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Cornell University Library 
 
 arV17306 
 
 A manual of physical measurements. 
 
 3 1924 031 246 998 
 
 olin.anx 
 
Library 
 
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 tine Cornell University Library. 
 
 There are no known copyright restrictions in 
 the United States on the use of the text. 
 
 http://www.archive.org/details/cu31924031246998 
 
MANUAL OF 
 
 PHYSICAL MEASUREMENTS 
 
Publi'dhed by the 
 
 McGra-w-Hflll Book. Company 
 
 ^Successors to i:Ke KockDeparhnentfi of tKe 
 
 McOrav Publishing Company Hill Publishing Company' 
 
 Publishers of &ooks for 
 Electrical World TheEngineenng and Mining Journal 
 
 Engineering Record American Machinist 
 
 Electric Railway Journal Coal Age' 
 
 MeiaHurgical and Chemical Engineering R>wer 
 
A MANUAL 
 
 OF 
 
 PHYSICAL MEASUREMENTS 
 
 BY 
 
 ANTHONY ZELENY, Ph.D. 
 
 Professor of Physics in the University of Minnesota 
 
 AND 
 
 HENRY A. ERIKSON, Ph.D. 
 
 Assistant Professor of Physics in the University of Minnesota 
 
 THIRD EDITION 
 
 mggraw-hill book company 
 
 239 WEST 39TH STREET, NEW YORK 
 6 BOUVERIE STREET, LONDON, E. C. 
 1913 
 K 
 
Copyright, 1903, 190§ and 1913 
 
 ET 
 
 ANTHONY ZELENT and HENKY A. ERIKSON 
 
 THE SCIENTIFIC PRESS 
 
 ROBERT DRUMMOND AND COMPANY 
 
 BROOKLYNj N. V- 
 
PREFACE 
 
 This manual is an outline of the laboratory experi- 
 ments given in the one-year and the two-year courses 
 in general Physics at the University of Minnesota. 
 The laboratory work, in these courses, supplements the 
 lectures and recitations, and occupies one two-hour 
 period a week for two and four semesters respectively. 
 
 The experiments given in a junior course of one 
 semester in electrical measurements are also included 
 in the section on electricity. 
 
 It is taken for granted that the student has acquired 
 a general knowledge of a subject before it is considered 
 in the laboratory and no attempt is made in the manual 
 at completeness in subject-matter or in explanations. 
 The work is done under the guidance of an instructor 
 who furnishes any additional information necessary. 
 
 The student should feel that acceptable results depend 
 upon his own ability to properly adjust the apparatus, 
 and he alone should plan and execute the details of the 
 experiments, subject of course to the criticism of the 
 instructor. 
 
 We here wish to thank Prof. J. Zeleny and Mr. Paul 
 E. Klopsteg for valuable suggestions, criticisms, and 
 assistance during the preparation of this manual. 
 
 Anthony Zeleny, 
 Heney a. Ekikson. 
 
INTRODUCTION 
 
 .Work in the Physical Laboratory brings the student 
 into first-hand touch with physical principles and physical 
 apparatus, and the impressions produced through the 
 senses furnish a solid foundation for further study. 
 
 The close attention to every detail and the exercise 
 of deliberate judgment which are required in every 
 experiment if a worthy result is to be obtained, tend to 
 produce a habit of accuracy which is of inestimable 
 value. This training is obtained only when every effort 
 is made to get the very best result that the time allowed 
 for the experiment and the apparatus employed will 
 permit. 
 
 Before observations on any experiment are begun, 
 the theory of the experiment should be mastered as 
 well as the functions of the various parts of the apparatus 
 which is to be used. Without such study, necessary 
 observations may be omitted or taken in the wrong- 
 way, and apparatus whose value depends upon its accuracy 
 may be injured permanently because of a lack of knowl- 
 edge about its delicate parts. Furthermore it is important 
 to determine the degree of accuracy of every result 
 which is obtained. 
 
 Errors. — For various reasons it is impossible to obtain 
 the absolute value of an unknown quantity. Any 
 measurement is affected, to a greater or less extent, by 
 errors which may be classified under the two heads: 
 constant and accidental. 
 
vm INTRODUCTION 
 
 As examples of constant errors may be mentioned the 
 following: 
 
 Physical errors, or errors which arise out of physical 
 sources, e.g., change of length of a steel tape with tem- 
 perature; instrumental errors, such as faulty construc- 
 tion or adjustment of apparatus; personal errors, or the 
 personal equation; blunders. 
 
 Constant errors can with proper care be eliminated 
 or corrected for. Under accidental errors are grouped 
 those which remain after the constant errors have been 
 taken into account. It is not possible to determine the 
 magnitude of the accidental error. The best that can 
 be done is to determine the probable limits within which 
 the true result lies. 
 
 A number of rules' and methods for finding these 
 limits have been developed in the Theory of Least 
 Squares. On page 213 of the appendix is given a resume 
 of those parts of the theory which apply to measurernents 
 made in the physical laboratory. 
 
 The rigid application of these rules in the elementary 
 laboratory is hardly justifiable. However, some simple 
 method for determining the first figure in the result 
 affected by the accidental error, is necessary and there- 
 fore the mastery of the following is required. 
 
 Error in a Single Reading. — Instruments should be 
 read to a fraction of their smallest division and hence the 
 last figure in the reading is an estimated quantity, the 
 accuracy of which depends upon the experience of the 
 observer and the size of the smallest division. Since 
 in general the estimation is liable to be incorrect by 
 only about one of the estimated parts the quantity so 
 estimated is in part accurate and must be included in 
 the result and is the last significant figure. 
 
 Error in an Average. — The average of several readings 
 is more reliable than a single reading. The average 
 
INTEODUCTION 
 
 IX 
 
 of three or more readings should therefore be obtained 
 whenever possible. In general the average may be 
 relied upon to one more figure than the least accurate 
 of the quantities averaged and this figure should there- 
 fore be retained as the last significant figure in the average. 
 
 Error in a Computed Result. — In a sum, difference, 
 product, and quotient as many figures should be retained 
 as are found in the least accurate of the quantities in- 
 volved. 
 
 Notation of Numbers. — All figures to the right of the 
 
 4 5 6 7 
 Time in Minutes 
 
 figure affected by the error are of no value and should 
 be omitted. 
 
 In very large or very small numbers, all digits includ- 
 ing the first affected by the error, are written down, 
 a decimal point is placed after the first or second digit, 
 and its proper position is indicated by some power of 
 ten. Thus 4. 189 XlO^ and 5.89X10"''. 
 
 Graphical Representation of Results. — The relation 
 between two varying quantities may be more clearly 
 shown by plotting them on coordinate paper, one quan- 
 tity being represented by distances parallel to the X 
 
X INTRODUCTION 
 
 axis and the other by distances parallel to the Y axis. 
 The choice of units to be used for laying off these quan- 
 tities, is arbitrary and need not be the same for the two 
 axes. In order to obtain, however, the most complete rep- 
 resentation the unit should be so chosen, where practi- 
 cable, that the smallest division on the coordinate paper 
 represents a unit of the last significant figure of- the 
 quantity. The unit selected should be recorded along 
 each of the axes. 
 
 The individual readings are not recorded on the 
 coordinate paper but are tabulated on a separate sheet. 
 The points are marked clearly by crosses or circles 
 surrounding the points. 
 
