LIBRARY UNIVERSITY OF CALIFORNIA. Deceived .....MAY 5 1894 , i8g -^ ^ ^Accessions No*5Z~. Class No. /">"* V CARL FRIEDRICH GAUSS. L- V -B-r* . . eoh, Bng. MAGNETIC MEASUREMENTS, WITH AN APPENDIX ON THE METHOD OF LEAST SQUARES. FRANCIS E. NIPIIER, A.M., M Professor of Physics in Washington University, President of the St. Louis Academy of Science. 'UITI7BRSIT7' D. VAN NOSTRAND, PUBLISHER, 23 MURRAY AND 27 WARREN STREETS. 1886. Library COPYRIGHT, 1886, BY D. VAN NOSTRAND. H. .-', HEWITT, PRINTER AND ILECTROTYPEB, 87 ROSE STREET, N. Y. Dept. Mech. Eng. PREFACE. DURING the last four or five years the writer has frequently been requested to furnish information re- lating to the practical details of a magnetic survey. The need of a brief hand-book to supplement the in- structions of the Coast and Geodetic Survey was felt by the writer while prepariug-rfor a similar survey of Missouri, and this little volume is offered to the pujb- lic in the hope that it may be of service to others contemplating similar work. At the same time the great and growing impor- tance of electrical and magnetic measurements will perhaps commend the volume to a wider circle of readers. The discussion of the method of Least Squares is an extension of an article in Weisbach's "Mechanics." F. E. N. ST. Louis, May 24, 1886. Bept Mecli Bng 1 . CONTENTS. PAE Introduction, 7 Declination, .......... 11 Determination of Scale Value of the Magnet, . - . . 13 Determination of Scale Value of Magnet C t . . 14 Determination of the Magnetic Axis, . . . . * 14 Magnetic Axis of (7 6 , . . . . . ; 15 Magnetic Declination at Jefferson City, Mo., .,- . 18 Inclination, . . . . . . . . . . 21 Magnetic Inclination at Jefferson City, Mo., . . 23 Pendulum Vibrations, . . . ..... ^ .\ 25 Moment of Inertia, . . . . . " ., . 33 Moment of Inertia of Ring Y, . . . ' . ; 36 Moment of Inertia of the Magnet, . . . 37 Computation for Moment of Inertia of Magnet (7e, . 40 Moment of Inertia of Magnet (7 6 , .... 41 Correction of the Oscillation Series for Torsion, . , 42 Temperature Correction for Magnetic Moment, . . * 44 Horizontal Intensity Oscillations, .... 46 Horizontal Intensity Computation, ... 47 Reduction of the Time of Oscillation to that of an Infinitely Small Arc, 48 Deflection Series for Intensity, ...... 51 Horizontal Intensity Deflections, .... 56 Determination of the Temperature Coefficient q, . . 57 Systems of Units, 65 Explanation of the Plates, 69 C CONTENTS. PAGK Appendix on the Method of Least Squares, ... 73 Properties of the Arithmetical Mean, ... 74 Observations on Two or More Quantities, ... 77 Weighted Observations, . . . 82 Graphical Methods, 85 Time of Elongation of Polaris, 90 Table I., 92 Tables II., III., 93 Azimuth of Polaris ut Elongation, . . . . 94 0? INTROTTOtlON. A FORCE is completely known when its direction and intensity have been determined. The direction of the lines of force in the earth's gra- vitation field is indicated by the plumb-line. The plumb-bob tends to move downwards along the line of force. The direction of the lines of force of the earth's magnetic field would be indicated by the direction taken by the magnetic axis of a magnetic needle sus- pended at its centre of gravity, so that it could move freely in any direction. When thus placed the opposite ends of the magnet-needle tend to move in opposite directions along the line of force. The magnetic axis of a needle is a line passing through its poles, as will be explained more fully later. The position of the lines of force is referred to two planes viz., the horizontal plane and the plane of the geographical meridian. The position of the lines of force is really determined by means of two magnet-needles, one of which is free to move in a horizontal plane, and the other in a vertical plane, since it is impossible mechani- cally to combine these motions in a single needle of suf- ficient delicacy. The first needle determines, at the point of observa- tion, the angle between the plane of the geographical meridian and the vertical plane containing the line of force at the point. This angle is called the declination. The second needle determines the angle between the 8 THEORY OF MAGNETIC MEASUREMENTS. line of force and a horizontal plane. This angle is called the inclination or dip. It remains to explain the method used in determining the intensity of force. In earlier times the weight of a unit mass was taken as the unit of force. The weight of a pound is not the same for different places on the earth, and hence this unit can only be used for rough work or for local deter- minations. According to Newton's law for attraction, the force with which either of two masses m and M attract the other, the distance between them being d, is a) u This equation is verified by the motions of the planets and of falling bodies. If M represents the mass pf the earth in pounds or grammes, m being the mass of a body at the surface, d being the radius of the earth, in feet or centimetres, then w represents the force (measured in units which are not yet supposed to be fixed) by which a spring separating the two bodies would be compressed. This force is usually called the weight of the mass m. It is equally the weight of the earth. The weight of the earth is always equal to that of the body weighed, and is therefore an indeterminate quantity. ' represents the force with which a unit mass would attract another unit mas*, the distance between them being unity. By experiments with Att wood's machine it is shown that if a force acts upon 'm units of mass, imparting an acceleration a, the force is represented by the expression, F=Kma, (2) where TTis the force which would impart a unit accele- ration to a unit mass, F being determined in any units. THEORY OF MAGNETIC MEASUREMENTS. 9 If the body m is free to move under the attraction of the earth, it receives an acceleration g. Hence the at- traction of the earth for the m units of mass is w = K mg. (3) The earth, having a mass M, likewise moves with an acceleration a' towards the body m, the force acting on the earth being therefore K Ma = w= Kmg, so that , m The value ^ is so small that a' is inappreciable. In equation (1) it is evident that the value of K' might be taken as the unit of force. The value K in (2) and (3) would not then be unity. This is not done, however, but the unit of force is so chosen that JTin (2) and (3) is unity. The unit force is then that force which can impart a unit acceleration to a unit mass (gramme or pound). In centimetre-gramme second (abbreviated C. Gr. S.) units this unit of force is called the dyne. Equation (3) then becomes / A \ where w is the weight of m units of mass expressed in the above-chosen unit. It also follows that w -* The left side of this expression is the weight of a unit mass. In English units the weight of a pound at Lon- don is 32.1912; at any other place the weight of a pound is g units of force, where g is the acceleration of 10 TflllOKY Oi 1 MAGNETIC MEASUREMENTS. a falling body at the place. In C. G. S. units the weight of a gramme at Paris is 980.94 dynes, and at any point it is g dynes. The weight of - units of mass is there- ^7 fore the unit force. The weight of a gramme at any point in the earth's field gives a measure of the force which acts upon the gramme at that point, tending to cause motion along the lines of force ; in other words, the value of g at any point is a measure of the strength of the earth's gravita- tion field at that point. Adopting the above unit of force, the value of K' in (1) may be calculated as follows in C. Gr. S. units : Con- sider the case of a gramme at the surface of the earth, M representing the mass of the earth in grammes, and d its mean radius in centimetres. Then in (1) m = l; Jf==6.14 x 10 27 ; rf=6.37 X 10 8 : w=981. Hence 981 X (6.37 X 10 8 ) 2 _ 1 6.14 x 10 27 "1.543 x 10 7 ' From this the mass which must be placed at any point in order to attract an equal mass with a force of one dyne can be computed, since rr, mm =A/ -TT, = 3928 grammes. Hence 3928 grammes would attract an equal mass at a distance of one centimetre, with a force which would impart to one gramme an acceleration of a centimetre per second, or to 3928 grammes an acceleration of cm. per sec. The 3928 grammes is called the as- 'THEORY OF MAGNETIC MEASUREMENTS. 11 tronomical unit of mass. If this is taken as the unit mass the astronomical equation of attraction becomes mM , w = -^-> ( 5 ) w being given in the ordinary unit of force the dyne, or some equivalent unit. The unit magnetic pole or the electro-static unit quantity of electricity is denned in accordance with this equation. The unit pole is that pole which will act upon an equal pole at a unit distance with a unit force. Dept. Mecli Bug-. DECLINATION. The magnetic needle commonly used for precise de- terminations is of the collimator form, consisting of a small, cylindrical shell of steel, one end of which is closed by a lens, the principal focus of which is on a scale, etched or photographed on a glass plate, which closes the other end. This scale should be divided de- cimally into about 100 parts, numbered continuously from one extremity to the other. The angle subtended at the middle of the magnet by one scale division should be between one and three minutes. This angle is called the scale value of the magnet. The magnet hangs in a stirrup, supported on a long fibre of raw silk, its position in the stirrup being fixed by small brass guide-rings around the magnet. The magnet is enclosed in a box,, from the top of which a glass tube extends upward, its top terminating in a graduated torsion-head to which the suspension-fibre is attached. The ends of the box are provided with 12 THEORY OF MAGNETIC MEASUREMENTS. windows, one admitting light from a mirror upon the scale, and the other for telescopic observation. It is usually better to use well-seasoned wood in the con- struction of the box, and to avoid the use of metal in all parts nearest to the magnet, as it is almost impossible to obtain brass free from iron. It is necessary to ex- amine all brass screws or fittings in a declinometer, if they are near the magnet, and, if found to be magnetic, they should be replaced by others, or proper correction made for their effect on the position of the needle and on the intensity of the field. This correction, although really a function of the strength of the field, may be taken as constant. In some forms of instrument the observing telescope is connected with the magnet-box, and mounted with it on the same azimuth circle, the centre of which is be- low the point of suspension of the magnet and in the same vertical. In others it consists of a transit, or alt- azimuth instrument, mounted on the fixed support which carries the magnet-box. In the former case a change in the pointing of the telescope introduces a torsion in the fibre. The latter instrument, which is due to Gauss, is preferable for field-work.] THEOKY OF MAGNETIC MEASUREMENTS. 13 DETERMINATION OF SCALE VALUE OF THE MAGNET. Let the telescope be focussed upon the centre of the magnet scale, in which case we may assume, in order to fix our ideas, that the optical axes of the telescope and the collimator magnet coincide. If the magnet be turned on its suspension-fibre through any angle, and the telescope be turned on its vertical axis through the same angle and in the same direction, the optical axes of the magnet and telescope will again be parallel. If the two instruments turned about a common vertical axis the optical axes will also be coincident ; but if they turn around parallel axes the optical axes will be paral- lel and not coincident in the second position. In both cases, however, the scale-reading in the first and second positions will be the same. This preservation of an un- changed scale-reading in the case mentioned is also true if the lines of collimation of the magnet and telescope are not parallel. The value of one division of the scale may therefore be determined by pointing the telescope successively on the principal divisions of the scale, taking the readings of the azimuth circle for each point- ing. If the magnet-box moves with the telescope the magnet must hang on its suspension-fibre, and the readings must be corrected for torsion in a manner to be hereafter explained. If the magnet-box does not move the magnet may be fastened in its normal posi- tion during the operation. If the number of pointings is odd, the circle-readings corresponding to divisions equidistant from the middle of the scale are reduced to the mean division by finding their means, as is shown in the third column of the table below. The mean reading of the middle division is here 338 53'. 4. 14 THEOKY OF MAGNETIC MEASUREMENTS. DETERMINATION OF SCALE VALUE OF MAGNET C 6 . Azimuth circle. Reading of DIFFERENCE F ROM MEAN. Scale. Mean middle of Verniers. division. Circle. Scale. 160 335 44'. 