GEODESY INCLUDING ASTRONOMICAL OBSERVATIONS, GRAVITY MEASUREMENTS, AND METHOD OF LEAST SQUARES BY GEORGE L. HOSMER Associate Professor of Topographical Engineering, Massachusetts Institute of Technology FIRST EDITION NEW YORK JOHN WILEY & SONS, INC. LONDON: CHAPMAN & HALL, LIMITED 1919 COPYRIGHT, 1919, BY GEORGE L. HOSMER Stanhope jprcss ?. H. GILSON COMPANY BOSTON, U.S.A. PREFACE In this volume the author has attempted to produce a text- book on Geodesy adapted to a course of moderate length. The material has not been limited to what could be actually covered in the class, but much has been included for the purpose of giving the student a broader outlook and encouraging him to pursue the subject farther. Numerous references are given to the standard works. Throughout the book the aim has been to make the underlying principles clear, and to emphasize the theory as well as the details of field work. The methods of observing and computing have been brought up to date so as to be consistent with the present practice of the Coast and Geodetic Survey. The chapters on astronomy and least squares are included for the sake of completeness but do not pretend to be more than in- troductions to the standard works. The student cannot expect to master either of these subjects in a short course on geodesy, but must make a special study of each. The author desires to acknowledge his indebtedness to those who have assisted in the preparation of this book, and especially to Professor J. W. Howard of the Massachusetts Institute of Technology for suggestions and criticism of the manuscript; to the Superintendent of the Coast and Geodetic Survey for valuable data and for the use of many photographs for illustrations; and to Messrs. C. L. Berger & Sons for the use of photographs of the pendulum apparatus and several electrotype plates. Tables XII to XVII are from electrotype plates from Breed and Hosmer's Principles and Practice of Surveying, Vol. II. G. L. H. CAMBRIDGE, April, 1919. iii 405416 TABLE OF CONtENTS CHAPTER I GEODESY AND GEODETIC SURVEYING TRIANGULATION ART. PAGE 1 Geodesy I 2 Geodetic Surveying I 3 Triangulation 2 4 Classes of Triangulation 2 5 Length of Line 3 6 Check Bases 4 7 Geometric Figure 6 8 Strength of Figure 6 9 Number of Conditions in a Figure 9 10 Allowable Limits of Ri and R* 10 11 Reconnoissance n 12 Calculation of Height of Observing Tower 11 13 Method of Marking Stations 19 14 Signals Tripods .' 18 15 Heliotropes 19 16 Acetylene Lights 23 17 Towers 25 18 Reconnoissance for Base Line 28 CHAPTER H BASE LINES 19 Bar Apparatus for Measuring Bases 31 20 Steel Tapes 31 21 Invar Tapes 32 22 Accuracy Required 33 23 Description of Apparatus 34 24 Marking the Terminal Points 36 25 Preparation for the Measurement 36 26 Measuring the Base 36 27 Corrections to Base-Line Measurements Correction for Grade 37 28 Corrections for Alignment 38 29 Broken Base 38 30 Correction for Temperature 39 v VI CONTENTS ART. PAGE 31 Correction for Absolute Length 39 32 Reduction of Base to Sea-Level 40 33 Correction for Sag 41 34 Tension 42 CHAPTER III FIELD WORK OF TRIANGULATION MEASUREMENT OF ANGLES 35 Instruments Used in Measuring Horizontal Angles 44 36 The Repeating Instrument 44 37 The Direction Instrument 46 38 The Micrometer Microscope 48 39 Run of the Micrometer 49 40 Vertical Collimator 52 41 Adjustments of the Theodolite 53 42 Effect of Errors of Adjustment on Horizontal Angles 54 43 Method of Measuring the Angles 56 44 Program for Measuring Angles 58 45 Time for Measuring Horizontal Angles 63 46 Forms of Record 64 47 Accuracy Required 65 48 Reduction to Centre , 65 49 Phase of Signal 67 50 Measures of Vertical Angles 68 CHAPTER IV ASTRONOMICAL OBSERVATIONS 51 Astronomical Observations Definitions 71 52 The Determination of Time 73 53 The Portable Astronomical Transit 74 54 The Reticle 76 55 Transit Micrometer 76 56 Illumination 78 57 Chronograph 78 58 Circuits 81 59 Adjustment of the Transit 81 60 Selecting the Stars for Time Observations 83 61 Making the Observations 84 62 The Corrections 87 63 Level Correction 87 64 Pivot Inequality 87 65 Collimation Correction 89 66 Azimuth Correction 89 67 Rate Correction 90 CONTENTS vii ART. PAGE 68 Diurnal Aberration 90 69 Formula for the Chronometer Correction 91 70 Method of Deriving Constants a and c, and the Chronometer Correction, Ar 92 71 Accuracy of Results 97 72 Determination of Differences in Longitude 97 73 Observations by Key Method x 99 74 Correction for Variation of the Pole 101 75 Determination of Latitude 101 76 Adjustments of the Zenith Telescope 103 77 Selecting Stars 103 78 Making the Observations 104 79 Formula for Latitude 105 80 Calculation of the Declinations 106 81 Correction for Variation of the Pole ! 106 82 Reduction of the Latitude to Sea-Level 107 83 Accuracy of the Observed Latitude 109 84 Determination of Azimuth no ' 85 Formula for Azimuth in 86 Curvature Correction 112 87 Correction for Diurnal Aberration 112 88 Level Correction 113 89 The Direction Method . 113 90 Method of Repetition 116 91 Micrometric Method 118 92 Reduction to Sea-Level Reduction to Mean Position of the Pole .... 120 CHAPTER V PROPERTIES OF THE SPHEROID 93 Mathematical Figure of the Earth 122 94 Properties of the Ellipse 123 95 Radius of Curvature of the Meridian 125 96 Radius of Curvature in the Prime Vertical 126 97 Radius of Curvature of Normal Section in any Azimuth 128 98 The Mean Value of Ra 131 99 Geometric Proofs 132 100 Length of an Arc of the Meridian 134 101 Miscellaneous Formulas 135 102 Effect of Height of Station on Azimuth of Line 136 103 Refraction . 139 104 Curves on the Spheroid The Plane Curves 139 105 The Geodetic Line 140 106 The Alignment Curve 143 107 Distance between Plane Curves 144 V1U CONTENTS CHAPTER VI CALCULATION OF TRIANGULATION ART. PAGE 108 Preparation of the Data 147 109 Solution of a Spherical Triangle by Means of an Auxiliary Plane Triangle 149 no Spherical Excess 149 in Proof of Legendre's Theorem 150 112 Error in Legendre's Theorem 152 113 Calculation of Spheroidal Triangles as Spherical Triangles 152 114 Calculation of the Plane Triangle 153 115 Second Method of Solution by Means of an Auxiliary Plane Triangle. .. 154 CHAPTER VII CALCULATION OF GEODETIC POSITIONS 116 Calculation of Geodetic Positions 158 117 The North American Datum 159 118 Method of Computing Latitude and Longitude 160 119 Difference in Latitude 160 120 Difference in Longitude 164 121 Forward and Back Azimuths 166 122 Formulae for Computation 168 123 The Inverse Problem 170 124 Location of Boundaries 171 125 Location of Meridian 172 126 Location of Parallel of Latitude 172 127 Location of Arcs of Great Circles ' 174 128 Plane Coordinate Systems 174 129 Calculation of Plane Coordinates from Latitude and Longitude 175 130 Errors of a Plane System 180 131 Adjusting Traverses to Triangulation 183 CHAPTER VIII FIGURE OF THE EARTH 132 Figure of the Earth 185 133 Dimensions of the Spheroid from Two Arcs 187 134 Oblique Arcs IQO 135 Figure of the Earth from Several Arcs 191 136 Principal Determinations of the Spheroid 193 137 Geodetic Datum 195 138 Determination of the Geoid 196 CONTENTS IX ART. PAGE 139 Effect of Masses of Topography on the Direction of the Plumb Line. . . 197 Laplace Points 201 140 Isostasy Isostetic Compensation 202 CHAPTER IX GRAVITY MEASUREMENTS 141 Determination of Earth's Figure by Gravity Observations 206 142 Law of the Pendulum 206 143 Relative and Absolute Determinations 206 144 Variation of Gravity with the Latitude 207 145 Clairaut's Theorem 210 146 Pendulum Apparatus 211 147 Apparatus for Determining Flexure of Support 216 148 Methods of Observing 219 149 Calculation of Period 221 150 Corrections 222 151 Form of Record of Pendulum Observations . . 232 152 Calculation of g 233 153 Reduction to Sea-level : . . . . 233 154 Calculation of the Compression 235 CHAPTER X PRECISE LEVELING TRIGONOMETRIC LEVELING x55 Precise Leveling 237 156 Instrument 240 157 Rods 240 158 Turning Points 241 159 Adjustments 241 160 Method of Observing 242 161 Computing the Results 243 162 Bench Marks 245 163 Sources of Error 245 164 Datum 249 165 Potential 250 166 The Potential Function 250 167 The Potential Function as a Measure of the Work Done 251 168 Equipotential Surfaces 252 169 The Orthometric Correction 254 170 The Curved Vertical .- . . 256 171 Trigonometric Leveling 257 X CONTENTS ART. PAGE 172 Reduction to Station Mark 257 173 Reciprocal Observations of Zenith Distances 258 174 When Only One Zenith Distance is Observed . 261 CHAPTER XI MAP PROJECTIONS 175 Map Projections 265 176 Simple Conic Projection 265 177 Bonne's Projection / 267 178 The Polyconic Projection 268 179 Lambert's Projection 271 180 The Gnomonic Projection 273 181 Cylindrical Projection 274 182 Mercator's Projection 274 183 Rectangular Spherical Coordinates 278 CHAPTER XII APPLICATION OF METHOD OF LEAST SQUARES TO THE ADJUST- MENT OF TRIANGULATION 184 Errors of Observation 279 185 Probability 280 186 Compound Events 280 187 Errors of Measurement Classes of Errors 281 188 Constant Errors 281 189 Systematic Errors 281 190 Accidental Errors 281 191 Comparison of Errors 281 192 Mistakes 283 193 Adjustment of Observations < 283 194 Arithmetical Mean 284 195 Errors and Residuals 284 196 Weights 284 197 Distribution of Accidental Errors 285 198 Computation of Most Probable Value 290 199 Weighted Observations 291 200 Relation of h and p 291 201 Formation of the Normal Equations 293 202 Solution by Means of Corrections 293 203 Conditioned Observations 294 204 Adjustment of Triangulation 295 205 Conditions in a Triangulation 296 206 Adjustment of a Quadrilateral 297 CONTENTS xi ART. PAGE 207 Solution by Direct Elimination 302 208 Gauss's Method of Substitution 302 209 Checks on the Solution 304 210 Method of Correlatives 304 211 Method of Directions 309 212 Adjusting New Triangulation to Points Already Adjusted 310 213 The Precision Measures 314 214 The Average Error 316 215 The Mean Square Error 316 216 The Probable Error 317 217 Computation of the Precision Measures, Direct Observations of Equal Weight : 319 218 Observations of Unequal Weight 321 219 Precision of Functions of the Observed Quantities 322 220 Indirect Observations 324 221 Caution in the Application of Least Squares 325 GEODESY CHAPTER I GEODESY AND GEODETIC SURVEYING TRIANGULATION 1. Geodesy. Geodesy is the science which treats of investigations of the form and dimensions of the earth's surface by direct measure- ments. The two methods chiefly employed in determining the earth's figure are (i) the measurement of arcs on the surface, combined with the determination of the astronomical positions of points on these arcs, and (2) direct observation of the variation in the force of gravity in different parts of the earth's surface. 2. Geodetic Surveying. Geodetic Surveying is that branch of the art of surveying which deals with such great areas that it becomes necessary to make systematic allowance for the effect of the earth's curvature. In making an accurate survey of a whole country, for example, the methods of plane surveying no longer suffice, and the whole theory of locating points and calculating their positions must be modified accordingly. Such surveys require the accurate loca- tion of points separated by long distances, to control the accuracy of subsequent surveys for details, such as coast charts and topo- graphic maps, or for national and state boundaries. The general method employed is that of triangulation, in which the location of points is made to depend upon the measurement of horizontal angles, the distances being calculated by trigonometry instead of being measured directly. This method was first applied to the measurement of arcs on the earth's surface by Snellius of Holland in 1615. : GEODETIC SURVEYING TRIANGULATION Although we may make this distinction when denning the terms it is not necessary to separate the two in practice. It is evident that geodetic surveys must be made before accurate dimensions of the earth can be computed; and, conversely, it is true that before geodetic surveys can be calculated exactly, the earth's dimensions must be known. Hence geodetic surveys are usually conducted with a twofold purpose: (i) for collecting the scientific data of geodesy, and (2) for mapping large areas, every survey depending upon data previously determined, but also adding to or improving the data already existing. For this reason the measurements are made with greater refinement than would be necessary for practical purposes alone. 3. Triangulation. A triangulation system consists of a network of triangles the vertices of which are marked points on the earth's surface. It is essential that the length of one side of some triangle should be measured, and also that a sufficient number of angles should be measured to make possible the calculation of all the remaining triangle sides. In addition to the measurements that are abso- lutely necessary for making these calculations it is important to have other measurements for the purpose of verifying the ac- curacy of both the calculations and the field-work. 4. Classes of Triangulation. Triangulation is divided, somewhat arbitrarily, into three grades, called primary, secondary, and tertiary, the classification depending upon the purpose for which the triangulation is to be used and upon the degree of accuracy demanded. The primary system is planned and executed for the purpose of furnishing a few well-determined positions for controlling the accuracy of all dependent surveys. Since the primary is usually the only tri- angulation which is employed in the purely scientific problems of geodesy, the selection of the primary points will be governed in part by the requirements of any geodetic problem that it is proposed to investigate. The secondary triangulation is some- what less accurate than the primary, and the lines are generally LENGTH OF LINE 3 shorter; it is often simply a means of connecting the primary with the tertiary system. Sometimes the secondary is extended into a region which is to be surveyed but which is not covered at all by the primary triangulation, and then it becomes the con- trolling triangulation of the region. The tertiary triangulation furnishes points needed for filling in details on the hydrographic or topographic maps. It is of a low order of accuracy as com- pared with the primary, but is amply accurate for controlling the surveys for detail. These tertiary, stations furnish the start- ing points for plane-table surveys, traverse lines, etc. All three classes of triangulation are not necessarily present in a survey unless it be a very extensive one. In surveys of minor import- ance there may be but one class of triangulation. 5. Length of Line. The length of line which may be used is determined largely by the character of the country to be surveyed. In California, where the mountains are high and the atmosphere is exception- ally clear, the network of triangulation known as the " Davidson quadrilaterals" (Fig. i) is composed of lines varying in length from 50 to over 150 miles; whereas in flat country, lines from 15 to 25 miles long are the most common. Although the progress of the triangulation is apparently more rapid when long lines are used, it is not necessarily economical to use very long sights. The time gained by having but few stations to occupy may be more than offset by the delays due to unfavorable atmospheric conditions. Furthermore, it may be necessary to introduce many additional stations in the detail surveys in order to reach all parts of the area to be mapped. The accuracy of triangulation is not appreciably lessened by using rather short lines. In planning the system an attempt should be made to use that length of line which will result in the greatest economy, taking into consideration the cost of reconnoissance, signal building, base-line measurement, and the measurement of the angles. 4 GEODESY AND GEODETIC SURVEYING TRIANGULATION 6. Check Bases. It has already been stated that at least one line in a system must be measured. In order to verify the accuracy of all the measurements, it is customary to introduce additional base lines into the triangulation at intervals varying from 50 to 500 miles. Mt.Shasta SCALE OF MTLE8 20 10 60 80 100 FIG. i. Primary Triangulation in California (Davidson Quadrilaterals). The lengths of these bases may be found by calculation of the tri- angles as well as by the direct measurement; this furnishes a most valuable check on the accuracy of the field work. In the triangulation of the United States Coast and Geodetic Survey the frequency with which these check bases should occur is de- CHECK BASES termined by the strength of the chain of triangulation as found by the method given in Art. 8. The factor RI (Equa. [a]) be- tween bases should be about 130 for primary work, although this may be increased to 200 if necessary. In the triangulation of New England there are three bases: (i) Cooper Mt.Blue, Nanticket West Hill FIG. 2. Primary Triangulation of New England. the Fire Island base, about 9 miles long, in the southern part of Long Island; (2) the Massachusetts base, about 10 miles long, near the Northeast corner of Rhode Island; and (3) the Epping base, about 5 miles long, in Maine. These base lines are shown as heavy lines in Fig. 2. The total length of the triangulation between the Epping and Fire Island bases is about 350 miles. 6 GEODESY AND GEODETIC SURVEYING TRIANGULATION The accuracy with which the triangulation was executed is indi- cated by a comparison of the measured and computed lengths. The length of the Epping base as calculated from the Fire Island base is 0.042 meter less than the measured length; the length of Epping base calculated from the Massachusetts base is 0.136 meter less than the measured length. 7. Geometric Figure. The geometric figure generally recognized as the best one for triangulation purposes is the quadrilateral, consisting of four stations joined by six lines, thus forming four triangles in which there are altogether eight independent angles to be measured. This figure furnishes a greater number of checks than any of the simple figures and therefore gives a good determination of length. The polygon having an interior station is also a strong figure. Figures which are more complex than these usually make the calculation troublesome and expensive, while simpler figures, like single triangles, result in diminished accuracy. In the work of the United States Coast Survey the primary triangulation is made up chiefly of complete quadrilaterals and partly of polygons having an interior station. In these figures all of the stations are supposed to be occupied with the triangulation instrument, but for secondary and tertiary triangulation some stations may be left unoccupied. 8. Strength of Figure. In deciding which of several possible triangulation schemes should be adopted it is essential to inspect the chain of triangles with a view to ascertaining which is the strongest geometric figure, that is, which one will give the calculated length of the final line with the least error due to the shape of the triangles. An estimate of the uncertainty in the computed side of a tri- angle is given by its probable error as found by the method of least squares. The square of the probable error (p) of a triangle side as computed through a chain of triangles is given by the equation f = 3 STRENGTH OF FIGURE 7 < in which d is the probable error of an observed direction, Nd is the number of directions observed, N e is* the number of geometric conditions that must be satisfied in the figure, and 8 A and d B are the differences in the log sines corresponding to a difference of i" in the angles A and B, A being opposite the known side and B opposite the computed side. A and B are known as the distance angles. The 2} indicates that the quantity in brackets is to be computed for each triangle in the chain and the sum of these numbers used in the formula. The factor ^ - depends upon the kind of figure chosen and the factor 2 [&A Z + &A&B + 5 B 2 ] depends upon the shape of the triangles of which the figure is composed; hence the product of the two is a measure of the strength of figure and is independent of the precision with which the angles themselves are measured. The strength R of any figure is therefore given by the equation * = ^ The smaller the value of this product the more favorable the geometric conditions, and the stronger the figure. If the value of this product be computed for every possible route through the triangulation system, there will result a mini- mum value (RI) for the best chain of triangles, a second best value C# 2 ), and a third and fourth, and so on. It will be found that the chain of triangles having the greatest influence in fixing the length of the final line is that corresponding to RI, or the best chain. The second-best chain will have some influence, and the third and fourth correspondingly less. Hence, in choosing be- tween two or more possible systems of triangulation which join a given base with some specified line, that route having the smallest RI is to be preferred, unless RI proves to be nearly the same for the different routes, in which case that chain having the smallest Rz would be chosen. As an example of the way in which the preceding method would 8 GEODESY AND GEODETIC SURVEYING - TRIANGULATION be applied, take the case of the quadrilateral shown in Fig. 3. Assuming the base AB to be already fixed in direction, the point C is then determined by observing the new directions AC and BC. D is fixed by the directions AD and BD. In addition to these four the directions CB, CA, CD, DC, DB, DA are all ob- served. This gives 10 observed directions as the value of N& FIG. 3. In determining the number of geometric conditions it is seen that there are four triangles, and that in each triangle the sum of the three angles must equal a fixed amount, 180 + the spherical excess of that triangle. It will be found, however, that if any three of these triangles are made to fulfill these conditions, the fourth will necessarily do so, and hence is not really independent; in other words, there are but three conditions dependent upon the closure of the triangles. In addition to these three angle conditions there is also a distance check; that is, the angles must be so related that the computed length of side CD is the same, no matter which pair of triangles is used in making the computation. The angles of the triangle may in each case add up to the correct amount, and yet the figure will not be a perfect quadrilateral unless this last condition is fulfilled. There are then, in all, four geometric conditions existing among the angles (N c = 4). Therefore the factor for the completed quadrilateral is N d - N c _ joj^j. = N d io = NUMBER OF CONDITIONS IN A FIGURE 9 In the triangle ADB the distance angles for computing the diagonal are DAB and ADB, that is, 71 and 71. The difference for i" for 71 is 0.72 in units of the 6th decimal place. The quantity in brackets in the formula is therefore (0.52 +0.52 + 0.52) = 1.56, or 2 to the nearest unit. In Table I these numbers are given for all combinations of angles which will occur in prac- tice, so that this factor may be found at once by entering the table with the two distance angles. For the triangle BDC the distance angles for computing the side DC are 93. and 38, the tabular number being 7. For this chain of triangles, then, RI = 0.6 X (2 -f 7) = 5.4. For triangle BA C the angles are 76 and 62, and the number equals 2. For triangle DC A the angles are 120 and 29, and the number equals n. Therefore R 2 = 0.6 X 13 = 7.8. If we compute CD through the triangles ACB and DCB, we find RZ = 15.6. Using triangles DBA and DC A, Ri = 30.6. In comparing the strength of this quadrilateral with that of any other figure, reliance would be placed mainly upon RI = 5. 4 and partly upon R 2 = 7.8. Following are the values of factor - d - for several figures lid frequently used in triangulation: single triangle, 0.75; quad- rilateral, 0.60; quadrilateral with one station on fixed line not occupied, 0.75; quadrilateral with one station not on fixed line not occupied, 0.71; triangle with interior station, 0.60; tri- angle with interior station, one station on fixed line not occupied, 0.75; triangle with interior station, one station not on fixed line not occupied, 0.71; four-sided figure with interior station, 0.64; five-sided figure with interior station, 0.67; six-sided figure with interior station, 0.68. (For additional cases see General In- structions for the Field Work of the Coast and Geodetic Survey, 1908; or Special Publication No. 26.) 9. Number of Conditions in a Figure. In determining the number of conditions in any figure it is well to proceed by plotting the figure point by point, and to write down the conditions as they arise, but it will be of assistance to 10 GEODESY AND GEODETIC SURVEYING TRIANGULATION have a check on the results obtained by this process. If n rep- resents the total number of angles measured, and 5 the number of stations, then, since it requires two angles to locate a third point from the base line, two more to locate a fourth point from any two of these three points, and so on, the number of angles required is 2 (s 2); and since each additional angle gives rise to a condition, the number of conditions will equal the number of superfluous angles, or N c = n 2 (s 2) = n 25 + 4. For example, in a quadrilateral in which one station is unoccupied there are six angles measured, and #,. = 6 8+4 = 2. The number of conditions may also be found from the equation N c = 2 / - k ~ 3 s + S u + 4, where / = the total number of lines, /i = the number of lines sighted in one direction only, s = the total number of stations, and s u = the number of unoccupied stations. In the preceding example this equation becomes #0 = 12-3-12 + 1+4 = 2. 10. Allowable limits of R^ and R Q . In the primary triangulation of the United States Coast and Geodetic Survey, the extreme limits for R and R^ between base nets are 25 and 80, respectively. These are reduced to 15 and 20 if this does not increase the cost over 25 per cent. For secondary triangulation the limits for RI and R% are 50 and 150; these are reduced to 25 and 80 if the cost is not more than 25 per cent greater. For tertiary triangulation the 50 and 150 limit may be exceeded if it appears necessary. As stated in Art. 6, when R t has accumulated to 130 between bases, a new base line should be introduced as a check on the accuracy of the calculated lengths. If the character of the country is such that a base cannot be located at this point, RI may be increased to 200 if necessary. CALCULATION OF HEIGHT OF OBSERVING TOWER II 1 1 . Reconnoissance. The work of planning the system is in many respects the most important part of the project and demands much experience and skill. Upon the proper selection of stations will depend very largely the accuracy of the result, as well as the cost of the work. No amount of care in the subsequent field-work will fully com- pensate for the adoption of an inferior scheme of triangulation. Three points in particular will have to be kept in mind in planning a survey: (i) the "strength" of the figures adopted; (2) the dis- tribution of the points with reference to the requirements of the subsequent detail surveys; and (3) the cost of the work. In de- ciding which stations to adopt it is desirable to make a prelimi- nary examination of all available data, such as maps and known elevations. If no map of the region exists, a sketch map must be made as the reconnoissance proceeds. While much information may be obtained from such maps as are available, the final de- cision regarding the adoption of points must rest upon an exami- nation made in the field. All lines should be tested to see if the two stations are intervisible. This may be done by means of field glasses and heliotrope signals. In cases where the points are not intervisible, owing to 'intervening hills or to the curvature of the earth's surface, it will be necessary to determine approxi- mately, by means of vertical angles or by the barometer, the elevation of the proposed stations and of as many intermediate points as may be required, and then to calculate the height to which towers will have to be built in order to render the proposed stations visible. If the height of the towers is such as to make the cost prohibitive, the line must be abandoned and another scheme of triangles substituted. 12. Calculation of Height of Observing Tower. After determining the elevations of the stations and the inter- vening hills along a line, as well as the distances between them, the height of the tower required may be found by the following method: The curvature of the earth's surface causes all points to appear lower than they actually are. A hill appearing to be 12 GEODESY AND GEODETIC SURVEYING TRIANGULATION exactly on the level of the observer's eye is in reality higher above sea-level than the observer. The light coming from the hill to the observer's eye does not, however, travel in a straight line, but is bent, or refracted, by the atmosphere into a curve which is concave downward and is approximately circular. The result is that the object appears higher than it would if there were n@ refraction. The amount of the apparent change in height due to refraction is found to be only about one-seventh part of the apparent depression due to curvature. Since these two correc- tions always have opposite signs and have a nearly fixed relation to each other, it is sufficient in prac- tice to calculate the correction to the difference in height due to both cur- vature and refraction, and to treat the combined correction as though it were due to curvature alone, since the curvature correction, being the larger, always determines which way the total correction shall be applied. In Fig. 4, A is the position of the observer,' looking in a horizontal di- rection toward point B. BC is the amount by which B appears lower than it really is, since A and C are both at the same eleva- tion (sea-level). By geometry, BC : AB = AB : BD ft or BC = BD Since BC is small compared with BD, the percentage error is small if we call AB = AC and BD = the diameter of the earth, whence (dist.) 2 FIG. 4. BC diameter (approx.). The light from B f (Fig. 5) follows the dotted curved path which is tangent to the sight line at A. The observer therefore sees B f CALCULATION OF HEIGHT OF OBSERVING TOWER at B. In order to find the relation of BB' to BC it is convenient to employ w, the coefficient of refraction, which is denned as the number by which the central angle AOB must be multiplied in order to obtain the angle BAB'-, therefore angle of refraction = 2 X m X BAC. Since these angles are small, distances BB' and BC are nearly proportional to the angles themselves, hence BB' : BC = BAB' : BAC and BB' = 2 m X BC. The net correction (B'C = h) is the difference between the two, that is h = BC - BB' __(dist.) 2 diam. _ (dist.) 2 diam. FIG. 5. 2 m (dist.) 2 diam. (i - 2m). The mean value of m is found to be about 0.070. Substituting this, and the value for the earth's diameter, and reducing h to feet, we have h (in feet) = K 2 (in miles) X 0.574, or K (in miles) = Vh (in ft.) X 1.32, in which K is the distance in miles. Values of h and K for distances up to 60 miles will be found in Table II. As an example of how this formula is applied, suppose it is de- sired to sight from A to B (Fig. 6), and that a hill C obstructs the line. At A draw a horizontal line AD and also a curve AE parallel to sea-level. The distance from the tangent to the dotted K 2 curve at C is , which for 46 miles is 1411.9 ft. Similarly, K 2 - at B, = 4708.0 ft. But since the ray of light from B to A 14 GEODESY AND GEODETIC SURVEYING TRIANGULATION is curved, B is seen at B f , or 659.2 ft. nearer to the tangent AD\ similarly, C appears to be 197.7 ft. nearer the tangent line. Therefore, in deciding the question of visibility we may compute the combined correction and say at once that the curve at C is D ~38, FIG. 6. 1214.2 ft. below AD, and at B is 4048.8 ft.* below AD. Adding 2300 ft. (the elevation of A) to each of these values of h, we obtain the (vertical) distances from the tangent line down to sea-level, namely 3514.2 ft. and 6348.8 ft. at C and B } respectively. Sub- 84 miles B' FIG. 7. tracting the elevations of C and B, we obtain 2464.2 ft. and 4548.8 ft. as the distances of points C and D below the tangent line AD. The three points are now referred to a straight line (the tangent), and the question of visibility is determined at once by similar * Since the table extends only to 60 miles, the value of h is first found for half the distance (42 mi.), and the result multiplied by 4. CALCULATION OF HEIGHT OF OBSERVING TOWER 15 triangles. In Fig. 7 it will be seen that the straight line from B f to A is || X 4548.8 = 2491.0 ft. below the tangent (opposite C), and consequently is 26.8 ft. lower than C. Twenty-seven-foot towers would therefore barely make B' visible from A . In order to avoid the atmospheric disturbances near the ground at C the* towers would really have to be carried up to a height of 40 ft. or even more. Of course the line of sight is not actually straight between A and B, as shown in the diagram; but this method of solving the problem gives the same result as though the curva- ture and refraction were dealt with separately and the sight lines all drawn curved. If it were required to find the heights of towers necessary to make it possible to sight from A across a water surface to Z>, we should proceed as follows: Suppose the elevation of A above the water surface is 20 ft. and that of D is 10 ft. From A we may draw a line tangent to the water-level at T (Fig. 8). Knowing FIG. 8. the height of A, we may find the distance AT from Table II. Subtracting this distance from AD, we find the distance TD. From this latter distance we may compute the height of the tan- gent line above the surface at D, and, finally, knowing the height of Z>, we find the distance of D below the tangent line. Now that the points are referred to a straight line, we have at once the height of tower required on D alone. If the two towers are to be of equal height, we may estimate the required height closely and then verify the result by a second computation, add- ing the assumed height of the tower to the elevation of A . If it is desired to keep the line of sight at least 10 ft. above the surface at every point in order to avoid errors due to excessive refraction, we may draw a parallel curve 10 ft. above the water surface and solve the problem as before. The difference in radii 16 GEODESY AND GEODETIC SURVEYING TRIANGULATION of the two curves will not have an appreciable effect on the com- puted values of h and K. 13. Method of Marking Stations. The importance of permanently marking a trigonometric sta- tion and connecting it with other reference marks cannot be easily overestimated, since by this means we may avoid the costly work of reproducing triangulation points which have been lost. When the station is on ledge, the point is best marked by making a fairly deep drill hole and setting a copper bolt into it. A triangle is chiseled around the hole as an aid in identifying the point. Other drill and chisel marks should be made in the vicinity, and their distances and directions from the center mark determined; these will serve as an aid in recovering the position of the center mark- in case it is lost. If the station is on gravel or other soft material, the station mark on the surface is usually a stone or concrete post, set deep enough to be unaffected by frost action and having a drill hole or other distinguishing mark on top. There is usually also a sub- surface mark, such as a second stone post, a bottle or a circular piece of earthenware, placed some distance below the surface mark, to preserve the location in case the latter is lost. The Coast and Geodetic Survey and the United States Geological Survey use cast metal discs provided with a shaft ready to place in concrete, and bearing an inscription giving the name of the organization and other information. (See Figs, ga and gb.) The following description and sketch are given to illustrate a description of a triangulation station. Triangulation Station " Beacon Rock." The station is in the town pf , , on a hill on the property of John Smith situated on the north side of the road from Bourne to Canterbury. It is reached by a trail which leaves the road at a point about 250 meters west of Smith's house. It is about 225 meters by trail to the station. The point is marked by a one-inch copper bolt set in a drill hole in the ledge and with a triangle chiseled around it, and by witness marks as shown in the accompanying sketch. The hill is somewhat wooded to the north and west, but there is a clear view in all other directions. FIG. ga. Triangulation Station Mark. (Coast and Geodetic Survey.) FIG. gb. Reference Mark. (Coast and Geodetic Survey.) 1 8 GEODESY AND GEODETIC SURVEYING TRIANGULATION DISTANCES AND AZIMUTHS FROM CENTER Station. Azimuth. Dist. to drill hole. 71. 3m. 41.0 m. 21.47 m. 101. 2 m. 78.3401. Holder 21 50' Bear Hill 121 16' Witness Mark Dayton . o / Witness Mark 283 05' Sheen Id. . . ^2< A.O' FIG. 10. Sketch of Triangulation Station. 14. Signals, Tripods. In order that the exact position of the station may be visible to the observer when measuring the angles, a signal of some sort is erected over the station. For comparatively short lines, less than about 15 miles, the tripod signal is often sufficient. (See Fig. u.) It is not expensive to build, saves the cost of a man to HELIOTROPES 19 attend signal lights (as is necessary with heliotropes or acetylene lights), and permits setting the instrument over the station with- out removing the signal. It usually consists of a mast of 4" x 4" spruce, with legs of about the same size. Three horizontal braces of smaller dimensions (2" X 3") tie the mast to the legs, and three longer horizontal braces are nailed to the legs. If the FIG. ii. Tripod Signal. signal is very large, additional sets of braces may be put on, to give greater stiffness. The size of the mast may be increased by nailing on one-inch boards, giving a mast 6" X 6". 15. Heliotropes. When sighting over longer lines it is necessary to use heliotrope signals if observing by day, and acetylene lights if observing by night. The heliotrope is simply a plane mirror with some device for pointing it so that reflected sunlight will reach the distant 20 GEODESY AND GEODETIC SURVEYING TRIANGULATION station. The two more common heliotropes are (i) the one in which the light is directed through two circular rings of slightly different diameters (Fig. 12), and (2) that known as the Steinheil heliotrope (Fig. 13). The ring heliotrope consists essentially of two circular metal rings, of slightly different diameters, mounted on a frame, and a mirror mounted in line with the two rings in such a manner that it can be moved about two axes at right angles to each other. For convenience in observing distant stations these two rings and the mirror are often mounted on the barrel of a telescope. The FIG. 12. Heliotrope. rings should be so mounted that the line between the centers of the rings may be adjusted parallel to the line of sight of the tele- scope. In using the heliotrope the axis of the rings is pointed by means of threads which mark the center of the openings, or by means of the telescope itself after the axis of rings and the line of sight of the telescope have been made parallel. Since the sun's apparent diameter is about o 32', the angle of the cone of rays reflected from the mirror is also o 32'. It is not necessary, therefore, to point the beam of light with great precision. If the central ray is nearly a quarter of a degree to one side of the station, there will still be some light visible to the observer at the distant station. On account of the rapidity of the sun's motion it is necessary to reset the heliotrope mirror at intervals of about one minute. HELIOTROPES 21 FIG. isa. Steinheil Heliotrope. \LL V To Station FIG. i3b. 22 GEODESY AND GEODETIC SURVEYING TRIANGULATION The Steinheil heliotrope consists of a mirror with both faces ground plane and parallel and so mounted that it can be moved about two axes at right angles to each other. One of these axes is coincident with that of a cylindrical tube which contains a small biconvex lens and a white surface (usually plaster of Paris) for reflecting light. This tube may be moved about two other axes at right angles to each other. A small circular portion of the glass in the center of the mirror is left unsilvered, so that light may pass through the glass plate down into the tube. In pointing the Steinheil heliotrope the cylindrical tube con- taining the lens must be pointed toward the sun, so that the light which passes through the hole in the mirror will pass through the lens, and, after reflection from the plaster surface, will again pass through the lens to the back surface of the mirror, there to be partly reflected and partly transmitted through the glass. Keeping the tube in this position, the mirror itself must be so turned that the spot of light made visible by this last reflection will appear to cover the hill or station to which the light is to be Sent. One form of heliotrope, in use by the Coast Survey, called a box heliotrope, consists of a pair of rings with a mirror mounted behind them, and with sights above the rings for pointing. A telescope is mounted to one side of and parallel to the heliotrope. The various parts remain in position in the box when in use. (Fig. 14.) The size of mirror used in any heliotrope must be regulated according to the length of line and the atmospheric conditions. Most heliotropes are provided with some arrangement for varying the size of the opening through which the light passes. If the exposed portion of the mirror subtends an angle of about o. /7 2 the amount of light will be sufficient for average conditions. This is equivalent to making the diameter of the opening about one-half inch for each ten miles. Different atmospheric condi- tions will require different openings. All heliotropes are provided with a second mirror, usually ACETYLENE LIGHTS 23 larger than the first, called the back mirror; this is to be used whenever the angle between the sun and the station is too great to permit sending the ray by a sing e reflection. The back mirror is set so as to throw light onto the first mirror and the heliotrope is then adjusted to the reflection of the sun as it appears in the back mirror. FIG. 14. Box Heliotrope. (Coast and Geodetic Survey.) 1 6. Acetylene Lights. In the triangulation along the ninety-eighth meridian, in 1902, the Coast and Geodetic Survey experimented with acetylene lights for triangulation at night. These experiments were suc- cessful, and, owing to the fact that the work could usually pro- ceed regardless of clouds, the use of lights resulted in greater 24 GEODESY AND GEODETIC SURVEYING TRIANGULATION economy than observations by daylight. The lamps used at first were ordinary acetylene bicycle lamps remodeled in the instrument division of the Survey. The front door of the lamp was removed and the ordinary lens replaced by a pair of condens- ing lenses 5 inches in diameter. When in use the lamp is secured to the platform by means of a screw, and may be moved both in FIG. 15. Acetylene Signal Lamp. (Coast and Geodetic Survey.) altitude and in azimuth. A small tube is fastened to the top of the lamp for pointing it toward the observer's station. The entire outfit, including a 5-lb. can of carbide, weighs but 2i| Ibs. (See Coast and Geodetic Survey Report for 1903, p. 824.) The recent practice of the Survey is to use automobile lamps in place of the bicycle lamps. (Fig. 15.) TOWERS 25 17. Towers. Where a line is obstructed by hills or woods, or where the curvature of the earth is sufficient to make the station invisible, it becomes necessary to construct towers. If there is much heavy timber about the station, placing the instrument station FIG. 16. Eighty-foot Tower. (Coast and Geodetic Survey.) on the ground may necessitate so much cutting that it will be more economical to construct a tower than to cut the timber. The form of tower now used by the United States Coast Survey is very light and slender as compared with the older ones. This kind of tower (Fig. 16) admits of more rapid construction and 26 GEODESY AND GEODETIC SURVEYING- TRIANGULATION can be built at a lower cost; it is sufficiently rigid to withstand all ordinary storms. The manner of framing the tower is shown in the cut (Fig. 17). When the ties are nailed on, the legs are sprung slightly into the form of a bow, thus giving additional stiffness to the structure. One side of the inner tripod, which is to support the instrument, is first framed on the ground. This side and the third leg of the tripod are raised into position by a fall and tackle and a derrick, which may be a tree or a section of one of the legs of the outer scaffold. The derrick should be at least two- thirds the height of the piece to be raised. After the tripod is raised and all braces nailed on, it is itself used as a derrick for hoisting the two opposite frames of the outer scaffold into position. The ties and braces of the other two sides are then nailed in place. It should be observed that the inner and outer structures are entirely separate, so that the movement of the observer on the platform of the scaffold will not disturb the instrument. The legs of the tripod and the scaffold are anchored by nailing them to foot pieces set underground. The outer tower is guyed with wire as a protec- tion against collapse in high winds. This kind of signal saves lumber, transportation, and cost of construction; it has a small area exposed to the action of the wind; the short ties have the effect of reducing the vibration due to wind, which is troublesome in large towers; the light keeper is placed above the observer (10 ft. or so) and can operate his lights without interfering with the observations. Another ad- vantage of these towers is that the amount of twisting due to the sun's heating is found to be exceedingly small. (For further details consult Coast Survey Report for 1903, p. 829.) The United States Lake Survey now uses a tower constructed entirely of gas pipe, which has proved to be more economical than timber. It is put together in sections and hoisted as it is built. The upper part of the structure is built first and is then hoisted from the ground by means of tackles ; the next section is then added on, all the work being done from the ground. This TOWERS *** t 0' Stand for lights 3' Top floor SIDE OF 60 FT. SCAFFOLD SIDE OF 60 FT. TRIPOD 1818 22190 PLAN OP SCAFFOLD AND TPIPOD Scale of Feet II 20 FIG. 17. Framing plan of 6o-ft. Tower. 28 GEODESY AND GEODETIC SURVEYING TRIANGULATION kind of tower is easy to construct, and the material is portable; the area exposed to wind is very small. Figs. 1 8 and 19 illustrate small towers built of green poles cut near the station. These towers were erected to enable the observer to see over the dense growth of timber. In the tower shown in Fig. 19 standing trees, stripped of their branch.es, were utilized for two of the legs of the outer scaffold. 18. Reconnoissance for Base Line. With the Invar tape apparatus, to be described in Chapter II, base lines may now be measured over much rougher ground than was formerly possible, when bar apparatus was used; still it is advantageous to have the base line located in as smooth and level country as possible, provided this does not require weak triangula- tion to connect the base with the main scheme of triangles. The network of triangles required in making this con- nection should be selected with the same care and according to the same principles as was described for primary triangulation. In some cases it is found practicable to use the side of a primary triangle for the base line. For example, in the triangulation extending from Texas to California the Stanton base, which is one of the primary lines (8 miles in length), was measured directly with the tape apparatus. FIG. 1 8. Twenty-five-foot Tower built of Green Poles. RECONNOISSANCE FOR BASE LINE FIG. 19. Forty-foot Tower built on trees in place. 30 GEODESY AND GEODETIC SURVEYING TRIANGULATION PROBLEMS, Problem i. What is the strength of the quadrilateral having all the angles equal to 45 ? In case one station on the base is not occupied with the instrument, what is the strength ? If one station not on the base is unoccupied, what is the strength ? Problem 2. Compare the strength of the three figures given in Fig. iga. FIG. Problem 3. Three hills A,B, and C are in a straight line. The distance from A to B is 10 miles and the distance from B to C is 15 miles. The elevations are A 600 ft., B = 550 ft., and C = 650 ft. respectively. Compute the height of a tower to "be built on C the top of which will just be visible from A. Problem 4. Four hills A, B, C, and D are in a straight line. The elevations are A = 810 ft., B = 775 ft., C = 1030 ft, D = 1300 ft. respectively. The distances of B, C, and D from A are 8 miles, 28 miles, and 38 miles. Find the height of towers on A and D to sight over B and C with a lo-ft. clearance. The two towers are to be of the same height. Problem 5. What angle is subtended by a six-inch mast at a distance of twelve miles? SL"' Problem 6. If a fourteen-inch mirror is used on a heliotrope at a distance of 150 miles, what is the apparent angular diameter of the light? oV"J CHAPTER II BASE LINES 19. Bar Apparatus for Measuring Bases. In nearly all the earlier base-line measurements (up to about 1885) the apparatus employed consisted of some arrangement of metal bars. Such apparatus was capable of yielding accurate results, but was cumbersome to use; consequently the base-line work was a comparatively expensive part of the survey. An account of the development of base-measuring apparatus will be found in Clarke's Geodesy and in Jordan's Vermessungskunde, Vol. Ill; descriptions of numerous forms used in this country will be found in the reports of the superintendent of the Coast and Geodetic Survey. 20. Steel Tapes. Experiments with the use of steel tapes for base-line measure- ments were made by Jaderin at Stockholm in 1885, by the Missouri River Commission in 1886, and by Woodward on the Coast and Geodetic Survey base at Holton, Indiana, in 1891. The use of steel tapes for this purpose was attended with such success that for twenty years they were very generally used, and by 1900 they had almost wholly superseded the bar apparatus in this country. The greatest practical difficulty encountered in the use of steel tapes for precise measurement is that of determinmg the true temperature of the steel when making the measurements in sun- light. The air temperature, as indicated by ordinary mercurial thermometers, is seldom the correct temperature for the tape, except during rainy weather or at night. For this reason it was found necessary to make all measurements of base-lines at night in order to secure the required accuracy. 31 BASE LINES 21. Invar Tapes. In 1906 the Coast Survey made a series of tests on six primary base-lines, using the ordinary steel tapes and also several new 50- meter tapes made of an alloy of nickel and steel called invar. This alloy was discovered by C. E. Guillaume, of the Interna- FIG. 20. Invar Tape on Reel. tional Bureau of Weights and Measures, Paris. The tapes were made by J. H. Agar-Baugh, of London. The alloy mentioned has a very low coefficient of expansion, roughly one-twenty-fifth that of steel,* and consequently has a great advantage over steel * The coefficient of steel is about o.oooon, that of invar is about 0.0000004, f r ACCURACY REQUIRED 33 for base-line measurement. The metal is more easily bent than steel, but with proper care in handling the tapes, and with the use of fairly large reels, there is little difficulty in making the measurements and in securing the required accuracy. The re- sults of these tests on the invar tapes may be summed up as follows: Measurements with invar tapes may be made during daylight with all the accuracy demanded in base-line work, whereas measurements with steel tapes must be made at night in order to secure the required accuracy. In working by daylight the errors of observation are smaller and the party can make greater speed than when working at night. On account of the small temperature coefficient of the invar tape any error due to the failure of the thermometers to indicate the true temperature of the tape has much less effect on the com- puted length when the measurements are made with invar than when they are made with steel. Since it is not necessary to standardize the invar tape in the field, as was always done with the steel tape, the cost of measure- ments made with the invar is materially less than that of measure- ments made with steel. The superiority of these tapes has been demonstrated by re- peated trials, and they are now used almost exclusively by the Coast Survey in making base measurements. 