / // 1 o / // Vizagapatam 17 41 22 17 40 26 6 36 6 21 35-44 40-03 Hydrabad... 17 30 14 17 29 22 2 48 i 35 28-29 31-53 Bombay . . . 18 53 49 18 52 28 5 25 -4 6^4- 50-74 Mangalore 12 52 14 i* 53 5 3 2 -2 4 52-61 5^-79 Bangalore . . . 13 o 41 13 i 29 2 14 o 39 20-62 20-75 Madras 13 4 4 J 3 4 5i 3 26 3 19 8-26 1490 FIGURE OF THE EARTH. 315 Let s be the distance of one of these stations from Bellary, then by (5), page 270, = 1. *""" ( i + 2 n cos 2 + n 2 )* (1 + f n O 2 cos 2 cos 2 a), \ 1 + M) where a is the azimuth. Hence, neglecting a term in nd 2 80, be S0 -- 20(1 + sin 2 <- sin 2 2< J 2 cos 2 cos 2 a) Sfl , which is the alteration of corresponding to very small alterations of c and n. If 8 CD be the corresponding variation in the calculated longitude, then, see page 277, equations (13) sin a 7 . 80 cos sin co = -- -80 = ^.- ; cos ^ sin cos \fr substituting in this last the above expression for 80, and re- placing the variations be, bn by their equivalents in u, v, we get a result of the form 8o> = where the values of A and can be at once written down and calculated numerically for each station in the above table. If in the spheroid E we replace U, v by *'= 1-0626, / =-0-6172, that spheroid becomes E'. If CD, be the longitude of one of the stations in the table calculated from the triangulation on the spheroid E n a/ its longitude as referred to ~E>' ; CD the same as referred to U, then o> co, = Au +J3v, o>' a>, = Au' + Bv', o> = a* Omitting degrees and minutes, we obtain the following results for the longitudes of the six stations on the spheroid E : Vizagapatam ...... 36-083 2.32001* 2-4008 v Hydrabad ......... 3<>543 ~ -5799 w 0-600 1 v Bombay ......... 48-170 +1-511214 + 1-5627 Mangalore ......... 55-528 +07419 t +0-7678 v Bangalore ......... 20-352 0-2339 u 0-2421 v Madras ......... 12885 1-1841 u 1-22561? 316 FIGURE OF THE EARTH. Suppose that at Bellary there is a deflection y to the east while that at one of the other stations is /: the geodetic longitude of this other station being l + Au + Bv on E and its observed longitude li', sec 0' y sec $ = 1 + A u -f- Bv. Thus y is expressed in terms of y by an equation y r = m + au + bv + cy. The sum of the squares of the j^'s at the seven longitude stations is then to be added to the U of the last paragraph, and the y treated as one of the #'s. We shall now write down the corrections x to the observed latitudes of the stations in the meridian arcs, pages 32-36, and the y's for the Indian longitude stations : Saxaford 3-081 7-0628 u 5-6800 v + 0-0962 x. North. Kona -* 7 -5-356 4 y -7-3508 u *j yy -4-9501 v yy i + 09965 Xi Great Stirling . . 6-140 6-7562 % -4-2763 v + 0-9968 x l Kellie Law -6-506 -6-3193 u 3-8104 v + 0-9969 Xi Durham -6-847 -5.7884 1* -3-277717 + 0-9972 Xi Clifton -7-851 -5-3177 u -2-8372 v +0-99740;, Arbury -4-151 -4-8748 u -2-4497 v + 0-9976 Xi Greenwich -4-530 4-6070 u -2-22851; + 0-99783?, Dunkirk -6-545 -4-4478 u 2-IO2I V + 0-99780:, Dunnose -6-969 4-2980 u 1-9860 v + 0-9979 a?, Pantheon 7-823 3-6613 u 1-5278 v + o 9982 Xi Carcassonne / O 6-114 O v VJ - ;> w -1-6368 u v 4 -0-4499 v + 0-9992 Xi Barcelona -4-177 0-9767 u -0-2235 v + 0-9995 xi Montjouy -0-822 0-9708 u -0-2216 v + o 9995 Xi Formentera o-ooo o-oooo u o-oooo v + I -OOOO Xi Fugler Stuor-oivi Tornea... Kilpi-maki Hogland Dorpat... Jacobstadt Nemesch Belin + 2-779 9-1041 u 10-0005 v + 0-9961 a;., + 1-260 - 8.3904 u - 8-8682 v +0.99632-, + 6-518 -7.3668 u - 7-3288 v + 0-9967 X? + 1-297 6-2198 u - 57382 v + 0-9972 Xi + 2-117 -5-3041 u - 4-5770 v + 0-9975 a;, + 1 -060 4-6913 u - 3-8572 v + 0-9978 X* + 4-807 4-0167 u - 3-U93^ + 0-9981 x 2 + 2-371 -3-35I6 u - 2-4506 v + 0-9984 Xi + 2-768 - 2-4149 u 1-6090 v + 0-9989 x. 2 FIGURE OF THE EARTH. 