7 188.7 80 150 336 09.0 164.4 70 140 336 34.0 139.4 60 130 336 57.5 115.9 50 120 337 19.5 93.9 40 110 337 43.0 70.4 30 100 338 06.0 47.4 20 90 338 29.5 23.9 10 80 338 52.5 338 52*'. 5 00.9 00 70 339 16.5 53.0 23.1 10 60 339 39.5 52.7 46.1 20 50 340 05.0 54.0 71.6 30 40 340 27.5 53.5 94.1 40 30 340 52.7 55.1 119.3 50 20 341 15.6 54.8 142.2 60 10 341 37.5 53.3 164.1 70 342 00.0 52.3 186.6 80 338 53'.4 Sum 1692.0 Sum 720 The scale value is therefore DBTBEMINATION OF THE MAGNETIC AXIS. The magnetic axis is a straight line joining the poles of the magnet. If the magnet is freely suspended the axis lies in the line of force. The magnetic axis is de- termined by taking scale-readings with the scale alter- nately erect and inverted. If the line of collimation of the telescope should happen to coincide with the mag- netic axis of the magnet, it would then be pointed in the plane of the magnetic meridian, and the reading of THEORY OF MAGNETIC MEASUREMENTS. 15 the erect and the inverted scale would be identical. The division-lines of the scale should always be accurately vertical when read. The position of the magnetic axis varies continually by the jarring incident to travel. A freshly-magnetized magnet is especially unstable, chang- ing even with variations of temperature. Declination magnets should be carried south end up, and should be kept away from other magnets. The axis should be de- termined at each station, or as often as experience shows to be necessary. This will, of course, depend upon the age of the magnet, the hardness of the steel, and the kind of treatment the magnet receives. The custom which some county surveyors have of remagnet- izing their compass-needles whenever they are dissatis- fied with their instrument is not a wise custom. MAGNETIC AXIS OF <7 6 . Magnet. SCALE. Mean. Alternate means, 1 and 3, 2 and 4, etc. Axis reads. Left. Bight. E I E I E I E 74.6 79.9 62.8 74.9 64.6 70.4 65.0 75.1 82.3 87.7 87.0 86.1 91.0 86.5 74.8 81.1 75.2 80.9 75.3 80.7 75.7 7s!6o 81.00 75.25 80.80 75.50 78.05 78.10 78.07 78.05 78.10 78.07 When the telescope is pointed on the division 78.07 of the scale its line of collimation is in the plane of the magnetic meridian. The left and right scale-read- ings in columns 2 and 3 are the extreme readings of the scale during an oscillation, it being assumed that the amplitude does not diminish. This is sufficiently pre- cise for heavy needles or small amplitudes. 16 THEOEY OF MAGNETIC MEASUREMENTS. The determination of magnetic declination involves a determination of the direction of a true north and south line, as shown by the azimuth circle, and the mean position of the magnetic axis of the needle for the day, or for a series of days, as read on the same circle. The daily swing of the needle in summer is, on the average, about 15', the north end of the needle being at its greatest eastern elongation at about 7.15 to 7.30 o'clock A.M. on normal summer days, and at its western elongation at 1.15 to 2 o'clock P.M. These hours vary somewhat with the season of the year and for different parts of the country. The mean position of the needle, as deduced from hourly observations throughout the day, is, on the average, within half a minute of the mean of eastern and western elongations. The mean for successive days frequently varies by five minutes, even in times of minimum magnetic disturbance. This mean position might be obtained by pointing on the axis reading of the magnet at the two elongations, and taking the mean of the azimuth circle-readings. It is better to point approximately on the axis at about 6.30 A.M., and, clamping the circle, to take readings of the magnet-scale at intervals of ten or fifteen minutes until after elongation has passed. Leaving the circle unchanged, if possible, a similar set of scale-readings should be taken, so as to include the afternoon elon- gation. These observations, then, show any abnormal changes in declination. The mean scale-reading of the two elongations is then found ; if it should happen to coincide with the magnetic axis the telescope would then be pointed in the mean magnetic meridian for the day. Should it not thus coincide the circle-reading must then be corrected by the small angle over which the telescope would sweep in turning from the mean THEORY OF MAGNETIC MEASUREMENTS. 17 scale-reading of the elongations to the magnetic axis. The sign of this correction will depend upon whether the scale is erect or inverted, whether it reads from left to right or the reverse when the scale is erect, and whether the telescope shows an erect or an inverted image. The silk fihre should be examined both before and after ach set of observations, in order to detect any torsion that may develop. Changes in atmospheric humidity are likely to develop torsion in a fibre, par- ticularly if it be a new one. The fibre should be no larger than is necessary to sustain the magnet without too frequent breaking. To examine for torsion, the magnet is removed from the stirrup and a brass cylin- der of the same weight substituted. The torsion-head should then be turned until this cylinder sets parallel to the magnet-box. Before the magnetic observations are begun the ver- niers of the transit should be set to and 180, and the instrument pointed on some well-defined object near the horizon. The lower clamp being secured, the upper one is released and the magnetic observations begun as explained. The mark should be far enough away so that it will be unnecessary to refocus when the telescope is pointed on the scale. In choosing a mark it is well to remember that objects easily visible in the evening may be invisible in the morning by reason of fogs or changes in shadows. The station should always be described by aid of sketches, and the distance and bearing of corner-stones or other available points of re- ference recorded. The following table shows the meth- od of recording the observations. Specimen blanks for magnetic observations can be obtained by observers on addressing the Superintendent of the U. S. Coast and Geodetic Survey, Washington, D. C. 1* 18 THEORY OF MAGNETIC MEASUREMENTS. MAGNETIC DECLINATION AT JEFFERSON CITY, MO. In orchard of Phil. E. Chappell. Mark spire of State House, about one mile distant. Date, Aug. 12, 1879. Instrument, Declinometer No. 3, U. S. C. and G. Survey. Magnet No. 1, scale erect. Scale value, 1'. 90. Mark reads, A, 18000'.0; B, 35958'.o at 6 A.M. Line of detorsion, 15. Azimuth circle set to A 363 56'. ; B 183 55'.0. Observer, F. E. N. Time. SCALE-BEADING. Mean. Remarks. Left. Right. A.M. 7 h 18 m 81 d .l 8 d .05 83.05 Removed torsion weight at 7 h 02 m A.M. 7 23 83.2 7 37 88 '.0 83 '.6 83.3 7 55 82 .4 84 .1 83.3 8 17 82 .9 83 .9 83.4 Max. East. 8 30 82 .5 83 .0 82.75 8 40 , 83.0 9 25 81 .9 82 .9 82.4 Removed magnet. Line of detorsion unchang- ed. Replaced magnet. P.M. Ih 15 m 78 d .2 79 d .O 78.6 Line of detorsion same. Azimuth circle not ? changed. 1 25 78 .3 78 .9 78.6 1 40 78 .2 78 .9 78.55 1 52 . . . . . 78.5 Max. West. 2 07 78 .2 79 .2 78.7 2 16 78.8 2 30 78 '.7 79 .1 78.9 Line of detorsion, 15. Mark reads, A 180 Ol'.O ; B 360 OO'.O. THEORY OF MAGNETIC MEASUREMENTS. 19 Mean reading E. and W. elongations. . 80 d . 9 Axis of magnet reads 80 .8 Reduction to axis -f 0.1 = + 0'.2 Azimuth circle reads. . A 363 55'. Magnetic south reads 363 55 .2 Mark reads 180 00'. Azimuth of mark. , S 175 28 .1 E True south reads 355 28.1 Magnetic declination E. of N 8 27'. 1 In illuminating the magnet- scale the direct solar ray should never, under any circumstances, be allowed to enter the magnet-box. An illumination from a white cloud or an illuminated sheet of paper is effective. In field-work, if a tent is not available, the whole instru- ment should be covered with a soft, heavj 7 cloth, and the tripod should similarly be protected against solar radia- tion. No vibrations of the magnet should be allowed, excepting small vibrations about a vertical axis. Larger vibrations may be checked by the end of the finger ; , but it is better to use a camel's-hair brush operated from the outside by means of a lever or spring, which must be so arranged that no air-currents are introduced into the magnet-box. In field-work the tripod must always be mounted firmly on stakes. Unless the observations of a survey are made during intervals of magnetic calms, it is necessary to establish a base station, where all ob- servations of declination as well as intensity are observed at the same time as at the field station. The methods for determining the true meridian are easily accessible, and it is not thought necessary to treat this part of the subject. For the work of a survey a precision of one minute of arc is sufficient. Star obser- 20 THEORY OF MAGNETIC MEASUREMENTS. vations are the most satisfactory. The method of equal altitudes requires more time than it is sometimes conve- nient to devote to it. The best method of determining the meridian is by observation of a circumpolar star on elongation. Table L, at the end of this volume, gives the time of occurrence of the elongation of the pole-star, correct within five minutes, for the years between 1885 and 1895. The time is local astronomical time. Table IV., at the close of this volume, gives the azimuth of Polaris at elongation (counted from the north) for the years 1885 to 1895, inclusive, and is accurate enough for all ordinary purposes. It should be observed, however, that in computing this table the mean declination of Polaris for the beginning of the year is necessarily used. To obtain the azimuth to the nearest tenth of a minute the apparent declina- tion for the date of observation should be employed, and the azimuth computed from the folio wing formula : costf sm A = - (6) where A = azimuth, $ = declination, and cp = latitude. The apparent place of Polaris is given in the American Ephemeris for every day in the year. The difference in the mean and apparent places may produce a difference of O'.o in the computed azimuth, and values of the azi- muth taken from the table are therefore subject to an error of that amount. The precision with which the true meridian must be determined depends upon the precision with which the magnetic meridian is determined or, in other words, upon the number of days of observation. In order to illuminate the field of the instrument THEORY OF MAGNETIC MEASUREMENTS. 21 where there is no axial illumination, a minute mirror may be mounted in front of the object-glass of the telescope. This mirror may be mounted on a ring clasping the tube. The position of the mirror can be made adjustable by a double-jointed rod attached to the ring. A bull's-eye lantern will then serve to throw light upon the mirror. It is, of course, always understood that the observing telescope is always kept in adjustment, the levels and line of collimation being carefully examined. For more care- ful determinations the telescope should be reversed ; but in the field-work of a survey this is usually unnecessary if the instrument is kept in adjustment. INCLINATION. The inclination or dip is the angle between the line of force and a horizontal line lying in the magnetic meridian. A needle accurately balanced on a horizon- tal axis directed in the magnetic prime-vertical, so that the needle moves freely in the plane of the magnetic meridian, will, when magnetized, set with its magnetic axis in the line of force at the point. The position of the needle is determined by means of a graduated circle, having its plane coincident with that of the magnetic meridian. The zero of graduation is a horizontal dia- meter. In some instruments the ends of the needle point to the scale divisions, which are read by means of magnifying-glasses. In such instruments the gradua- tion is not closer than to ten minutes. In other instru- ments the observation is made by means of compound microscopes having radially-placed threads, which are set on the marked ends of the needle, the position being determined by means of verniers. The circle of the instrument is placed in the mag- 22 THEORY OF MAGNETIC MEASUREMENTS. netic meridian by means of a long, horizontal compass- needle set parallel to some mark drawn on the box. It may also be done by taking four readings of the azimuth circle of the instrument, with the inclination-needle adjusted to a dip of 90, as follows : 1st, with the circle facing south (magnetic) and needle (marked side) facing south ; 2d, circle south, needle north ; 3d, circle north, needle south ; 4th, circle north, needle north. The mean azimuth circle reading for these four positions, 90, gives the circle reading for the true meridian. The line upon which the setting compass is adjusted may be drawn on the box after the meridian has been thus de- termined by the second method, the dip-needle having, of course, been first removed. The results of the two methods should be occasionally compared. The vertical circle being thus set in the magnetic meridian, with its divided side east, and with the marked side of the needle (face) east, two or three readings of the two ends of the needle are taken, the needle being slightly lifted so that the readings are independent. The needle is then reversed in its bearings so that it faces west, and readings are again taken as before. The circle is then reversed, so that the position is then " circle west face east." After taking readings in this position the needle is again reversed in its bearings, the position being "circle west face west." The polarity of the needle is then reversed, and the above readings are again taken, beginning with the last posi- tion. The table which follows will show the method of recording and reducing in order that the instrumental errors may be investigated. It will be observed that, in the case shown in the table, the centre of gravity of the needle is not in the axis of the needle, but slightly dis- placed towards the marked end. The errors of eccen- THEORY OF MAGNETIC MEASUREMENTS. tricity are corrected by the reversals described, but they should always be reduced to a minimum. MAGNETIC INCLINATION AT JEFFERSON CITY, MO. Orchard of Phil. E. Chappell. Aug. 12, 1879. Instrument used, Barrow No. 9. Needle No. 3. Observer, F. E. N. POLARITY OF MARKED END SOUTH. CIBCLE EAST. CIRCLE WEST. Face east. Face west. Face east. Face west. S. N. S. N. S. N. S. N. 65 56' 42 40 65 46',0 66 01' 65 46 47 71 02' 70 56 71 00 71 00' 70 57 71 00 65 36' 33 35 65 40' 40 41 70 50' 71 03 03 70 48' 71 00 71 00 65 51'.3 70 59 .3 70 59'.0 65 34'.6 65 40 / .3 70 58 .6 70 56'.0 65 48'. 