22. Accuracy Required. It is found that there is little, if any, advantage in measuring a base-line with a precision greater than one part in 500,000, since to do this would give the base-line a greater precision than could be maintained in the angle measurements. There is little difficulty, however, in obtaining a higher precision; the bases measured by the Coast Survey in 1906 and 1909 show a precision of one part in 2,000,000 or better. It is customary to divide bases into sections of about a kilometer in length, and to measure each section twice. If the two results show a discrepancy greater 34 BASE LINES than 2o wm VK (K being the number of kilometers in the section), the measurements are repeated until they do agree within this limit; if the first two results agree within this limit, no additional measurements are taken. This procedure is consistent with the requirement that the base be measured with a precision of at least i in 500,000, but that no attempt be made to increase the precision much beyond this limit. 23. Description of Apparatus. The invar tapes are usually about 53 meters long, with two graduations 50 meters apart. In some tapes a length of one decimeter at each end of the 5o-meter length is subdivided into millimeters for convenience in reading. Intermediate points on the tape, such as the 25 meter point, are marked by single lines. The tape is about \ inch X -^ inch in cross section and weighs about 25 grams per meter. This metal is softer than steel and has to be wound on a reel of at least 16 inches diameter in order to avoid permanent bends in the tape and consequent changes in length. (Fig. 20.) In use it is supported at the ends and usually at one intermediate point. The tension is applied by means of a spring balance reading to 25 grams, the tension ordinarily used being 15 kilograms. An apparatus used for applying the tension and similar to that used by the Coast Survey is shown in Fig. 21. The point of the iron bar holding the spring balance is pushed into the ground, and the upper end is moved right or left to align the tape. The adjustable clamp makes it possible to raise or lower the balance so as to bring the end of the tape to the right grade. The spring balance employed is a com- mercial article and is constructed to read correctly when held in a vertical position and with the weight hung on the hook. When the balance is used in a horizontal position, the true tension is greater than the indicated tension. The correction to be applied to the scale readings is found by suspending known weights on a cord passing over a pulley and secured to the hook of the balance when held in a horizontal position. The thermometers used with this apparatus are graduated to half degrees and are provided DESCRIPTION OF APPARATUS 35 FIG. 21. Tension Apparatus. 36 BASE LINES with spring clamps so that they may be readily fastened to the tape for making readings, or removed from it when it is being carried forward. 24. Marking the Terminal Points. The ends of the base line to be measured are marked in the same manner as triangulation points, that is, by bolts set in drill holes in stone monuments or by special castings set in concrete; the points are tied in by several measurements to prevent the position being lost. There is usually also a sub-surface mark (see Art. 13). Intermediate points on the line are often marked by stone or Qoncrete posts. 25. Preparation for the Measurement. The first step in measuring the base is to run the line out roughly with transit and tape and clear the ground from obstruc- tions; at the same time the measuring stakes are set in position. These may be 4" X 4" stakes set exactly one tape-length apart and high enough so that the tape is everywhere clear of the ground. On top of each stake is placed a strip of copper or zinc upon which is scratched the reference marks used in making the measurements. Next, the slope of each tape-length is deter- mined by taking level readings on the tops of all the stakes. The intermediate stakes (one or three in number) are set in line, and nails for supporting the tape are placed at the proper grade. 26. Measuring the Base. t The actual measurement is begun by stretching the tape over the first pair of stakes; the zero end of the tape is placed over the end mark of the base, either by means of a transit set at one side of the line or by a special device called a cut-ojff cylinder. The tape is aligned by means of field glasses or by a transit set on line and the tension is then applied. When the zero graduation of the tape is exactly over the end mark and the tension is correct, the position of the forward (50 meter) end is marked on the metal strip, and the temperature is read on all the thermometers. The tape is then carried forward and the process repeated until the measurement of the section is completed. If there is a short CORRECTIONS TO BASE-LINE MEASUREMENTS 37 measurement at the end of, the line, this may be taken with an ordinary metric steel tape graduated its whole length. When- ever it is necessary to set forward or backward on one of the metal strips in order to bring the reference mark on the milli- meter scale, this fact is recorded; it is also indicated on the metal strips, which are all preserved as a part of the permanent record. Measurements of bases made in this manner can be made at the rate of about 2 kilometers per hour. If the wind is blowing, it may be found necessary to use three intermediate supports in order to maintain the required standard of accuracy. If the first two measurements of any one section of the base show a discrepancy not exceeding 2O mm X VK, the mean is considered as sufficiently accurate and no further measurements of this section are made. 27. Corrections to Base-Line Measurements. Correction for Grade. Where the slope is determined by direct leveling, the most convenient formula for computing the horizontal distance is one B d FIG. 22. involving the difference in elevation of the ends of the tape. In Fig. 22, let h be the difference in elevation of the end points A and B, and let / be the length and d the required horizontal distance. Then Corr. for grade = C g = I d = I - 38 BASE LINES But l- !-_--- - * - /(i - - . . . ) Therefore C. If , h 1 -7i + 8?+r--; [I] If the slope has been found in terms of the vertical angle a, the correction may be computed by the expression C g = 2 I sin 2 J a = I vers a. [2] In good base-line work the errors in length due to errors in deter- mining the grade should never exceed one part in one million. 28. Corrections for Alignment. The errors in aligning a straight base-line can easily be kept so small as to be negligible. If any point is found, however, to be out of line by an amount sufficient to affect the length, the cor- rection may be computed by Formula [i], 29. Broken Base. Sometimes it is desirable or necessary to break a base into two parts which make a small (deflection) angle with each other. If the two sections are measured with the usual precision, and if the angle also is accurately measured, the length may be computed c FIG. 23. as follows: let a and b, -Fig. 23, be the measured lengths, and 6 the angle between them, and let c be the desired base, then from the triangle we have c 2 = a 2 + b 2 + 2 ab cos 0. 2 Putting for cos 6 the series i h , there results 2 c 2 = ( a + b) 2 - abd 2 . CORRECTION FOR ABSOLUTE LENGTH 39 Placing the factor (a + b) 2 outside the brackets and extracting the square root, abe 2 "1* '**'* -**] or c = a + b ~ rr (sin i') 2 , 2 (a + o) where is in minutes of arc. Substituting the value of sin i', , 060 2 c = a -f- 6 0.000,000,042,308 - [3] (log. 0.000,000,042,308 = 2.62642 10). 30. Correction for Temperature. The temperature correction may be computed if we know the coefficient of expansion, the actual temperature of the tape and the standard temperature, and the measured length of line. If k is the coefficient, t the observed temperature, /o the standard temperature, and L the measured length, then Temperature correction = +kL (t fo). [4] The temperature correction is often expressed as a term in the tape equation, as shown in the following article. 31. Correction for Absolute Length. The length of the tape is usually expressed in the form of an equation, such as ^5i6 = 5o m + (12.382""* 0.016""*) -f (0.0178""* it 0.0007""*) (* - 2 5-8 C.), [5] meaning that tape number 516 is 12.382""* more than 50"* long at a temperature of 25. 8 C., and that 0.016""* is the uncertainty of this determination. The quantity 0.0178 is the temperature change for i for a 50** length, and 0.0007 is the uncertainty in this number. (The temperature coefficient for this tape is 0.000,000,356.) BASE LINES According to the present practice, tapes are standardized at Washington * under exactly the same conditions, in regard to tension, temperature determination, and manner of support, as those which are to govern the field measurements. By this means all uncertainty in the absolute length and in the tension correction is kept within narrow limits. 32. Reduction of Base to Sea-Level. In order that all triangulation lines may be referred to the same surface it is customary to employ the length of the line at sea-level between the verticals through the stations. In Fig. 24, let B represent the measured base at elevation h above sea-level (supposed spherical) , and b the length of base reduced to sea-level, R a being the radius of curvature of the surface (see Art. 97 and Table XI). Then, since the arcs are proportional to their radii, b_ = Ra ~D ~D I / JJ J\ a -J- fl and Therefore the reduction to sea-level is [6] If there is a great difference in elevation in different parts of the base, the line should be divided into sections and the mean value of h found for each section. Then B in the formula is taken as * United States Bureau of Standards, Washington, D. C. t See footnote on page 51. CORRECTION FOR SAG 41 the length of the section in question. The logarithm of the mean radius of curvature in latitude 45, which may be used for short sections, is 6.80470. Question. Is it necessary to reduce each triangulation line separately to sea- level? 33. Correction for Sag. Between any two consecutive points of support the tape hangs in a curve known as the catenary, its form depending upon the weight of the tape, the tension applied, and the distance between the points of support. In Fig. 25 let / be the horizontal distance between the supports, the two being supposed at the same level; let n be the number of FIG. 25. such spans in the tape-length, t the tension, and w the weight of a piece of tape of unit length. Also let v equal the (vertical) sag of the middle point of the tape below the points of support. Since the curve is really quite flat under the tension actually employed in field-work, the length of the catenary will be sen- sibly equal to that of a parabola whose axis is vertical and which passes through the points A, B, and C. The equation of this p parabola is x 2 = y, and the length of curve, found by the 4V usual method of the calculus, is 2 s = I -\ + . The difference 2 s I between the length of curve AB and the chord AB is approximately / 8 * r i 25 / = - X W 42 BASE LINES If we consider the forces acting on the tape at the point C, and take moments about the point of support A, we have wl I X - = v t. 2 4 Therefore v = ~ [b] o t Substituting in [a] the value of v found in [b], we find that the shortening of this section of tape due to sag is 8 /wP\* I wl\ 2 For n sections, we have nl = L, whence Correction for sag = C 8 = ( ) [7] 24 \ * / 34. Tension. The modulus of elasticity of the tape due to the tension applied equals the stress divided by the strain. If a = the elongation and L the length, and if / equals the tension and S the area of the cross section, then the modulus of elasticity E is given by LL Sa The elongation is -C,-||, . [8] where C t is the correction for the increase in length due to tension. Evidently the difference in length due to a change from tension to to tension t is a = (t t ). oA The value of E must be found by trial, applying known ten- sions and observing a directly. TENSION 43 To allow for slight variations in tension, such as those due to the failure of the spring balance to give the desired reading the instant the scale of the tape is read, the correction may be derived as follows: Since the effective length of the tape depends both upon the elongation due to tension and upon the shortening due to sag, and since these both involve /, the variation may be found by differentiating the expression Li = L + C t - C. Lil_L /^V 2 SE 24 U / ' regarding t as the independent variable. The differentiation gives This is the correction due to small variations in /. This quantity may be found satisfactorily by actual tests, varying / by known amounts and observing the change in length directly. It was once the practice to compare the tape with the standard when it was supported its entire length, and to calculate the sag and tension corrections to obtain the effective length when sup- ported at a few points. The present practice of comparing the tape under the same conditions that are to exist in the field- work eliminates all uncertainty in these computed corrections. PROBLEMS Problem i. Derive the equation of the parabola stated in Art. 33. Compute the length of the parabola between the points of support A and B. Problem 2. The difference in elevation of the ends of a 5o-meter tape is 7.22 ft., obtained by leveling. What is the horizontal distance? Problem 3. A base line is broken into two sections which meet at an angle of i 59' 3i".6. The lengths of the two segments are 1854.275 meters and 3940.740 meters. What is the distance between the terminal points? Problem 4. The length of a base line is i7486 m .58oo measured at an altitude of 34.16 meters. The latitude of the middle point of the base is 38 36'. The azi- muth of the base is 16 54'. What is the corresponding length of the base at sea- level? CHAPTER III FIELD-WORK OF TRIANGULATION MEASUREMENT OF HORIZONTAL ANGLES 35. Instruments Used in Measuring Horizontal Angles. Instruments intended for triangulation work are of two kinds: the direction instrument, first designed in England by Ramsden in 1787, and the repeating instrument, first used in France about 1790. The former is the one chiefly used at the present time for primary triangulation; the repeating instrument, on account of its comparative lightness and simplicity, is much used on triangulation of lesser importance. Triangulation instruments are larger than ordinary surveying transits, the diameter of the circles varying in different instru- ments from 8 to 30 inches. It is found, however, that small circles can be graduated so accurately that little or nothing is gained by using circles more than from 10 to 1 2 inches in diameter. Furthermore, the smaller circles are less affected by flexure than the larger circles. All triangulation instruments except the very smallest are built with three leveling screws and are used on solid supports, like stone piers, or on the tripods of observing towers. Small instruments intended for work of a lower grade of accuracy may be used on their own tripods. 36. The Repeating Instrument. The repeating instrument has an upper and a lower plate arranged exactly as in the surveyor's transit, and the graduated circle is read by two or more verniers graduated to 10" or to 5". Verniers reading finer than 5" are not practicable, and depend- ence must be placed upon the repetition principle for securing greater precision. Fig. 26 shows a repeating instrument having an 8-inch circle which is read by two verniers to 10 seconds. The THE REPEATING INSTRUMENT 45 FIG. 26. Repeating Instrument. (C. L. Berger & Sons.) 46 FIELD-WORK OF TRIANGULATION telescope of this instrument has an aperture of if inches and a magnifying power of 30. Since an instrument of this kind is likely to be used in sighting on pole signals, the cross-hairs are usually arranged in the form of an X, the pole bisecting the angle between the hairs when the pointing is made. Single vertical hairs would not be practicable except on short lines and wide signals, as the width of the ordinary hair is so great that it completely obscures the pole on long distances. 37. The Direction Instrument. The direction instrument has but one horizontal circle, read by two or more microscopes instead of verniers. The circle can be turned about the axis and clamped in any desired position. The motion of the telescope and the microscopes is entirely in- dependent of the motion of the circle; the latter can be shifted while the upper part of the instrument (called the alidade) re- mains clamped. It is evident that a repeater could be used as a direction instrument, but that a direction instrument could not be used for measuring angles by the repetition method- Fig. 27 shows a i2-inch theodolite with microscopes reading to seconds. The circle of the direction instrument is usually graduated into 5' spaces. The direction of the line of sight of the telescope is read by first noting the degrees and 5' spaces in a small index microscope, and then accurately measuring the fractional parts of the 5' spaces by means of the three equidistant micrometer microscopes. The micrometers can usually be read to seconds directly, and to tenths of a second by estimation. The mean of the three micrometer readings is taken as the true reading, and this is added to the reading of the index microscope to obtain the direction. The telescope of the 1 2-inch theodolite used by the Coast Sur- vey has an aperture of 2.4 inches, a focal length of 29 inches, and magnifying powers of 30, 45, and 60. The circle is graduated to 5' and reads to seconds by means of three microscopes. A camel's-hair brush (inside the cover plate) sweeps over the graduations. The base is made very heavy, and the bearing THE DIRECTION INSTRUMENT 47 FIG. 27. Twelve-inch Theodolite. (Coast and Geodetic Survey.) 4 8 FIELD-WORK OF TRIANGULATION surfaces of the centers are glass hard. The centers on this instru- ment are very long. The upper parts of the instrument are made chiefly of aluminum, in order to diminish the weight bearing upon the centers. This design produces an instrument of exceptional stability. Direction instruments are used chiefly on long lines and in connection with heliotropes or lights. For this reason the cross- hairs usually consist of two vertical hairs, set so as to subtend an angle of from 10" to 20", and two horizontal hairs, set much farther apart and used merely to limit the portion of the vertical hairs to be used in pointing. 38. The Micrometer Microscope. The construction of the micrometer microscope is shown in Fig. 28. The head of the screw is graduated into 60 divisions E-S FIG. 28. corresponding to seconds of angle. As the screw head is turned the two parallel hairs in the field of the microscope are moved in a direction parallel (tangent) to the edge of the graduated circle. The distance between these hairs is just sufficient to leave a small white space on each side of a line of graduation when it is centered between the two hairs. The pitch of the screw and the focal length of the objective of the microscope are so related that five whole turns of the screw will carry the hairs from one gradua- tion to the next. The number of whole turns of the screw may RUN OF THE MICROMETER 49 be counted on a notched scale visible in the field of view of the microscope. The fraction of a space to be measured is that lying between the zero point of the notched (comb) scale and the graduated line last passed over by the zero point. Strictly speaking, the zero point is that position of the hairs in the zero notch at which the scale on the screw head will read exactly zero. The position of the hairs for a zero reading of the screw may be adjusted by holding fast the graduated ring on the screw head and turning the milled edge screw head which moves the hairs. The microscope inverts the image of the graduated circle so that graduations increasing in the direction of azimuths will appear to increase from left to right in the field of view of the microscope. The readings on the screw head increase as the screw is turned left-handed, and the hair lines move in the direction of decreasing graduations over the circle. To measure the space between the zero of the microscope and the last line passed over, it is only necessary to turn the screw until the graduation in question bisects the space between the hairs, and then to read the comb scale and the scale on the screw head. This reading is to be added to the number of the gradu- ated line, to obtain the direction as shown by this microscope. For example, if the screw is turned two revolutions (two notches) and ten divisions in order to center the 47 05' mark between the hairs, the reading of this microscope is 47 05' + 2' 10" = 47 of 10". A complete set of readings of one direction would consist of readings of each of the three microscopes on both the preceding and the following graduations, six readings in all. 39. Run of the Micrometer. If the microscope is perfectly adjusted with respect to the graduated circle, and if the latter is perfectly plane, then five whole revolutions of the screw should carry the hairs from one line to the next, and the reading of the screw should be the same on all lines. Since this condition is rarely fulfilled, there is ordi- narily a small difference in the forward and backward readings, called the error of run of the micrometer. 50 FIELD-WORK OF TRIANGULATION The forward reading F is the reading taken when the threads are moved from the zero position (Fig. 29) to the preceding mark (25' in Fig. 29a). The back reading B is the one taken on the following (30') mark, Fig. 29b. The graduations on the screw- head decrease as the threads move from 25' to 30'. If the FIG. 29. Field of Micrometer Microscope. FIG. 2ga. Forward Reading. FIG. 290. Back Reading. micrometer screw is turned so that the threads move from its zero to the 25' mark, then the reading F is to be added directly to the circle reading. In the figure the reading is 201 25' + i' 26.2" = 201 26' 26.2". Without assuming anything in regard to the actual value of one turn of the screw, the value may be RUN OF THE MICROMETER 51 computed by dividing the angular space between graduations by the number of turns or divisions recorded in passing from one graduation to the next. If R = the value of one revolution, then R _ 300" = 300" 300 + F B 300 + r where r is the run of the micrometer in seconds (divisions) as indicated by the differences of the forward and backward read- ings, positive if F is greater than B. If the screw turns more than five times in passing from the 25' line to the 30' line, the reading B, on the 30' line, will be smaller than F, since the screw readings are decreasing. This makes F B = r positive. Hence the denominator of the above fraction is greater than 300, and the value of one turn is less than unity, as it should be ac- cording to the assumption. If F is the forward reading in any given case, it must be con- verted into arc by multiplying it by the value of one turn, since F is simply a certain number of turns and divisions, not the true number of minutes and seconds, Therefore True reading = F (- \3oo Since r is small (say 2" or 3"), it is permissible to write R = 300 and the true reading = F ( i -. . . j = F-F - (a) 300 This formula corrects the forward reading only, and assumes * By actual division =i x + x z x 3 + - . If* is small enough so that x* and the following term may be neglected, then - = i x. 52 FIELD-WORK OF TRIANGULATION that the bisection and reading are perfectly made. If the back reading is corrected in a similar manner, the result is True reading = 300" - (300" - B) (i - - The first 300" is the space between 25' and 30'. The factor (300" B) is the space between zero and the 30' mark deter- mined by the B reading. It should be remembered that when the micrometer is turned to the 30' mark, the readings are de- creasing; therefore the direct reading does not give this space, but 5 minus this space. Simplifying this expression we have True reading = B + r - B (b) 300 Since there is no reason for preferring either the forward or the backward reading, the mean is used as the best value. The mean of (a) and (b) is F + B r F + B ' r - -r - - x 2 2 2 300 p IT? If - = m, then the correction to m, the mean of the two 2 readings, is Corr. = m [10] 2 300 A general table may be computed for different values of m and r, so that no special computation is necessary when correcting a direction. It is good practice to determine r from all the F and B readings, and to employ this average value when making the corrections. 40. Vertical Collimator. In centering a signal over a station, placing a mark under a new signal, or centering the theodolite over the station mark, the Coast Survey observers sometimes employ the vertical collimator shown in Fig. 30. The instrument is adjusted by means of spirit levels, which revolve around a vertical axis Like those of a transit. ADJUSTMENTS OF THE THEODOLITE 53 A telescope may be placed in coincidence with the vertical axis of the collimator, and its line of sight adjusted to point vertically downward. With the instrument in this position the observer may obtain a point which is vertically above the center mark of the station. FIG. 30. Vertical CoIIimator. (Coast and Geodetic Survey.) 41. Adjustments of the Theodolite. The adjustment of the levels attached to the alidade is made by means of reversals about the vertical axis of the instrument, exactly as with the engineer's transit. The adjustment of the stride level is tested by placing it on the horizontal axis, reading both ends of the bubble, and then re- versing the level and reading again. The adjusting screws of the stride level should be turned so that the bubble moves half- 54 FIELD-WORK OF TRIANGULATION way back from the second position to the first. When the stride level is so adjusted that it reads the same in either position, it is in correct adjustment, and the horizontal rotation axis may then be leveled by moving the adjustable end of the axis until the bubble is in the center of its tube. Of course the two adjust- ments may be made simultaneously. If desired, the stride level may be used also to make the vertical axis truly vertical. The adjustment of the line of sight in a plane perpendicular to the horizontal axis may be made by reversals about the hori- zontal axis as in testing an engineer's transit; or it may be made by sighting an object, lifting the telescope out of its bearings, and, after reversing the axis, replacing it in the bearings. If the object is no longer in the line of sight, the reticle is brought half- way back from the second position toward the first. The test of the adjustment of the microscopes is made by measuring the run of each micrometer, taking first a forward and then a back reading. In case the run of a microm- eter is greater than about 3", it should be adjusted by changing the distance from the objective to the reticle, and then moving the whole microscope so that the graduations are again in focus. If the image of the division is greater than 5 whole turns of the screw, the objective should be moved toward the eyepiece, and then the whole microscope moved away from the circle. Moving the objective away FIG. 31. from the micrometer lines diminishes the angle be- tween the two lines of sight corresponding to the 5 turns, and reduces the size of the image of the division (Fig. 31). It will usually require a series of trials to perfect this adjust- ment. 42. Effect of Errors of Adjustment on Horizontal Angles. The effect of errors due to the inclination of the horizontal axis to the horizon, and those due to the imperfect adjustment for collimation (line of sight), are not independent of each other. These errors are usually so small, however, that it is permissible EFFECT OF ERRORS OF ADJUSTMENT 55 to compute their effect separately, as though only one existed at one tune. In Fig. 32, Z is the true zenith and Z' the point where the vertical axis of the instrument prolonged pierces the celestial sphere. 5 is a point whose altitude is h. Assuming that the horizontal axis makes an angle i with the horizon, and that all other errors are zero, then from the figure it will be seen that we may write sinZ' si n _ sin i sin Z'S ' or, with sufficient accuracy, where h is the angular altitude of the point sighted. FIG. 32. It appears, then, that for each point sighted there should be a correction to the circle reading equal to i tan h. Triangulation points are usually so nearly on the horizon, and by careful atten- tion to the leveling the error i may easily be kept so small, that there is seldom any necessity for applying the correction except for such observations as those on a circumpolar star for azimuth. In the preceding paragraph it is assumed that the vertical axis is truly vertical, the graduated circle being horizontal, while the horizontal axis is not horizontal. If the two axes are at right angles to each other, but the vertical axis is inclined to the true vertical by a small angle i, it may be shown, by a diagram similar to Fig. 32, that the same correction applies to this case also. 56 FIELD-WORK OF TRIANGULATION The error of a horizontal direction due to an error of collima- tion may be computed as follows : Let the error in the sight line be represented by c\ then, when the axis of collimation (Fig. 33) traces out the great circle ZN, the line of sight traces out the parallel circle SA , which is c seconds from ZN. If S be any point FIG. 33. toward which the cross-hair is pointing, and if arc SN be drawn perpendicular to ZN, the error in direction, or the angle at Z, is found from the equation sinZ _ sine sin N sin ZS ' or, since Z.N = 90, Z = c sec h. [12] Each direction should therefore be corrected by the quantity c sec h. On account of the small value of c in a well-adjusted instrument this correction is necessarily small; furthermore, it is usually eliminated from the final result by the method employed in making the observations. 43. Method of Measuring the Angles. In measuring angles with the repeating instrument the common practice has been to measure the angle six times, beginning with the left-hand signal of a pair and measuring toward the right, and then, after reversing the telescope both in altitude and in azimuth, to measure six times from right to left. The recent practice of the Coast Survey has been to measure first the angle itself by six repetitions, left to right, with the telescope direct, then the METHOD OF MEASURING THE ANGLES 57 explement (360 minus the angle) six times, moving the alidade in the same direction as before, left to right, the telescope being reversed. This brings the vernier nearly back to the same read- ing as by the previous method, but it differs in the mechanical operation. If there is any systematic effect on the angle, due to the action of clamps or to drag on the centers, it is eliminated from the final result, provided such errors are the same for a large as for a small angle. The reversal of the telescope in the preceding programs is in- tended to eliminate the errors of adjustment of the line of colli- mation and of the rotation axis of the telescope. It does not eliminate errors due to imperfect leveling. The measurement of angles in both the left-to-right and the right-to-left direction is designed to eliminate possible twist in the support of the instru- ment, upon the assumption that this twist, takes place at a uniform rate. In order to eliminate errors due to faulty graduation of the circle, the initial reading for different sets of observations may s O be shifted by ^ . where m is the number of sets taken and n mn is the number of verniers. For example, in- taking four sets with a two-vernier instrument, the vernier would be set ahead 45 each time. Errors in the graduation of the verniers may be eliminated in a similar manner by changing the vernier setting th part of a circle division at the beginning of each new set. m For four sets, on a 10' graduation, the first setting might be zero, the second 45 02' 30", the third 90 05' oo", and the fourth 135 07' 30". With the direction instrument the method of measurement consists in first pointing the telescope at some conspicuous signal, selected as the first of the series around the horizon, and reading all the microscopes, then turning the telescope to the other signals in order and reading all the microscopes at each pointing. After the last pointing has been completed and the microscopes read, 58 FIELD-WORK OF TRIANGULATION the telescope is reversed, the pivots remaining in the same bear- ings, and the series is repeated, the signals being sighted in the reversed order. The horizontal circle remains clamped during the entire process. The above measurements constitute a single "set." As many sets may be taken as are required to give the necessary accuracy. To eliminate systematic errors of gradua- tion and errors of the micrometers, the circle reading is advanced for each, new set, as explained later in the " Instructions for Primary Triangulation." It should be observed that the ac- curacy depends upon the circle's remaining undisturbed in azimuth during each set. In making bisections, either when pointing on the signal or when reading the microscopes, the observer should proceed as rapidly as he can without making careless pointings and without danger of making mistakes. Much time spent in perfecting settings and in watching them to see if they are correct appears to reduce slightly the accidental errors of observation, but does not really increase the accuracy of the work, as shown by the final results of the triangulation. The longer the time that is permitted to intervene between pointings, the greater the oppor- tunity for the circle to shift its position or change its temperature ; and the effects of these changes are probably greater than the accidental errors of pointing and reading. 44. Program for Measuring Angles. 'Various programs of observations have been devised, with a view to eliminating or reducing the errors in horizontal angles. The principal errors which have to be considered in planning the field work of the triangulation, and the methods adopted for eliminating them, are as follows: i. Errors due to non-adjustment of the theodolite. These are all eliminated by the use of the instrument in the direct and reversed positions, except that due to erroneous leveling. The leveling of the plate and the horizontal axis must be carefully attended to in setting up the instrument, and must be corrected whenever it becomes necessary. This may be done at any time PROGRAM FOR MEASURING ANGLES 59 between sets of angles or, if a repeater is used, at any time when the lower clamp is loose. 2. Errors arising from imperfections of graduation. These are practically eliminated by distributing the readings uniformly around the circle. 3. Errors of eccentricity of the circle and alidade. These errors are almost wholly eliminated by reading two or more equidis- tant microscopes or verniers. 4. Errors due to twisting of the tripod under the action of the sun's rays. The twist is eliminated by reversing the direction of the measurements, provided the rate of twist and the speed of measuring the angles are both uniform. The rate of twist on some towers has been found to be about one second of angle per minute of time. On the slender towers used on the gSth meridian triangulation, the twist was so small that it could not be detected by an examination of the measurements. 5. Errors due to irregular refraction of the atmosphere and to difficult seeing. These will be partly eliminated by taking a large number of measurements; if the results indicate that the neces- sary precision is not being obtained, it will be best to wait until favorable. This can be judged best by the " probable error " of the direction. 6. The personal error of the observer. The personal error is partly eliminated by measuring the angle a large number of tunes. 7. Errors due to temperature and wind. Errors due to fluctua- tion of the temperature of the instrument, and to vibrations caused by wind, may be reduced by shielding the instrument from the sun and wind, either by a tent or by a temporary build- ing. The following list of instructions to observers is taken from the Coast and Geodetic Survey Special Publication No. 19 (1914), and represents the present practice of that Survey. i. Instruments. In general, direction instruments of the highest grade should be used in triangulation of this class. Repeating theodolites are to be used only when the station to be occupied is in such a position as to be difficult of occupation 6o FIELD-WORK OF TRIANGULATION with a direction instrument or when there is doubt of the instrument support being of such a character as to insure that the movement of the observer about the in- strument does not disturb it in azimuth. Such stations usually occur on lighthouses and buildings. 2. Number of observations Main scheme Direction instrument. In making the measurements of horizontal directions measure each direction in the primary scheme 16 times, a direct and reverse reading being considered one measurement, and 1 6 positions of the circle are to be used, corresponding approximately to the following readings upon the initial signal: Number. Reading. Number. Reading. I O OO 40 9 128 oo 40 2 3 IS 01 SO 30 03 10 10 ii 143 oi 50 158 03 10 4 6 45 04 20 64 oo 40 79 oi 5o 12 13 14 173 04 20 192 oo 40 207 oi 50 8 94 03 10 109 04 20 15 16 222 03 10 237 04 20 3. When a broken series is observed, the missing signals are to be observed later in connection with the chosen initial or with some other one, and only one, of the stations already observed in that series. With this system of observing no local adjustment is necessary. Little time should be spent in waiting for the doubtful signal to show. If it is not showing within, say, one minute of when wanted, pass to the next. A saving of time results from observing many or all of the signals in each series, provided there are no long waits for signals to show, but not otherwise. 4. Standard of accuracy. In selecting the conditions under which to observe primary directions, proceed upon the assumption that the maximum speed con- sistent with the requirement that the closing error of a single triangle in the primary scheme shall seldom exceed three seconds, and that the average closing error shall be but little greater than one second, is what is desired rather than a greater ac- curacy than that indicated with slower progress. This standard of accuracy used in connection with other portions of these instructions denning the necessary strength of figures and frequency of bases will in general insure that the probable error of any base line, as computed from an adjacent base, is about i part in 88,000, and that the actual discrepancy between bases is always less than i part in 25,000. 5. Rejections Direction observations. The limit for rejection of observations upon directions in the main scheme shall be 5 seconds from the mean. No observa- tion agreeing with the mean within this limit is to be rejected unless the rejection is made at the time of taking the observation and fof some other reason than simply that the residual is large. A new observation is to be substituted for the rejected one before leaving the station, if possible without much delay. ii. Vertical measures in main scheme. At each station in the main schema PROGRAM FOR MEASURING ANGLES 6l vertical measures are to be made over all lines in the main scheme radiating from it. These vertical measures should be made on as many days as possible during the occupation of the station, but in no case should the occupation of the station be pro- longed in order to secure such measures. Three measures, each with the telescope in both the dire.ct and the reversed positions, on each day, are all that are required. These measures may be made at any time between 11.00 A.M. and 4.30 P.M., except that in no case should primary vertical measures be made within one hour of sun- set. It is desirable, however, with a view of avoiding errors due to diurnal varia- tion of refraction, to have a fixed habit of observing the verticals in the main scheme at a certain hour, as, for example, between 2 and 3 P.M. If the vertical measures at a station are made by the micrometric' method, double zenith distance measures shall be made on at least two of the lines radiating from that station. 13. Marking of stations. Every station, whether it is in the main scheme or is a supplementary or intersection station, which is not in itself a permanent mark, as are lighthouses, church spires, cupolas, towers, large chimneys, sharp peaks, etc., shall be marked in a permanent manner. At least one reference mark of a perma- nent character shall be established not less than 10 meters from each station of the main scheme and accurately referred to it by a distance and direction. Such ref- erence marks shall preferably be established on fence or property lines, and always in a locality chosen to avoid disturbance by cultivation, erosion, or building. It is desirable to establish such reference marks at all marked stations. At all stations where digging is feasible both underground and surface marks which are not in con- tact with each other shall be established. ' Wood is not to be used in permanent marks. . 14. Descriptions of stations. Descriptions shall be furnished of all marked stations. For each station which is in itself a mark, as are lighthouses, church spires, cupolas, towers, large chimneys, sharp peaks, etc., either a description must be furnished, or the records, lists of directions, and lists of positions must be made to show clearly in connection with each point by special words or phrases if neces- sary the exact point of the structure or object to which the horizontal and vertical measures refer. Every land section corner connected with the triangulation must be fully described. The purpose of the description is to enable one who is un- familiar with the locality to find the exact point determined as the station and to know positively that he has found it. Nothing should be put into the description that does not serve this purpose. A sketch accompanying the description should not be used as a substitute for words. All essential facts which can be stated in words should be so stated, even though they are also shown in the sketch. 15. Abstracts and duplicates. The field abstracts of horizontal directions and vertical measures are to be kept up and checked as the work progresses, and all notes as to eccentricities of signals or instrument, of height of point observed above ground, etc., which are necessary to enable the computation to be made, are to be incorporated in the abstracts. As soon as each volume of the original record has been fully abstracted and the abstracts checked, it is to be sent to the Office, the corresponding abstracts being retained by the observer. A duplicate of the de- scription of stations is to be made. If the original descriptions of stations are 62 FIELD-WORK OF TRIANGULATION written in the record books, a copy of these descriptions compiled in a separate book may be considered the duplicate and should then be marked as such. A duplicate of the miscellaneous notes mentioned above may also be made if con- sidered desirable. No other duplicates of the original records are to be made. Pencil originals should not be inked over. 16. Number of observations Main scheme Repeating theodolite. If a te- peating theodolite is used for observations in the main scheme, corresponding to those indicated in paragraph 2, make the observations in sets of six repetitions each. ' For each angle measured follow each set of six repetitions upon an angle with the telescope in the direct position immediately by a similar set of six on the explement of the angle with the telescope in the reversed position. It is not necessary to re- verse the telescope during any set of six. Make the total number of sets of six repetitions on each angle ten five directly on the angle and five on its explement. Measure only the single angles between adjacent lines of the primary scheme and the angle necessary to close the horizon. With this scheme of observing no local adjustment is necessary, except to distribute the horizon closure uniformly among the angles measured. The limit of rejection corresponding to that stated in para- graph 5 shall be for a set of six repetitions 4" from the mean. 19. Field computations. The field computations are to be carried to hun- dredths of seconds in the angles, azimuths, latitudes, and longitudes, and to seven places in the logarithms. The field computation may be stopped with the com- pletion of the lists of directions for all stations and objects, and the triangle side computation for the main scheme and supplementary stations, unless there are special reasons for carrying it further. The computation to this point should be kept up as closely as possible as the work progresses, to enable the observer to know that the observations are of the required degree of accuracy. No least square ad- justments are to be made in the field. All of the computation, taking of means, etc., which is done in the record books and the lists of directions should be so thoroughly checked by some person other than the one who originally did it as to make it unnecessary to examine it in the Office. The initials of the person making and checking the computations in the record books and the lists of directions should be signed to the record as the computation and checking progress. Investigations of the accumulated error in the azimuth of a chain of triangles indicate that there is a systematic tendency of the triangulation to twist in azimuth, due to unequal heating of the different parts of the theodolite by the sun. In day ob- servations on arcs running north and south there appears to be a greater accumulated error in azimuth on the east side of the chain than on the west side. This is apparently due to the fact that the observations were made chiefly or wholly in the afternoon. Observations made at night show less difference between the two PROGRAM FOR MEASURING ANGLES sides of a chain of triangles. The errors due to this cause may be diminished by making the instrument out of metal having a lower coefficient of expansion, such as nickel-iron, and by in- creasing the proportion of night observations. The unequal heating effect may also be diminished in day observations by turning the circle 180 in azimuth between sets. The following set of pointings, to be substituted for that on p. 60, is designed to accomplish this purpose. CIRCLE READINGS FOR INITIAL DIRECTIONS.* Posi- tion. Telescope direct. Teles :opj reversed. Posi- tion. Telescope direct. Telescope reversed. I 00 40 180 oo 40 9 128 OO 40 308 oo 40 2 195 oi 50 15 oi 50 10 323 oi 50 143 01 50 3 30 03 10 210 03 10 II 158 03 10 338 03 10 4 225 04 20 45 04 20 12 353 04 20 173 04 20 5 64 oo 40 244 oo 40 13 192 oo 40 12 OO 40 6 259 oi 50 79 oi 50 14 27 oi 50 207 oi 50 7 94 03 10 274 03 10 15 222 03 IO 42 03 10 8 289 04 20 109 04 20 16 57 04 20 237 04 20 For a method of correcting azimuths for the accumulated twist of triangulation, see page 202. 45. Time for Measuring Horizontal Angles. It was formerly the practice to measure angles only during that part of the day when signals appear steady, that is, during the latter part of the afternoon and sometimes in the early morning. In 1902 the Coast Survey parties were instructed to observe from 3 P.M. until dark, on heliotropes, and then to continue, with the use of acetylene lights, until n P.M. The criterion to be used in deciding whether conditions were favorable was not the appear- ance of the signals themselves, but the variations of the measures of the angles. The results showed that angles can often be measured with sufficient accuracy at times when the appearance of the signals would indicate poor conditions. From the results of this season's work it became evident that night observations * From Coast and Geodetic Survey Special Publication No. 19. 6 4 FIELD-WORK OF TRIANGULATION are somewhat more accurate than those made in daylight. Ob- serving at night is also more economical than observing in the day on heliotropes, because at night the observer is less dependent upon weather conditions (see Art. 16). 46. Forms of Record. The following are forms of record which may be used for hori- zontal angles of triangulation. HORIZONTAL ANGLES. DIRECTION INSTRUMENT. Station, Corey Hill. Date, May 21, 1907. Observer, A. N. Recorder, W. R. N. Inst. No. 31. Set No. 2. Station observed. Time. Tele- scope. Micro. Circle. Run. Mean. Cor. for run. Cor'd meas. / F. B. Blue Hill Prospect h m 4 30 Dir. Dir. A B C A B C 15 oi 138 30 51-5 54-o 49.0 '50-5 53-7 48.5 0.6 0-3 51-2 2O. 2 Si-5 20.9 22. 18.1 50.9 20.5 21.5 18.0 20.3 20. o HORIZONTAL ANGLES. REPEATING INSTRUMENT. Station, Corey Hill. Date, May 21, 1907. Observer, J. N. B. Instr. B. & B., No. 1567. Station. Time. Tel. Rep. Ver. A. B. Mean. Angle. Mean. Blue Hill h m 3 20 D O OO OO /t OO 00 orn orn to Prospect P.M. I 123 28 10 20 IS 6 *2o 49 40 40 40 123 28 l6.7 R 20 49 40 40 40 6 OO IO 10 10 123 28 15 .O 123 28 15.8 * Note. Since the angle is over 120 degrees the A vernier has passed 360 degrees twice in the six repetitions. In computing the mean we divide the 720 degrees by 6 mentally and write down 12 , then divide the 20 degrees by 6, add the whole degrees to 120, and then divide the minutes and seconds. Observe that when six repetitions are used, the remainder, when dividing the degrees by 6, gives the first figure of the minutes, i.e., 20 degrees 4-6 = 3 degrees in the mean, plus 2 degrees to be carried to the minutes column giving 20 minutes. Similarly in dividing the minutes by 6 the remainder is the tens place in the seconds. REDUCTION TO CENTER 47. Accuracy Required. As stated in paragraph 4 on p. 60 the degree of accuracy re- quired on the Coast Survey triangulation is such that the error of closure of a triangle shall seldom exceed 3" and shall average about i". The following list, taken at random from a longer list in Special Publication No. 19, will indicate the degree of accuracy actually obtained in the work of the Coast Survey. Section. Probable error of an observed Average closing error of a Max. cor. to direction. Maximum closing error of a triangle. triangle. Nevada California 0.23 O."C7 O.6o I C7 New England rto 26 O 7< I 17 2 O2 Eastern Oblique Arc. 30 o 78 O 74 2 73 Holton Base net. . O 34 O 7Q o 84 2 28 Atlanta base to Dauphin Island-base ' 0.36 I .IO 0.84 2 6q Lampasa base to Seguin base. 0.45 I-I3 1.96 3 31 Calif. Washington Arc o-53 I .22 2.03 6-35 48. Reduction to Center. In case certain lines from any station are obstructed, it may become necessary to set the instrument over a point at one side of the center, called an eccentric station, and to measure the angles at this new point. These angles are measured with the same degree of pre- cision as though the instru- ment were at the center. Before such angles can be used for solving the tri- angles, they must be reduced J to the values they would have if the instrument were placed at the center. The data nec- essary for the calculation include the approximate distances (D) to the points sighted, the distance from the center mark to the in- strument (d) , called the eccentric distance, and the angle at the in- strument between the center mark and each of the signals sighted. FIG. 34. 66 FIELD-WORK OF TRIANGULATION In Fig. 34, let C be the center, E the instrument, and 5 one oi the signals. The angle CES = a (called the azimuth), measured right-handed from the center to the distant signals, may be calculated for each signal by combining angles already measured, provided the line EC has been connected with any one signal by means of an angle. The angle S is the change in the direction, or azimuth, of the triangulation line due to the eccentricity of the instrument station. Solving the triangle for S, we have j w K U u cminoi^T-' FIG. 43. Chronograph Record. separate circuit having a battery of only one cell, in order to avoid injury to the mechanism, and operates the chronograph circuit through the points of a relay. The transit micrometer operates on the make-circuit, which is converted into breaks by a relay. If a key is used, it replaces the micrometer relay and breaks the circuit when the key is pressed. Fig. 43 shows a portion of a chronograph record. 