CONTENTS 
 
 PAGE 
 
 Preface v 
 
 Introduction vii 
 
 Errors in results vii 
 
 Error in a single reading viii 
 
 Error in an average viii 
 
 Errors in computed results ix 
 
 Notation of numbers ix 
 
 Graphical representation of results ix 
 
 MECHANICS 
 
 Measurement of length 1 
 
 " " volume 2 
 
 " " radius of curvature 6 
 
 " " mass 8 
 
 Ratio of the lengths of the balance arms 12 
 
 Time of vibration of an oscillating body 13 
 
 Bending of a rod 15 
 
 Elongation of a wire 17 
 
 " " spiral spring 18 
 
 Hooke's law and simple harmonic motion 19 
 
 Torsion of a rod 20 
 
 Moment of inertia of a body 23 
 
 Experimental test of equation I = Io+Md?; 24 
 
 Parallelogram of forces 25 
 
 Angular acceleration 27 
 
 Experimental test of equation Fr = oe I - 28 
 
xii CONTENTS 
 
 PAGE 
 
 Angular velocity 29 
 
 Experimental test of equation Fs = Jco^I 30 
 
 Momenta before and after inelastic impact 31 
 
 Acceleration of gravity — simple pendulum method 32 
 
 Acceleration of gravity — compound pendulum method 34 
 
 FLUIDS 
 
 Boyle's law 35 
 
 Specific gravity of solids 35 
 
 Specific gravity by Mohr's balance 36 
 
 Calibration of an hydrometer 38 
 
 Surface tension 39 
 
 HEAT 
 
 Atmospheric pressure 41 
 
 Calibration of thermometer 43 
 
 Charles' law 46 
 
 Boyle's and Charles' Laws Combined 48 
 
 Coefficient of linear expansion 49 
 
 Radiating power 50 
 
 Absorbing power 52 
 
 Specific heat 52 
 
 Latent heat of fusion 54 
 
 " " " vaporization 55 
 
 Vapor pressure of water at different temperatures 57 
 
 Relative humidity 58 
 
 Mechanical equivalent of heat 61 
 
 ELECTRICITY AND MAGNETISM 
 
 Electrostatics , 64 
 
 Magnetic field .'. 66 
 
 Adjustment of magnetometer 67 
 
 Adjustment of telescope 68 
 
 Measurement of angles of deflection 70 
 
 Torsion in fiber ; 71 
 
 Ratio of torsion 72 
 
CONTENTS xiii 
 
 PAGE 
 
 Horizontal intensity of the earth's magnetic field 75 
 
 Magnetic moment and pole strength 80 
 
 Horizontal intensity — comparison method 81 
 
 Inchnation and intensity of earth's field 81 
 
 Adjustment of tangent galvanometer 82 
 
 Constant of tangent galvanometer by calculation 84 
 
 Calibration of ammeter by means of tangent galvanometer .... 87 
 
 Correction factor of an ammeter 89 
 
 Ayerton shunt 91 
 
 Relation between current and deflection on moving coil galvan- 
 ometer 93 
 
 Making of ammeter from a galvanometer 97 
 
 Difference of potential — calorimeter method 98 
 
 Proof of Ohm's law— " " 101 
 
 Testing of voltmeter— " " 101 
 
 Making of voltmeter from a galvanometer 102 
 
 Electromotive force of cells 103 
 
 Electro-chemical series 104 
 
 Resistance — substitution method ^ 105 
 
 ' ' — fall of potential method 106 
 
 Resistance of uniform wire is proportional to its length 108 
 
 Resistance — ^Wheatstone wire bridge method 110 
 
 ' ' — ^Wheatstone box bridge method 112 
 
 Temperature coefficient 114 
 
 Measurement of various resistances 115 
 
 Adjustment of a resistance 115 
 
 Measurement of high resistance 116 
 
 Specific resistance of an electrolyte 117 
 
 Electromotive force — Ohm's method 118 
 
 " " — Poggendorff's method 119 
 
 " " — potentiometer method 120 
 
 Calibration of voltmeter — potentiometer method 122 
 
 " "mil-ammeter— " " 125 
 
 " "ammeter— " " 126 
 
 " " " —voltmeter method 129 
 
 Theory of baUistic galvanometer 130 
 
 Constant of a ballistic galvanometer 132 
 
 Damping correction for a ballistic galvanometer 134 
 
 Determination of the constant of a ballistic galvanometer 135 
 
 Quantity of electricity • • 138 
 
 Capacity of mica condenser — absolute method 140 
 
xiv CONTENTS 
 
 FAGS 
 
 Constant of ballistic galvanometer — condenser method 142 
 
 Capacity of Leyden jar 143 
 
 Capacity — comparison method 144 
 
 Residual charges 145 
 
 Electromotive force — condenser method 145 
 
 Insulation resistance — method of leakage 146 
 
 " " — direct deflection method 148 
 
 Work done in cutting lines of force 149 
 
 Lines of induction within a solenoid 151 
 
 Coefficient of mutual induction — calculated 154 
 
 Relation between quantity of induced electricity and the cur- 
 rent in the primary 156 
 
 Relation between the quantity of induced electricity and the 
 
 resistance in the secondary circuit 157 
 
 Coefficient of mutual induction 157 
 
 self-induction — calculated 158 
 
 " — Zeleny's method 160 
 
 " " — Anderson's method . . . '. 163 
 
 Comparison of two coefficients of self-induction 168 
 
 Absolute determination of resistance 170 
 
 Figure of merit by three methods '. 171 
 
 Number of fines in a magnetic field 174 
 
 Horizontal component of earth's magnetic field — earth inductor 
 
 method 175 
 
 Inclination of earth's magnetic field — earth inductor method . . 178 
 
 Intensity of the earth's field 179 
 
 Magnetic properties of iron — ^baUistic galvanometer method . . . 179 
 
 SOUND 
 
 Velocity of a transverse wave along a cord 183 
 
 Experimental test if ii=x;\/~" 184 
 
 Velocity of sound in air 185 
 
 asoUd 186 
 
 Frequency of a fork by stationary waves 188 
 
 by graphical method 189 
 
CONTENTS XV 
 
 LIGHT 
 
 PAGE 
 
 Spherical mirrors 190 
 
 Focal lengths of radii of curvature of mirrors 192 
 
 Lenses ' 192 
 
 Focal lengths of lenses 193 
 
 Magnifying power of microscopes and telescopes 194 
 
 Adjustment of spectrometer 200 
 
 Angle of incidence is equal to angle of reflection 203 
 
 Angle of prism 204 
 
 Refractive index of glass 205 
 
 Wave-length of hght (two methods) 206, 208 
 
 Polarized light 209 
 
 APPENDIX 
 
 Theory of errors 213 
 
 TABLES 
 
 Conversion factors 219-221 
 
 Time 221 
 
 Logarithm 221 
 
 Data with regard to earth ■ 221 
 
 Velocity 221 
 
 Data with regard to various elements 222 
 
 Numerical value of error factor 226 
 
 Alloys 227 
 
 Gases 227 
 
 Density of water 228 
 
 Boiling point of water 229 
 
 Barometric corrections 229, 230 
 
 Pressure of mercury vapor 230 
 
 ' ' water vapor 231 
 
 Dry and wet bulb hygrometer 232 
 
 Wave-lengths 233 
 
 Index of refraction 233 
 
 Elastic constants 233 
 
 Compressibihty 234 
 
 Electrical units 234 
 
XVI CONTENTS 
 
 FAGIl 
 
 Definition of electrical units by London conference (1908) 236 
 
 Weston normal cell 238 
 
 Relation between electrical units 238 
 
 Squares, cubes and reciprocals 239 
 
 Logarithims of trigonometric functions 240 
 
 Trigonometric functions 241 
 
 Logarithms of numbers 242 
 
 Approximate expressions 244 
 
 Index 245 
 
MECHANICS 
 
 EXPERIMENT 1 
 
 TO MEASURE A LENGTH IN CENTIMETERS AND IN 
 INCHES DIRECTLY BY MEANS OF A SCALE 
 
 Apparatus. — A metric scale; an English scale; a cyl- 
 inder whose length is to be measured. 
 
 Method and Manipulation. — Place the scale against 
 the side of the cylinder so that the two are parallel, and 
 so that the scale extends beyond the ends of the cylinder. 
 Do not try tp adjust until a line on the scale is even 
 with an end of the cylinder. The distance that the 
 cylinder extends into a division must be determined 
 by estimating to tenths. A division is the space be- 
 tween the centers of two adjacent lines on the scale. 
 Divide this distance mentally into halves and the half 
 into which the cylinder or object extends into five parts, 
 each of which then is a tenth of the whole division. 
 
 The difference between the readings obtained for the 
 two ends, reckoned in each case from the zero end of 
 the scale, gives the length of the cylinder. 
 
 In repeating observations place the cylinder each time 
 opposite a different portion of the scale. 
 
 Obtain an average length in inches and in centimeters 
 by taking several readings on each scale. 
 
 Solve for the number of centimeters in an inch. 
 
 Sources of Error. — The scale and object may not be 
 parallel. 
 
 The point read on the scale may not be exactly oppo- 
 
2 _ MECHANICS 
 
 site the end of the object. This error is generally due 
 to parallax, i.e., the eye is not in a line perpendicular 
 to the scale at the point where the reading is taken. 
 Refbbence. — Stewart & Gee, Vol. I, pp. 1-46. 
 
 EXPERIMENT 2 
 TO DETERMINE THE VOLUME OF A CYLINDER 
 
 Method. — The length and diameter of the cylinder 
 are measured, and from these quantities the volume is 
 computed. 
 
 Apparatus. — Vernier caliper; micrometer caliper; mag- 
 nifying glass; the cylinder used in Exp. 1. 
 
 I. To MEASURE THE LENGTH OP THE CYLINDER WITH 
 THE VERNIER CALIPER. 
 
 The vernier caliper, Fig. 1, consists of a graduated 
 
 11^ 
 
 W 
 
 Fig. 1. 
 
 limb a, with a fixed jaw perpendicular to it, 
 and a second movable jaw c, parallel to the 
 first and provided with a series of lines e, 
 called a vernier. The movable jaw can be 
 clamped anywhere along the limb a. Since 
 the initial line on the vernier (at the left in the figure) 
 coincides with the zero line on the scale of the hmb when 
 the jaws are together, the position of this line at any time 
 gives on the scale the separation of the jaws. When 
 
MECHANICS 3 
 
 the initial line falls between two divisions on the scale, 
 the fraction of a division is not estimated by the eye, 
 but is measured by means of the vernier. 
 The Vernier. 
 
 Let Z=the length of a division on the scale, 
 V = the length of a division on the vernier, 
 n= the number of divisions on the vernier. 
 
 The number of divisions n, on the vernier equals in 
 length (n-1) divisions on the scale of the limb. 
 
 In Fig. 2 the initial line of the vernier coincides with 
 line 4 on the scale. Line 1 is, then, the distance (l-v) 
 from line a; line 2 is the distance 2(l-v) iroin b; etc. 
 
 n-/ E Limb 
 
 III! 
 
 iiTun 
 
 Z3« t 
 
 i l l 1 1 1 
 
 ] \ 
 
 ♦ Vernier 
 
 " ^ 
 
 1 MM 
 
 1 lllllll 
 
 II 
 
 1 
 
 
 
 III! 1 ill 
 
 
 
 
 1 
 
 10 
 
 
 Fig. 2. Fig. 3. 
 