317 Ssuprunkowzi ... ~ryy + 5-377 / ~T~ ' 1-2301 u ~tJ v *J " - 0-7175 v yyy "* + 0-9994 x, Wodolui + 4-008 0-6085 u 0-3282 v + O-QQQ7 X, Staro Nekrassowka + 0-000 o-oooo u, 0-0000 V yyy/ ^2 + I- OOOO X 2 Shahpur -4-141 -8-5624 u + 5-1019 v + 0-9973 *s Khimnana -0-250 - 7-9691 u + 4-9884 v + o 9976 0-3 Kaliana + 3-369 7-6619 u + 4-9 I2 3 + 0-99770-3 Garinda -1-979 -7 0917 u + 4-7415 v + 0-9980 x s Khamor + 2-216 63122 u + 4-445Q V + O-QQS^ 3\ Kalianpur O-Q33 * T* T"Tpy v + 4-I7QO V w yy^o *i + 0-008 K a:, Fikri v yoo -2-174 4-9698 u T * / y w ' + 3.7827 w yy^o ^3 + 0-9988 0-3 Walwari + 5-506 - 4-5109 u + 3-5137 V + 0-99890-, Damargida + 2-647 - 3-545 i + 2-8861 r + 0-99920-3 Darur + 6-086 2-8652 u + 2-3968 v + 0-9994 0-3 Honur -1.748 -2-4183 u + 2-0569 v + 0-99950-3 Bangalore + 5-175 1-7264 u + 1-5020 v + O-OQQ7 X, Patchapaliam + 0.425 1-0050 u + 0-8933 v yyy / + 0-9998 0-3 Kudankulam + o-ooo o-oooo u + O-OOOO V + i-oooo 0-3 Vizagapatam . . . + 0-6126 - 2-2103 u 2-2872 v + 0-9870 y Hydrabad + 2.1487 -0-5530 u - 0-5723 y + 0-9880 y Bombay -3-9452 + 1.4298 u + 1-4785 V + 0-98*1 y Mangalore -2-8457 + 0.7232 u + 0.74851; + 1-0099 y Bangalore 0-2611 0-2279 u - 0-2358 v + 1-0094 y Madras + 4-5052 -I-I535 - 1-1938 v + 1-0091 y Bellary + o-oooo + o-oooo u + o-oooo -y + i -oooo y Cape Point 0-325 I -660 2 U + 0-2558 v + 0-9993 #4 Zwart Kop + 0-833 1-6150 u + 0-2535 * + 0-99930-, Royal Observatory -0-755 -1.50991* + 0-2470 v + 0-99930-4 Heerenlogement , . . + 0-304 0-8030 u + 0-1672 v + 0-9996 x t North End + o-ooo o-oooo u + O.QOOO + I-OOOO X 4 Cotchesqui + 0-582 -1-1224 u + 1-0852 v + I -0000 X s Tarqui + o-ooo o-oooo u + O-OOOO V + i-oooo 0-5 Let each of the corrections be now equated to zero and the whole treated by the method of least squares. After elimina- ting- the y and the *r's the remaining- equations are = +56-6615+ 301-7624 u + 126-9252 v, =16-9677+ 126-9252 + 221-43070, the solution of which gives u = 0-2899, v = -f 0-2428. (12) 318 FIGURE OF THE EARTH. The values of the corrections to the observed latitudes are as in the following table : STATION. CORRECTION. STATION. CORRECTION. Saxaford North Rona Great Stirling ... KellieLaw Durham Clifton Arbury + 1-453 + 0-081 -0-710 - 1 -089 -!-453 -2-486 + 1-180 Shahpur Khimnana Kaliana Garinda Khamor Kalianpur Fikri ... a -3-550 + 0-141 + 3-652 -1-904 + 1-993 -1-392 2 -Q4Q Greenwich Dunkirk + 0-778 I-2K2 Walwari Damargida ... + 4-532 + 1-240 Dunnose Pantheon Carcassone 1-691 2-617 1.228 Darur Honur Bangalore + 4-362 -3-684 Barcelona Monti ouy . + 0-573 + 3.Q2'7 Patchapaliam Kudankulam T fyv$ 2-204 1-128 Formentera Fuglenaes Stuor-oivi + 4-524 + 0029 I-A23. Vizagapatam Hydrabad Bombay + 0-649 + 2-I2I Tornea L 4^0 + 3QII Mangalore 4-050 2-Q2/1 Kilpi-maki T.2C8 Bangalore - y*^ Hogland Dorpat Jacobstadt Nemesch Belin Kremenetz Ssuprunkowzi Wodolui -0-422 -1-483 + 2-247 0-221 + 0-108 -2-232 + 2-588 + I-I 33 Madras Bellary Cape Point Zwart Kop Eoyal Observatory Heerenlogement. . . North End oo + 4-499 -0050 0-161 + 0983 -0637 + 0-198 0-380 Staro Nekrassowka -2-973 Cotchesqui Tarqui + 0-586 -0*585 The sum of the squares of the corrections being 285-763 ..., the probable error of a single latitude is Moreover, if we write A> B for the absolute terms of the J FIGURE OF THE EARTH. 319 last written equations (12) they may be put in the form = ^ + -0043667 A 0025030 B, = v 0025030 ^ + -0059508 B, so that the probable, error of au + fiv is 1-645 (-004367 a 2 --005006 a/3 + -005951 /3 2 )*. Now c involves 2085 #, and a involves 2085 u + 2022 v\ hence their probable errors are respectively 227 and 245 feet. Moreover, for the number representing what is called the ellipticity, since it involves 8-44 v a + c 292-96 1-07 Finally the values of a and c are these a = 20,92^202, c = 20,854,895, their ratio ._ . c : a = 292-465 : 293-465. 7. An examination of the corrections to the observed latitudes in the table given above, does not lead us to suppose that any of the arcs are badly represented by the spheroid just deter- mined,, that is to say, they appear to conform well to' the mean figure. But to enquire more particularly into this point : suppose that for one of the arcs taken by itself only, we calculate that curve, &, either elliptic or of the more general character which best represents that arc. Then by least squares we get A', B', C', and also a certain correction f to be added to the observed latitude < of the southern point S. Then the normal at that point of & which corresponds to 8 is inclined to the equator at the angle < + f. The coordinates of a point of <& in latitude $ are of = (A'ff) cos + J (B'-C') cos 34> + i +' equal respectively to the values of x t y for $ = $ + , then IT and K are so determined that <& and CH' coincide at & The normal distance between these curves in the latitude < is f = (af-a?) cos $ + (/-y) sin , which expresses the distance by which a point in (55' is farther from the centre of the earth than the corresponding point in . Let. A'- A = B) = then The values of for the large arcs are given in the adjoining table. In the case of the Anglo-French and Bussian arcs, the ANGLO-FEENCH. EUSSIAN. INDIAN. Lat. C Lat. C Lat. C ft. o ft. o ft. 60 - 2-7 70 + 4-1 32 - 4-2 58 - 3-6 68 + 3-8 30 + 3-8 56 - 1-8 66 + 3-i 28 + 8-3 54 + 1-9 64 + 2-0 26 + 9-3 52 + 6-8 62 + 0-8 24 H- 6-9 50 + H-8 60 -05 22 + 2-1 48 + 16-1 58 -1-8 2O - 4-3 46 + 18-8 56 -2-9 18 ii. i 44 + 18.9 54 -3-7 16 -167 42 + 15-7 5 2 -4-0 14 19-6 40 + 8-i 50 -3-7 12 -185 48 -2-7 10 -ii. 8 FIGURE OF THE EARTH. 