59. Adjustment of the Transit. In placing the transit on the supporting pier before adjusting it in the meridian, the base of the instrument must be placed so 82 ASTRONOMICAL OBSERVATIONS nearly in the meridian that all further adjustment in azimuth may be made by the adjusting screws provided for this purpose. The foot plates should then be cemented to the pier. The tele- scope is focused as in an engineer's transit first the eye- piece, then the objective. A distant terrestrial object may be used for the first trial, but the final focusing should be done at night on the stars. A difference is usually noticed between the focus required by day and that found at night when artificial light is used. The striding level and the horizontal axis may be adjusted simultaneously by placing the level in position, reading both ends of the bubble, then reversing it, end for end, and taking another set of readings. Half the displacement of the bubble may be corrected by adjustment of the level and half by leveling the axis. The verticality of the threads or the micrometer line is tested by rotating the telescope slightly about its horizontal axis and noting whether a fixed object remains continuously on the thread as it traverses the field of view. Adjustment is made by rotat- ing the diaphragm or the micrometer box until this condition is fulfilled. The collimation is adjusted by placing the middle line of the reticle or the mean position of the micrometer line as nearly as possible in the collimation axis. To test this, point the wire on some object, reverse the telescope in its supports (axis end for end), and see if the object is still sighted. If it is not, bring the wire halfway back by means of the lateral adjusting screws. The finder circles should be tested to see if they read zero when the collimation axis is vertical. Point on some object, level the bubble, and read the circle. Reverse the telescope, point on the same object, and repeat the readings. The mean reading is the true zenith distance, and half the difference between the "two readings is the error of adjustment. Set the vernier to read the true zenith distance, sight the object again, and then center the bubble by means of the adjusting screws. SELECTING THE STARS FOR TIME OBSERVATIONS 83 To place the line of collimation in the meridian, first determine a rough chronometer correction by leveling the axis and setting the circles for the zenith distance of some star which is near the zenith and which is about to culminate. If the (sidereal) chronometer is nearly regulated to local sidereal time, the right ascension of such a star will be nearly the same as the chronom- eter reading. If the chronometer is not regulated at all, it may be set approximately right by calculating the sidereal time cor- responding to the mean time as indicated by a watch. An error of one or two minutes will not cause great inconvenience, as all that is necessary is to identify the star and begin observing before it has passed. The time, at which this star will pass the middle vertical thread must necessarily be very close to the true sidereal time (right ascension of star), because near the zenith the effect of the azimuth error on the observed time is very small. The difference between the right ascension of the star and the chronometer reading is an approximate value of the chronometer error. Using this value of the chronometer error, calculate the chronometer time when some slowly-moving (circumpolar) star will pass the meridian. When this calculated time arrives, point the middle thread or the micrometer thread on the star, using the azimuth adjustment screws. This places the instrument nearly in the meridian. A repetition of the whole process (on a different pair of stars) will give a still closer approximation. It is not necessary or desirable to spend much time in re- ducing the errors of azimuth, level, and collimation to very small quantities. They should be so small as to cause no inconvenience in making the observations and in computing the results, but since they must be determined and allowed for in any case, the final result is quite as accurate if the errors themselves are not extremely small. 60. Selecting the Stars for Time Observations. There are two general methods of selecting the stars to be used for a time determination. The older method requires observa- tions on ten stars, five with the axis of the telescope in one posi- 84 ASTRONOMICAL OBSERVATIONS tion (say illumination or clamp east) and five with the axis re- versed (illumination or clamp west). In each half -set one of the stars is a slow-moving one, that is, one situated near the pole. Of the remaining four stars in each half -set two should preferably be north of the zenith and two south of the zenith, and in such positions that their azimuth errors balance each other, that is, their A factors (see Art. 66) should add up to zero. In the more modern method, used with the transit micrometer, twelve stars are employed, six in each position of the axis. None of these is near the pole, but their positions are so chosen as to make the algebraic sum of their A factors nearly equal to zero. By the older method the error in azimuth adjustment is more accurately determined, but with a proper selection of stars the value of the azimuth correction need not be determined so ac- curately, because it has a relatively small effect upon the com- puted chronometer correction. In preparing for observations a list of stars should first be made out, giving the name or number of each star, its magnitude, right ascension, declination, and zenith distance, together with the star factors depending upon its position, as explained later. The declination of the stars chosen should be such that the algebraic sum of the A factors is less than unity. It is desirable that the list contain as many stars per hour as possible, but sufficient time must be allowed for reading the stride level, reversing the instrument, making records, etc. The telescope should be re- versed before each half-set. In preparing this list the zenith distance of a star is computed by the relation r = - 5, [i 7 ] where f is the zenith distance (positive if south of the zenith), < is the latitude, and 5 is the declination (positive for stars north of the equator). 61. Making the Observations. In beginning the observations, set the vernier of the finding circle at the zenith distance of the first star and bring the bubble MAKING THE OBSERVATIONS 85 to the center of its scale by moving the whole telescope. The clamp had better not be used if the telescope can be relied upon to remain in position when undamped. When the star appears in the field, bring it between the two horizontal hairs by tapping the telescope with the finger. Set the micrometer line on the star and keep it bisected until the observations (4 turns of screw) are completed. If the instrument is not provided with a microm- eter, the observer simply presses the observing key as the star passes each of the vertical threads. When the observations are made by the key method, the observer attempts to press the key as soon as possible after the star is actually bisected by the wire. In doing this he makes an error which tends to become constant as the observer gains in experience. This is known as his per- sonal equation. Since the personal equation depends chiefly upon the rapidity and uniformity with which the observer is able to record his observations, rather than upon his ability to bisect the star's image, the use of the transit micrometer very nearly elimi- nates this error. After half the stars in one set have been observed, the axis should be reversed, end for end, in the supports. The striding level should be read one or more times during each half-set. If the pivots are not truly circular in section, the average inclina- tion of the axis may be found by taking level readings with the telescope set at different zenith distances, both north and south. The striding level should be used with great care, because the level corrections may be relatively large and cannot be eliminated by the method of observing, as in case of the collimation error and, to some extent also, the azimuth error. Following is a record of a set of observations as read from the chronograph sheet, together with the readings of the striding level. (See United States Coast and Geodetic Survey Special Publication No. 14, p. 21.) 86 ASTRONOMICAL OBSERVATIONS Station, Key West. Date, Feb. 14, 1907. Instrument, transit No. 2, with transit micrometer. Observer, J. S. Hill. Recorder, J. S. Hill. Chronometer, Sidereal 1824. Star: S. Monocer. ^ 6 Aurigae 18 Monocer. f Geminor. f Geminor. 63 Aurigae Clamp: W W W W W W Level W E W E W E d d d d d d N62.o 20. o 861.2 19 4 N6i.5 19 5 17.7 59-5 17.7 59 6 17.7 59-7 +44-3 -39-5 +43-5 - -40.2 +43-8 -40.2 +4-8 +3-3 +3-6 d Computation of level constant: Mean N + 4 20 8 + 3.30 + 3 75 X( 5-039 = + 0.146 = *w h m h m h m h m h m h m 635 6 39 642 6 46 65 S 7 04 I sin PS cos 5 or P = c sec 5 = cC. [22] The collimation factor C will be found in Table III. 66. Azimuth Correction. The error of setting the instrument in the meridian is measured by the constant a, the azimuth of the axis of collimation expressed in seconds of time. This constant is derived from the varia- tions in the observations themselves. In Fig. 46, P is the pole, Z the zenith, and S the star. In the triangle PZS, P is the re- quired correction, and S'ZS is a, the azimuth error. Applying the law of sines., sinP _ sing sin S'ZS "cos 6' or P = a sin f sec 5 = a-A. [23] The azimuth factor A may be taken from Table III. The con- 90 ASTRONOMICAL OBSERVATIONS stant a is positive when the plane of the axis of collimation is east of south. A is positive for all stars except those between the zenith and the pole. FIG. 46. 67. Rate Correction. In order to compute these corrections it is necessary to reduce all observations of the chronometer correction to some definite epoch, for example, the mean of all the observed times, so that variations in the chronometer correction itself will not affect the determination of the transit errors. This is done by applying the correction R=(t- To] r h , [24] where / is the chronometer time of transit, To is the mean epoch of the set, and rh is the hourly rate of the chronometer, positive if losing, negative if gaining. 68. Diurnal Aberration. The motion of the observer due to the diurnal motion of the earth makes all stars appear farther east than they actually are; in other words it apparently increases their right ascensions. The amount of the correction is expressed by the equation K = o s .o2i cos sec 5. [25] FORMULA FOR THE CHRONOMETER CORRECTION 91 This formula may be derived as follows: the velocity of a point on the earth's equator (toward the east) is 0.288 mile per second. For any other latitude the velocity is 0.288 cos mile per second. The velocity of light is 186,000 miles per second, and the angular Equator FIG. 47- displacement (*') of the star toward the east point of the horizon is therefore equal to tan" 1 ^. The effect on the ob- 186,000 served time is the angle K at the pole, Fig. 47. Hence sin/c _ sin/ sin 90 cos 5 or K = o".3i9cos0sec6 = o*.o2i cos< sec 6. Values of this correction will be found in Table IV. 69. Formula for the Chronometer Correction. The true sidereal time, or right ascension of the star, is given by the equation a = i + AT + K + R + Aa + Bb + Cc, [26] in which / is the mean of the observed transits and AT is the chronometer ' correction. Since the corrections for aberration, rate, and inclination may be found directly, they are applied to / at once. If we call /i the value of / thus corrected, then a /i = Ar + Aa + Cc, or AT = (a - /O - Aa - Cc. [27] 92 ASTRONOMICAL OBSERVATIONS 70. Method of Deriving Constants a and c, and the Chro- nometer Correction, AT. The method shown in the following table is the one used when the observations are made with the transit micronometer and when the latitude is less than 50. For greater latitudes the observations are reduced by the method of least squares. COMPUTATION OF TIME SET. [Station, Key West, Florida. Date, Feb. 14, 1907. Set, 2. Observer, J. S. Hill. Computor, J. S. Hill.] Star. Clamp. a - t. it. c. A. Cc. Aa. AT = (-)- Cc-Aa. 1. S Monocer... 2. ^ 5 Aurigae... 3. 18 Monocer. . 4. 8 Geminor. 5. f Geminor.. 6. 63 Aurigae . . . 7. i Geminor.. 8. Can. Min. 9. a Can. Min. 10. /3 Geminor. 11. v Geminor.. 12. Geminor.. +15.00 +15.08 +15.04 +15.03 +15-00 +15-02 +14-43 +14-45 +14-45 +14.41 +14.42 +14-47 o.oo +0.08 +0.04 +0.03 o.oo +O.02 -0.57 -0.55 -0.55 -0.59 -0.58 -0.53 + I.O2 +1.38 + I.OI + I.2I +1.0? + 1-30 +0.26 -0.45 +0.37 0.20 +0.07 -0.34 0.07 +0.28 +0.33 o. 0.19 -0.05 5 +0.27 +0.36 +0.26 +0.32 +0.28 +0.34 -0.30 0.27 0.26 -0.30 -0.32 0.29 +O.O2 0.03 +0.03 O.OI o.oo O.O2 O.OO +O.OI +O.OI o.oo O.oi 0.00 5 +14.71 +14.75 +14.75 +14.72 +14.72 +14.70 +14.73 +14.71 +14.70 +14.71 +14.75 +14.76 1. 3 2. 3 5- 2 6. 5 9- 4 10. 9 .00 St + 3. loc + 0. .00 8t + 3.89^ 0. .12 S/ + 2. 75 c 0. Mean AT = + 14. 727 0.04 = 0.13 = 0.09 = o 0.13 = o 0.12 = + 2.61 = o (2) X 0.707 (i) + (S) (6) X 0.920 (8) + (9) 3. 3. 4. 3.00 dt 3.47 c 0.34 a E +1.74 = 7. i 8'. 4 12. - ii. St = 0.274 from (10} 5 AT= + 15.00 0.274 = + 14.726 1.63 = .82 - 5.38 c i. 32 -5. 38 c +2.73 = +2.73 = (3)Xo.6o7 (4) + (7) from (8) 14. 0.82 +1.02 0.99 a w 0.13 = 16. 0.820.83+0.560^. +1.63 = 13- c = + 0.262 from (12) IS- ajp = +0.071 17. a E = +0.036 5 +O.O2 O.O2 0.02 +0.01 +0.01 +0.03 O.OO +O.O2 +0.03 +0.02 0.02 0.03 The serial numbers in the lower part of the table show the order of the different steps of the computation. Equation i is METHOD OF DERIVING CONSTANTS 93 obtained by taking the terms corresponding to the three southern- most stars (that is, Nos. i, 3, and 5), substituting the sums of these numbers in the equation AT -j- Cc -f- A a (a ti) = o, and treating this result as though it were the equation for a single star. Equations 2, 3, and 4 are found in a similar manner. This gives four equations for the twelve stars, two for each half-set. Since there are now as many equations as there are unknowns, the quantities c, aw, &E, and AT may be found by solving these equations simultaneously. Notice that in this solution 15* has been dropped from AT, and that 5t is the small correction which must be added to 15* to obtain AT. The following method of deriving the constants and the chronometer correction without employing least squares is applicable when the two groups of stars have A factors which are not so nearly balanced, or where the list of observed stars con- sists of one slowly-moving (azimuth) star and several time stars in each half-set. This method gives, by a series of approxima- tions, very nearly the same result that would be obtained by the method of least squares. The various steps in the computation are shown in tabular form in Fig. 48. The formulas on which the method is based are as follows: For each star we may write an equation of the form a - ^ = AT + Aa + Cc. [28] Then for the east and west groups we have (a - ti) w = AT + A w a w + C w c, )' xv (a - t,] E = AT + A w a E + C E c. \ Assuming at first that a E and a w are equal, we find an approxi- mate value of c by subtracting the second equation from the first. Solving for c, we find (a ti) E In the above example, 10. 25 10.17 - ' +0.03. 1.42 + 1.34 94 ASTRONOMICAL OBSERVATIONS < a a g. 10 w w r* r* "^t r* O O M O OO M W g 2 $ % 8"^ ci o d o ci M* a* ! ^ a 8 ro ro f5 1 ro P? P? ??'+ j O O O . . w . . w . i : f 4 4 * - *" * M Kj ro * M- t tO M P! O O O O O Hi M K a oo * <*5 P? P? P? CO i d + ey II t 1 * oo ^f vo ^ ^ O P* O VO M *^ ti o "N* M ro *g oo w ^O O O O VQ ^O O. O i i i O PI M fe X*l 1 S A . . "?+ i (s o* 10 ii >H co - M H Q . (N p? p? p? rj- o ^0 to o o o o 5/8 Jo S o.'R w & . 5 "^ *? N 1 + Hr ** &fe HI P4 fO ^t ^f OOOOO 00000 o 1 o 5 3" M b | s^ PJ ^ ^ J? p? P? 8. 3 *&o?o3 r^ *t to o oo o. M 00 P O O 0. 0> i . dfe ^ ^ ^ " ^ tod^d ^^o. I -U 551$ 5 5^ H a a a a 1 ft 3 2 S- S i *l "? r^tOMt^ 0.00. toOPi^rto W O M O fO to M mood tn -^ cr N i + i w*a bj P fO r3 fO * w oo vb TJ- o> to ^* 0.^ . PJ a a a d + 10 .0 P* .0 6 0> || P- 0> 00 * PI o o M S 12 *3 {7 1 d PO . * f 4 1 5 *8 ^ r- V "! to o A | M 1 "? "+ r? w ir> ro it . w p? P? P7 H o H 1:4 O O> 10 O <> OOO Mt^^J-MOt S f ^ f S ft S" {I * So o n t-H pq & ^ PQ METHOD OF DERIVING CONSTANTS 95 M 1? C w II ^ I OO *^ -i ^^ ^^ *^ *? M *^ ^T d o . d o i I +i'n'i - = " KI fe; O O vj <3 lOt^rfQ w 00 -00 M M M M OOOO OOOO . O^ CT* O^ O 1 ^ O^ O^ CT> O^ tt 1 + 1 1 + 1 1 T + + f j ookoo oooo d o d M d d o M ^ M + ^ 1 +++ 1 +++ H VO wj t~ Tt" OO rJ-CO M MOOt-tTj- CniOPOO^ < - * d d d d M d o" o* 1 + 1 + ++ 1 + tf MMMCI MMH1C4 O oooo oooo dodo o d d d 1 1 1 1 ++++ d rtt^rfM <> CO 10 o o> o o 0* 0* M CT> CS OO M M M M 1 1 1 1 M M M M I HHHH.EB&** W W ^ ^ . . . . . - . Lrf ' |2 C * M * rrt " S rrt ' rt? '.'.'.'. '.'.'.'. CO ' CO CO ' CO O^H^IH C^lnOjH Z^ - - ~ Z^ - g '.'.'.'. '.'.'.'. _^ co to '_j3 to '^ to -3 ^^ *TJ ~y '.j Q "j -i y _^ 0) .... CD 0} .... P*M O O ^ j-Jf^<^ 'xoj.dd.T3 *xojddB ^si -pe 96 ASTRONOMICAL OBSERVATIONS Using this approximate value of c, the last terms in Equations (a) are computed and subtracted from (a h) in each case, leaving the equations in the form (a - /i - C e ) = &T + A w a w . Taking each half-set separately, and also grouping the azimuth star and the time stars separately, we have for the next group (a - ti - C c ) = AT + (A w a w ) az, (a - /i - C c ) = AT + (A w a w ) time, and a similar pair of equations for the second position of the axis. From Equa. (b) we derive (a ti C c )az. (a h Cjtime aw= ~ ~A~ ~A~ "-az. ** time In the example, 10.08 10.21 a E = ^ = -0.78 0.96 0.03 20.74 19.21 and aw = *-* = 0.76. -2.05 +0.03 Employing these approximate values of a E and a w , the A a cor- rections are computed and subtracted, giving the value in the column headed a t\ C c Aa. For the time stars these values are 19.23 and 19.19. Since these values do not agree for the two positions of the instrument, the value of c is evidently in error. A second approximation must be made by treating the difference of these numbers (0.04) as an error in c and obtaining a correction to c by the same process that was used in finding c in the first instance, that is, r* * I 9- I 9 ~~ I 9- 2 3 Correction to c = - z ~ = 0.014. 1.42 + 1.34 Hence c = +0.03 0.014 = +0.016. With this improved value of c new values of a# and aw are com- puted as before. The second values are a E = 0.768 and a w = 0.772. Using these values, the chronometer corrections are found to agree, and hence no further approximation is necessary. DETERMINATION OF DIFFERENCES IN LONGITUDE 97 The azimuth and collimation corrections are now found for each star, as shown in the upper part of the table. The mean of the AT's for all the stars is the chronometer correction for the mean of the observed times. The residuals (v) are computed by sub- tracting AT for each star from the mean of the AT's for that group. These should add up nearly to zero. Whenever the most accurate results are desired, the computa- tion may be made by the method of least squares. For the details of this method see Coast and Geodetic Survey Special Publication No. 14, p. 41. 71. Accuracy of Results. The error in the computed value of AT 1 due to accidental errors alone may be kept within a few hundredths of a second. Ob- servations made by the key method may be subject to a large constant error, the observer's personal equation, which may be several times as large as the accidental error. Observations made with the transit micrometer are nearly free from personal errors. 72. Determination of Differences in Longitude. The determination of the difference in longitude of two stations consists in measuring the difference between the sidereal times at the two places. The method almost exclusively used for accurate longitudes in places where a telegraph line is available is that in which the times are compared by electric signals sent over the telegraph line. Wireless apparatus may be used for this purpose, but it has not as yet come into general use, probably because it is not as economical as the ordinary lines. The method used at present by the Coast Survey differs considerably from the old method, owing to the introduction of the transit micrometer. According to the usual program each observer, provided with transit, chronometer, and chronograph, determines the local sidereal time by the method previously described; then the two chronometers are compared by means of arbitrary signals, which are sent over the telegraph line and recorded simultaneously on 9 8 ASTRONOMICAL OBSERVATIONS OBSERVATIONS BY KEY METHOD 99 both chronographs; and, finally, each observer again determines the local sidereal time. According to the Coast Survey instructions (Spec. Pub. No. 14) each half -set should consist of from 5 to 7 stars (preferably 6), all of these to be time stars (no azimuth star). The algebraic sum of the azimuth factors (A) should be less than unity. Four half-sets are observed during an evening, and the telescope axis is reversed before each half-set. The observers do not exchange places during the occupancy of the station, as was formerly the practice. Observations on three or four nights usually give the desired accuracy. Fig. 49 shows the switchboard and the arrangement of the electric circuits required in longitude observations. When the observer is making observations for time, the circuit is arranged as shown in Fig. 42. Fig. 50 shows the circuit as arranged during the exchange of arbitrary signals. These signals are made by tapping the signal key in the main-line circuit. Half of these signals are sent by the eastern observer, half by the western, in order to eliminate the error due to the time of transmission of the signal. The chronometers mark the record sheets while the signals are being sent, so that the time of each signal may be read from each chronograph sheet. The difference in longitude is found from interpolated chronometer corrections. 73. Observations by Key Method. If the transit micronometer is not used, the selection of stars must be modified so as to allow more time between observations. Since the observations will be subject to the personal errors of both observers, it is important that the observers should exchange places at the middle of the series, so that their relative personal equation will enter the latter half of the observations with its algebraic sign changed. The arrangement of the circuits is shown in Fig. 51, in which an observing key replaces the relay and circuit of the transit micrometer. IOO ASTRONOMICAL OBSERVATIONS Battery Battery Signal Relay Telegrapher's & Signal Key Main Line FIG. 50. Electrical Connections Exchange of Signals Transit Micrometer Method. Battery Signal Relay Telegrapher's & Signal Key Main Line FIG. 51. Electrical Connections Exchange of Signals Key Method. DETERMINATION OF LATITUDE COI 74. Correction for Variation of the Pole. The periodic variation of the position of the pole affects all observations for longitude and must be allowed for by applying the corrections given in tables published annually by the Inter- national Geodetic Association. (See Art. 81, p. 106.) 75. Determination of Latitude. The method which has been chiefly used in this country for determining astronomical latitudes for geodetic purposes is that known as Talcott's (or the Harrebow-Talcott) Method. The instrument employed is the zenith telescope, illustrated in Fig. 52. The principle involved is that of measuring, not the absolute zenith distances of stars, as is done with the meridian circle, but the small difference between the zenith distances of two stars which are on opposite sides of the zenith. By a proper selection of stars this difference in zenith distance may be made so small that the whole angular distance to be measured comes within the range of the eye-piece micrometer, which for most instru- ments is about half a degree. A sensitive spirit level attached to the telescope serves to measure any slight change in the in- clination of the vertical axis of the instrument between the two observations on a pair of stars. The accuracy of the results obtained by this method is superior to that of every other field method, and compares favorably with the results obtained with the largest instruments. The horizontal axis of the telescope is very short as compared with that of the transit instrument; small errors in the inclination of the axis, however, have very little effect upon the results; a close adjustment is therefore unnecessary. Since the instrument is used in the plane of the meridian and must be quickly turned from the north side to the south, or vice versa, the horizontal circle is provided with stops which are adjustable, so that the telescope may be quickly changed from one side of the zenith to the other. The micrometer, placed in the focal plane of the eye- piece, is set so as to permit of measuring small angles in the verti- cal plane. The head of the screw is graduated to read to about 102 : ASTRONOMICAL OBSERVATIONS FIG. 52. Zenith Telescope. (Coast and Geodetic Survey.) SELECTING STARS 103 o".5 directly and to o".o5 by estimation. The spirit level has an angular value of one (2 wm ) division equal to about i".5. 76. Adjustments of the Zenith Telescope. When the instrument is in perfect adjustment, the plate levels should be central in all azimuths as the telescope is turned about the vertical axis. The leveling may be perfected by use of the more sensitive latitude level. The horizontal axis must be at right angles to the vertical axis. The movable micrometer threads must be truly horizontal. They may be adjusted by a method similar to that used in adjusting the engineer's level by swinging the telescope horizontally through a small angle and observing whether the thread remains on a fixed point. The collimation adjustment should be made in the same manner as in a transit, but is not of so great importance. Allowance must be made for the eccentricity of the telescope when making the colli- mation adjustment. The value of one turn of the micrometer may be determined approximately by observations upon a close circumpolar star near its elongation. The most satisfactory way, however, is to derive the value of one turn from the latitude observations themselves, by the method of least squares. The value of one division of the latitude level may be determined by means of a level trier, or it may be found by varying the inclina- tion of the telescope and employing the eye-piece micrometer to determine the amount of this inclination by observations on a terrestrial mark. When in use the instrument is mounted on a wooden or con- crete pier. It is usually protected by a tent or other temporary shelter. In order to make the observations, it is necessary to have a chronometer regulated to local sidereal time with an error not exceeding one second of time. 77. Selecting Stars. The list of stars in the American Ephemeris will not ordinarily be sufficient for latitude observations, on account of the exacting nature of the conditions. It will be necessary to consult such IO4 ASTRONOMICAL OBSERVATIONS star catalogues as Boss's Preliminary General Catalogue of 6188 stars for the Epoch 1900, or one of the Greenwich catalogues. In order to keep the zenith distances within the required limits, it will often be necessary to observe on stars which are much fainter than those used for time observations. The pairs of stars selected should, if possible, differ by less than 2o m in their right ascension and by less than 20' in their declinations. The actual zenith distance of a star should not exceed 45. Following is a speci- men star list for zenith telescope observations. OBSERVING LIST (FORM i). [St. Anne, 111., June 25, 1908. Zenith telescope No. 4. Search factor = 2 = 82 03'.] 41 01^.3. 0) d , 6 A A . *.. Jo M rt ^ 9 "SI ^J CH ll> g-|g -o- in B "3 S i.| h^ -^ * II "U.S I i ' IH S ^3 = 5, + f,, and from the northern star it is EZ = ES n - ZS n , or d> = 8 n f n . FIG. S3- The mean of the two values of is 106 ASTRONOMICAL OBSERVATIONS If we let n a and s 8 = the level readings for the southerly star, n n and s n = the level readings for the northerly star, d = the angular value of one division of the level, r 8 and r n = the refraction corrections, M 8 and M n = the micrometer readings, and R = the value of one turn of the micrometer, then the latitude is determined by the equation - 2 (M, - M n ) -R + - {(n, + O - (s. [30 This formula applies .when the zero of the level scale is in the center of the tube. If the zero is at the eye-piece end of the tube, the level correction is 7 + - { (n, - n n ) + (s, - s n )}. 4 If for any reason the observations are not made when the star is exactly on the meridian, another term must be added to the above formula; this will be of the form -f- f (m a + m n ) when m, and m n are the reductions of the measured zenith distances to the true zenith distances. (See Special Publication No. 14, p. 119.) For the application of least squares to the computation of latitude see Chauvenet, Spherical and Practical Astronomy ; Hay ford, Geodetic Astronomy; and Coast and Geodetic Survey Special Publication No. 14. 80. Calculation of the Decimations. When the stars selected are not found in the Ephemeris, it will be necessary to calculate the apparent declinations for the date of the observation. Formulae and tables for making these reductions will be found in Part II of the Ephemeris. See also Coast and Geodetic Survey Special Publication No. 14, p. 116. 81. Correction for Variation of the Pole. The observed latitude may be in error by several tenths of a second, owing to the fact that the observed value necessarily REDUCTION OF THE LATITUDE TO SEA-LEVEL 107 refers to the position of the pole at the date of the observation, whereas the fixed value of the latitude of a place is that referred to the mean position of the pole. Fig. 54 shows the plotted positions of the pole for every o.i year during the period 1900.0 to 1906.0 (Jordan). The coordinates of the instantaneous pole 0.30 o.io 0.00 0.10 0.20 0.30 -0.20 SCALE OF FEET 10 20 30 FIG. 54. Motion of the North Pole, 1900 to 1906. and data for correcting observed values are published annually by the International Geodetic Association, and observations may be referred to the mean pole by employing these tables. 82. Reduction of the Latitude to Sea-Level. In order that all latitudes may refer to the same level surface, they are all reduced to their values at sea-level. If we suppose a io8 ASTRONOMICAL OBSERVATIONS lake surface, in the northern hemisphere, to be at a great height above sea-level, then it may be shown that the northern end of this lake surface is actually nearer to the surface of the sea than is the southern end of the lake surface. If we imagine a series of such surfaces at varying heights above sea-level, it is obvious that the vertical is a curved line, since it must at every point be normal to the level surface passing through that point. Evi- dently this curved line is concave toward the earth's rotation axis. To correct an observed latitude at elevation h to the cor- responding latitude at sea-level, it is necessary to apply the correction A< = o".o52 h sin 2 <, where h is in thousands of feet, formula becomes If h is expressed in meters, the 0.000171 h sin 2 (See Art. 170, p. 256.) Values of this correction will be found in Table VII. Below is an example of the form of record and com- putation of latitude from Special Publication No. 14.) RECORD OF LATITUDE OBSERVATION. [Station, St. Anne. Date, June 25, 1908. Chronometer, 2637. Observer, W. Bowie.] Star Micrometer. Level. Chro- nom- Chro- Meri- No. of pair. number Boss Nor S. eter time of nometer time of observa- dian dis- Re- marks. cat. Turns. Div's. North. South. culmi- tion. tance. nation. 42 .6 4623 N 24 88.2 9-2 71.6 103.8 18 13 18 II 4651 S 16 66.0 42.2 8-7 * 18 18 39 * + i6t 103.2 71.0 * These columns used only when star is observed off the meridian. t This is the continuous sum, up to this pair, of the south minus the north micrometer turns. ACCURACY OF THE OBSERVED LATITUDE LATITUDE COMPUTATION 109 Date. Catalogue. Micrometer. Level. Merid- ian dis- tance. Declination. Star No. Nor S. Reading. Diff. Z. D. N. S. Diff. 1908. June 25 4623 4651 N S 24 88.2 16 66. c *. d. 8 22.2 09.2 7 I.6 42.2 103.2 42.6 103.8 08.7 71.0 d. -1.05 S. 64 21 59.53 17 46 48.62 Sum and half sum. Corrections. Latitude. Remarks. Micrometer. Level. Refrac- tion. Meridian. o / // 82 08 48.15 41 04 24.08 / a [-3 03-56 a -o-39 O.o6 41 Ol 20.07 Value of one division of latitude level: Upper i".6co Lower 1.364 Mean 1.482 Value of one turn of micrometer = 44". 650 83. Accuracy of the Observed Latitude. The latitude may be determined by this method with a prob- able error of from o".3 to 0^.4 from one pair of stars. The final value for the latitude of the station determined from as many pairs of stars as can be observed on one night may be found with an error of from 0^.05 to o".io (or 5 to 10 feet). It is not consid- ered advisable to observe the same pair of stars on several nights, as was formerly the practice, owing to the comparatively large errors in the declinations themselves. The present practice is to observe each pair but once and to observe such a number of pairs that the uncertainty of the final latitude is not greater than o".io. In view of the fact that nearly every latitude is affected by a station error which may amount to several seconds, and that the real object of the observation is to determine this station error, it is better to determine a large number of latitudes with the degree of accuracy above mentioned than to attempt to diminish HO ASTRONOMICAL OBSERVATIONS the error of observation and occupy but a small number of stations. This results in the practice of occupying stations but one night, unless for some reason it is apparent that the required accuracy will not be reached without additional observations. 84. Determination of Azimuth. When determining an azimuth for the purpose of orienting a triangulation system, the observer usually has a choice of several methods, all of them capable of yielding the required accuracy, for example, (i) measuring the angles between a circumpolar star and the triangulation lines. by means of the direction in- strument, (2) measuring from a triangulation station to a cir- cumpolar star with the repeating instrument, or (3) measuring from a circumpolar star to an azimuth mark with the micrometer of a transit instrument. In all determinations of azimuth it is necessary to know the local time in order to compute the azimuth of the star. This must be found by special observations, unless, as is often the case, the longitude is being determined at the same time and the chronometer correction is already known. For the purpose of orienting the primary triangulation it is necessary to determine the azimuth with an error -not exceeding o".$o. At Laplace stations (coincident triangulation, longitude, and azi- muth stations), where the accumulated twist of the chain of tri- angles is to be determined, it is desirable to determine the azi- muth within o".3o or less. It is also desirable that the instru- ment station and the azimuth mark should both be triangulation stations. When horizontal angles are being measured at night, the azimuth observation is made a part of the same program by including pointings on a circumpolar star with the -regular series of pointings on lights at the triangulation stations. An azimuth found by this method is more accurate than one determined by means of an auxiliary point and subsequently connected with the triangulation by means of a horizontal angle measured by daylight. On account of the slow apparent motions of stars near the pole, nearly all accurate azimuth observations are made on close cir- FORMULA FOR AZIMUTH III cumpolars, since errors of the latitude and the time have less effect on the result than for stars farther from the pole. The stars ordinarily used for azimuth observations are shown in Fig. 55- * 51 Cephei Xll' FIG. 55. Circumpolar Stars. 85. Formula for Azimuth. In general all these methods consist in calculating the azimuth of the star at the instant of observation and combining this azimuth with the measured horizontal angle from the star to the station. The azimuth of a circumpolar star is found by the formula ,, sinJ r T tanZ = : > [33] cos tan 5 sin cos t where Z is the azimuth measured from the north toward the east, and / is the hour angle. If Equa. [33] be divided by cos < tan 5, then tanZ = cot 6 sec < sin t i cot 5 tan cos t = cot 6 sec sin / ( - - ) \i a/ [34] 112 ASTRONOMICAL OBSERVATIONS If values of - are tabulated * this formula will be found more i a convenient than Equa. [33]. 86. Curvature Correction. In computing the azimuth of the star it would evidently be inconvenient to apply the formula to each separate pointing on the star, on account of the large amount of computation. It is simpler and sufficiently accurate to calculate the azimuth of the star at the mean of the observed times of pointing, and then to correct the computed azimuth for the small difference between this azimuth and the mean of all the azimuths. The correction for this difference is T .. j o Curvature Correction = tanZ- z\ > he] n ** sin i in which n = the number of pointings and r = the interval of time (in seconds) between the ob- served time and the mean. The sign of the correction is such that it always decreases the angle between the star and the pole. For the derivation of this formula see Hayford's Geodetic Astronomy, p. 213. The correc- tion may also be written in the form tan Z [6.73672] - V T 2 . (See Doolittle's Practical Astronomy, p. 537.) 87. Correction for Diurnal Aberration. On account of the motion of the observer, due to the earth's rotation, the star is apparently displaced toward the east. The correction to the computed azimuth for the effect of this apparent displacement is given by the expression ^ f AI // cosZcos<6 r ,, Corr. for Aberra. = o .32- [36] COS n This correction is always positive for an azimuth counted clock- wise. For the derivation of this formula see Doolittle's Practical Astronomy, p. 530. * For a table of values of log - see Special Pub. No. 14. THE DIRECTION METHOD 113 88. Level Correction. If the horizontal axis is not level when a pointing is made on the star, the observed direction must be corrected by the following quantity: Lev. Corr. =-[(w + w') - (e + e')] tan h. [37] 4 For proof of this formula see pp. 55 and 87. If the level is graduated from one end to the other, Lev. corr. = - [(w - w r ) + (e - e'}} tan h, [38] 4 where w and e are read before, and w' and e' are read after, the reversal of the striding level. If the azimuth mark is not near the horizon, it is necessary to apply a similar correction to the observed direction of the mark. The correction is to be added algebraically to readings which increase in a clockwise direction. 89. The Direction Method. In observing for azimuth by this method the observations are carried out almost exactly as in measuring the angles of a tri- angulation, except that the chronometer is read whenever a point- ing is made on the star, and level readings to determine the inclination of the axis are made just before or just after pointing on the star. The altitude of the star should be measured at least twice during the observations. In observing Polaris in connec- tion with a number of triangulation stations, it is best to take the pointing on the star last. From twelve to sixteen sets should be made with the direction instrument, in order to secure the necessary precision. Following is an example of the form of record and computation of an azimuth by the method of directions. 114 ASTRONOMICAL OBSERVATIONS HORIZONTAL DIRECTIONS [Station, Sears, Tex. (Triangulation Station). Observer, W. Bowie. In- strument, Theodolite 168. Date, Dec. 22, 1908.] Position. Objects observed. d 6 H Q * g JJ S Backward. 6*52 g g a Jj a &c S3 Q+ 3 Remarks. h m " i Morrison . . 8 19 D A o o 35 35 i division of the 5 4i 41 striding level C 36 34 37.0 = 4"- 194 R A 180 oo 36 35 B 32 31 C 35 34 33.8 35-4 oo.o Buzzard... D A 53 30 43 42 B 41 42 C 34 33 39 2 R A 233 30 39 37 B 34 32 C 38 38 36.3 37-8 02.4 Allen D A 170 14 61 62 B 57 55 C 61 59 59-2 R A 350 14 50 49 B 63 60 C 53 53 54-7 57-0 21.6 Polaris.... D A 252 01 54 53 W E h m s B 54 53 9.3 28.0 i 48 35-5 C 51 51 52-7 27-7 9-i i 51 06.0 18 4 o ^ 18 Q i 49 50.8 R A 72 OI 09. 09 24-9 6-3 B 02 OI 13-0 31-7 C 10 08 06.5 29.6 H. 9 -13-5 25.4 - 7.0 THE DIRECTION METHOD COMPUTATION OF AZIMUTH, DIRECTION METHOD. [Station, Sears, Tex. Chronometer, sidereal 1769. = 32 33' 31" Instrument, theodolite 168. Observer, W. Bowie.] Date 1908, position Dec. 22, i 2 3 Chronometer reading I 49 50 8 2 oi 33 o 2 16 31 o 2 43 28 8 Chronometer correction 4 37 . 5 4 37 5 4 37 4 Sidereal time a of Polaris i 45 13 3 I 26 41 . 9 i 56 55-5 i 26 41.9 2 II 53.6 i 26 41 8 2 38 51-5 I 26 41 8 t of Polaris (time) o 18 31.4 o 30 13.6 o 45 ii. 8 i 12 09 7 t of Polaris (arc) 4 37' 5i" o 7 33' 24" o 11 17' 57" o 18 02' 25" 5 5 of Polaris . ... 88 49 27 4 log cot & 8 . 31224 8 31224 8 31224 8 31224 log tan 9.80517 9.80517 9 80517 9 80517 log cos t 9 99858 9 99621 9 99150 078ll log a (to five places) 8.11599 8.11362 8.10891 8 09552 log cot S 8 312243 8 312243 8 312243 log sec o 074254 o 074254 o 074254 log sin / . . ... 8.907064 9 118948 9 292105 94QOQ24 log-i-.. 0.005710 0.005679 o 005618 o 005445 i a log (-tan A) (to 6 places) A = Azimuth of Polaris, from north* Difference in time between D. andR... . 7.299271 o 06 50.8 m s 2 3O 7-5III24 o ii 09.2 m s 2 CO 7.684220 o 16 36.9 m 5 3 18 7.882866 o 26 15.0 m s i "*8 Curvature correction O o o o Altitude of Polaris = h o t II 33 46 33 46 33 46 ri 46 d - tan h = level factor 0.701 o 701 0.701 o 70! 4 Inclination "J" 7 7 2 7 o I 8 Level correction 4 9 5 o 4 9 I 3 Circle reads on Polaris 252 01 29 6 86 58 ii 2 281 54 27 o 116 45 48 6 Corrected reading on Polaris Circle reads on mark 252 01 24.7 170 14 57 o 86 5806.2 5 15 58 2 281 54 22.1 200 17 42 4 116 45 47.3. 35 18 45 4 Difference, mark Polaris 278 13 32 3 278 17 52 o Corrected azimuth of Polaris, from north * o 06 50 8 O II 09 2 o 16 36 9 o 26 15 o 180 oo oo.o 180 oo oo.o 180 oo oo.o 180 CO oo.o Azimuth of Allen . . 98 06 41 5 98 06 42 8 08 06 43 4 08 06 4"? I (Clockwise from South) To the mean result from the above computation must be applied corrections for diurnal aberra- tion and eccentricity (if any) of Mark. Carry times and angles to tenths of seconds only. * Minus, if west of north. t The values shown in this line are actually four times the inclination of the horizontal axis in terms of level divisions. n6 ASTRONOMICAL OBSERVATIONS 90. Method of Repetition. In observing by the repetition method the program given on p. 57 is followed, with the addition of readings of the chronom- eter and the stride level, taken when the telescope is pointing at the star. The altitude of the star should be measured, if possible, but may be computed from the known time if necessary. The verniers are read only at the beginning and end of a half set, as when measuring the angles of a triangulation. Following is an example of the form of record and computation of an azimuth by the method of repetition. RECORD AZIMUTH BY REPETITIONS. [Station, Kahatchee A. State, Alabama. Date, June 6, 1898. Observer, O. B. F. Instrument, lo-inch Gambey No. 63. Star, Polaris.] [One division striding level = 2. "67.] Objects. Chr. time .on star. 3 o 1 1 1 0> rt Level read- ings. W. E. Circle readings. Angle. ' A B S h m s at it Mark D o 178 O3 22 ^ 2O 21 . 2 Star 14. 4.6 3O I 4? IO 7 i /O w o *^ J-T" tj.\J \s . ^ J.W . f 9-2 5-9 49 08 2 52 Si D 3 9.6 5.6 5-2 i?.o 56 10 R 4 11.3 4.0 7-8 7.4 Set No. 5.. 14 59 12 5 IS oi 55 R 6 8.7 6.6 IOO 16 2O 2O 2O 72 57 50.2 ii. 9 3-4 14 54 17-7 68.2 53.6 + 14-6 Star IH O4 44 R i II Q 3 A X 3 V/f. if if * y O * H 1 8.5 6.8 07 18 2 09 54 R 3 7-9 7-3 II. 2 4.1 Set No. 6.. 14 IS D 4 9.0 6.1 5-9 9-6 16 14 5 15 18 24 6 5-9 9-6 9.1 6.2 Mark D 177 27 oo OO OO 72 51 46.7 15 ii 48.2 69-4 53-i + 16.3 METHOD OF REPETITION 117 COMPUTATION AZIMUTH BY REPETITIONS [Kahatchee, Ala. = 33 13' 40" '.33.] Date 1898 set June 6 ; June 6 6 Chronometer reading 14 C4 17 7 15 ii 48 2 Chronometer correction 311 31 I Sidereal time 14 53 46.6 iq ii 17 I a of Polaris I 21 2O . 3 I 21 2O 3 t of Polaris (time) I"? 32 26 3 1 3 40 l6 8 / of Polaris (arc) 6 of Polaris .... 203 06' 34". 5 88 41: 46 q 207 29' 12". o log cot 5 8 33430 8.33430 lop tan O O77S3S O O77<3I 4.6 7 Corrected angle. .. ... 72 "?7 48 7 72 r , x = ===== [41] a (i e 2 ) sin r , and y = ) /. " [42] vi - ^sm 2 95. Radius of Curvature of the Meridian. To find the radius of curvature of the meridian (7O> a Ppty the general formula dx? dy x & / = --- - ^ 3; a 2 * The relation i + tan 2 = sec 2 is used in this transformation. T^ / \ From (i) ^ 3; a 2 126 PROPERTIES OF THE SPHEROID Differentiating this equation, we have d\ dx 2 VI , x 2 b*\ = - I -v 4- i a 2 / V r y a 2 / JL a 2 / } Therefore R m = - [a 4 / + I i e L sin" 5 < i e L sm* 6 4 a 2 cos 2 M be computed, it will be found to be p = i - * cos sec \f/ According to Meunier's theorem the radius of curvature of the normal section equals the radius of curvature of this central section divided by the cosine of the angle between the two planes, * The negative sign indicates only the direction of bending. It is customary to regard the value of R m as positive. RADIUS OF CURVATURE IN THE PRIME VERTICAL 127 that is, by cos (< t). Hence the radius of curvature of the prime vertical section is N. To show this geometrically, let A and B in Fig. 58 be two points on the same parallel of latitude. The normals to the surface at* A and B always intersect at H on the minor axis. Let C be a point on the prime vertical section through A, and also on the meridian of B. The normals at A and C intersect at some point K above H. K is approximately the center of curvature of the FIG. 58. arc AC. When the meridian PBC is taken nearer to A, points A and C approach each other, the intersection of their normals approaches the true center of curvature, and the length CK approaches the true radius of curvature. But the nearer C approaches J., the nearer it approaches B. Hence CK must ultimately coincide with AH; that is, H is the point toward which the center of curvature is approaching and the normal N is the radius of curvature of the prime vertical section at A . From Fig. 56 it is evident that N=-^ [44] Values of log N will be found in Table X. 128 PROPERTIES OF THE SPHEROID ^^ The normal terminating in the teai*q) axis is n = r -, [45] , Vi-e 2 sin 2 The radius of the parallel of latitude ( = x) is given by R p = N cos <. [46] 97. Radius of Curvature of Normal Section in any Azimuth. Having found the radii of curvature of the two principal sections, it now remains to find a general expression for the radius of cur- vature in any azimuth, and it will be shown that this may be expressed in terms of the two radii already found. The equation of the spheroid is or bV + b V + flV = a?b\ (a) In Fig. 59 the Zi-axis coincides with the polar axis of the spheroid. If M be any point on the meridian ZiM, and MY any section cut by a plane through MH (the normal) making an angle a with the meridian, then the equation of the spheroid may be transformed so as to refer to the origin C and the new Z axis CM. Let the coordinates of any point P be #1, y\ t z\, and let the new coordi- nates be x, y, z. Then, from Fig. 59, the relation of the new, coordinates to the old is given by xi = OG = OC + x + 2 cos + y cos a sin = Ne* cos + x + 2 cos -f y cos a sin <, yi = y sin a, Zi = 2 sin y cos a cos 0. Substituting these values in (a), b 2 (Ne* cos -f s + 2 cos + 3; cos a sin <) 2 + #y sin 2 -f a 2 (s sin < y cos cos 0) 2 = a 2 Z> 2 , which is the equation of the spheroid referred to the new axes. If x is made equal to zero, then P will be on the curve M Y, and the equation becomes the equation of this plane section, that is, RADIUS OF CURVATURE OF NORMAL SECTION 129 b 2 (Ne* cos + z cos < + y cos a sin ) 2 + #y sin 2 a + a 2 (z sin y cos a cos ) 2 the equation of the ellipse MY. FIG. 59. To determine the radius of curvature at M it is necessary to find -7- and - - and to substitute these values in the general dy dy* formula for radius of curvature. Expanding the last equation, collecting terms, and dividing through by a 2 , y 2 [i - # (i - cos 2 a cos 2 0)] + z 2 (i - e 2 cos 2 0) yz (i # cos a sin cos #) + 2 y (i e 2 ) # e 2 cos a sin cos + 22e 2 (i - e 2 ) .^.cos 2 ^ = (i - e 2 ) (a 2 - 130 PROPERTIES OF THE SPHEROID or, in abbreviated form, y*A + z*B - yzC + 2yD + 2zE = F. Differentiating this equation, y being taken as the independent variable, dy dy dy Differentiating again, 75 2B y A/\ 2 ^dz (- r ) - 2C \dl cos cf>) 2 (i e 2 )Ne 2 cos a sin N (i - e 2 ) _ (i e 2 ) (sin 2 a + cos 2 a) + e 2 cos 2 a. (i sin N(i- e 2 ) (i e 2 ) sin 2 a + cos 2 cos 2 a e 2 sin 2 sin 2 a -\ ^-r cos 2 a (i e 2 sin 2 i e 2 in 2 o: + N CQ& a NR m THE MEAN VALUE OF 131 Substituting these differential coefficients in the usual formula for radius of curvature, we have r.,.1 L47J N cos 2 a + R m sin 2 a If a = o, the radius of curvature of the meridian; and if a = 90, then R^ = = = N, Rm the radius of curvature of the prime vertical. Values of log Ra for different latitudes and azimuths will be found in Table XI. 98. The Mean Value of R a . The mean value of Ra at any point for all azimuths from o to 360 may be found as follows: if the angular space about any point M be divided into a large number of small parts, each equal to da and each expressed as a fraction of a radian, then the num- ber of such parts in a radian will be , and the number in a cir- da cumference will be ^~ . If the value of R a be computed for each da of these azimuths, then the sum of these values of R a , divided by their number, is the mean value; that is, mean R a = I R a JQ 2TT NR 2Tr Jo N cos 2 a + R m sin 2 a tffc. TT JQ IT Jo N cos 2 a + R m sin 2 a da. 132 PROPERTIES OF THE SPHEROID To integrate this quantity, substitute a new variable, / = tanaV/-?, from which dt = l/T?- T~- Bv dividing both * N ' N cos 2 a numerator and denominator by N cos 2 a and factoring NR m , the integral may be put in the form I T * Rm I j P 2 /-^r, ^"LZ^i^Z mean # a = - v^ m ]V I , . N cos 2 a which, by substitution, becomes mean R a = - V^ m A^ f - 7T Jo I *_ + / 2 = - VR m N [tan- 1 7T [48] The mean radius of curvature is, therefore, the geometric mean of the radii of curvature of the two principal sections. 99. Geometric Proofs. Geometric proofs of the last two formulae will be found in- structive. To find R a geometrically, imagine a tangent plane at the point M and also a parallel plane at an infinitesimal dis- tance below M. This second plane will cut from the surface a small ellipse. It has already been shown that the radius of curvature of the prime vertical section is ^V. In Fig. 60 the points A, M, and B are on the circle whose radius is N and whose center is the point H on the axis. By similar triangles, M C : CA = CA : CK. Since MC is infinitesimal, GEOMETRIC PROOFS similarly, for a section in the meridian ur b * = ; 2 -K-m and, in general, for any section, MC =-^-- 133 The coordinates of the point P (Fig. 61) are x = s siri a and y = s cos a. FIG. 61. Substituting these in the general equation of the ellipse, s 2 sin 2 a s 2 cos 2 a ~^~ ~v~ ' But, from the preceding equations, - 2 = and a 2 N R hence or NR m N cos 2 a + R m sin 2 a [47] PROPERTIES OF THE SPHEROID To show geometrically that the mean value of R a = observe that, as before, mean R a -T 2 IT JQ R a -da and, from the preceding paragraph, Therefore But Therefore mean K n = 2 IT JQ b 2 da. i r 2 * I s 2 da = area of ellipse = irab. 2 JQ n mean R a = - X irab X -^ TT 0* Therefore aR, b mean R a = VNR m . [48] 100. Length of an Arc of the Meridian. Any small arc of the meridian ellipse may be regarded as an arc of a circle whose radius is R m , the error being very small for short arcs. The length, therefore, is or, if d" arc i". [49] If the arc is so long that the value of R m varies appreciably, it is necessary to find 5 by integrating the expression a (i e 2 ) ds = * ': dd> between the limits . Integrating, ) d. In order to integrate the terms of the series in parenthesis we simplify the expression by means of the following relations: sin 2 = \ \ cos 2 0, sin 4 = f \ cos 2 + \ cos 4 0, sin 6 = ye if cos 2 + y g cos 4 -^ cos 6 0. Integrating and substituting the limits, 0i and 2 , we have 5 = a (i e 2 ) { A (02 0i) \ B (sin 2 02 sin 2 0i) + 1 C (sin 4 02 - sin 4 0i) . . . } , [50] in which A = 1.0051093, B = 0.0051202, and C = 0.0000108. (See Jordan's Handbuch der Vermessungskunde, Vol. Ill, p.. 226; and Crandall's Geodesy and Least Squares, p. 163.) 101. Miscellaneous Formulas. The following formulas, relating to the ellipse, are given here for convenience of reference. The geocentric latitude may be found from the expression tan \l/ = * = (i e 2 ) tan = tan 0. [51] x a* The maximum difference between and ^ is about o n' 40", at latitude 45. At the equator and at the poles the difference is zero. The reduced latitude, (see Art. 94, p. 123), may be found from the geodetic latitude by means of the relation a tan = b tan [52] which is readily proved from Fig. 57. PROPERTIES OF THE SPHEROID The compression of the spheroid, that is, the flattening at the poles, is expressed by f- (sal The length of a quadrant of the meridian is given by [54] FIG. 62. 102. Effect of Height of Station on Azimuth of Line. Since the normals drawn from two points on the surface do not in general lie in the same plane, there will be an error in the observed horizontal direction of a station, depending upon its height above the- surface of the spheriod. This error may be likened to the error of sighting on an inclined range-pole; the * From the equation for the length of a meridian arc, we have for the quadrant JT q = a (i - c 2 ) f ( i + ^ t 2 (i - cos 2 ) + ^ e 4 (3 - 4 cos 2 + cos 4 ) ) '. In the triangle MEE' sin 5 EE' EE' sin EE'M EB + BM EB , . (approx.), or EB where 0' is the latitude of B'. 