 If now the vernier is moved until line 1 coincides with the 
 line a, the initial line of the vernier is the distance {l~v) 
 from line 4. In this case the reading of the initial line 
 is 4:-\-{l-v). If line 2 coincides with h, the initial Hue 
 is 2{l-v) from 4; etc. It is seen that the fraction of a 
 division the initial line is away from 4= (Z-v) multiplied 
 by the number of the vernier line which exactly coin- 
 cides with a line on the scale. The quantity {l-^) is 
 called the " least count " of the vernier. The value of 
 {l-v) is determined in the following manner: 
 vn = l{n—V) = ln—l (by construction of the vernier), 
 
 ln—vn=l • 
 
 I 
 {l-v) = - 
 n 
 
MECHANICS 
 
 Fig. a. 
 
 b 
 
 Fig. 6. 
 
 The least count thus equals the length of a division 
 on the scale divided by the number of divisions on the 
 vernier. The reading of the vernier in Fig. 3 is 4.23. 
 
 [On some verniers n divisions do not equal (n-1) 
 divisions on the scale but some other whole number. 
 The value of the least count in these cases is determined 
 by similar reasoning.] 
 
 When no line on the vernier exactly coincides with 
 a line on the scale, i.e., when two adjacent lines of the 
 vernier are within a division on the scale, a fraction of 
 the least count must be estimated. 
 
 When the vernier lines are at equal distances from 
 
 the scale lines, as 
 shown in Fig. a, the 
 fraction of the least 
 count is 0.5. In Fig. b, 
 when one line appears 
 twice as far from a 
 scale line as the other, 
 the reading is 0.3 of the least count; in Fig. c it is 
 0.2. 
 
 Manipulation. — Find the least count of each vernier 
 on the caliper. Bring the jaws together and note the 
 reading of the initial line when the jaws touch. 
 
 The difference in the reading when the jaws are 
 together and the reading when the object is between 
 them, is the length of the object. Obtain the length 
 of the cylinder in terms of each scale on the caliper. 
 Express the average length in centimeters. 
 
 Sources of Error. — 1. Jaws may not be parallel. 
 
 2. The object measured may not be parallel to the 
 scale. 
 
 3. Too great pressure may bend jaws or change dimen- 
 sions of the object measured. 
 
 C 
 
 Fig. c. 
 
MECHANICS 5 
 
 II. To MEASURE THE DIAMETER OF THE CYLINDER WITH 
 THE MICROMETER CALIPER. 
 
 The micrometer caliper, Fig. 4, consists of a uniform 
 screw moving in a fixed nut. 
 
 The nut d, has a scale /, and carries the arm a. The 
 head of the screw b, with its divisions at e, is called the 
 micrometer head. The end of the screw c, can be 
 moved against any object placed in the opening ni. 
 
 Fig. 4. 
 
 Examine the caliper and determine the following 
 quantities : 
 
 1. Length of a division on the hnear scale on the nut. 
 
 2. The pitch of the screw. 
 
 3. Number of divisions on the micrometer head. 
 
 4. The distance advanced by the screw when it is 
 turned through one division on the micrometer head. 
 
 5. The fraction of a millimeter that can be obtained 
 by estimating. 
 
 Manipulation. — Determine the zero reading of the 
 micrometer. Do not force the screw. Stop when the 
 head first touches the object. A friction head g, if 
 provided, insures the appHcation of the proper pres- 
 sure. Turn until the friction head begins to move, or, if 
 there is a ratchet attachment, until it clicks. 
 
 Place the cylinder on the table when clamping it 
 
6 
 
 MECHANICS 
 
 between the jaws of the caliper. This practically in- 
 sures getting the cylinder perpendicular to the jaws. 
 
 Measure the diameter at several points along the 
 length of the cylinder and determine the average diameter. 
 
 Sources of Error. — 1. The pitch of the screw may not 
 be known accurately, and may vary in different parts. 
 
 2. Change of temperature of the caliper due to the 
 heat of the hand may change the zero reading during 
 the experiment. 
 
 III. Having obtained the length and diameter, 
 
 COMPUTE THE VOLUME OF THE CYLINDER IN CUBIC CENTI- 
 METERS. 
 
 EXPERIMENT 3 
 
 TO DETERMINE THE RADIUS OF CURVATURE OF A 
 SPHERICAL SURFACE BY MEANS OF A SPHE- 
 ROMETER 
 
 Apparatus. — A spherometer; a piece of plate glass; 
 
 a sheet of paper; dividers; 
 metric scale; a spherical 
 surface. 
 
 The spherometer consists 
 of a screw with a microm- 
 eter head. This screw 
 turns in a nut which is 
 supported by three legs at 
 equal distances from each 
 other. The screw and legs 
 end in blunt points. The 
 former is in the center of 
 the triangle formed by 
 the latter. A vertical scale 
 
 attached to the nut is used to determine the number of 
 
 whole revolutions of the micrometer head. 
 
 Fig. 5. 
 
MECHANICS 
 
 Determine the following quantities: 
 
 1. Length of a division on the vertical scale. 
 
 2. Pitch of the screw. 
 
 3. Value of a division on the micrometer head. 
 Method and Manipulation. — The gpherometer is 
 
 placed on the plate glass, and the micrometer head is 
 turned until the screw just touches the plate, and is thus 
 in the same plane as the tips of the legs. To deter- 
 mine this condition, press very lightly on the nut with 
 two fingers, one over each of two legs, and tap with the 
 other hand over the third leg. If the spherometer hob- 
 bles the screw is too low, and if it does not, it is likely 
 to be too high. Find the point where the spherometer 
 will not hobble, but will do so if the micrometer head 
 is turned through a tenth 
 of a division. The read- 
 ing of the scale and micro- 
 meter at the former posi- 
 tion gives the zero reading. 
 
 The spherometer is then 
 placed on the spherical 
 surface. The micrometer 
 head is again turned 
 until the four points touch 
 the surface. The scale 
 and micrometer are read, 
 and the distance a of 
 Fig. 6, through which the 
 screw moved from its 
 zero position, is deter- 
 mined. 
 
 Turn the screw until 
 the four points are again in the same plane. 
 
 Place the instrument on a piece of paper, and press 
 lightly until the four points make small dents in the 
 
 Fig. 6. 
 
8 MECHANICS 
 
 paper. By aid of dividers and scale, the average dis- 
 tance DB between the legs and the point of the screw 
 is determined 
 
 The radius of curvature is calculated from the dis- 
 tances a and DB. 
 
 Suggestion. — The vertical scale usually has the zero 
 mark at the center. This is liable to confuse beginners. 
 It is better to call the highest mark on the scale zero. 
 
 EXPERIMENT 4 
 TO DETERMINE THE MASS OF A BODY 
 
 Apparatus. — The analytical balance. 
 
 The arrangement of parts is as shown in the figure. 
 The beam is usually supported by an agate knife-edge 
 on an agate surface. The beam can be raised by means 
 of a milled head in front of the case so that the knife- 
 edge is not in contact with the supporting surface. 
 
 There are two adjustments that should be understood. The 
 ten-sitiveness of the balance is dependent upon the distance be- 
 tween the knife-edge and the center of gravity of the beam. 
 The shorter this distance is, the greater the sensitiveness. This 
 distance can be changed by raising or lowering the nut that is 
 on the vertical screw above the beam. 
 
 Attached to the beam is a horizontal screw with a movable 
 nut. By the moving of this nut, the beam may be adjusted so 
 that the pointer comes to rest at the center of the lower scale. 
 
 On the beam, or attached to it, is a scale on which 
 the distance between the central knife-edge and the 
 edge supporting the scale pan is usually divided into 
 ten major parts. Smaller divisions, if present, are usu- 
 ally not used. Over this scale hangs a rider which 
 can be placed at any point on the scale. This rider 
 weighs ten milligrams and is, therefore, when placed 
 at the first division on the scale, equivalent to one 
 
MECHANICS 9 
 
 milligram placed in the scale pan; when at the second, 
 to two milligrams; etc. Brass weights are used for 
 
 Fig. 7. 
 
 denominations ranging from the largest down to one 
 gram. For fractions of a gram down to .01 gram or ten 
 milligrams, platinum or aluminum weights are used. 
 
10 MECHANICS 
 
 Less than ten milligrams is obtained by means of the 
 rider. 
 
 Method and Manipulation. — Release the beam care- 
 fully, and if it is at rest, set it in vibration by raising 
 and again releasing. Note at what place the pointer 
 turns, counting the divisions from the left end of the 
 scale and estimating to tenths of a division. Take five 
 successive turning points. Call a, c, e, the readings on 
 one side, and b, d, the readings on the other. Then the 
 reading on the scale at which the pointer would stop is 
 
 b+d a-\-c+e 
 
 ^- 2 • 
 
 This is called the zero point. It is necessary to take 
 one more reading on one side because of the decrease 
 in the vibrations due to friction. 
 
