GEODESY

the geoidis the surface which coincides with the ocean surfaces and their hypothetical continu- ations inland in imaginary channels; all tidal movement being considered suppressed. the study of the form of the geoid, fre- quently referred to as the “ figure of the earth,” is a main ob- ject of geodesy. it has been prosecuted by various agencies in sepaiated portions of the land arcas. even in 1926 only a small fraction of the total area of the globe has been examined. such results as have been obtained may be co-ordinated by the assumption that the geoid does not differ greatly from a spheroid (cllipsoid of revolution); and the assumption of an ellip- soid of three unequal axes is not at present considered useful. “determination of the figure of the earth’’ may be used con- veniently in the restricted sense of determination of the elements of the most suitable spheroid; while the science of geodesy em- braces further the study of the actual geoidal form in detail, ancl even the explanation cf this form. figures of the larth—during the progress of the science and the growth of observation results, various figures of the earth have been obtained. kiach survey department has mace use of that figure which at the time seemed best. a variety of spheroids are now in use as reference figures, on which the calculations of the corresponding surveys are based. there is growing inconven- ience in this, and it would be ideal if all the separate geodetic surveys could express their results in terms of a single sphcroid. although there are technical difficulties in the relation of the origins of separate surveys to any unique spheroid, it was nore the less decided at the mecting of the international union of geodesy and geophysics at madrid in 1924 that one spheroid should be internationalised and that, so far as practicable, results should be expressed in terms of it. the international spheroid is that determined by j. f. hayford in 1909.! equatorial radius. ; 6,378,388 =18 metres reciprocal of flattening. : 2907 -0+0°5 polar semi-diameter . ; - =2 6,356,909 metres the above figures were derived from observations made in the u.s.a. only; and the introduction of the principle of isostasy (¢.2. ) was the novel feature of the work. remembcring that the entire u.s.a. areca of 3,026,789 sq. m. is but 1/66th of the entire earth’s area, 197,000,000 sq. m., some doubt might be felt as to their univer- sal applicability and their stated probable errors, an independent consideration by ihelmert? of hayford’s figures yielded probable errors+35 metres and +o-8, further, helmert * in i915, from a discussion of numcrous gravity results from places scattered over the globe found 1/296-7 +0-6 for the flattening, which 1s a very strong corroboration of hayford’s value. the reference spheroid.—that the geoid should be a spheroid was a very plausible conjecture; for a spheroid is a possible boundary for a rotating gravitating fluid mass—the flattening depending on the internal density distribution. when a solid mass is considered in place of a fluid mass some modification in the resultant form may be expected; at the same time the development of isostasy supports the view that the earth be- haves in many respects as though it were viscous, tending to the form it would assume if fluid. however, long before such evi- dence became available, it was tacitly assumed that the geoil was actually a spheroid. nineteenth century observations were reduced by clarke and others, treating any discrepancies from spheroidal form as due to observation error. this was justiiiable when data were scant, and the instruments, with which they had been obtained, were of poor precision. more recent data from greatly improved instruments, how- ever, leave no room for doubt that there are measurable differ- 1figures throughout the text refer to notes at the end of the article, 167 ences between the geoid and any adopted spheroid. geodesists study the actual form of the geoid, expressing it with reference to a spheroid, which is accordingly named the ‘ reference sphe- roid.”’ when the reference spheroid is in close correspondence with the geoid, the mutual separation is small at all points. this facilitates many computations: for some purposes observed horizontal angles, which are geoidal angles, may be treated as spheroidal angles: as has practically always been done, though in a few cases small correction was really necessary. the general origin.—suppose that the geoidal form were known throughout, and apply to this the international (or any other) spheroid. how is the latter to be orientated? the minor axis of the spheroid may be set parallel to the actual axis of rotation of the earth, whose direction is known unequivocally to the precision of astronomical telescopes. this answers the question of orientation, the location of the spheroid remains at choice. any point on the spheroid may be brought into coincidence with corresponding point on the geoid; but ey ove potut can be so treated, which may be called the ‘‘general origin.’”