CALIBRATION OF PRISMATIC COMPASSES

Abstract

Apart from “stickiness” of the suspension and looseness of the sights, prismatic compasses are subject to three internal sources of error:- <list list-type="roman-lower"> <list-item>

Collimation error. This may be caused by <list list-type="alpha-lower"> <list-item>

magnetic axis not being parallel to the zero line of the graduated circle;</list-item> <list-item>

line of sight not passing through the axis of rotation.</list-item> </list> It is unnecessary to aftempt to distinguish between the above faults, which introduce constant errors into the compass readings.</list-item> <list-item>

Eccentricity error. This is caused by the axis of rotation failing to pass through the centre of the graduated circle. This introduces an error into the compass readings of E sin θ cosec I°/R where E is the eccentricity, R the radius of the graduated circle and θ the angle between the line of sight and the line joining the centre of the circle to the axis of rotation. Eccentricity error is completely eliminated by observing both forward and back bearings, but this is not always practicable.</list-item> <list-item>

Irregular division of the graduated circle. This error is negligible in any modern compass.</list-item> </list>     

A CHANCE CORRECTION OF A TRAVERSE

Abstract

It sometimes happens that from a point on a line of theodolite traverse two fixed points are visible. In the absence of a visit to at least one of these points, B or C, or a precise knowledge of a bearing, it is not possible to fix absolutely the station, say A, of the traverse. Nevertheless, the fact remains that if the angle subtended by the fixed points is measured and found to be α, say, the station A must lie on an arc of a circle through BC “capable of” this angle α. Is there any assumption which is justifiable under these circumstances?     

INFLUENCE OF GEOIDAL TILT ON AZIMUTH

Abstract

If the horizontal circle of a theodolite is tilted, the azimuth of a point in the field will be affected. The tilt may be a mere question of dislevelment but it may be also due to a deviation of the vertical which the level on the instrument cannot measure. In either case the effect on azimuth or bearing is of the same nature, which is worth consideration for more than one reason. That it has previously been investigated elsewhere is no valid reason against renewed consideration here, particularly as the consequences are probably unknown to some readers.     

POINT TO POINT COMPUTATION OF GEOGRAPHICAL COORDINATES OF LONG LINES

Abstract

The computation of geographical coordinates in a geodetic triangulation is usually carried out using Puissant's method, in which the assumption is made the sphere radius ν (the radius of curvature of the spheroid perpendicular to the meridian) not only touches the spheroid along the whole small circle of latitude ?,but also, since ρ (the radius of curvature in meridian) is very nearly equal to ν it makes such close contact with the spheroid that the lengths of sides and angles of a geodetic triangle may be considered identical on both sphere and spheroid.     

THE ORTHOMORPHISM OF THE STEREOGRAPHIC PROJECTION

Abstract

When developing the argument leading to the stereographic solution of the spherical triangle and its application to field astronomy (Empire Survey Review, Vol. 2, No. 10, October, 1933, p. 226) A. J. Potter rendered a very useful service in demonstrating how proofs of the two practically useful properties of the stereographic projection can be provided along lines that demand no more than simple geometry in their development. The proof advanced for the unique property that any circle on the. sphere remains a circle in projection is at once simple and complete; but in the attempt to prove that the projection is orthomorphic in the sense that angles everywhere remain true there is the difficulty that the argument was developed for what must be regarded as a special case in that the point was located on the great circle through the origin of the projection normal to the plane of the projection. Treatment of the problem along similar lines for other points away from the central meridian does not seem to admit of such ready solution and the alternative approach suggested here, while still not demanding. anything beyond simple geometry for its understanding, affords a proof for a general case.     

SOME EFFECTS OF PULLEY ECCENTRICITY IN BASE MEASURING APPARATUS

Abstract

The comprehensive paper on the suspension of tapes by M. Hotine in the January, 1939, issue of the Empire Survey Review (v, 31, 2) did not contain any reference to this question, as was pointed out by A. J. Morley in a letter published on page 261 in the same volume (v, 34, 261). A brief analysis has been made by F. Yates of the theoretical effects of pulley eccentricity and misalignment (“Gold Coast Survey Department Records” VoL III, 1931, page 43) but I have not seen any further reference to the subject and have recently experienced the effects of such a defect in our own apparatus, so the followingnotes nlay be of interest. Before proceeding to details I will describe briefly those parts of the apparatus which are considered here and give a short summary of the whole paper.     