 Raise the beam, place the body to be weighed in the 
 left pan, and add weights to the right until the pointer 
 swings within the scale. Add weights only when the 
 beam is raised, and try the effect of each addition by 
 very carefully lowering the beam' and noting which way 
 the pointer begins to move. Take readings for turning 
 points and average as in the case above. This gives 
 the point of rest (R). If the point .of rest happens to 
 be the same as the zero point, the mass of the weights is 
 equivalent to that of the body. If, however, the point 
 of rest is less than the zero point, the mass of the weights 
 is too large, and if the point of rest is larger, the mass 
 of the weights added is too small. In the last two 
 cases we must determine what weight would have to 
 be subtracted or added in order to bring the pointer to 
 the zero reading. In order to do this we must know 
 what is called the sensibilitij of the balance, i.e., the 
 
MECHANICS 11 
 
 deflection of the pointer that one milligram will produce. 
 To obtain this, change the weights by one milligram, 
 by means of the rider, and determine, in the same manner 
 as above, another point of rest which we will call the 
 sensibility point (S). The difference betAveen the poijil 
 of rest and sensibility point gives the sensibility. 
 
 The weight to be added or subtracted is numerically 
 equal to the difference between the zero point and the 
 point of rest, divided by the sensibility. 
 
 Suggestions. — In getting the sensibility, add one mil- 
 ligram if the point of rest is larger than the zero point, 
 and subtract if it is less. 
 
 Always use the largest weights permissible. This 
 facilitates the weighing, as it reduces the number of 
 separate weights needed and reserves the smaller weights 
 for the final adjustment. 
 
 The method described above is usually spoken of as 
 the method of vibrations. 
 
 Errors and Precautions. — Before changing the weight, 
 and when through weighing, always raise the beam of 
 the balance. This should be done when the pointer is 
 at the center of the scale. 
 
 Since air currents influence the swing of the beam, 
 readings must not be taken immediately after closing 
 the case of the balance. 
 
 Take the reading for the turning point when the 
 eye, the pointer and its image in the mirror back of 
 the scale, are in the same line. This avoids the error 
 due to parallax. 
 
 Always handle the weights with the forceps. Never 
 touch them with the fingers. 
 
 When through with a weight, always return it to the 
 proper place in the case. 
 
 As the zero point is affected by temperature changes 
 and slight shifting of the parts of the balance, a zero 
 
12 MECHANICS 
 
 point must be determined after the load has been re- 
 moved. The average of the first and last zeros is used 
 in the interpolation. 
 
 Reference. — Stewart & Gee, Vol. I, pp. 63-94. Kohlrausch, 
 pp. 31^3. 
 
 EXPERIMENT 5 
 
 RATIO OF THE LENGTHS OF THE BALANCE ARMS 
 
 Method. — Place a body having a mass of about fifty 
 grams in the left pan, and obtain the apparent mass, 
 using the method of vibrations. Then obtain the ap- 
 parent mass when the body, is in the right pan. 
 
 li w = the mass of the body, 
 
 a = the apparent mass of the body when in the 
 
 left pan, 
 6 = the apparent mass of the body when n the 
 
 right pan, 
 r = the length of the right arm, 
 l = the length of the left arm, 
 
 then wl = ar. 
 
 and ii)r = bl 
 
 r lb 
 
 Remarks. — If the left pan is used for the body, the 
 apparent mass must be multiplied by the ratio j in 
 
 order to obtain the correct mass, since w = aj. 
 
 V 
 
 If the ratio is not known, the correct mass can be 
 obtained by resorting to double weighing as above and 
 taking the square_root of the product of the apparent 
 masses, i.e., w = \/ab. 
 
MECHANICS 
 
 13 
 
 Besides the inequality of the balance arms, there are two other 
 main sources of error that are not discussed here but which should 
 be considered in exact weighing. One is the error due to difference 
 in the buoyant force of the air on the weights and the body when 
 their volumes are not equal, and the other is the inaccuracy of the 
 weights used. 
 
 EXPERIMENT 6 
 TIME OF VIBRATION OF AN OSCILLATING BODY 
 
 Apparatus. — An iron cylinder suspended by means 
 of a wire; a watch. 
 
 Method. — The vibrations of a torsional pendulum 
 are isochronous for a large angle of vibra- 
 tion provided the supporting wire is not 
 twisted beyond the elastic limit. In the 
 case of a pendulum or a vibrating magnet 
 this angle must be small. 
 
 The t me between two turns, A and B, 
 Fig. 8, is obtained. This time divided by 
 the number of intervening vibrations gives 
 the desired period. 
 
 When the vibrations are slow the time of a 
 turn is determined in the following manner : 
 A white vertical line is marked on the 
 vibrating body. Two vertical points are 
 placed so that they are in line with the 
 white line on the body when it is at rest. 
 The body is set in vibration and the time of 
 day is recorded at every instant that the 
 white mark passes the line of the verticals. To 
 do this, the observer taps with a pencil the 
 instant of passage and the assistant notes 
 the time to a fraction of a second, and records the same. 
 
 The time of about fourteen of these passages is 
 taken, the direction of the first being noted. 
 
 Fig. 8. 
 
14 
 
 MECHANICS 
 
 By referring to Fig. 8, it is seen that the average of 
 the times of the 7th and 8th, the 6th and 9th, etc., gives 
 the time of the turning point A, each average being of 
 course hable to the mean of the errors in the two observa- 
 tions. To reduce this error, the mean of all these averages 
 is taken as the time of the turning point A and hence 
 this value of the time is based on all the observations 
 in the set. 
 
 After an interval of about fifteen minutes from the last 
 passage of the first set, another similar set is taken, the 
 direction of the first vibration being noted. 
 
 This second set is treated in the same 
 manner as the first, hence giving the time 
 of a second turning point B. 
 
 The number of vibrations n between A 
 and B must next be determined. To do 
 this, find the approximate time t of one 
 vibration by dividing the difference between 
 the times of the first and next to the last 
 passage in each set by the number of their 
 intervening vibrations. Do likewise for the 
 second and the last passage of each set, and 
 average all the four values thus obtained. 
 Then divide the difference between the times 
 of A and B by t. If A and B are on the 
 same side of the center, n must be a whole 
 and even number. If, then, the difference 
 between the times of A and B divided by t 
 gives a number plus a fraction, n must be 
 the nearest even number. 
 
 The difference between the times of A and 
 Fig. 9. B divided by n gives the desired period. 
 
 When the vibrations are rapid, i.e., when the 
 interval between two successive passages is short, and hence 
 not ample for observing and recording the time of each. 
 
 -3B 
 
MECHANICS 15 
 
 the above method must be modified. In this case, it is 
 better to note the time of turning instead of the time of 
 passage. The observer taps at every 2d, 5th, or 10th 
 turn of the pedulum, depending on the rapidity of the 
 vibrations. Care must here be exercised in counting the 
 intervening turns. Only those on one side are counted. 
 
 Take the time of about five turns. If this number 
 is taken, the mean of the 1st, 2d, 3d, 4th, and 5th gives 
 the time of turn A based on all the observations in the 
 set. 
 
 Continuing on from the last turn in the first set, the 
 observer counts about one hundred turns, announcing the 
 approach of the last by counting aloud, and tapping at 
 the 100th. Then a set similar to the first is taken, from 
 which the time of turn B is obtained as in the case of A . 
 
 The difference between the times of A and B divided 
 by the number of intervening vibrations gives the de- 
 sired period. 
 
 EXPERIMENT 7 
 BENDING OF A ROD— HOOKE'S LAW 
 Apparatus. — A rod supported by a fulcrum at each 
 
 Fig. 10. 
 
 end has a pan attached at its center. Over the middle 
 of the rod is a micrometer, by means of which the strain 
 is measured. 
 
16 
 
 MECHANICS 
 
 Method. — The micrometer is read when the point of 
 the micrometer screw just touches the top surface of 
 the rod. Care must be exercised that the same degree 
 of contact is obtained each time. To repeat a reading 
 turn the micrometer back and reset. Obtain in this 
 way an average reading for each of several loads. 
 
 Graphic Interpretation of Observations. — The rela- 
 tion between strain and stress is best shown by the use 
 of coordinate paper. Plot the stresses as abscissas, and 
 the strains as ordinates, choosing such units as will 
 
 r~ 
 
 I 
 
 I 
 
 I 
 
 I 
 
 i! 
 
 1 1 
 
 Jt-lL 
 
 
 
 
 :: 
 
 , : 
 
 II 1 II 1 1 
 
 
 
 it 
 
 ■ V 
 
 ■ <f 
 :t 
 
 ■ Y 
 
 
 ■■■■■■■■■■■■■■- ■■■■■i- 
 
 ■-■;."• -i J 
 
 lUfl nTTnTTrTn 
 
 :::S' -- 
 
 ■ ■ ■ t - ■ ■ 
 
 ■-t - ■ 
 
 -■-t - 
 
 
 ;i 
 
 . L 
 ■1 
 
 :l 
 
 \\ 
 -A 
 - 1 
 
 
 * a —- 
 
 4 
 
 1 mw 
 
 
 -i 
 
 > 
 
 
 Fig. 11. 
 
 make the two largest quantities about equal to the 
 length of the axes of reference. 
 
 In Fig. 11 suppose that the strain c is produced by 
 the stress a. Mark the point at which the vertical and 
 horizontal lines from the extremities of a and c meet. 
 Find similar points for all of the observations taken. 
 
 As observational errors cause the points to vary 
 from their true position, a uniform curve drawn so as to 
 leave as many points on one side as on the other will 
 best represent the result. 
 