?” when this has been done, some portions of the geoid will lie without the spheroid, and other portions will le within, the two surfaces will intersect in a number of closed curves, on one of which will lie the general origin. the normals to the geoid —verticals—will not coincide with the corresponding normals to the spheroid except at points along certain closed curves, quite distinct from the curves of intersection of the two surfaces. the form of these curves will change if the general origin is changed. the inclination of geoidal and spheroidal normals at corresponding points is what has generally been called the “ detlection of the plumb-line.” it will be clear now that the deflection of the plumb- line at a point depends on the choice of spheroid of reference and of general origin. it is not an absolute quantity, but dependent on the reference system. when this latter has been selected, there is a unique value of the deilection at any point which has been geodeti- cally connected with the origin. in practice, however, it is necessary to assume separate origins for cach detached survey, and accordingly the dellections of surveys on separate origins cannot be regarded as absolute quantities. in this respect the international spheroid can- not be considered as unique as yet. the selected figure —meanwhile the study of the geoid in each detached survey may be continued. at any point of the carth’s surface, astronomical observations combined with w.t. signals enable both latitude and longitude to be uniquely determined, independently of any assumed reference figure. a number of such points are connected by linear measure and triangulation, and values, called geodetic, of latitude and longitude are com- puted o the selected figure. the differences of astronomic and geodetic values are the deflection components. if these are sufhi- ciently numerous—which may be judged by the smoothness of deflection variation—the separation of the geoid and spheroid can be integrated. an alternative method of finding this separation is based on the measurement of vertical angles between the triangulation stations combined with spirit levelling and deflection results. the observed vertical angles are reduced to the spheroidal vertical by application of the appropriate deflections. if the terrestrial re- raction* % of each ray is computed, it becomes merely a_geo- metrical problem to calculate the differential spheroidal heights. spirit levelling, which with its short rays intimately follows the gcoidal surface, yields the geoidal heights. both spheroidal and geoid- al heights are thus known, and hence the separation of geoid from spheroid. ordinary triangulated heights, uncorrected for the deflection, are not the same as spirit levelled heights, apart from the question of precision; and they certainly are not geoidal heights. geoidal heights are what enter into practical problems; but from the geodesist's point of view they do not mean much until the form of the geotd, to which they refer, has been determined. base line measurements —modern bases are measured with wires hanging in catenary undcr constant tension, a system in- troduced by e. jiderin of stockholm. his original plan was to use two wires having different expansion coefficients, whence the temperature could be determined and the necessary correc- tions applied. with guillaume’s invention of “invar” this became unnecessary. a bright wire of small gauge (1-65 mm. is the usual diameter) ts not much affected by the sun's radiant heat, and so its temperature does not differ much from that of the air. with the low expansion coefficient of invar, 4x1077 per degree centigrade, an error of 2°°5 causes an error in length of only 1 in ro¢, the method of using invar 168 wires has been closely studied by benoit and guillaume’ and their procedure may be confidently followed. it is vastly simpler than the old methods with compensated bars and is also much more accurate; further, much rougher country can be negotiated and so, longer bases become possible. for the standardisation of wires at the observatory, before and after the measurement of the base, different methods of laying out a length of 24 mctres, which is the usual length of the wires, have been employed. an apparatus designed by sir david gill for india was fully described in fngineering 1915. the ultimate standard of tength in this is a nickel bar of il section, one metre long. stand- ards for ordinary use are h bars of invar, one metre and four metres in length. invar has been observed to undergo a secular change in length which continues for many years. investigating this, guillaume found that the instability is due to the presence of carbon which gradually forms cementite, fes;c, with the iron. the addition of chromium, which has a greater affinity for carbon than iron has, prevents this and an invar with ten-fold increased stability has been produced. triangulation —theodolites with circles of 12 or 10 in. are usually employed for the highest class geodetic triangulation. mr. connolly of the india stores department considers even that the fullest instrumental accuracy can be incorporated in an 8-in. theodolite fitted with a large objective telescope. luminous signals, heliotropes by day and lamps by night, are essential. vertical angles should be taken between 1 and 3:30 p.m., when the refraction is usually a minimum and most regular. none the less there is frequently at this time poor definition, and horizontal angles should be observed at night or by day within three hours of sunrise or sunset. control.