138 PROPERTIES OF THE SPHEROID Now EH' = OH' - OH = (N r - n f ) sin 0' - (N - n) sin j^V = N'# sin 0' - Ne 2 sin 0. Wf5 ^ Therefore 5 = - (N'e 2 sin 0' - #e 2 sin 0) ('N' . \ sin <' sin 0)> AT in which may be put = i with small error. Then 6 = e 2 cos ' (2 cos - (0 + ') sin V where A = <' ; Then 6 = ^cos 2 ^' X A< (approx.). = e 2 cos 2 0' - cos a / 9 9 /\ 8 2 2 ,f (i e 2 sin 2 ) 2 = e L cos 2 ^ cos a * ; ~ LZ - *(i - e 2 ) The factor g >m differs but little from unity (i e ) may be considered equal to unity in this equation. , e 2 5 cos 2 0' cos a ( . Then 5 = - (a) The linear distance BB' = hd, and the correction to the azi- muth (x) 3.t point A is given by hd sin a s arc i" /?e 2 cos 2 X sin o; cos a a arc i" arc i" e 2 - sin 2 a cos 2 r , [55] as given by Clarke (Geodesy, p. 112). This may be written x" = k h sin 2 a cos 2 0' [56] REFRACTION 139 *2 where 2 \ the dimensions being in meters. The logarithm of k is 6.03920. When the signal is NE or SW of the observer the azimuth must be increased to obtain the correct azimuth at sea-level; if the signal is NW or SE the observed azimuth must be decreased. If, when deriving the above equation, we place the fraction n O / / p = i, the formula for x" should have a replaced by N. I r For = 45, a = 45, and h = 1000 meters, the value of x" is o".o547. This is much smaller than the probable error of an observed direction (see p. 65), and is therefore negligible except for great heights. This correction has been applied to angles measured in the primary triangulation of the California and Texas arc and the California and Washington arc. It is too small to affect the triangulation of the eastern half of this country. Questions. What influence does the height of the observer have upon the result? Why does the distance not enter into the formula? Which one of the two approximations is more accurate, that giving a in the denominator, or that giving .AT ? 103. Refraction. Inasmuch as the refraction acts in the vertical plane at any point, and the vertical plane changes its direction as the ray pro- ceeds along the line, it is evident that there must be some hori- zontal displacement of the object sighted, due to the refraction. Investigations show that this error is quite inappreciable for all lines that can actually be observed. 104. Curves on the Spheroid. The Plane Curves. When a theodolite is set up at any point A and leveled, its vertical axis is made to coincide with the direction of the normal at A , which, except for local deflections, coincides with the direc- tion of the force of gravity at A. If another theodolite is set up at B, in a different latitude and a different longitude, it is evident 140 PROPERTIES OF THE SPHEROID that these vertical axes are not in the same plane, since their normals (plumb lines) never intersect. The greater the latitude, the lower the point where the normal intersects the polar axis. It is clear that the line marked out on the surface of the spheroid by the line (or, rather, plane) of sight of the first theodolite is not the same as the line marked out by the vertical plane of the other theodolite. If A is southwest of B, then the curve cut by the plane of the theodolite at A is south of that cut by the plane of sight of the theodolite at B. This may be seen from the fact that both planes contain the chord AB; and since the normal at A is higher at the polar axis, the curve itself must be lower (farther south) . 105. The Geodetic Line. Another curve which holds an important place in the theory of geodesy is known as the geodetic line. This is the shortest line that can be drawn on the surface of the spheroid between two given points. It is not a plane curve, but has a double cur- vature, A characteristic property of the curve is that the oscu- lating plane * at any point on the curve contains the normal to the surface at that point. In most cases the geodetic line is found to lie between the two plane curves and has a reversed curvature. Fig. 63 is a photograph of a model, the semi-axes of which are a = 6 inches and b = 3.5 inches. The two plane curves are shown and between them, with the curvature slightly exaggerated, is the geodetic line. In order to obtain a clear conception of the nature of the ge- odetic line, let us imagine that a transit instrument is set at point A (Fig. 64), leveled, and then sighted at point B. Then it is moved to point B, set up, and leveled again, and a back sight is taken on A] point C is then fixed by reversing the telescope. When the sight is taken to A, the sight line traces out the plane curve BbA ; and when point C is sighted, it traces out BbC. The * The osculating plane may be considered to pass through three consecutive points of the curve. In reality it is the limiting position approached by the plane as the distance between the three points decreases indefinitely. THE GEODETIC LINE 141 FIG. 63. Plane Curves and Geodetic Line. FIG. 64. 142 PROPERTIES OF THE SPHEROID instrument is then taken to C and the process repeated. It should be observed that the (vertical) sight plane of the instru- ment coincides with the normal to the surface at each station. If the points A, B, C, D are imagined to approach nearer and nearer, so that AB, BC, etc., become infinitesimal elements of the curve, the plane which contains three consecutive points of the curve also contains the normal to the surface. If we imagine the instrument to move along this line, it is seen that the vertical plane of the instrument twists so that it always contains the normal. One of the characteristic properties of the geodetic line is shown by the equation R p sin a = k, a constant [57] R p being the radius of the parallel and a the azimuth of the ge- odetic line at any point. This equation may be derived analyti- cally by the methods of the calculus of variations (see Clarke, Geodesy, p. 125) or by geometric construction (see Jordan, Ver- messungskunde, Vol. Ill, p. 395). From this equation it will be seen that when a is a maximum (90), sin a = i and R p = k. The constant of the equation is therefore the radius of the parallel of latitude beyond which the geodetic line does not pass. When a is a minimum, R p is a maximum, that is, R p = a, the equatorial radius of the spheroid. This shows that in general a geodetic line cutting the equator at any angle a may go northward up to some (limiting) parallel of latitude (corresponding to R p = k) t but will not pass north of this parallel. In the southern hemi- sphere it will reach a limit ( 0) having the same numerical value. Such a geodetic line, when traced completely around the spheroid, will not in general return exactly on itself, but will pass the initial point on the equator in a slightly different longitude and then proceed to form another loop around the spheroid. Except for a few particular cases the geodetic line lies between the two plane curves and divides the angle between them in the ratio of about 2 to i, as shown in Fig. 65. THE ALIGNMENT CURVE 143 If the terminal points P and Q are in nearly the same latitude, the geodetic line may cross the plane curve. It is important to bear in mind that the lengths of these different curves on the spheroid differ by quantities that are quite inappreciable in practice. The differences in length are far shorter than the distances by which the curves are separated at their middle points (Art. 107), and even these latter are negligible in practice. Also the angle by which the azimuth of the geodetic differs from the azimuth of the plane section is much smaller than can be measured. FIG. 65. It should be noted that the geodetic line itself cannot be sighted over directly, because it is not a plane curve, and that the geodetic triangle can be obtained only by computation. 1 06. The Alignment Curve. Another curve which may be drawn on the surface is denned in the following manner: if the theodolite be supposed to move from A to B, keeping always in line between the two points (that is, the azimuths of A and B 180 apart), and the instrument being always leveled, its path will be a curve which lies very close to the geodetic line and generally between the two plane curves. This is called the alignment curve. It is possible to define other curves * between these two points. * See Coast Survey Report for 1900, p. 369. 144 PROPERTIES OF THE SPHEROID Such curves are of theoretical value only, since the lengths of all such lines on the earth's surface differ from each other by quanti- ties too small to measure. The two-plane curves, however, are separated by a distance which is quite appreciable. 107. Distance between Plane Curves. The maximum separation of the two plane curves may be computed approximately as follows: the angle (d') between the FIG. 66. two planes is very nearly equal to the angle d multiplied by sin a, since d is the angle measured in the plane of the meridian, whereas the angle desired (5', Fig. 66) is that perpendicular to the planes of sight. Therefore 5' = se 2 cos 2 cos a sin a N (see equation (a), p. 138). The distance of the chord AB (Fig. 67) below the surface (D) at its middle point is given by 2 2 or, approximately, D DISTANCE BETWEEN PLANE CURVES The curves are separated at their middle points by the hori- zontal distance n*' s 2 ^ s # cos 2 cos a sin a - e 2 cos 2 cos a sin a. [58] D FIG. 67. The difference in azimuth may be computed approximately by finding the angle between the two tangents to the curve drawn from one of the stations and prolonged half the distance (Fig. 68). The terminal points of these tangents will be at a distance D above the surface and will be separated by a distance 2 D8'. The angle between these two lines is nearly 2 P8' k s arc i" 2 5 s e 2 cos 2 # cos a sin a isarci" e 2 cos 2 cos a sin arc i [59] 146 PROPERTIES OF THE SPHEROID For the oblique boundary line between California and Nevada * 5 = 650,000 m., (400 mi.), m = 37 oo', a = 134 33'; whence Dd' = 1.8 meters and the difference in azimuth = 2^. . 2D8' FIG. 68. For the western boundary of Massachusetts s = 80,930 m., (50 mi.), m = 42 24', a = 195 12'; this gives Dd' = 0.0015 meter and Aa = o".oi6. PROBLEMS Problem i. Prove by the process outlined in the first paragraph of Art. 96 that the radius of curvature of the prime vertical section of the spheroid is N, the nor- mal terminating in the minor axis. Problem 2. A model of the spheroid has an equatorial diameter of 12 ins. and a polar diameter of 7 ins. Compute the correction to reduce to "sea-level" the azimuth of a line in latitude 45, the azimuth being 45 and the elevation of object being one inch above the surface of the spheroid. Problem 3. What will be the maximum separation of two plane curves drawn on the model described in problem 2 if s = 7.5 ins., mean < = 30, a = 45? (Use the approximate formula.) * See Coast Survey Report for 1900, p. 368. CHAPTER VI CALCULATION OF TRIANGULATION 108. Preparation of the Data. From the records of the field-work of the triangulation we ob- tain a value for each angle, supposed to be freed from the errors of the instrument, eccentricity of station, phase of signal, eleva- tion of signal, etc. Before these angles are employed for solving the triangles, they should be examined to see if they satisfy any geometric conditions existing among them. If at any station two or more angles and their sum have been measured, then these angles must be so corrected that they exactly equal their sum. If the horizon has been closed, the measured angles must be ad- justed so that their sum equals 360. If the angles have been measured with different degrees of precision, as, for example, with different instruments or a different number of sets or of repetitions, the different angles should be given proper weights; and if the best possible values are desired, the angles at each station should be adjusted by the method of least squares. After the station adjustment, as it is called, has been completed, the triangles must be examined to see if the sum of the three angles in each triangle fulfills the requirement that this sum shall equal 180 plus the spherical excess of the triangle. The verticals at the three triangulation stations are not parallel to each other, because the surface is curved. Consequently the sum of the angles will exceed 180 by an amount which, on a spherical surface, would be exactly proportional, and which, on a spheroidal surface, is nearly proportional to the area of the tri- angle. As was shown in the preceding chapter (Art. 102), the error in the direction of an object, due to the fact that the earth is spheroidal 147 148 CALCULATION OF TRIANGULATION instead of spherical, is extremely small, even when the object is several thousand meters above sea-level. Hence it follows that if the vertices of a spheroidal triangle are projected vertically onto the surface of a tangent sphere,* the errors thus produced in the horizontal angles of the triangle will be much less than the errors in the measurement of the angles, because the points on the sphere and those on the spheroid are separated by compara- tively short distances. This enables us to compute spheroidal triangles as spherical triangles and greatly simplifies the com- putation. The lengths of the triangle sides will be practically the same on the two surfaces. In this connection it is well to bear in mind that if the topog- raphy of the earth's surface were represented on an 1 8-inch globe the total variation in elevation would scarcely be greater than the thickness of a coat of varnish. The elevation of the geoid above the spheroid would be very much smaller than this, and the distance between the spheroid and the tang^^ sphere at any station would usually be still smaller. This will give some idea of the minuteness of the errors under discussion. It should be remembered that, whereas the triangulation stations themselves are at va- rious heights above sea-level, these are all supposed to have been projected down vertkally onto the spheroid before beginning the computation of the triangle. The points of which we shall speak in discussing the solution of the triangles and the geographical positions of the stations are these points on the spheroidal surface and not the original station points. * The sphere is supposed to be tangent at the center of gravity of the triangle to be computed. FIG. 68a. SPHERICAL EXCESS 149 In solving triangles by the methods given below, the following approximations have been made, and it is assumed that in all cases the resulting errors are negligible. i. The reduction to sea-level reduces the observed direction to that corresponding to the geoid (or actual surface), not the spheroid, as is assumed. '2. The effect of local deflection of the plumb line is not allowed for. 3. The effect of atmospheric refraction on the direction (hori- zontal refraction) is neglected. 4. The reduction of the observed direction (plane curve) to that of the geodetic, or shortest, line is omitted. There are in reality eight triangles formed by the plane curves, which are treated as if they were identical (see Art. 104). 109. Solution of a Spherical Triangle by Means of an Auxiliary Plane Triangle. The direct solution of the triangles of a net as spherical triangles j^uld be unnecessarily complicated. This may be avoided by employing a principle known as Legendre's Theorem, namely, that if we have a spherical triangle whose sides are short com- pared with the radius of the sphere, and also a plane triangle whose sides are equal in length to the corresponding sides of the spherical triangle, then the corresponding angles of the two tri- angles differ by approximately the same quantity, which is one- third of the spherical excess of the triangle. no. Spherical Excess. The spherical excess of a triangle is directly proportional to its area, as shown in spherical geometry. Hence, if A' is the area of any triangle, R is the radius of the sphere, S is the surface of the sphere, and e is the spherical excess of the triangle; then, since the spherical excess of the tri-rectangular triangle is - , 150 CALCULATION OF TRIANGULATION ie 2 A f 7 ;= A' Therefore e = To express e in seconds of arc, divide by arc i", and we have A' be sin A r, -, 6 = F^7 > = 2 ^arci"' where b, c, and A are two sides and the included angle of the tri- angle, a and b being in linear units. The sphere which is tangent to the spheroid at the center of gravity of the triangle, and which has the same average curva- ture, is a sphere of radius = VR m N', whence be sin A , . r , , e = T) *r - 7-. = mbcsmA. [6i\ 2R m Nsuci" The quantity - - = m is given for different latitudes 2 R m N arc i in Table XII. The latitude to be used in finding m is the mean of the latitudes of the three vertices of the triangle. Questions, Is this auxiliary plane triangle the same as the chord triangle formed by joining the points by straight lines? Are the two similar in shape? in. Proof of Legendre's Theorem. To prove Legendre's theorem, let A', B' and C be the angles of the spherical triangle, and A, B, and C those of the plane tri- angle; the sides of the plane triangle are a, b, and c, and those of the spherical triangle are a'R, b'R, and c'R, then, in the plane triangle, tf + c 2 - a 2 cos A = -- - -- > (a) or - a* - b* - c* PROOF OF LEGENDRE'S THEOREM 151 In the spherical triangle, A f cos a' cos 6' cos c' cos A = ; 1-7; -, sin 6 sm c Expanding each sine and cosine (omitting terms of higher order than the fourth), a' 2 a' 4 / b '*^ b '*\( C ' 2 -L- C '*\ i 1 i 1 lli 1 1 . , 2 24 \ 2 24/ \ 2 24/ k (- a' 2 + b" 1 + c' 1 ) - A (6'* + C' 4 - o' 4 ) - j 6V 2 = ft ( - a' 2 + ft' 2 + c' 2 ) - A (&'" + c' 4 - a' 4 ) - i 6V 2 ] I +, t? C ' c /4 _ fl /4 + 6 & y 2 26V 2 4 6V - a'W + 6 /4 + 2 6V 2 - a ; V 2 + 1 2 6V whence cos .4' = + 2 a r V /2 + 2 2 6V' 6 4 6V From (a), (6), and (c) cos ^4' = cos A \ 6V sin 2 A. Let # be the difference between A and A 1 '. Then cos x = i and sin x = x" arc i" (nearly), since x is small, and cos A' = cos (A + #) = cos A sin ^Iz" arc i /; = cos ^4 i 6V sin 2 A ; that is, x" arc i" sin ,4 = \ 6V sin 2 ^. r r // 6Vsinyl Therefore x" = - 77 > 6 arc i IJ2 CALCULATION OF TRIANGULATION or, since b f = - and c' = > /v /v. be sin ^4 r , = 6 IP are i" ' [62] It will be noticed that this is one-third of the spherical excess as found in Equa. [60]. The same result would also be found for angles B and C. 112. Error of Legendre's Theorem. The error in Legendre's theorem * as applied to the sphere may be studied by carrying out the above series so as to include terms of higher powers than the fourth. Jordan (Vermessungskunde) gives a numerical example showing the amount of this error in a triangle of which the side AC is about 65 miles in length; the angles are shown below: ,4 ' = 40 39'-3<>".38o B' = 86 13 58 .840 C' = 53 06 45 .630 180 oo' i4".85o Denoting the spherical angles by A', B' } C f , and the correspond- ing plane angles by A, B } C, the differences are as follows, the first column containing the values derived from Legendre's theorem in its ordinary form, the second containing the smaller terms which are usually neglected. Approx. Exact. A' A 4".Q5ooi8 4.950036 B' B 4 ..950018 4.949997 C' C 4 .950018 4.950021 113. Calculation of Spheroidal Triangles as Spherical Tri- angles. It is customary to assume that the differences between the spherical and spheroidal triangles are negligible when the actual points are projected down onto a tangent sphere of radius VR m N. Clarke, in his Geodesy, shows the error of this assumption in the case of a triangle having a side over 200 miles long, the result being as follows: * See Coast Survey Special Publication No. 4, p. 51. CALCULATION OF THE PLANE TRIANGLE 153 A' B' C' e' Spheroidal 98 44' 37".096| 58 16' 4 6".S994 2 3 OO' I2". 73 03 A B C e Spherical 98 44' 37"-i899 58 16' 4 6".4737 23 oo' I2". 7 6 3 4 The preceding example indicates that in triangles composed of lines such as can be sighted over on the earth's surface the error involved in computing spheroidal triangles as spherical triangles is negligible in practice. 114. Calculation of the Plane Triangle. After the spherical excess has been computed, the angles of an auxiliary plane triangle may be found by applying Legendre's theorem, that is, by deducting one- third of the spherical excess from each spherical angle. The difference between the sum of these plane angles and 180 is the error of measurement and may be distributed equally among the three angles unless a least- square adjustment is to be made. In any case this method of distributing the error may be used for a preliminary determina- tion of the distances. The lengths of the triangle sides are now found by plane trigonometry. Since all three angles of a tri- angle will usually be known, the only formula that will be used, except in rare cases, is the sine formula, a _ sin A b ~ sinB A convenient arrangement of this computation, used by the Coast and Geodetic Survey, is shown in the following table. The spherical excess of the triangle in this case is o".86, which give? i ".2 as the error of closure of the triangle. Stations. Observed angles. Correc- tion. Spheri- cal angles. Spheri- cal excess. Plane angles and distances. Loga- rithms. Blue Hill to Prospect o / n 61 47 18 8 it n 18 4 n 22723. 08 m. t II 4.356 4673 Blue Hill 35 45 15.4 0.4 15 o o 3 35 45 *4 7 9 766 6415 Prospect 82 27 27.9 0.4 27.5 0.3 82 27 27.2 9.9962261 Observatory to Prospect . . . Observatory to Blue Hill. . 180 oo 02.1 15067.13 25563.20 4.178 0306 4.4076152 154 CALCULATION OF TRIANGULATION US. Second Method of Solution by Means of an Auxiliary Plane Triangle.* Another method of solution which has been used to some ex- tent in Europe is as follows: Let ABC (Fig. 69) be the spherical triangle and A'B'X an auxiliary plane triangle having two of its angles, a and j3, equal to the corresponding angles in the spherical triangle. Evidently the third angles will not be equal. A' FIG. 69. Let a' and b' in the plane triangle be the sides corresponding to a and b. In the spherical triangle we have . a sm- sm a R sin/3 . b' sm- and in the plane triangle sin a a sin |3 b' for all values that may be given to a' and b'\ whence . a a sin , = L = ^ b ~ b'~ b'' sm- R * See Jordan, Vermessungskunde, Vol. Ill, 39. SECOND METHOD OF SOLUTION 155 This equation is satisfied if we place ^L -. ' i V . b and = sin The general expression for any triangle side may be written --sin- s' being the side of an auxiliary plane triangle corresponding to the side s of the spherical triangle. Taking logs of both members, v ^ i ^ ^^ log- = log sin- = log! - -=. - J\. K. Vtv A Now, since 2 3 (where M = log e 10 = 0.4342945, the modulus of the common logarithms), we may write , s' , . s , s log - = log sin- = log- \ Mf ^_Y / *\ OT ' Ms 2 , .?: , 5' Ms 2 Therefore log - - log - = / r,. i or log s- logs =- 2 > 1 6 3J which is the correction to the log of the triangle side. * The next term -3- 51 o.ooo ooo oooi for a distance of 100 kilometers. loo K* 156 CALCULATION OF TRIANGULATION In calculating this correction, R 2 should be replaced by R m N. Values of these corrections will be found in Table XIII for the argument log s. Example. Stations. Spherical angles. Distances. Logarithms. Blue Hill to Prospect. .... 1 II 22,723 .08 4 -35^ 4673 Correction g s' 4 -2c6 4.664 Observatory 61 47 18 4 O CX4 Q2IS Blue Hill ?s 4^ i^ .0 9 .766 6423 Prospect 82 27 27.5 Q.QQ6 2262 s' 4 I?8 O3O2 Correction 4 Observatory to Prospect. . 15,067.13 4.178 0306 4.407 6141 Correction II Observatory to Blue Hill . . 25,563.20 4.407 6152 Notice that after the base of the first triangle has once been reduced by subtracting the correction, the computation of the whole chain of triangles may be carried out, using the spherical angles only. It is not necessary to add the corrections to the logarithms of the computed sides until their true values are to be found. PROBLEMS Problem i. Compute the area in square miles of a triangle on the earth's sur- face having a spherical excess of i", assuming that the earth is a sphere of radius 3960 miles. Problem 2. Compute the sides of the following triangles: Correction to angles from figure adjustment. Error of closure of triangle. Corrected spherical angles. Spherical Station. (a) Mt. Ellen Tushar Wasatch Wasatch to Mt. Ellen; azimuth, 333oi'o8".65; back-azimuth, 153 25'o5".oo; dist. 123,556.70 meters; logarithm, 5.0918663. Latitude of Wasatch, 39 06'- 54".364; longitude, 111 27' n".9i5. (&) Uncompahgre -f-o' / .i7 Mt. Waas o .10 Tavaputs +o .58 -o". 7 o ) ( 49 36' 36".88 ) +o .98 +o". 22 \ 55 56 26 .70 o .06 ) ( 74 27 30 .75 ) ( 31" 54' 6i".S7 ) -fo".6s 98 16 41 -16 I 46".i5 f 49 48 63 .42 ) PROBLEMS 157 Mt. Waas to Uncompahgre; azimuth, 288 01' 25".7i; back-azimuth, 109 07'- o6".n; dist. 162,928.01 meters, logarithm, 5.2119958. Latitude Mt. Waas, 38 32' 2i".444; longitude, 109 13' 38".3O2. Problem 3. Position of point B j ^" 39 , T 3, ^"'ctfi Position of point C Azimuth B to C 353 17' 2i".8i; dist. 40232.35 meters; (^"=4.6045754); back-azimuth 173 19' 24 / '.64. The spherical angles are 4 57 53' 14^.39 (A is east of BC.) B 62 23' 3 i". 4 o C 59 43' i7"-93 Compute the spherical excess and solve the triangle. , Problem 4. Position of pt. L\ latitude 42 26' i3".276, longitude 70 55'52".o88. Distance L to N, 3012.0 meters (log = 3.478 8600). Azimuth L to N, 314 34' oo"; back-azimuth, 134 35' 03". Position of pt. N, latitude 42 25' 04".764, longitude 70 54' i8".2 3 2. Angle at L, 36 15' 07"; at N, 63 44' 59"; at E, 79 59' 5?"- (E is east of LN.) Compute the spherical excess and solve the triangle. Problem 5. The observed angles of a triangle and their corrections as found by adjustment are as follows: Angle. Corrections. Sand Hill 40 57' 2 8".i 3 -o". 3S Rutherford 54 22 59 .51 o .61 Miller 84 39 35 .03 o .44 The position of Rutherford is latitude = 37 08' 57"-928 N, longitude = 98 06' 3i".6i8 W. The position of Miller is latitude = 37 02' 20^.963 N, longitude 97 55' 43"-9o8 W. The azimuth from Miller to Rutherford = 127 28'- i7"-95; back-azimuth 307 21' 47".3o. Distance in meters, 20139.64; logarithm, 4.3040518. Solve the triangle. Problem 6. Show that the substitution of .Equa. (b) p. 150 in Equa. (c) p. 151 is permissible under the assumptions made in Arts. 109 and in. CHAPTER VII CALCULATION OF GEODETIC POSITIONS 1 1 6. Calculation of Geodetic Positions. In geodetic surveys covering large areas the positions of the triangulation points are expressed by means of their latitudes and longitudes. Over limited areas . a system of rectangular spherical coordinates may be used to advantage, but for such areas as have to be surveyed in this country the latitude and longitude system is preferable. Before the latitude and longitude of one triangulation station can be calculated from the coordinates of another station, it is necessary to know the dimensions of the spheroid which is taken to represent the earth's figure, and also to fix definitely the lati- tude and longitude of some specified station, as well as the azimuth of the direction to some other triangulation station. This selected position and direction determine the relative posi- tion of the whole survey with respect to the adopted spheroid, and constitute what is known as the geodetic datum. The surveys of different countries may be computed on different spheroids or may be located inconsistently on the same spheroid. The different portions of a survey of the same country will be located inconsistently on the same spheroid until they have been con- nected by triangulation. The two spheroids which have been most extensively used for geodetic surveys are (i) that computed by Bessel in 1841, and (2) that by Clarke in 1866. The Bessel spheroid was computed from data obtained chiefly on the continent of Europe, and conse- quently conforms closely to the curvature of that portion of the earth. This spheroid is still in general use in Europe. Clarke's spheroid of 1866 was computed from arcs distributed over a much 158 THE- NORTH AMERICAN DATUM 159 larger portion of the earth's surface; it shows a greater amount of flattening at the poles than the Bessel spheroid, and conse- quently assigns a flatter curvature to the. surf ace in the latitude of Europe and of the United States. The Bessel spheroid was employed by the Coast Survey in the earlier years. As the sur- veys gradually extended, the errors due to using this spheroid became more and more apparent, until finally, in 1880, it was decided to change to the Clarke spheroid. The latter conforms much more nearly to the curvature of the surface in the United States. 117. The North American Datum.* In 1901 the United States Coast and Geodetic Survey adopted what was then called the United States Standard Datum, by assigning to the station Meade's Ranch the following position on the Clarke spheroid: Latitude, 39 13' 2 6".686 Longitude, 98 32' 3o".5o6 Azimuth to Waldo, 75 28' 14". 52 In 1913 this datum was adopted by the governments of Canada and Mexico, and it is now known as the North American Datum. In deciding upon a geodetic datum it was necessary to con- sider two important points: first, the datum should be so chosen as to reduce to a minimum the labor of recomputing the geodetic positions; second, it must place the triangulation system in such a position that no serious error will occur in any part of the sys- tem. At the time this datum was selected there was a large number of triangulation points located along the Atlantic Coast. By selecting a position for Meade's Ranch consistent with the old datum upon which this triangulation was calculated, a large amount of recomputation was avoided. At the same time it was apparent that this also placed the triangulation very near to its theoretically best position. * See Coast Survey Special Publication No. 24, p. 8, or Special Publication No. 19, P- 80. i6o CALCULATION OF GEODETIC POSITIONS 118. Method of Computing Latitude and Longitude. Assuming that the latitude and longitude of a station (A) are known, as well as the distance and azimuth to a second station (B), we will now develop the formulae * necessary to compute the geodetic latitude and longitude of the second point In doing this we shall have to solve the differential spherical tri- angle formed by joining the two points with the pole. FIG. 70. 119. Difference in Latitude. In Fig. 70, P' is the pole of the spheroid. P is the pole of a sphere tangent to the spheroid along the parallel of latitude through A. The radius of the sphere is N, and its center is at H. Let A be the known station and B the unknown station. * These formulae were first given by Puissant; see his Traite de Geodesic, Vol. I; see also Coast and Geodetic Survey Report for 1894, and Special Publication No. 8. DIFFERENCE IN LATITUDE l6l The angular distance of A from the pole is 7; the unknown dis- tance of B is 7' ; a is the arc AB ; a is the azimuth; and = 180 a. If 7' is computed by a direct solution of the spherical triangle ABP, the required precision can be reached only by the use of about ten-place logarithms. It is more convenient, and quite as accurate, for such short lines as occur in practice, to employ for- mulae giving the difference in latitude, that is 7 7'. The formula for the direct > solution of 7' in the spherical tri- angle is cos 7' = cos 7 cos a + sin 7 sin a cos e. (a) Since 7' is a function of 0, its value may be expressed as a con- verging series by means of Maclaurin's formula, giving 7 =7', 7 = 90 - 0, , = 180 - a, Equation (j) becomes 0' = a cos a + ~~ sin 2 a tan <^> 2 ~ T ( x +3 tan 2 0) sin 2 a cos a ... (&) o In order to transfer the coordinates of the triangulation points from the sphere to the spheroid, it should be noticed that if the radius of the sphere is N (the normal) and its center is at H (Fig. 70), and the polar axes of the sphere and spheroid coincide, then the parallels of latitude through A coincide, the spheroid being tangent to the sphere along this parallel; also, the latitude (0) DIFFERENCE IN LATITUDE 163 will be the same for both surfaces, and the distances and azimuths of AB on the two will differ by inappreciable quantities. We may therefore put a = , where s is the distance in linear units. Then (k) becomes 0-0' = a -f ^sin 2 atan0--sin 2 cosc*(i +3 tan 2 0). The difference in latitude should be measured, however, on a curve of radius R m , since it is measured along a meridian. The linear difference in latitude is nearly the same for the two sur- faces, and the angular difference in latitude will vary inversely as the radii; that is, (0 - 0') # = A0" R M arc i". is taken as half the difference in latitude, 60; that is, ,. 50 arc i" a

K K O K* when s is the length of any line on the surface. 1 66 CALCULATION OF GEODETIC POSITIONS If is an angle expressed in seconds, then the last equation K becomes /c"\ 2 M( S -} arc 2 i" , s , . s \R/ log--logsm- R R 6 Taking logs of both members, i tA-ct ri \ i Mf arc 2 1 "\ , . fs"\ log (diff. of logs) = log - - -f 2 log \ / \K/ Applying this formula first to AX", log (diff. of logs) = 8.2308 + 2 log AX". (q) Apply the formula to , and, observing that the second term is log (diff. of logs.) = 8.2308 + 2 log s + 2 log A' (r) = 5. 2488* + 2 log S. (s) This correction is to be subtracted because arc 7 is greater than sin &) In Table XIII the corrections are tabulated to show the values of log 5 and log AA" for the same log diff. The correction for log s is negative and that for log A\" is positive. The algebraic sum of the two corrections is to be added to log AX r/ . The method of making these corrections is illustrated in the example on p. 1 70. The new longitude X' is given by V = X + AX". [67] 121. Forward and Back Azimuths. Owing to the convergence of the meridians the forward and re- verse azimuths of a line will not differ by exactly 180, as in plane * Based on the value 8.5090 for log A'. FORWARD AND BACK AZIMUTHS 167 coordinates. The amount of this convergence is computed as follows: In the triangle PAB, Fig. 70, by Napier's analogies, 2 COS | (7 + T) Substituting, and noting that A -f B + Aa = 180, and that an increase in AX causes a decrease in A a, 2 sin |0 + *0 whence - tan i A = tan i AX 2 2 COS ^ (0 2 A< cos 2 Therefore Putting for J Aa the series .^^L\ A0 cos I 2 / I A X.^l-irtaniAA.^T+ . COS M 3 2 cos A^ 2 J L 2 J and for tan 5 AX the' series then ^T OS A0 2 J 1 ^ sin0 m AX 3 sin0 OT ^ AX 3 sin 3 24 ,A0 cos -^ cos - cos 3 - 222 1 68 CALCULATION OF GEODETIC POSITIONS AX 3 Multiplying by 2 and factoring out > 24 sin0 OT _i_ , A , 3 / sin0 m _ sin 3 cos 3 1 A0/ cos ^ A0 12 Vcos J A cos 3 1 A0> Placing cos \ A0 = i in the small term and reducing A a and AX to seconds -of arc, - A " = AX " coffe + ^ (AX " )3 Sin * C S2 ** ^ '" I = AX" sin m sec ^ + (AX") 3 - F, [68] in which F is an abbreviation for ^ sin m cos 2 $ m arc 2 1" and is given by its log in Table XIV. This F term amounts to only o".oi when log AX" = 3.36. . . . The back azimuth a is given by a = a + A + 180. [69] In calculating the geodetic position of a point, the azimuth of the line to that point is to be found from the known azimuth of the fixed side of the triangle by using the corrected spherical angle, not the plane angle of the auxiliary triangle. The com- putations of 0' and X' may be verified by computing the position from two sides of the triangle and noting whether the same $' and X' are obtained from the two lines. The reverse azimuths are checked by noting whether their difference equals the spher- ical angle at the new station. In this manner the calculation of each triangle may be made to check itself. 122. Formulae for Computation. For convenience of reference the working formulae are here brought together. AX = A', s. sin a sec 0' [66] * The value of A< may be made more accurate by the addition of the following term: - \ s 2 k - E + | s 2 cos 2 a k E + \ s 2 cos 2 a sec 2 A 2 - k arc 2 i", in which k =s 2 sin 2 a C. FORMULA FOR COMPUTATION 169 (or, log AX" = logs + CiogA>-Cio gs +log sin a + log 4' + logsec 0'), - Ac* = AX" sin f(0 + 0') sec i A0 + (AX") 3 . F, [68] in which /? = s cos a B, 50 = s cos a 5 + s 2 sin 2 a C hs 2 sin 2 a E. The position of the new point and the reverse azimuth are then given by 0' = + A0, [65] X' = X + AX, [67] a = a + Ac* + 180. [69] The arrangement of the computation is illustrated by the fol- lowing example. The two pages show the two computations of a position in the same triangle. In the first page of the computation, the known station is Waldo and the position of Bunker Hill is to be found. Since the value of Aa depends upon AX and AX depends upon 0', the three parts of the solution must be carried out in the order indicated. In computing A0, take out B, C, D, and E for the given latitude 0. The (60) used in the D term is usually taken as the algebraic sum of the first two terms of the series; if the E term is large, it should be included also. The h in the E term is the first (B) term alone. The algebraic signs of the functions of a are important and should be carefully attended to. When computing AX, 0' is known and the factor log A' must be taken out for this new latitude 0', not for 0. The primes are inserted to call attention to this. To correct for the difference between the arc and the sine, enter Table XIII with log AX and logs as arguments. The algebraic sum of the two values of "log. diff." is the correction to be applied to log AX. The value of Aa is found last. The values of 0' and X' are checked by noting whether the same values are obtained from the two computations. The two reverse azimuths should differ by the spherical angle at the new station, which checks the computations of Aa. i yo CALCULATION OF GEODETIC POSITIONS a Z. a Aa a' Waldo to Meade's Ranch Meade's Ranch and Bunker Hill Waldo to Bunker Hill 255 i?' I?' 1 - 52 86 20 54 .50 341 38 12 .02 +4 43 .09 Bunker Hill to Waldo Third angle 180 161 42 55 .11 38 08 34 .02 A0 *' 39 09' 55". 645 -17 39 -209 Waldo s = 34,407 . 64 meters Bunker Hill X AX X' 98 49' So". 128 -07 29 .652 38 52 16 .436 98 42 20 .476 5 cos a B h ist term 2d term 3rd and 4th terms -A* i (0 + *0 4.5366549 9-97730I8 8.5109150 5* sin 2 a C 3d term 4th term 5 sin a 9-07331 8.99674 i 31553 (60)2 D Arg 5 AX 6.0499 2.3832 -h s z sin 2 a E (AX)3 F AX sin J (0 + 0') sec i (A0) -Aa 3- 0249 n 8.0700 6.0871 3.0248717 1058". 9409 o .2429 9.38558 +0.0271 0.0015 8.4331 21 +03 7 . 1-820 n 7-959 7.872 1059 -1838 + - 0256 +0.0256 4.5366549 9. 498 3680 n 8.5091469 0.1087088 5.831 2 . 652 877 n 9-799043 1059.2094 39 01' 06". 04 A' sec 0' AX 2. 652 8786 n 18 Corr. -18 2.451 921 n -283". 09 2.6528768 n 449" -652 123. The Inverse Problem. Not infrequently it is required to find the distance and mutual azimuths between two stations whose latitudes and longitudes are known. If we place x = s sin a and y = s cos a, then, from Equa. [66] and [64], we have AX cos , or s = ffN = NAX" cos < arc i" + %N (AX" cos arc i") 3 tan 2 0, [74] which gives the distance AB corresponding to any difference in longitude AX. If in Equa. [64] we place a = 90, ' 2 #12. arc i" The offset P from the prime vertical (tangent) for any distance from the initial point is [75] 174 CALCULATION OF GEODETIC POSITIONS' Since P varies as s 2 , the offsets for equidistant intervals along the line may be readily calculated. The direction of the pole from any point (x) on AB is given by PxA = 90 + Aa, in which it is sufficiently accurate to take Aa = AX sin m . [76] Since the numerical value of A a increases directly as AX, it will be sufficient to take the increments of A a as proportional to s. If the arc of the parallel is a long one, it is advisable to break it into sections, and to establish a new point at the beginning of each section by direct latitude observation. (See United States Northern Boundary Survey, Washington,, 1878.) 127. Location of Arcs of Great Circles. The general method of laying out arcs not coincident with the meridian is that of determining astronomically the latitudes and longitudes of the terminal points, and then running a random line between them. The direction and distance between the terminals may be found by Formulae [70] to [73] for the inverse solution of the geodetic problem. The azimuth is determined by observation at intermediate points. The error of the random line is corrected in the usual way. For long arcs triangulation would be substituted for direct measurement. (See Appendix 3, Coast Survey Report for 1900, "The Oblique Boundary Line between California and Nevada.") 128. Plane Coordinate Systems. When all the points to be located in a survey are comprised within a relatively small area, such as a city or a metropolitan district, the calculations are greatly simplified by the use of plane coordinates. If there are reliable triangulation points already established within the area, these will naturally be used as a basis for the new survey, or at any rate to check the new triangulation. In establishing a system of plane coordinates it is necessary to decide first upon the positions of the coordinate axes. These CALCULATION OF PLANE COORDINATES 175 will naturally be a meridian and a great circle at right angles to it; or, more properly speaking, they will be straight lines tangent to these two circles at their point of intersection, all points being supposed to lieHn the plane denned by these two lines. The origin of the system must be denned in terms of the coordinates of some specified point of the survey (geodetic datum, p. 158). Unless this is done, the origin will not be the same when derived from different points, and ambiguity will exist regarding the true position of the origin. The origin may be taken as coincident with the selected triangulation point, as in the case of the survey of Boston, Massachusetts, and Baltimore, Maryland; or it may be the intersection of a selected meridian and parallel as derived from the assigned latitude and longitude of some station. In Springfield, Massachusetts, for example, the origin is the inter- section of the 42 04' parallel and the 72 28' meridian, as de- termined by the published latitude and longitude of the United States Armory flagpole. The direction of the meridian must be defined as making a certain angle with a specified Jine of the sur- vey, preferably one which passes through the fundamental point. The point at which the plane is tangent to the spheroid must not be confused with the (o, o) point of the system. The former should be within the area surveyed, preferably at its center, in order to avoid large spherical errors. The latter may be taken at any convenient distance outside the area by assigning to the tangent point large values of x and y, in order to avoid negative values in the coordinates of the survey points. The tangent point is on the sphere as well as on the plane; the (o, o) point is not necessarily on the sphere. 129. Calculation of Plane Coordinates from Latitude and Longitude. In calculating the plane coordinates of a point, we may apply Formulae [70] to [73] for the inverse solution of the geodetic problem, one of the points being the origin (tangent point) whose coordinates are < and X, and the other the triangulation point the coordinates of which are ' and X'. The x and y there given are I 7 6 CALCULATION OF GEODETIC POSITIONS the plane coordinates desired. If the coordinates of many points are to be transformed, it will prove to be more convenient to use specially prepared auxiliary tables and to modify the calcula- tions as follows. In Fig. 72 P is the triangulation point whose latitude and longitude are known, and whose coordinates x and y with refer- ence to the origin are desired. For such distances as are likely FIG. 72. to occur in a plane system it may be assumed that PE = PD; that is, x equals the length of the arc of the parallel PD. The ordinate y = PC may be taken as PA (the difference in latitude) plus BC* (the offset from great circle to parallel). From For- mula [70], [77] x = PD = AX" If x is to be expressed in feet, COS0' A' * = AX". = x 3.2808$. [7 g| (See Table A.) * If P is south of the origin, the offset must be subtracted. CALCULATION OF PLANE COORDINATES I 77 TABLE A. VALUES OF LOG + 0.515 9842* Distance west of origin in feet = x = AX" X H Lat. '. LogH. Lat. *'. LogH. P. P. 570 572 574 576 1 II 42 10 1.8768536 / II 42 20 I.87S 7103 i 19 19 19 19 2 38 38 38 38 30 7966 30 6530 3 57 57 57 58 4 76 76 77 77 II 7396 21 5957 5 95 95 96 96 30 6825 3 5383 6 114 114 H5 "5 7 133 J 34 134 134 12 6255 22 4809 8 152 153 153 154 9 171 172 172 173 30 5684 30 4235 10 190 191 191 192 13 5114 23 3661 ii 209 2IO 2IO 211 12 228 229 230 230 3 4543 30 3086 13 247 248 249 250 14 266 267 268 269 14 397i 24 2512 15 285 286 287 288 30 3400 30 1937 16 304 305 306 307 17 323 324 325 326 15 2828 25 1362 !8 342 343 344 346 19 36i 362 364 365 30 2256 30 0787 20 38o 38i 383 384 16 1684 26 I .875 O2I2 21 399 400 402 403 22 418 419 42i 422 3 III2 30 1.8749636 23 437 439 440 442 24 456 458 459 4 6! 17 1.8/6 0541 27 9o6l 25 475 477 478 480 3 1.8759968 30 8485 26 494 496 497 499 27 513 5i5 517 5i8 18 9396 28 7910 28 532 534 536 538 29 55i 553 555 557 30 8823 30 7334 30 570 572 574 576 i9 8250 29 6757 30 7677 3 6181 20 1.875 7I>3 30 1.8745604 This is the form adopted by the city of Springfield, Mass., lor its coordinate system. i 7 8 CALCULATION OF GEODETIC POSITIONS TABLE B. VALUES OF 0.515 9842 - log B Dist. N. of Origin in Feet = A0" X K + x 2 Dist. S. of Origin in feet = A" X K x 2 Lat. Log. K. Lat. Log. K. P. P., Diff. i' = 12.8. 42 10 2 .OO5 2891 42 2O 2.005 3 I0 9 n I O n 22 5 30 2988 30 3116 2 23 5 II 2994 21 3122 3 I 24 5 30 3000 3 3129 4 I 25 5 12 3006 22 3135 5 I 26 6 30 3013 3 3141 6 I 27 6 13 3019 23 3U7 7 I 28 6 30 3026 30 3154 8 2 29 6 14 3032 24 3160 9 2 3 3039 30 3167 10 2 15 3045 25 3i73 ii 2 30 3052 30 3180 12 3 16 3058 26 3186 13 3 30 3064 30 3i93 14 3 i? 3070 2 7 3i99 15 3 30 3077 30 3205 16 3 18 3083 28 3211 17 4 3 3090 3 3218 18 4 i9 3096 29 3224 19 4 3 3103 30 3231 20 4 20 2.005 3109 30 '2.005 3237 21 4 The difference in latitude PA is converted into feet by multi- .28o8{ B tan plying A0" by -~ a . (Table B .) The offset EC (Formula [75]) = A [79] The factor a "J , in feet, may be taken from Table C which was calculated by the formula [80] * For another method of calculating this offset, see an article entitled " A Method of Transforming Latitude and Longitude into Plane Coordinates," by Sturgis H. Thorndike, Journal Boston Society Civil Engineers, Vol. 3, No. 7, September, 1916. CALCULATION OF PLANE COORDINATES 179 TABLE C. VALUES OF LOG (ft.) = log C - log B - 0.515 9842 Offset from parallel = log L + 2 log x Lat. Log. L. Lat. Log.L. P. P. Diff. i' = 25.4. o / // 42 10 2-33 46o / // 42 20 2-33 7H I O 24 IO 30 473 30 727 2 I 25 II II 486 21 739 3 I 26 II 30 499 30 752 4 2 27 II 12 S 12 22 765 5 2 28 12 30 525 3 778 6 3 29 12 13 537 23 790 7 3 30 55 30 803 8 4 14 562 24 815 9 4 3 575 3 828 10 - 4 15 587 25 840 ii 5 30 600 30 853 12 5 16 612 26 865 13 6 30 625 30 878 14 6 17 638 27 892 IS 6 3 651 30 905 16 7 18 663 28 917 17 7 30 676 30 930 18 8 i9 689 2 9 942 i9 8 30 702 30 955 20 8 20 2-33 7H 3 2.33967 21 9 22 9 23 10 Example. As an illustration of how this method would be applied, let us sup- pose that it is desired to compute the plane coordinates of A Powder horn in a system whose origin is the dome of the State House, Boston, Massachusetts. We first compute A<" and AX" and then apply formulae [78], [79] and [80] as shown. Powderhorn Lat. 42 24' 04".683 Long. 71 01' 52".oo6 State House 42 21 29 .596 71 03 51 .040 log x 2 = 7.9 l8 3 logZ, = 2.33752 " 2 o8 35 "' 87 logA<" = 2.190 5754 \ogK = 2.005 3 I2 9 AX log Offset 0.23935 1.7352 ft. log 4.195 8883 15699.59 ft. 1-74 59-034 " = ii9".o 34 log AX" = 2.075 6710 logZ7 = 1.875 2422 log* = 3.9509132 x = 8931.27 ft. East of State House. y = 15701.33 ft. North of State House If it is preferred to make the conversion from AX to # always on the same parallel of latitude, that of the origin, a table may be calculated, giving the length of each minute (i' to 10') and each i8o CALCULATION OF GEODETIC POSITIONS second (i" to 60") of arc on this parallel; the difference in longi- tude may be taken out, by parts, from this table. If this is done, however, it is necessary to make allowance for the convergence of the meridians between, the two parallels by solving for the dis- tance AB = y sin 6 (Fig. 72). The convergence 6 = AX" sin m and its sine may be tabulated for different values of AX and $ m . If the triangulation point is north of the origin, AB is to be sub- tracted; if south, it is to be added. 130. Errors of a Plane System. . In order to investigate the errors of a plane coordinate system like the preceding, let us assume that a line starts from the origin o, Fig. 73, in an azimuth a, and follows the surface of a sphere of G FIG. 73. radius ^/R m N (for latitude ) for a distance 5 meters, to point A ; and that another line OA' ', having the same azimuth and length, lies in the plane which is tangent to the sphere at o. The point A f in the plane then represents the point A on the sphere as de- termined by a direct measurement from the origin. The defects of the plane system as a means of representing points on a sphere will be shown by the error in reproducing point A' by following different routes, such, for example, as traversing due north and then due west on the sphere, or due west and then due north. If a perpendicular AF (an arc of a great circle) be let fall from ERRORS OF A PLANE SYSTEM 181 A (Fig. 73) to the meridian through o, its length will be deter- mined by .a . s sin = sin sin a, R R where a is the perpendicular distance in meters and R is the radius of the sphere. For the corresponding distance on the plane, a = s sin a. Distinguishing the plane and spherical values of a by sub- scripts, p and s, the difference in length may be found as follows : + dp a* = s sin a Rsiif 1 (sin a sin ) o|f^3 Rssina . s 3 . s 3 . , = 5 . sma ___ + _ sma __ sm3a+ . . . = --sin a cos 2 a -f- . 6 R Assuming that = 40, a = N 45 W, and s = 20,000 meters (about 12 miles), then a p a, = o m .on6. If another such line were to extend 2o,ooo m , N 45 E, to B, the terminal points A and B would then be 0*^.0232 farther apart if calculated on a plane than if calculated on the sphere.* If the survey proceeds from o northward to the point F, where the great circle from A, perpendicular to the meridian, inter- sects that meridian, and then westward along this great circle to A, the point A would be reached without error, if the measure- ments were perfect. The point computed on the plane would not agree, however, with A' as already established. The excess of the spherical distance b 8 , along the meridian to the foot of the perpendicular F, over the plane distance b p is found as follows: * This does not refer to the chord-distance AB, but to the distance on the spherical surface. 182 CALCULATION OF GEODETIC POSITIONS In the spherical right triangle, tan-=r = tan cosa. K. K. Then b, b p = R tan 1 (tan cos a J s cos a s 3 cos a sin 2 a Assuming the same data as before, we find that in order to reach A, on the sphere, we must run N 14142.15886 meters and then W 14142.12400 meters. Since in this case 5 sin a = scosa = 14142. 135 63 w , such a traverse, when computed on the plane, gives a point o m .o2323 N and o m .on63 E of point A' . A similar traverse running west to point G (Fig. 73) and then north to A Would give a point o m .on63 S and o m .o2323 W of point A r . The relative positions are shown (actual size) in Fig. 74. A' From O direct From O north then west FromO west then north FIG. 74. FIG. 75. The maximum discrepancy in the traverse is then about o w .o5, or about two inches. This would appear as an error of closure of the traverse OF AGO even if there were no error whatsoever in the measurements themselves. The difference in length between an arc of the parallel and an ADJUSTING TRAVERSES TO TRIANGULATION 183 arc of the great circle is found as follows: In Fig. 75, J AB = r sin = R sin - . Replacing the sines by their series in terms of 2 2 the arcs, r ( --- ) = R ( --- -V The difference between \ 2 48 / \2 48; r AX, the arc of the parallel, and Re, the arc of the great circle, is 24 24 AX 3 D AX 3 cos 3 0, N = R cos -- R -- - (approx.) 24 24 since 6 = AX cos 0, nearly. Therefore r AX - R0 = ^ R cos AX 3 (i - cos 2 0) = 2T R OX") 3 . arc 3 i" cos sin 2 0. In order to compare this with the previous examples, we must put AX /r = ii92".4, which corresponds to the distance between A and B. The error r AX RO is found to be o m .oi86 for the total arc, or 0^.0093 for the half arc. The difference between the length of the parallel and the x coordinate is therefore o m .on6 0^.0093 = o m .oo23. These results indicate that a plane system may be extended over an area twelve miles in radius without involving errors of computation as great as the errors of measurement, and also that the formulae given may be used whenever it is safe to use plane coordinates. 131. Adjusting Traverses to Triangulation. Whenever a traverse is to be run from one triangulation point to another, or if the circuit is to return to the original triangula- tion point, some method must be provided to allow for the effect of convergence of the meridians. The most obvious method is to refer all bearings in the traverse to the direction of the initial meridian, taking no account of true bearings at any other point of the survey. This method is subject to very small errors, far within the limit of accuracy of the field measurements, unless the area is much greater than that ordinarily covered by a traverse. 