 If, as in Fig. 11, the curve is a straight line, the re- 
 lation is that one quantity varies directly as the other. 
 
MECHANICS 
 
 17 
 
 Since c, d, and a, b, are homologous sides of similar 
 triangles, c : d :: a : b, i.e., the strains are to each other 
 as the stresses, as is stated in Hooke's law. 
 
 References. — Properties of Matter, by Poynting & Thom- 
 son, p. 85. Experimental Elasticity, by Searle, p. 100. 
 
 EXPERIMENT 8 
 
 J°L 
 
 ELONGATION OF A WIRE 
 
 Apparatus. — A metal frame is suspended from the 
 cei.ing. The wire to be used is attached 
 to the upper cross bar of this frame, and 
 a pan for weights is attached to its lower 
 end. Just above the pan, clamped to the 
 wire, is a projecting index. 
 
 To the lower end of the frame is at- 
 tached a micrometer screw which is so 
 placed that the point of the screw can be 
 made to touch the under surface of the 
 projecting index. 
 
 Method. — Place a weight in the pan so 
 that the wire becomes straight. Read the 
 micrometer when the point of the screw 
 just touches the index. Readjust and 
 read, several times. Lower the screw, 
 and add a weight. Again adjust, and 
 read the micrometer, several times. Con- 
 tinue until four weights in addition to the 
 zero load have been added. 
 
 Plot the readings on coordinate paper 
 as in the last experiment. F g l'' 
 
 Young's Modulus (ilf) is the ratio of 
 the stress per unit area to the strain per unit length. 
 
18 
 
 MECHANICS 
 
 Measure the length of the wire, and by means of a 
 micrometer caliper obtain the diameter of the wire at 
 several places. 
 
 Compute Young's Modulus from the ratio of strain 
 to stress shown by the curve. 
 
 EXPERIMENT 9 
 
 ELONGATION OF A SPIRAL SPRING 
 
 Apparatus. — A spiral spring, carrying a pan for weights, 
 is suspended in front of a mir- 
 rored scale. 
 
 Method. — Readings are 
 taken on the mirrored scale 
 opposite the point of the lower 
 hook when no weights are in 
 the pan, and after each addi- 
 tional weight until about four 
 have been added. Each read- 
 ing must be taken when the 
 eye is in line with the point of 
 the hook and its image in the 
 mirror. 
 
 Plot the readings obtained, 
 
 and determine if Hooke's law holds within the limits of 
 
 observational error. 
 
 Fig. 13. 
 
 Repbbbnce. — Properties of Matter, by Poynting & Thomson, 
 p. 103. 
 
MECHANICS 19 
 
 EXPERIMENT 10 
 
 HOOKE'S LAW AND SIMPLE HARMONIC MOTION 
 
 Apparatus. — Same as in Exp. 9. 
 
 Method and Manipulation. — In the preceding experi- 
 ment it was found that 
 
 Stress „ 
 
 5r: — — =K, a. constant. 
 
 strain 
 
 When a spring carrying a constant load is elongated 
 beyond the point of rest, the upward contracting force 
 is greater than the force of gravity acting downward 
 upon the attached mass. If the spring is shortened, 
 the force of gravity is greater than the contracting 
 force. At the point of rest the two forces are equal. 
 
 For any displacement x of the spring from the point 
 of rest, there is a force tending to restore it that is equal 
 to ma, where a is the acceleration it gives to the attached 
 mass m. 
 
 From the last experiment 
 
 — = /t, or a = Ki X. 
 z 
 
 This is the condition for harmonic motion for which 
 the general equation is 
 
 
 If the mass m produces a static elongation s, then 
 when the mass m is vibrating vertically and is at a dis- 
 placement equal to s, the acceleration is g; therefore, at 
 this instant 
 
 X _s 
 
 a g. 
 
20 MECHANICS 
 
 The equation for the time thus becomes 
 
 Determine from the curve obtained in Exp. 9 the 
 elongation s that a mass m will produce. 
 
 Substitute the value of s in the equation, and com- 
 pute the time t. Using the mass m that produced the 
 elongation s, cause it to vibrate vertically, and deter- 
 mine the time of vibration. This time should agree 
 within the limits of observational error with the com- 
 puted result. 
 
 EXPERIMENT 11 
 
 TORSION OF A ROD 
 
 Apparatus. — A rod is firmly attached at one end to 
 
 the center of an iron cylinder and 
 
 at the other to a graduated disc on 
 
 which the angle of torsion may be 
 
 read. Attached to the circimiference 
 
 of the disc is a cord to the free end 
 
 of which a weight pan is connected. 
 
 Method and Manipulation. — Take the reading on the 
 
 disc when the pan is empty, and after each additional 
 
 load until several have been added. Care must be taken 
 
 to avoid parallax and friction between the various parts. 
 
MECHANICS 21 
 
 Plot the readings obtained, and determine if Hooke's 
 law holds. 
 
 Rbfbeencb. — Properties of Matter, by Posmting & Thomson, 
 p. 78. 
 
 EXPERIMENT 12 
 HOOKE'S LAW AND SIMPLE HARMONIC MOTION 
 
 Apparatus. — Same as in Exp. 11, and in addition a 
 watch. 
 
 Method. — Since, as found in the preceding experiment, 
 the strain is proportional to stress, the rod, when set 
 in motion as a torsional pendulum, will vibrate with 
 simple harmonic motion. In order to compute the time 
 of vibration the general equatiori for the period of a 
 simple harmonic motion, 
 
 1 = 2%^ —, 
 
 must be altered so as to involve quantities that can in 
 this case be readily measured. 
 
 For an angular displacement 6, a point at a distance 
 r from the axis has a linear displacement x = rQ. Like- 
 wise if a is the angular acceleration, the linear acceler- 
 ation at a distance r is a = ra, 
 
 a rot. x' 
 
 t = 2%. 
 
 The moment of the couple necessary to produce a 
 displacement 6 is T%, where T is the moment of tor- 
 sion, i.e., the moment of the couple necessary to pro- 
 duce a displacement of a radian, but 
 
 r9 = «/ (Duff, p. 89.) 
 
22 MECHANICS 
 
 where I is the moment of inertia of the system. 
 
 £ = i = -L 
 ■■ a a T' 
 
 and t 
 
 = 2x^. 
 
 If, therefore, 7 and T are known, the period t can be 
 determined. • 
 
 To obtain T determine from the curve in Exp. 11 the 
 number n of circumference divisions in the angle of 
 torsion produced by a mass m. This angle 9, in radians, 
 
 equals ^2%, where N is the number of di^asions in the 
 
 whole circumference. If r is the radius of the 
 graduated disc, then 
 
 .. 1 Q . 
 
 Compute / from the dimensions of the cyl- 
 inder. 
 
 Substitute the values of T and I in 
 
 equation and solve for t. 
 
 Remove the rod with its attached disc 
 
 ] and cylinder, and suspend from bracket so as 
 
 Fig 15 ^ form a torsion pendulum as shown in Fig. 
 
 15. Determine the time of vibration. This 
 
 should check with the computed time if the vibration is 
 
 simple harmonic. 
 
MECHANICS 23 
 
 EXPERIMENT 13 
 
 MOMENT OF INERTIA OF AN IRREGULAR BODY ABOUT 
 AN AXIS THROUGH ITS CENTER OF INERTIA 
 
 Method. — When the body is suspended by a wire 
 whose moment of torsion is T, its time of vibration as a 
 torsional pendulum is 
 
 « = 2u^y 
 
 t = 2%ylyj, (See Exp. 12.) 
 
 If a body of known moment of inertia I' is suspended 
 from the same wire the time of vibration , . 
 
 t' = 2'K. 
 
 I = —I' 
 
 Vr 
 
 o 
 
 Apparatus. — The body whose moment of in- pj^ ^g 
 ertia is to be determined; a body whose moment 
 of inertia can be calculated; a steel wire with proper 
 attachments. 
 
 Manipulation. — Determine the time of vibration for 
 each of the bodies to an accuracy comparable Avith the 
 accuracy in the other measurements. Compute I' from 
 the mass and dimensions of the body. Solve for I. 
 
24 MECHANICS 
 
 EXPERIMENT 14 
 TO TEST EXPERIMENTALLY THE EQUATION I=Io+Md2 
 
 Method. — The moment of inertia / of a body may be 
 determined experimentally by adding it to a torsional 
 pendulum of known moment of inertia /', and known 
 time of vibration t' , and obtaining the new time of 
 vibration t. 
 
 Thusi' = 2TrJy> < = 2xJ: 
 
 T+T 
 
 <2 — /'2 
 
 If two similar bodies of such form that their moments 
 of inertia can be computed from 
 their masses and dimensions be 
 placed symmetrically on opposite 
 sides of the torsional pendulum so 
 as not to disturb its position, the 
 equation 7 = 7o+-^^ci^ can be tested 
 experimentally. 
 
 Apparatus. — The torsion pendu- 
 
 jj-jQ ^7 lum used in Exp. 12; two small 
 
 cylinders. 
 
 Manipulation. — Obtain by computation from its 
 dimensions the moment of inertia I' of the torsional pen- 
 dulum, and determine its time of vibration. 
 