—triangulation emanating from a fixed point with observed base and azimuth yields positions of all stations and lengths and azimuths of all sides. errors are unavoidably gen- erated. their probable amounts are given by the formulae of | de graaff hunter ® now indicated. these involve the quantity m=(1+f)m ¥18;/, in which / is the average length of side; m (ferrero’s error of mean square of an angle) = ¥ y(a*/3n), where a is the triangular error and 2 is the number of triangles; and f, which lies between 0 and 1/6, depends on the type of figures in the series. further, if o is the starting point and a he station at which errors are sought: r=oa and s=curved distance, oa measured medially through the series, both expressed in units of 100 miles, then p.e. in seconds of azimuth at a=1-'"575 ¥ zal’s, p. e. in 7th figure of log. side at a=33-:2 ¥ 2m?s, p. e. in feet in northing or casting at a=4-03 ¥ z( me r’ds). the summation © is in each case for a set of series for which values of m differ. the first two of these formulae give the means of determining at what intervals control of triangulation is desirable. clearly an extra base will control the error developed in side length: to control the azimuth it is necessary to form a laplace point, at which azimuth and longitude are determined astronomically. a numerical example will illustrate the method. it is desirable to introduce a control when the p.e. attains 3 times the error of the control. two bases are involved, one at each end. if the p.i. of a base is i in 108 then the p.e. of the control is ¥2 in 108 the p.e. of side reaches thrice this amount when 33-2m ¥s=3 v¥ 2-107 log. (1,108). taking m = 0-2, corresponding to very high class triangulation, we find s is 772 miles. next, if aa, aa, are differences astronomic minus geodetic azimuths; l, i.,, a, a, triangulated values of longitude and latitude of a and o, and t the difference between the local times at a and o, then laplace’s equation 1s aa.coseca, — aacoseca = (l-l,) — 15t which serves to determine aa, a quantity to be subtracted from the astronomic azimuth to give the correct geodetic azimuth, the p.e. of an astronomic azimuth is taken as 0’’-2 for high class work; that of tas o*-03. hence the p.e. in azimuth determined from laplace's equation is v¥ [(o-2 sin a cosec a,)?+(0-2)?-+ (0-45 sin a)?] if we take x=a,=45° this become 07:42 and 1-575 x0-2 ¥s=3xx 0-42, whence s is 1600 miles. if the time error could be reduced to o*-o1 the value of s would be almost halved and the precision of azimuth would be about the same as that of side control. it is to be pointed out that there is little gain in controlling the side length with great accuracy while the equally important azimuth control is much less precise. hitherto the ruling criterion has been that of azimuth. the conclusion is that until time observations are more precise, the highest class triangulation is not likely to be much improved by the controls available unless its extent is considerably greater than 1,000 miles. geodesy determinations of height-—the precision of spirit levelling is so great as to justify the recognition of the lack of parallelism of the various level surfaces, each of which is approximately spheroidal. it is nowadays customary in levelling of high pre- cision to apply to the observed differences of height the correc- tion 78 for the convergence of these surfaces, that is to say, the orthometric correction; and to publish the orthomctric heights. as regards differences of height found by triangulation much improvement is called for. observed vertical angles are re- ferred to the local geoidal vertical; they require correction to the reference spheroid vertical, as well as for refraction, after which the height differences can easily be calculated. refraction.—when reciprocal observations have been made at two points aj, a», if e:, w:, 61 are respectively the angle of eleva- tion, the refraction and the deflection at a, towards as, and similarly with changed suffixes for as, then fi +61— a: + f2+ 62 — an +c=0, c being the angle between spheroidal verticals at a;, a», com- puted from the triangulation. the ordinary practice is to ignore 6, and 6. and to assume that a= @2=q; whence 2q=e,+f2.