184 CALCULATION OF GEODETIC POSITIONS PROBLEMS Problem i. Calculate the latitude and longitude of point A, Problem 3, Chapter VI, from both lines, and the back azimuths AB and AC. Problem 2. Calculate the latitude and longitude of point E, Problem 4, Chapter VI, and the back azimuths EL and EN. Problem 3. Calculate the portion of Sand Hill in Problem 5, Chapter VI. Problem 4. What will be the error of closure of a survey which follows the cir- cumference of a circle whose radius is 20,000 meters (on the earth's surface) if the survey is calculated as though it were on a plane, the latitude of the center being 40 N. and the measurements being exact? CHAPTER VIII FIGURE OF THE EARTH 132. Figure of the Earth. The term "figure of the earth " may have various interpreta- tions, according to the sense in which it is employed and the de- gree of precision with which we intend to define the earth's figure. When we say that the earth is spherical, we mean that the sphere is a rough approximation to the true figure, sufficiently close for many purposes. We adopt the sphere to represent this figure because it is a simple surface to deal with mathematically. When a closer approximation is required, we employ the spheroid, or ellipsoid of revolution. This figure is so near the truth that no closer approximation has ever been needed in practical geodetic operations, although an ellipsoid (three unequal axes) or an ovaloid (southern hemisphere the larger) may be nearer the truth. All the surfaces mentioned are regular mathematical sur- faces, substituted for the true surface on account of their sim- plicity. In defining the true figure it is necessary to distinguish be- tween the topographical surface and that surface to which the waters of the earth tend to conform because they are free to adjust themselves perfectly to the forces acting upon them. It is this latter surface with which we are chiefly concerned in geodesy; the land surface is not referred to except in such ques- tions as the effect of topography upon the direction and in- tensity of gravity. The true figure^ called the geoid, is defined as a surface which is everywhere normal to the force of gravity, that is, an equipotential surface; and of all the possible surfaces of this class it is that particular one which coincides with the mean surface of the oceans of the earth. Under the continents 185 i86 FIGURE OF THFv EARTH it is the surface to which the waters of the ocean would tend to conform if allowed to flow into very narrow and shallow canals cut through the land. It is necessary to suppose these canals narrow and shallow in order that the quantity of water removed may not modify the figure over the ocean areas. Some idea of the relation of the spheroid, the geoid, and topo- graphical surface may be gained by an inspection of Fig. 76. It will be seen that the geoidal surface coincides with the surface of the ocean, and that it intersects the spheroid at some distance out from the shore line. The inclination of the normal to the FIG. 76. plumb line (station error) shows the angle between the two sur- faces at this point. The surface of the geoid may be represented conveniently by means of contour lines referred to the spheroid as a datum sur- face. In Fig. 77, which shows contours of the geoid within the limits of the United States proper, that portion of the contours shown in full lines is taken from a map published by the Coast and Geodetic Survey in " Figure of the Earth and Isostasy " (1909); the remaining portions (dotted) were sketched in by eye, following in a general way the topography of the continent. Such a map conveys no real information about the elevations of the geoid except along the full lines, but is given simply to show how the contours would be used in representing the geoid. When we speak of the spheroid as the ''figure of the earth " we DIMENSIONS OF THE SPHEROID FROM TWO ARCS 187 mean that particular spheroid which best represents the earth as a whole, or which most closely fits some specified area. The dimensions of such a spheroid are not to be regarded as fixed, but are subject to revision with each accession of new data. Such a spheroid necessarily depends upon a large amount of data, and the calculations for fixing its dimensions are long and compli- cated, involving the adjustment of many observations by the method of least squares. 127 122 117 112 107 102" 97" " 8T 82 77 72 3 67 FIG. 77. Contours of the Geoid. The principal methods of determining the spheroid are (i) by the measurement of arcs, which may be portions of meridians, of parallels, or of great circles; (2) by means of areas containing several astronomical stations rigidly connected by triangulation ; and (3) by observations of the force of gravity. 133. Dimensions of the Spheroid from Two Arcs. The simplest method by which the dimensions of the spheroid can be determined is by the measurement of two meridian arcs. The length of each arc and the latitudes of the terminal points of each must be measured. If the earth were a perfect spheroid, and if there were no errors of measurement, the two arcs would determine exactly the elements pf the spheroid. i88 FIGURE OF THE EARTH In the equation of the ellipse there are two constants to be determined, and it will be shown that the determination of the curvature of the meridian ellipse at two points will enable us to compute these constants and consequently all the other elements of the ellipse. In Fig. 78, suppose that the lengths of the two FIG. 78. meridian arcs have been measured by triangulation and that their lengths are s and s', and that the differences of the latitudes of their terminals are A< and A0', respectively. The radii of curvature of the ellipse at the middle points of the arcs are a (i - e 2 ) and R m = R m ' = a (i - i 2 ) in which < and <' refer to the middle points of the arcs and a and e are unknown. If the two arcs are regarded as arcs of circles whose radii are to be found, then and RJ = arc i arc i DIMENSIONS OF THE SPHEROID FROM TWO ARCS 189 are the two radii of curvature, A< being in seconds. The shorter the arcs, the less the error involved in assuming that they are circular. Equating the two values of R m and R m f , we have s a(i -e?) , , s' a (i - A7^ = (l -^si Dividing (a) by (b) and solving for e 2 , Having found e 2 from Equa. [81], the equatorial radius a may be computed by substituting the value of e 2 in either (a) or (6). The value of b may then be found from the relation 6 2 = a 2 (i - e 2 ). (c) The compression / is given by r d b r i / - [531 The length of a quadrant of the meridian 'may be found by applying Equa. [54], Chapter V. In this method of determining the elements of the spheroid it should be observed that there are just enough measurements to enable us to solve the equations, and no more. All errors of measurement enter the result directly; we should not, therefore, expect to derive very accurate values from two arcs. As an illustration of the preceding method let us take the Peruvian Arc and a portion of the Russian Arc, the data for which are as follows: 190 FIGURE OF THE EARTH PERUVIAN ARC Station. Astr. lat. Dist. in meters between the parallels of latitude. Tarqui o / // S 3 04 32 068 } Cotchesqui N o 02 *! 387 \ 344,740.5 RUSSIAI $ ARC (Northern End) Tornea N 6^ AQ A.A. <7 } Fuglenaes tl u -> 4-y 44-i>/ c N 7O AO II 23 1 53984L7 Substituting in Formulae 81, (a) and (c), the resulting values are e 2 = 0.0065473, b 6,356,440 m. 134. Oblique Arcs. If an arc (ABj Fig. 79) is inclined to the meridian at a small angle, it may be utilized to determine the curvature of the meridian as follows: Referring to Equa. (n), Chapter VII, it is seen that the dif- ference in latitude of the terminal points of the line is given by the series for A0". Hence the length of the meridian arc is given by A0". R m arc i", and A0" R m arc i" = s cos a s 2 sin 2 a tan 2 N + s 3 sin 2 a cos a (i + 3 tan 2 0) . [82] Each line of a chain of triangles may be projected onto the meri- dian, and its length found by this formula. The length and dif- ference in latitude of the end points are thus found, and the projection treated as though it were a measured meridian arc. FIG. 79. FIGURE OF THE EARTH FROM SEVERAL ARCS IQI The sum of all these short arcs may then be treated as a single arc to be combined with another similar arc in the computation of a and e. 135. Figure of the Earth from Several Arcs. When several arcs are to be used to determine the elements of the spheroid, there are more data than are necessary for the direct solution as given in Art. 133. The arcs usually consist of several sections; that is, the latitudes of several stations along the same meridian are observed and the distances between them are determined by the triangulation. The problem is one of combining all these measurements by the method of least squares in order to obtain the most probable values of the elements. Only the outline of the method can be given here. From Equa. [49] we have for the length of a meridian arc s = A0 R m arc i", which is sufficiently accurate for short arcs. For long arcs a more accurate expression is necessary. Suppose that an arc consists of several sections, the latitude of the initial point being 0i, the second 02, etc., and that the meridian distances between the stations are s, si, etc. From the first two latitudes Instead of finding a and e* directly, it is more convenient to assume approximate values of these quantities and to compute the most probable corrections. Let us assume the equations a = a + 5# and e 2 = e? + 5e*. Let Ro be the value of R m corresponding to ei = - r, in these terms. Substituting in (i), . x q . 9 v ^ + (i- f sin 2 0)- S PRINCIPAL DETERMINATIONS OF THE SPHEROID 193 If we place x = da, y = d G which is known as the Laplace equation. Triangulation stations at which the astronomic longitude and azimuth have been ob- served are called Laplace points. The geodetic and astronomic longitudes in the United States are subject to probable errors of less than o".5. The astronomic azimuths are also determined with about the same accuracy. 202 FIGURE OF THE EARTH The geodetic azimuths, however, as carried through the tri- angulation, are subject to an error about ten times as great. The triangulation may therefore be greatly strengthened by correcting the geodetic azimuths at Laplace points by means of the above equation. The manner of correcting the geodetic azimuth is illustrated by the following example, taken from Supplementary Investiga- tion in igog of the Figure of the Earth and Isostasy. U. S. Standard longitude of Parkersburg \ \ = 88 01' 49^.00 Astronomic " _ " " = 88 01 48 .30 A G in longitude o".7o A G in azimuth = (0.70) ( sin ) = +o .44 Astronomic azimuth Parkersburg to Denver = 143 16 15 .55 True geodetic azimuth Parkersburg to Denver =143 16 15. n U. S. Standard azimuth Parkersburg to Denver = 143 16 15 .64 Correction to U. S. Standard azimuth = ~o"-53 140. Isostasy Isostatic Compensation. For many years it has been known that the estimated and observed values of the station error are not in even approximate agreement, and it has long been suspected that the explanation would be found in the fact that the densities of the material immediately beneath the surface are unequal, regions of deficient density lying beneath mountain ranges, and regions of excessive density lying beneath low areas and under the ocean bottom. It is supposed that at some depth the excess above the surface is compensated by the defect below the surface, and vice versa. This condition is given the name isostasy. It appears that the theory was first clearly stated by Major C. E. Button in 1889, and since that time it has been the subject of much study. In 1909 and 1910 there were published by the Coast and Geodetic Survey the results of a very extensive investigation conducted by Professor J. F. Hayford, then Inspector of Geodetic Work and Chief of the Computing Division. The investigation was based primarily upon the computation of the topographic deflections at a large number of astronomical stations in the United States. The best topographic maps available- were used for this purpose. ISOSTASY ISOSTATIC COMPENSATION 203 These computed deflections were then compared with the known (observed) deflections at these same stations as found from the triangulation and astronomical observations. In substantially all cases the computed deflection was found to exceed the ob- served deflection by a large amount, although the two were usually of the same algebraic sign. Computations were then made to test the theory that this condition called isostasy actually exists. The condition known as isostasy may be stated as follows: the mass in any prismatic column which has for its base a unit area of the horizontal surface lying at the depth of compensation, for its edges vertical lines (lines of gravity), and for its upper limit the actual irregular surface of the earth (or the sea surface if the area in question is beneath the ocean) , is the same as the mass in any other similar prismatic column having a unit area on the same surface for its base. Such prismatic columns have different heights but the same mass, and their bases are at the same depth below the geoidal (sea-level) surface. Computations were made assuming different depths of com- pensation, for the purpose of finding at what depth the computed deflections (taking isostasy into account) most nearly agree with the observed deflection. It was found that the compensation was most nearly complete (more than -$ complete) at a depth of about 122 kilometers, or about 76 miles. It should be observed that, while the densities in the prismatic columns tend to compensate, the resultant deflection of the plumb line is not zero, for the portions of the column nearest the station have a much greater influence than the distant portions. The tendency is to throw all the zeniths outward from the continental dome, assigning to the surface a curvature which is greater than it should be. Thus, if isostasy is not taken into account, the dimensions of a spheroid computed from such data will be too small. This investigation not only included a determination of the most probable depth of compensation, and a substantial proof of the validity of the theory in so far as it applies to the 204 FIGURE OF THE EARTH United States, but also included a determination of the most probable dimensions of the spheroid for that area. In this calcu- lation the area method was employed. The dimensions of the spheroid resulting from this investigation are as follows: = 6,378,388 i8 m , b = 6,356,909, j = 297.0 0.5. The general conclusions in regard to the existence of isostasy within the limits of the United States were later confirmed by the results of a similar investigation of the compensating effect upon observed values of the force of gravity determined with the pendulum. The results of these investigations will be found in the follow- ing publications of the United States Coast Survey: John F. Hayford, The Figure of the Earth and Isostasy from Measurements in the United States, 1909. John F. Hayford, Supplementary Investigations in 1909 of the Figure of the Earth and Isostasy, 1910. John F. Hayford and William Bowie, The Effect of Topography and Isostatic Compensation upon the Intensity of Gravity, Special Publication No. 10, 1912. William Bowie, The Effect of Topography and Isostatic Com- pensation upon the Intensity of Gravity, Special Publication No. 12, 1912. William Bowie, Investigation of Gravity and Isostasy, Special Publication No. 40, 1917. PROBLEMS Problem i. Compute the dimensions of the spheroid from the following arcs. Name. Lat. of middle point. Amplitude. Length in feet. Peruvian (Delambre's) English S i 31 oo N <<2 3 = c e cos <. The component of C+ directly opposed to G is c e cos 2 (vertically upward). Hence # = G c e cos 2 0. [89] * See Jordan's Handbmh der Vermessungskunde, Vol. Ill, p. 627. 7) 2 t The centrifugal force may be expressed by - , where v is the velocity of a par- ticle at the equator. The distance moved by a particle in one rotation ( = i sidereal day = T seconds) is zirr. Hence the centrifugal force = ( ^r) r = coV, where &> is the angular velocity. T = 86,400 sidereal seconds = 86,164.09 mean solar seconds. VARIATION OF GRAVITY WITH THE LATITUDE 209 Substituting in [89] the value of G at the equator, g* = ge + c e - c e cos 2 = g e + c e sin 2 = go + (g P - e)sin 2 0; that is, g = g e (i + g -^ L ^ 2 \ ge sn [88] In order to obtain an accurate numerical expression for g^, of the same general form as the above, we may write fr = .(i+ sin 2 0) and then determine the value of B which is in best agreement with all observed values of g. For such a formula Dr. Helmert * published, in 1884, the equation go = 978.0x30 (i + 0.005310 sin 2 <), [90] in which go is supposed to be the value at sea-level and the unit is dynes of force, or centimeters of acceleration. This may be expressed for convenience in terms of go at lati- tude 45. Since sin 2 45 = J, and since and 2 sin 2 = i cos 2 , sin 2 = \ J cos 2 < = #45 B 2 which becomes go = 980.597 (i 0.002648 cos 2 0). * Helmert, Hohere Geodasie,Vo\. II, p. 241. [91] 210 GRAVITY MEASUREMENTS In 1901 Dr. Helmert gave the more accurate forms go = 978.046 (i + 0.005302 sin 2 0.000007 sin 2 2 ) [92] and go = 980.632 (i 0.002644 cos 2 + 0.000007 cos 2 2 ), [93] in which the number 0.000007 ( = i-^0 ^ s a coefficient found theoretically from assumptions regarding the internal structure of the earth. These formulae refer to the absolute value of g at Vienna. To refer to the " Potsdam system/' to which all values of g observed in the United States are referred,* the equations must be written g = 978.030 (i + 0.005302 sin 2 0.000007 si" 2 2 0) [94] and go = 980.616 (i 0.002644 cos 2 ^+0.000007 cos2 2 0)- [95] In the Coast Survey Special Publication No. 1 2, entitled " Effect of Topography and Isostatic Compensation upon the Intensity of Gravity " (second paper) the following formula is given: go = 978.038 (i + 0.005302 sin 2 0.000007 sin 2 2 0)> [96] equivalent to go = 980.624 (i 0.002644 cos 20+ 0.000007 cos2 2 0)> which is Helmert's formula of 1901 corrected by 0.008 dyne. The constants in these equations were derived from observations in the United States only. In Special Publication No. 40, a study is made of observations in the United States, Canada, Europe and India. The formula resulting from this investigation is go = 97 8 -39 ( x + 0.005294 sin 2 - 0.000007 sin2 0)> [97] 145. Clairaut's Theorem. The relation between the flattening of the spheroid at the poles * The American observations for g were referred to Greenwich (England), Paris (France), and Potsdam (Germany) by observations made in 1900 by G. R. Putnam, (see Coast Survey Report for 1901). PENDULUM APPARATUS 211 and the values of g p and g e is expressed by Clairaut's theorem, published in 1743, namely, in which c e is the centrifugal force at the equator. In this formula the terms of the second order have been omitted. If these terms are included, the formula becomes = ._ B __.. B __ B [98a] 2 ge \ 3 W T 4 ge 21 21 / in which and 4 are coefficients to be determined from the observations (Helmert, Hohere Geodasie, Vol. II, p. 83). It is by means of this equation that the form of the earth is com- puted from gravity observations. 146. Pendulum Apparatus. Nearly all of the observations of gravity for geodetic purposes are made with pendulums of invariable length, by the relative method. The description of apparatus in the following articles will be limited to one type, the half -seconds invariable pendulum apparatus as designed and constructed by the United States Coast Survey. The first half-seconds invariable pendulum with electrical apparatus for determining the period appears to have been devised by Sterneck (Austria) in 1882. In 1890 T. C. Mendenhall, then Superintendent of the Coast and Geodetic Survey, designed an apparatus of this kind but differing in many details, however, from any previous design. This apparatus has been used ever since that time in substantially the same form excepting the addition of the interferometer for determining the flexure. This apparatus includes three half-second pendulums, each about 248""* long and having an agate plane at the point of suspension. The agate plane rests on a knife-edge support (angle of 130) attached to the pendulum case in which the pendulums are enclosed when they are swung. The use of the blunt angle on the knife edge and the placing of the plane (rather than the 212 GRAVITY MEASUREMENTS PENDULUM APPARATUS 213 knife edge) on the pendulum are designed to secure greater permanence of length, upon which the accuracy of the method depends. The pendulums are made of an alloy of copper and aluminum and weigh 1 200 grams each. The three are of slightly .different lengths so that they will have different periods. Their FIG. 85. Dummy Pendulum (with thermometer), Regular Pendulum, and Leveling Pendulum. (C. L. Berger and Sons.) form (Fig. 85) is such as to give strength and at the same time offer but little resistance to the air. In addition to the three observing pendulums there is a dummy pendulum, of the same size and shape but carrying a thermometer packed in filings of the same metal. There is also a small pendulum provided with a spirit level for leveling the knife edge. Pendulums made of 214 GRAVITY MEASUREMENTS invar metal are now (1919) being constructed by the instrument division of the Coast and Geodetic Survey so that it will be possible to make gravity observations on mountain peaks and other places where the control of temperature is difficult. The use of this metal will make it unnecessary to construct a " con- stant temperature room." The pendulums are swung in an air-tight case from which the air may be nearly exhausted by means of a pump. Levers are provided for lowering the pendulum onto the knife edge and for FIG. 86. Flash Apparatus. starting and stopping the pendulum. Inside the case is a manom- eter tube for registering the air pressure, and also an additional thermometer. Levels are provided for leveling the case, and there is a graduated scale under the pendulum for reading the arc of oscillation. In the most recent work of the Coast Survey the pendulum receiver has been enclosed in a felt and leather case to prevent fluctuations in temperature. The observations are made by comparing the times of oscilla- tion of the pendulums with the half-second beats of a break- circuit (sidereal) chronometer connected electrically with the "flash apparatus " used for observing the coincidence. PENDULUM APPARATUS 215 The flash apparatus (Fig. 86) consists of a shutter a operated by the armature of an electromagnet b in the circuit and a mirror c behind the shutter which reflects light through the slit d to two small mirrors e, which reflect it into an observing telescope/; one of the small mirrors is attached to the pendulum and the other to the knife-edge support. In the most recent form of the flash apparatus, the observer looks down through a vertical tel- escope and sees the flash reflected by a prism. This arrange- ment is more convenient for the observer than the older form because the pendulum receiver is usually mounted on a very low support. When the pendulum is at rest and the shutter open, a beam of light from a lamp * at one side of the apparatus strikes the mirror c at an angle of 45 and passes through the slit; it is reflected from both mirrors at e and appears to the observer as two horizontal bright slits side by side. The mirrors may be adjusted so that these slits appear to be at the same height, so as to form one continuous band. If the pendulum is set swing- ing, the reflected image now appears to travel up and down, while the image from the other mirror is stationary. If the shutter is closed and allowed to open only for an instant at the end of each second (or each two seconds), the observer sees that at each successive opening of the shutter the moving image has changed its position relative to the fixed image. This is due to the fact that the period of the pendulum is longer than the sidereal second and the pendulum has made .slightly less than one complete (double) oscillation. By watching the flashes and noting the chronometer readings when they coincide, the ob- server obtains the number of seconds between two successive coincidences. During this interval the pendulum has evidently lost just one oscillation on the (half -second) beats of the chronom- eter. In the interval between two successive coincidences the pendulum has made one less than twice as many oscillations as * An electric bulb placed inside the flash box is now used instead of the oil lamp. 2l6 GRAVITY MEASUREMENTS the chronometer has beat seconds. During the interval between any two coincidences the number of oscillations is twice the number of seconds (s) less the number of coincidence intervals (n). Hence the time of one oscillation (P) is given by An examination of this formula will show that an error in noting the times of coincidence produces a relatively small error in P, and for this reason the method is almost independent of the observer's errors. On account of the variation of g (and consequently of P) with the latitude of the station, it is necessary to use a mean-time chronometer at stations situated near the pole, because the period of the pendulum approaches so closely to the sidereal half-second that the coincidence intervals are inconveniently long. In case a mean- time chronometer is used, the formula becomes [100] 2 s + n 147. Apparatus for Determining Flexure of Support. Observations with pendulums mounted on a very flexible sup- port show plainly that when a pendulum is set swinging, it com- municates motion to the case and the support and sets them oscillating, and this oscillation in turn affects the observed period of the pendulum. The apparatus now used to measure the effect of this flexure is one which operates on the principle of the interferometer.* This is an optical device (Fig. 87) consisting of a lamp and lens arranged so as to furnish a beam of sodium light; a glass plate arranged so as to separate the beam of light into two parts, one of which is transmitted, the other reflected; two mirrors, one in the path of each beam of light; and a telescope for observing the image. When the different parts of the appara- * A description of the interferometer will be found in the Coast Survey Report for 1910. CORRECTIONS 217 ' 8 Q -8 > ^ 2l8 GRAVITY MEASUREMENTS tus are properly adjusted, dark and light bands will appear in the field of the telescope, owing to interference of the sodium-light waves of the two beams. One of the mirrors is mounted on the pendulum receiver, while the rest of the apparatus is on an inde- pendent support in front of it. When the pendulum is set swinging, it sets the case in motion, and this in turn moves the mirror, causing a slight variation in the length of the path of one of the beams of light. This causes the interference bands to shift back and forth; the amount of shift may be estimated by observing the motion of the bands over a cross-hair or a scale in the field of the telescope. It is usually observed by noting the FIG. 88. scale readings of both edges of some band in each of its two posi- tions (before and after shifting). The movement of the edges of a band divided by the width of the band (in scale divisions) gives the movement in units of the width of a band. Fig. 88 represents the interference (dark) bands and the scale divisions in the field of the telescope. Tests made with the pendulum mounted on supports of dif- ferent degrees of flexibility will show the relation between the observed movement of the fringe bands and the resulting error in the period of the pendulum. In the Coast Survey tests the re- sults showed that a movement equal to the width of one band produced a change of 173 in P in units of the seventh decimal place. This is more conveniently expressed as follows: o.oi F METHODS OF OBSERVING 2IQ produces a change of 1.73 in P, where F is the width of a band. This constant was determined with the pendulum swinging through an arc of 5 mm on the scale, and all observed flexures must be reduced to this arc before correcting P. 148. Methods of Observing. The receiver should be mounted on a solid support such as a cement or brick pier, the foot screws cemented to the pier, and the instrument sheltered as in case of astronomical observations. It is important that the instrument should be so sheltered that the temperature will not fluctuate rapidly. The apparatus should be leveled by means of the spirit level on the outside of the case and then the knife edge should be leveled by means of the leveling pendulum. In moving the pendulums great care should be used to protect them from injury and to prevent any foreign matter from adhering to them. The accuracy of the results will depend upon the permanency of length, and any injury due to fall, or change of period due to change in the mass, will affect the period and vitiate the results. The pendulums should not be touched with the hands, but should be lifted by means of a special hook made for this purpose. The flash apparatus, chro- nometer, and interferometer should be placed upon supports that are entirely independent of the pendulum support. Various programs of observing have been tried, but the follow- ing has been chiefly used by observers of the Coast Survey. Each of the three pendulums is swung first in the direct and then in the reversed position, making six swings each of eight hours' dura- tion. The error of the chronometer is obtained by star-transit observations (Arts. 52-71) made just before the beginning and at the end of the series. The following table will indicate more clearly the order of operations. Star Observations 9-10 P.M. Start Pendulum No. i 10 P.M. Reverse No. i 6 A.M. Start No. 2 2 P.M. Reverse No. 2 10 P.M. Start No. 3 6 A.M. Reverse No. 3 2 P.M. Star Observations 9 P.M. Stop Pendulum No. 3 after star observations 220 GRAVITY MEASUREMENTS If star observations are lost at the end of the set, the swings are continued until star observations are obtained. At the begin- ning and end of each swing several coincidences are observed. At the end of each swing several more are observed. Very little time is lost between swings, so that they are almost continuous between star observations. For this reason the variations in the rate of the chronometer are almost entirely eliminated from the mean result of all the swings. Since 1913 the Coast Survey observers have obtained the chronometer corrections from the Naval Observatory time sig- nals instead of by direct observations. This results in a great saving of tune and cost. Another change in the regular pro- gram, recently introduced, is to swing the pendulums for twelve hours instead of eight, and in the direct position only, instead of direct and reversed. After a pendulum is placed in position on its support, the case closed, and the air exhausted until the pressure is about 6o mm , the observer lowers the pendulum until it rests upon the knife edge, starts it swinging through an arc of about o 53', and notes the arc on the scale. To observe coincidences, the observer switches in the chronometer and the flash apparatus and then watches the flashes to see when they are approaching coincidence. As the two approach he notes the hours, minutes, and seconds on the chronometer when the advancing edge of the moving flash touches the first edge of the fixed flash. A few seconds later he notes when the receding edge of the moving flash touches the second edge of the fixed flash. The mean of the two gives the true time of coincidence of centers more accurately than it could be ob- served directly. Such observations are made on several succes- sive coincidences, the flash moving alternately upward and downward. By combining the up and the down observations, errors of adjustment are eliminated. After a few of these have been recorded, the observer cuts out the chronometer and leaves the pendulum swinging for a period of nearly eight hours. Im- mediately after the observations for coincidences are completed, CALCULATION OF PERIOD 221 the temperatures are read on the two thermometers, and the pressure is read on the manometer tube. At the end of the eight-hour period the observer again observes a few coincidences as well as the arc (now diminished to about o 20'), the pressure, and the temperatures. It is not necessary that he continue observing throughout the whole eight-hour period, because the few observations already referred to make it possible to estimate correctly the number of coincidences which must have occurred between the observed times. It is customary to take the ob- servations with two or more chronometers as a check. This description applies to the 8-hour program outlined above. If the pendulums are swung for a 1 2-hour period it is necessary to start each pendulum with a somewhat larger arc (i2f) in order that it may have a sufficient amplitude at the end of 12 hours to enable the observer to read the coincidences of the flash conveniently and accurately. It is desirable that the temperature of the apparatus be kept as nearly uniform as possible, and that there be little vibration. In order to allow the pendulum time to assume the temperature of the receiver the next pendulum to be swung is placed inside the case before it is used in the observations. While the case is still in position the observer must place the interferometer in position and observe the movement of the interference bands while the pendulum is swinging. 149. Calculation of Period. After the observations are complete and the time observations and the chronometer rates are computed, the time of one oscilla- tion for each pendulum in each position is found as follows: divide the total number of seconds in an S h interval by the num- ber of seconds found for one coincidence interval (see example), to obtain the number of intervals that have occurred during the swing. Since this must be a whole number, there will be no difficulty in determining it correctly. Then reverse the process, dividing the total interval by the number of coincidence intervals, to obtain the accurate value of the number of seconds (s) in one 222 GRAVITY MEASUREMENTS coincidence interval. The uncorrected period of the pendulum is found by P = [101] 2S I for a sidereal chronometer, Table G, or P = ? [102] 2S + for a mean-time chronometer. 150. Corrections. This period must then be corrected to reduce it to its value at assumed standard conditions, namely, Infinitesimal arc, Temperature 15 C., Pressure 6o mm at o C., True sidereal time, and Inflexible support. The correction to reduce P to its value for an infinitesimal arc is PM sin ( log sin $' a formula given by Borda, in which P = the period, M = the modulus of the common system of logarithms, and < and 0' = the initial and final arcs. The temperature correction is T being the observed temperature centigrade and a the co- \ efficient to be found by trial, (a = +0.000008 34). The pressure correction is - Pr 1 , [105] i + 0.00367 rj K in which Pr = observed pressure in mm, T = temperature centigrade, and K = coefficient to be found by trial. CORRECTIONS 223 The constant 0.00367 is the coefficient of expansion of air for iC. The rate correction is given by the expression + 0.000011574 RP, [106] where R = daily rate of chronometer on sidereal time, + when losing and when gaining. The coefficient is the reciprocal of the number of seconds in one day. The flexure correction is computed by dividing the observed movement of the fringe band (in scale divisions) by the width of a band and then reducing this to an arc of 5 mm by dividing by the observed arc and multiplying by 5. The result is the displace- ment for a 5 rom arc in terms of the width of a band. This dis- placement, multiplied by the coefficient (173 mentioned before), gives the correction to be subtracted from P. 224 GRAVITY MEASUREMENTS TABLE D. REDUCTION OF SCALE READING IN MILLIMETERS TO MINUTES OF ARC Scale. i.o mm. 2.0 mm. 3.0 mm. 4.0 mm. 5-o mm. mm. 0.0 12 23 35 46 58 O.I 13 24 36 48 59 O.2 14 26 37 49 60 0-3 15 27 38 50 61 0.4 16 28 39 63 -5 17 29 4i 52 64 0.6 19 30 42 53 65 0.7 2O 3 1 43 55 66 0.8 21 32 44 56 67 0.9 22 34 45 57 68 TABLE E. ARC CORRECTIONS (ALWAYS SUBTRACTIVE) FOR HALF-SECOND PENDULUMS Arc at Beginning Arc at end. 90'. 85'. 80'. 75'. 70'. 65'. 60'. 55'. 50'. 45'- 40'. 35'- 30'. 25'- 20'. 5 10 12.0 II. 10. 9-0 8.1 7-3 6.5 5.8 5-0 4-3 3.6 3 2.4 1.9 1.4 15 14-4 13.3 12.2 ii. i IO.O 9.0 8.0 7-2 6.3 5.4 4.6 3-9 3-2 20 25 16.9 10 t 15-6 17.8 14-3 16.4 13-0 ICO ii. 8 TO *1 10.7 9.6 8.6 7.6 6.6 5-7 6 Q 4-9 4-1 30 A y *o 21.7 20.1 I8. 5 i-O . VJ 17.0 A O- / 15.6 14.2 12.9 ii. 6 10.4 9.2 u. y 8.1 35 24.1 22.4 20.7 19.2 17.6 16.1 14.6 13 2 ii. 8 40 26.5 24.7 22.9 21.2 19-5 17.9 16.3 14.8 13.3 45 29.0 27.1 25.2 23-4 21.6 19.9 18.2 50 31.5 29. 4 27.4 25-5 23.6 21.8 20. o 55 34 i 32.0 29.8 27.8 25.8 60 36.7 34-4 32.2 30.0 27.9 65 39-4 37-0 34.6 70 42.1 39-6 37-1 75 44-9 80 47-7 85 90 In practice it is convenient to combine Tables D and E into a single table computed for such intervals that little interpola- tion is necessary. FORM OF RECORD OF PENDULUM OBSERVATIONS 225 TABLE F. CORRECTION FOR PRESSURE Temp. C. 50 mm. 55 mm. oo mm. 65 mm. 70 mm. 75 mm. 80 mm. 85 mm. 90 mm. o + 10 +5 o -5 10 -15 20 -25 -30 I + 10 +5 o -5 10 -15 20 -25 -^30 2 + 10 +5 -5 - 9 -14 -19 -24 29 3 II 6 +1 4 9 14 19 24 29 4 II 6 +1 4 9 14 19 24 29 5 II 6 +1 4 9 14 19 24 28 6 II 6 +1 4 9 14 19 24 28 7 II 6 2 3 8 13 18 23 28 8 II 6 2 3 8 13 18 23 27 9 12 7 2 3 8 13 17 22 27 10 12 7 2 3 8 13 17 22 27 ii 12 7 2 3 7 12 17 21 26 12 12 7 2 2 7 12 17 21 26 13 12 1 3 2 7 12 17 21 26 14 12 8 3 2 7 II 16 21 26 IS ' 13 8 3 2 6 II 16 20 26 16 13 8 3 2 6 II 16 20 25 17 13 8 4 6 II 15 2O 25 18 13 8 4 6 IO IS 2O 24 19 13 9 4 5 IO IS 2O 24 20 13 9 4 5 IO IS 2O 24 21 14 9 4 5 IO 14 19 24 22 14 9 4 5 10 14 19 23 23 14 9 5 5 9 14 19 23 24 14 9 5 O 4 9 14 18 23 25 14 10 5 o 4 9 13 18 22 26 14 10 5 +1 4 9 13 18 22 27 H 10 +1 4 8 13 17 22 28 + 15 +10 +6 +1 - 4 - 8 13 -17 22 2 9 + 15 + 10 +6 +1 - 3 -18 12 -17 21 30 + 15 + 10 +6 +1 - 3 - 8 12 -17 21 Body of table gives corrections (in yth decimal place of sec- onds) to period of half seconds pendulum. 226 GRAVITY MEASUREMENTS TABLE G. PERIODS OF QUARTER METER PENDULUM NOTE : To obtain period to 7th decimal place, prefix .50 or .500 to figures in the table. Body of table gives o 220O 2300 2400 2500 2600 2700 2800 2900 3000 3100 o 11,390 10,893 10,438 10,020 9634 9276 8944 8636 8347 8078 I 8 4 8 9 34 16 30 73 4i 33 44 75 2 79 84 30 12 26 70 38 30 42 72 3 74 79 25 08 23 66 35 27 39 70 4 69 74 21 04 19 63 32 24 36 67 5 11,364 10,870 10,417 10,000 9615 9259 8929 8621 8333 8064 6 58 65 12 9996 12 56 25 18 30 62 7 53 60 08 92 08 52 22 15 28 59 8 48 55 04 88 04 49 19 12 25 57 9 43 5i I0 >399 84 OI 46 16 09 22 54 10 n,338 10,846 io,395 9980 9597 9242 8913 8606 8320 8052 ii 33 4i 9 1 76 93 39 10 03 17 49 12 28 37 86 72 90 35 06 OO 14 46 13 22 32 82 68 86 32 03 8597 II 44 14 17 27 78 64 82 28 oo 94 08 4i is 11,312 10,822 io,373 9960 9578 9225 8897 859i 8306 8039 16 07 18 69 56 75 22 94 88 03 3B 17 02 13 65 52 7i 18 9i 85 OO 33 18 H,297 08 61 48 68 15 87 82 8297 3i i9 92 04 56 44 64 12 84 79 95 28 20 11,287 io,799 10,352 9940 9560 9208 8881 8576 8292 8026 21 82 94 48 36 57 05 78 73 89 23 22 7 6 90 43 32 53 OI 75 70 86 20 23 72 85 39 28 49 9198 72 68 84 18 24 66 80 35 25 46 95 68 64 81 15 25 11,261 10,776 10,331 9921 9542 9i9i 8865 8562 8278 8013 26 56 7i 26 17 38 88 62 59 75 10 27 5i 67 22 13 35 84 59 56 73 08 28 46 62 18 09 3i 81 56 53 70 05 29 4i 57 14 05 27 78 53 5o 67 03 30 11,236 io,753 10,309 9901 9524 9174 8850 8547 8264 8000 31 3i 48 05 9897 20 7i 46 44 62 7997 32 26 44 OI 93 17 68 43 4i 59 95 33 21 39 10,297 89 13 64 40 38 56 92 34 16 34 92 85 09 61 37 35 53 90 35 11,211 10,730 10,288 9881 95o6 9i58 8834 8532 8251 7987 FORM OF RECORD OF PENDULUM OBSERVATIONS 227 WHEN PENDULUM IS SLOWER THAN CHRONOMETER Top and left-hand arguments combined give interval s = ten coincidence intervals. t = period in seconds. 3200 3300 3400 3500 3600 3700 3800 390 4000 4100 4200 7825 7587 7364 7153 6954 6766 6588 6418 6258 6105 5960 o 22 85 62 5i 52 64 86 17 56 04 58 I 2O 83 59 49 So 62 84 15 55 02 57 2 17 80 57 ' 47 48 60 82 14 53 OI 55 3 IS 78 55 45 46 59 81 . 12 52 6099 54 4 78l2 7576 7353 7H3 6944 6757 6579 6410 6250 6098 5952 5 IO 74 5i 4i 42 55 77 9 48 9 6 Si 6 08 7i 49 39 4i 53 76 07 47 95 5 7 05 69 46 37 39 5i 74 05 45 93 48 8 03 67 44 35 37 49 72 04 44 92 47 9 7800 7564 7342 ' 7133 6935 6748 6570 6402 6242 6090 5945 10 7798 62 40 3i 33 46 69 OO 4i 89 44 ii 96 60 38 29 3i 44 67 6399 39 87 42 12 93 58 36 27 29 42 65 97 38 86 4i 13 9 1 55 34 25 27 40 63 96 36 84 40 14 7788 7553 7331 7123 6925 6738 6562 6394 6234 6083 5938 15 86 5i 29 21 23 37 60 92 33 81 37 16 83 48 27 18 21 35 58 9i 3i 80 35 i7 81 46 25 16 I 9 33 56 89 30 78 34 18 78 44 23 14 18 3 1 55 87 28 77 33 19 7776 7542 732i 7112 6916 6730 6553 6386 6227 6075 593i 20 74 39 19 IO 14 28 5i 84 25 74 30 21 7i 37 16 08 12 26 50 82 24 72 28 22 69 35 14 06 IO 24 48 81 22 7i 27 2 3 66 32- 12 04 08 22 46 79 20 70 26 24 7764 7530 7310 7102 6906 6720 6544 6378 6219 6068 5924 25 62 28 08 oo 04 19 43 76 17 66 2 3 26 59 26 06 7098 O2 17 4i 74 16 65 21 27 57 23 04 96 OO 15 39 73 14 64 20 28 7754 752i OI 94 6898 13 38 7i 13 62 19 29 7752 7519 7299 7092 6897 6711 6536 6369 6211 6061 5917 30 So 16 97 90 95 IO 34 68 IO 59 l6 31 47 14 95 88 93 08 32 66 08 58 !<: 3 2 45 12 93 86 9i 06 3i 64 07 56 13 33 42 IO 9i 84 89 04 29 63 05 55 12 34 7740 7508 7289 7082 6887 6702 6527 6361 6204 6053 5910 35 228 GRAVITY MEASUREMENTS TABLE G (Cow.)- PERIODS OF QUARTER METER PENDU- NOTE : To obtain period to 7th decimal place, prefix .50 or .500 to figures in the table. Body of table gives 2200 2300 2400 2500 2600 2700 2800 2900 3000 3100 36 06 25 84 78 02 54 3i 30 48 85 37 OI 20 80 74 9498 5i 28 27 45 82 38 11,196 16 75 70 95 48 24 24 43 80 39 91 ii 7i 66 9i 44 21 21 40 77 40 11,186 10,707 10,267 9862 9488 9141 88l8 8518 8237 7974 4i 81 02 63 58 84 38 IS 15 34 72 42 76 10,698 58 54 81 34 12 12 32 69 43 7i 93 54 50 77 3i 9 09 29 67 44 66 88 50 46 73 27 06 06 26 64 45 11,161 10,684 10,246 9842 9470 9124 8803 8503 8224 7962 46 56 79 42 39 66 21 00 00 21 59 47 Si 75 38 35 62 18 8797 8498 18 57 48 46 70 33 3i 59 14 94 95 16 54 49 4i 66 29 27 55 ii 90 92 13 52 50 11,136 10,661 10,225 9823 9452 9108 8787 8489 8210 7949 5i 3i 56 21 19 48 04 84 86 08 47 52 26 52 17 16 45 01 81 83 05 44 S3 21 47 12 12 4i 9098 78 80 02 42 54 16 43 08 08 38 94 75 78 8i99 39 55 n, in 10,638 IO,2O4 9804 9434 9091 8772 8475 8i97 7936 56 06 34 IO,2OO 9800 30 88 69 72 94 34 57 OI 29 10,196 9796 27 84 66 69 9i 32 58 11,096 25 92 92 23 81 63 66 89 29 59 91 20 88 88 20 78 60 63 86 26 60 I I, 086 10,616 10,183 9785 9416 9074 8757 8460 8183 7924 61 82 ii 79 81 13 7i 54 57 81 21 62 77 07 75 77 09 68 5i 54 78 19 63 72 02 7i 73 06 65 47 52 75 16 64 67 10,598 67 69 O2 61 44 49 73 14 65 11,062 IQ.593 10,163 9766 9398 9058 8741 8446 8170 79ii 66 57 89 59 62 95 55 38 43 67 09 67 52 84 54 58 92 5i 35 40 65 06 68 47 80 50 54 88 48 32 37 62 04 69 42 75 46 So 84 45 29 34 59 01 70 11,038 10,571 10,142 9747 938i 9042 8726 8432 8i57 7899 FORM OF RECORD OF PENDULUM OBSERVATIONS 229 LUM WHEN PENDULUM IS SLOWER THAN CHRONOMETER Top and left-hand arguments combined give interval s = ten coincidence intervals. / = period in seconds. 3200 3300 3400 3500 3600 3700 3800 3900 4000 4100 4200 o 38 05 86 80 85 OI 26 60 O2 52 09 36 35 03 84 78 83 66 99 24 58 00 5o 07 37 33 OI 82 7 6 8! . 97 22 56 6199 49 06 38 3 7498 80 74 80 95 21 55 97 47 05 39 7728 7496 7278 7072 6878 6693 6519 6353 6196 6046 5903 40 26 94 76 70 76 92 17 52 94 44 O2 4i 23 92 74 68 74 90 16 5o 93 43 oo 42 21 9<> 72 66 72 88 14 48 9i 42 5899 43 18 87 70 64 70 86 12 47 90 40 98 44 7716 7485 7267 7062 6868 6684 6510 6345 6188 6039 5896 45 14 83 65 60 66 83 09 44 87 37 95 46 ii 80 63 58 64 81 07 42 85 36 93 47 09 78 61 56 62 79 05 40 84 34 92 48 06 76 59 54 61 77 04 39 82 33 9i 49 7704 7474 7257 7052 6859 6676 6502 6337 6180 6031 5889 50 02 72 55 50 57 74 00 36 79 30 88 5i 7699 69 53 48 55 72 6499 34 77 28 86 52 97 67 Si 46 53 70 97 32 76 27 85 53 95 65 48 44 51 69 95 3i 74 26 84 54 7692 7463 7246 7042 6849 6667 6494 6329 6173 6024 5882 55 90 60 44 40 47 65 92 28 7i 23 81 56 88 58 42 38 46 63 90 26 70 21 80 57 85 56 40 36 44 61 88 24 68 2O 78 58 83 7680 54 7452 38 7236 34 7032 42 6840 60 6658 87 6485 , 2 3 6321 67 6165 18 6017 77 5875 g 78 49 34 30 38 56 83 20 64 IS 74 61 76 47 32 28 36 54 82 18 62 14 73 62 73 45 30 26 34 52 80 1 6 61 12 7i 63 7i 43 28 24 32 51 78 15 59 II 70 64 7669 7440 7225 7022 6831 6649 6477 6313 6158 6010 5868 65 66 38 23 20 29 47 75 12 56 08 67 66 64 36 21 18 27 45 73 IO 55 07 66 67 62 34 19 17 2 5 44 72 08 53 05 64 68 59 32 17 15 23 42 70 07 52 04 63 69 7657 7429 7215 7013 6821 6640 6468 6305 6150 6OO2 5862 70 230 GRAVITY MEASUREMENTS TABLE G (Cow.). PERIODS OF QUARTER METER PENDU- NOTE : To obtain period to 7th decimal place, prefix .50 or .500 to figures in the table. Body of table gives o 2200 2300 2400 2500 2600 2700 2800 2900 3000 3100 71 33 66 38 43 77 38 23 29 54 96 72 28 62 34 39 74 35 20 26 5i 94 73 23 57 30 35 70 32 17 23 49 9i 74 18 53 26 3i 67 29 14 20 46 89 75 11,013 10,548 10,122 9728 9363 9025 8711 8418 8i43 7886 76 08 44 17 24 60 22 08 15 4i 84 77 04 40 13 20 56 19 05 12 38 81 78 10,999 35 09 16 53 16 02 09 35 79 79 94 3i 05 12 49 12 8699 06 33 76 80 10,989 10,526 IO,IOI 9709 9346 9009 8696 8403 8130 7874 81 84 22 10,097 05 42 06 93 OI 28 72 82 79 18 93 OI 39 02 90 8398 25 69 83 74 13 89 9697 35 8999 87 95 22 67 84 70 09 85 94 32 9 6 84 92 2O 64 85 10,965 10,504 10,081 9690 9328 8993 8681 8389 8lI7 7862 86 60 10,500 77 86 25 9 78 86 14 59 87 55 10,495 73 82 21 86 75 84 12 57 88 5i 9i 68 79 18 83 72 81 09 54 89 46 87 64 75 14 80 69 78 06 52 90 10,941 10,482 10,060 9671 93" 8977 8665 8375 8104 7849 9i 36 78 56 68 08 74 62 72 OI 47 92 3i 73 52 64 04 70 60 70 8098 44 93 27 69 48 60 OI 67 56 67 9 6 42 94 22 65 44 56 9297 64 54 64 93 40 95 10,917 10,460 10,040 9653 9294 8961 8650 8361 8091 7837 96 12 56 36 49 90 57 48 58 88 34 97 08 52 32 45 87 54 44 56 85 32 98 03 47 28 4i 83 Si 42 53 83 30 99 10,898 43 24 38 80 48 39 50 80 27 100 10,893 10,438 10,020 9634 9276 8944 8636 8347 8078 7825 FORM OF RECORD OF PENDULUM OBSERVATIONS 231 LUM WHEN PENDULUM IS SLOWER THAN CHRONOMETER Top and left-hand arguments combined give interval s = ten coincidence intervals. t = period in seconds. 3200 33oo 3400 3500 360O 3700 3800 3900 4000 4100 4200 O 55 27 13 II 19 38 67 04 49 OI 60 71 52 25 ii 09 18 37 65 02 47 oo 59 72 5o 23 09 07 16 35 63 oo 46 5998 58 73 48 21 07 05 14 33 61 6299 44 97 56 74 7645 7418 7205 7003 6812 6631 6460 6297 6142 5995 5855 75 43 16 02 OI 10 3 58 96 4i 94 53 76 4i 14 oo 6999 08 28 57 94 40 92 52 77 38 12 7198 97 06 26 55 2 38 9i 5i 78 36 10 96 95 05 24 53 9i 36 89 49 79 7634 7407 7194 6993 6803 6622 6452 6289 6i35 5988 5848 80 3i 05 92 9i OI 21 So 88 34 87 47 81 29 03 90 89 6799 19 48 86 32 85 45 82 27 01 88 87 97 17 47 85 3 84 44 83 24 7399 86 85 95 16 45 83 29 . 82 42 84 7622 7396 7184 6983 6794 6614 6443 6281 6128 598i 5841 85 20 94 82 81 92 12 42 80 26 80 40 86 17 92 80 79 IO 40. 78 24 78 38 87 IS 90 78 77 88 09 38 77 23 77 37 88 13 88 76 75 86 07 37 75 22 75 36 89 7610 7386 7174 6974 6784 6605 6435 6274 6l20 5974 5834 90 08 83 72 72 82 03 33 72 18 72 33 9i 06 81 70 70 81 O2 3 2 70 17 7i 3 2 9 2 03 79 6 7 68 79 6600 30 69 16 69 3 93 OI 77 65 66 77 6598 28 67 14 68 29 94 7599 7375 7163 6964 6775 6596 6427 6266 6112 5967 5828 95 96 72 61 62 73 95 25 64 ii 65 26 96 94 70 59 60 7i 93 2 3 62 IO 64 25 97 92 68 57 58 70 9i 22 61 08 62 23 98 90 66 55 56 68 89 20 59 06 61 22 99 7587 7364 7153 6954 6766 6588 6418 6258 6105 5960 5821 IOO 232 GRAVITY MEASUREMENTS 151. Form of Record cf Pendulum Observations. Following is a specimen record of a single swing made with ''Apparatus B," belonging to the Coast Survey. Station: Sawah Loento, Sumatra. Date: May 7, 1901. Observer: G. L. H. Chronometer: Bond 541 (sid.) Pendulum B 4, Direct, on Knife edge / Observed coincidences. Pressure. Temperature. Arc. h m s mm. (C). mm. D 9 59 03 U 10 02 12 27-5 D 05 ii 27.5 22 . 6 4-5 = 52' U 08 18 55-o D II 12 U 14 19 D 4 54 42 U 58 12 28.0 D 5 oo 43 28.0 28.8 0.9 = 10' U 04 08 56.0 D 06 42 U 10 06 55-5 4-2 51.3 atoC. Ther. error 25.70 -30 Total interval (mean) 6^ 55 43* = 24,943*. Approximate length of coincidence interval = 3"* 01* = 181*. Number of coincidence intervals = 138. Length of one coincidence interval = 180.75. Period (uncorrected) = 0.5013869. Uncorrected Period Corr. for Arc " " Temp. " " Press. " " Rate (No. 541) " " Flexure Corrected Period = REDUCTION TO SEA-LEVEL 233 152. Calculation of g. After the period has been corrected for instrumental errors, the value of gravity (g) may be found by comparing the period (P) with that of the same pendulum at some point where the value of g is known, say at Washington. If the value at Wash- ington is g wj then E>2 * w r i S=--Jf&- I 10 ?] Evidently it is of the greatest importance that the period should not change during a series of observations made for the purpose of comparing P at different stations. The pendulum should be swung at frequent intervals at the base station, to test its in- variability; in any case it should be swung at the beginning and end of every series. Example. Suppose that the mean corrected period of a set of pendulums at a station is 0.5012480, and at Washington, the base station, is 0.5007248, and that g w is taken as 980.111 dynes. Then, by formula [107], g = 978.066 dynes. 153. Reduction to Sea-Level. The value of gravity found in the manner just described is the value at the station, assuming the length of the pendulum to be invariable and the chronometer correction to be correct. In comparing values at different stations, however, it is essential to reduce the observed value to the value at sea-level. A formula long used for this purpose is one devised by Bouguer when re- ducing observations made along the Peruvian arc in 1749. This formula is in which H is the elevation of the station above sea-level, r is the radius of the earth, 5 is the density at the surface, and A is the mean density of the earth. The first term of this formula allows for the decrease in gravity due to height alone; the second term, for the increase in attraction due to the topography beneath the station. 234 GRAVITY MEASUREMENTS The correction for height of station is derived from the law of gravitation, namely that the force of attraction varies inversely as the square of the distance; whence Therefore go = g (i + ^) - [109] The correction for topography is based upon the assumption that it is due to the attraction of a cylinder whose axis is vertical and whose height is small compared with its width. The at- traction on a unit mass at the station is shown by Helmert (Hohe. Geocasie t V(A. II, pp. 142 and 164) to be Ag = 2 irkdH. (a) The attraction of the sphere on the same mass is . (b) r 3 Dividing (a) by (b) and multiplying by g, Adding both corrections ([109] and [no]) and remembering that the two are of opposite sign, 2H 3 5 H n = a 4- a -- a . . r ^ r g 2 A r Another method of reduction which has been much used is to omit the last term of Bouguer's formula, and correcting for height only. In this case the correction to g is ^ . 2 H r , Corr. = H -- g, [112] or Corr. = +0.0003086 H (meters). [1120] * See also Clarke, Geodesy, p. 325. For an additional term for irregularity in topography see Coast Survey Report for 1894, p. 22. CALCULATION OF THE COMPRESSION 235 This method was introduced because the former method showed large disagreement between observed and computed values. The second, or ''free-air," method showed better agree- ments, indicating a compensation due to variations of density beneath. The method employed by Professor Hayford in the Coast Survey investigation shows that still better agreement is obtained by the introduction of the assumption of isostasy. The results corrected by this method show a close general agreement, but in certain localities there is evidence that the isostatic adjustment is imperfect for example, near Seattle in the United States and at certain places near the Himalayas in India. 154. Calculation of the Compression. By employing a large number of observed values of g the most probable values of the constants g e and g p may be found. From these data the compression may be derived by applying Clairaut's formula, 2 f a P z 6 6 /2TT\ 2 The value of c e is f 1 a, where T = 86164.09 seconds and a is the equatorial radius. Using Clarke's value of a, the resulting value of c e is found to be c e = 0.033916, ana using for g e the value 978.038,* we obtain Then for the compression, we have a b i a 297.1 If the more accurate form [980] of Clairaut's equation is em- ployed, the result is a b _ i a 298.2 * See Coast Survey Special Publication No. 12. 