 Place the two cylinders on opposite sides of the 
 pendulum so that their centers are equidistant from 
 and in line with the wire, and obtain the time of vibra- 
 tion with the same degree of accuracy as above. 
 
 By means of the values /', t and t', thus obtained, 
 compute I, the moment of inertia of the cylinders about 
 the axis around which they rotated. 
 
 Obtain the mass and dimensions of the cylinders 
 
MECHANICS 
 
 25 
 
 and their distance d from the axis of rotation. The 
 value of I obtained by substituting these values in the 
 equation I = Io+M(P should agree to within the limits 
 of observational error with the experimental result. 
 Reference. — Duff, p. 67. 
 
 EXPERIMENT 15 
 
 TO PROVE EXPERIMENTALLY THAT/? = Va^ + b^ + 2ab COS 6 
 
 —TO CONSTRUCT THE PARALLELOGRAM OF 
 
 FORCES— TO SHOW THAT SF COS (]) = 
 
 Apparatus. — A vertical board is provided with two 
 spiral springs supported by vertical arms. 
 
 Method and Manipulation.— 1. R = y/ (^ -\rW' ^2ah cos 6. 
 Determine K, the number » o, 
 
 of grams necessary to pro- 
 duce an elongation of one 
 centimeter, for each spring 
 by measuring its length I 
 before and its length V 
 after the weight W is at- 
 tached to the spring so 
 that it hangs vertically. 
 To obtain the length of 
 the spring, place one point 
 of a divider in the depres- 
 sion in the supporting pin, 
 and the other even with 
 the end of the wire that 
 projects near the lower 
 end. The distance be- 
 tween the points of the F^°- 18- 
 dividers is obtained in centimeters from a steel scale . Then 
 
 W 
 
 K = 
 
 I'-V 
 
26 MECHANICS 
 
 Connect the springs and the weight W by means of a 
 fine thread as shown in Fig. 18. Measure the length 
 of the springs while under tension. 
 
 The elongation of each spring times K, gives the forces 
 a and b. 
 
 To determine cos 6 attach a paper to the vertical 
 board and carefully place two dots beneath each arm of 
 the string. Lines drawn through these dots will give 
 the direction of the forces a, h, and R. From some point 
 on one of the lines A and B drop a perpendicular to the 
 other. The quotient of the proper sides of the triangle 
 formed gives the cosine desired. 
 
 The values of a, b and cos 6 substituted in the equa- 
 tion to be tested should give a value of R that checks 
 withT^. 
 
 2. ParaUehgram o/ /orces.— To construct the parallelo- 
 gram of forces lay off on line A the force a by choosing 
 a convenient number of units of force to the unit of 
 length used. Likewise lay off force b, using the same 
 unit as before. With these two lengths as sides construct 
 the parallelogram. The vertical diagonal of this paral- 
 lelogram should have the direction of the force R, and 
 the length of this diagonal times the number of units of 
 force to the unit length chosen for laying off the forces 
 a and b, should check with the value of W. 
 
 3. 2F cos 4>. — Through the point o draw a line cd in 
 any direction. From any point on each of the lines 
 representing the forces a, b, R, drop a perpendicular to 
 the line cd, and, as above, determine the cosine of the 
 angles formed at o. By means of these cosines compute 
 the components along cd of the forces a, b, and R, and 
 show that 'SF cos ^=0, 
 
MECHANICS 
 
 27 
 
 EXPERIMENT 16 
 ANGULAR ACCELERATIOIir 
 
 Apparatus. — A heavy disc is mounted so that it may- 
 be rotated about its axis. A tuning fork is placed so 
 that its vibrations may be 
 traced by means of a stylus 
 on the smoked face of the disc. 
 The other face of the disc has 
 a circle that is graduated to 
 degrees. On each side of the 
 disc is a magnifying glass fitted 
 with a cross-hair. By the aid 
 of these, the length of a given 
 number of wave lengths on the 
 tracing may be determined in 
 degrees on the graduated cir- 
 cle. 
 
 Method and Manipulation. — By means of a small 
 amount of wax attach the string, and wind it on the 
 disc. To the free end of the string, attach a weight of 
 about 100 grams. Adjust the disc so that the stylus 
 will trace the vibrations of the fork on the smoked 
 surface. Let the weight start the cylinder from rest, 
 and then move the fork slowly so that the tracings will 
 not overlap. As the velocity of the disc increases, 
 the traced vibrations become longer. Let 6, 6', 9", etc., 
 equal consecutive angular lengths of forty vibrations. 
 Let n be the frequency of the fork. The lengths 6, 
 6', 6", etc., are the consecutive angular distances the 
 disc traveled during consecutive intervals of forty 
 
 Fig. 19. 
 
 The difference k between them is. 
 
 vibrations or — sec. 
 n 
 
 , i.- 40 40 
 
 therefore, the angular acceleration per — sec. per — sec. 
 
 n n 
 
28 MECHANICS 
 
 Obtain an average value for k from several measure- 
 ments, and compute a, the angular acceleration per 
 second per second. 
 
 EXPERIMENT 17 
 TO TEST EXPERIMENTALLY THE RELATION Fr = al 
 
 Apparatus. — Same as in Exp. 16. 
 
 Method and Manipulation. — Obtain a, the positive 
 angular acceleration of the disc in the manner explained 
 in Exp. 16. 
 
 In like manner obtain ai, the negative acceleration, 
 i.e., the acceleration when the speed of the disc was 
 decreasing because of friction. 
 
 Since the weight M descends Avith an acceleration 
 a, it acts on the cylinder with a force 
 
 P=M{g-a), 
 or since a = ar 
 
 P = M ig-ar) 
 
 where g is the acceleration of gravity and r is the radius 
 of the disc on which the string supporting the weight 
 is wound. 
 
 Of this force an amount equal to Mair is required to 
 overcome the frictional resistance, and the remainder is 
 effective in producing motion. Therefore the force re- 
 quired in the equation to be tested is 
 
 F=M[g-{a+o^i)r]. 
 
 This F times the radius of the drum gives tJie first 
 member of the equation. 
 
 The value of the moment of inertia / is computed 
 from the expression for the moment of inertia of a cyl- 
 inder about its axis. 
 
MECHANICS 29 
 
 I multiplied by a, gives the value of the second mem- 
 ber which should check with the first. 
 
 Instead of correcting for friction by determining the 
 negative acceleration, one may offset the resistance due 
 to friction by adding a weight to the string that is just 
 sufficient to cause the disc to move before the main 
 weight is added. In this case ai, above becomes zero. 
 
 EXPERIMENT 18 
 ANGULAR VELOCITY 
 
 Apparatus. — Same as in Exp. 16. 
 
 Method and Manipulation. — Wind the string about the 
 disc, and note the height of the weight. Release the 
 disc, and the instant the weight strikes the floor let 
 the stylus trace the vibrations of the fork on the smoked 
 surface. Repeat several times. The length of one of the 
 traced vibrations is the angular distance a particle on the 
 disc traveled during one period of the fork. 
 
 If = the length of a vibration in degrees, 
 n = the frequency of the fork, 
 wi = the angular velocity in degrees per second, 
 then 0)1= n9. 
 
 In getting a value for 6, measure the length of several 
 vibrations and divide by the number. 
 
 Since on account of friction the velocity of the disc 
 is not uniform, care must be taken to measure the 
 vibrations, in each case, corresponding to the instant 
 the weight strikes the floor. 
 
 From wi determine co the angular velocity in radians 
 per second. 
 
30 MECHANICS 
 
 EXPERIMENT 19 
 TO SHOW THAT Fs = J w^I 
 
 Apparatus. — Same as in Exp. 16. 
 
 Method and Manipulation. — The object is to show 
 that the force F producing rotation, from rest, in a body 
 multiplied by the distance s through which it acts is 
 equal to the kinetic energy possessed by the body when 
 the force ceases to act, i.e., 
 
 Fs-- 
 
 where w = the angular velocity, 
 
 7 = the moment of inertia of the body. 
 
 F is the effective force, and must be found as in Exp. 17. 
 If F is taken equal to Mg then the quantity §(ru)2M 
 must be added, i.e., 
 
 Fs = |(o27+|r^w2M 
 
 where M is the mass of the descending weight and r 
 the radius of the disc. 
 
 Obtain w as in Exp. 18, after having offset friction by 
 adding a supplementary weight as directed in Exp. 17, 
 and in addition measure the distance s. Compute / 
 from the mass and dimensions of the disc. Substitute in 
 the last quantity, and see if the quantity holds within 
 the limits of observational error. 
 
MECHANICS 
 
 31 
 
 EXPERIMENT 20 
 
 TO PROVE THAT THE MOMENTUM IS THE SAME AFTER 
 AS IT IS BEFORE AN INELASTIC IMPACT 
 
 Apparatus. — Two metal cylinders are suspended from 
 ceiling by parallel wires as shown in Fig. 20. One of 
 the cylinders is provided with fine points that during col- 
 lision penetrate the lead surface of the other thereby pre- 
 venting the two from separating. 
 
 Method and Manipulation. — Adjust the cylinders until 
 
 Fig. 20. 
 
 they are horizontal and in line -mth each other. Place 
 the table so that when the cylinders are swinging the ver- 
 tical rod beneath B moves parallel and close to the wire 
 cd. The wire cd is leveled by adjusting so that it is 
 even with the surfaces of the mercury contained in 
 two communicating glass tubes. 
 