6 70 59'.2 65 37'.4 70 57'.3 68 23'.9 68 17'.4 68 20'.6 POLARITY OF MARKED END NORTH. CIRCLE WEST. CIRCLE EAST. Face west. Face east. Face west. Face east. S. N. S. N. S. N. S. N. 66 56' 52 47 67 02' 66 59 53 72 22' 12 15 72 22' 13 17 66 51' 52 54 66 58' 59 59 72 23' 20 22 72 24' 22 22 66 51'.6 66 58'.0 72 16'.3 72 17'.3 66 52'.3 66 58'.6 72 21'.6 72 22'.6 66 54'.8 72 16'. 8 66 55'. 4 72 22'. 1 69 35'.8 69 38'.7 69 37'.3 Resulting Inclination, 68 58'.9 Time of beginning, 11^ 15 A.M. Time of ending, lli> 45" A.M. Magnetic meridian reads 16 24', set by compass. Left series, 68 59'.8. Right series 68 58'. 0. 24 THEORY OF MAGNETIC MEASUREMENTS. It is best to reverse the polarity of the needle before making the first determination at any station. This is done by placing the needle upon its side upon a block, into which it fits with its upper side nearly flush with the surface of the block. A hole in the centre of the depression serves to admit the axle. Two bar-magnets are used in magnetizing the needle. Opposite poles are brought down upon the centre of the needle on oppo- site sides of the axle, the magnets being inclined to an angle of 40 to 45 with the horizontal plane. Preserv- ing this inclination, the magnets are simultaneously moved in opposite directions until they leave the needle. The magnets are then lifted several inches and brought down as before at the axle ; but they should not be al- lowed to touch each other. The stroke should be made with uniform speed. The supporting block should have a raised guide along one side, so that the strokes may be parallel to the geometrical axis of the needle, in order to avoid eccentricity in the position of the magnetic axis. For the first magnetization at any station three strokes may be made on each side of the magnet. For the subsequent reversal at the station four strokes may be made. The magnetism of the needle diminishes somewhat as the result of the shocks incident to travel, and this is the reason for the difference in the number of strokes. After magnetization the needle should be at once placed in position ; but it should not be used for about ten minutes, as the position of the magnetic axis is likely to fluctuate, and will give very discordant re- sults. In the table of reductions it will be observed that the positions in the left half are reproduced in the right half, so that the two means should agree, inde- pendently of the instrumental errors. THEOKY OF MAGNETIC MEASUREMENTS. PENDULUM VIBRATIONS. The determination of magnetic intensity or the strength of the earth's magnetic field is made by a mag- net which is oscillated as a magnetic pendulum. It therefore becomes necessary to give an exposition of pendulum vibrations, in order to make the subject intel- ligible. It will further simplify the treatment if the ordinary gravitation pendulum is first discussed, since the unit of force is usually defined in terms of the weight of a given mass of matter. By (4) the weight of a pound at any point is g. This is the force on a pound at any point in the earth's gravitation field. It may also be called the strength of the earth's field at the point. In a study of the earth's gravitation field it is therefore neces- saryto find the value of g at a sufficient number of points. The direction of the lines of force are at once indicated by a plumb-line. A simple pendu- lum is a heavy par- ticle having m units of mass, suspended by a line without mass to a fixed sup- port. If deflected, it tends to fall, and is constrained to slide in a circle about the point of suspension as centre. The particle slides down an in- clined plane the angle of which continually changes, 26 THEORY OF MAGNETIC MEASUREMENTS. being always a, where a is the angle of deflection. The action of the field on the pendulum in a vertical direction is w = mg. The component along the path at any point is mg sin a, the acceleration at the point being g sin a. The velocity acquired in falling from A to P is that due to MN, the vertical height of A above P ; hence v* = 2gMN. (7) Calling L the radius of the circle or the length of the pendulum, and denoting the chord CP by s and the chord CA by a, we have, by similar triangles, hence and by (7) (8) It is evident that v will have the same value whether the sign of s is -f- or ; and for any one value of s there are two values of v, which are numerically equal but of unlike sign. Hence the velocity at P and P' will be the same, and it will be the same whether the ball is ascend- ing or descending in its path. A and B being the ex- tremes of the excursion, the velocity is zero at those points, since s=a. At G the velocity is greatest, since s = 0. The velocity at C is The time of vibration that is, the time required to traverse the arc AB can be easily obtained when the THEOKY OE MAGNETIC MEASUKEMEtfTS. arc is small so that it does not sensibly differ from its chord. Thus, let AB represent the arc AGE of the previ- ous figure. With C as a centre and A C, or 0, as a ra- dius, describe a circle, and suppose a point to traverse B this circle with an uniform velocity of ->/ At any instant let the point be at Q. Its velocity resolved parallel to AS will be Fcos TQN=Vcos CQP since CP=s and CQ = a. | This is the same velocity which the pendulum has at P. Hence the point will move around the arc AQB in the same time which the pendulum requires to oscillate from A to B. Since the point moves over the arc A QB = Tta with the uniform velocity of y -y, it follows that the time required, which is the time of vibration of the pen- dulum, is (10) THEORY OF MAGNETIC MEASUREMENTS. This formula will give the time with very considerable precision, if the arc AB is not over 3 or 4. Formula (10) applies directly to the simple pendulum, but it also holds for the compound pendulum. It, how- ever, becomes necessary to define what is meant by the length of the pendulum, since the particles which com- pose it are not at equal distances from the axis of sus- pension. The particles nearest to the axis of suspension tend to oscillate in a shorter time than those near the lower extremity. Hence points near the axis have a longer time, and those near the bottom a shorter time, than they would have if vibrating alone around the same axis. Hence a series of points must exist which vibrate precisely as they would if they were unconnected with the system. These points are all in a straight line parallel to the axis of suspension, and constitute what is called the axis of oscillation. The perpendicular distance be- tween the two axes is the length of the compound pendulum. It is the length of the simple pen- dulum, which would make its vibration in the same time. Let S and represent the axes of suspension and of oscil- lation, G the centre of gravity of the pendulum, and dm any small element of mass. LetSO=L; 8G=K; Sm=r. Denote the /_ &SC by 6, and / mSC by (6 -f a). The planes of these angles are at right an- gles to the axes and S, and, in general, do not coincide. The pendulum might be supposed condensed upon THEORY OF MAGNETIC MEASUREMENTS. 29 the vertical plane containing G, and at right angles to the axes 8 and 0. If this change comes about by moving the particles horizontally, a thin plate will result which will have the same properties as the ori- ginal pendulum. The planes of the angles 6 and 6 -J- a would then coincide. At any instant the linear acceleration of is g sin 6. The angular acceleration of and of every other point of the system is y sin 6. If the element dm were dis- connected from the system its linear acceleration at this instant would be (/sin (6 -\- a). The force required to produce this acceleration on dm is F=dmgam (6 + a). (11) When connected with the system the real linear accele- f* ration is j g sin 0. The force required to produce this acceleration is F' = dm 7 j-gsm6. (12) The difference F' Fis a force which must be applied to dm in excess of its tangential weight component, in order to give it its actual acceleration as part of the system. The moment of this force about S is r a r (F r F) = y g sin 6 dm g sin (6 + a) dm. The integral of this expression for the entire pendulum is necessarily zero ; hence y sin 6 / dmr* = /dmg.r sin (6 -j- a). In this expression the integral in the first member is the 30 THEORY OJ? MAGNETIC moment of inertia 1 of the entire pendulum, or J/A 2 , where A is the radius of gyration. The integral in the second member is the moment of the weight of the entire pendulum, which is the same as though the whole mass were at the centre of gravity. Hence the last equation becomes L i The fact that #.sin 0, the real linear acceleration of 0, cancels from this expression, shows that (13) is true, independent of the position of the pendulum. The time of vibration of a compound pendulum is then found by substituting this value of L in (10), from which <=M/lF- (14) The denominator of (14) is the moment of the force tending to produce rotation when the pendulum is de- flected 90 from its position of repose. The line through the centre of gravity, and at right angles to the two axes, is then at right angles to the lines of force of the earth's gravitation field. It is evident that some point must exist in the pendulum, at which if its en- tire mass were concentrated the moment of inertia about $ would remain unchanged. This point is called the centre of gyration, and its distance from $ is called the radius of gyration. Denoting this radius by A, the value of / becomes /= J/A 2 . Hence by (13) the rela- tion between A, L, and K is K = L.K, (15) TttEOEY OF MAGNETIC or the radius of gyration is a mean proportional between the distances of the centre of gravity and of the axis of oscillation from the axis S. For the simple pen- dulum these distances are all equal. If the compound pen- dulum consists of a thin rod having its axis of suspension intersect- ing the axis of figure at right angles, the ex- pression for the length L will have the form jq a + ~ the distances .TTand K being the distances of the cen- tre of gravity of the two parts of the bar, separated by a plane through 8 and at right angles to the axis of figure. The corresponding radii of gyration are X and A/, the squares of both being positive. If MK= M'K\ the value of L, and hence also the value of t, becomes infinite. This is the condition of a balanced lever. If the rod is of uniform section, this condition is realized when the point S is midway between the extremities of the bar. A bar of steel thus suspended will not oscillate as a gravitation pendulum, but when magnetized it will oscillate as a magnetic pendulum. Let m represent the quantity of magnetism in each end of the magnet, measured in the units already de- fined (p. 11). Let F represent the action of the earth's magnetic field (or the earth-magnet) on a unit quantity of magnetism. Then the force acting on each end of the THEORY OF MAGNETIC MEASUREMENTS. . 