236 GRAVITY MEASUREMENTS By studying a large number of gravity observations in all parts of the world Helmert obtained the value a b i a 298.3 0.7 In the publication entitled Effect of Topography and Isostatic Compensation upon the Intensity of Gravity the authors give a b i a 298.4 1.5 In the most recent report on gravity work (Coast Survey Special Publication No. 40, 1917), the compression calculated from the observations in the United States, Canada, Europe and India is a b I r -, a ~ 297.4 By employing Equa. [88] the value of g may be computed for each station on the assumption that the earth is a spheroid. A comparison at each station of the observed and computed values of gravity indicates to what extent the geoid departs from the spheroid at each point. Problem i. PROBLEMS Compute from the following data: Station. go- Latitude. Umanak, Greenland 082 ?Qi? + 7O 4O 2Q Sawah Loento, Sumatra Q?8 O57 oo 41 40 Problem 2. If the coincidence intervals are 5 during an 8-hour swing, what will be the error ih P due to an error of i 8 in noting the time of a coincidence? CHAPTER X PRECISE LEVELING TRIGONOMETRIC LEVELING 155. Precise Leveling. The term precise leveling is applied to the operation of deter- mining differences in elevation of successive points on the earth's surface with instruments and methods which, though similar to those used in ordinary leveling, are more refined and capable of yielding a much higher degree of precision. In order to secure the greatest possible accuracy, it is necessary to modify our con- ception of the nature of a level surface and to introduce certain corrections which are ordinarily negligible. It should be ob- served that since the line of sight of the instrument is always theoretically perpendicular to the direction of gravity at each station, it lies in a plane which is tangent to the geoid, not to the spheroid. In tracing out a level line by means of the spirit level we are following the curvature of the geoidal surface. The term precise leveling has for many years been applied to all leveling of a fairly high degree of precision, but there have been various limits of precision prescribed by the different or- ganizations carrying on the work. The accuracy obtainable has been so greatly increased through recent developments in instru- ments and methods that in 1912 a new class of leveling, known as leveling of high precision, was established by the International Geodetic Association; it is to include every line, set of lines, or net, which is run twice in opposite directions, on different dates, and whose errors, both accidental and systematic, computed in accordance with formulas stated in the resolution,* do not exceed dbi mm per kilometer for the probable accidental error and dbo.2 mm per kilometer for the probable systematic error. * See Coast Survey Special Publication No. 18, p. 88. See also Report of In- ternational Geodetic Association for 1912. 237 238 PRECISE LEVELING TRIGONOMETRIC LEVELING Many different instruments have been used in the past for precise leveling, some of the "wye " type and some of the "dumpy " type. All precise levels, however, have certain characteristics in common: namely, (i) a telescope of high mag- nifying power, mounted on a heavy tripod: (2) a sensitive spirit level; (3) a slow-motion screw for centering the bubble; (4) stadia PRECISE LEVELING 2 39 wires for determining the length of sight; and (5) a mirror or other optical device for viewing the bubble from the eye end of the telescope. Before the year 1 899 the precise leveling of the United States Coast Survey was done with a wye level and target rods. The target was not set exactly on the level of the instrument, but FIG. Spa. Precise Level. (C. L. Berger and Sons.) was set approximately, and corrections to this approximate read- ing were determined, using the micrometer screw to measure the small vertical angles. Since 1899 * a dumpy level of new design has been substituted for the wye level, the self-reading rod * For a discussion of this change in methods see Coast Survey Report for 1899, p. 8, and for a description of the new instrument see Coast Survey Report for 1900, p. 521, and for 1903, p. 200. 240 PRECISE LEVELING TRIGONOMETRIC LEVELING -a adopted, and the micrometer screw used only for centering the bubble. This new instrument and method have been adopted by several other branches of the government service. 156. Instrument. The new instrument, sometimes called the prism level, is designed to reduce, so far as possible, any errors arising from unequal heating of the different portions of the instrument. (Fig. 89.) The tele- scope barrel is made of an alloy of iron and nickel having a low coefficient of expansion (0.000004 P er i C.). The level vial is set into the telescope tube as low as possible without interfering with the cone of rays from the object glass. This diminishes the effect of differential expansion of the parts support- ing the level. At one side of the telescope is another (similar) tube containing a pair of prisms which, together with a mirror mounted above the telescope, enable the observer to view the ends of the bubble with the left eye at the same time that he looks at the rod with the right eye. The arrange- ment of mirror and prisms is such that there is no parallax caused by the glass in the level or the mirror. The instrument is provided with the usual small levels for the approximate leveling of the base. 157. Rods. The rods used are of the non-extensible type, graduated to centimeters and marked so that they may be read directly by the observer through the telescope, the millimeters being estimated. (Fig. 90.) The rods are in the form of a cross (in sec- tion) ; they are treated with paraffin to make them proof against moisture. Metal plugs are inserted three meters apart for verifying the length of the ADJUSTMENTS 241 rod. Each rod has a spirit level attached, to show when it is vertical, and also a thermometer, which is read at each sight. 158. Turning Points. Foot-pins are carried by all leveling parties, to be used when other turning points are not available. These are about one foot long, with a depression at the top in which to hold the rod. A rope run through a small hole is provided for pulling up the pin. Most of the leveling of the Coast Survey is carried along railroad lines, and the top of a rail is the usual turning point. 159. Adjustments. The adjustments of the level are nearly the same as those of the ordinary dumpy level. The rough levels are adjusted so as to remain in the center when the telescope is revolved about the vertical axis. The axis of the long bubble tube is adjusted parallel to the line of sight of the telescope whenever it is much in error. This adjustment is tested each day by taking four readings, like those used in the "peg " method, except that the shorter sights are 10 meters in length and the longer sights are of the usual length, (say ioo m ). From these four readings a factor C is computed, which is the ratio of the correction for any reading to the corresponding rod interval. The difference in the sums of the foresight and backsight at any set-up is to be multiplied by this factor C. To find an expression for C, call % and ih the rod readings for the nearer sights, and di and d% the rod readings for the distant sights, Si and s 2 the nearer stadia intervals, and Si and 5 2 the dis- tant stadia intervals, the subscripts referring to the first and second instrument positions. Then the true difference in eleva- tion from the first set-up is (ni + Csi) - (4 + C5i), and for the second set-up, (4 + C&) - (2 + Cs z ). 242 PRECISE LEVELING TRIGONOMETRIC LEVELING Equating and solving for C, /i Oh *t - (di + 4) (& + &)-(* + *) C is -f- if the line of sight is inclined downward. Below is table showing a determination of C (from Coast Survey Report for 1903). DETERMINATION OP C. 8.20A.M., AUGUST 28, 1900 (Left-hand page.)' (Right-hand page.) Number of station. Thread reading, backsight. Mean. Thread interval. Rod. Thread reading, foresight. Mean. Thread interval. A B Corr. fo 1515 1528 1542 2252 2357 2462 r curv. and r 1528.3 2357-0 0461 . 7 13 14 27 105 105 2IO 419 52 W W 0357 0462 0566 1276 1288 1301 36; 0461 .7 1288.3 1528.3 105 104 20 9 12 13 25 2818.7 ef. -0.8 2816.6 2817.9 367 2817.9 0-1-3 ( 0.004= C If the value of C is less than 0.005, the instrument should not be adjusted. If between 0.005 an d o.oio, the observer is advised not to adjust. If over o.oio, the adjustment should be made. The adjustment is made by moving the level rather than the cross-hair ring, to avoid moving the line of sight from the optical axis. 160. Method of Observing.* It is customary to use two rods, the one that is held for a fore- sight on a certain turning point being kept at the same turning point for a back sight. The instrument is set up and leveled, * The General Instructions for Precise Leveling will be found in Coast Survey Special Publication No. 22, p. 29. COMPUTING THE RESULTS 243 and all three hairs are read on the back rod, the level being kept central at each reading. As soon as possible thereafter the three hairs are read in a similar manner on the forward rod. The readings are estimated to miUimeters. The temperature on the rod thermometer is read at the same time. The level should be shaded from the sun in order to avoid unequal heating of its parts. In selecting instrument and rod points, the observer must keep the difference in length of the forward and backward sight less than 10 meters on any one set-up and less than 20 meters for the accumulated difference at any time. The readings of the upper and lower (stadia) wires enable the recorder to determine the difference in distance at each set-up. The maximum length of sight allowable is 150, a distance reached only under exception- ally favorable conditions. At odd-numbered stations the back sight is taken first; at even-numbered stations the fore sight is taken first. This results in the same rod being read first each time. Lines between bench marks are divided into sections of from one to two kilometers each. Each of these sections is run for- ward and backward. If the two differences in elevation so de- termined are found to differ by more than 4 mm VK (K = kilo- meters), both runnings must be repeated until such a check is obtained. Lines may be run with such care that it is seldom necessary to repeat, but the maximum economy appears to be reached when from 5 to 15 per cent of the sections have to be re- run. On page 244 is a set of notes used in leveling with this in- strument (Coast Survey Report, 1903). The most recent practice is to record the readings directly on adding machines carried with the leveling outfit. This results in a saving of time and in avoiding many mistakes in recording and adding. 161. Computing the Results. In computing the results of precise leveling, corrections are applied for the nonadjustment of the level, for curvature and 244 PRECISE LEVELING TRIGONOMETRIC LEVELING refraction, for error in length of rod, for error due to temperature of rod, and for the orthometric correction. The curvature and refraction corrections are usually taken from tables (Coast Survey Report, 1903). The length of rod is tested at the office at the be- ginning and end of the season, and variations during the season are tested in the field by means of a steel tape. The temperature correction is derived from tables, the argument being the ob- served temperatures. SPIRIT LEVELING (Left-hand page.) (Right-hand page.) Date: August 29, 1900. From B.M. : 68. To B.M. : O Sun : C. Forward. Backward: Wind : S.T. (Strike out one word.) Thread Thread XT- -{ read- Thread Sum of Rod read- Thread Sum of JNO. OI station. ing, Mean. [ inter- inter- and ing Mean. inter- inter- back- val. vals. temp. fore- val. vals. sight. sight. 43 0674 99 V 2683 99 0773 0773.0 99 38 2782 2782.3 100 0872 198 2882 199 0925 106 w 2415 103 44 1031 1030.3 104 35 2518 2518.0 103 1135 210 408 2621 206 405 0484 9 8 V 2510 96 45 0582 0582.3 99 35 2606 2606.0 96 0681 197 605 2702 192 597 0398 97 W 2859 96 46 0495 0495.0 97 34 2955 2954-7 95 0592 194 799 3050 191 7 88 1627 26 V 1006 29 47 1053 1053-3 27 34 1035 1034-7 28 1080 53 852 1063 57 845 11895.7 3933-9 -7961.8 2 : 25 P.M. SOURCES OF ERROR 245 162. Bench Marks. The bench marks used in precise leveling are of various types. Wherever it is practicable, the metallic plates shown in Fig. 91 are used to mark the points, but nearly all of the kinds of bench marks which are used by engineers are used also in this class of work. The distance between benches is not allowed to exceed 15 kilometers; every 100 kilometer section should -have at least 20 bench marks, a good average distance being 2.5 kilometers. In cities the old bench marks are often utilized for the precise levels. 163. Sources of Error. The sources of error which it is particularly necessary to study in this class of work are (i) unequal effects of temperature changes in the instrument, (2) gradual rising or settling of the instrument or rods, (3) variations in refraction of the air, (4) unequal lengths of sights, (5) errors in length and temperature of rod, and (6) convergence of level surfaces. TABLE H. TOTAL CORRECTION FOR CURVATURE AND REFRACTION Distance. Correction to rod reading. Distance. Correction to rod reading. m. m. mm. m. mm. o to 27 0.0 1 60 -1.8 28 to 47 O.I 170 2 .1 48 to 60 0.2 180 2-3 61 to 72 -0-3 190 -2.6 73 to 8 1 -0.4 200 -2.8 82 to 90 -0-5 2IO -3-0 91 to 98 -0.6 2 2O -3-3 99 to 105 -0.7 230 -3-7 106 to 112 -0.8 240 4.0 113 to 118 -0.9 250 -4-3 119 to 124 .0 260 -4-7 125 to 130 .1 270 -5-o 131 to 136 .2 280 -5-4 137 to 141 - -3 2 9 -5-8 142 to 146 - -4 300 -6.2 147 to 150 - -5 246 PRECISE LEVELING TRIGONOMETRIC LEVELING CORRECTION 247 TABLE I. DIFFERENTIAL CORRECTION FOR CURVA- TURE AND REFRACTION Mean length of sight in rod interval in milli- meters. Difference of sights in rod interval in millimeters. 2 4 6 s 1C 12 14 (6 IS 20 22 24 26 28 30 32 34 36 38 40 4 44 46 48 50 5~ 54 5658 10 20 30 40 50 60 TO 80 90 100 no 120 130 140 ISO 160 170 180 190 200 210 22O 230 240 250 260 270 280 290 300 3io 320 330 340 350 360 400 440 480 520 .0 -0 .0 c .0 c c .c .0 .0 c .c .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .c c Q c ~ ,0 .0 c c .0 c c Q .0 c .0 .0 ,c .0 .0 .0 .0 .0 .0 .c .0 c c .X . I .X .1 .X .X .1 .1 .1 .1 .1 I I .1 .1 .1 .1 ^ I .1 .X .1 .1 .1 .1 .1 . .2 .2 .2 .2 .1 .1 - .1 ,1 .1 .1 _^_ .2 .2 .2 .2 .1 .1 .1 .1 .1 .1 .1 .1 .1 _ .1 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .2 .0 .0 .0 .c .c .c .0 .0 .0 .0 .0 Q .0 .0 .0 z .0 -0 .0 .0 .0 .0 ^0 .0 .0 c .0 .0 .1 .1 .1 .1 .1 .1 .1 .1 .X .1 .1 .1 .1 .1 .X .1 .1 .1 .1 .X .X .X .1 .X .c .c c I I .0 I .1 I .1 . I .1 .1 .1 .1 .1 .1 I .1 .1 .1 I .1 .1 .1 .1 .1 .X .1 .1 .1 .1 .1 .1 .1 .1 .1 .1 .1 .1 .1 .1 .1 I I .1 .1 .1 .1 .1 .1 I .1 .1 .1 .1 .1 .1 .1 2 .2 .1 .1 .1 .1 .1 .1 .2 .2 .2 .1 .1 .1 .1 .1 .c .c .c c .c .c ,c .c .c .c .0 c .0 .0 .0 .0 I I I .1 .0 .0 .c .0 .c .0 .0 -C c .c .c .c .0 .0 .0 .0 .0 .c .0 .0 1 .c .0 .c .0 .0 .0 ."6 .0 .0 c I I I I I I I I I I I I I .1 I I I I I I . I I I I I I I I , I I I I I I I .1 .1 I I I .1 I I I .1 .1 I .1 .1 I .1 ;I . I ' .1 I .1 .1 " .2 .2 .2 .2 .2 .2 .2 .2 .2 .1 .1 .1 .1 .1 .1 .1 .1 .1 .X I _I 2 2 2 2 .1 I -2 2 -2 .2 .2 .2 2 3 3 3 3 3 3 ^ .2 .2 3 .3 3 3 .3 3 _i -4 4 4 -2 -3 3 -3 3 -3 3 -3 3 -3 3 3 3 -3 3,3 .3J4 4 4 4 .4 4 -4 .1 .2 .2 .2 2 2 .2 c .c .in .0 .0 .c z .0 .0 .0 .c .c .c .c .c .c .c ' z .0 I .0 .0 .0 .0 c .1 I I I 2 2 2 2 2 2 3 .2 .2 2 .2 2 .2 2 2 .2 3 3 3 2 .2 .2 -2 3 3 3 '2 2 2 3 3 3 3 2 2 2 3 3 3 3 2 2 3 3 ^ 2 3 .3 3 } 3 3 3 .3 3 ^ I I I .1 I I _I 2 T 2 2 2 2 2 2 .2 .2 2 2 2 3 3 3 _d 4 4 4 4 4 4 4 248 PRECISE LEVELING TRIGONOMETRIC LEVELING TABLE J. CORRECTION FOR TEMPERATURE (IN MILLIMETERS) Temp. C. Difference of elevation in meters. i 2 3 4 5 6 7 8 9 IO ii 12 13 14 I O.O O.O O.O O.O O.O o.o O.O O.O O.O o.o o.o 0.0 0.0 O.I 2 O.O o.o O.O o.o O.O o.o O.I O.I O.I O.I O.I O.I O.I O.I 3 o.o o.o O.O o.o O.I O.I O.I O.I O.'l O.I O.I O.I 0.2 0.2 4 o.o o.o O.O O.I O.I O.I O.I O.I O.I 0.2 O.2 O.2 O.2 0.2 5 0.0 o.o O.I O.I O.I O.I O.I O.2 O.2 O.2 O.2 O.2 o-3 0-3 6 o.o o.o O.I O.I O.I O.I O.2 O.2 O.2 O.2 0-3 0-3 o-3 0-3 7 o.o 0. O.I O.I O.I O.2 O.2 O.2 O.2 0-3 0-3 0-3 0.4 0.4 8 l 0.0 0. 0. O.I O.2 O.2 O.2 0-3 0-3 o-3 0.4 0-4 0.4 0.4 9 0.0 O. 0. O.I 0.2 O.2 O.2 0-3 0-3 0.4 0.4 0.4 o-5 0-5 10 o.o O. 0. 0.2 0.2 O.2 0-3 0-3 0.4 0.4 0-4 0-5 0.6 ii 0.0 0. 0. O.2 O.2 0-3 0.3 0.4 0.4 0.4 o-5 o-5 0.6 0.6 12 0.0 0. 0. 0.2 0.2 0.3 o-3 0.4 0.4 0-5 o-5 0.6 0.6 o-7 13 o.o 0. 0.2 0.2 0-3 0.3 0.4 0.4 o-5 0-5 0.6 0.6 0.7 0.7 14 O.I 0. 0.2 0.2 0-3 0.3 0.4 0-4 0.6 0.6 o-7 0.7 0.8 IS O.I O. 0.2 0.2 o-3 0.4 0.4 0-5 0-5 0.6 0.7 0.7 0.8 0.8 16 O.I 0. 0.2 0-3 o-3 0.4 0.4 0-5 0.6 0.6 0.7 0.8 0.8 o-9 17 O.I O. 0.2 0-3 0.3 0.4 0-5 o-5 0.6 0.7 0.8 0.8 0-9 o-9 18 O.I O. O.2 0-3 0.4 0.4 0.6 0.6 o-7 0.8 o -9 -9 i .0 19 O.I 0.2 O.2 0-3 0.4 0.5 5 0.6 0.7 0.8 0.8 o-9 .0 i .1 20 O.I O.2 O.2 0-3 0.4 o-5 0.6 0.6 0.7 0.8 o-9 I .0 .0 i .1 21 O.I O.2 O.2 o-3 0.4 o-5 0.6 0.7 0.8 0.8 0-9 i .0 .1 I .2 22 O.I O.2 0-3 0.4 0.4 o-5 0.6 0.7 0.8 0.9 i .0 i .1 .1 I .2 23 O.I O.2 o-3 0.4 0-5 0.6 0.6 0.7 0.8 o-9 I .0 i .1 .2 1-3. 24 O.I 0.2 0-3 0.4 0-5 0.6 0.7 0.8 0.9 i .0 i.i 1.2 .2 25 O.I O.2 o-3 0.4 0-5 0.6 0.7 0.8 0-9 I .0 i .1 1.2 3 i-4 26 27 O.I O I O.2 O 2 o-3 O 7 0.4 O 4 o-5 r\ f 0.6 0.6 0.7 0.8 0.8 o-9 I O I .0 i .1 I .2 3 I / 28 2Q O.I O . I O.2 O . 2 w -6 O A 0.4 Io s ^ 3 0.6 o 6 0.7 0*7 0.8 0.8 0.9 /-i c\ 1 .0 1 .1 I .2 1-3 4 - 5 1.6 *y 30 O.I O.2 w .q. 0.4 u o 0.5 0^6 - / 0.7 0.8 u.y I .O 1 .1 I .2 i-3 i-4 1.6 31 O.I 0.2 0.4 0-5 0.6 0.7 0-9 I .O 1 .1 I .2 i -4 r -5 1.6 i -7 32 O.I 0-3 0.4 o-5 0.6 0.8 o-9 I .O I .2 1-3 i .4 1 .5 i .7 1.8 33 O.I o-3 0.4 o-5 0.7 0.8 0.9 I .1 I .2 i .4 1.6 i .7 1.8 34 i < O.I O I 0.3 Ct 1 0.4 OA 0-5 0.6 o-7 f\ i-j 0.8 o 8 i .0 I .1 I .2 i-4 i-5 1.6 1.8 i-9 6 j 36 O.I u -6 o-3 4 0.4 0.6 u - / 0.7 \J . O 0.9 i .0 I .2 I .A 1.6 i-7 2 . 2 .0 37 O.I 0.3 0.4 0.6 o-7 o-9 I.O 1.2 1-3 I .5 1.6 1.8 1 -9 2 .1 38 O.I 0.3 0.5 0.6 0.8 0.9 i .1 I .2 i-4 I .5 i .7 1.8 2 .0 2.1 39 O.2 0.5 0.6 0.8 0.9 i .1 I .2 i-4 1.6 1-7 i .9 2 .O 2 .2 40 O.2 o-3 -5 0.6 0.8 i .0 i .1 1-3 i-4 1.6 1.8 1.9 2 .1 2 .2 41 O.2 0.3 0.5 0.7 0.8 I .0 i .1 1-3 i .5 1.6 1.8 2.0 2 .1 2-3 42 0.2 0.3 0.5 0.7 0.8 i .0 I .2 1.5 1.7 1.8 2.0 2.2 2-3 43 0.2 0.3 0.5 0.7 0-9 i .0 I .2 I .4 i .5 i .7 i-9 2 .1 2 .2 2.4 44 O. 2 0.3 0.5 0.7 o-9 i .1 1.2 i-4 1.6 1.8 2 .1 2-3 2-5 45 O.2 0.3 0.5 0.7 0-9 i.i 1-3 i-4 1.6 1.8 2 .O 2 .2 2-3 2-5 DATUM 249 164. Datum. The datum for precise levels is mean sea-level, or the surface of the geoid, as found from tidal observations. This is assumed to be correctly given by the mean of the several " annual means " as derived from tidal observations for sea-level. The heights of the tide are recorded automatically on a self-registering gauge. (See Cut.) The vertical motion of the float is reduced (the ratio FIG. Qia. Self-Registering Tide Gauge. (Coast and Geodetic Survey.) depending upon the range of tide) by passing the connecting wire and cord over a series of pulleys, and is communicated to a recording pencil which marks on a sheet of paper passing over a revolving drum. The drum is revolved at a uniform rate by clock mechanism. The height of the water is referred to a bench mark in the vicinity. Observations of the tide should be ex- 250 PRECISE LEVELING TRIGONOMETRIC LEVELING tended over a period of at least one year in order to determine sea-level with sufficient precision for this class of leveling. In the tidal records at some stations there appear to be small systematic variations in the annual means extending over periods of several years; but, taking the records as a whole, the variations do not seem to follow any particular law, and they are treated as acci- dental. (See Coast Survey Special Publication No. 26.) 165. Potential. In order to investigate the nature of the orthometric correction, due to the convergence of level surfaces, it will be necessary to consider first some of the elementary mechanical principles of the earth's gravitation and rotation. Whenever two attracting bodies are separated, work is done upon them and energy is stored up; that is, the potential energy of the system is increased. The change in potential energy is measured by the amount of work done. When the bodies are an infinite distance apart, the potential energy is a maximum ; when the bodies are in contact, the potential energy of the system is zero. If the masses are free to move, they will always move in such a direction as to diminish the potential energy of the system. If we imagine a unit mass placed at any point P in space and attracted by a mass M, and if the potential energy of the unit mass be measured by the work done upon it to move it from P to infinity, this quantity of potential energy is a property of the given point P; in other words, it is a function of the coordinates of P. It is called the potential at that point. It is not necessary that there should actually be a unit mass at the point, but the conditions are such that if a unit mass were placed at P, it would have this amount of potential energy. It should be observed that the increase of potential energy is measured by the fall in potential. 1 66. The Potential Function. If an attracting body M be divided into small elements, and the mass Aw of each element be divided by its distance from a THE POTENTIAL FUNCTION 251 point P, the limit of the sum of all these fractions, as the elements are made smaller, is called the value at P of the potential function due to M, or simply the potential of P. Calling this function V, then of, if Aw is of density 6 and has the coordinates x', /, z', and P has the coordinates x, y, z, then x' - *Y + (y'- y) 2 + (' - z) 2 ]* The integration over the entire mass gives the value of the po- tential function at P.* 167. The Potential Function as a Measure of Work Done. The amount of work required to move a unit mass (concen- trated at a point) from a point PI to another point P 2 , by any path (Fig. 92), against the attraction of a mass M, is equal to the fall in potential V\ F 2 , where Vi and F 2 are the values of the potential function at the points PI and P 2 . To show this, let r\ and r 2 be the distances from the center of M to the points PI and P 2 . FIG. 92. The work done in moving a unit mass through a small space dr equals the force ( J times the space dr. But the force (^~\ at any point A equals at that point, since V (for a unit mass) = - . Hence the work r r dV j T/ T/ r i = - J ri *r Jr = Fl - F2; [II9] that is, the work done equals the fall in potential. * See Peirce, Theory of the Newtonian Potential Function. 252 PRECISE LEVELING TRIGONOMETRIC LEVELING If the point P 2 is moved to an infinite distance, F 2 become zero, and the potential at P\ then equals the work done in moving the unit mass from PI to infinity; or it is the work done by it in moving from infinity to the point P\. 1 68. Equipotential Surfaces. A level surface, or an equipotential surface, is one having at every point the same gravity potential. It is everywhere per- pendicular to the direction of gravity.* The mean surface of the ocean is such a surface. The surface of any lake is also an equipotential surface. From the proof given in the preceding article it is evident that if there are two such equipotential sur- faces, the difference in potential is the work done upon a unit mass in moving it from one surface to the other. This difference in potential is independent of any particular points on the sur- faces and of the path followed in passing from one to the other; for example, the work done in raising a unit mass from sea-level to the south end of a lake is the same as the work done in raising a unit mass from sea-level to the north end of the lake. Since the work done is the force (w) times the distance (dti) through which it acts, it is evident that w X dh is a constant between two level surfaces. Also, since g varies as the weight (force), g X dh is a constant between these two surfaces. The force of gravity is less at the equator (Art. 144) than at the poles, on account of the action of the centrifugal force. Hence we should expect to find that a given level surface is farther from sea-level at the equator than it is at a point nearer the pole. If several such surfaces be drawn (Fig. 93), they will be seen to converge toward the pole. They are all parallel to each other at the equator and at the poles, and have their greatest difference in direction at < = 45. Since g is about one-half of one per cent less at the equator than * It may be proved that if there is a resultant force at a point in space due to attracting masses, this force acts in the direction of the normal to the equipotential surface through the point (see Peirce, Theory of the Newtonian Potential Function, p. 38). It should be kept in mind that the " force of gravity" is the resultant of the force of attraction and the centrifugal force. EQUIPOTENTIAL SURFACES 253 at the pole, the height h between surfaces is about one-half of one per cent greater at the equator. Hence, if a level surface were icoo meters above the sea-surface at the equator, it would be only 995 meters above sea-level at the pole. A surface at half the elevation would converge (very nearly) half as much. In the line of levels run from San Diego to Seattle the convergence was found to be about i \ meters, showing that at high elevations this error is by no means a negligible one in precise leveling. E O FIG. 93. It is evident that if a series of bench marks is established along a meridian (in the northern hemisphere), and all are placed at the same elevation, using the ordinary methods, those at the northern end of the line lie nearer to sea-level than those at the southern end of the line. It becomes necessary, then, to revise the definition of elevation. If the ordinary definition of elevation is retained, and no allow- ance made for convergence of level surfaces, then different results for the elevation of a point will be obtained, according to which path is followed. If we measure vertically upward from A to B (Fig. 94), and then level by means of the water surface BC, we obtain a greater height for point C than we should if we leveled by water from A to D and then measured vertically upward from D to C. If a correction is applied, however, to allow for the con- vergences of these surfaces, the result is that different portions 254 PRECISE LEVELING TRIGONOMETRIC LEVELING of the lake surface have different elevations, which is apparently absurd if the true nature of the level surface is not understood. In order to avoid this apparent difficulty another method some- times employed is to number all the surfaces with a serial number (called the Dynamic Number), so that all points on the same surface will have their elevation expressed by the same number. This number is defined as the work required to raise one kilogram from sea-level to the given surface, the unit being the kilogram- FIG. 94. meter at sea-level in latitude 45. The United States Coast Survey has adopted the method of applying to ordinary elevations the correction for convergence, called the Orthometric Correction. The Standard Elevations of the Coast Survey in Special Pub- lication No. 1 8 are given by the Orthometric Elevation. 169. The Orthometric Correction. Let W be the work (in absolute units) required to raise a unit mass from sea-level to a point at elevation h, and let H be the dynamic number of the surface through the point, defined by the quotient W -f- #45, where #45 is the value of g at sea-level in latitude 45 (Equa. [p6a], p. 210). Then, since g X dh is constant for two level surfaces separated by height dh, W I gdh = 45 / (i 0.002644 cos 2 . . . ) /o t/o dh in which the integration takes place along the curved vertical. THE ORTHOMETRIC CORRECTION 255 Integrating, W = g 4 s |(i - 0.002644 cos 2 4>) h . . . 1 . [120] The dynamic number W = H = = h (i 0.002644 cos 2 . . . ). [121] To find the correction to the elevation due to a change in the latitude, differentiate the last equation with respect to as the independent variable, and we obtain o = dh 0.002644 ( 2 h sin 2 d + cos 2 < dh . . . ) = dh(i 0.002644 cos 2 <) + 0.005288 /r sin 2 J<, 0.005288 /^ sin 2 d T -, and dh = [.122] i 0.002644 cos 2 = (0.005288 h sin 20) (1+0.002644 cos 2< . . . )^arci',* [123] the factor arc i' being introduced to reduce d to minutes of arc. A more definite idea of the magnitude of this correction may be gained from the following example. Assuming that the ele- vation of Lake Michigan is 177 meters at Chicago, latitude 41 53', what is the elevation of the lake at Milwaukee, in latitude 43 3'? In tne formula, h = 177, d = 70', and = 42 28'; the computed values of dh is 0.0190, and the lake level at Mil- waukee is therefore 176.9810 meters. Tables for computing the orthometric correction will be found in Coast Survey Special Publication No. 18, pp. 54-56. The relation between the dynamic numbers and the ortho- metric elevations is illustrated in the following table, which is an extract from the special publication just mentioned. Station. Latitude. Orth. elev meters. Dyn. number. Smithland, La / 10 ">"> 14.7729 14.7545 Meridian Miss 72 22 IO4. Q4Q4 104.8292 Amblersburg, W. Va Summit, Cal. 39 23 24. 20 494.9221 Il6^ 434^ 494.6287 1164.1008 Riordan, Ariz. ... 34'>2 2213 .8lI2 * For additional terms, neglected in the above formula, see Coast and Geodetic Survey Special Publication No. 18, p. 49. See also Ch. Lallemand, Nivellement ie Haute Precision, Encyclopedic des Travaux Publics, Paris, 1912. 256 PRECISE LEVELING TRIGONOMETRIC LEVELING 170. The Curved Vertical. In view of what has been said regarding the change in the direction of level surfaces with an increase in elevation, it is clear that the vertical line is curved, being concave toward the pole, and therefore that any observation for latitude made at a point above sea-level is referred, not to the true normal to the surface at sea-level, but to the direction of that portion of the vertical which is at the elevation (h) of the station. In order to deter- mine the amount of the correction to reduce the observed latitude FIG. 95. to its value at sea-level, refer again to Equa. [122], p. 255. An inspection will show that the denominator of this fraction is usually not far from unity; and since the correction desired is itself quite small, we may assume dh = 0.005288 h sin 2 < d. [124] The correction to the observed latitude is the difference in the slope of the two surfaces (sea-level and the level of station) measured in the plane of the meridian. From Fig. 95 it is seen that the angle between the level surface through S and a surface parallel to sea-level drawn through S is dh -f- Rd <. But, by Equa. [124], dh 0.005288 h sin 2 Rd~ R Reducing this to seconds of arc, dh _ 0.005 2 88 /? sin 2 ~ R&rci" REDUCTION TO STATION MARK 257- Since R arc i" = 101.3 feet (very nearly), the correction to the latitude may be written 0^.0522 h sin 2 t [125] where h is in thousands of feet; or, if h is in meters, the correction is 0.000171 h sin 2 <. [126] 171. Trigonometric Leveling. The method of measuring the vertical angles between triangu- lation stations has already been described in the chapter on field- work. From the field note-book we have the several measures of the angles, the height of the instrument, and also of the point sighted in each case above the station marks. The elevation of one station above sea-level is assumed to be known, and that of the other is to be computed. Before this can be done, the angle must be reduced to the value it would have if the instrument and the point sighted were coincident with the station marks. 172. Reduction to Station Mark. From the diagram (Fig. 96) it is evident that if i is the height of the instrument at A , and o that of the object sighted at B, and FIG. 96. 5 the distance between stations, obtained from the triangulation, then the correction to the vertical angle at A is Four places in the logarithms are sufficient in computing this correction. 258 PRECISE LEVELING TRIGONOMETRIC LEVELING This reduction need be made only in case of reciprocal obser- vations, that is, observations of the vertical angle from both ends of the line. In case of observations from one station only, the quantity i o, in meters, can be applied directly to the com- puted difference in elevation. When a sight is taken from one station PI to another station P 2 , the verticals of the two stations do not (in general) intersect, because they lie in different planes. If we imagine a plane which is parallel to both verticals, and then project both verticals onto this plane, we obtain the result shown in Fig. 97. FIG. 97. 173. Reciprocal Observations of Zenith Distances. In Fig. 97, PI and P 2 represent the two instrument stations; their elevations above sea-level are PiSi = hi and P 2 5 2 = fe. The ray of light is assumed to take the form of a circular curve, RECIPROCAL OBSERVATIONS OF ZENITH DISTANCES 259 whose radius is determined by the coefficient used in the calcu- lation. The two measured zenith distances are ft and fo. The angle of refraction is Af = rPiP 2 = TP 2 Pi = m 9, where m is the coefficient of refraction, and the central angle PiOP 2 . The radius of curvature of the section SiS 2 is R a} ap- proximately equal to OSi, or to 062 . The quantity to be computed is the difference in elevation fa hi, which may be found by solving the triangle PiP^.* In the triangle PiP^, P*L* = fa hi, the desired difference in elevation; P\L L-^Oj .j5.C, [129] in which ^ = i+f, ^a the correction for elevation of the station of known elevation, WHEN ONLY ONE ZENITH DISTANCE IS OBSERVED 261 the correction for the difference in elevation, and C = i H -- : , 12 R a 2 the correction for distance. The logarithms of A , B, and C are given in Tables K, L, and M, for the arguments hi, log 5 tan - , and log 5, respectively. 174. When only one Zenith Distance is Observed. From (g) and (ti) we have f The refraction angle is Af = m8, where m is the coefficient, to be obtained from the best obtainable values, and which is ap- proximately equal to 0.071 ; substituting mB in the above equation we have and tan - = tan (90 + (0.5 - ) 9" arc i" - f ,) since 0" = Putting this 5 term = k, we have tan p^ 1 ) = tan [90 + k - ft], (n) Substituting in [129] from (), h, - h = s tan [90 +k-h]A.B-C, [130] in which A, B, and C have the same meaning as before, except that B is given for the argument log [s tan (90 + k ft)]. Example. Zenith Distance of Mt. Blue from Farmington, 87 of i8".8; dis- tance, 15,519 meters; m = 0.071; instrument 2.20 meters above station mark; point sighted 4.40 meters above station mark; elevation of Farmington, 181.20 meters. 262 PRECISE LEVELING TRIGONOMETRIC LEVELING m 0.071 (0.5 w) 0.429 log R a " sini' log = 9.6325 log ^ = 4-1909 colog R a sin i" = 8. 5092 2.3326 6.8052 K 4.6856 i . 4908 90 oo oo 215".! = Q 3 '35".i f = 87 07' 18 .8 + 2 5 6 / i6 // . 3 tan log s A B C Red. to Sta. Diff. Eleva. 796.51 meters 2.20 " 794- 3 1 Elev. Farmington 181 . 20 Elev. Mt. Blue 975.51 TABLE K* hi. Log A, units of fifth place of decimals. ft* Log A, units of fifth place of decimals. *L Log A, units of fifth place of decimals. ftfr Log A, units of fifth place of decimals. Meters. Meters. Meters. Meters. O IS4I 3IS6 4770 O II 22 33 73 1688 3303 4917 I 12 23 34 220 1835 3449 5064 2 J 3 24 35 367 1982 3596 5211 3 14 25 36 SH 2128 3743 5357 4 IS 26 37 661 2275 3890 5504 5 16 27 38 807 2422 4036 5651 6 17 28 39 954 2569 4183 5798 7 18 29 40 IIOI 2715 4330 5945 8 19 30 4i 1248 2862 4477 6091 9 20 31 1394 3009 4624 10 21 32 i54i 3156 4770 * In these tables log Ra is taken as 6.80444, Spheroid of 1866. the mean radius in latitude 40 on WHEN ONLY ONE ZENITH DISTANCE IS OBSERVED 263 Table K gives the values of log A , the correction factor for the elevation of the known station, by showing the limiting values of the elevation fe, between which log A may be taken as o, i, 2, 3, etc., units of the fifth place of decimals. Log A is positive, except in the very rare case where hi corresponds to a point below mean sea-level. TABLE L Log 5 tan i Log 5 tan i Log s tan J (T - Ti) or log Log B, units (T, - TO or log Log B units (Ti - Ti) or log Log B units 5 tan (90 + * of fifth place s tan (90 + k of fifth place 5 tan (90 + * of fifth place -r i) (* in of decimals. - f i) (* in of decimals. - Ti) ' (s in of decimals. meters.) meters.) meters.) o 2.167 3-397 3.685 i 9 17 2.644 3-445 3-7II 2 10 18 2.866 3-489 3-735 3 II 19 3.011 3.528 3.758 4 12 IO 3.121 3 .565 3-779 5 13 21 3.208 3-598 3-8oo 6 14 22 3.281 3-629 3.820 7 IS 23 3-343 3-658 3-839 8 16 24 3-397 3-685 3-857 Table L gives the values of log B, the correction factor for approximate difference of elevation by showing the limiting values of log [s tan J (ft - ft)] or log r s tan (90 + k - ft)] be- tween which log B may be taken as o, i, 2, 3, etc., units of the fifth place of dec'mals. Log B has the same sign as the angle i (ft - ft) or 90 + k - ft; for example, if log [s tan (ft -ft)] lies between 3.565 and 3.598 and J (ft ft) is positive, logB = +0.00013, but if \ (ft ft) is negative then log B = 0.00013, i.e., 9.99987 10, the former way of writing being usually more convenient in practice. 264 PRECISE LEVELING TRIGONOMETRIC LEVELING TABLE M Logs (s in meters). Log C, units of fifth place of decimals. Logs (sin meters). Log C, units of fifth place of decimals. o.ooo 5-297 4 4.875 5-352 I 5 5-"3 5-395 2 6 5.224 5-432 3 7 5-297 5-463 Table M gives the value of log C, the correction factor for dis- tance between stations, by showing the limiting values of log 5 between which log C may be taken as c, i, 2, 3, etc., units of the fifth place of decimals. Log C is always positive. PROBLEMS Problem i. Calculate the orthometric correction for a line extending 2 north- ward from a point in latitude 45 N at an elevation of 1000 meters. Problem 2. Compute the correction for reducing to sea-revel a latitude observed at an elevation of one mile in latitude 45 N. Problem 3. Vertical angle from S to B, +2 24' 58".Q4. Vertical angle from B to S, 2 35' 34". 20. Elevation of S = 108.87 meters; distance, 23,931.6 meters; log R , 6.8052. Compute the elevation of B. CHAPTER XI MAP PROJECTIONS 175. Map Projections. Whenever we attempt to represent a spherical or a spheroidal surface on a plane some distortion necessarily results, no matter how small may be the area in question. The problem to be solved in constructing topographic or hydrographic maps is to find a method which will minimize this distortion under the existing conditions. The number of projections which have been devised is very great; for the description and the mathe- matical discussion of the properties of these projections the reader is referred to such works as Thomas Craig's Treatise on Projections, United States Coast and Geodetic Survey, 1882; The Coast and Geodetic Survey Report, 1880; C..L. H. Max Jurisch, Map Projections, Cape Town, 1890; G. James Morrison, Maps, Their Uses and Construction, London, 1902; and A. R. Hinks, Map Projections, Cambridge, 1912. . In this chapter we shall consider only those projections which are used for such maps and charts as are of importance in geo- detic surveys and in navigation. 176. Simple Conic Projection. In this projection the map is conceived to be drawn on the surface of a right circular cone which is tangent to the sphere or the spheroid along the middle parallel of latitude. The apex of the cone lies in the prolongation of the axis of the spheroid. From Fig. 98 it is evident that the distance TA from the apex to the parallel through A is equal to N cot 0. If the cone is developed on a plane surface we shall have a sector whose center is T and whose radius is N cot . (Fig. 99.) All other parallels of latitude on the map will be circles drawn about 265 266 MAP PROJECTIONS the same center T, and all meridians will be represented by straight lines passing through T. The spacing between the parallels of latitude is obtained by laying off distances along the central meridian which are proportional to the distances between the same parallels on the spheroid. The position of the meridians is found by subdividing the middle parallel into spaces which are proportional to the lengths of the arcs of the FIG. 98. same parallel on the spheroid. Straight lines are then drawn from the center T through these points of sub-division. Any meridian or any parallel may be assumed for the central meridian and middle parallel of the map. It is evident from the above that this is not a true projection, that is, the points are not those that would be obtained by projecting from the center of the sphere onto the cone. If the scale of the map is such that the position of the center T cannot be represented on the paper, the curves may be laid off by plotting certain points by means of their rectangular coordinates as described later under the poly conic projection. It is evident that the meridians and parallels of a conic projection intersect at right angles in all parts of the map, as they do on the sphere. The scale of the map is not correct, BONNE'S PROJECTION 267 however, except along the middle parallel. For a map having a great extension in the longitude and but little in the latitude^ the conic projection is fairly accurate. Fig. 100 shows a com- pleted conic projection covering the area of the United States. 110 105 100 95 90 85 FIG. 100. Simple Conic Projection. 80 ; 75 ' 177. Bonne's Projection. This projection is a modification of the simple conic and meets the objection that the scale of the latter becomes inaccurate as the distance from the middle parallel increases. The parallels of latitude are concentric circles as before, but each parallel is sub-divided into spaces which are proportional to the corre- sponding spaces on that parallel on the spheroid. The central meridian and all parallels are therefore correctly sub-divided. The meridians are obtained by joining the points of sub-division on the parallels. The meridians in this projection are all curved, except the central one, and they intersect the parallels 268 MAP PROJECTIONS nearly, but not quite, at right angles (Fig. 101). The distortion in this projection is very small, and for small areas it is practi- cally a perfect projection. It has been much used in Europe. 130 125 115 110 105 100 95 90 85 80 75 70 66 60 120 105 100 95 90 85 FIG. 101. Bonne's Projection. 80 70 178. The Polyconic Projection. The idea of using several cones, or the polyconic projection, is due to Mr. F. R. Hassler, the first superintendent of the Coast Survey. Each parallel of latitude shown on the map is de- veloped on a cone tangent along that parallel. The radius (TA) for any parallel (latitude 0) is Ncot^r, and the angle between two elements of the cone when developed is approxi- mately 6 = (d\) sin $, as will be evident from Fig. 102. In constructing the map the degrees of latitude are laid off along the central meridian, the spacing corresponding to the distances on the spheroid. The points where the meridians intersect the parallels are plotted from their rectangular co- THE POLYCONIC PROJECTION 269 ordinates, the coordinate axes being in each case the central meridian and a line at right angles to it drawn through the latitude in question. The coordinates themselves are found as follows: In Fig. 103, let A be the intersection of some meridian and parallel which are to be drawn on the map. Then the FIG. FIG. 103. radius TA = N cot may be computed from the known lati- tude of A, and the angle 6 may be computed from the known difference in longitude between O and A by the equation 6 = (d\) sin 0. Then for x and y we have and TA sin = N cot < sin (d\ sin y = TAversO = - :vers0 sin & ztan- 2 = x tan \ (d\ sin [132] Values of these numbers will be found in Tables XVI and XVII. 270 MAP PROJECTIONS 8 8 LAMBERT'S PROJECTION 271 It is evident that the parallels and meridians do not intersect at right angles except at the central meridian. The meridian and parallels are both curved, as in Bonne's projection, but since the lower parallels are flatter there is a separation of the parallels which becomes more marked toward the east and west margins of the map. For this reason this map becomes less and less accurate as the longitude is extended. In mapping areas which extend principally north and south, it is superior to other projections. It is in general use in the United States for Government maps. Fig. 104 shows a polyconic projection covering the area of the United States. There is one disadvantage in the Polyconic and the Bonne's projections, namely, that if two maps of adjoining areas are to be placed side by side they cannot be placed exactly in con- tact because the limiting (common) meridian curves in opposite directions on the two maps. In the simple conic and in the Lambert projection, to be described in the next article, the meridians are straight and this difficulty does not exist. 179. Lambert's Projection. The Lambert projection having two standard parallels was invented about the middle of the eighteenth century, but has recently been brought into prominence through its use in the French battle maps. The fundamental notion is that of a cone tangent along the middle parallel of the map, the radius of this parallel (on the map) being N cot <, and the angle between the central meridian and any other meridian being (d\) sin 0. This would give a map in which one parallel, and only one, is correctly divided. We may, however, modify the projection so as to have two standard (correct) parallels. This is done by reducing the scale (multiplying by a constant) and is practi- cally equivalent to employing a cone which cuts the spheroid in the two standard parallels. The other parallels are so spaced that the scale of the map is the same for all azimuths at any one place, that is, the scale along a meridian is the same as the scale in an east and west 272 MAP PROJECTIONS plane. A projection having this property is said to be "con- formal." It may be proved that this condition is true if the spacing between parallels is H , where /3 is the arc of the 6 p0 _80 9 J70 eO'SO'tp^gplO 10 20 30 40 70 60 60 40 30 20 10 10 20 FIG. 105. Lambert Projection. This projection may be extended indefinitely in an east and west direction without error. The error becomes greater and greater as the map is extended to the north and south. In this respect it is just the contrary of the Polyconic Projection. THE GNOMONIC PROJECTION 273 For a complete description of this projection, together with tables for projecting maps, see United States Coast Survey Special Publications 47 and 52. 1 80. The Gnomonic Projection. In the gnomonic, or central, projection the projecting point is at the center of the sphere and the plane of the map is tangent to the sphere at some selected point. Every plane through the center cuts the sphere in a great circle and cuts the map in a straight line; hence every great circle is represented by a straight line and every straight line on the map must represent a great circle. Fig. 106 shows the Atlantic Ocean projected on a plane tangent at = 30 N and X = 30 W. 70 60 60 40 30 20 10 10 FIG. 106. Gnomonic Projection or Great-circle Chart. The meridians and the equator are of course represented by straight lines. The parallels of latitude are conic sections, in this case hyperbolas. The parallels are best constructed by employing the equations of the curves and plotting points by means of coordinates. 274 MAP PROJECTIONS The gnomonic projection is used almost exclusively for deter- mining the positions of great circles for the purposes of naviga- tion. By joining any two places by a straight line the great- circle (or shortest) track is at once shown. The latitudes and longitudes of any number of points on this track may be read off the chart and, if desired, may be transferred to any other chart and the curve sketched in. The point where the great circle approaches most nearly to the pole is found at once by drawing from the pole a line perpendicular to the track. The foot of this perpendicular is the vertex, or point of highest latitude. 181. Cylindrical Projection. If a cylinder is circumscribed about a sphere so as to be tan- gent along the equator, and if points be projected onto the cylinder by straight lines from the center, the cylinder, when developed will give a map in which the meridians and parallels are all straight lines, the relative distances between points being approximately correct near the equator but distorted in high latitudes. The meridians will all be parallel to each other. The parallels of latitude will be parallel to each other and will be spaced wider and wider apart as the latitude increases. Evi- dently the scale of the map is different for different latitudes. It is also true that at any point the scale along a meridian is not the same as the scale along a parallel. Such a projection is of no practical value, but it aids in understanding the Mer- cator chart which is described in the next article. 182. Mercator's Projection. A modification of the above projection, known as Mercator's, consists in so spacing the parallels of latitude that the relation between increments of latitude and longitude on the chart is the same as the relation between increments of latitude and longi- tude at the corresponding point on the earth's surface, or ap- proximately, i' lat. on chart: i' long, on chart = i' lat. on spheroid: i' long, on spheroid. If this relation is preserved, it will be found that any line of constant bearing (loxodrome or rhumb line) will be represented by a straight line on the chart. MERCATOR'S PROJECTION 275 In Fig. 107 let AB on the earth's surface be represented by A'E' on the chart (actual size). In order that the two lines may have the same bearing it is necessary that or __ = _R m d dx CB R p d\ dy dx R p d\ R m d. (a) Pole CHART FIG. 107. In other words, since the longitude has been expanded (in the ratio -J by the method of constructing the chart, it is neces- sary to expand the latitudes in the same ratio in order to preserve the scale and give AB the same bearing. Now since dx is rep- resented as large as the corresponding arc on the equator, we have dx ad\ a_ R p d\~ R p d\~ R p Substituting in (a), we obtain 276 MAP PROJECTIONS or, since R p = N cos < dy = 1 ^ L -.ad Ncos cos (; e 2 sin 2 <) cos(i Multiplying e 2 by sin 2 + cos 2 0, the integral may be sep- arated into two, giving, after multiplying numerator and de- nominator by cos <, rcos _ C+ ecosd(t> cos 2 JQ i e 2 sin 2 i sin 2 i e sm where M 0.4342945, the modulus of the common logarithms. Employing the formulae, , i + sin x and ! ; = tan the equation may be expressed sin r i [133] in which y is in the same linear units as a. In order to express y in nautical miles or minutes of arc on the equator * it is necessary to multiply by - , giving, air * The Nautical Mile contains 6080.20 ft.; this is not identical with the number of feet in one minute of arc on the earth's equator. For a discussion of this matter, see Appendix 12, Coast Survey Report for 1881. MERCATOR'S PROJECTION - 277 , [134] or y = 7915.705 log tan U5H J - 22^.945 sin #-0.051 sin 3 <. [135] ico- 120 120 i 100 J 80" FIG. 108. Mercator Chart. Also x = 60 X X, [136] the unit being the nautical mile. Values of y, called meridional parts, will be found in works on navigation. This chart is much used by navigators because it possesses the property that the bearing of any point B from a point A as 278 MAP PROJECTIONS measured on the chart is the same as that bearing on which a vessel must sail continuously to go from A to B. The track cuts all meridians on the globe at the same angle, just as a straight line on the chart cuts all meridians at the same angle. This track is not the shortest one between A and B, but for ordinary distances the length differs but little from that of the great-circle track. In following a great-circle track the navi- gator transfers to the Mercator chart a few points on the great- circle obtained from his great-circle chart, by means of their latitudes and longitudes and then sails on the rhumb lines between consecutive plotted points. Fig. 108 shows a Mercator chart. 183. Rectangular Spherical Coordinates. A system of rectangular spherical coordinates, used in Europe, consists in referring all points to two great circles through some selected origin, one of them being the meridian, the other the prime vertical. Within small areas these coordinates are prac- tically the same as rectangular plane coordinates. When the area is so great that the effect of curvature becomes appre- ciable, small corrections are introduced, so that the form of the plane coordinates is retained without loss of accuracy. Such a system is very convenient when connecting detail surveys with the triangulation, particularly for local surveyors who may not be familiar with geodetic methods of calculating latitudes and longitudes. The method is not well adapted to mapping very large areas. (See Crandall's Geodesy, p. 187.) CHAPTER XII APPLICATION OF METHOD OF LEAST SQUARES TO THE ADJUSTMENT OF TRIANGULATION 184. Errors of Observation. Whenever an observer attempts to determine the values of any unknown quantities, he at once discovers a limit to the precision with which he can make a single measurement. In order to secure greater precision in his final result than can be obtained by a single measurement, he resorts to the expedient of making additional measurements, either under the same con- ditions or under different conditions. Under these circumstances it will be observed that the results are discordant and that the same numerical result almost never occurs twice.