 By means of a vertical mirrored scale obtain the 
 height /f. Raise cylinder ^4 and attach it at E. Obtain 
 the height H'. Release A. After collision the two 
 cylinders move on together, and the vertical rod beneath 
 B ■will leave at a definite point the paper index / placed 
 on the wire cd. By means of a thread draw the two 
 
32 MECHANICS 
 
 bodies over until the vertical rod just touches the paper 
 index and obtain the heights H". 
 
 The horizontal velocity of A just before impact is 
 v = y/2g{H' — H) and the velocity of A and B after 
 impact is v' = V2g{H"-H) . 
 
 If the momentum is the same before and after impact, 
 
 mV2giH'-H) = {m+m'W2g{E" - H) , 
 or 
 
 mVH'-H= {m+m')VH"-H. 
 
 From the data obtained, compute each member of this 
 equation, and compare. 
 
 The height of the t'vC'o cylinders is best obtained by 
 measuring the height of A and of B and averaging the 
 two. 
 
 Repeebnce. — Properties of Matter, by Poynting & Thomson, 
 p. 109. 
 
 EXPERIMENT 21 
 
 TO DETERMINE THE ACCELERATION OF GRAVITY BY 
 THE SIMPLE PENDULUM USING THE METHOD OF 
 COINCIDENCE FOR DETERMINING THE TIME OF 
 SWING 
 
 Apparatus. — The pendulum consists of a metal sphere 
 suspended by means of a fine wire. A point beneath the 
 sphere just touches a small mercury surface when the 
 pendulum is in the position of rest. The circuit of which 
 the pendulum is a part contains a battery and sounder. 
 At the end of every second the circuit is closed for an 
 instant by means of a relay actuated by the laboratory 
 clock and if the pendulum touches the mercury at this 
 instant the sounder will click. 
 
 Method and Manipulation. — Determine the distance 
 I from the under side of the supporting clamp to the 
 center of the sphere. 
 
MECHANICS 
 
 33 
 
 
 
 
 
 
 
 — 
 
 ] 
 
 @ 
 
 
 HH 
 
 
 
 Cause the pendulum to swing through a small ampli- 
 tude, and, when the sounder clicks, begin counting the 
 swings. Record on paper 
 the counts corresponding to 
 every click and continue 
 counting to five hundred. 
 Since the mercury surface 
 has an appreciable width, 
 the sounder will respond 
 several times for each unison 
 with the clock. The average 
 of a set may be taken as the 
 time of exact unison. ^'° ^^■ 
 
 If the pendulum is longer than a seconds pendulum, 
 the number of swings plus one that it makes between 
 two consecutive sets of coincidences will represent the 
 interval in seconds. Therefore if we subtract the aver- 
 age of the counts for the first set of coincidences from 
 the similar average of the last set, and add to this differ- 
 ence a number equal to the number of spaces between 
 sets, we shall obtain the time in seconds from the middle 
 swing of the first to the middle swing of the last set. 
 This time divided by the number of swings made by the 
 pendulum gives the time t of one swing. In a similar 
 way combine the second set with the second from the 
 last, and likewise the third with the third from the last, 
 and average. By means of I and t obtain g from the 
 equation 
 
34 MECHANICS 
 
 EXPERIMENT 22 
 
 ACCELERATION OF GRAVITY BY THE PHYSICAL 
 PENDULUM METHOD 
 
 Apparatus. — ^A long rectangular bar, that has near 
 one end an axis perpendicular to its plane. The bar 
 is suspended in such a manner that the ends of 
 the axis of suspension rest equally on the sup- 
 porting surface at each side. 
 
 Method and Manipulation. — Obtain the 
 time of vibration to an accuracy of 0.1 per cent. 
 
 Measure the length and width of the bar, 
 also the distance from the upper end to the 
 under side of the axis. 
 
 ja. 
 
 If i = the time of vibration, 
 
 7 = the moment of inertia of the pendulum 
 
 about the axis of suspension, 
 h=the distance from the axis to the center 
 of gravity of the pendulum, 
 Fig. 22. M = the mass of the bar, 
 
 VI 
 then 1 = 2% 
 
 and 
 
 VMgh 
 
 4x^7 
 ''Mhf 
 
 Compute 7 from the dimensions of the bar, and obtain 
 g by substituting the obtained data in the equation. 
 
 Refekence. — ^Properties of Matter, by Po3rntmg & Thomson, 
 p. 7. 
 
FLUIDS 
 
 EXPERIMENT 23 
 
 BOYLE'S LAW— THE VOLUME OF A GAS VARIES INVERSELY 
 AS ITS PRESSURE OR PV = A CONSTANT 
 
 Apparatus. — For description of apparatus see Exp. 30. 
 
 Method and Manipulation. — The temperature of the 
 water in the jacket is kept constant. The volume cor- 
 responding to the different pressures is determined as 
 directed in Exp. 30. From these values the constant is 
 computed. 
 
 EXPERIMENT 24 
 
 SPECIFIC GRAVITY OF SOLIDS 
 
 Apparatus. — Specific gravity bottle; chemical balance; 
 distilled water; the solid whose specific gravity 
 is to be determined. 
 
 Method. — Obtain the mass m of the sub- 
 stance. Fill the bottle with distilled water 
 to a fixed point in the capillary tube. Wipe 
 dry and obtain its mass M. Place the sub- 
 stance in the bottle and have the water 
 extend to the same point in the capillary tube 
 and obtain the mass M'. 
 
 The specific gravity 
 
 s = 
 
 m 
 
 m+M-M'' 
 
 Fig. 23. 
 
 Precautions. — The water must be boiled 
 in order to free it from air. The tube leading from the 
 
 35 
 
36 
 
 FLUIDS 
 
 bottle must be closed to prevent evaporation. As the 
 volume of the bottle and the density of the water change 
 with temperature, care must be exercised to have the 
 temperature of the water the same at each filling. 
 
 EXPERIMENT 25 
 
 TO DETERMINE THE SPECIFIC GRAVITY OF A LIQUID 
 BY MEANS OF A MOHR'S BALANCE 
 
 Apparatus. — ^A Mohr's balance consists of a beam 
 
 Fig. 24. 
 
 supported on steel knife edges. One half of the beam 
 is graduated to tenths. Attached to the other half 
 is a counter weight so that the beam will balance in air 
 when a sinker S is suspended at the end of the tenth 
 
FLUIDS 37 
 
 division of the graduated arm. A rider is also provided 
 which, when attached to the beam at the end of the 
 tenth division, will cause the beam to balance when the 
 sinker is immersed in water at 15° C. By Archimedes' 
 principle, this rider is then equal in mass to the mass of 
 the water displaced. Four other riders are also provided 
 which have masses equal to 1.0, 0.1, 0.01, 0.001 of the 
 mass of the rider mentioned. The sinker is provided 
 with a thermometer so that the temperature at which 
 it is used may be determined. 
 
 Method and Manipulation. — Suspend the sinker in 
 the liquid the specific gravity of which is desired, and 
 add riders until a balance is obtained. Then, if the 
 riders are distributed as shown in Fig. 24, where a is the 
 unit rider, b the 0.1 and c the 0.001, the specific gravity 
 is 1.2608 if the temperature of the liquid is 15° C. If, 
 however, the temperature of the liquid is t°, and the 
 reading of the balance is Wt, the density S; of the liquid is 
 
 , ^ Si5U>,/l+al5 
 '~ Wis \l+at° 
 
 and the specific gravity is 
 
 „ ^ ^sWi /l+al5 
 SU)15 \l + (xt° 
 
 where a = the coefficient of expansion of the sinker, 
 S =the density of water at 4° C, 
 S 15 = the density of water at 15° C, 
 Wi5 = the reading of the balance when sinker is 
 immersed in water at 15° (_wi5 = l by con- 
 struction of balance). 
 
 Since 3 and Sjs are nearly equal, the above equation 
 
 reduces to 
 
 /l + al5\ 
 
 'S = ^'ll+7<^;' 
 
38 
 
 FLUIDS 
 
 and since a is small we have approximately 
 
 it = S=w,. 
 
 The distribution of the riders on the graduated arm that 
 causes the beam to balance when the sinker is immersed 
 gives therefore the approximate value of the specific 
 gravity of the liquid at any temperature. 
 
 Determine the approximate and the exact specific 
 gravity of the provided solution. 
 
 EXPERIMENT 26 
 
 TO. CALIBRATE AN HYDROMETER SCALE OF 
 EQUAL DIVISIONS 
 
 Apparatus. — An hydrometer is an instrument for 
 determining directly the density or specific 
 gravity of a liquid. It consists of a 
 glass tube which is weighted with mer- 
 cury or lead at the lower end and which 
 has a thinner portion at the other that 
 is graduated in equal divisions or in un- 
 equal divisions that give directly the 
 density or specific gravity. 
 
 Method and Manipulation. — ^Prepare 
 two solutions of such densities as will 
 give a reading near each end of the scale 
 when the hydrometer is immersed in 
 them. Let ri and r2 be the two readings. 
 Determine by means of a Mohr's balance 
 density Si and S2 of the two solutions. 
 Determine also the mass m of the hydro- 
 meter. Then, since by Archimedes' prin- 
 
 m m 
 
 ciple T" and "r~ are the volumes of the 
 
 01 02 
 
 liquid displaced in each case, the volume of the stem 
 
 Fig. 25. 
 