+c. the ratio q‘'c=k has been called the coefficient of refraction, andas k=} — } it can be computed when a ray has been observed at both ends. there are cases when 6,+6 is by no means negligible in com- parison with «,-+-e2; moreover there is little justification for the assumption w, =w.2, unless the two stations are at the same height. refraction has a diurnal change, falling to a minimum about the time of maximum temperature, moreover the value of the minimum is pretty constant from day to day. it is in this constancy that lies the value of observing vertical angles between the hours of it and 3:30 p.m. consideration 4°> of the physical laws leads to a formula which represents the refraction usually met with at these hours. at other hours the refraction ts ordinarily larger by an amount which varies as the defect of temperature from maximum and as the cose- cant of the angle of elevation of the ray. the physical cause © is that the temperature lapse-rate is nearly constant within the height limits of the ray, and docs not vary much during the day except comparatively close to the ground. the refractive effect of a plane stratum of air close to the ground can be expressed in terms of the air densities of its upper and lower surfaces. some confirmation of this view is obtained by considering barometric height readings. further research is required, and more observations, with full particulars of local deflection, are needed for investigation. in cases where the data are complete, the refraction docs not appear so in- tractable as has gencrally been supposed. astronomical latitudes —for the observation for latitude the zenith telescope and the talcott-ilorrebow method seemed to have superseded all others (see 11.610); but the method of equal altitudes is worthy of attention. several special instru- ments have been designed for this observation, all of which determine latitude and time simultaneously; whereby 1n associa- tion with wireless time signals, deflection in both meridian and prime vertical can be found. the prismatic astrolabe of mm. claude and drtencourt, 4 ¥, 3, which is made in two sizes, has a horizontal telescope with an equilateral prism mounted in front of the objective. the edges of the prism are horizontal, and one face, that nearest to the objective, is vertical. a mercury bath is suitably placed below and slightly in front of the prism. light from a star of approximately 60° altitude falls normally on the upper face of the prism, and is reflected inter- nally at the lower face towards the lower half of the olsjective. at the same time light from the same star falls on the mercury, ts re- flected into the prism through the lower face, then internally at the upper face, and so to the upper half of the objective. in both cases after reaching the objective, the light is brought to a focus and two images of the star are formed. these approach one another, and the instant at which they pass is timed —this instant corresponding toa fixed altitude depending on the angle of the prism, imperfection in the focus of the telescope and the refraction. the actual altitude 1s usually regarded as an unknown; corrected for refraction it should be a constant for one instrument. provisional values for this, for the latitude and for the time being assumed, the result of one observation can be plotted as a straight line on a chart with time and aseca as co-ordinates. four stars, one in each quadrant, forma set of observations and yield four such lines which should touch a circle of radius equal to the discrepancy of the altitude from that provisionally assumed, both multiplied by cos a. the co-ordinates of the centre of this circle represent the time and latitude sought, and the constancy of its radius is a measure of pre- geodesy cision of the observation. if there is too much wind the partially shielded mercury surface is disturbed, and observation becomes tm- possible. the adjustments are very simple, as is also the observation when the observer is practised; and there is practically no recording. computations and preparation of star programme are not excessive. excellent results have been obtained with this instrument. in the discussion of a paper jackson stated that the images are elongated, being in reality short spectra; so that stars of different colour would give different results. other users of the astrolabe have not found this to be appreciable. practical drawbacks are that each star yields only one observation and that the time estimate is bur- dened by personality. to overcome the latter l. fave!