321 curve & is elliptic : in the case of the Indian arc 66 latitude stations have been used to determine the curve not restricted to the ellipse : its equation is f = 20932184-167963-6 cos 20 + 28153-2 cos40. Here we see the local form of the meridian sea-level in India with reference to the mean figure of the earth. Sup- posing- there is no disturbance of the sea-level at Cape Comorin, then from that point northwards a depression sets in, attaining a maximum at about 14 latitude of nearly 20 feet: thence it diminishes, disappearing at about 21. An elevation then commences, attaining at 26 about 9 feet : then this elevation diminishes and becomes a small depression at 32. This deformation may or may not be due to Hima- layan attraction ; at any rate we have here an indication that that vast table-land does not produce the disturbance that might have been anticipated. But although as far north as Kaliana on the meridian of the great arc the effect of the Himalayas is not perceptible, yet at Banog 057 / north of Kaliana, the deflection northwards is about 30". The Anglo-French arc shows a deformation nearly as large as the Indian. After all, in either case, the quantity f is as small as could well be expected. 8. With the values of a and c which we have obtained we find the following. If #, y be the coordinates of a point in lati- tude <, that is, its distance from the axis of revolution and from the plane of the equator x 20944044 cos 17865 cos 30-f 23 cos 50, y = 20837084 sin0 17789 sin 3^ + 23 sin 5. Again, , p being the radii of curvature in the direction of the meridian and perpendicular to the same % = 20890564 106960 cos20 + 228 cos4<, p = 20961932 35775 cos2<+ 46cos4<; Y 322 FIGURE OF THE EARTH. log 1 1 // = 7-994477820 + -002223606 cos 20 000001897 cos 40, log- J_- = 7-992994150 + -000741202 cos20 & p sin 1 000000632 cos 40. If 8, 5' be the lengths of a degree of latitude and of a degree of longitude b - 364609-12-1866.72 cos 20-f 3-98 cos 40, 5' = 365542-52 cos0 .311-80 cos 30+0-40 cos 50, the unit throughout being the standard foot of England. The length of the meridian quadrant is the first term of above multiplied by JTT. Expressed in inches, the ten-mil- lionth part of the quadrant is 39 in -377786, whereas the length of the legal metre is 39 in -370432. CHAPTER XIV. PENDULUMS. IF a heavy particle suspended from a fixed point by a fine thread, inextensible and without weight, be allowed to make small oscillations in a vertical plane under the influence of gravity, then I being the length of the thread and g the acceleration due to gravity, the time of a small oscillation in vacua is expressed by the formula where a is the maximum inclination of the thread to the vertical, and the unit of time one second. The time here expressed is the interval between two consecutive passages of the lowest point. When a does not exceed one degree the term in a 2 is almost insensible. Supposing the case of in- definitely small vibrations, if the same simple pendulum be swung in two different places at which the intensities of gravity are ^, /, then the corresponding times t, t f of vibration are connected by the relation * :<"=/:*. Or if n, n' are the numbers of vibrations made in a mean solar day ^:^ = ,,:/. Therefore, if a simple pendulum makes at the equator n vibrations per diem, the number n' that it will make in latitude $ is given by the equation * = a {1 +(!-*) sin 2 0}, (2) Hence, if n, n' be actually observed, m and < being known, e the ellipticity of the earth becomes known. Y 2 324 PENDULUMS. So also if A be the length of the pendulum which makes one vibration per second at the equator, A' the length of the pendulum making one vibration per second in latitude