5 magnet is Fm. The points of application of this force are assumed to be at two points called poles, as the points of application in the gravitation pendulum are assum- ed to be at the centres of gravity. The poles are merely the cen- tres of the forces act- ing on the magnet. Their position is not related in any simple manner to the geo- metry of the magnet, but depends upon the law of magnetic distribution. The distances of the poles from S will therefore remain undetermined, and may be denoted by L. The force Fm acting on each pole of the magnet is the analogue of the force gM acting on the centres of gravity of the pendulum. But in the magnet the signs of m at opposite ends of the magnet are unlike, so that the moment of the force acting on the magnet and tending to produce rotation when the magnet lies at right angles to the lines of force is not zero, but 2FmL, since the signs of m and L reverse simultaneously. The force is applied as a couple. In a field of unit strength this moment becomes 2mL, which is usually called the magnetic moment of the magnet. This quantity will hereafter be denoted by M. Hence the time of vibration of a magnet free to oscillate through the position of repose (which is the direction of the lines of force) becomes, from (14), ~ (16) THEORY OF MAGNETIC MEASUREMENTS. 33 The needle, thus suspended like a dipping-needle, measures the total force. The suspension of a needle in this manner presents great mechanical difficulties. A needle hung on a fibre of silk, and constrained to vibrate in a horizontal plane, is very much more sensi- tive. Such a needle determines the horizontal com- ponent of the force F, from which F is readily calcu- lated if the inclination is known. For the horizontal needle the time of vibration be- comes where H=Fcos d, in which d is the angle of inclina- tion and H the horizontal component of the total force F. MOMENT OF INEKTIA. The moment of inertia / can be determined by com- putation when the vibrating mass has a regular geo- metrical form, but it is usually better to use the indi- rect method due to Gauss. The magnet is first allowed to vibrate freely, and its time of vibration, t, is deter- mined. The magnet is then loaded with a known mass of non-magnetic substance, so arranged with reference to the axis of vibration that its moment of inertia is known. Let it be /'. In the first experiment the time of vibration is repre- sented by (17). In the second experiment it is repre- sented by the equation v-& The observations must be corrected for torsion, and 34 THEORY OF MAGNETIC MEASUREMENTS. unless the temperature of the two series is the same the observations for the free magnet must be reduced to that of the loaded series. The method of making these corrections will be explained later. The observations should be made in a room where the temperature can be held uniform. If the determinations of t and t' are made within a short interval of time, the values of HM may be assumed the same in the two equations, after the corrections for torsion and temperature have been ap- plied. It is somewhat better to make alternating de- terminations of t and f , in order to eliminate changes in H. The two equations give, when thus corrected, (19) It is most convenient to add the moment of inertia T in the shape of an accurately-turned ring of brass or gun-metal, the dimensions of which are accurately de- termined at some known temperature. This ring is mounted upon the magnet, the plane of the ring being horizontal, and the axis of the ring being in the axis of rotation. This is accomplished by first suspending the magnet in a horizontal position, and, pointing the tele- scope on its scale, the adjustment of the ring must be such as to reproduce the same pointing on the scale. The formula for the moment of inertia of the ring may be deduced as follows : Calling dl' the moment of inertia of an elementary ring of mass dm of radius r, thickness h, and radial width dr, the value of dT is d,T = dm. r* = %7rrJi.dr.D. r\ where D is the density of the material of which the ring is composed. Integrating between the limits r' and r ff , THEOKY OF MAGNETIC MEASUREMENTS. 35 1 = 2nliD r" r\lr =. J r' = g (" + ")> (20) where w is the mass of the ring, the external and inter- nal radii being r" and r'. The following example will illustrate the method of finding the numerical values of /' as function of the temperature : INERTIA RING Y OF THE U. S. COAST AND GEODETIC SURVEY. The radii of the ring as determined by Mr. Schott were, at 62 F., r' = 1.1715 inch =0.09762 ft, r*= 1.4219 " =0.11846" w= 812.93 grains. The coefficient of expansion a was taken 0.000,010 for 1 F. For the centigrade degree it is, of course, f of this quantity. Denoting the radii at 62 F. and at 6 F. by r 62 and r e> the values of the radii for a temperature are '. = *'[! + (0-62)] and neglecting the square of a (6 62), the values of r' and r"* are 36 THEORY OF MAGNETIC MEASUREMENTS. The errors due to these approximations at a tempera- ture of 100 ft are about one-thirtieth of the error of measurement of the radii, assuming that the lengths in decimals of a foot are correct to the fifth decimal place. Substituting these values in equation (20), it becomes a/j f = I (>" + r'\,) [1 + 8 (& - 62)], which for computation would be put in the form fiiy I'e - f ('" + "..) (1 - 124) + aw (r"\, + r'\,) 6, from which the following table has been computed. MOMENT OF INERTIA OF RING Y. FOOT-GRAIN UNITS. 0F I'e log. I'Q 60 9.5771 0.98123 70 9.5790 0.98132 80 9.5810 0.98141 90 9.5829 0.98150 100 9.5848 0.98158 Degrees. p. p. log. 1 1 2 2 3 3 4 3 5 4 6 5 7 6 8 7 9 8 uepu. iviecn. Jung. THEORY OF MAGNETIC MEASUREMENTS. 37 MOMENT OF INERTIA OF THE MAGNET. The magnet being oscillated at a temperature 6, its time of vibration t becomes When loaded with the inertia ring, and at a tempera- ture 6' for magnet and ring, the time of vibration will be The thermometer should be placed within the magnet- box and read through a window in the same. The mag- netic moment varies with the temperature, and therefore the value of t must be corrected to the temperature 0'. Both t and t' should also be corrected for the torsion of the suspension-fibre. The manner of making these cor- rections is explained later. This being done, the two values of HM being assumed equal in (21) and (22), the result is **& -*)-"'- If 6 and & do not differ more than one or two de- I e grees, the value of -j- may be taken as unity. The value of I fi , is then If 6' is the higher temperature, it will be observed that the effect of calling unity is to make the result- ** ing value of I Q , too small. If this difference is thought 38 THEORY OF MAGNETIC MEASUREMENTS. to be appreciable, it may be assumed that the deter- r\ i /D/ mined value corresponds to a temperature . But /& it is very easy to make the experiments at a suffi- ciently constant temperature, so that no correction is needed. This is best done by making the experiments in some dry basement-room from which artificial heat is excluded, making use of temperature fluctuations due to changes of temperature of the external atmo- sphere. The magnet and ring should be continu- ally on or in the magnetometer with the thermometer, and the magnetometer-box should be left open during the intervals between the observations. In handling the ring and magnet the fingers should be covered with non-conducting material, as rubber or rubber-cloth. If artificial heat is used the temperature changes should be slow, and the desired temperature should be main- tained constant for some hours before the observation, in order to allow the magnetic condition of the magnet to become stable. The temperature should not be raised above the highest summer heats (shade) which are to be experienced in summer work. At least twenty determinations of I e should be thus obtained at varying temperatures 8. As the change in / is simply due to expansion, and is very small, the func- tion may be considered a linear one, and may be repre- sented by the equation Iog/=log7 +(8-8)41, (24) where log 7 is the value of log / at a temperature 6 Q , the mean temperature of the series, and z//is the change in log / per degree of change in temperature. The value of log / is the mean of the values log I given in THEORY OF MAGNETIC MEASUREMENTS. 39 the third column of the following table. It then re- mains to find AI, for which the 23 observations furnish 23 equations. The value of AI may be determined by graphical methods. The simultaneous values of log I Q and 6 are plotted, and by aid of a thread the position of the line represented by (24) is determined, the con- stants for which are thus easily determined by well- known methods of analytical geometry. The constants may also be determined by means of the method of least squares. When the computations are properly arranged this involves little labor, and the calculation is made in the table as an example of the method. The discussion of this method is given in the appendix. Let it be required to assign a value to AI in order to most nearly satisfy the 23 equations. If any value at random be assigned to AI, then in general the value of log / o - log I 9 H- (0 - ) A I will not be zero. If its value be denoted by e, and if for convenience we put and there will be 23 equations of the form e=y -{- u AI. The values of e should all be small numerically, some being minus and some plus. The sum of the values e a or J2e a for the twenty-three equations should be a mini- mum. As is shown in the appendix, the value of AI which will make 2e* a minimum is u where 2 uy is the sum of the twenty-three products of the simultaneous values of u and y, 2u* being the sum 40 THEORY OF MAGNETIC MEASUREMENTS. of the values u*. This computation is given in the final columns of the table. The resulting value is = 0.000021 and hence log I e = 1.22294 -f- 0.000021 ( 0- 64.8), (25) from which the small table for the log moment of inertia of the magnet at various temperatures has been calcu- lated. COMPUTATION FOR MOMENT OF INERTIA OF MAGNET <7 6 . Date. Log. I e y u uy - 1880-81 Oct. 18 68.7 1.22326 -0.00032 + 3.9 -0.00125 15.21 Nov. 3 65.5 1.22295 + 1 + 0.7 + 0.00001 0.49 " 10 58.4 1.22587 - 293 - 6.4 +0.01875 40.96 Dec. 8 65.0 1.22535 - 241 + 0.2 -0.00048 0.04 " 31 540 1.22465 - 171 -10.8 + 0.01847 116.64 Jan. 1 48.5 1.22238 + 56 -16.3 -0.00913 265.69 2 51.0 1.22239 + 55 -13.8 -0.00759 190.44 3 53.0 1.22227 + 67 -11.8 -0.00791 139.24 4 51.1 1.22126 + 168 -13.7 -0.02302 187.69 4 50.4 1.22080 + 214 -14.4 -0.03082 207.36 ' 13 63.0 1.22540 - 247 - 1.8 + 0.00445 3.24 ' 17 52.0 1,22260 :+ -34 -12.8 -0.00435 163..84 ' 18 48.0 1.22135 "+ 159 -16.8 -0.02671 282.24 " 19 53.5 1.22248 + 46 -11.3 -0.00520 127.69 " 26 66.5 1.22346 52 + 1-7 -0.00088 2.89 " 28 61.5 1.22188 + 106 - 3.3 -0.00350 10.89 Feb. 2 61.2 1.22241 + 53 - 3.6 -0.00191 12.96 " 4 65.5 1.22223 + 71 + 0.7 + 0.00050 0.49 5 68.1 1.22177 + 117 + 3.3 +0.00386 10.89 " 23 99.7 1.22511 - 217 + 34.9 -0.07573 1218.01 " 26 103.5 1.22224 + 70 +38.7 + 0.02709 1497.69 March 4 96.0 1.22138 + 156 +31.2 + 0.02867 973.44 " 29 86.5 1.22425 - 131 + 21.7 -0.02843 470.89 Means. . 64.8 1.22294 -0.12511 5938.92 THEOEY OF MAGNETIC MEASUREMENTS. 41 The following table for the log of the moment of inertia has been calculated from (25). The table of proportional parts gives the differences from 1 to 9. For instance, if this magnet were oscillated at a tempe- rature 87, the log of the moment of inertia of the mag- net would be 1.22341. MOMENT OF INERTIA OF MAGNET <7 6 , FROM (25). e Log. IQ 50 1.22263 60 1.22284 70 1.22305 80 1.22326 90 1.22347 100 1.22368 Degrees. p. p. log. 1 2 2 4 3 6 4 8 5 11 6 13 7 15 8 17 9 19 pt. Mech 42 THEOEY OF MAGNETIC MEASUREMENTS. CORRECTION OF THE OSCILLATION SERIES FOR TORSION. If a brass weight having the same moment of inertia as the magnet were suspended on the silk fibre, the torsion of the fibre would cause it to oscillate slowly. The time of vibration would be expressed by an equation of the form (14) or (16), viz. : where #, the directive constant depends in this case upon the length, diameter, and material of the fibre. A horizontal magnet having the same value of / would, therefore, have the time of vibration, * 4/-wW ( 3 ?) If the magnet were not affected by torsion its time of vibration would be (28) hence from (37) and (28) (29) It is here assumed that the line of detorsion lies in the magnetic meridian, so that the fibre does not influ- ence the position of the needle when it is at rest. PI In order to find the value of -7717-, turn the torsion- HM head, to which the upper extremity of the fibre is at- tached, through, say, 90; the magnet will follow through an angle v, where v will be usually about six to ten THEOKY OF MAGNETIC MEASUREMENTS. 43 minutes. The number of minutes t>f twist in the fibre will be 5400 v. Since v is small, the moment of the earth's force tending to bring the magnet into the meridian will be HMv (really HM sin v) ; the moment of the force of torsion of the fibre tending to deflect the magnet still more is (5400 v) d. Hence for equili- brium or v This in (28) gives, since TT is small, 5400 + 7, 5400 This gives the necessary correction when v is mea- sured. In determining v the torsion-head is first turn- ed 90 in a + direction, and then backwards in a di- rection 180, and then again in a -f- direction 90, which should reproduce the original scale-reading. The dif- ferences between the scale-readings should correspond to v, %v, and v, which, added together and divided by four, gives the value of v. If the twist is not all re- moved from the fibre, the diiferences for the -[- and the position will not be equal. The observations and computation for torsion will be found in a specimen table of oscillations, p. 46. 