® described equipment to measure the personal equation, but up to 1925 this has not proved completely successful. an alternative instrument, free from two of the above noted objections is the circumzenithal apparatus of mm. nust and fric.6 in this the light from the star enters the instrument through a parallel plate of glass and falls on a half silvered mirror, being split into two portions. one portion proceeds to a mercury bath, and after reflection there and at a second mirror passes again through the half sitvered mirror. thence both portions enter the observing tele- scope. the mercury is completely screened from the wind. by means of an achromatic reversible prism placed in front of the plate at which light enters the instrument, a star can be observed at three slightly different altitudes. an additional interior prism in front of the objective, which causes the formation of four images, allows 7 observations to be made on the same star at each altitude. a simple bent telescope, in which reflection occurs at a mercury bath, has been designed and used by de la baume pluvinal. ob- serving 10 stars on the same evening at the paris observatory, he ob- tained values of latitude and longitude differing by 0’’-2 and 0*-03 from the well established observatory values. in egypt, wade has employed a simple method of finding differential latitudes at places a few hundred metres apart. longttudes —several of the large wireless stations send out special time signals, which can be picked up at very great dis- tances by suitable apparatus. these provide a basis for longitude determination when the local time has been observed. the very powerful station at bordeaux is an example; and the precise times of the signals are determined by the “ service horaire de l’observatoire de paris."". the signals are of the ‘ rhythmic ”’ type, in which 300 dots are sent out uniformly spaced over 293 seconds. at the end of the set the precise times of those of the previous day are signalled. very precise comparison can be made by noting at what exact seconds the incoming signals coincide with the beats of the observer’s clock: alternatively, when a_ recording receiver is used, all the 300 dots are recorded and subsequently measured. owing to the time of duration of the signals and of the clock beats not being precisely equal, there is some personality in the estimate of instant of coincidence. to obviate this thrum !8 devised a method in which the incoming signals and those of the clock extinguish one another in the telephone when they overlap. difference of opinion exists as to whether the registration or the audition method is the more precise. time signals have been picked up at several of the larger observa- tories day by day for years. . sampson’ pointed out and dis- cussed serious discrepancies of the order of 0°-1 in the relative times. the complete explanation is still to be found. on the other hand longitude circuits recently formed by the u.s.c. and g.s. are stated to have closing errors of the order of 0*-o1, suggesting that field in- struments and conditions are superior to the large instruments and conditions at the fixed observatories, a world project of longitude was put forward by general ferrie in 1921. a mixed commission of the international unions of astronomy and of geodesy and geophysics was formed, and the project has been discussed at rome, 1922, madrid, 1924, and cambridge, 1925. details have been advanced sufficiently for a decision to execute the scheme in 1926 to be arrived at. it is hoped to fix very accurately the longitude of numerous points over the earth, thus helping to co-ordinate geodetic surveys. at the same time much expcrience in the highest precision work of this nature will be obtained. a com- prehensive account of clock and wireless installations for this pur- pose is given by the bureau des longitudes.”® gravity.—the absence of means of measuring the “sway ” or flexure of the stand of pendulum apparatus was perhaps the principal cause of uncertainty in the older pendulum observa- tions. two methods have been devised and used for a number of years; and a further advance in eliminating the effect of sway has been made by vening meinesz. of the two methods referred to, one is in vogue in the u.s. and consists in actually measuring the movement of the stand by aid of an interferometer2tm it is claimed that the error in ‘“¢g” from this determination rarely exceeds 2 in 107. the other method, due to schumann, 2! provides a means of mounting a second pendulum on the same stand. the first pendulum is set swinging, and from 169 time to time the amplitude of the second pendulum, originally at rest, is noted. formulae expressing the effect of sway in terms of oscillations are given. the precision appears to be much the same as that of the first method. both of these methods regard the sway as a constant during the course of the observations at a station, lasting perhaps three days. vening meinesz *, 28, 24, solves the problem more completely, and also overcomes the effects of carth movements by always em- ploying simultaneously two isochronous pendulums. these are mounted on one stand so that the distance between their points of support is invariable, and are set swinging in different phases. denoting the angles of elongation by @, @ then the angle 6,—4, may be regarded as the angle of elongation of a hypothetical pen- dulum which is free from the effects of sway; as also of various irregular perturbations, so far is