THE STEREOGRAPHIC SOLUTION OF THE SPHERICAL TRIANGLE

Abstract

In the course of his stimulating and suggestive paper in your recent issue, No. ro, pp. 226–38, Mr. A. J. Potter writes on p. 233 “but there is no simple construction by which X can then be found”, and again on p. 237 “a direct construction, if there be such”. This cheerful challenge invites the construction of a circle centred on a given line, passing through a given point thereon, and touching a given circle, and I have found the lure of Mr. Potter's gauntlet as irresistible as its recovery has proved delicate. In order to shoulder responsibility and by no means to claim highly improbable originality, let me confess that the problem is new to me and the two constructions I offer are my own; I venture to hope that Mr. Potter may consider one or other of them not unworthy of his epithet “simple”, though I freely admit the aptitude of his empiric procedure to its purpose. The proofs are not long, but for fear of overshooting my welcome I offer them to anyone for the asking; and for the same reason my diagrams are small and therefore mere.     

DETERMINATION OF AZIMUTH AND LATITUDE FROM OBSERVATIONS OF A SINGLE UNKNOWN STAR BY A NEW METHOD

Abstract

The purpose of this paper is to describe a new and easy method of determining the (astronomical) latitude and azimuth at any place and to explain the line of approach and the formulae. It will be seen that the method should be useful to a wide circle of land surveyors. One of its principal advantages is that identification of the star is not necesssary and it can be used when no star chart or star catalogue is available.     

A CURVE OF TRANSITION

Abstract

The original object of a transition curve was to ease the change from a straight line of communication to a circular line, or vice versa. It may even be laid out to connect one circular curve with another of different radius. In this manner the discomfort and even the danger to travellers in vehicles traversing such lines of way are obviated, abrupt variations of acceleration being overcome. The first curve to be used for this purpose was the elastica, introduced by Froude about 1842, and it is worthy of note that in their initial lengths all transition curves conform closely to this earliest form of easement. It is indeed obvious that, since the circle is a curve of the second order, an easing curve must be of a higher order ; the curve of the next higher order is of the third degree, and of all such curves the cubic parabola is the simplest, its equation being a²y ==x³.     

COMPENSATION AS A CONSEQUENCE OF TWO KINDS OF ROCK STRUCTURE

Abstract

The following points occurred to me when reading the interesting paper on crustal equilibrium in E.S.R. No. 23. The principle of compensation or isostasy necessarily involves the idea of two different kinds of rock structure—one strong, the other weak or in extreme cases fluid; for example, there is the familiar case of the strong iceberg resisting change of shape in the liquid sea. In dealing with crustal problems of the earth then, we should make up our minds which part is to be considered as strong, e.g. the granite crust, and which part as weak or fluid, e.g. material at a depth x km. (say roo km.); by weak or fluid I mean that a possibility exists of horizontal movement.     

EVOLUTION OF LAND RECORDS

Abstract

Owing to delays at this end in the preparation of the paper on “The direct use of vertical photographs in Zanzibar for cadastral purposes”, for which apology is due to Mr. Chambers as well as to readers, its publication has been regretfully deferred. We are using the opportunity to publish the Bibliography which could not conveniently appear with our opening paper last April, and to accompany this with an outline of some early and feudal developments in land records which we hope may help to interest a wider circle of land surveyors in this feature of cadastral work. For experience in many countries has impressed on us that the effective record of rights and duties pertaining to land, particularly in parts of the world less stably patterned agriculturally and more closely divided than England, has suffered grievously from a divorce between the legal and the technical aspects of cadastral record and between its operation and effects as sought on parchment in legislatures and the administrative offices of governments and as realized in daily practice by landholders and cultivators on the ground itself.     

LONG LINES ON THE EARTH

Abstract

The Arc of the Geodesic.—In the first part of this paper a method was given for computing the azimuth of a geodesic. The method gives the convergence of the geodesic correctly up to the second power of e the eccentricity. The formula (9), however, also depends on the assumption that σ, the arc-length of the geodesic, can be obtained with sufficient accuracy from the Supplemental Dalby Theorem, that is to say, by a purely spherical computation. It is, therefore, needful to show that this supposition is justifiable; a means must in fact be indicated for verifying the assumption.     

THE GEODIMETER: AN INSTRUMENT FOR THE ACCURATE MEASUREMENT OF DISTANCES BY HIGH-FREQUENCY LIGHT VARIATIONS

Abstract

The constant K in equation (12) represents distance expended through time lags in the instrument itself, and, although the value of K can be calculated from electrical data, this would not be very satisfactory and it would be better to determine it directly by means of observations over a line of known length. In addition, the point from which K would be reckoned is not a convenient one for actual field measurements. Instead of this, it is more convenient to choose an index mark on the instrument itself and referall measurements to this and thence to the mark over which the instrument is set up.     