, and dg the increment in the same in passing vertically to a height h above the sea : a is the radius of the equator. Considerable labour was ex- pended in calculating the effect on the pendulum of the attraction of the hills on which each station is situated. o I! Fig. 63. Generally, as in Bouguer's formula, it has been considered sufficient to allow for the attraction of an indefinitely ex- tended plateau of uniform height h as though its surfaces were planes : but if we take into account the curvature of the 340 PENDULUMS. earth's surface, this attraction should clearly be increased : the effect of the curvature may be thus obtained. OG8 being a radius of the earth drawn through the station S, and GH a section of the sea level, SXK an arc of a concentric circle, so that KH = SG = /, we require the attraction at S of the solid generated by the revolution of SKGH round SG. Let SK = &, while SX measured along the circle is = x : if also XP measured towards the centre of the earth = y, then the radius of the circle GH being 3 4 56 85987 08 Kaliana 29 30 55 86029-33 Bangalore S. 13 o 41 85986 47 Dataira 28 44 5 86028-57 Mangalore* 12 51 37 8598889 Usira 26 57 6 86o23;-5o Aden* 12 46 53 85991-68 Pahargarh 2 4 56 7 86015-30 Pachapaliam 10 59 40 85984.77 Kalian pur 24 7 ji 86014-87 Alleppy* 9 29 39 85985-90 Ahmadpur 23 36 21 86012-62 Mallapatti 28 45 85983-34 Calcutta* 22 32 55 86012-73 Minicoy Id. y T*/ 8 17 i 85987-02 Badgaon 20 44 23 86005-13 Kudankolam* . . . 8 IO 21 85982-99 Somtana ... 19 5 o 86000-60 " Punnce* 8 o 28 85982 88 y * Coast stations. PENDULUMS. 343 3. SAWITSCH. STATIONS. LAT. VIBRA- TIONS. STATIONS. LAT. VIBRA- TIONS. Q 1 II O 1 II Tornea 65 5 43 8659033 Wilna 54 41 2 86549 37 Nicolaistadt ... 63 5 33 86578 25 Belin 52 2 22 86538-73 St. Petersburg 59 56 30 86568 68 Kremenetz ... 50 6 8 86531-51 Reval 59 26 37 86567-41 Kamenetz 48 4 39 86524.74 Dorpat 58 22 47 86563-21 Kiscbinef 47 i 30 86519-18 Jacobstadt . . . 5<5 30 3 8655474 Ismail 45 20 34 86511 32 FREYCINET. 4. DUPEKREY. Q I II / // Paris 48 50 I4N 86406 oo Paris 48 50 i 4 N 86406-85 Toulon 43 7 20 86385-46 Mowi* 20 52 7 86315-41 Rawakf o i 348 86279-35 Isle of France 20 9 56 86315 97 Isle of France 20 9 238 86315-87 Port Jackson 33 5 1 34 86351-96 Port Jackson 33 5' 40 86351-21 Falkland Island 5* 35 18 8641464 Falkland Island 5i 3i 44 864i8'i2 * Sandwich Islands. Near New Guinea 5. LUTKE. / // O 1 II Petersburg . . . 59 56 3i 86273-08 Bonin Island* 27 4 12 86163-14 Sitka 57 2 58 86261-44 Guamt 13 26 21 86121 78 Petropaulowski 53 o 53 86240-83 Ualan % 5 21 16 8611663 Greenwich . . . 51 28 40 86240-18 Valparaiso . . . 33 2 30 86169-23 Off S.E. Coast of Japan. f- Ladrone Islands. Caroline Islands. 6. BIOT & ARAGO. Dunkirk O 1 II 51 2 IO 86m4.-oo Fi^eac / // 4.4. ^6 4.Z 86^0^ QI Clermont Bordeaux 45 46 4 8 44 5 26 86510-50 8650663 Formentera ... S^ 39 56 86485.00 344 PENDULUMS. The numbers in Table I agree with those of Baily, M.R.A.S., vol. vii, pp. 96, 97, with the exception that the vibration numbers of Foster and Sabine are all (excluding* those at Port Bowen and Altona) increased by 0-14 in order that at London the mean may be 86400. At four stations common to these two observers, viz. London, Ascension, Trinidad, Maranham, the differences between the vibration numbers are (Foster Sabine) -f-0-28, 0-30, +0-46, 0-45, and the mean for the two observers is taken in each case. The Indian observations are given in the second table and the Russian in the third : the last, taken from M. R. A. $., vol. xliv, page 31 4, are converted into vibration numbers. The results of Duperrey and Freycinet are contained in the fourth table. These observers have four stations in common : if we take their results as given in Baily's Memoir, pp. 91, 92, and multiply Duperrey's vibration numbers by a factor of which the logarithm is 9-9810785, we have the numbers given in the table. The mean of the results at the four common stations is used in the subsequent calculation. The fifth table contains Lutke's results : the sixth those of Biot and Arago converted into vibration numbers ; Eecueil d? observations geodesiques, &c., par MM. Biot et Arago, 1821, page 573. The selection of Kew instead of London as a reference point or base for the Indian series was unfortunate, greatly dimin- ishing the weight of those observations in the determination of the figure