44 THEORY OF MAGNETIC MEASUREMENTS. TEMPERATURE CORRECTION FOR MAGNETIC MOMENT. The magnetic moment of a magnet diminishes slight- ly for an increase of temperature, and the change may be assumed to take place in accordance with the equa- tion : X t .=.Jf l [l-q(ff-0)-] > (31) where q is a very small quantity, representing the frac- tion of itself by which M Q diminishes when heated through one degree. The corrections heretofore dis- cussed having been made, the times of vibration at the two temperatures will be / T~ (32) (33) Dividing one of these equations by the other, and re- placing M , by its value in (31), the resulting equation is : which gives the time of vibration of the magnet at a temperature 6, if the time of vibration at 0' is known, and in terms of the coefficient q-. ; The method of de- termining this quantity will be explained later. It should be observed that the temperature correction is by far the most difficult and important of the correc- tions to be made in magnetic observations. In field- work it is particularly troublesome. It is better in such cases to use a tent with double walls and roof or with a THEOEY OF MAGNETIC MEASUEEMENTS. 45 large fly, so as to keep the sunlight from the inner tent. It should be open on opposite sides, in order to admit free circulation of air, although it is not good to have the magnet-box too much exposed to wind. In the de- flection observations, to be discussed later, the tempera- ture of the deflecting magnet must be carefully deter- mined. The formula for oscillations becomes, therefore, = ^-> (35) where /a __ The temperature 6 is usually that of the deflection series. The value / in (35) is of course I Qf , or the moment of inertia at the temperature 6' at which the magnet was actually oscillated. This is easily shown by assuming as sufficiently precise, the equation I v = /. [1 + c (ff - (7)], and giving the values I 9 and I 9 , to this quantity in equa- tions (32) and (33). The proof easily follows. . Mech THEOKY OF MAGNETIC MEASUKEMENTS. HORIZONTAL INTENSITY-OSCILLATIONS. Date Aug. 12, 1879. Station, Jefferson City, Mo., ChappelPs orchard. Instrument, University Magnetometer, Magnet 6V Watch, Jiirgensen No. 10890 ; loses 3 a per day. Observers, F. E. N. and J. W. S. No. Oscill. Watch reads. Temp. Extreme scale-readings. Time of 100 oscill. 81 70 d .5-90 d .O o Oh 1 Qm 4.1 s K 10 14 49 .7 10osc,=68 8 .2 20 15 58 .0 68.3 30 17 06 .2 68 .1 40 18 14 .3 68.1 50 19 22 .6 68.3 Mean 68 .2 100 oscill 100 25 04.0 81 ll m 22 8 .0 ll m 22 8 .5 110 26 12.2 11 22 .5 120 27 20.7 11 22 .7 130 28 29.0 11 22 .8 140 29 37.0 11 22.7 150 QH AX. noo a 82 74 d .6-86 d .8 Means 6' 81.3 11 22 .63 1 scale div. = 2'. 345. Torsion circle. Scale. Mean. Diff. Logs. 180 + 90 74.6 68.1 86.8 90.0 80.7 79.0 1.7 v = 4'.l 5400 + v 3.73272 -180 74.2 91.0 82.6 3.6 1.7 5400 (a.c.) 6.26761 + 90 71.9 90.0 80.9 7.0 Mean v = 1.75 div. Correction 0.00033 THEORY OF MAGNETIC MEASUREMENTS. 47 HORIZONTAL INTENSITY COMPUTATION. Observed time of 100 oscill ..................... 682 s . 63 Time " 1 " .................... 6.8263 Correction for rate .................... +0 .0002 f = 6 .8265 q q (6 r 0) l-2(0'-0) 0.00048 *' 5400 + v' 5400 a.c. / 3 / MR M H Logs. +4.1 0.83419 1.66838 0.00033 9.99914 0.00197 0.99803 TtU 1.66785 M= 0.7440 8.33215 0.99430 1.22340 0.54985 9.87158 0.67827 Deflection series, 12th Aug., at 8.45 A.M. = 77.2. /is from an old table, and was cor- M rect at that date for temp, of 0' = 81.3. j=[ ME M 9.19331 0.54985 9.74316 9.87158 48 THEORY OF MAGNETIC MEASUREMENTS. SEDUCTION" OF THE TIME OF OSCILLATION TO THAT OF AN INFINITELY SMALL ARC. The time of vibration for an infinitely small arc has already been shown to be HM The complete time of vibration of any oscillating body in which the force which draws it towards its po- sition of repose is proportional to the sine of the angle of displacement, is .) where a is the total arc described. This formula or its equivalent can be found in any good treatise on analyti- cal mechanics. The formula may be written The value where t is the observed time of vibration. of the expression within the parenthesis, denoted by A, is given in the table. It is never necessary with a colli- mator magnet, or with one employing a reflecting scale according to the origi- nal method of Gauss, to make the value of a over 40' to 1, so that this correction may ordinarily be neglected altogether. In case the correction is to be made, the first term of the series only need a A 0.00000 1 00 2 02 3 04 4 08 5 12 6 17 7 23 8 30 9 39 10 0.00048 THEORY OF MAGNETIC MEASUREMENTS. 49 be used, and the sine may be replaced by the arc. The formula then becomes <-<(-). where a is the mean of the arcs of the first and last os- cillations. An equivalent expression is tl aa ^ 1 ~ where a! and a" are the arcs of the first and last oscil- lations. The observations should be so arranged that the amplitude does not diminish more than one-third during the determination in order to apply these latter formulae. In order to obtain the corrected value t* in equation (36), if correction is to be made for amplitude, as well as torsion and temperature, the latter formula becomes t 2 where the arcs are, of course, expressed in circular mea- sure. In the example given in the blank the person observ- ing the magnet-scale called " time " on every tenth oscil- lation, and the assistant read the watch, observing the second-hand by means of a magnifying-glass. The time may also be taken by a single observer, if pro- vided with a chronometer having a jumping second or half-second hand. The beat is taken up and carried in mind just before the oscillation to be timed occurs. The second is best divided by noting the position of the middle division of the scale at the beats before and after its transit of the cross-hair. It is not necessary to count the oscillations between 50 and 100, as the time 50 THEORY OF MAGNETIC MEASUREMENTS. of occurrence of the hundredth beat can be calculated, with a precision sufficient to enable one to recognize it with certainty, after the first half of the series is made, as is shown in the blank, where the time of 100 oscil- lations, calculated from the first half of the series, is ll m 22 3 .0, which, added to the watch-reading of the initial observation, 9* 13 41". 5, gives 9 h 25 m 03 9 .5 as the calculated time of occurrence of the hundredth os- cillation. The observation for time should, of course, be made when the magnet is at the middle of its swing. Formula (35) gives the value of H in terms of the un- known quantity M. If this quantity does not change in time, (35) may be used in making relative determina- tions. This method may be used with an old magnet, if it is protected from the influence of other magnets and from mechanical shocks. The slight decrease in the value of M which may be expected may be de- tected by oscillating before and after the tour, at a base station, the change being assumed constant. It is, however, always better to determine the value of M fifteen or twenty times in the course of a summer. This is done by obtaining an independent equation involving M and H, as will now be shown. This method is due to Gauss. Bept.Mecli.Biig. THEORY OF MAGNETIC MEASUREMENTS. 51 DEFLECTION SERIES FOR INTENSITY. The magnet used in g N Eig. 6 the oscillation series is re. r <--< -*r ll _ -m *.m J -mbS placed by a small magnet, n 5, Fig. 6. The oscillation or "intensity" magnet, as it is sometimes called, is then mounted on a horizon- tal bar, with its longitudinal axis at right angles to the plane of the magnetic meridian, the prolongation of its axis bisecting the axis of the suspended needle, as is shown in Fig. 6. This causes a deflection of the needle, the angle of deflection (for a given distance r between the centres of the magnets) being greater as the value of H is less. In some magnetometers the telescope and magnet-box are mounted on a common azimuth circle, and the whole instrument turns about the vertical axis of the circle. The telescope is always adjusted on the central scale-division. The angle of deflection is thus read on the azimuth circle. In others the instrument re- mains in position in the plane of the magnetic meri- dian, the angle of deflection being read on the magnet- scale. The latter form was that originally devised by Gauss. These instruments bear to each other the same relation as the sine and the tangent galvanometer. The discussion for both forms of instruments will be given. Let m be the strength of thfpoleS of intensity mag- net N 8, 21 the distance between' Tts poles, and let m' and W be the analogous values for the needle n s. Let r be the distance between the centres of the mag- nets. The distance I' being small compared with r, the repulsion of N on n will be approximately j ^- a , THEOKY OF MAGNETIC HEASUKEMENTS. and the attraction of S on n will be mm' The differ- ence between these expressions will approximately repre- sent the resultant repulsion of N S on n, or =- mm' mm , 1 +.. .J since higher powers of may be omitted. Hence the moment of the couple acting on n s is p + ^ P+- ) The needle is deflected through a small angle, , and comes to rest when the deflecting moments of the magnet N S and the earth's field are equal. The moment of the force due to the earth's field is M'H sin u. The force of N S on n s being, for the tangent magnetometer, assumed to act at right angles to the magnetic meridian, the moment of this force will be 21' F cos u. For equilibrium M'H sin u ZMM^ r' COS U. THEOKY OF MAGNETIC MEASUREMENTS. 53 This equation involves several approximations not made in the great discussion of Gauss. The determi- nations are, however, subject to errors of adjustment and to errors dependent on the position of the magnetic axes of the magnets, which do not coincide with the geometrical axes. In practice, therefore, the value of P 2J 9 in the small term 2 ,- is replaced by a constant, P, and the errors of adjustment and of the approximations are thrown upon this quantity. The equation then be- comes (38) which is the same result as is obtained by the more gene- ral discussion of Gauss. At each station the value of u is observed for two distances, r and r'. The two equations thus obtained can be combined by eliminating -== if the two determi- nations are made near together in time, and thus a value of P is obtained which will ^satisfy the two equa- tions. The residual errors of adjustment and of eccen- tricity are thus thrown upon the quantity P, which al- ways turns out to be small, if r and r' are properly chosen. The sign of P is sometimes -f- and sometimes . For obtaining an average value of P it is custo- mary to make at least twenty observations, and for re- ducing the observations of a season it is sufficient to take an average value of P as determined at all the stations. When the sine magnetometer is used the deflecting magnet is at right angles to the needle when the read- ing is taken. The moment of the deflecting magnet on 54 THEORY OF MAGNETIC MEASUREMENTS. the needle is then 2l'F instead of 21' F cos u. Hence the final equation becomes (39) If the two deflection series are made at different tem- peratures they must be reduced to a common tempera- ture. Let 6" = the mean temperature of one series, and 6 that of the other. Let u' be the observed angle of deflection in the first series : it is required to find the angle u of deflection if the temperature had been 0, or that of the other series. For the tangent instru- ment the two equations become M n r P 2? =lr' tea M p -i - - t + . . .J. Dividing one equation by the other, and reducing by equation (31), the result is tan u' tan =_____. (40) The oscillation series is then also reduced to a tempe- rature 6, as has been already explained. If 6 and 6" do not differ more than two or three de- grees, the mean of the two values -^ unreduced for temperature may be taken for a temperature \(Q + ^")> to which the oscillation series is then reduced. In the sine magnetometer no twist is developed in the sustaining fibre, but in the tangent instrument a tor- sion correction is needed, which is deduced as follows : 1. The twist having been removed from the fibre by THEORY OF MAGNETIC MEASUREMENTS. 55 means of the brass torsion-weight, the small magnet being suspended, if the torsion-head is turned through 90 the magnet is displaced through an angle v. The number of degrees of twist in the fibre is 90 v. Hence 1 of twist in the fibre will displace the magnet v 90 v 2. The initial conditions, being as in 1, deflect the magnet through an angle u' by means of some other magnet. But for the torsional effect of the fibre the deflection would be a little greater, the angle being u. Hence a twist of u' degrees in the fibre causes a dis- placement of u u' degrees of the magnet, or a twist 7/ T - - 77 ' of 1 displaces the magnet --- ; degrees. Hence v u u' 90 v~ u'~~ or u u' + u' _ v + 90 v ~~i7~ 90 v and u__ 90 1 v_ u' . 