TRANSVERSE MERCATOR FORMULAE

Abstract

The Transverse Mercator Projection, now in use for the new O.S. triangulation and mapping of Great Britain, has been the subject of several recent articles in the “Empire Surpey Review. The formulae of the projection itself have been given by various writers, from Gauss, Schreiber and Jordan to Hristow, Tardi, Lee, Hotine and others—not, it is to be regretted, with complete agreement, in all cases. For the purpose for which these formulae have hitherto been employed, in zones of restricted width and in relatively low latitudes, the completeness with which they were given was adequate, and the omission of certain smaller terms, in the fourth and higher powers of the eccentricity, was of no practical importance. In the case of the British grid, however, we have to cover a zone which must be considered as having a total width of some ten to twelve degrees of longitude at least, and extending to latitude 61 °north. This means, firstly, that terms which have as their initial co-efficients the fourth and sixth powers of the longitude ω (or of y) will be of greater magnitude than usual, and secondly that tan² ? and tan⁴ ? are likewise greatly increased. Lastly, an inspection of the formulae (as hitherto available) shows a definite tendency for the numerical co-efficients of terms to increase as the terms themselves decrease—e.g. terms in η⁴, η⁶, etc.     

TESTING A THEODOLITE FOR ACCURACY

Abstract

When a theodolite is used to measure an angle, the result will be subjected to certain instrumental and personal errors which affect the measurement. Such errors may be accidental or systematic. Those of the former type, which follow no law and which may with equal probability occur at any graduation, are more easily eliminated, since, if a very large number of readings is taken, it is probable that the errors will cancel out and that the mean will approximate very closely to the correct figure. Systematic errors are usually due to instrumental defects and rnay be expressed as a function of the reading itself; it is the object of the manufacturer to eliminate these as far as possible, since cancellation by reiteration or by repetition is not to be expected wholly.     

THE VARIATION OF THE ATMOSPHERIC COEFFICIENT OF REFRACTION WITH THE NATURE OF THE SOURCE OF LIGHT

Abstract

In working out vertical heights on the Akuse-Kete Krachi chain of triangulation in the Gold Coast a fairly considerable difference was found between values of the coefficient of refraction obtained from observations taken during the day and those taken at night, the mean values being 0.069 for daylight observations to heliographs and 0.087 for night observations to lamps. This difference no doubt is due mainly to the condition of the atmosphere during the day differing from its condition during the night rather than to any effect due to different sources of light. A new chain has recently been observed in Western Ashanti, and the index of refraction for the daylight observations again gave a lower value than that obtained from the night observations, the figures being 0.073 and 0.099 respectively. For the night work three different sources of light were used, hurricane lamps for short lines, Tilley vapour-pressure lamps for lines of intermediate length, and McCaw acetylene signalling lamps by Watts for long lines. It occurred, therefore, to the writer to examine the results to see if the mean values of the index of refraction showed any variations for the different light sources, since it seemed reasonable to suppose that the constitution of the light emitted from each source would be different and hence that the coefficient of refraction might vary.     

THE ART OF ORIGINAL CIRCULAR DIVIDING (1650–1850)

Abstract

The problem of dividing the circle into equal parts has occupied the minds of astronomers and instrument makers from the earliest times, but little is known of the methods adopted by such renowned observers as Hevelius or Tycho Brahe, who are said to have divided their own instruments (I). It is possible, however, to trace the various steps by which progress was made and to appreciate the urgent need that was felt, particularly throughout the eighteenth century, for improvement in the accuracy of astronomical instruments, if only to satisfy the increasing demands for navigational charts and for the means of determining the position of a ship at sea.     

THE DOUBLE IMAGE TELEMETER

Abstract

With the double image telemeter distances are measured optically by placing a prism in front of a portion of the object glass, so that two overlapping images are obtained. The image seen through the prism has moved through a fixed angle determined by the prism, and if a graduated staff is sighted the linear shift of image can be measured by reading the position of the displaced zero against the staffimage as seen through the uncovered portion of the objective. The distance from staff to instrument can then be calculated from the linear image shift and the deflection angle produced by the prism by the usual formulae of tacheometry.     

THE CONFORMAL TRANSFORMATION OF A NETWORK OF TRIANGULATION

Abstract

In a recent issue of this Review, an example is given of the conformal transformation of a network of triangulation using Newton's interpolation formula with divided differences. While the application of the method appears to be new, attention should be drawn to the fact that Kruger employed Lagrange's interpolation formula in a discussion and extension of the Schols method in a paper which was published in the Zeitschrift für Vermessungswesen in 1896. A reference to this paper was given at the end of the paper, “Adjustment of the Secondary Triangulation of South Africa”, published in a previous issue of the E.S.R. (iv, 30, 480).     

THE SURVEY FRAMEWORK OF NIGERIA

Abstract

Angular and Linear Errors.—In the results which follow, the number of stations quoted on the chains is exclusive of all the stations on the base-extension nets at Minna, Rijau, Chafe, and Naraguta. The total number of stations can therefore be obtained by adding the numbers on all the chains and the base-extension nets and deducting the number of points common to other chains. The quoted fractional misclosures in length are the actual misclosures obtained and the theoretical fractional misclosures which might be expected to be developed through the chains from the probable errors of the adjusted angles.