of the earth. For Kew is not in connection with any of the earlier pendulum stations. Great advantage would have been gained had the Indian eeries been extended to include at least two stations of the Sabine and Foster series, and also St. Petersburg. We may, however, though the link is not so strong as might be desired, utilize the observations of Goldingham connecting London and Madras, and thus append the Indian series to the English series *. Again, in order to connect the Indian series with the Russian, the Russian pendulums which had been used by M. Savvitsch were sent out to India and swung at a few stations, and finally at Kew. Heaviside's result at this point being, according to Sawitsch, 440 1 -7170, may be considered 1 An extension of the Indian Pendulum Series is under consideration. G PENDULUMS. 345 as one of the series of Table 3. This however is an 'absolute ' result, and therefore not strictly one of Sawitsch's series. It may be taken as a connection but a somewhat weak one. We may check the two connections just explained in the following manner. In the equation . , T / Madras \ / Kew \ /St. Petersburg^ Greenwich = London ( ^ j I I ^-^ ) ( - v 2 ) V London / VMadras / V Kew / Greenwich \ ,St. Petersburg/ taking London as 86400, take the first ratio on the right side from Goldingham, the second from the Indian series, the third from the Russian, the fourth from Lutke ; then we get for Greenwich a vibration number within a small fraction of a vibration the same as London. This is fairly satisfactory : unfortunately, however, notwithstanding the observations of Sabine, Foster, and Baily, the exact difference of Greenwich and London is not well determined, some of the results being positive, some negative : the amount is probably not more than half a vibration. If, e being the earth's ellipticity, we put 77 = -| m e, and if w , n be the vibration numbers of an invariable pendulum in latitudes 0, <, we know that n 2 = 2 ( 1 + r 7 sin 2 <). The problem before us is to determine rj and thence e. Let rj Q = -0052022 be an approximate value of rj, and m being ^-f-y, put 77 = 1* +TOOOO' 290 10000 290 + 8. 4y Let N be the approximate vibration number of a pendulum at the equator, N + z the true number, N the vibration number of the same at a station in latitude <, N+ x the same vibration number corrected for local disturbance, then which is easily put in the form . , 1 , , ' " ' N ' 20000 N ' ZN 2N 346 PENDULUMS. Here z is one or two units, and we shall put unity for its coefficient. Let further sn 20000 N then to each station corresponds an equation of the form x = z + ay + m, z remaining the same in any one series of observations ; but differing for different series. We shall determine y and the z's so as to make the sum of the squares of the a?'s be a mini- mum. Let 2H = ( z i + a \y + m \f + fa + a \y + m i) 2 + ( z \ + (* 2 + a ^y + %) 2 + (** + &c., &c., &c. ; and from ^ = eliminate the 2 J s by means of dy da dQ, dQ, -j = 0, -j- = 0, -y = . . . , dz l dz 2 dz s and the result is the sum of the quantities ii = |(i 2 ) - f ("O 2 ! y+ K i) - f W W, (10) ? 1 ^ l &C. } &C., equated to zero ; an equation which gives at once y. Here i n is the number of stations in the n ih series. The sum of the squares of the coefficients of the m's as they enter into the expression for y is the reciprocal of the coefficient of y in this final equation. We shall now give the results of this calculation. In the first instance, supposing all six series independent, PENDULUMS. 347 the equations (10) become Ti = 78-7593y+ 2-8664, T 2 = 7-8295 y 2-5995, J 3 = 2-31 38y- 4-9886, T 4 = 6-4696^ + 14-3964, J 5 = 6-9186y + 7-1597, T" 6 = 0-4307y 0-3743, =102-7215^+16*4601 ; /. y 0-160, 77 = -0051862, 1 e = 2887 Secondly; assuming the connection of the English and Indian series: 1^2= 105-4021 ^48-2878, T 9 = 2-31 38y 4-9886, J 4 = 6.4696^+14-3964, r 5 = 6-9186^+ 7-1597, F 6 = 0-4307^ 0-3743, =121-5348y 32-0946; .