90 v ~ 1 _ Jl ~ ^90* Reducing to minutes, v , , n * where v is in minutes. The torsion correction, there- fore, has the same form as in the oscillation series. The form for the observation of the deflection series is here given. The deflecting magnet C 9 is first placed on the west end of the bar, then on the east end, etc., the direction which the north end points being, shown USI7BESIT7 k. * 0V 56 THEORY OF MAGNETIC MEASUREMENTS. HORIZONTAL INTENSITY-DEFLECTIONS. Date Aug. 12, 1879. Station, Jefferson City (ChappelPs). Magnet C 6 deflecting, Magnet Cn suspended. Instrument, Uni- versity Magnetometer. Observer, F. E. N. ,.. North end. Time. Temp. Scale- reading. Alternate Means. Diffs. r. m W E W E 8 h 29 m 8 33 8 36 8 38 75 76 76.5 76 33.0 126.6 33.0 126.5 33.00 126.55 93.60 93.55 0> II 8 34 75.9 93.57 I E W E W 8 8 8 8 30 32 35 40 75.5 76 76.5 77 125.9 33.8 125.8 33.5 125.85 33.65 9205 92.15 92.10 8 34 76.2 Means 76.0 2u 92.835 Computation, = |r 3 tan u (1 - J. Torsion Scale, circle. Mean. Diffs. u = 46.417 ldiv.= 2'.89 5400 + v 5400 u = 134' .43 ) 2 14' 43 j" Logs. 1.66668 0.46090 0.00091 180 270 90 180 79.7 83.6 75.7 79.7 3.9 7.9 4.0 79.9 *79.5 2.12849 Met 3.95 in v = tan u r z M H 8.59241 0.90309 9.69897 9.99887 9.19334 5400 +'v 5400 (a.c.) Logs. 3.73330 6.26761 Correction 0.00091 THEORY OP MAGNETIC MEASUREMENTS. 57 in the second column. The order of the experiments is indicated in the time column, and is so arranged as to eliminate the hourly change in declination. The method of reduction will be sufficiently apparent by in- spection of the blank. A second series with r = 1.75 ft. at a mean temperature 6 of 78.4 gave for a corrected value of u 3 20'. 66. These two series were used in calculating the value of P in equation (38), which was found to be -f 0.0083. The value of P used in the final reduction is, however, a mean for the work of a season. The resulting value of log TV for the second position was 9.19326. Hence for the two series the mean tempera- ture 6 is 77.2,and the mean value ^is 9.19331, which JJ. are used in the oscillation blank. DETERMINATION OF THE TEMPERATURE COEFFICIENT q. The temperature coefficient has been used in correct- ing both the oscillation and the deflection series. Either of these correction formulae may be used in determin- ing the value of q if all the other values in the equation are directly observed. If the oscillation series is used the magnet-box may be surrounded by a copper water- jacket, with windows, closed with double walls of mica or glass, to admit of the proper illumination and to en- able one to read the thermometer, which should be placed within the magnet-box. The copper vessel must be first examined to see if it acts magnetically, and, if so, its position must remain unchanged during the de- termination. The jacket is first filled with ice-water, 3 58 THEORY OF MAGNETIC MEASUREMENTS. and ice-water is allowed to slowly run through the jacket in such a manner as to secure as uniform a cir- culation as possible. The temperature should be held constant for at least an hour before observations are begun. The temperature need not fall below 40. After the ordinary oscillation determination, as shown in the blank, the ice-water is drawn off and warm water or steam is passed through the jacket. The change in temperature should be gradual, and the higher tempera- ture (maximum summer-heat in the shade) should again be maintained for an hour. Unless these precautions are taken fallacious values of q will result. If another instrument is not available for the simultaneous deter- mination of H (relative determinations only are needed), the lower temperature should again be reproduced with similar precautions, in order to eliminate changes in H; the mean of the two determinations at the mean of the lower temperatures (which may differ two or three de- grees) being combined with that at the higher. Since the magnet is oscillated at the two temperatures, the value of / for those temperatures must be used. The equations for the two temperatures are : . H'M RI a' Dividing one equation by the other, and combining with (31), from which THEOEY OF MAGNETIC MEASUREMENTS. 59 If the determinations at 6 and 0' have alternated, as TT explained above, the ratio /may be assumed unity. Its value may, however, be determined by a deflection magnetometer, which may easily be extemporized in any good laboratory; and this is strongly recommended. This instrument should be in a room of constant tem- perature. The ratio H tan u' H' tan u if the tangent magnetometer is used [eq. (38).] The angles u f and u must, however, be corrected for change in declination, which may be done by removing the de- flecting magnet. The value of q may also be determined by the method of deflections. It is also best in this method to use a second instrument to determine changes in declination. The change in H should also receive a correction as be- fore. The deflection-magnet is put in position on the deflection-bar and surrounded with a copper water- jacket. The deflected needle and declination are simul- taneously read at the lower temperature. If a second instrument is not available, the angle of deflection must be determined by removing the deflection-magnet from the bar. It is then more difficult to control its tempe- rature, but with proper care and patience the method will give good results. If the sine magnetometer is used the deflection for- mula (39) gives the equation hence m t M fi , M sin u' sin u sn u (43) 60 THEORY OF MAGNETIC MEASUREMENTS. By (31) M , M ^ * = - q (6' _ 8). (44) Since u' and u differ very little from each other, sin u' sin u = cos u (u' u) ; hence, by (43) and If u' u = d, measured in scale divisions, and the value of one scale division in minutes be s, the arc of 1' in terms of radius being L, then the above equation becomes 2 = cotw 0^70, (45) where u is the angle of deflection at the lower tempera- ture 6. If the tangent magnetometer is used equation (38) gives, as in the previous case, = _ _ M B tan u If %' and u are small (about two degrees), u u' being not over two or three minutes, which is about its usual value, the numerator tan u' tan u may be written u' u. Under these conditions the expression for q is the same as that deduced for the sine magnet- ometer (45). In case the above approximation is not admissible the formula becomes tan u tan u' The following practical example will show the method of determination : THEOKY OF MAGNETIC MEASUREMENTS. 61 Determination of q for magnet <7 6 of U. S. C. and G-. Survey, March 11, 1881. F. E. N., observer. Magnet 6 deflecting (7 17 , which is suspended in the University magnetometer, mounted on south pier of the clock-room of Washington University. Declinometer No. 3, U. S. C. and G. Survey, with magnet No. 1 sus- pended, was mounted on north pier, in order to correct for hourly change. Both magnets had been suspended for a week in order to render the fibres constant, torsion being corrected. The scale values of the magnets were, (7 17 , 2'. 89 ; No. 1, T.902 ; and the scales are so mounted that when the easterly declination increases, the scale- reading of No. 1 increases, while that of (7 17 decreases. At 2 o'clock P.M. the scales read : 79.0. No. 1, 77.95. At 3 P.M. magnet C 6 was put in place, deflecting (7 n , r being twenty-one inches. It was surrounded with a copper jacket of ice-water, which was fed by a drip of ice-water from a piece of ice during the night. The next morning the scales read as follows : SCALE-BEADING. Temperature of No. 1. 17 . (7.. Room. 8 h 30 m 82.0 149.15 67.5 70 8 45 81.6 149.5 67.4- 70 9 40 80.55 150.0 67.5 70 10 15 79.9 150.2 67.6 70 Mean, 67.5. At 10.15 the ice-water was gradually removed and THEORY OF MAGNETIC MEASUREMENTS. hydrant-water substituted, which was gradually warmed with a Bunsen flame. The readings were then : Hour. SCALE-BEADING. Temp, of ' No. 1. Cfo ll h 55 m 77.3 150.9 103 71 12 10 77.4 151.2 104 72 20 77.4 151.0 105 72 85 77.5 151.0 104 72 50 77.5 150.9 105 72 1 00 77.5 150.9 104 72 1 17 77.6 151.1 105.3 72 Mean, 104.3. At 1.18 P.M. the doors of the water-bath were opened and G 6 was allowed to cool down slowly, the water being gradually replaced by ice-water as before. The read- ings were then : SCALE-BEADING. Hour. Temp, of Ct- Temp, of Room. No. 1. cw 4 h OO m , 78.3 151.5 66 72 14, 78.5 151.8 66 72 25 78.65 151.8 66.8 72.2 36 78.65 151.75 67.5 72.5 45 79.0 151.7 67.8 72.2 Mean, 66.8. THEOKY OF MAGNETIC MEASUREMENTS. 63 At 4.55, G 9 being removed, the suspended magnets read : No. 1, 78.95. #, 79.35. The mean reading of No. 1 was 78. 7, and the read- ings of (7 17 were corrected to this, as is shown in the following table. In applying the correction to the scale-reading of (7 17 its sign is reversed, by reason of Magnet No. 1. Correc- tion in scale div. of C 17 . Reading of C7 17 . Mean Reading. Correc- tion to Mean. Observed. Corrected. t-H 82.0 -3.3 +2.2 149.15 151.4 ! 81.6 80.55 -2.9 -1.85 + 1.9 + 1.2 149.5 150.0 151.4 151.2 151.25 79.9 -1.2 +0.8 150.2 151.0 77.3 + 1.4 -0.9 150.9 150.0 77.4 + 1.3 -0.86 151.2 150.3 HH t 1 77.4 + 1.3 -0.86 151.0 150.2 1 77.5 + 1.2 -0.8 151.0 149.2 150.04 1 77.5 + 1.2 -0.8 150.9 150.1 - 77.5 + 1.2 -0.8 150.9 150.1 77.6 + 1.1 -0.7 151.1 150.4 78.3 + 0.4 -0.26 151.5 151.24 a 1 78.5 78.65 78.65 79.0 +0.2 + 0.05 + 0.05 -0.3 -0.13 -0.03 -0.03 +0.19 151.8 151.8 151.75 151.7 151.67 151.77 151.72 151.89 151.66 64: THEORY OF MAGNETIC MEASUREMENTS. the fact that the scales read in opposite directions, as stated. For the lower temperature 6, Series I. and III., the mean readings are : 6 Series I., 67.5 " III., 66.8 Means 67.15 151.46 For the higher temperature 6', 6' Cn reads Series II., 104.3 150.04 Hence a change of 37.15 F. in the temperature of magnet C t produces a change of 1.42 scale divisions in the reading of <7 17 . The angle of deflection u at the lower temperature is determined from the simultaneous readings before and after the experiments, as follows : Before Series I. After Series HI. No. 1 read 77.95 78.95 Mean of series.. 78.7 78.7 Correction to mean -fO.75 0.25 Keduced to scale of <7 17 0.49 +0.16 (7 17 read 79.0 79.35 # corrected 78.51 79.51 C n during deflection 151.25 151. 66 Angle of deflection 72. 74 div. 72. 15 div. Hence the mean angle of deflection u at the lower temperature is 72.45 scale divisions of (7 n , or 3 29'.4. The value of q is therefore computed as follows : THEORY OF MAGNETIC MEASUREMENTS. 65 V 5 = 2.89 d=1.42 L = 0.00029 -0 = 37.15 o = 0.00047 Logs. 0.461 0.152 6.464 cot. 1.215 a.c. 8.430 6.722 The observer should then compute a table of the values of log [1 (6' 6) q~\ for values of 6' 6 be- tween -f- 9 and 9, and a table of proportional parts for tenths of a degree, to facilitate the reduction of the oscillation series. The value of q depends on the nature and hardness of the steel of which the magnet is composed. In the above case a previous determina- tion some years before gave for this magnet a value q = 0.00048. SYSTEMS OF UNITS. In the government surveys in England and America the fundamental units taken have been the foot, the grain, and the second. In most scientific measurements the fundamental units used are now the centimetre, gramme, second. It is therefore of some importance to show how the results in one system are to be expressed in the other. As a preliminary a few simple illustrations of the theory of physical units will be given, in order to make the subject clear. For additional information the reader is referred to Everett's " Units and Physical Constants." If any length is measured in feet the length may be said to be I' times the length of one foot. 3* 66 THEORY OF MAGNETIC MEASUREMENTS. If L' represent the length of one foot, the whole dis- tance may be written I'L 1 . If the same distance be measured in centimetres, I being the number of centi- metres and L the length of one, then this distance may also be written IL. As the distance is the same, what- ever the system of units used, it follows that IL = 1'L'; and hence J = |^r. (47) Here I and I' are the numerical quantities, which are usually called the " lengths," the one being in centi- Tt metres, the other in feet. The ratio -j- is the length of a foot in centimetres, or the ratio of the dimensions of these two units. By direct comparison this ratio has been found to be 30.4797. Any length expressed in feet is converted into the equivalent expression of the same length in centimetres by multiplying by this num- ber. Density is defined to be the mass per unit of volume of any body. It is expressed by the ratio , where v is the volume of the mass m. If M represents the magni- tude of the unit mass, and L the magnitude of the unit length, the magnitude of the unit volume being then Z 3 , then any density would be completely represented by the expression If M represents the magnitude of the gramme, and L that of the centimetre, the unit density would be a gramme per cubic centimetre. If M represents the THEOKY OF MAGNETIC MEASUREMENTS. 