* The ques- tion at once arises, then, What are the best values of the un- known quantities which it is possible to obtain from these measurements ? The method of least squares has for its main objects (i) the determination of the best values which it is possible to obtain from a given set of measurements, and (2) the determination of the degree of dependence which can be placed upon these values, or, in other words, the relative worth of different deter- minations; (3) it also enables us to trace to their sources the various errors affecting the measurements and consequently to increase the accuracy of the result by a proper modification of the methods and instruments used. The method is founded * This is only true, however, when the observer is taking each reading with the utmost possible refinement. If, for example, angles are read only to the nearest degree, the result will always be the same no matter how many times the measure- ment may be repeated; but if read to seconds and fractions, they will in general all be different. 279 2 8o ADJUSTMENT OF TRIANGULATION upon the mathematical theory of probability, and upon the assumption that those values of the unknowns which are ren- dered most probable are the best that can be obtained from the measurements. 185. Probability. If an event can happen in a ways and fail in b ways, and all of these ways are equally likely to occur, the probability that the event will happen in any one trial is expressed by the fraction - , and the probability that it will fail is expressed by a + b a -\- b Since it must either happen or fail, the sum of the two prob- abilities represents a certainty. This sum is 7 H = i. a + b a + b Therefore the probability of the happening of an event is repre- sented by some number lying between o and i, the larger the fraction the greater the probability of its happening. For ex- ample, a die may fall so that any one of its six faces is uppermost, and all of these six possibilities are equally likely to occur; the probability of any one of its faces being up is . 186. Compound Events. If a certain event can happen in a ways and fail in b ways, and if a second, independent, event can happen in a' ways and fail in b' ways, and all are equally likely to occur, then the total number of ways in which the events can take place together is (a + b) (a f + b'). The number of ways in which both can hap- pen is aa', and the probability of its happening is : - ( , ,. (a + b) (a -\-o ) For example, the probability of double six being thrown with a pair of dice is J X J = ^V ^ t * s evident that the probability of the simultaneous occurrence of two events is the product of the probabilities of the occurrence of the component events. In a similar way it may be shown that the probability of the simultaneous occurrence of any number of independent events is the product of their separate probabilities; that is, if PI, PZ, P 3 . . . are the probabilities of the occurrence of any number COMPARISON OF ERRORS 281 of independent events, the probability of their simultaneous occurrence is P = p, x P 2 x P z . . . , [137] 187. Errors of Measurement Classes of Errors. Every measurement of a quantity is subject to error, of which the following kinds may be distinguished. 1. Constant Errors. 2. Systematic Errors. 3. Accidental Errors. 188. Constant Errors. A constant error has the same effect upon all observations in the same series of measurements. For instance, if a steel tape is o.oi ft. too long, this error affects every 100 ft. measurement in just the same way. 189. Systematic Errors. A systematic error is one of which the algebraic sign and the magnitude bear a fixed relation to some condition. For ex- ample, if the measurements with the tape are made at different temperatures, the error resulting from this variation of tem- perature is systematic and may be computed if the tempera- tures and the coefficient of expansion are known. 190. Accidental Errors. Accidental errors are not constant from observation to ob- servation; they are just as likely to be positive as negative; in general they follow the exponential law of error, as will be explained later (Art. 197). The error of placing a mark opposite to the end graduation of the tape is of this class. 191. Comparison of Errors. There is in reality no fixed boundary between the accidental and the systematic errors. Every accidental error has some cause, and if the cause were perfectly understood and the amount and sign could be determined, it would cease to be an accidental error, but would be classed as systematic. On the other hand, errors which are either constant or systematic may be brought 282 ADJUSTMENT OF TRIANGULATION into the accidental class, or at least made to partially obey the law of accidental error, by so varying the conditions, instru- ments, etc., that the sign of the error is frequently reversed. If a tape has o.oi ft. uncertainty in length, this produces a constant error, in the result of a measurement. If, however, we use several different tapes, each with an uncertainty of o.oi ft., this error may be positive or negative in any one case. In the long run these different errors tend to compensate each other like accidental errors. In the class of systematic errors would be placed such errors as those due to changes in temperature, light, and moisture, or change in the adjustments of instruments. These errors may be computed and allowed for as soon as we know the law governing their action, or they may be partially eliminated by varying conditions under which the measurements are made. Under the constant class comes the observer's error, which tends to become constant with increased experience in observing. This error may be allowed for as soon as its magnitude and sign have been determined, or it may be eliminated by the method of observation. Certain errors in the instrument may have a constant effect on the result; these may be dealt with in the same manner as the personal error. It should be noticed that after the constant error or the systematic error has been elimi- nated, there still remains a small error due to the fact that the magnitude of the constant error itself was not perfectly deter- mined or that its elimination was imperfect. This remaining error must be regarded as an error of the accidental class, since its magnitude is unknown and it is just as likely to be positive as negative. Under accidental errors are included all those which are sup- posed to be small and just as likely to be positive as negative. They are due to numerous unknown causes, each error being in reality the algebraic sum of many smaller errors. Under this class may be noted errors in pointing with a telescope, errors in reading scales and estimating fractions of scale divisions, and ADJUSTMENTS OF OBSERVATIONS 283 undetected variations in all of the conditions governing syste- matic errors. 192. Mistakes. These are not errors, but they must be considered in connec- tion with the discussion of accuracy of observations. They in- clude such cases as reading one figure for another, as a 6 for a o, or reading a scale in the wrong direction, as reading 46 for 34. 193. Adjustment of Observations. When the number of measurements is just sufficient to de- termine the quantities desired, then there is but one possible solution, and the results must be accepted as the true values. When additional measurements are made for the purpose of increasing the accuracy of the results, this gives rise to discrep- ancies among the different measurements of the same quantities, since each is subject to errors. The method of least squares enables us to compute those values which are rendered most probable by the existence of the observations and in view of the discrepancies noted; it cannot, however, tell us anything about the existence of constant errors, unless new observations made under different conditions reveal new discrepancies. For example, if a pendulum is swung and certain small variations in the last decimal place of the period are noticed, these may be regarded as due to small errors in the running of the chronom- eter and to accidental errors of observing; but if the pendulum case be mounted on a support whose flexibility is very much greater than that of the first, and larger variations are now observed, it becomes apparent that an error of the systematic class is affecting all our observations, though it does not appear at all in the first observations, because all the measurements were affected alike. An investigation of the law governing this error, and the determination of its magnitude and sign, enable us to correct the result for such part of the error as we are able to determine. There remains in the result, however, an accidental error, namely, the error in the measurement of the flexure correction. 284 ADJUSTMENT OF TRIANGULATION 194. Arithmetical Mean. The formulae employed in adjusting observations are usually made to depend upon the axiom that if a number of observations be made directly upon the same quantity, all made under the same conditions and with the same care, the most probable value of the quantity sought is the arithmetical mean of all the separate results; that is, if the results of the observations are MI, M z , Mz, . . . M n , the most probable value of the quan- tity, MQ, is given by [I38] It is to be carefully noted that this is not the true value, M, but simply the most probable value under the circumstances; if additional measurements be made, MO changes correspond- ingly in value, because we know more about its real value than we did at first. 195. Errors and Residuals. It now becomes necessary to distinguish between errors and residuals. The error is the difference between any measured value and the true value. Its magnitude can never be known, because the true value can never be known. The residual is the difference between a measured value and the most probable value. This is a quantity which may be computed for any set of observations. In a set of very accurate observations which are free from constant and systematic errors the residual is a close approximation to the true error. It may be shown that for the case of direct observations the algebraic sum of the residuals is zero; that is, if we compute Vi = Mi M , ih = M 2 M , etc., then ^v = o, where Vi, % . . . are the residuals. 196. Weights. In case the measurements are of different degrees of relia- bility, they are given different weights. The weight of an observation may be regarded as the number of times the ob- servation is repeated and the same numerical result obtained. DISTRIBUTION OF ACCIDENTAL ERRORS 285 It expresses the relative worth of different measured values. Weights are purely relative and may be computed on any base desired. To say that two measurements have weights 2 and i respectively, is the same as saying that they have weights \ and \. ' From the above definition it is apparent that the weighted mean is expressed by that is, the weighted mean is found by multiplying each ob- servation by its weight, adding the results, and dividing by the sum of the weights. Multiplying an observation (Mi) by its weight (pi) is the same as taking pi observations each equal in value to MI. 197. Distribution of Accidental Errors. An inspection of the results of a large number of measure- ments will show that (1) H- and errors are equally numerous. (2) Small errors are much more numerous than large ones. (3) Very large errors seldom occur. The curve which expresses the law of variation of such errors will be of the form shown in Fig. 109. In accordance with (i) 286 ADJUSTMENT OF TRIANGULATION the curve is symmetrical; in accordance with (2) its maximum is at the axis of F; from (3) it is evident that the curve cuts the axis of X at some distance from 0. The manner in which observations are affected by accidental errors is shown by the "shot apparatus " shown in Fig. no. A FIG. no. "Shot Apparatus." large number of small shot, representing observations, are allowed to drop through an opening in the middle of the case. If there were no obstructions the shot would fall directly into the central (vertical) compartment. Between the opening and the vertical compartments a number of pegs are interposed, each representing a source of error or deflection of the shot from its natural course. v The shot are therefore diverted some- DISTRIBUTION OF ACCIDENTAL ERRORS 287 what from a straight course and arrange themselves in the different columns in the manner shown. The curve joining the tops of the columns is seen to resemble closely the "curve of error." In order to obtain a formula expressing the law of error we suppose the curve asymptotic to the axis of X, and write the equation of the curve in the general form y=f(x), [140] where x represents the magnitude of an error and y the fre- quency with which this error occurs on a large number of measure- ments; / represents some unknown function of x. It is neces- sary to assume that the number of observations is very large; otherwise the supposed balancing of + and errors will be imperfect. The true error x can never be known, but the distribution of the residuals about the most probable value will evidently follow the same general law, so we may write also y=f(v) [141] as the law to which the residuals must conform. This equation also expresses the probability of the occurrence of a residual v. If we let the total area between the curve and the axis of X be represented by unity, then the probability that a certain residual will fall between the limits v and v + dv will be represented by the area included between the curve, the X axis, and the two ordinates at v and v + dv, since in the long run the number in a given column will be proportional to the probability expressed by the ordinate at that point, that is, ydv = f (v) dv. [142] If we suppose n observations of equal weight, giving the results MI, M 2 , . . . M n , to be made on any functions of the unknowns Zi, zi, . . . , z, giving the residuals vi, %, . . . , v n , then the probability of the occurrence of these residuals is / (v\) dv, dv . . . f (v n ) dv. The probability of the simultaneous 288 ADJUSTMENT OF TRIANGULATION occurrence of these residuals is the product of the separate probabilities, that is, P = / Oi) do X / (%) dv X . . . / (v n ) ((ii)\ n = a, maximum. 1^1 V / L J J It is evident that P is a maximum when ' v\ + % 2 + ^n 2 = a minimum, [152] that is, when the sum of the squares of the residuals has its least value. Equa. [147] express the conditions necessary to make P a maximum or to make the sum of the squares of the residuals a minimum. Since the function F means multiplication by a con- stant, Equa. [147] become ? . o. dv n _ [lS3l fan Cj%tft w RELATION BETWEEN H AND P 291 These equations are equal in number to the number, q, of un- known quantities, and their simultaneous solution gives the most probable values of the unknown quantities. They are usually called Normal Equations. 199. Weighted Observations. If the observations are of different weights, each observation equation should be used (Art. 196) the number of times denoted by its weight. Hence, in forming the normal equations we should multiply each observation equation by the coefficient of the unknown and by the weight of the equation. The normal equations in this case are as follows: = o. = o. = o. [154] This same result will be obtained if we first multiply each ob- servation equation by the square root of its weight. This shows that multiplying a set of equations by the square roots of their weights reduces them all to observations of weight unity (equal weights). 200. Relation between h and p. If the n observations have weights pi, p z , . . . , and the con- stant h is hi, fa, . . . for these observations, then p = feer-W . fee-*"* . . . = fefe . . . n and /*iW + feW + is to be a minimum. [156] The conditions for this minimum are dl' o. =0. [157] 2Q2 ADJUSTMENT OF TRIANGULATION Equas. [154] and [157] express the same conditions. Hence pi ' pz ' h\ : h? : . . . , showing that the weight of an observation varies as the square of the constant h for the observation. Consequently the more accurate the observation the greater the value of h. Example. As an illustration of the manner of applying these equations to the computation of the most probable values of the unknowns, suppose that at a tri- angulation station (Fig. in), the angles have been measured as shown. Denoting the most probable values of these angles by 0i, 2 , and z 3 , the measurements are given by the following equations: 02 = 3 = Zl + 22 = Z 2 + Z 3 = Z 2 + Z 3 = 31 10 i.o, 40 50 10 .0, 42 10 IQ .7, 72 OO 26 .O, 4 10 46 .0, 83 00 30 .2. Denoting by vi, vz, etc., the residuals of the different measurements, these may be written 2 40 z 3 - 42 zi + z 2 72 + z 2 + 3 114 Z 2 + Z 3 - 8 3 10 17 .o = DI, 5O IO .0 = 1)2, 10 19 .7 = V S , OO 26 .O = 4, 10 46 .0 = 1)5, 00 30 .2 = Z> 6 , n. which are called observation equations. If we apply equations (153), differentiating each v with respect to the three unknown quantities in succession, we obtain the normal equations. 3 Zl + 2 2 + Z 3 217 2l' 29".0 = O, 2 0i + 4 2 + 2 3 310 01 52 .2 = O, Zi + 2 2 + 3 3 239 21 35 .9 = o. Solving these simultaneously, we obtain 01 = 31 10' i6". 4 S, 02 = 40 50 09 -875, z 3 = 42 10 19 .90. These are the most probable values of the angles. SOLUTION BY MEANS OF CORRECTIONS 293 201. Formation of the Normal Equations. It should be observed that since the observation equations are linear in this case, the differential coefficients are equal to the numerical coefficients. Hence, to form the normal equations we may proceed as follows : For each unknown, form a normal equation by multiplying each observation equation by the numerical coefficient of the unknown in that equation, adding these results and placing the sum equal to zero. This rule is simply a statement in words of what is expressed in Formula [153] as applied to linear equations. If the observations are of different weights, the only change in the above rule is that each observation equation is multiplied by its weight as well as by the coefficient of the unknown. In regard to the observation equations it should be understood that they are not like ordinary equations. They are often written, however, with zero hi place of the v in the right hand member. Observation equations cannot be multiplied by any number or combined with each other (except when forming nor- mal equations) ; for if this is done, the weight of the observation is thereby changed. 202. Solution by Means of Corrections. If the independent terms * in the observation equations are large, it will often save labor in the calculations if we place the unknown quantity Zi equal to an approximate value M\ plus a correction Zi, Z 2 = M 2 -f 22, etc. Substituting these values in the original observation equations, we obtain a new set of equations in terms of the corrections and in which the independent terms will be small. By forming normal equations and solving as be- fore, we find the most probable values of the corrections. Adding these corrections to the approximate values, we find the most probable values of the unknown quantities themselves. * The independent term in any equation is that term which does not contain any of the unknowns. 294 ADJUSTMENT OF TRIANGULATION Example. In the example just solved, suppose we assume for the approximate values the results of the direct measurements, and let zi, z 2 , etc., represent the most probable corrections. Then the observation equations become zi = o, z 2 = o, z 3 = o, Zl + Z 2 + l".O = O, zi + z 2 + z 3 + o .7 = 0, z 2 + z 3 o .5 = 0. Forming the normal equations as before, we have 3 zi + 2 z 2 + z 3 + i"-7 = o, 2 Zi + 4 Z 2 + 2 Z 3 + I .2=0, Zl + 2Z 2 + 3Z 3 + O .2 = O. The solution of these equations gives zi = -o". 55 , z 2 = o .125, Z 3 = +0 .20, which, added to the values observed directly, give the same results as before. 203. Conditioned Observations. If the quantities sought are not independent of each other, but are subject to certain conditions, the solution must be modified accordingly. Each observation gives rise to an observation equation, and each condition may be expressed by a condition equation. The solution may be effected by eliminating, between the two sets of equations, as many unknowns as there are equa- tions of condition. From the remaining equations we may form the normal equations and solve for the most probable values of the unknowns. Substituting these values back in the original condition equations, we obtain the remaining unknowns. Example. The three angles of a triangle are A = 61 07' 52".oo, B = 76 50'- 54 ; '.oo, and C = 42 01' i2".i5. The spherical excess is 02". n. The weights assigned to the measured angles are 3, 2, and 2, respectively. These angles are subject to the fixed relation A + B + C = 180 oo' 02".! i. Letting ti, v 2 , t> 3 be the most probable corrections to the observed values, the observation equations are vi = vi, wt. 3 * = *, " 2 V 3 = V 3 , " 2 and the condition equation is vi + n + vs- 3"-96 = o. (<*) = +o".9o, ADJUSTMENT OF TRIANGULATION 295 Eliminating v 3 , there remain *>i = PI, wt. 3 V2 = V, "2 *=-*-* + 3" 96 " 2 Forming the normal equations and solving, * = +o". 99 , V2 = +1 .485. Substituting these values in equation (d), These corrections, added to the measured angles, give the adjusted angles, as follows: A =61 07' 52".oo, B = 76 50 55 -48, C = 42 01 13 .64. Notice that the discrepancy is distributed inversely as the weights. This will always be the case when each unknown is directly observed, and there is but one equation of condition; that is, the correction to the first is 1,1,1 X +3 and the correction to the second is The correction to the third is the same as the correction to the second. 204. Adjustment of Triangulation. The adjustment of the angles of a triangulation net naturally divides itself into two parts: (i) the adjustment for the dis- crepancies arising at each station, and (2) the adjustment of the figure as a whole. According to theory these should all be adjusted simultaneously in order to obtain the most probable values of the angles. The usual practice, however, is to deal with the two separately. The local, or station, adjustment is made first if the method of observing is such that a local adjust- ment is required. If the observations are made in accordance with the program given in Art. 44 (sec. 2, Coast Survey in- structions), no station adjustment is necessary. If the angles are measured by the repetition method and the horizon is closed, the error is distributed in inverse proportion to the weights (see Art. 203). If there are conditions existing among the angles, 296 ADJUSTMENT OF TRIANGULATION due to measuring sums of the different single angles, the adjust- ment may be effected by expressing these as condition equations and then forming normal equations and solving, as in the ex- ample, p. 294. This method of making the local adjustment first is justified, not only on the ground of saving labor, but also because of the well-known fact that the most serious errors are those due to eccentricity of signal and instrument, phase of signal, refraction, etc., which do not appear to any large extent in the local ad- justment but which do appear in the figure adjustment. If we compute the precision of angles from the discrepancies noted at each station, and then estimate from these values the error of closure to be expected in the triangle, we find that these are smaller than the errors of closure actually occurring, showing the presence of constant errors, which do not appear in the local adjustment. 205. Conditions in a Triangulation. The geometric conditions connecting the angles in a net are of two classes: (i) those which express the relation among the angles of a triangle or other figure, and (2) those which express the relation existing among the sides of the figure. If we plot, for example, a quadrilateral figure, starting from one side as fixed, we shall find that if the sum of the angles in three of the triangles equals their theoretical sums, all sums in the other triangles will also (necessarily) equal their theoretical amounts, namely, 180 + e" . This shows that of all the possible angle equations which might be written for this figure only three are really independent. In order to determine the number of angle equations in any net, let s be the total number of stations, s u the number of stations not occupied, / the total number of lines in the figure, and /i the number of lines sighted over in one direction only; then the number of angle equations in the figure is + s + i. [159] ADJUSTMENT OF A QUADRILATERAL 297 In a triangle it is necessary that all stations should be occu- pied and that all lines should be sighted over in both directions, in order to have one angle equation, that is, If a new station is added, it must be occupied and the two lines sighted over in both directions, in order to yield a new angle equation. If this is done, the quantity / s is increased by 2 1 = 1. If a line is drawn between two stations already located, / is increased by i and there is a new angle equation corresponding. For each new line sighted in one direction only, / is increased by i and h is increased by i, so that the total is unchanged. The number of side equations in a net may be estimated as fellows: Starting with one line as fixed, it is evidently neces- sary to have two more sides in order to fix a third point. Hence, in order to plot a figure, we must have at least 2 (s 2) lines hi addition to the base, that is, 2 5 3 lines in all. Any addi- tional lines used must conform to those already used, in order to give a perfect figure; hence the number of conditions giving rise to side equations will equal the number of superfluous lines, that is, / 2 s + 3, where / is the total number of lines and s is the number of sta'tions. It should be observed that while the side equation is primarily a relation among the sides, it is also a relation among the sines of the angles, and this fact en- ables us to adjust the figure by altering the angles. 206. Adjustment of a Quadrilateral. For any quadrilateral figure in which all of the (eight) angles have been measured there may be found three equations which express the condition that the triangles must all "close." There are more than three equations which may be formed; but if any three of these equations are satisfied, the others necessarily follow and hence are not independent. There will also be one side equation expressing the condition that the length of a side (AB\ when computed from the opposite side (CD), is exactly 298 ADJUSTMENT OF TRIANGULATION the same, no matter which pair of triangles is employed in the computation. In selecting the three-angle equations we may take any three triangles and write an equation for each expressing the con- dition that the sum of the three angles equals 180 + e" . It is advantageous in this case to avoid triangles having small angles. In selecting the side equation it is well, however, to select one involving small angles, so as to give large coefficients of the corrections. If the angle equations were, also chosen so as to involve the small angles, the solution would be likely to prove unstable, on account of the equality of some of the coefficients. B FIG. 112. FIG. 113. A convenient method of writing a side equation is to select some point, called the pole, and write the three directions from it to the other stations in the order of azimuths. For example, taking the pole at A, Fig. 112, write first AB-AD-AC. Then from this write the ratios AB AD AC AD'AC'AB' the method of forming which is evident. If we now replace ADJUSTMENT OF A QUADRILATERAL 299 each line by the sine of the angle opposite to it in the triangle which is indicated by the fraction, and place the whole equal to unity, we have smADB sinACD sin ABC _ . , sinABD smADC smACB It may be shown, by solving the different triangles and elimi- nating the sides, that this equation expresses the condition that the length of AB as computed from CD is the same no matter which route is followed in the computation. Problem. Prove by a direct solution of the triangles in Fig. 112 that Equation [160] is true. Designating the angles by means of the numbers shown in Fig. 113, the equation becomes sin 2 sin (4 + 5) sin 8 7 - TT ; - ; - = I. sin (i -f 8) sin 3 sin 5 Before this equation can be used, however, it is practically necessary to reduce it to linear form, since an application of Equa. [153] to any but linear equations would be complicated. Suppose our equation to be put in the general form sin (Mi + vi) sin (M 3 + q,) f , ' in which the angle is written as an approximate value M plus a small correction v. Taking logs of both members and then applying Taylor's theorem, we have, neglecting squares and higher powers, log sin Mi + (log sinMi) i* + oJVL\ - (logsinM 2 + (logsinM 2 ) fy + j = o. [163] The quantity - (log sin MI) is the variation per i" in a 300 ADJUSTMENT OF TRIANGULATION table of log sines, the correction v being in seconds. Hence, placing 61 = - (log sin Mi), etc., we have - to + to - to + + log sin Mi - log sin M 2 + = o. [164] The algebraic sum of the log sines represents the amount by which they fail to satisfy the condition equation. Placing this sum equal to /, the side equation given above becomes to + 5 4+ 5^4+5 + to (8l+8&l+8 + to + to) 1 = 0. [165] Example. Let us suppose that the measured angles are (Fig. 113), 1. 6io7'52".oo 2. 38 28 34 .90 3. 38 22 19 .10 4. 42 or 12 .15 5. 29 14 32 .85 6. 70 21 59 .20 7. 49 26 21 .85 8. 30 57 7 -io These angles are supposed to have been adjusted for local conditions. To form the angle equations, take the triangles ABD, ADC, and ABC for which the values of the spherical excess are i".36, i".77 and i".o2, respectively. The computation is shown in tabular form as follows: 1 + 8 9204'5 9 ".io 2 38 28 34 .90 7 49 26 21 .85 179 59 55 -85 180 oo 01 .36 3 38 22' r 9 ".io 4 + 5 71 15 45 .00 6 70 21 59 .20 1 80 oo 03 .30 180 oo 01 .77 5 29 14' 32 ".8 5 6 70 21 59 .20 7 49 26 21 .85 8 30 57 07 .10 180 oo 01 .00 180 oo 01 .02 +0".02 ADJUSTMENT OF A QUADRILATERAL This gives for the three angle equations (i + 8) + 2 + 7 = 180 oo' oi". 3 6, 3 + (4 + S) + 6 = 180 oo 01 .77, 5 + 6 + 7 + 8 = 180 oo 01 .02, or, written as corrections, fi-B + % + V7 - 5.51 = o, t>3 + f 4+5 + V6 +1-53 =0, V 6 + V 6 + V 7 + 8 O.O2 = O. To form the side equation, take the pole at A. Then we have giving AB AD AC_ AD' AC' AB' sin (4 + 5) sin 8 sin 5 sin (i + 8) sin 3 or log sin 2+log sin (4+5)+log sin 8 log sin (1+8) log sin 3 log sin 5=0. The computation of the constant term of this equation is given in the following table. The log sines of those angles appearing in the numerator, together with their diff. for i" (in units of the 6th place of decimals) are placed in the left-hand column, and those in the denominator are placed in the right-hand column. The constant / is the difference in the sums of the log sines. Angle. log sine (+). Diff. i". Angle. log sine ( ). Diff. i". 2 4+5 8 9.7939242 9.9763501 9.7112329 + 2.65 +0.72 +3-51 1+8 3 5 9.9997129 9.7929268 9.6888702 -0.08 + 2.66 +3.76 9.4815072 9.4815099 72 -27 Therefore / = -2.7. The side equation becomes 2.65 % + 0.72 t; 4 +s + 3.51 t> 8 + 0.08 fli+8 2.66 v 3 3.76 1> 6 - 2.7 = o. Since the observations are direct, all of the observation equations take the form The eight observation equations and the four condition equations are now written, and we are ready to adjust the quadrilateral. 302 ADJUSTMENT OF TRIANGULATION 207. Solution by Direct Elimination. If we select for the four independent unknowns ?; 2 , %, %, and %, and express the four conditions in terms of these, we have 3- OI 5^ - 5.282% -4.751% + 6.609% + 0.2351, +4.282% + 5.751% - 5.609% +3.725, % = +4.015% - 5.282% - 5.751% + 5.609% - 5.255, v 7 = -4-015 % + 5-282 ** + 4-75 1 v$ - 6.609 ^8 + 5.275. % = . V*. From these we form the following normal equations (Art. 198, Equa. [153]): 111. b ft Const. +58.450 -75.534 -79-581 +91.501 - 75-534 + IO2 .036 + 105.193 -123.474 - 79.581 + 105.193 + 112.292 -127.388 + 9I-50I 123.474 -127.388 + 151.282 -56.527 + 70.329 + 75.589 -83-678 The simultaneous solution of these equations will give the most probable values of the corrections. 208. Gauss's Method of Substitution, In solving a large number of equations simultaneously it is convenient to use some definite system of eliminating the un- knowns, in order to avoid labor and the danger of mistakes. Let us suppose that the observation equations are of the form d\x + biy + ciz + /i = vi, and that the normal equations are represented by [aa] x + [ab] y + [ac] z + [al] = o, o, o, >66] GAUSS'S METHOD OF SUBSTITUTION 303 . in which the brackets indicate the sum 01 all the terms found by multiplying the numerical coefficients according to the rule on p. 293. If the first normal equation be divided by [aa] and solved for x, the result is [aa] [aa] [aa] Substituting this in the second equation, we have This is usually abbreviated [bb - 1] y + [be i] z + [bl i] = o. [168] Substituting this in the third equation, we have [be i] y + [cc i] z -f [cl i] = o. [169] These two equations, [168] and [169], are called the "first reduced normal equations." Solving [168] for y, [be i] [bl i] . [M-i] [bb-i] whence [cc 2] z + [cl 2] = o, [170] i i r 1 f 1 1 00 * I 1 r-t 1 in which [cc 2] = [cc i] 7- - 7 [fc i] i] and The solution of [170] gives the value of z. By substituting this in [168] and [169] the value of y may be found. Finally, from [166] the value of x may be found. An inspection of [166] will show that all coefficients below and to the left of a diagonal drawn from the x term of the first equation to the z term of the third equation are duplicates of the others. These may be omitted in writing the equations. 304 ADJUSTMENT OF TRIANGULATION Solving the equations in Art. 207, p. 302, by the method of substitution just described, we obtain the following results: v 2 = +1.80, Z> 3 = -0.19, v 5 = -0.07, % = -0.75. Substituting these values in the condition equations, p. 301, we find for the remaining unknowns, *>i +8 = +2.05, v 7 = +1.67, Z>4+5 = -0.51, 1>6 = 0.83. The final angles are as follows: 1. 6io7'54."8o 2. 38 28 36. 70 3- 38 22 l8. QI 4. 42 01 ii. 71 5. 2Q 14 32. 78 6. 70 21 58. 37 7. 49 26 23. 52 8. 30 57 06. 35 The above is an example of a rather unstable solution of normal equations. It requires a relatively large number of significant figures to give the corrections to two places of decimals. 209. Checks on the Solution. In practice it would not be advisable to proceed in the solution of a large number of equations without some safeguard against mistakes of computation. A valuable check consists in adding to the normal equations an extra term which is merely the sum of all the coefficients of i, %, etc., and treating this term like any other term of the equation. This is illustrated later in the example on p. 313. 210. Method of Correlatives. When there are many condition equations, the method of sub- stitution is likely to prove laborious. If, as is usually the case in triangulation, the observations are direct and equal in number to the number of unknowns, the "Method of Correlatives " will METHOD OF CORRELATIVES 305 be found preferable. By this method we eliminate one unknown for each condition equation, employing for this purpose the method of undetermined multipliers. Suppose that we have made m direct observations, MI, M 2 , . . . , M m , of m different quantities, of which the most probable values are 2i = Mi + i, 22 = Af 2 +%,..-, 2m = M m + V m . Let these m unknowns be connected by the following n con- ditions equations: a&i + a^Vz . . . a m v m + /i = o, blVi + 62% . &mflm + k = O, the a's being the coefficients in the first equation, the b's those of the second, etc. The quantities /i, k, etc., represent the amounts by which the observations fail to satisfy the condition equations. If the original condition equations are not linear in form, they must be made so by a method similar to that given on p. 299. Since the most probable values of the v's are to be found, we must have fli 2 + *>2 2 + = a minimum, [172] or PI dvi + v 2 dih + = o [172] for all possible values of dvi, d^ etc. Hence it must hold true for the equations 0i dvi + 02 dvz + = o, [173] &i dvi + fe dvz + = o. obtained by differentiating [171]. The number of these equa- tions is n. The number of terms in [172] is m, m being greater than n. Let the first equation in [173] be multiplied by ki, the second by k 2 , etc., and Equa. [172] by i. The products are then added, giving + bih + -vi) dvi =o. [174] 306 ADJUSTMENT OF TRIANGULATION The ^'s are to be so determined that this equation will hold true. This equation will be satisfied if the coefficient of each differential in it is placed equal to zero, that is, if - i = i Substituting these values of vi, i>2, etc., from Equa. [175] in Equa. [171], we obtain h [aa] + k 2 [ab] +& [al] + /i = o, fe W + k 2 [bb] + - k n [bl] + h The solution of these equations gives the values of ki, fe, k 3 , etc., which are the correlatives of the condition equations. By substituting these values in Equa. [176] the v's are found. Since the form of Equa. [176] is the same as that of normal equations, it is evident that they may be solved by the method of substitu- tion. In case the observations are of different weight, the minimum equation would be Mi 2 + />2% 2 + pmVm 2 = a minimum, [177] and the other equations would be modified accordingly. Example. As an illustration of the method of correlatives we will use the same quadrilateral that was adjusted by the method of direct elimination. The obser- vation equations are eight in number and all of the form The four condition equations are ( fli+s + % tions -j v s + 04+5 I V 6 + Z> 6 + - 5.51 = o, Angle equations \ v 3 + v*+ b + v 6 + 1.53 = o, Z>6 + V7 -\~ V 8 O.O2 = O. Side equation, 2.65 V2 + 0.72 04-H, + 3.51 V 8 + 0.08 fi+8 2.66 vs 3-7 6 ^6 2.70 = o. METHOD OF CORRELATIVES The coefficients of the corrections are then tabulated as follows: 307 V. a. b. c. d. Vl+3 + 1 +0.08 V 2 + 1 + 2.65 9f + 1 + 1 t>3 + 1 -2.66 V<+5 + l +0.72 fe + 1 + 1 fs + 1 -3.76 v s + 1 +3-51 Substituting these in the angle equations, (i + 8) = +2.058 - 0.094 X 0.08 = +2.05 2 = +2.058 0.094 X 2.65 = +1.81 7 = +2.058 0.406 = +1.65 +5.51 Check (4 + 5) = 6 -0.435 + 2.66 X 0.094 = -0.435 + -7 2 X 0.094 = -0.435 0.406 = 0.19 0.50 0.84 -1.53 Check 5 = 0.4064 3.76 X 0.094 = 0.05 6 = 0.435 ~~ 0.406 = 0.84 7 = +2.058 4.06 = +1.65 8 = 0.406 + 3.51 X 0.094 = 0.74 +0.02 Check The coefficients of the correlative Equa. [176] are tabulated as follows : aa. ab. ac. ad. bb. be. bd. cc. cd. dd. fl+S + i +0.08 0.0064 V 2 + i + 2.6 S 7.0225 V 7 + i + 1 +1 V 3 + i -2.66 7.0756 V4+5 + i +0.72 0.5184 *>6 + i +1 +1 | +1 -3.76 14.1376 t>8 +1 +3-51 12.3201 Sum +3 + 1 + 2-73 +3 +1 1.94 +4 -0.25 +41.0806 308 ADJUSTMENT OF TRIANGULATION The correlative equations are therefore I. 3 h + o + 3 + 2.73 ki 5.51 = o II. o + 3 & + k 3 1.94 4 + 1-53 = III. ki + kz + 4 ka 0.25 ki 0.02 = o IV. +2.73 ki 1.94 h 0.25 h + 41-0806 h 2.7 =o The solution of these equations gives for the correlatives, ki = +2.058, kz = -0.435, 3 = 0.406, 4 = 0.094. Applying these equations to the measured angles, we obtain the final angles. Measured Angles. Correction. Seconds Corrected. (i + 8) 92 04' 59"-io + 2 ".o5 2 38 28 34 .90 + i .81 7 49 26 21 .85 + i .65 oo .00 Check 3 38 22' i 9 ".ia - o".i 9 i8". 9 i (4 + 5) 71 15 45 .00-0 .50 44 .50 6 . 70 21 59 .20 o .84 58 .36 oi".77 e" = oi .77 oo .00 Check 5 29 14' 32".8s - o".o S 32 ".8o 6 70 21 59 .20 o .84 58 .36 7 49 26 21 .85 + i .65 23 .50 8 3 57 07 .10 o .74 06 .36 or .02 e" = oi .02 oo .00 Check The check of the side equations is as follows: 2.65 X 1.81 = 4.80 0.08 X 2.05 = 0.16 0.72 X 0.50 = 0.36 +2.66 X 0.19 = 0.49 3.51 X 0.74 = 2.60 +3-76 X 0.05 = Q.20 +1.84 -0.85 -0.85 +2.69 (Should equal 2.70.) METHOD OF DIRECTIONS 309 If the sums of the log sines are again computed (see p. 301), using the corrected seconds, they will be found to equal 9.481 5090 for both columns. 211. Method of Directions. The method of correcting the directions instead of the angles is particularly applicable when the measurements have been taken by the method of directions, Art. 43. In the United States Coast Survey office it is the usual practice to employ this method of adjusting, whether the observa- tions were made by the direction method or by the method of repe- tition. In the quadrilateral adjusted in Arts. 208-210, let us denote the directions by the numbers i to 12 (Fig. 114) and the corrections to those directions by the same num- bers, (i), (2), etc., enclosed in parentheses. Each angle is ex- pressed as the difference of two directions; that is, the angle 4 + 5 means the angle between the directions marked 4 and 5. The four condition equations are the same as before except as to the change in notation. Angle Equations 10 FIG. 114. 1-53 = o. - (4)+ (6) -0.02=0. Side equation, -5.31 (n) + 2.65 (12) + 3.04 (7) + 0.72 (9) - 3.51 (2) + 3-59 (3) - 0-08 (i) + 2.66 (10) - 3.76 (8) - 2.7 = o. If CD were a fixed line obtained by a previous adjustment, the corrections (9) and (10) would be omitted. The angle equations could be simplified in this case by selecting two equations which involve angles depending upon those two directions. 3 io ADJUSTMENT OF TRIANGULATION The first table for the coefficients of the corrections is given below. Direction. a. b. C. d. I I +0.08 2 . J -3-51 3 + 1 +1 +3-59 4 I . I + 1 I +1 7 I """" I -r3-4 8 +1 -3.76 9 + 1 +0.72 10 I + 2.66 ii I + 1 -5-31 12 +I + 2.65 The remainder of the work, that is, the calculation of co- efficients ^aa, ^ab, etc., and the solution of the numerical equa- tions, is carried out as in the preceding example (Art. 210). The solution of the normal equations gives the corrections to the directions. The correction to any angle is the difference of the corrections to the directions of its sides. 212. Adjusting New Triangulation to Points already Adjusted. In the quadrilateral shown in Fig. 115 the triangle BDE is sup- posed to have been previously adjusted. Point C is determined by the directions i, 2, and 3 in connection with the directions along the sides of the fixed triangle, and also by directions 4, 5, ADJUSTING NEW TRIANGULATION 3 11 and 6. The directions to be found are i, 2, 3, 4, 5, and 6. The directions as taken from the field-notes are as follows: Point sighted. Direction after local adjustment. Corrected seconds. AtC ' / // D 00 00.00 B 123 49 24.97 E 207 52 33-50 > " B At D / // A o oo oo.oo 00.67 C 296 57 55-83 E 311 12 14.48 12 .69 B 258 27 57.39 57-i8 AtE . / // // D o oo oo.oo 01.32 C 13 38 27.54 B 81 28 43-98 43-05 AtB F o oo oo.oo 01. 06 E 122 32 11.29 12.56 C 150 38 41.62 D 168 19 14.81 15-48 In taking directions from this table, the corrected seconds should be used whenever an adjustment has been made. The number of angle equations in the figure is _/J ~s + i, or 6 4 + 1 =3. The number of side equations is / 2 s + 3, or 6 8+3 = 1. Since, however, the exterior triangle is already adjusted, there will be but two angle equations needed in the adjustment. For these two angle equations take the triangles DCE and BEC\ then -(a) + (i) - (3) + (/) - (6) + (4) = o and - (i) + (b) - ( 5 ) + (6) - (c) + (2) = o. 312 ADJUSTMENT OF TRIANGULATION But since the exterior lines are not to be changed, (a), (/), (b), and (c) are all zero. The absolute terms in the angle equations are found as fol- lows: -(a) + (i) i 3 38'26".22 -(3) + (/) 14 14 16 .86 -(6) + (4) 152 07 26 .50 180 oo 09 .58 180 oo oo .02 -d) + (&) 6 7 59'i5".5i -(5) + (6) 84 03 08 .53 -(c) + (a) 28 06 29 .06 J 79 59 S3 -io 180 oo oo .08 +6". 9 8 For the side equation take the pole at C. sin (-(2) + (<*)) sin(-(i) + (6)) sin ( -.(3) + (/)) sin(-(*) + (3)) 'sin (-(c) + (2)) 'sin(-(a) + (i)) " Tabulating the log sines, log sin (+) dit. I" -(2) + (rf) I7 4 o'33".86 9-4823521 +66.I -(i) + (6) 67 50 15 .51 9.9666666 + 8.6 -(3) + (0 U 14 16 .86 9.3908478 +83.0 8.8398665 log sin ( ) -() + (3) 38 29' s8".6 S 9.7941460 +26.5 (c) + (2) 28 06 29 .06 9.6731464 +39.5 (a) + (i) 13 38 26 .22 9.3726010 +86.7 8.8398934 8665 . constant = 269 The side equation is therefore +6.61 X - (2) + 0.86 X - (i) + 8.30 X - (3) - 2.65 X (3) - 3.95 X (2) - 8.67 X (i) - 26.9 = o. Carrying out the same process as outlined in Art. 210, we have the following: ADJUSTING NEW TRIANGULATION TABLE OF COEFFICIENTS. 313 Direc- a. b. c. Sum. aa. ab. ac. as. bb. be. bs. cc. cs. tion. I +i i - 9-53 - 9 53 +i i - 9-53 - 9-53 +i + 9-53 +9-53 90.8209 90.8209 2 +i 10.56 - 9.56 +i 10.56 -9-56 111.5136 100.9536 3 i -10.95 -II-9S +i +10.95 +11.95 119.9025 130.8525 4 +i + i +i + i 5 i i +i +i 6 i +i +i -i +i Total.... +4 2 +1.42 +3-42 +4 -1.03 +0.97 322.2370 322.6270 From these sums we derive the correlative equations. CORRELATIVE EQUATIONS Number. *> * *, Const. Check. Sum. I 2 3 +4 2 +4 + 1-42 - 1-03 +322.24 + 9-56 - 6.98 26.9 + 12.98 6.01 +295.73 + 3-42 + 0.97 +322.63 It should be observed that the "constant" terms are taken directly from the condition equations. The "sum" term con- tains the sum of the coefficients of the &'s. The "check" term is the algebraic sum of the constant and sum terms. The solution is given in detail in the following table: The different operations are indicated in the left-hand column. The factors by which the equations are multiplied are in the right-hand column. 2 +4 -1.03 -6.98 6.01 Factor IX - i +0.71 +4-78 +6.49 +1 4 2 II +3 -0.32 2.20 +0.48 3 +322.24 26.9 + 295-73 IX ' 42 0.50 - 3-39 - 4 .6l -0-355 4 3 0.03 - 0.24 + 0.05 III +321.71 -30.53 + 291.17 ADJUSTMENT OF TRIANGULATION The preceding table is an abbreviated form of the method of substitution explained in Art. 208. The correlatives are found as follows: I. II.- III. Const. k, k z *i +9-56 +0.135 -1.487 2.20 0.03 -30.53 , _ 30.53 _ , ' 32L70 2 = 123 = +0.7433 t 8.208 2 .230 R\ 2.OS2 +8.208 Calculating the corrections for the correlatives, I. 2. 3- 4 5- 6. fel 2.052 + 2.052 2 .052 + 2.052 k, - o . 743 + 0-743 -0-743 +0-743 k 3 -0.904 I .OO2 -1.039 -3.699 O.262 + I.OI3 -2.052 -0-743 + 2-795 Applying these corrections to the directions, we have the final adjusted values Dir. No. Observed directions. Correction. Corrected seconds. 4 01 II O OO OO.OO // -2.05 // 57-95 6 123 49 24.97 207 52 33-50 -0-74 + 2.80 24.23 36.30 I 2 3 13 38 27.54 150 38 41.62 296 57 55-83 3.70 0.26 + I.OI 23.84 41.36 56.84 213. The Precision Measures. Referring to the equation of the curve of error, Art. 197, y = ke- h ** 9 [149] THE PRECISION MEASURES 315 we see that there are two constants to be determined for any particular set of observations. These two constants are not independent, however, as will be shown. The total area be- tween the curve and the X axis was taken equal to unity; there- fore ft or k I e~ JQ ~ h * xZ Q from which f e JQ 2k In order to integrate this expression let / = hx and dt = h dx. Then f e~*dt = f e~ h ^hdx. JQ JQ Multiplying this equation by . f~ e~*dt= f e~ ht dh, JQ JQ we have 2 rf" /V*> = / / e- h * (l + x *>hdxdh Jo JQ dx C e~" (1+xl) ( - 2 A) (i +x 2 ) dh = - r * = -[tan- 1 *] = - 2 J I + X 2 2 L Jo 4 Therefore f e~*dt = JQ 2 V* h and = -"7' 2 2Jfe or K = => [178] which shows the relation between the two constants. 316 ADJUSTMENT OF TRIANGULATION The equation of the curve of error may now be written y = ^7=e-*\ [179] VTT 214. The Average Error. The average error (r/) is the arithmetical mean of the errors, all taken with the same sign. To derive an expression for the average error, we see from equation (142) that / (x) dx is the probability that an observation will fall between the limits x and x + dx\ that is, it represents the proportion of all the errors that will probably fall within these limits. Hence, if n observa- tions are made, the number in this strip will be nf (x) dx. The sum of all the observations will be or 2n r o n I xf(x)dx, /_00 ,00 / xf(x)dx. Jo The average error equals the sum of the errors divided by the number, that is, C" vf( J XJ(X _2_y ^7 dx (-2h 2 x)dx 215. The Mean Square Error. The mean square error (/*) of an observation is the square root of the arithmetical mean of the squares of the errors. Since the number of errors between x and x + dx is nf (x) dx, the sum of the squares of these errors is nx 2 f(x)dx. THE PROBABLE ERROR 317 The sum of the squares of all the errors is n F *?f(x)dx. *J 00 Therefore n? = -^ f e~ hSxt ^dx. (d) VTT ^-o -^ / e~ h2x2 dx = i, or If we differentiate this with respect to h as the independent variable, we obtain -2k (" f**a*dx=-~- (e) Substituting (e) in (d), M = -V [181] h V2 216. The Probable Error. The probable error (r) of an observation is an error such that one half the errors of the series are greater than it and the other half are less than it; that is, the probability of making an error greater than r is just equal to the probability of making an error less than r. The probability that an error of an observation will fall be- tween the limits x and x + dx is / (x) dx. The probability that the error will fall between the limits +r and r is given by r2 h /'+ r /(*)=-^; [187] Vn whence Therefore /i = y n i To find no, the mean square error of the mean value, we have, by Equa. (/), m ffZ$ I l8 9l From Equa. [184], y / = 0.6745 V -^ [190] ro = - 6745 ' and To find the average error (t;) of a single observation, we see that, from Equa. [188], n On the average the values of these residuals will be Adding and dividing by n, ,v ./n i ^x .In \ - = y fs = v ~ >?. n n n n OBSERVATIONS OF UNEQUAL WEIGHTS 321 Therefore rj = ~* , [192] v n (n - i) and i?o = =^=' ^93] The probable error is sometimes computed from the average error in order to avoid computing the squares of the residuals. From Equa. [183], 0-8453 X y r = , . * , [i94] v n (n i) and *-. [195] w Vw i Evidently the mean error may also be computed from 77. 218. Observations of Unequal Weights. If the observations have unequal weights, let pi, p%, etc., be the weights; then MI = -= , etc. By Art. 199, if each observation is multiplied by the square root of its weight, the observations are all reduced to weight unity. The residuals are therefore etc. Applying Formulae [188] to [195] to these residuals, we have 322 ADJUSTMENT OF TRIANGULATION Also, from which r = 0.6745 n i = 0.6745 770 [200] [201] [202] [203] [204] r = 0.8453 I, [205] ri = 0.8453 n, [206] r = 0.8453 w- [207] 219. Precision of Functions of the Observed Quantities Suppose that a quantity M is denned by M = M i + Afa, where Ifi and M 2 are independent and are observed directly. Let the mean square error (m.s.e.) of M\ be MI> and let that of M 2 be M2, the m.s.e. of the function M being denoted by pp. If we suppose the errors in the determination of MI to be Xi, Xi", Xi", . . . , and those of M% to be x^ , xj' , fy'", . . . , then the real errors of If, computed from the separate observations on MI and Af 2 , will be PRECISION OF FUNCTIONS OF THE OBSERVED QUANTITIES 323 and ;-y_fa' *)' + (*" +*"?+ ' '' n Xi 2 + 2 XiXz + #2 2 But the XiXz terms will cancel out, because in the long run there will be as many -f- as products x\x^ of the same magni- tude. Therefore MF 2 = Ml 2 + rf. [208] From Equas. [183] and [184] it is evident that r F 2 = r? + rf [209] and 7j F 2 = 77i 2 + rfe 2 . [210] Let us suppose that the function is denned by M = aiMi, where di is a constant; then the real errors of M will be > > 01 2 5X 2 and ui/ = ^* - = /iv ui . or MF = fl lMl . [211] By combining [208] with [211] it is clear that if M = aiMi -f azMz + dzMz + -, then rf = 5aV, [212] [213] [214] Suppose that the function is of the general form indicated by Let M i = 0i -f Wi, Af 2 = (h + ^2, etc., in which ai is a close approximation to Afi, 02 is a close approximation to Jl/2, and Wi and mz are small corrections such that their squares may be neglected. We may regard mi and ^2, etc., as containing the real errors of Afi, Af 2 , . . . , and A*I, M2, may be considered 324 ADJUSTMENT OF TRIANGULATION as the mean square errors of mi, nh, etc. Substituting in (g), we have M = / (Oi + mi), ( . . n\ M = M'+nii- -- \-nh- -- h , (ft) 0#i OO2 in which the terms containing the squares and higher powers of mi, nh, . . . have been omitted. Then the m.s.e. of M is the same as the m.s.e. of the terms in (/?). By Equa. [212], this is or, with sufficient accuracy, JdMj , 2 [dMj , ^ = 4wJ +M2 LaM 2 J + -'-- [2I5] Similarly, , + -"' [2I6] 2 , , and [2I7] . It should be observed that in the preceding cases the unknowns are supposed to be independent of each other. If the quantities MI, M%, etc., are functions of the same variable, a different pro- cedure is necessary. Also, in case the unknowns are subject to any number of con- ditions, the computation of the precision measure of any function must be so modified as to take into account the effect of these conditions. 