-. y = 0-264, j] = -0052286, 1 292-2 Thirdly ; assuming the connection of the Indian and Rus- sian only : Ii = 78-7593y+ 2-8664, T 2 , 3 = 55-0711^81-1328, Z 4 = 6-4696^ + 14-3964, T 5 = 6-9186^+ 7-1597, Y 6 = 0-4307y 0-3743, =147-6493^-57-0846; /. ^=0-387, 7;= -0052409, 1 e = 293-3 348 PENDULUMS. Fourthly ; assuming the connection of the English, Indian, and Russian : F 1 , 2 ,3 = 137-7645y 92-8100, F 4 = 6-4696^+14-3964, Y s = 6-9186y+ 7-1597, Y 6 = 0-4307y 0-3743, = 151-5834^-71-6282; /. y = 0-472, rj = -0052494, 1 e = 294-0 We may safely conclude that e lies between the limits indicated in the first and fourth solutions ; and comparing the second and third solutions with the ellipticity shown at page 319 it would appear that as far as can be ascertained from our data, the ellipticity resulting from pendulum ob- servations does not differ sensibly from that obtained from terrestrial measurements. To each of the above solutions there is a corresponding system of quantities #, indicating the apparent excess or defect of gravity at each station of observation. If we take the system corresponding to the first solution and compare the #'s of London and Kew, we find at the former a defect of 0-91 vibrations per diem and at Kew an excess of +5-15, a difference of 6.06 vibrations, in fact between two points only ten miles apart and nearly in the same latitude. This seems inadmissible ; and we are compelled to fall back on Golding- ham's observations at Madras as connecting the Indian series with London. The table on the opposite page shows the excess vibrations, or excess of gravity, at the different stations according to the second solution 1 * ~ 292-2 " The points marked with an asterisk were not used in the calculation. PENDULUMS. 349 STATIONS. EXCESS VIBRA- TIONS. STATIONS. EXCESS VIBRA- TIONS. STATIONS. EXCESS VIBRA- TIONS. Spitsbergen. . . + 3-09 Toraea + 3-3i Greenland ... 0-50 Kew + 2-89 Nicolaistadt - o-35 Port Bo wen + 1-08 More"* 22-08 St. Petersburg + O-4.8 Hammerfest ' " 7 -1-41 Meean Meer - 3-97 Reval r \j j|u + O-9I Drontheim ... -3-55 Ismail! a - 1-08 Dorpat + 0-4I Unst . + 1-75 Mussoorie* ... - 6-06 Jacobstadt ... ~ I-36 Portsoy + 1-67 Dehra* Q-3O Wilna O-O6 Leith Fort ... + 1-13 Nojli y O w - 4 .82 Be"lin - 0-74 Altona + 1-09 Kaliana - 4.09 Kremenetz ... 0-50 Clifton 0-09 Dataira - 2.25 Kamenetz . . . + 0-81 ArburyHill... + 083 Usira - i-54 Kischinef ... - 0-82 London -0.21 Pahargarh ... - 3-54 Ismail - 2-07 Shanklin ... -0-36 Kalianpur ... - '-54 New York ... + 0-20 Ahmadpur ... - 2 -33 Paris - 3-29 St. Bias ... -3-70 Calcutta ... + o-79 Toulon - 1-83 Jamaica + I-3I Badgaon 1-92 Mowi + 4-80 Trinidad -2-66 So in tan a 2-21 Rawak - 2-61 Porto Bello... + 3-85 Bombay + 2-84 Isle of France + 7.16 1 Sierra Leone + 0.66 Damargida ... - 4-43 Port Jackson - 0.38 Galapagos . . . + 2-43 Kodangal . . . - 2-46 Falkland ... - 3-85 St. Thomas... + 6-86 Cocanada . . . + 0-30 P. G. Lout ... + 4-53 Namthabad. . . - 3-4i Petersburg ... - 0-13 Para -1-50 Madras 1.28 Sitka - i-66 Maranham ... -3-45 Bangalore N. - 3-3^ Petropaulows. + i-59 Fernando . . . + 8-22 Bangalore S. - 3-82 Greenwich ... - 2-23 Ascension ... + 6-15 Mangalore ... I-I2 Bonin Island* + 11-79 Bahia -0-98 Aden + 1-81 Guam + 488 St. Helena ... + 9-32 Pachapaliam - 2-27 Ualan* + 9-93 Rio Janeiro... -1-41 Alleppy + 0-91 I Valparaiso ... - 2-41 Paramatta ... -0-44 Mallapatti ... - 1-65 C. Good Hope 0-80 Minecoy Id. + 3-49 Dunkirk ... + 1-96 Monte Video -i-43 Kudankolam - o-43 Clermont ... - 1-03 Staten Island + 2-90 Punnoa - o-53 Bordeaux ... 1-20 Cape Horn ... + 1-67 Figeac 102 S. Shetland... 4-3-90 I Formentera... + 1.29 350 PENDULUMS. The probable error of an equation or the probable ir- regularity of gravity expressed in seconds per diem is and that of y is 0-18, so that 1 e = 292-2l-5 These probable errors are however too small on account of the stations omitted from the calculation. During the progress of the pendulum observations in India General Walker called attention first in his Yearly Report, 1866, and again in subsequent Reports up to 1874 to the broad fact which was gradually being brought to light, viz. that there is a very decided diminution of intensity of gravity as approach is made to the Himalayas, and that at coast stations, and especially at the Island of Minicoy, there is an excess of gravity. In the Report 1874, pp. 20, 21, we find notice of the crowning result of Captain Basevi's investiga- tions that with which his life so sadly terminated that at the summit of the Himalayas there is a singularly great defect of gravity. These facts are visible in the figures contained in the last table. Kaliana 1 was fixed on by Sir G. Everest as the nearest ap- proach that should be made to the base of the Himalayas for reliable geodetic observations, and in our table we see that at that station and all north of it there is a large defect of gravity, attaining at More an amount of 22 vibrations. It is very remarkable that this is precisely the amount of the correction that had been applied for the attraction of the mountains, so that the apparent vertical attraction of the three miles of earth crust between More and the sea level is zero. And in fact at most of the other high stations the residual 1 An Account of the Measurement of two Sections of the Meridional Arc of India, by Lieut.