67 kilogramme, and L the decimetre, the unit density would be a kilogramme per cubic decimetre. The unit density in these two cases is identical, and is that of water. If M represents the kilogramme, and L the centimetre, the unit density would be a thousand grammes per cubic centimetre, and would be a thousand times that of water. The density of water in kilogrammes per M cubic centimetre is 0.001. Hence the expression -=- 8 Ju represents the magnitude or " dimensions " of the unit of density, in precisely the same sense that M represents the magnitude of the unit mass, and L that of the unit length. If, then, D is the density in grammes per cubic centi- metre, and D' the same density in pounds per cubic foot, we shall have an equation similar to the one lead- ing to (47), viz. : or If, for example, D be taken as unity, which is that of water (grammes per cubic centimetre), the equivalent density in pounds per cubic foot will be M_ (L'\> -M'\L Here ^ gramme = M' pound Keducjng, the value of D' is found to be 62.425, which irtlie density of water in pounds per cubic foot. 68 THEOKY OF MAGNETIC MEASUREMENTS. Eecurring now to the equations for determining //, viz.: M , .* ff = K *. it is required to find the conversion factor, which will reduce the value H measured in foot-grain-second units to C. Gr. S. units. It is to be observed that the value of P is so determined as to satisfy the two equations M r = ir 8 tan u 1 Solving these equations for P, its value is found to be p r* tan u r" tan u' r tan u r' tan u' ' Since the tangents in this expression are independent of any system of units, its value can only be changed by a change in the unit of length. Hence the dimensions p of P are L\ The ratio -5- will therefore be independent of the system of units used. It will easily be seen that the same is true of the torsion and temperature correc- tions which have been already discussed. Solving the two general equations for H, H=A where 1/5- / 27T 3 = v^7^?y THEOET OF MAGNETIC MEASUREMENTS. 69 The value of A is independent of any change in the system of units. / is measured by a mass into a dis- tance squared, r* is the cube of a distance. The unit of time is to be the second in both cases. Hence if H' be the horizontal component of the strength of the earth's field measured in the English units, and H the same strength in metric units 0. G-. S., B=(*,fir. Substituting the values of the unit ratios, log H= 8. 863778 -flog H' log H' = 1.336222 + log H. EXPLANATION OF THE PLATES. Plate I. represents the U. S. Coast Survey magnet- ometer, having the deflection-magnet in position for ob- servation. The shorter deflection-magnet is suspended to the torsion-head. The instrument can be used either as a sine or a tangent instrument. It can also be used as a declinometer. In place of the observing telescope of I. a small alt-azimuth instrument like that shown in II. may be used, being mounted on a table-tripod with the magnetometer. Plate III. represents the form of dip-circle now com- monly used. The vertical circle is about six inches in diameter, reading by opposing verniers to minutes. Two simple' microscopes serve to read the verniers, 70 THEORY OF MAGNETIC MEASUREMENTS. while two compound microscopes moving with the ver- niers are pointed upon the marked ends of the needle. These plates were kindly furnished by the eminent in- strument-makers, Fauth & Co., of Washington, D. C., being reproduced from their catalogue. PLATE L THEOEY OF MAGNETIC MEASUBEMENTS. 71 PLATE II. 72 THEORY OF MAGNETIC MEASUREMENTS. PLATE III. APPENDIX ON THE REDUCTION OF OBSERVATIONS BY THE METHOD OF LEAST SQUARES. If a blacksmith were to repeatedly measure the length of a piece of iron by means of his two-foot rule, his mea- surements would all agree. If the distance between two fine lines on the bar of iron is measured with the high- est attainable accuracy by means of a measuring engine, the results of separate and independent measurements do not agree except by accident, and the tendency to disagree is found to increase with the delicacy of the determination. It is therefore manifest that the true length of the bar can never be obtained. It is conceded universally that the arithmetical mean of the observed values is the best result that can be obtained, when all known corrections have been made and the observa- tions are all equally worthy of confidence, or, in other words, when they have equal weight. Really the obser- vations might be weighted differently by different ob- servers, if one happened to notice something affecting the value of an observation which should escape the attention of the other. Unsuspected causes may thus affect consecutive observations in a different degree, so that observations to which one observer might give equal weight might be differently weighted by another. It is impossible to reproduce exactly the same condi- 74 APPENDIX. tions, and to make consecutive observations in the same manner. In fact, the differences in the results obtained are due to such causes. When the observer does not know of some specific reason for attaching less im- portance to an observation than to others, it should be given equal weight, even if it is discordant. PROPERTIES OF THE ARITHMETICAL MEAN. Let a single quantity be measured n times, the observation values being x lf x^ x a , . . . x n . Let &I+ *+*+ v n =^z- If ^ represent the arith- metical mean, then 2x = 2v W ft If x be subtracted from each of the observation values, the resulting differences or residuals will be sometimes positive, sometimes negative. The smaller the residuals are numerically, the greater the precision of the measurements, and the greater the degree of con- fidence to which the mean is entitled, provided all con- stant errors affecting all measurements alike have been previously eliminated. Indicating these residuals by r, the n observations would give residuals as follows : etc., etc., etc., By adding, 2x nx =2r (2) THEORY OF MAGNETIC MEASUREMENTS. 75 By (1) it follows that the first member of (2) is equal to zero, hence 2r=Q. . (3) The arithmetical mean renders the sum of the resi- duals zero. Any other number thus treated would give residuals the sum of which would be greater or less than zero. If the separate observations were all pre- cisely alike, each residual would be zero. If the individual equations for the residuals are squared, the resulting values of r a are all positive. A little consideration will enable one to see that the sum of the squares of the residuals obtained from the arith- metical mean will be less than when the residuals are formed with any other number. This is, in fact, easily shown. The squared equations are : etc., etc., etc., r n * = x' n *-2x x n -}-x* Adding these equations, the result is 2r 9 = 2x* - 2x 2x + n X Q \ If any other value K were taken instead of the arith- metical mean, the residuals would have different values. Let them be called p. The last equation would then read The value of K is to be found, which will render 2p* a minimum. This value is found from the condi- tion = 22x -f 2nK= 0, 76 APPENDIX. from which If the different observations are not deserving of equal weight, the reduction must be so made that the values deserving of most confidence shall have most to do with determining the result. If the variation of the magnetic needle were deter- mined by four measurements to be 9 24.1 with weight 4 9 25.0 " " 5 9 24.8 " " 5 9 22.6 " " 4 the best value would be obtained by adding in the first and fourth values each four times, the second and third each five times, dividing the sum by the sum of the weights, or 18. In such a case the weights of the observations might be determined by the judgment of the observer, who is able to cite some specific reason for attaching less im- portance to some observations than to others. That the result of an observation is discordant is not of itself an important reason, and should be accepted with very great caution, although discordant observations will really affect the weight of the resulting mean. If the numbers to be weighted are the means of seve- ral equally good observations, their weights would in each case be represented by the number of observations. The weight of such a mean is also determined by its probable error, as is shown in more extended treatises, in which the theory of probability is applied to errors of observation. THEOEY OF MAGNETIC MEASUREMENTS. 77 OBSERVATIONS ON TWO OR MORE QUAN- TITIES. Let it be assumed that observations have been di- rectly made on three variable quantities u, v, y, which some graphical or other method has shown to be related, as is indicated in the following equation : y = au -f- bv. (4) Considering this as representing a physical relation, a and b are constants, the true values of which cannot be determined. The value of v might be u*, so that the second member of the equation would represent the first two terms of a series. It is required to assign values to a and b which will most nearly agree with the observations made on y, u, and v. Each set of simultaneous observations will give an equation of the form y = w.a--v.b, (5) where y, u, and v now become numerical quantities or coefficients. Such equations are called " observation equations" or "equations of condition." The latter term is in more general use, but the former term seems preferable. Any two of the observation equations would deter- mine the values of a and b. But, on account of the un- avoidable errors of observation, some other set of two would give somewhat different values of a and b. The values obtained from any two would necessarily satisfy those equations, but they would not in general satisfy the other equations. In other words, if for each obser- vation equation the value y au bv is calculated, as- signing to a and b any values determined as above, the value would not in general be zero. As in the similar 78 APPENDIX. case where the arithmetical mean was treated, each equation would take the form etc., etc., etc.., r n =y n u n a v n b where the residuals would not in general be zero for any values of a and b that could be used. Evidently any values whatever might be assigned to a and b, and a set of residuals would result. Values might be cho- sen so that the residuals might all be positive or they might all be negative. It is evident that the values of a and # which will most nearly satisfy all the equations THEORY OF MAGNETIC MEASUREMENTS. 79 will give small residuals, some of which will be positive and some negative. Without farther discussion it will be assumed that the best values of a and b will make the sum of the squares of the residuals a minimum. The problem is then to assign values to a and I which will make 2r* a minimum. The values a and b are then to be treated as independent variables, and with the vari- able 2v* they determine a surface. The conditions for the minimum point on this surface are: The first equation is the condition for the minimum point in any section of the surface parallel to the plane determined by the axes of 2r* and b. The second is the condition for minimum on any section at right angles thereto. If the two conditions are simultaneously im- posed it determines a minimum point in the surface. To find the value of -5V 3 , the residuals are squared, which gives the equations : **i f = y ' - 2u^ . a - 2V& . J + w/ . a' + %uj), . ab + v? . b* y'~ Zu,y, . a - 2v,y, .b + u,'.a* + 2u,v, . ab + v, 1 . b* etc. = y* 2u n y n . a 2v n y n .b + u n \a* + 2u n v n . ab + v n a . b* Adding these equations, the result is (6) 80 APPENDIX. This is the equation of the surface. The intersection of this surface by the plane of the axes 2v* and I is de- termined by introducing into it the condition aQ. Ita equation is therefore which is the equation of a conic section. Making 5 = in this equation, the value of the intercept on the axis 2r* is 2y*. The minimum point on this section is given by from which 2vy The sign of b will here depend upon the sign of the numerator. It is not necessary to further discuss this surface, aa it is only intended to point out the nature of the relation with which we are dealing. The conditions for the minimum point on tho surface are obtained from (6). They are : The first equation determines a minimum point on any section at right angles to the b axis. The second determines a minimum point on any section at right angles to the a axis. If the two equations are com- bined by elimination, the values of a and b may be ob- tained, which determine the minimum point on the surface. THEORY OF MAGNETIC MEASUREMENTS. 