220. Indirect Observations. The computation of the precision of the adjusted values in the case of indirect observations is more complicated than in the CAUTION IN THE APPLICATION OF LEAST SQUARES 325 case of direct observations, because it is necessary' to know the weight of each of the unknowns, and this can only be found by the solution of equations similar to the normal equations. It may be shown that if there are n observations on q un- knowns, ther = V -2_, [2l8] n-q where /* is the m.s.e. of an observation of weight unity. If p z is the weight of an unknown, then the m.s.e. of this un- known is / X~* o [219] Similarly, r = 0.6745 V ^" > [220] and 77 = ** [222] [223] hn (ti - q) 221. Caution in the Application of Least Squares. In applying the preceding principles it should be kept in mind that the ordinary adjustment by the method of least squares deals with the accidental errors only and can tell us nothing about the constant or systematic errors which may affect the results of observation. The " probable error " may therefore be far from the true error because such constant errors are present. We should think of the precision measures as indicating the de- viation of the result from the mean result of a large number of such observations, rather than its deviation from the true value. It is usually true that the constant or the systematic errors are 326 ADJUSTMENT OP TRIANGULATION far more serious than the accidental errors; the observer should be continually on the watch for constant errors which may affect his result. So long as the conditions under which a measure- ment is made remain exactly the same the systematic errors are likely to be the same and are therefore not observed. The presence of such errors is most likely to be observed when the conditions are varied as much as possible. If observations are made at different temperatures, or under different conditions of illumination, or with different instruments, the variations of the results are usually greater than when the conditions are not changed. These variations indicate the presence of systematic errors and often enable the observer to estimate their magnitude. The computation of the most probable value improves the result with respect to the accidental errors, but leaves the more serious form of error untouched. The futility of multiplying observations and adjusting them for the purpose of removing the small accidental errors, and at the same time failing to remove the large constant error, may be illustrated by the results ob- tained by a marksman who holds his rifle steadily and places all his shots in a small group, but whose rifle sights are so far out of alignment that his shots all strike far from the bull's-eye. Of what use is the large number of shots under those circumstances? An adjustment of his results by least squares would correspond to an attempt to find the center of his group of shots, and would tell nothing about the distance from the bull's-eye. A study of the causes of the error so that he could make an adjustment of his sights would accomplish more toward hitting the mark than an infinite number of shots find under the original conditions. Of course the comparison is quite untrue in one respect; the marksman knows where his mark is, while the observer can never know the true value of the quantity he is measuring. While the method of least squares may not show directly the presence of constant errors, a study of the precision of the results, and a knowledge of the law governing the behavior of accidental errors, may enable the observer to detect the presence of constant CAUTION IN THE APPLICATION OF LEAST SQUARES 327 error, or at least to decide whether it is probably present, and consequently to so modify his methods of observing as to reduce the effect of such constant error. Variations in the result which are greater than the error of observation shown by the precision measures is likely to mean that systematic error is present. This tracing of errors to their sources, and the consequent modification of instruments and methods, may constitute the most important application of least squares. REFERENCES Following are a few references to extended works on the sub- ject of Least Squares. BARTLETT, The Method of Least Squares (an Introductory Treatise). CHAUVENET, Treatise on the Method of Least Squares. (Theory Applications to Astronomy.) CRANDALL, Geodesy and Least Squares. (Applications to Geodesy.) MERRIMAN, Treatise on the Method of Least Squares. UNITED STATES COAST AND GEODETIC SURVEY, Special Publication No. 28. (Prac- tice of the United States Coast and Geodetic Survey.) WRIGHT AND HAYFORD, Adjustment of Observations. (Applications to Geodesy.) PROBLEMS Problem i. The following angles are measured at station O. AOB = 31 10' i 5 ".6 weight (i) BOC = 19 21 17 .4 " (i) AOC = 50 31 33 .5 " (2) COD = 38 50 16 .o " (2) BOD = 58 ii 32 .o " (i) AOD = 89 21 51 .5 (i) Adjust the angles. Problem 2. The angles of a triangle are as follows: A 5353'38".94 wt. (3) B 79 22 56 .17 (4) C 46 43 29 .27 " (2) The spherical excess is 2". 83. Adjust the triangle. 328 ADJUSTMENT OF TRIANGULATION Problem 3. The angles of a quadrilateral are as follows, the numbers correspond- ing to those in Fig. 113. The weights are all unity. The spherical excess may be neglected. 1. 2 3 3l'l2".5 2. 37 01 22 .5 3- 67 35 38 -3 4. 51 51 26 .7 5. 29 56 50 .o 6. 30 35 33 -2 7- 72 37 35 -o 8. 46 49 47 -5 The sum angles are 8 + 1 7o2i'os".o 2 + 3 104 37 oo. o 4 + 5 81 48 20 .8 6 + 7 103 13 08 .4 Adjust the quadrilateral. FORMULA AND TABLES FORMULA SERIES a* * * _ + ___ = * + ? + + + 6 40 112 ~3 ,r5 r 7 tan-* * = *- - + --- 35 7 BINOMIAL THEOREM (a + 6) = a w + wa" 1 -^ + W ( ^ 2 ~ ^ g m MACLAUREN'S THEOREM TAYLOR'S THEOREM # & V V LOGARITHMIC SERIES OTHER SERIES i i x i 330 FORMULA 331 ELLIPSE AND SPHEROID *-*^. Ra -*sin s 40* NR m N cos 2 a + R m sin 2 a Mean radius = p = ^NR m - CONSTANTS logio X = M \Og e X. M = modulus of system of common logarithms = 0.434 2945. log M *= 9.637 7843- K = 3.141 592 65. log = 0.497 1499. * = 57.29577. log = 1.758 1226. 180 X 60' = 3437.747. log = 3.536 2739 180 X 60' X 60 206 264.8. log = 5.314 4251. (Approx.) IT III arc i" sm i tan i arc i" = o.ooo 004 848 137. log = 4.685 5749. 77 = 206 264.806 = number of seconds in the radian, arc i" = about 0.3 inch at distance of one mile. CLARKE SPHEROID (1866) a = 6 378 206.4 meters. log = 6.804 6985. 6 = 6 356 583-8 meters. log = 6.803 2238. (Clarke's value of meter, 3.280 8693 feet.) a = 6 378 276.5 legal meters. log = 6.804 7033. b = 6 356 653.7 legal meters. log = 6.803 2285. (Q. S. legal meter, 39.37 inches or 3.280 8333 feet.) 332 FORMULAE COAST SURVEY SPHEROID (1909) = .6 378 388 18 meters. j = 297.0 0.5. b = 6 356 909 meters. RELATION BETWEEN tJNITS OF LENGTH (Legal) Meters in one foot = 0.304 8006. log = 9.484 0158. Feet in one (legal) meter = 3.280 8333. log = 0.515 9842. Inches in one (legal) me ter = 38.37. TABLES 333 TABLE I. TABLE FOR DETERMINING RELATIVE STRENGTH OF FIGURES IN TRIANGULATION ! o 10 12' 14 16 18 20 2'2' 24 : 28 C 30 35 40 -' 4.5 50' U* BO* 85* 70 c 7.5 c s:/ j 90 10 428 359 12 359 295 253 14 315 253 214 187 16 284 225 187 162 143 18 262 204 168 143 126 113 20 245 189 153 130 113 100 \il 22 232 177 142 11 103 91 81 74 24 221 167 134 111 95 83 74 67 61 26 213 160 126 104 89 77 6s Ijl ,50 .51 28 206 153 120 99 83 72 83 57 51 47 43 30 199 148 115 94 79 68 M 53 48 43 40 33 35 188 137 106 85 71 60 .52 46 41 37 33 27 23 40 179 129 99 79 65 54 47 41 36 32 29 23 !',< 16 45 172 124 93 74 60 50 43 37 32 M 2.5 20 16 13 11 50 167 119 89 70 57 47 39 34 29 26 23 18 14 11 I 8 ! 55 162 115 86 67 54 44 37 32 27 24 21 16 12 10 8 7 5 60 159 112 83 64 51 42 35 M 25 22 19 14 11 9 7 5 4 4 65 155 109 80 62 49 40 3:.l 2s 24 21 18 13 10 7 6 t 4 3 2 70 152 106 78 60 48 38 32 27 23 19 17 12 9 7 5 4 3 2 i 1 75 150 104 76 58 46 37 30 25 21 u 16 11 S 6 4 g 2 2 i 1 1 80 147 102 74 57 45 36 29 24 2(1 17 1.5 10 7 .5 4 3 2 1 i 1 85 145 100 73 55 "43 34 M 23 U 16 14 10 7 ,5 3 l" *2 1 i d 90 143 98 71 54 42 33 27 22 u 16 13 g 6 4 3 2 1 1 i u 95 140 96 70 53 41 32 M 22 is 1.5 13 y 1 4 3 2 1 100 138 95 68 51 40 31 25 21 17 14 12 8 6 4 3 2 1 105 136 93 67 50 39 30 25 20 17 14 12 8 ,5 4 2 2 1 110 134 91 65 49 38 30 24 1!> 16 13 11 7 .5 3 2 2 1 1 115 132 89 64 48 37 29 23 1!. 1.5 13 11 7 .5 3 2 2 1 120 129 88 62 46 36 28 22 is 1.5 12 10 7 .5 3 2 2 125 127 86 61 45 35 27 22 u 14 12 10 7 .5 4 3 2 130 125 84 59 44 34 26 21 17 14 12 10 7 5 4 3 135 122 82 58 43 33 26 21 17 14 12 10 7 .5 4 140 119 80 56 42 32 25 2(1 17 14 12 10 8 6 145 116 77 55 41 32 25 21 17 1.5 13 11 g 150 112 75 54 40 32 26 21 U 16 U 13 152 111 75 53 40 32 26 22 1" 17 16 154 110 74 53 41 33 27 23 21 U 156 108 74 54 42 34 28 2.5 22 158 107 74 54 43 35 30 27 160 107 74 56 45 38 33 162 107 76 59 48 42 164 109 ' 79 63 54 166 113 86 71 168 122 98 170 143 334 TABLES TABLE II. CORRECTION FOR EARTH'S CURVATURE AND REFRACTION Dist. Corr. Dist. Corr. Dist. Corr. Miles. 1 Feet. 0.6 Miles. 21 Feet. 253.1 Miles. 41 Feet. 964.7 2 2.3 22 277.7 42 1012.2 3 5.2 23 303.6 43 1061.0 4 9.2 24 330.5 44 1111.0 5 14.4 25 358.6 45 1162.0 6 20.6 26 388.0 46 1214.2 7 28.1 27 418.3 47 1267.7 8 36.7 28 449.9 48 1322.1 9 46.4 29 482.6 49 1377.7 10 57.4 30 516.4 50 ' 1434.6 11 69.4 31 551.4 51 1492.5 12 82.7 32 587.6 52 1551.6 13 97.0 33 624.9 53 1611.9 14 112.5 34 663.3 54 1673.3 15 129.1 35 703.0 55 1735.8 16 146.9 36 743.7 56 . 1799.6 17 165.8 37 785.6 57 1864.4 18 185.9 38 828.6 58 1930.4 19 207.2 39 872.8 59 1997.5 20 229.5 40 918.1 60 2065.8 TABLES 335 TABLE III. SHORT TABLE OF FACTORS FOR REDUCTION OF TRANSIT OBSERVATIONS Top Argument = Star's Declination (5). Side Argument = Star's Zenith Distance (f) . [For factor A use left-hand argument. For factor B use right-hand argument. For factor C use bottom line.J r Ifr 15 20 25 30 35 40 45 50 55 60 65 70 r 1 5 10 0.02 0.09 p. 17 0.02 0.09 0.18 0.02 0.09 0.18 0.02 0.09 0.19 0.02 0.10 0.19 0.02 0.10 0.20 0.02 0.11 0.21 0.02 0.11 0.23 0.02 0.12 0.25 0.03 0.13 0.27 0.03 0.15 0.30 0.03 0.17 0.35 0.04 0.21 0.41 0.05 0.25 0.51 89 85 80 15 20 25 0.26 0.34 0.42 0.26 0.35 0.43 0.27 0.35 0.44 0.28 0.36 0.45 0.29 0.38 0.47 0.30 0.40 0.49 0.32 0.42 0.52 0.34 0.45 0.55 0.37 0.48 0.60 0.40 0.53 0.66 0.45 0.60 0.74 0.52 0.68 0.85 0.61 0.81 1.00 0.76 1.00 1.24 75 70 65 30 35 40 0.50 0.57 0.64 0.51 0.58 0.65 0.52 0.59 0.67 0.53 0.61 0.68 0.55 0.63 0.71 0.58 0.66 0.74 0.61 0.70 0.78 0.65 0.75 0.84 0.71 0.81 0.91 0.78 0.89 1.00 0.87 1.00 1.12 1.00 1.15 1.29 1.18 .36 .52 1.46 1.68 1.88 60 55 50 45 50 55 0.71 0.77 0.82 0.72 0.78 0.83 0.73 0.79 0.85 0.75 0.82 0.87 0.78 0.85 0.90 0.82 0.89 0.95 0.86 0.94 1.00 0.92 1.00 1.07 1.00 1.08 1.16 1.10 1.19 1.27 .23 .34 .43 1.41 1.53 1.64 .67 .81 .94 2.07 2.24 2.40 45 40 35 60 65 70 0.87 0.91 0.94 0.88 0.92 0.95 0.90 0.94 0.97 0.92 0.96 1.00 0.96 1.00 1.04 .00 .05 .09 1.06 1.11 1.15 1.13 1.18 1.23 .22 .28 .33 1.35 1.41 1.46 .51 .58 .64 1.73 1.81 1.88 2.05 2.14 2.22 2.53 2.65 2.75 30 25 20 75 80 85 0.97 0.98 1.00 0.98 1.00 1.01 1.00 1.02 1.03 1.03 1.05 1.06 .07 .09 .10 .12 .14 .15 1.18 1.20 1.22 1.26 1.29 1.30 .37 .39 .41 1.50 1.53 1.55 1.68 1.72 1.74 1.93 1.97 1.99 2.29 2.33 2.36 2.82 2.88 2.91 15 10 5 90 1.00 1.02 1.04 1.06 1.10 1.15 1.22 1.31 1.41 1.56 1.74 2.00 2.37 2.92 TABLE IV. DIURNAL ABERRATION (*c) Lati- Declination = 5. = <*> 10 20 30 40 50 60 70 75 80 85 0.02 0.02 0.02 0*02 0*03 0*03 0*04 0*06 0*08 0*12 0.24 10 0.02 0.02 0.02 0.02 0.03 0.03 0.04 0.06 0.08 0.12 0.24 20 0.02 0.02 0.02 0.02 0.03 0.03 0.04 0.06 0.08 0.11 0.23 30 0.02 0.02 0.02 0.02 0.02 0.03 0.04 0.05 0.07 0.10 0.21 40 0.02 0.02 0.02 0.02 0.02 0.03 0.03 0.05 0.06 0.09 0.18 50 0.01 0.01 0.01 0.02 0.02 0.02 0.03 0.04 0.05 0.08 0.15 60 0.01 0.01 0.01 0.01 0.01 0.02 0.02 0.03 0.04 06 0.12 70 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.02 0.03 0.04 0.08 80 0.00 0.00 0.00 0.00 0.00 0.01 0.01 0.01 0.01 0.02 0.04 336 TABLES TABLE V. CORRECTION TO LATITUDE FOR DIFFEREN TIAL REFRACTION = % (r - r'). (The sign of the correction is the same as that of the micrometer difference.] One-half diff.of zenith distances. Zenith distance. 10 20 25 30 35 40 45 0.0 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.5 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.02 , 1.0 0.02 0.02 0.02 0.02 0.02 0.03 0.03 0.03 1.5 0.03 0.03 0.03 0.03 0.03 0.04 0.04 0.05 2.0 0.03 0.03 0.04 0.04 0.04 0.05 0.06 0.07 2.5 0.04 0.04 0.05 0.05 0.06 0.06 0.07 0.08 3.0 0.05 0.05 0.06 0.06 0.07 0.08 0.09 0.10 3.5 0.06 0.06 0.07 0.07 0.08 0.09 0.10 0.12 4.0 0.07 0.07 0.08 0.08 0.09 0.10 0.11 0.13 4.5 0.08 0.08 0.09 0.09 0.10 0.11 0.13 0.15 5.0 0.08 0.09 0.10 0.10 0.11 0.13 0.14 0.17 5.5 0.09 0.10 0.10 0.11 0.12 0.14 0.16 0.18 6.0 0.10 0.10 0.11 0.12 0.13 0.15 0.17 0.20 6.5 0.11 0.11 0.12 0.13 0.14 0.16 0.19 0.22 7.0 0.12 0.12 0.13 0.14 0.16 0.18 0.20 0.23 7.5 0.13 0.13 0.14 0.15 0.17 0.19 0.21 0.25 8.0 0.13 0.14 0.15 0.16 0.18 0.20 0.23 0.27 8.5 0.14 0.15 0.16 0.17 0.19 0.21 0.24 0.29 9.0 0.15 0.16 0.17 0.18 0.20 0.23 0.26 0.30 9.5 0.16 0.16 0.18 0.19 0.21 0.24 0.27 0.32 10.0 0.17 0.17 0.19 0.20 0.22 0.25 0.29 0.34 10.5 0.18 0.18 0.20 0.21 0.23 0.26 0.30 0.35 11.0 0.18 0.19 0.21 0.22 0.25 0.28 0.31 0.37 11.5 0.19 0.20 0.22 0.23 0.26 0.29 0.33 0.39 12.0 0.20 0.21 0.23 0.25 0.27 0.30 0.34 0.40 12.5 0.21 0.22 0.24 0.26 0.28 0.31 0.36 0.42 13.0 0.22 0.22 0.25 0.27 0.29 0.33 0.37 0.44 13.5 0.23 0.23 0.26 0.28 0.30 0.34 0.39 0.45 14.0 0.23 0.24 0.27 0.29 0.31 0.35 0.40 0.47 14.5 0.24 0.25 0.28 0.30 0.32 0.36 0.41 0.49 15.0 0.25 0.26 0.29 0.31 0.34 0.38 0.43 0.50 15.5 0.26 0.27 0.29 0.32 0.35 0.39 0.44 0.52 16.0 0.27 0.28 0.30 0.33 0.36 0.40 0.46 0.54 16.5 0.28 0.29 0.31 0.34 0.37 0.41 0.47 0.55 17.0 0.29 0.29 0.32 0.35 0.38 0.43 0.49 0.57 17.5 0.29 0.30 0.33 0.36 0.39 0.44 0.50 0.59 18.0 0.30 0.31 0.34 0.37 0.40 0.45 0.51 0.60 18.5 0.31 0.32 0.35 0.38 0.41 0.46 0.53 0.62 19.0 0.32 0.33 0.36 0.39 0.43 0.48 0.54 0.64 19.5 0.33 0.34 0.37 0.40 0.44 0.49 0.56 0.65 20.0 0.34 0.35 0.38 0.41 0.45 0.50 0.57 0.67 TABLES 337 TABLE VI. CORRECTION TO LATITUDE FOR REDUCTION TO MERIDIAN [Star off the meridian but instrument in the meridian. The sign of the correction to the latitude is positive except for stars^sputh of the equator and subpolars.] s 10 15 20 22* 24 26* 28 30* ' !2* 3- 36* 38* S 1 " " " " i 0. t 01 0.01 0.01 89 2 0.01 0.01 0.01 0.01 .01 0. 01 0.01 0.01 88 3 0.01 0.01 0.01 0.01 0.01 0.01 (I .01 0. a 0.02 0.02 87 4 0.01 0.01 0:01 0.01 0.01 0.02 .02 0. )2 0.02 0.03 86 5 0.01 0.01 0.01 0.01 0.02 0.02 0.02 .02 o. a 0.03 0.03 85 6 0.01 0.01 0.01 0.02 0.02 0.02 0.03 .03 0. 0.04 0.04 84 7 o.oi 0.01 0.02 0.02 0.02 0.03 0.03 .03 0. J4 0.04 0.05 83 8 0.01 0.02 0.02 0.02 0.03 0.03 0.03 .04 0. M 0.05 0.05 82 9 0.01 0.02 0.02 0.02 0.03 0.03 0.04 .04 0. 0.5 0.05 0.06 81 10 0.01 0.02 0.02 0.03 0.03 0.04 0.04 .05 0. M 0.06 0.07 80 12 0.01 0.01 0.02 0.03 0.03 0.04 0.05 0.05 .06 0. 06 0.07 0.08 78 14 0.01 0.01 0.03 0.03 0.04 0.04 0.05 0.06 .07 0. 07 0.08 0.09 76 16 0.01 ft. 02 0.03 0.03 0.04 0.05 0.06 0.07 .07 n. M 0.09 0.10 74 18 0.01 0.02 0.03 0.04 0.05 0.05 0.06 0.07 .08 D. JO 0.10 0.12 72 20 0.01 0.02 0.04 04 0.05 0.06 0.07 0.08 .09 0. 10 0.11 0.13 70 22 0.01 0.02 0.04 0.05 0.05 0.06 0.07 0.09 .10 0. 11 0.12 0.14 68 24 0.01 0.02 0.04 0.05 0.06 0.07 0.08 0.09 .10 0. 12 0.13 0.15 66 26 0.01 0.02 0.04 0.05 0.06 0.07 0.08 0.10 .11 0. 12 0.14 0.15 64 28 0.01 0.03 0.05 0.05 0.07 0.08 0.09 0.10 :12 0. 13 0.15 0.16 62 30 0.01 0.03 0.05 0.06 0.07 0.08 0.09 0.11 .12 0. 14 0.15 0.17 60 32 0.01 0.03 0.05 0.06 0.07 0.08 0.10 0.11 .13 0. 14 0.16 0.18 58 34 0.01 0.03 0.05 0.06 0.07 0.09 0.10 0.11 .13 0. 15 0.16 0.18 56 36 0.01 0.03 0.05 0.06 0.07 0.09 0.10 0.12 .13 0. 1.5 0.17 0.19 54 38 0.01 0.03 0.05 0.06 0.08 0.09 0.10 0.12 13 0. 1.5 0.17 0.19 52 40 0.01 0.03 0.05 0-07 0.08 0.09 0.11 0.12 14 16 0.17 0.19 50 45 0.01 0.03 0.05 0.07 0.08 0.09 0.11 0.12 14 16 0.18 0.20 45 S 40* 42* 44* 46* .48* 50* 52* 54< r 5( 58* 60* d 1 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.( i 01 ).02 0.02 89 2 0.02 0.02 0.02 0.02 0.02 0.02 0.03 0.( 1 09 i ).03 0.03 88 3 iO.02 0.03 0.03 0.03 0.03 0.04 0.04 0.( 4 04 1 ).05 0.05 87 4 0.03 0.03 0.04 0.04 0.04 0.05 0.05 0.( 8 M ).06 0.07 86 5 0.04 0.04 0.05 0.05 0.05 0.06 0.06 0.( 7 07 1 ).08 0.09 85 6 0.05 0.05 0.06 06 0.07 0.07 0.08 0.( I bo ( UO 0.10 84 7 0.05 0.06 0.06 0.07 0.08 0.08 0.09 0.1 10 i >.ll 0.12 83 8 0.06 0.07 0.07 0.08 0.09 0.09 0.10 OJ 1 12 i 1.13 0.14 82 9 0.07 0.07 0.08 0.09 0.10 0.11 0.11 0.1 2 0. 13 ( .14 0.15 81 10 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.1 4 0. 1,5 ( .16 0.17 80 12 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.1 1 0. 17 t .19 0.20 78 14 0.10 0.11 0.12 0.14 0.15 0.16 0.17 O.I '.i 0. 20 I .22 0.23 76 16 0.12 0.13 0.14 0.15 0.17 0.18 0.20 0.2 1 0. 23 I .24 0.26 74 18 0.13 0.14 0.16 0.17 0.18 0.20 0.22 0.2 I 0. 2.5 C .27 0.29 72 20 0.14 0.15 0.17 0.19 0.20 0.22 0.24 0.2 1 0. 28 ' C .29 0.32 70 22 0.15 0.17 Q. 18 0.20 0.22 0.24 0.26 0.2 B 0. 30 .32 0.34 68 24 0.16 0.18 0.20 0.21 0.23 0.25 0.27 0.2 9 0. H 1 .34 36 66 26 0.17 0.19 0.21 0.23 0.25 0.27 0.29 0.3 1 0. U 1 .36 0.39 64 28 0.18 0.20 0.22 0.24 0.26 0.28 0.31 0.3 3 0. IS n .38 0.41 62 30 0.19 0.21 0.23 0.25 0.27 0.30 0.32 0.3 I 0. n r .40 0.42 60 32 0.20 0.22 0.24 0.26 0.28 0.31 0.33 0.3 | 0. 3ft .41 0.44 58 34 0.20 0.22 0.24 0.27 0.29 0.32 0.34 0.3 ; 0. 40 .42 0.45 56 36 0.21 0.23 0.25 0.28 0.30 0.32 0.35 0.3 s 0. 41 .44 0.47 54 38 0.21 0.23 0.26 0.28 30 0.33 0.36 0.3 0. 41 .44 0.48 52 40 0.21 0.24 0.26 0.28 0.31 0.34 0.36 0.3 0. 42 45 0.48 50 45 0.22 0.24 0.26 0.29 0.31 0.34 0.37 0.4 i 0. 43 .46 0.49 45 338 TABLES TABLE VII. REDUCTION OF LATITUDE TO SEA LEVEL [The correction is negative in every case.] 5 85 10 80 15 75 20 70 25 65 30 60 35 55 40 50 45 Feet. 100 Meters 30 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.01 200 61 0.00 0.00 0.01 0.01 0.01 0.01 0.01 0.01 0.01 300 91 0.00 0.01 0.01 0.01 0.01 0.01 0.01 0.02 0.02 400 122 0.00 0.01 0.01 0.01 0.02 0.02 0.02 0.02 0.02 500 152 0.00 0.01 0.01 0.02 0.02 0.02 0.02 0.03 0.03 600 183 0.01 0.01 0.02 0.02 0.02 0.03 0.03 0.03 0.03 700 213 0.01 0.01 0.02 0.02 0.03 0.03 0.03 0.04 0.04 800 244 0.01 0.01 0.02 0.03 0.03 0.04 0.04 0.04 0.04 900 274 0.01 0.02 0.02 0.03 0.04 0.04 0.04 0.05 0.05 1000 305 0.01 0.02 0.03 0.03 0.04 0.05 0.05 0.05 0.05 1100 335 0.01 0.02 0.03 0.04 0.04 0.05 0.05 0.06 0.06 1200 366 0,01 0.02 0.03 0.04 0.05 0.05 0.06 0.06 0.06 1300 396 0.01 0.02 0.03 0.04 0.05 0.06 0.06 0.07 0.07 1400 427 0.01 0.02 0.04 0.05 0.06 0.06 0.07 0.07 0.07 1500 457 0.01 0.03 0.04 0.05 0.06 0.07 0.07 0.08 0.08 1600 488 0.01 0.03 0.04 0.05 0.06 0.07 0.08 0.08 0.08 1700 518 0.02 0.03 0.04 0.06 0.07 0.08 0.08 0.09 0.09 1800 549 0.02 0.03 0.05 0.06 0.07 0.08 0.09 0.09 0.09 1900 579 0.02 0.03 0.05 0.06 0.08 0.09 0.09 0.10 0.10 2000 610 0.02 0.04 0.05 0.07 0.08 0.09 0.10 0.10 0.10 2100 640 0.02 0.04 0.05 0.07 0.08 0.09 0.10 0.11 0.11 2200 671 0.02 0.04 0.06 0.07 0.09 0.10 0.11 0.11 0.11 2300 701 0.02 0.04 0.06 0.08 0.09 0.10 0.11 0.12 0.12 2400 732 0.02 0.04 0.06 0.08 0.10 0.11 0.12 0.12 0.13 2500 762 0.02 0.04 0.07 0.08 0.10 0.11 0.12 0.13 0.13 2600 792 0.02 0.05 0.07 0.09 0.10 0.12 0.13 0.13 0.14 2700 823 0.02 0.05 0.07 0.09 0.11 0.12 0.13 0.14 0.14 2800 853 0.03 0.05 0.07 0.09 0.11 0.13 0.14 0.14 0.15 2900 884 0.03 0.05 0.08 0.10 0.12 0.13 0.14 0.15 0.15 3000 914 0.03 0.05 0.08 0.10 0.12 0.14 0.15 0.15 0.16 3100 945 0.03 0.06 0.08 0.10 0.12 0.14 0.15 0.16 0.16 3200 975 0.03 0.06 0.08 0.11 0.13 0.14 0.16 0.16 0.17 3300 1006 0.03 0.06 0.09 0.11 0.13 0.15 0.16 0.17 0.17 3400 1036 0.03 0.06 0.09 0.11 0.12 0.15 0.17 0.17 0.18 i 3500 1067 0.03 0.06 0.09 0.12 0.14 0.16 0.17 0.18 0.18 TABLES 339 TABLE VII (Con.). REDUCTION OF LATITUDE TO SEA LEVEL (The correction is negative in every case.] 5 85 10 80 15 75 20 70 25 65 30 60 35 55 40 50 45 Feet. 3600 Meters. 1097 0.03 0.06 0.09 0.12 0.14 0.16 0.18 0.18 0.19 3700 1128 0.03 0.07 0.10 0.12 0.15 0.17 0.18 0.19 0.19 3800 1158 0.03 0.07 0.10 0.13 0.15 0.17 0.19 0.20 0.20 3900 1189 0.04 0.07 0.10 0.13 0.16 0.18 0.19 0.20 0.20 4000 1219 0.04 0.07 0.10 0.13 0.16 0.18 0.20 0.21 0.21 4100 1250 0.04 0.07 0.11 0.14 0.16 0.19 0.20 0.21 0.21 4200 1280 0.04 0.07 0.11 0.14 0.17 0.19 0.21 0.22 0.22 4300 1311 0.04 0.08 0.11 0.14 0.17 0.19 0.21 0.22 0.22 4400 1341 0.04 0.08 0.11 0.15 0.18 0.20 0.22 0.23 0.23 4500 1372 0.04 0.08 0.12 0.15 0.18 0.20 0.22 0.23 0.23 4600 1402 0.04 0.08 0.12 0.15 0.18 0.21 0.23 0.24 0.24 4700 1433 0.04 0.08 0.12 0.16 0.19 0.21 0.23 0.24 0.24 4800 1463 0.04 0.09 0.13 0.16 0.19 0.22 0.24 0.25 0.25 4900 1494 0.04 0.09 0.13 0.16 0.20 0.22 0.24 0.25 0.26 5000 1524 0.05 0.09 0.13 0.17 0.20 0.23 0.24 0.26 0.26 5100 1554 0.05 0.09 0.13 0.17 0.20 0.23 0.25 .26 0.27 5200 1585 0.05 0.09 0.14 0.17 0.21 0.23 0.25 0.27 0.27 5300 1615 0.05 0.09 0.14 0.18 0.21 0.24 0.26 0.27 0.28 5400 1646 0.05 0.10 0.14 0.18 0.22 0.24 0.26 0.28 0.28 5500 1676 0.05 0.10 0.14 0.18 0.22 0.25 0.27 0.28 0.29 5600 1707 0.05 0.10 0.15 0.19 0.22 0.25 0.27 0.29 0.29 5700 1737 0.05 0.10 0.15 0.19 0.23 0.26 0.28 0.29 0.30 5800 1768 0.05 0.10 0.15 0.19 0.23 0.26 0.28 0.30 0.30 5900 1798 0.05 0.11 0.15 0.20 0.24 0.27 0.29 0.30 0.31 6000 1829 0.05 0.11 0.16 0.20 0.24 0.27 0.29 0.31 0.31 6100 1859 0.06 0.11 0.16 0.20 0.24 0.28 G.30 0.31 0.32 6200 1890 0.06 0.11 0.16 0.21 0.25 0.28 0.30 0.32 0.32 6300 1920 0.06 0.11 0.16 0.21 0.25 0.28 0.31 0.32 0.33 6400 1951 0.06 0.11 0.17 0.21 0.26 0.29 0.31 0.33 0.33 6500 1981 0.06 0.12 0.17 0.22 0.26 0.29 0.32 0.33 0.34 6600 2012 0.06 0.12 0.17 0.22 0.26 0.30 0.32 0.34 0.34 6700 2042 0.06 0.12 0.17 0.22 0.27 0.30 0.33 0.34 0.35 6800 2073 0.06 0.12 0.18 0.23 0.27 0.31 0.33 0.35 0.35 6900 2103 0.06 0.12 0.18 0.23 0.28 0.31 0.34 0.35 0.36 7000 2134 0.06 0.12 0.18 0.23 0.28 0.32 0.34 0.36 0.36 340 TABLES TABLE VII (Con.). REDUCTION OF LATITUDE TO SEA LEVEL [The correction is negative in every case.) X. <6 * X 5 85 10 80 15 75 20 70 25 65 30 60 35 55 40 50 45 Feet. Meters. // ; // // " n 7100 2164 0.06 0.13 0.19 0.24 0.28 0.32 0.35 0.36 0.37 7200 2195 0.07 0.13 0.19 0.24 0.29 0.33 0.35 0.37 0.38 7300 2225 0.07 0.13 0.19 24 0.29 0.33 0.36 0.37 0.38 7400 2256 0.07 0.13 0.19 25 0.30 0.33 0.36 0.38 0.39 7500 2286 0.07 0.13 0.20 0.25 0.30 0.34 0.37 0.38 0.39 7600 2316 0.07 0.14 0.20 0.25 0.30 0.34 0.37 0.39 0.40 7700 2347 0.07 0.14 0.20 0.26 0.31 0.35 0.38 0.40 0.40 7800 2377 0.07 0.14 0.20 0.26 0.31 0.35 0.38 0.40 0.41 7900 2408 0.07 0.14 0.21 0.26 0.32 0.36 0.39 0.41 0.41 8000 2438 0.07 0.14 0.21 0.27 0.32 0.36 0.39 0.41 0.42 8100 2469 0.07 0.14 0.21 0.27 0.32 0.37 0.40 0.42 0.42 8200 2499 0.07 0.15 0.21 0.27 0.33 0.37 0.40 0.42 0.43 8300 2530 0.08 0.15 0.22 0.28 0.33 0.37 0.41 0.43 0.43 8400 2560 0.08 0.15 0.22 0.28 0.34 0.38 0.41 0.43 0.44 8500 2591 0.08 0.15 0.22 0.28 0.34 0.38 0.42 0.44 0.44 <600 2621 0.08 0.15 0.22 0.29 0.34 0.39 0.42 0.44 0.45 8700 2652 0.08 0.16 0.23 0.29 0.35 0.39 0.43 0.45 0.45 8800 2682 0.08 0.16 0.23 0.29 0.35 0.40 0.43 0.45 0.46 8900 2713 0.08 0.16 0.23 0.30 0.36 0.40 0.44 0.46 0.46 9000 2743 0.08 0.16 0.23 0.30 0.36 0.41 0.44 0.46 0.47 9100 2774 0.08 0.16 0.24 0.30 0.36 0.41 0.45 0.47 0.47 9200 2804 0.08 0.16 0.24 0.31 0.37 0.42 0.45 0.47 0.48 9300 2835 0.08 0.17 0.24 0.31 0.37 0.42 0.46 0.48 0.48 9400 2865 0.09 0.17 0.24 0.31 0.38 0.42 0.46 0.48 0.49 9500 2896 0.09 0.17 0.25 0.32 0.38 0.43 0.47 0.49 0.50 9600 2926 0.09 0.17 0.25 0.32 0.38 0.43 0.47 0.49 0.50 9700 2957 0.09 0.17 0.25 0.32 0.39 0.44 0.48 0.50 0.51 9800 2987 0.09 0.17 0.26. 0.33 0.39 0.44 0.48 0.50 0.51 9900 3018 0.09 0.18 0.26 0.33 0.40 0.45 0.48 0.51 0.52 10000 3048 0.09 0.18 0.26 0.33 0.40 0.45 0.49 0.51 0.52 TABLES 341 TABLE VIII. FOR CONVERTING SIDEREAL INTO MEAN SOLAR TIME [Increase in Sun's Right Ascension in Sidereal h. m. s.] Mean Time = Sidereal Time C. Sid. Hrs. Corr. Sid. Min. Corr. Sid. Min. Corr. Sid. Sec. Corr. Sid. Sec. Corr. I m s o 9 .830 I s 0.164 31 s 5-079 I s 0.003 31 s 0.085 2 o 19 .659 2 0.328 32 5.242 2 0.005 32 0.087 3 o 29 .489 3 0.491 33 5.406 3 o .008 33 o .090 4 o 39.318 4 0.655 34 5-57 4 o .on 34 0.093 5 o 49 .148 5 o .819 35 5-734 5 0.014 35 0.096 6 o 58 .977 6 0.983 36 5.898 6 0.016 36 0.098 I i 8.807 i 18.636 I .147 11 6.062 6.225 I o .019 0.022 37 38 O.IOI o .104 9 i 28.466 9 474 39 6.389 9 o .025 39 o .106 10 i 38.296 10 .638 40 6-553 10 O.O27 40 o .109 II i 48.125 ii .802 41 6.717 ii 0.030 41 O.II2 12 13 i 57 -955 2 7.784 12 13 .966 2.130 42 43 6.881 7-045 12 13 0.033 '35 42 43 O.II7 14 2 17 .614 14 2.294 44 7.208 14 0.038 44 .120 15 2 27.443 IS 2-457 45 7-372 15 0.041 45 0.123 16 2 37.273 16 2 .621 46 7.536 16 0.044 46 o .126 11 2 47.102 2 56.932 17 18 2 .785 2.949 7.700 7.864 11 0.046 0.049 47 48 0.128 O.I3I 19 3 6 .762 19 3 -"3 49 8.027 19 0.052 49 0.134 20 3 16.591 20 3-277 50 8.191 20 0-055 50 0-137 21 3 26.421 21 3-440 51 8-355 21 0.057 51 0.139 22 3 36.250 22 3.604 52 8.519 22 0.060 52 o .142 23 3 46.080 23 3-768 53 8.683 23 0.063 53 0.145 24 3 55 -909 24 3-932 54 8.847 24 o .066 54 0.147 25 4.096 55 9 .010 25 0.068 55 0.150 26 4-259 56 9.174 26 0.071 56 0.153 27 28 4-423 4-587 57 58 9.338 9.502 27 28 0.074 o .076 11 0.156 0.158 29 30 4-751 4-915 is 9.666 9.830 29 30 0.079 0.082 59 60 o .161 o .164 342 TABLES TABLE IX. FOR CONVERTING MEAN SOLAR INTO SIDEREAL TIME [Increase in Sun's Right Ascension in Solar h. m. s.] Sidereal Time = Mean Time + C. Mean Hrs. Corr. Mean Min. Corr. Mean Min. Corr. Mean Sec. Corr. Mean Sec. Corr. I m s o 9 .856 I s o .164 31 s 5-93 I s 0.003 31 0.085 2 o 19.713 2 0.329 32 5-257 2 o .005 32 0.088 3 o 29.569 3 0-493 33 5-42i 3 0.008 33 o .090 4 o 39 .426 4 0.657 34 5.585 4 .Oil 34 0.093 5 o 49 .282 5 0.821 35 5-750 5 o .014 35 o .096 6 o 59 -139 6 0.986 36 5.9I4 6 o .016 36 0.099 7 8-995 7 i .150 37 6.078 7 o .019 37 .101 8 18.852 8 i -3!4 38 6 .242 8 .022 38 o .104 9 28.708 9 1.478 39 6.407 9 0.025 39 o .107 10 38 -565 10 1-643 40 6-57 1 10 o .027 40 O.IIO ii 48.421 ii i .807 4i 6-735 ii 0.030 4i O .112 12 58-278 12 1.971 42 6 .900 12 0.033 42 O.II5 13 2 8.134 13 2.136 43 7.064 13 0.036 43 o .118 14 2 17.991 14 2.300 44 7.228 14 0.038 44 o .120 15 2 27.847 15 2.464 45 7-392 i5 o .041 45 0.123 16 2 37.704 16 2.628 46 7-557 16 0.044 46 o .126 17 2 47.560 i7 2-793 47 7.721 i7 0.047 47 0.129 18 2 57-417 18 2-957 48 7.885 18 0.049 48 0.131 iQ 3 7-273 19 3.121 49 8.049 19 0.052 49 0.134 20 3 17-129 20 3-285 50 8.214 20 0.055 50 0.137 21 3 26.986 21 3-45 5i 8.378 21 0.057 5 1 o .140 22 3 36-842 22 3.614 52 8.542 22 0.060 52 o .142 23 3 46.699 23 3.778 53 8.707 23 0.063 53 0.145 24 3 5 6 -555 24 3-943 54 8.871 24 0.066 54 0.148 25 4.107 55 9-035 25 0.068 55 0.151 26 4.271 56 9.199 26 o .071 56 0.153 27 4-435 57 9-364 27 0.074 57 0.156 28 4 .600 58 9-528 28 0.077 58 0.160 29 4.764 59 9.692 29 0.079 59 0.162 30 4.928 60 9-856 30 o .082 60 o .16^1 TABLES 343 TABLE X. LENGTHS OF ARCS OF THE PARALLEL AND THE MERIDIAN AND LOGS OF N AND R* [Metric Units.] Latitude. Parallel. Value of 1. Meridian. Value of 1. LogN. LogRm. Meters. Meters. 00 111,321 110,567.2 6.8046985 6.8017489 30 1,361 567.3 6987 7493 1 00 1,304 567.6 6990 7502 30 1,283 568.0 6996 7519 2 00 1,253 568.6 7003 7543 30 1,215 569.4 7012 7573 3 00 1,169 570.3 7025 7610 30 1,114 571.4 7040 7654 4 00 1,051 572.7 7057 7704 30 110,980 574.1 7076 7761 5 00 110,900 110,575.8 6.8047097 6.8017824 30 0,812 577.6 7120 7894 6 00 0,715 579.5 7146 7971 30 0,610 581.6 7174 8054 7 00 0,497 583.9 7203 8144 30 0,375 586.4 7235 8240 8 00 0,245 589.0 7270 8343 30 0,106 591.8 7307 8452 9 00 109,959 594.7 7345 8568 30 9,804 597.8 7385 8690 ID 00 109,641 110,601.1 6.8047428 6.8018819 30 9,469 604.5 7474 8954 11 00 9,289 608.1 7520 9094 30 9,101 611.9 7570 9241 12 00 108,904 615.8 7620 9395 30 8,699 619.8 7673 9555 13 00 8,486 624.1 7729 9720 30 8,265 628.4 7786 9892 14 00 8,036 633.0 7845 6.8020070 30 107,798 637.6 7907 0254 15 00 107,553 110,642.5 6.8047970 6.8020443 30 7,299 647.5 8035 0639 16 00 7,036 652.6 8102 0839 30 6,766 657.8 8171 1047 17 00 6,487 663.3 8242 1258 30 6,201 668.8 8315 1477 18 00 5,906 674.5 8389 1701 30 5,604 680.4 8465 1930 19 00 5,294 686.3 8544 2165 30 4,975 692.4 8624 2404 20 00 104,649 110,698.7 6.8048705 6.8022649 30 4,314 705.1 8789 2900 21 00 3,972 711.6 8874 3155 30 3,622 718.2 8960 3415 22 00 3,264 725.0 9049 3680 .30 2,898 731.8 9139 3950 344 TABLES TABLE X (Con.) LENGTHS OF ARCS OF THE PARALLEL AND THE MERIDIAN AND LOGS OF N AND R m IMetric Units.] Latitude. Parallel. Value of 1. Meridian. Value of 1. LogN. Log Rm. Meters. Meters. 23 00 102,524 110,738.8 6.8029231 6.8044225 30 2,143 746.0 9323 4504 24 00 1,754 753.2 9418 4788 30 1,357 760.6 9514 5077 25 00 100,952 110,768.0 6.8049612 6.8025370 30 0,539 775.6 9711 5667 26 00 0,119 783.3 9812 5968 30 99,692 791.1 9914 6274 27 00 9,257 799.0 6.8050017 6584 30 8,814 807.0 0121 6897 28 00 8,364 815.1 0227 7215 30 7,906 823.3 0334 7536 29 00 7,441 831.6 0443 7862 30 6,968 840.0 0552 8190 30 00 96,488 110,848.5 6.8050663 6.8028522 30 6,001 857.0 0774 8857 31 00 95,506 865.7 0888 9197 30 5,004 874.4 1002 9539 32 00 4,495 883.2 1117 9883 30 3,979 892.1 1233 6.8030231 33 00 3,455 901.1 1350 0582 30 2,925 910.1 1468 0935 34 00 2,387 919.2 1586 1292 30 1,842 928.3 1706 1651 35 00 91,290 110,937.6 6.8051826 6.8032012 30 0,731 946.9 1947 2375 36 00 0,166 956.2 2069 2741 30 89.593 965.6 2192 3109 37 00 9i014 975.1 2315 3479 30 8,428 984.5 2439 3850, 38 00 7,835 994.1 2564. 4224 30 7,235 111,003.7 2689 4599 39 00 6,629 013.3 2814 4976 30 6,016 023.0 2940 5354 40 00 85,396 111,032.7 6.8053067 / 6.8035734 < 30 4,770 042.4 3194 6115 < 41 00 4,137 052.2 3321 6496 30 3,498 061.9 3448 6878 42 00 2,853 071.7 3576 7262 30 2,201 081.6 3704 7646 43 00 1,543 091.4 3832 8031 30 0,879 101.3 3960 8416 . 44 00 80,208 111.1 4089 8802 30 79,532 121.0 4218 9188 45 00 78,849 111,130.9 6.8054347 6.8039574 30 8,160 140.8 4476 9960 TABLES 345 TABLE X (Con.)- LENGTHS OF ARCS OF THE PARALLEL AND THE MERIDIAN AND LOGS OF N AND R [Metric Units.] Latitude. Parallel. Value of 1. Meridian. Value of 1. LogN. LogRm. 1 Meters. Meters. 46 00 77,466 111,150.6 6.8054604 6.8040346 30 6,765 160.5 4732 0731 47 00 6,058 170.4 4861 1117 30 5,346 180.2 4989 1502 48 00 4,628 190.1 5118 1887 30 3,904 199.9 5246 2270 49 00 3,174 209.7 5373 2653 30 2,439 219.5 5500 3034 50 00 71,698 111,229.3 6.8055628 6. 80434*16 30 0,952 239.0 5754 3796 51 00 0,200 248.7 5880 4175 30 69,443 258.3 6006 4552 52 00 8,680 268.0 6131 4928 30 7,913 277.6 6256 5302 53 00 7,140 287.1 6380 5674 30 6,361 296.6 6504 6044 54 00 5,578 306.0 6627 6413 30 4,790 315.4 6749 6779 55 00 63,996 111,324.8 6.8056870 6.8047144 30 3,198 334.0 6991 7506 56 00 2,395 343.3 7111 7866 30 1,587 352.4 7230 8223 57 00 0,774 361.5 7348 8578 30 59,957 370.5 7465 8929 58 00 9,135 379.5 7582 9279 30 8,309 388.4 7697 9624 59 00 7,478 397.2 7811 9968 30 6,642 405.9 7924 6.8050307 60 00 55,802 111,414.5 6.8058037 6.8050644 30 4,958 423.1 8148 0977 61 00 4,110 431.5 8258 1307 30 . 3,257 439.9 8366 1633 62 00 2,400 448.2 8474 1956 30 1,540 456.4 8580 2274 63 00 0,675 464.4 8685 2590 30 49,806 472.4 8789 2900 64 00 8,934 480.3 8891 3208 30 8,057 488.1 8992 3510 65 00 47,177 111,495.7 6.8059092 6.8053809 30 6,294 503.3 9190 4103 66 00 5,407 510.7 9287 4393 30 4,516 518.0 9382 4678 67 00 43,622 525.3 9475 4959 30 2,724 532.3 9567 5235 68 00 1,823 539.3 9658 5506* 30 0,919 546.2 9747 5772 346 TABLES TABLE X (Cow.). LENGTHS OF ARCS OF THE PARALLEL AND THE MERIDIAN AND LOGS OF N AND R m [Metric Units.] Latitude. Parallel. Value of 1. Meridian. Value of 1.. LogN. Log R m . , Meters. Meters. 69 00 40,012 111,552.9 6.8069834 6.8056034 30 39,102 559.5 9919 6290 70 00 38,188 111,565.9 6.8060003 6.8056542 30 7,272 572.2 0085 6788 71 00 6,353 578.4 0165 7029 30 5,421 584.5 0244 7264 72 00 4,506 590.4 0321 7495 30 3,578 596.2 0396 7719 73 00 2,648 601.8 0468 7938 30 1,716 607.3 0539 8153 74 00. 0,781 612.7 0608 8361 30 29,843 617.9 0676 8563 75 00 28,903 111,622.9 6.8060742 6.8058759 30 7,961 627.8 0805 8950 76 00 7,017 632.6 0867 9135 30 6,071 637.1 0927 9314 77 00 5,123 641.6 0984 9487 30 4,172 645.9 1040 9653 78 00 3,220 650.0 1093 9814 30 2,266 653.9 1145 9968 79 00 1,311 657.8 1195 6.8060118 30 20,353 661.4 1242 0258 80 00 19,394 111,664.9 6.8061287 6.8060394 30 8,434 668.2 1330 0523 81 00 7,472 671.4 1371 '0646 30 6,509 674.4 1409 0763 82 00 5,545 677.2 1446 0873 30 4,579 679.9 1480 0976 83 00 3,612 682.4 1513 1074 30 2,644 684.7 1544 1163 84 00 1,675 686.9 1571 1248 30 10,706 688.9 1597 1325 85 00 9,735 111,690.7 6.8061620 6.8061395 30 8,764 692.3 1642 1459 86 00 7,792 693.8 1661 1517 30 6,819 695.1 1678 1567 87 00 5,846 696.2 1692 1611 30 4,872 697.2 1705 1648 88 00 3,898 697.9 1715 1679 30 2,924 698.6 1723 1702 89 00 1,949 699.0 1728 1719 30 975 699.3 1731 1729 90 00 111,699.3 6.8061733 6.8061733 TABLES 347 TABLE XL TABLE OF LOGARITHMS OF RADII OF CURVA- TURE OF THE EARTH'S SURFACE IN METERS FOR VARIOUS LATITUDES AND AZIMUTHS [Based upon Clarke's Ellipsoid of Rotation (1866) .J Azimuth. lat. llat. 2 lat. 3 lat. 4 lat. 5 lat. 6 lat. Meridian. 6.80175 6.80175 6.80175 6.80176 6.80177 6.80178 6.80180 5 177 177 178 178 179 180 182 10 184 184 184 185 186 187 188 15 195 195 195 196 197 198 199 20 209 209 210 210 211 212 214 25 227 228 228 228 229 230 232 30 248 249 249 250 250 251 252 35 272 272 272 273 273 274 276 40 296 297 297 297 298 299 300 45 322 322 322 323 324 324 325 50 348 348 348 348 349 350 351 55 373 373 373 373 374 374 375 60 396 396 396 396 397 398 398 65 417 417 417 418 418 418 419 70 435 435 436 436 436 437 437 75 450 450 450 450 451 451 452 80 461 461 461 461 462 462 463 85 468 468 468 468 468 469 469 90 470 470 470 470 471 471 472 Azimuth. 6 lat. 7 lat. 8 lat. 9 lat. 10 lat. 11 lat. 12 lat. Meridian. 6.80180 6.80181 6.80183 6.80186 6.80188 6.80191 6.80194 5 182 184 186 188 190 193 196 10 188 190 192 194 197 200 202 15 199 201 203 205 207 210 213 20 214 215 217 219 222 224 227 25 232 233 235 237 239 242 244 30 252 254 256 257 260 262 264 35 276 277 278 280 282 284 287 40 300 301 303 304 306 308 310 45 325 326 328 329 331 333 335 50 351 352 353 354 356 358 359 55 375 376 377 379 380 382 383 60 398 399 400 401 403 404 406 65 419 420 421 422 423 424 426 70 437 438 439 440 441 442 443 75 452 452 453 454 455 456 457 80 463 463 464 465 466 467 468 85 469 470 470 471 472 473 474 90 472 472 473 474 474 475 476 348 TABLES TABLE XI (Cow.). TABLE OF LOGARITHMS OF RADII OF CURVATURE OF THE EARTH'S SURFACE IN METERS FOR VARIOUS LATITUDES AND AZIMUTHS [Based upon Clarke's Ellipsoid of Rotation (1866).] Azimuth. 12 lat. 13 lat. 14 lat. 15 lat. 16 lat. 17 lat. 18 lat. o Meridian. 6.80194 6.80197 6.80201 6.80204 6.80208 6.80213 6.80217 5 196 199 203 206 210 215 219 10 202 206 209 213 217 221 225 15 213 216 219 223 227 231 235 20 227 230 233 236 240 244 248 25 244 247 250 254 257 261 265 30 264 267 270 273 276 280 284 35 287 289 292 295 298 301 305 40 310 313 315 318 321 324 327 45 335 337 339 342 344 347 350 50 359 361 364 366 368 371 373 55 383 385 387 389 391 394 396 60 406 407 409 411 413 415 417 65 426 427 429 430 432 434 436 70 443 444 446 447 449 451 453 75 457 458 460 461 463 464 466 80 468 469 470 471 473 474 476 85 474 475 476 478 479 480 482 90 476 ' 477 478 480 481 482 484 Azimuth. 18 lat. 19 lat. 20 lat. 21 lat. 22 lat. 23 Mat. 24 lat. 6 Meridian. 6.80217 6.80222 6.80226 6.80232 6.80237 6.80242 6.80248 5 219 224 228 234 239 244 250 10 225 230 234 239 244 250 255 15 235 239 244 249 254 259 264 20 248 252 257 262 266 271 277 25 265 269 273 277 282 287 292 30 284 287 292 296 300 305 309 35 305 308 312 316 320 324 329 40 327 330 334 338 341 345 350 45 350 353 357 360 364 367 371 50 373 376 379 382 386 389 392 55 396 398 401 404 407 410 413 60 417 419 422 424 427 430 432 65 436 438 440 443 445 448 450 70 453 454 456 459 461 463 465 75 466 468 470 472 473 476 478 80 476 478 479 481 483 485 487 85 482 483 485 487 489 490 492 90 484 485 487 489 490 492 494 TABLES 349 TABLE XI (Con.). TABLE OF LOGARITHMS OF RADII OF CURVATURE OF THE EARTH'S SURFACE IN METERS FOR VARIOUS LATITUDES AND AZIMUTHS [Based upon Clarke's Ellipsoid of Rotation (1866).] Azimuth. 24 lat. 25 lat. 26 lat. 27 lat. 28 lat. 29 lat. 30 lat. Meridian. 6.80248 6.80254 6.80260 6.80266 6.80272 6.80279 6.80285 5 250 256 262 268 274 280 287 10 255 261 267 273 279 285 292 15 264 270 276 282 288 294 300 20 277 282 288 293 299 305 311 25 292 297 302 308 313 319 325 30 309 314 319 324 330 335 340 35 329 333 338 343 348 353 358 40 350 354 358 362 367 372 377 45 371 375 379 383 387 391 396 50 392 396 399 403 407 411 415 55 413 416 420 423 426 430 434 60 432 435 438 442 445 448 451 65 450 453 455 458 461 464 467 70 465 468 470 473 475 478 481 75 478 480 482 484 487 489 492 80 487 489 491 493 495 498 500 85 492 494 496 498 501 503 505 90 494 496 498 500 502 504 507 Azimuth. 30 lat. 31 lat. 32 lat. 33 lat. 34 lat. 35 lat. 36 lat. Meridian. 6.80285 6.80292 6.80299 6.80306 6.80313 6.80320 6.80327 5 287 294 300 307 314 322 329 10 292 298 305 312 319 326 333 15 300 306 313 320 326 333 340 20 311 317 324 330 337 343 350 25 325 331 337 343 349 355 362 30 340 346 352 358 364 370 376 35 358 363 369 374 380 385 391 40 377 382 386 392 397 402 407 45 396 400 405 410 414 419 424 50 415 419 423 428 432 436 441 55 434 437 441 445 449 453 457 60 451 455 458 462 465 469 472 65 467 470 473 476 480 483 486 70 481 484 486 489 492 495 498 75 492 494 497 500 502 505 508 80 500 502 505 507 510 512 515 85 505 507 510 512 514 517 519 90 507 509 511 514 516 518 521 350 TABLES TABLE XI (Cow.). TABLE OF LOGARITHMS OF RADII OF CURVATURE OF THE EARTH'S SURFACE IN METERS FOR VARIOUS LATITUDES AND AZIMUTHS [Based upon Clarke's Ellipsoid of Rotation (1866).] Azimuth. 36 lat. 37 lat. 38 lat. 39 lat. 40 lat. 41 lat. 42 lat. Meridian. 6.80327 6.80335 6.80342 6.80350 6.80357 6.80365 6.80373 5 329 336 344 351 359 366 374 10 333 340 348 355 363 370 378 15 340 348 355 362 369 376 384 20 350 357 364 371 378 385 392 25 362 368 375 382 388 395 402 30 376 382 388 394 401 407 413 35 391 397 402 408 414 420 426 40 407 412 418 423 429 434 440 45 424 429 434 439 444 449 454 50 441 445 450 454 459 464 468 55 457 461 465 469 474 478 482 60 472 476 480 484 487 491 495 65 486 489 493 496 500 503 507 70 498 501 504 507 510 514 517 75 508 510 513 516 519 522 525 80 515 517 520 523 525 528 531 85 519 522 524 527 529 532 534 90 521 523 526 528 531 533 536 Azimuth. 42 lat. 43 lat. 44 lat. 45 lat. 46 lat. 47 lat. 48 lat. Meridian. 6.80373 6.80380 6.80388 6.80396 6.80404 6.80411 6.80419 5 374 382 389 397 404 412 420 10 378 385 393 400 408 415 423 15 384 391 398 406 413 420 428 20 392 399 406 413 420 427 434 25 402 408 415 422 429 436 442 30 413 420 426 433 439 446 452 35 426 432 438 444 450 456 462 40 440 446 451 457 462 468 474 45 454 459 464 470 475 480 485 50 468 473 478 482 487 492 496 55 482 486 490 495 499 503 508 60 495 499 502 506 510 514 518 65 507 510 514 517 520 524 528 70 517 520 523 526 529 532 536 75 525 528 531 534 536 539 542 80 531 534 536 539 542 544 547 85 534 537 540 542 545 548 550 90 536 538 541 544 546 549 551 TABLES 351 TABLE XI (Con.)- TABLE OF LOGARITHMS OF RADII OF CURVATURE OF THE EARTH'S SURFACE IN METERS FOR VARIOUS LATITUDES AND AZIMUTHS [Based upon Clarke's Ellipsoid of Rotation (1866).] Azimuth. 48 lat. 49 lat. 50 lat. 51 lat. 52 lat. 53 lat. 54 lat. Meridian. 6.80419 6.80426 6.80434 6.80442 6.80449 6.80457 6.80464 5 420 428 435 443 450 458 465 10 423 430 438 445 453 460 467 15 428 435 442 450 457 464 471 20 434 441 448 455 462 469 476 25 442 449 456 463 469 476 482 30 452 458 465 471 477 484 490 35 462 468 474 480 486 492 498 40 474 479 485 490 496 . 501 506 45 485 490 495 500 505 510 515 50 496 501 506 510 515 520 524 55 508 512 516 520 524 528 533 60 518 522 526 530 533 537 541 65 528 531 534 538 541 545 548 70 536 539 542 545 548 551 554 75 542 545 548 551 554 557 559 80 547 550 553 555 558 561 563 85 550 553 555 558 560 563 566 90 551 554 556 559 561 564 566 Azimuth. 54 lat. 55 lat. 56 lat. 57 lat. 58 lat. 59 lat. 60 lat. Meridian. 6.80464 6.80471 6.80479 6.80486 6.80493 6.80500 6.80506 5 465 472 479 486 493 500 07 10 467 474 481 488 495 502 09 15 471 478 485 492 498 505 11 20 476 483 489 496 502 509 15 25 482 489 495 501 508 514 20 30 490 496 502 508 514 519 25 35 498 503 509 515 520 525 31 , 40 506 512 517 522 527 532 37 45 515 520 525 530 534 539 43 50 524 528 533 537 542 546 50 55 533 537 541 545 548 552 56 60 541 544 548 552 555 558 62 65 548 551 555 558 561 564 67 70 554 557 560 563 566 569 72 75 559 562 565 568 570 573 75 80 563 566 568 571 573 576 78 85 566 568 570 573 575 578 80 90 566 569 571 574 576 578 80 352 TABLES TABLE XI (Cow.)- TABLE OF LOGARITHMS OF RADII OF CURVATURE OF THE EARTH'S SURFACE IN METERS FOR VARIOUS LATITUDES AND AZIMUTHS [Based upon Clarke's Ellipsoid of Rotation (1866).] Azimuth. 60 lat. 61 lat. 62 lat. 63 lat. 64 lat. 65 lat. 66 lat. Meridian. 6.80506 6.80513 6.80520 6.80526 6.80532 6.80538 6.80544 5 07 14 20 26 32 38 44 10 09 15 22 28 34 40 45 15 11 18 24 30 36 42 47 20 15 21 27 33 39 44 50 25 20 26 31 37 42 48 53 30 25 30 36 41 46 51 56 35 31 36 41 46 51 56 60 40 37 42 46 51 56 60 64 45 43 48 52 56 60 64 68 50 50 54 58 62 65 69 73 55 56 60 63 67 70 74 77 60 62 65 68 72 75 78 81 65 67 70 73 76 79 82 84 70 72 74 77 80 82 85 87 75 75 78 80 83 85 87 90 80 78 80 83 85 87 89 91 85 80 82 84 86 88 90 92 90 80 83 85 87 89 91 93 Azimuth. 66 lat. 67 lat. 68 lat. 69 lat. 70 lat. 71 lat. 72 lat. Meridian. 6.80544 6.80550 6.80555 6.80560 6.80565 6.80570 6.80575 5 44 50 55 61 66 70 75 10 45 51 56 62 66 71 76 15 47 53 58 63 68 72 77 20 50 55 60 65 70 74 78 25 53 58 62 67 72 76 80 30 56 61 65 70 74 78 82 35 60 64 69 73 77 81 84 40 64 68 72 76 80 83 87 45 68 72 76 79 83 86 89 50 73 76 79 83 86 89 92 55 77 80 83 86 89 91 94 60 81 84 86 89 91 94 96 65 84 87 89 92 94 96 98 70 87 90 92 94 96 98 6.80600 75 90 92 94 96 98 6.80600 01 80 91 93 95 97 99 01 02 85 92 94 96 98 6.80600 01 03 90 93 95 97 98 00 02 03 TABLES 353 TABLE XII. VALUES OF LOG m FOR COMPUTING SPHERI- CAL EXCESS. (METRIC SYSTEM.) Latitude Log m Latitude Log m Latitude Log m o / 18 oo I .40639- 10 / 33 i .40520 10 / 48 oo i .40369 - 10 18 30 636 33 3 516 48 30 364 19 oo 632 34 oo 5" 49 oo 359 19 30 629 34 3 506 49 3 354 2O 00 626 35 o 5oi 50 oo 349 20 30 623 35 30 496 5 3 344 21 OO 619 36 oo 491 51 oo 339 21 30 616 3 6 3 486 5i 3 334 22 00 612 37 oo 482 52 oo 329 22 30 608 37 3 477 52 3 324 23 oo 605 38 oo 472 53 oo 319 23 3 601 38 30 467 53 3 3H 24 oo 597 39 462 54 oo 309 24 30 594 39 3 457 54 3 304 25 oo 590 40 oo 452 55 oo 299 25 3 586 40 30 446 55 30 295 26 oo 582 41 oo 441 56 oo 290 26 30 578 4i 3 43 6 56 30 285 27 oo 573 42 oo 43 1 57 oo 280 27 3 569 42 3 426 57 30 276 28 oo 565 43 421 58 oo 271 28 30 560 43 3 416 5830 266 29 oo 556 44 oo 411 59 oo 262 29 30 SS 2 44 3 406 59 3 *57 30 oo 548 45 400 60 oo 253 30 3 544 45 3 395 60 30 249 31 oo 539 46 oo 390 61 oo 244 31 30 534 46 30 385 61 30 240 32 oo 53 47 oo 380 62 oo 235 3 2 3 i .40525 47 3 i -40375 62 30 i .40231 (The above table is computed for the Clarke spheroid of 1866.) 354 TABLES TABLE XIII. CORRECTION TO LONGITUDE FOR DIFFER- ENCE BETWEEN ARC AND SINE logs(-). log difference. logdX(+) logs(-). log difference, log d\ (+). 3 -876 o .000 oooi 2-385 4-871 o .000 0098 3-38o 4 .026 02 2-535 4.882 103 3-391 4.114 03 2 .623 4.892 1 08 3.401 4.177 04 2.686 4.903 114 3-412 4.225 5 2.734 4.913 119 3.422 4 -265 06 2-774 4.922 124 3-431 4.298 07 2.807 4.932 130 3-441 4.3 2 7 08 2.836 4.941 136 3-450 4-353 09 2.862 4-95 142 3-459 4.376 10 2.885 4-959 147 3.468 4.396 II 2-905 4.968 153 3-477 4-4I5 12 2.924 4-976 160 3-485 4-433 13 2.942 4-985 166 3-494 4-449 14 2-958 4-993 172 3-502 4-464 15 2-973 5.002 179 3-5ii 4.478 16 2.987 5 .010 186 3-5I9 4.491 17 3 .000 5-oi7 192 3-526 4.503 18 3.012 5-025 199 3-534 4.526 20 3-35 5-033 206 3-542 4.548 2 3 3.057 5.040 213 3-549 4-570 25 3.079 5-047 221 3.556 4.591 27 3 .100 5-054 228 3-563 4.612 30 3.121 5 -062 2 3 6 3-57i 4-631 33 3.140 5.068 243 3-577 4.649 36 3-158 5-075 2 5 1 3-584 4-667 39 3-I76 5 .082 2 59 3-591 4.684 42 3-193 5.088 267 3-597 4.701 45 3-210 5-095 275 3.604 4.716 48 3-225 5 .102 284 3.611 4-732 52 3.241 5.108 2 9 2 3.617 4.746 56 3-255 5 -ii4 300 3-623 4.761 59 3.270 5 -120 39 3.629 4-774 63 3.283 5 -126 318 3-635 4-788 67 3.297 5-132 327 3-641 4.801 7i 3-310 5-138 336 3-647 4-813 4-825 6 3.322 3-334 5-144 5-15 345 354 3-653 3-659 4-834 8 4 3-343 S-iS 6 364 3-665 4.849 8 9 3.358 5 .161 373 3.670 4.860 94 3-369 5-167 383 3.676 TABLES 355 TABLE XIV. LOGARITHMS OF FACTORS FOR COMPUTING GEODETIC POSITIONS Lat. Log A Log LogC Log> LogE 1 18 oo 10 8.509 5862 10 8.512 2550 10 0.91816 IO 2 .1606 20 5.7317 10 5836 2474 o .92243 2 .1641 5 -7337 20 5811 2397 o .92667 2 .1675 5 -7358 30 5785 2320 0.93088 2.1709 5-7379 40 5759 2243 o .93505 2 .1742 5-7400 50 5733 2165 o .93919 2-1775 5 -7422 19 oo 5707 2086 o -9433 2.l8o8 5-7443 10 5681 2006 0.94737 2 .1840 5 -7464 20 5 6 54 1927 o .95142 2 .1872 5 7486 30 5627 1847 o .95544 2 .1903 5 -7508 40 5600 1766 o -95943 2 .1934 5 -753 50 5573 1684 o .96339 2 .1965 57552 20 00 5546 1602 o .96733 2 .1996 5-7574 10 55i8 1519 0.97123 2 .2O26 5 .7597 20 5490 1435 0.97511 2 .2055 5 -7619 30 5462 I35i 0.97896 2 .2084 5 -7642 40 5434 1267 o .98279 2.2II3 5 -7664 50 5406 1182 o .98659 2 .2142 5-7688 21 00 5377 1096 o .99037 2 .2I7O 5-77" 10 5348 IOIO 0.99412 2 .2198 5-7734 20 S3 20 0924 0.99785 2 .2226 5-7757 30 5290 0836 1 .00156 2 .2253 5 .778o 40 5261 0748 I .00524 2 .2280 5.7804 50 5232 0660 I .00890 2 .2307 5 .7828 22 00 5202 O57r I .01253 2 .2333 5 -7851 10 5172 0481 1.01615 2 -2359 5 -7875 20 5142 0391 1 .01974 2.2385 5-7899 30 5112 0301 1 .02331 2 .2411 5 -7924 40 5082 O2 10 1 .02686 2 .2436 5 -7948 50 505 1 0118 I .03039 2 .2461 5 -7972 23 oo 5020 8.512 0026 1 .03390 2 .2485 5 -7997 10 4990 8-5H9934 1 .03739 2 .25IO 5 .8021 20 4959 9840 1 .04086 2 .2534 5.8046 30 4927 9747 I .04431 2-2557 5.8071 40 4896 9 6 53 1 .04775 2 .2581 5-8096 50 4865 9558 1 .05116 2 .2004 5.8121 24 oo 4833 9463 1 .05456 2 .2627 5 .8146 10 4801 9367 1 .05794 2 .2650 5.8172 20 4769 9271 1 .06130 2 .2672 5-8i97 30 4737 9174 1 .06464 2.2694 5 -8223 40 4704 9077 1 .06797 2 .2716 5 -8249 50 4672 8979 1 .07128 2 -2738 5 -8274 60 8.509 4639 8.5118881 I -07457 2 .2759 5-8300 356 TABLES TABLE XIV (Continued) Lat. Log A LogS LogC Log> LogE o / 25 oo 8.509 4639 8.5118881 07457 2 .2759 5 -8300 IO 4606 8783 07785 2 .2780 5 -8326 20 4573 8684 .08111 2 .2801 5 -8352 30 4540 8584 08435 2 .2822 5 -8379 40 457 - 8484 .08758 2 .2842 5 -8405 50 4473 8383 .09080 2 .2862 5 -8431 26 oo 4439 8283 .09400 2 .2882 5-8458 10 4406 8181 .09718 2 .2902 5 -8485 20 4372 8079 .10036 2 .2922 5 -8512 3<> 4337 7977 I035I 2 .2941 5 - 8 539 40 433 7874 .10666 2 .2960 5 -8566 50 4269 7771 .10979 2 .2978 5 -8593 27 oo 4234 7667 .11290 2 .2997 5 .8620 10 4200 75 6 3 .11600 2 .3015 5 .8647 20 4165 7458 .11909 2 .3033 5 -8675 30 413 7353 .12217 2 .3051 5 -8702 40 4094 7248 12523 2.3069 5 -8730 50 4059 7142 .12829 2 .3086 5.8757 28 oo 4024 7036 .I3I3 2 2 .3104 5 -8785 IO 3988 6929 13435 2.3I2I 5-8813 20 3952 6822 13737 2 -3137 5 -8841 30 39i7 6714 .14037 2.3154 5 -8870 40 3881 6607 14337 2.3170 5-8898 50 3845 6498 14635 2.3187 5 -8926 29 oo 3808 6389 14932 2 .3203 5 -8955 IO 3772 6280 .15228 2.3218 5 -8983 20 3735 6171 15522 2 .3234 5 -9012 30 3699 6061 .15816 2 .3249 5 -9041 40 3662 595 .16109 2 .3264 5.9069 50 3625 5840 .16401 2 .3279 5.9098 30 oo 3588 5729 .16692 2 .3 2 94 5-9127 10 355i 5 6l 7 .16981 2 .3309 5-9I57 20 35i4 5505 .17270 2 .3323 5 -9186 30 3476 5393 17558 2 .3337 5-9215 40 3439 5281 17845 2 -335I 5 -9245 50 34oi 5168 .18131 2 .3365 5 -9274 31 oo 3363 554 .18416 2 -3379 5 -9304 IO 3325 4941 .18700 2 .3392 5 -9334 20 3287 4827 18983 2 .3405 5 -93 6 3 30 3249 47i3 .19266 2 .3418 5 -9393 40 3211 459 s .19548 2 -343 1 5 -9423 50 3173 4483 .19828 2 -3444 5 -9453 60 8.5 9 3134 8.5114368 . 20108 2 .3456 5 -9484 TABLES 357 TABLE XIV (Continued) Lat. LogA LogB LogC LogD LogE o / 32 oo 8.5093134 8.5114368 .20108 2 .3456 5 -9484 10 3096 4252 .20387 2.3469 5 -95I4 20 3057 4i3 6 .20666 2 .3481 5-9544 30 3018 4020 .20944 2 -3493 5 -9575 40 2980 3903 .21220 2 .3504 5-9605 50 2940 3786 .21496 2 .3516 5 -9636 33 oo 2901 3669 .21772 2 -3527 5 -9667 10 2862 3551 .22047 2 -3539 5.9698 20 2823 3433 .22321 2 .3550 5 -9729 30 2784 3315 22594 2 .3561 5.9760 40 2744 3*97 .22866 2 -357I 5 -9791 SO - 2704 3078 23138 2 .3582 5 -9822 34 oo 2665 2959 .23409 2 .3592 5 -9853 IO 2625 2840 .23680 2.3602 5-9885 20 2585 2720 2395 2 .3612 5-99i6 30 2545 2600 .24219 2 .3622 5 -9948 40 255 2480 .24488 2 .3632 5.9980 50 2465 2360 .2475 6 2 .3642 6.0011 35 oo 2425 2239 .25024 2 -3651 6.0043 IO 2384 2118 .25291 2.3660 6.0075 20 2344 1997 25557 2.3669 6 .0107 30 2304 i87S 25823 2 .3678 6 .0140 40 2263 J754 .26088 2.3687 6.0172 50 2222 1632 .26353 2.3695 6 .0204 36 oo 2l82 1510 .26617 2 .3704 6 .0237 10 2141 1387 .26881 2 .37" 6.0269 20 2IOO 1265 27145 2 .3720 6 .0302 30 2059 1142 .27407 2 .3728 6 .0334 4 20l8 1019 .27670 2 -3735 6 .0367 50 1977 0895 27932 2 -3743 6.0400 37 oo IO 1936 1895 0772 0648 .28193 .28454 2 -375 2 .3758 6.0433 6 .0466 20 1853 0524 .28715 2 .3765 6 .0499 30 1812 0400 28975 2 .3772 6 .0533 40 1771 0276 .29234 2 -3779 6.0566 50 1729 0151 .29494 2 .3785 6.0600 38 oo 1687 5.511 0027 29753 2 .3792 6.0633 IO 1646 8.5109902 .30011 2 .3798 6.0667 20 1604 9777 .30269 2.3804 6 .0701 30 1562 9652 30527 2 .3810 6.0734 40 1521 9526 30785 2 .3816 6.0768 50 1479 9401 .31042 2 .3822 6.0802 60 8.509 1437 8.5109275 .31299 2.3827 6.0836 358 TABLES TABLE XIV (Continued) Lat. Log A LogS LogC LogD Log o / 39 oo 8.509 U37 8.5109275 i .31299 2 .3827 6 .0836 10 1395 9149 i ^1555 2 .3832 6 .0871 20 1353 9023 i .31811 2 -3838 6.0905 3<> 1311 8897 i .32067 2 .3843 6 .0939 40 1269 8771 i -32323 2 .3848 6 .0974 50 1227 8644 i .32578 2 .3852 6.1008 40 oo 1184 8517 i -32833 2 .3857 6.1043 IO 1142 8391 i .33088 2 .3861 6.1078 20 1 100 8264 i .33342 2 .3866 6.1113 30 1057 8i37 i -33596 2 .3870 6.1148 40 1015 8010 i -33850 2 .3874 6.1183 50 0973 7883 i .34104 2 .3878 6.1218 41 oo 0930 7755 i -34358 2 .3882 6.1253 IO 0888 7628 i .34611 2 .3885 6.1289 20 0845 7500 i .34864 2 -3889 6.1324 30 0803 7373 L35II7 2 .3892 6.1360 40 0760 7 2 45 i -35370 2 .3895 6.1395 50 0718 7117 i -35623 2 .3898 6.1431 42 oo 0675 6989 i .35875 2 .3901 6.1467 IO 0632 6861 i .36127 2 .3903 6.1503 .20 0590 6733 i -36379 2 .3906 6.1539 30 0547 6605 i .3663! 2 .3908 6.1575 40 0504 6477 i .36883 2 .3910 6 .1612 50 0461 6348 i .37135 2 .3913 6.1648 43 oo 0419 6220 i .37386 2 .3914 6.1684 10 0376 6092 i .37638 2 .3916 6.1721 20 333 59 6 3 i .37889 2 .3918 6.1758 30 0290 5835 i .38141 2 -39*9 6.1795 40 0247 5706 i .38392 2 -392I 6.1831 50 0204 5578 i -38643 2 .3922 6.1868 44 oo 0162 5449 i .38894 2 .3923 6.1905 10 0119 53 2 o i .39145 2 .3924 6.1943 20 0076 5*92 i .39396 2 .3925 6.1980 30 5090033 5 6 3 i .39648 2 ^925 6 .2017 40 .5089990 4935 i .39898 2 .3926 6 .2055 5o 9947 4806 i .40149 2 .3926 6 .2092 45 oo 9904 4677 1 .40400 2 .3926 6 .2130 10 9861 4548 i .40651 2 .3926 6.2168 20 9818 .4420 i .40902 2 .3926 6 .2206 30 9776 4291 I-4II53 2 .3926 6 .2244 40 9733 4162 i .41404 2 .3925 6.2283 So 9689 4034 i -41655 2 .39 2 5 6.2321 60 .508 9647 5103905 i .41906 2 .3924 6.2359 TABLES 359 TABLE XIV (Continued) Lat. Log A Log B LogC LogD LogE o / 46 oo 8.5089647 8.5I039 5 .41906 2 .3924 6 .2359 10 9604 377J 42157 2 .3923 6 .2398 20 95 6 I 3648 .42409 2 .3922 6 .2436 30 95i8 3519 .42660 2 .3921 6 .2475 40 9475 3391 .42911 2 .3920 6.2514 50 9433 3262 .43I 6 3 2 .3918 6-2553 47 oo 939 3134 43414 2 -39l7 6.2592 10 9347 35 .43666 2 .3915 6 .2632 20 934 2877 43917 2 -3913 6.2671 30 9261 2749 .44169 2.3911 6.2710 40 . 9 2 i9 2621 .44421 2.3909 6.2750 50 9176 2493 .44673 2 .3906 6 .2790 48 oo 9133 2364 .44926 2 .3904 6.2830 10 9091 2236 45!7 8 2 .3901 6.2870 20 9048 2108 45431 2 -3898 6.2910 30 9005 1981 45683 2 .3895 6 .2950 40 8963 1853 45937 2.3892 6.2990 50 8920 I7 2 5 .46190 2-3889 6 .3031 49 oo 8878 1598 .46443 2 .3886 6.3071 10 8835 1470 .46696 2. 3 882 6.3112 20 8793 1343 4695 2 .3878 6.3I53 30 875 1216 .47204 2 .3875 6 .3194 40 8708 1088 47459 2 .3871 6-3235 50 8666 0962 47713 2 .3866 6.3276 50 00 8623 0835 .47968 2 .3862 6.3318 IO 8581 0708 48223 2 -3858 6 -3359 20 8539 0581 .48478 2 ^853 6 .3401 30 8497 0455 .48734 2 -3848 6 -3443 40 8455 0328 .48989 2 .3843 6.3485 50 8413 O202 .49246 2 -3838 6-3527 51 oo 837i 5.5100076 .495 02 2 .3833 6.3569 IO 8329 8.5099950 49759 2 .3828 6.3612 20 8287 9825 .50016 2 .3822 6 .3654 30 8245 9699 50273 2.3817 6.3697 40 8203 9574 50531 2.3811 6.3740 50 8161 9448 .50789 2 -3805 6.3782 52 oo 8120 ' 93 2 3 .51048 2 -3799 6 .3826 10 8078 9198 51307 2 .3792 6.3869 20 8036 9074 .51566 2.3786 6.3912 30 7995 8949 .51826 2 .3779 6 .3956 4 7953 8825 .52086 2 -3773 6.4000 50 7912 8701 52347 2 .3766 6 .4043 53 oo 7871 8577 .52608 2 -3759 6.4088 IO 7829 8453 .52869 2 -375I 6.4132 20 7788 8329 53i3i 2-3744 6.4176 30 7747 8206 53393 2 .3736 6.4221 4 7706 8083 53656 2 .3729 6 .4265 50 7665 7960 53919 2.3721 6.4310 60 8.508 7624 8.509 7838 1-54183 2.3713 6-4355 TABLES -o o M t>. Tf oo oo co oo oo O ONOO CO 00 to t^. O CONO t^ t^oo oo oo 00 00 00 00 00 O\ O - cotr> f ON ON ON O Tf Tf Tf IO oo oo oo oo tOOv N tOOO to to to to to oo oo oo oo oo M Tf t>. ON V> CM tOOO M IOOO CM CM CM CO CO CO to to to to to to oo oo oo oo oo oo Tf t^. O O ON Tf O I s * Tj- M CO NO to CO NO t^ t^ t^- t^ ON PI T}- t^. ON t^OO 00 00 00 ON ON ON ON O 3-0^2? 00 00 00 00 00 t^ O CM toco O IH CM CM CM CM CO 00 00 00 00 00 CO O 00 t^ to to Tj-Tj-tONO t- 00 O W Tj- t>. ON N tOOO M Tf t^ O *5NO ON 1 ** CO Tf NO 00 O CM CM CM CM CO (NJ Tj-NO 00 O co co co co T}- ON ON ON ON ON CS (S CS CS CN (N to t^ ON M IO IO IONO NO ON ON ON ON ON IO t^ O O CONO O co co co co co CM CM CM CM CM ON O\ ON ON ON t^OO O M co CO CO Tf Tf Tf CN) 0 10 to NO t^-00 ON O Tf Tf Tf Tf IO to to 10 10 10 to to to to >o M CM CO Tf 10 to to to to to IO IO IO VO to \o to to to to VO t^OO ON O w IO to tO IONO NO to IO 0 10 VO 10 10 10 coco coco T}- O to H ^ co ON to CM CO to M t>. Tf M t^ Tf O t^ co O * CO co co co co tO tONO NO *-. NO NO NO NO NO co co co co co t^OO ON ON O ON ON ON ON O NO NO NO NO t^- co co CO co co M M M CO CO CO co co co co co H Tf t>. ON CM IOOO M rj- t-* O CONO ON M 10 ON CM to ON CM IOOO CM tOOO H 00 00 00 00 00 oo oo co oo oo oo"co"oo'oo"co > to to to to to oo oo oo oo co to to to to to to oo oo oo oo oo oo 3 tnvo t^oo os