-Colonel Everest ; pp. xli, xlii. At Banog the observed azimuth is affected to the extent of 20" by Himalayan attraction. PENDULUMS. 351 discrepancy is diminished or removed if we omit the cor- rection for the attraction of the table-land lying between the station and the sea-level. It would seem then that these pendulum observations have established beyond question the fact previously indicated by the astronomical observations of latitude in India that there exists some unknown cause, or distribution of matter, which counteracts the attraction of the visible mountain masses. If it be considered too bold a speculation to surmise that there may be vast cavities under great mountain masses, then the most probable explanation is to be sought in the hypothesis of Archdeacon Pratt and this view of the matter is favoured by General Walker in his preface to the pendulum volume l . 1 Account of the Great Trigonometrical Survey of India, vol. v, pp. xxxii, xxxiii. NOTES AND ADDITIONS. NOTE, page 36. The recently published second part of the Mem. du Dep. Gen. de la Guerre contains an account of the determination of the astronomical amplitude of the Algierian arc. The chain contains some 66 principal triangles, with determina- tions of latitude and azimuth at the extreme stations Nemours, towards Morocco, Bone on the frontier of Tunis, and at Alger (Algiers), near the centre of the chain. The three astronomical differences of longitude corresponding to Bone-Alger, Alger-Nemours, Bone-Nemours, were inde- 352 NOTES AND ADDITIONS. pendently determined by the electro-telegraphic method, with the results shown in the following table : STATIONS. No. OF DAYS. BY DIRECT OBSERVATION. CORRECTED FOR PERSONAL EQUATION. PROB. ERROR. m. s. in. s. s. Bone-Alger . . . 7 18 51-222 18 51.392 001 1 Alger-Nernours 9 19 35-119 J9 34-949 + O-OII Bone-Nemours 8 38 26-498 38 26-328 0-013 The sum of the first two longitude intervals should be equal to the third : the actual discrepancy amounts to only 3 -013 which is exceedingly satisfactory. In 1867, the theodolite displaced finally the repeating circle, and with a theodolite, or 'azimuth circle' as they call it, of the very simplest construction, the western portion of the Algierian chain was completed. The length of the arc of parallel, reduced to the latitude of 36, is stated to be as follows : m. Bone-Alger 425234-7 Alger-Nemours 441139-8 These results being dependent on an assumed figure of the earth cannot however in their present shape be used in an investigation of the figure of the earth. If we take as resulting from the calculations at page 322, 295826-4 feet as the length of a degree of the parallel of 36, we get the following contrast : STATIONS. ASTR. AMP. GEODETIC AMP. G-A Bone-Alger ... Alger-Nemours Bone-Nemours o / // 4 42 50-79 4 53 44- 20 9 36 3499 O / II 4 42 57- 8 7 4 53 3289 9 3 6 3 -76 + 708 -11-31 - 423 These differences seem to point to a considerable attraction to the west at Algiers. The trigonometrical stations in the Algierian chain are, with a few exceptions, marked by well-built pillars of stone generally conical frust^a in form having a vertical axial aperture communicating with the centre-mark of the station. NOTES AND ADDITIONS. 