this the case that apparatus so arranged has been used for determining gravity at sea in a sub- marine; and determinations have been made in holland in places where the ground is very unstable. observation is much as usual in pendulum work, the ray of light passing from the mirror of the first pendulum to that of the second and thereby recording the differential angular movement of the two. in consideration of its success at sea, vening meinesz’s meth- od supersedes the earlier work of hecker and duftield * with hypsometers and barometers. the possibilities opened up are very extensive. helmert’s formula of 1g01 has been used widely in pendulum work. it is £0= 978-030 (1-+0-005302 sin?a—o-000007 sin?2a), \ being the latitude. bowie, 2 p. 196, taking isostasy into ac- count, found 978-039 as the equatorial value, and -005294 as the coefficient of sin?a, while couchman?? found helmert’s value too small by -or1 in the case of indian stations. in 1915 wlelmert? gave a formula involving the longitude 3 = 978-052 (1-7 -005285 +7 sin?’a—-o00007 sin?2x +-oo0018 +4 cos?a cos2(l+17° +6) this is based on 3,000 stations reduced by borass. it implies an elliptic equator with major axis in 17° w. longitude, whose half length is 230 metres greater than the semi-minor axis—com- pare clarke’s 3-axial figure results, with 8° 15’ w. as the longi- tude and 465 metres excess length of the semi-major axis. the eetvos gravity torsion balance ** measures the rate of change of gravity. the change per cm. along meridian given by the formula is 8-1x10° c.g.s. units in latitude 45°; and as the instrument is incredibly sensitive, it is capable of measuring 110% and less, and easily detects irregularities. nowadays it is used in prospecting for oil. (see geolocy; isostasy.) references.—(1) j. f. hayford, supplementary investigation in 1909 of the figure of the farth and isostasy, coast and geodetic survey, united states (1910); (2) f. r. helmert, sitsungsberichte der kon. preuss. akademie der wissenschaften (1911); (3) f. r. helmert, neue formein fiir den verlauf der schwerkraft im meeres- miveau berm festlande (1915); (4) j. de graalf hunter, formulae for atmospheric refraction, etc., survey of india professional paper no. 14 (1913); (5) j. r. benoit et c. e. guillaume, la mesure rapide des bases geodesiques (1908); (6) j. de graaff hunter, the earth's axes and triangulation, survey of india professional paper no. 16 (1918); (7) c. lallemand, nivellement de haute precision (1886); (8) precise levelling in india, operations of the great trigonometrical survey, vol. 19 (1910); (9a) j. de graaff hunter, gravity survey; (gb) tdem, trigonometrical eights and refraction; (10) j. de graaff hunter, ‘ diurnal change in atmospheric refraction,” budletin geodesique, no. 2, april 1923, app. 12; (11) comptes rendus de l’ academie des sciences, vol. 171 (nov. 1920); (12) a. claude et j. f. l. driencourt, description et usage de t'’astrolube d@ prisme (1910); (13) john ball and h. knox shaw, a iandbook of the prismatic astrolabe (1919); (14) ‘ the prismatic astrolabe,’’ geo- graphical journal, vol. 54, p. 29 (1919); (15) louis fave, “ appareil destine 4 determiner |’@quation personclle dans les observations a vastrolabe, a prisme et a faciliter l’instruction des observateurs, bulletin geodestque, no. 2, april 1923 (1924); (16) l. benes, le nouvel appareil circumzenithal de am. nusl et fric; (17) e. b. hl. wade, ‘a method of determining small differences of latitude,” helwan observatory, bulletin no. 27; (18) e. a. thrum, “ reception of wireless time signals at the adelaide observatory,’ monthiy notices of the reyal astronomical society, geophysical supplement, vol. i, p. 55 (1923); (19) r. a. sampson, afonthly notices of the reyal astronomical society (june 1918, may 1920, nov. 1920, jan. 1922, april 1925); (20) reception des signaux [oraires, publies par je bureau des longitudes (paris, 1923); (20a) w. hf. burger, the measurement of the flexure of pendulum supports with the interferometer, report for 1910 coast and geodetic survey, app. no. 6; (21) r. schumann, ‘* ueber die verwendung zweier pendel, etc.,’’ zeitschrift flir mathematik und physik, 44th year, parts 2, 3 170 (leipzig, 1899); (22) f. a. vening meinesz, projet d'un nouvel appareil pendulaire; (23) f. a. vening meinesz, ‘‘ the determina- tion of gravity at sea in a submarine.” geographical journal, vol. 65, p. 501 (1925); (24) f. a. vening meinesz, observations de penduie duns les pays-bas, 1913-1921 (delft, 1923); (25) w. g. dutfield. determination of gravity at sea,” report of the committee of the rritish association (1919); (26) wm. bowie, ‘the timportance of isostasy in geodetic research,” builetin geodesique no. 5 (1925); (27) if. j. couchman, phe pendaluin observations in india ane burman, 1908-1913, survey of india professional paper no. 15 (igt5); (28) h. shaw and fe. lancaster-jones, ‘ ketves torsion balance,” proceedings of the physical society, vol. 35, pp. 151, 204 (1923). (j. pe g. il.)