81 These values are : 2v* 2 uy 2uv 2 vy _ uy ^ uv ^ uv. J (8) The calculations are made as is shown in the sub- joined table, in which the observed quantities are given in the first three columns, the quantities called for by equations (8) being the sums of the succeeding columns. No. u* uy vy 2 uv Equations (7) are called "normal equatiom" the first being called the normal equation for a, and the second the normal equation for b. It will^be observed that they may be obtained from the n observation equations (5) as follows : To find the normal equation for a, each observation equation is multiplied through by the co- efficient of a in that equation. The sum of the result- ing equations is the normal equation for a. Similarly the normal equation for 1) is obtained by using the co- 82 APPENDIX. efficients of b as multipliers, as will be seen by inspect- ing (5) and (7). This process is really equivalent to giving weights to each equation equal to the coefficient of the quantity whose normal equation is to- be formed. These weighted equations are then added together, giving a single equa- tion. If this equation were divided through by n it would give a properly weighted mean equation. This division by n is, of course, unnecessary in solving the equations for a and b, as it would not affect their values. WEIGHTED OBSERVATIONS. If the observations are not of equal weight they must be additionally weighted. If y# u lt v y are each the mean of p v equally good observations, y^ u^ v y the means of p^. observations, etc*, the normal equations be- come, as will be easily seen, (9) ^ = b 2pv* -f a 2puv 2 pvy = By elimination the values of a and t are : "i ~~ pv* 2puv 2puv * 2 pvy 2puv2puy I 2puv J The computations are made exactly as in the preced- ing case, excepting that an additional column is added THEORY OF MAGNETIC MEASUREMENTS. 83 just after the first, containing the weight assigned to each set of values u, v, y. These weights may be as- signed wholly on the judgment of the observer, if this is possible, or they may simply be the number of obser- vations of which each is the mean, modified, if deemed proper, as the judgment of the observer may decide. Where each set of values u, v, y is the mean of a large number of observations, the theory of probabilities enables one to calculate the weights from the probable errors, but ordinarily this is a wholly useless refine- ment. If the original function has the form y = a + bv, (11) the value of u in (4) becomes unity. (In fact, (4) may be put into this form by dividing through by u. The 11 v quantity - would then replace y, and the quantity would replace v in all subsequent equations, and u would be replaced by unity.) Putting u = l or 2u* =n in (8), the equations be- come a= n 2 vy 2v 2y (13) The manner of calculating is apparent from the table following equation (8). If the original function has the form y = lv, (13) the value of a becomes zero. There can be no normal equation for , and the first of equations (7) must dis- 84 APPENDIX. appear. In the normal equation for b in (7) the condi- tion a must also be introduced. The value of b becomes then <"> It will be observed that this result is different from that obtained by taking the average values of the quan- tity -, which would be - 2 -. This is simply due to J v n v the fact that (14) is the result of weighting the observa- tion equations in proportion to the magnitude of the observed quantities or coefficients y and r in the various equations. Finally, if observations are made on a single quantity, so that the function has the form y = l), (15) the value of v in (13) and (14) becomes unity and 2v* = n. Hence (14) becomes J =?' ( 16 > or the best value of the quantity ft is the arithmetical mean of the observations a result which has already been agreed upon under that head in the early part of the discussion. The additionally weighted equations corresponding to (12) are 2 pv* 2 py ^2pv^2pvy "] PV \ (17) 2sp 2 pvy 2pv 2 py THEOKY OF MAGNETIC MEASUREMENTS. 85 For (14) weighted equations give the value of b, and finally, for weightedL observations on a single quan tity, (18) becomes as was shown in the special example given on p. 76. GRAPHICAL METHODS. The preliminary investigation of any new function is always made by graphical methods. In a majority of cases met in practice the graphical method is sufficient. In order to be able to make use of this method, the computer must be familiar with the analytical geome- try, so that, from the curve which is obtained by plotting the observed values of the variables, he can form an idea of the mathematical relation sought. By far the greater number of cases which are met in physical in- vestigation are represented by the equation y = ~bx n . Here y and x are the physical variables, and b and n are unknown constants, the values of which are to be determined so as to best satisfy the equations. If n 0, then y = b, or the function is that given in (15). If n = 1, then y is directly proportional to x, and the plotted values of y and x will give a straight line pass- ing through the origin. If n= 1, the equation is an inverse proportion, and the curve will be an equilateral 86 APPENDIX. hyperbola. If n = 2 or , the curve will be a parabola, etc. In some cases it may be necessary to add a con- stant term to the second member, so that the equa- tion will take the form of (11). To take a special case, in order to make the manner of reduction well understood : Days. log M. y d 9.83989 - 0.00260 + 134 6 990 - 261 + 128 20 990 - 261 + 114 23 9.84055 - 326 + HI 35 9.83894 - 165 + 99 50 762 - 33 + 84 60 868 - 139 + 74 64 846 - 117 + 70 74 901 - 172 + 60 86 766 - 37 + 48 106 728 + 1 + 28 129 778 - 49 + 5 146 622 4- 107 - 12 156 714 + 15 - 22 165 784 - 55 - 31 169 700 + 29 - 35 171 623 4- 106 - 37 184 656 + 73 - 50 195 679 + 50 - 61 202 504 + 225 - 68 218 680 + 49 - 84 228 568 + 161 - 94 234 451 + 278 - 100 245 437 + 292 - Ill 373 242 + 487 - 239 Means: 134 9.83729 In 1865-6 Professor Wm. Harkness made a series of intensity determinations, and deduced the log. moment of his magnet at the several temperatures of observa- THEOEY OF MAGNETIC MEASUREMENTS. 87 tion.* These values were reduced to the mean tempe- rature of his series by means of equation (31). The re- sults are given in the annexed table, where the first column gives the number of days from his first observa- tion, on Oct. 24, 1865, and the second column the value of log M at a temperature 75. 8 F. The values in the two columns being plotted, the points thus determined are shown on the diagram (p. 89), It is manifest that if any assumption regarding the de- crease in log M be made, it must be that of uniform de- crease. The equation representing this relation will be log M = log M ad, where log M Q is the value of log M at any assumed date, and d is the number of days from the assumed date to that of any other observation, a being the daily change in the value of log M. If the mean of all the values of log M be taken as log Jf , it gives the value of the quan- tity log M for the mean date of the series, which is 134 days after the first observation. The straight line rep- resenting the observations must run through the point determined by these two mean values. This line is also to be so drawn as to give weight to other points in pro- portion to their distance from the point representing the mean values. The equations of condition become of the form log M log M ad = 0, where log J/" = 9.83729, and where d is estimated in days from the 134th day, which corresponds to March 17', 1866. * " Smithsonian Contributions to Knowledge," vol. xviii. p. 55 of his memoir. 88 APPENDIX. Calling log M log M=y, the equations of condi- tion become y ad = 0. ' These values of y and d are given in the third and fourth columns of the table. In order to form the nor- mal equation for a, each observation equation, of which there are twenty-five, is multiplied through by d, the coefficient of a, and the resulting equations are added. The normal equation becomes Performing the calculations, the value of u will be found to be and hence the original equation becomes log M = 9.83729 - 0.0000195 d. It is evident that, after having plotted the values of log J/and d, the position of the line can be determined with a precision sufficient for most purposes by means of a fine thread, which is laid through the points in such a way as to agree with them as nearly as possible. The position of this line is shown on the diagram. If desired, one point on the line may be determined with precision by obtaining the means of the observa- tions as in the first two columns of the table. After the line is drawn the slope of the line, or the value of (), is then found by measuring on the diagram the co-ordi- nates x', y' and x", y" of any two points, which should, of course, be as far apart as possible. In this case y' - y" a=2- n 7. x" x THEORY OF MAGNETIC MEASUREMENTS. 89 Mean /\ Date P 90 APPENDIX. The co-ordinates are, of course, to be measured in terms of the scales used in plotting. Such graphical solutions are in the large majority of cases sufficient for all purposes. They should in all cases precede any more exact mathematical solution, in order that one may see whether the observations are sufficiently precise to warrant a more exact solution, or whether the assumed equation is in harmony with the observations. TIME OF ELONGATION OF POLARIS. EXPLANATION OF TABLES I., II., AND III. The following tables (pp. 92-93) give the astronomi- cal times of elongation of the Pole star for the 1st and 15th of each month from 1885 to 1895. They are com- puted for a latitude of 40 and a longitude of 6 hrs from Greenwich. From them the local astronomical time of elongation to the nearest minute, for any latitude be- tween 25 and 55, may be obtained by applying the correction given in Table II. The correction for differ- ence of longitude is insignificant, amounting to 0'M5 for a difference of one hour, to be subtracted for places west of the 6th meridian, and added for places east. To obtain the time of elongation for any date not given in the tables, subtract 3 m .94 from the tabular time of elongation for every day elapsed, if the tabular date is the smaller, or add the same correction if the tabular date is the larger. This correction may be obtained from Table III. The astronomical day begins at noon, and is twelve hours behind the civil date. From noon to midnight THEOKY OF MAGNETIC MEASUEEMENTS. 91 the astronomical and civil dates are the same ; from midnight to noon the civil date is one greater. Thus Jan. 12, 14 h 40 m , astronomical time is Jan. 13, 2 h 40 m A.M. civil time. Example Eequired the time of eastern elongation of Polaris on Aug. 8, 1888, for a place whose latitude is 44 30'. Time of elongation, 1888, Aug. 1 (by Table I.).. 10 h 38 m Correction for latitude (by Table II.) . . 1.2 " 7 days (by Table III.) -27.6 Time of elongation Aug. 8, 1888 10 h 9 ra APPENDIX. TABLE I. EASTERN ELONGATIONS. 188C. 1887. 1888. 1889. 1890. 1891. 1892. 1893. 1894. 1895. h m h m h m h m h m h m h m h in h m h m Apr. 1. 1838 1839 1836 1838 1839 1840 1837 1838 1839 1840 " 15. 1743 1744 1741 1743 1744 1745 1742 1743 1744 1745 Mayl. 1640 1641 1638 1640 1641 1642 1639 1640 1641 1643 " 15. 1545 1546 1544 1545 1546 1547 1544 1545 1546 1548 June 1. 1438 1440 1437 1438 1440 1441 1438 1439 1440 1441 " 15. 1344 1345 1342 1343 1345 1346 1343 1344 1345 1346 July 1. 1241 1242 1239 1241 1242 1243 1240 1241 1242 1244 " 15. 1146 1147 1145 1146 1147 1148 1145 1146 1147 1148 Aug. 1. 1040 1041 1038 1039 1041 1042 1039 1040 1041 1042 " 15. 945 946 943 944 946 947 944 945 946 947 Sept. 1. 838 839 836 838 839 840 837 838 839 841 " 15. 743 744 742 743 744 745 742 743 744 746 Oct. 1. 640 641 639 640 641 642 639 641 642 643 WESTERN ELONGATIONS. 1886-7 1887-8 1888-9 1889-90 1890-1 1891-2 1892-3 1893-4 1894-5 hm h m h m h m h m h m h m h m h m Oct. 1. 1830 1831 1828 1829 1831 1832 1829 1830 1831 " 15. 1735 1736 1733 1734 1736 1737 1734 1735 1736 Nov 1. 1628 1629 1626 1628 1629 1630 1627 1628 1629 " 15. 1533 1534 1531 1532 1534 1535 1532 1533 1534 Dec. 1. 1430 1431 1428 1429 1431 1432 1429 1430 1431 " 15. 1335 1336 1333 1334 1336 1336 1334 1335 1336 Jan. 1. 1227 1229 1226 1227 1228 1229 1226 1227 1229 " 15. 1132 1133 1131 1132 1133 1134 1131 1132 1133 Feb. 1. 1025 1026 1023 1025 1026 1027 1024 1025 1026 " 15. 930 931 928 929 930 932 929 930 931 Mar. 1. 835 832 833 834 835 832 833 835 836 " 15. 739 736 738 739 740 737 738 739 741 Apr. 1. 632 630 631 632 633 630 631 633 634 THEOltY OF MAGNETIC MEASUREMENTS. 93 TABLE II. Latitude. Correction. 25 +l m .9 28 + 1 .6 31 + 1 .2 4 +0 .8 37 + .4 40 +0 .0 43 -0 .5 46 -1 .0 49 -1 .6 52 -2 .3 55 -3 .0 TABLE III. No. days. Correction. 1 3.9 2 7.9 3 11.8 4 15.8 5 19.7 6 23.6 7 27.6 8 31.5 9 35.5 10 39.4 11 43.3 APPENDIX. 1 q o e q 1 p g H O o q i> TJ; i^ o oo o^o o o oq q to TO T* q 05 os q q o* -^ cq os TO oq TO os to n< TO s TO 5 oq so - * : :.' o