353 The bases at Bone and Oran (near Nemours) are about 1 kilometres in length ; the length of either of these bases, as calculated (through the intervening 88 triangles) from that of the other, differs about 16 inches from the measured length. The azimuth circle, or theodolite, constructed by M. Brunner, of Paris, and used at the stations of the grand quadrilateral, Mulhacen, Tetica, Filhaoussen, M'Sabiha, has a diameter of 16 inches, and is read by four micrometers. The tftlps^flpe is 24 inches in focal length and 2 inches in aperture. The electric light forming the signals at the stations just named was placed in the focus of a reflector 20 inches in diameter and 24 inches focal length. This reflector is a concavo-convex lens of glass, of which the convex surface is silvered, the radii of the surfaces are so related that spheri- cal aberration is destroyed and the reflector is practically paraboloidal. The emergent cone of white light has an am- plitude of 24', which is sufficient to cover any little errors in directing the axis of the lens on the distant station. This direction is of course effected by a telescope and special mechanism, insuring the greatest precision. In the longitude observations connecting Tetica and M'Sabiha, however, a refracting lens of eight inches diameter was found sufficient for throwing the electric light a distance of 140 miles. The revision of the French meridian chain of Delambre and Mechain was commenced in 1870, at the base of Perpignan in the south of France, and has been completed as far as the base of Melun near Paris, an extent of 6 30'. A few only of the old stations have been refound, thus the work is entirely new. M. Perrier has adopted the system of night observations, the light being a petroleum lamp in the focus of a refracting lens of eight inches diameter and two feet focal length. The old system of using church towers as trigonometrical stations has been abandoned, and in the woody and difficult country between Bourges and Melun the work was carried on by scaffoldings eighty and a hundred feet high. In this district the chain of triangles is double. The close of the triangles (or the error of the sum of the observed angles) indicates great precision in the work the greatest error being I" '20, and the average 0"-53. A a UJL 354 NOTES AND ADDITIONS. At the central and important station of Puy de Dome a local attraction of 7"-0 in latitude has been detected. NOTE, page 59. With reference to (13), it should be remarked that it is necessary to leave the absolute terms symbolical, only if , g, h ... are liable to have any numerical values. Otherwise, for a single set of numerical values of those coefficients, A, , C...can be obtained by elimination from (12), and then, as appears from line 6, page 59, S is given by NOTE, page 157. As Kater's value of the metre, viz. 39 in -37079, is still fre- quently adopted as the real length, the following remarks on the value given at page 157, namely: Mttu = 39^-37043 O in -00002 may be useful. Kater obtained his value from comparisons between a certain English scale and two metres brought from Paris. One of these, a platinum metre, is certified to have been compared by Arago with ' a standard metre 3 very slender authority; and about the authority for the second metre nothing is said. With respect to the comparisons, we have no information as to the errors of the thermometers used with the bars, they do not appear to have been inves- tigated ; nor is there any reference to any precaution taken for avoiding the bugbear ' constant error,' which is or should be the first and last anxiety of every observer. The length he obtained is not of course in inches of the present standard yard. The comparisons however made at Southampton in 1864 between the standard yard and this same platinum metre, lead to the result that the 'standard metre' to which Arago made reference had a length = 39 in -37046. The sufficiency and consistency of the authorities on which is based the value given at page 157, are doubtless beyond question. AND ADDITIONS. 355 NOTE, page 165. Suppose that each of the segments of a " base line is mea- sured n times, and let the results of the several measurements of the first segment be s\, *" 1} s"\... ; the mean = o- 1 . For the second segment let the measurements give s 'z> * x/ 2> s " '?. > the mean = a 2 , and so on for each of the segments. Then, i being the number of segments, the adopted length of the base line is 0i + -f V. This agrees with (17) when a and b are small. / sr FOURTEEN DAY USE RETURN TO DESK FROM WHICH BORROWED This book is due on the last date stamped below, or on the date to which renewed. Renewed books are subject to immediate recall. 3 1956 LI LD 21-100m-2,'55 (B139s22)476 General Library University of California Berkeley