FLUIDS 39 
 
 m m 
 
 between r-i and r^ is r" — 7~ and the volume per division 
 
 01 02 
 
 is if 8i<S2 
 
 m / 1 1 
 
 1^ — 
 
 r2-ri\Si S2 
 
 Compute the value of k. 
 
 Solving the above equation for S2, and dropping the 
 subscript in the case of S2 and ?-2, we obtain an equation 
 which may be used in computing the density correspond- 
 ing to any point r on the scale, namely 
 
 mil 
 
 m—kliir—riY 
 
 Determine S for each division on the scale, and plot 
 these values as ordinates and the sca,le divisions as 
 abscissas. 
 
 If a Baum^ hydrometer is used compare the values 
 thus obtained with the attached direct reading scale. 
 
 EXPERIMENT 27 
 SURFACE TENSION 
 
 If h is the he height to which a liquid rises in a capil- 
 lary tube, T the surface tension in dynes per centimeter 
 length of film, and r the radius of the tube, then 
 
 2 cos a' 
 For water 
 
 2 • 
 
 Apparatus. — A glass tube; distilled water; a mirrored 
 scale; a microscope with stage micrometer. 
 
40 
 
 FLUIDS 
 
 ^fi- 
 fe 
 
 
 Fig. 26. 
 
 Manipulation.— After thoroughly cleaning the glass 
 tube as directed below, heat it in a Bunsen flame until 
 it becomes very pliable, then quickly 
 draw it into a capillary tube of about 
 .5 mm. diameter. Break the tube so 
 as to obtain a portion about 15 centi- 
 meters long and clamp it vertically so 
 that it extends into a vessel filled to the 
 brim with the liquid. Obtain the height 
 h by taking the reading a on the scale 
 for the surface of the liquid in the tube 
 and likewise the reading h for the sur- 
 face of the liquid in the vessel. 
 
 The diameter 2r of the capillary tube 
 is measured by means of the stage 
 micrometer. Cut the tube at the point of the water 
 meniscus. Attach the tube to the stage and focus the 
 microscope so that the end is seen clearly. When the 
 cross-hair is tangent to the bore, read the micrometer, 
 then move the stage so as to make the cross-hair tangent 
 to the other side, and again read. Measure two diameters 
 at right angles to each other. Use the average of the 
 two. I 
 
 Precautions. — The glass tube must be thoroughly 
 cleaned. First wash with chromic acid or caustic hy- 
 drate solution, and clean with water and cloth; then 
 wash with nitric acid and rinse thoroughly with dis- 
 tilled water. Wipe dry with clean cloth. 
 
 Before taking a reading, draw the liquid almost to 
 the top of the tube and allow it to settle, or push the 
 tube into the liquid and partly withdraw. 
 
 Reference. — Fropertiea of Matter, by Poynting & Thomson, 
 p. 140. 
 
HEAT 
 
 EXPERIMENT 28 
 ATMOSPHERIC PRESSURE 
 
 The pressure of the atmosphere is measured by the 
 weight of the column of pure mercury at 0° C. that it 
 can support. 
 
 If Ho is the height of the mercury column the pres- 
 sure in dynes per square centimeter is 
 
 Hodg 
 
 where d is in the density of mercury (13.596). 
 
 If the temperature is t, and if the reading H for 
 the height of the column is obtained by means of a scale 
 that is correct at the temperature t' the true height 
 
 Ht = ll+^(t-t')]H, 
 
 where ^ is the coefficient of expansion of the scale. 
 
 - If the height of the mercury column is Ht when its 
 
 temperature is t, its height at zero degrees 
 
 where a is the coefficient of expansion of mercury. 
 
 The height of the column including both corrections is 
 
 41 
 
42 
 
 HEAT 
 
 If the scale is correct at 62° F. and H and t are 
 obtained in English and Fahrenheit units, the equation 
 becomes 
 
 Ho = 
 
 [l + i^(f-62) l 
 l+'z(f-32) 
 
 % 
 
 where a = .0001001 and & = . 00001043. 
 
 The tension of the mercury vapor in the vacuum 
 above the mercury causes a small depression, the cor- 
 rection for which may be obtained from 
 A tables. See Table 23. 
 
 The downward pull due to the meniscus of 
 
 the mercury also causes a depression. This is 
 
 _ reduced by using a tube of large diameter 
 
 1(25 mm). The correction may be obtained 
 from tables arranged for that purpose. Fre- 
 quently the scale is shifted to offset this error. 
 Apparatus. — Fortin's barometer. The con- 
 struction of this barometer is as indicated 
 in Fig. 27. The scale attached to the metal 
 casing is adjusted so as to measure from the 
 end of the ivory point projecting down from 
 the roof of the cistern. The mercury surface 
 in the cistern can be raised or lowered by 
 means of the screw underneath. The scale' is 
 provided with a vernier, and on the side oppo- 
 site and connected to it is a slide whose lower 
 edge is on a level with that of the vernier. By 
 means of the head c the vernier and slide may 
 be moved up or down. A thermometer is 
 attached to the metal casing of the barometer. 
 Manipulation. — Record the temperature, and adjust 
 the surface of the mercury in the cistern so that it just 
 touches the point. Then adjust the vernier so that the 
 top of the meniscus is in hne with the lower edges of the 
 
 Fig. 27. 
 
HEAT 
 
 43 
 
 vernier and slide. Record the reading for the height, 
 and correct it for expansion of scale and mercury by 
 means of the equation given above. 
 
 Obtain correction from Ihe table arranged for the 
 barometer, and compare with computed correction. See 
 Table 22 of Appendix. 
 
 EXPERIMENT 29 
 CALIBRATION OF A THERMOMETER 
 
 The errors that this calibration is designed to correct 
 are those due to the inaccuracies of the zero and boiling 
 points on the thermometer scale and 
 the irregularity of the bore of the tube. 
 On account of these errors the thermom- 
 eter scale does not accurately represent 
 temperatures. 
 
 Apparatus. — Thermometer; barom- 
 eter; jar of finely divided ice; a boiler 
 for holding the thermometer in steam. 
 
 Manipulation. — Immerse the ther- 
 mometer up to the zero mark in 
 melting ice. After the mercury becomes 
 stationary, note the reading z on the 
 thermometer scale, avoiding parallax 
 by use of a mirror. Then place the 
 thermometer in the boiler as shown 
 in Fig. 28, so that the bulb is about 
 two centimeters above the water sur- 
 face. Allow the water to boil, and 
 when the mercury surface becomes 
 stationary, note the reading a on the scale. Determine 
 the barometric height, and from it determine the tem- 
 perature A° at which water boils. The boiling tem- 
 perature of water at 760 mm. (29.921 in.) pressure is 
 
 C 1 
 
 Fig. 28. 
 
44 
 
 HEAT 
 
 100° C. It decreases 1° C. for about 26.7 mm. (1.05 in.) 
 decrease in pressure. 
 
 A° = 100*- 
 
 760 -g 
 26.7 ■ 
 
 l°{ 
 
 l{ 
 
 Separate a thread of mercury tiiat is about 20 degree 
 divisions long. This is best done by hold- 
 ing the thermometer in a horizontal posi- 
 tion and giving it a light and quick tap 
 at the upper end. If the thread that 
 separates is not of the proper length join 
 it to the rest and note the position on 
 the scale where they come together. This, 
 owing to a small air-bubble adhering to the 
 glass, is in general the place where the 
 mercury separates when the thermometer is 
 again tapped even though mercury has been 
 forced by the point. Then warm or cool 
 the bulb until the desired length is above 
 the joining point and tap. If unsuccess- 
 ful, repeat the process. Having separated a 
 thread of the desired length, place one end 
 even with the zero on the scale, and read 
 the fraction di that the other end is above 
 or below the 20 mark. Then place the 
 lower end at 20 and read the difference d2 
 at 40. Continue thus until the differences 
 ds at 60, di at 80, and ds at 100, have 
 been obtained. Repeat by placing the upper 
 end even with 100 and reading the differ- 
 ence at 80, etc. 
 
 Consider the quantities as indicated in 
 Fig. 29. 
 
 Let 100-a = p. 
 the number of degrees the temperature of 
 
 -d4 
 
 -ds 
 
 ■dz 
 
 -d, 
 
 Fig. 29. 
 
 Let r = 
 
HEAT 45 
 
 the mercury must be raised to increase its volume 
 an amount equal to the volume of the calibrating 
 thread. 
 
 The markings on the thermometer tube are sufficiently 
 accurate so that the difference between the length of 
 one division on the scale and the actual length of expan- 
 sion for one degree is so small that it may be neglected. 
 Several divisions, however, do not represent the same 
 number of degrees, for the differences accumulating 
 make an appreciable error. -Let the values obtained 
 for the small distances z, p, and di. . .ds represent de- 
 grees. Then the number of degrees necessary to expand 
 the volume of the mercury by the volume of the tube 
 from 0° to a is 
 
 z+l°-di + r-d2+r-d3+l°-di+r-d5-p 
 
 or 5l°-\-z — di — d2 — dz — di — ds — p. 
 
 But, experimentally, it is known that A° expands the 
 mercury from 0° point to a. 
 
 :. A° = 5r+z—di — d2 — d3 — di — ds—p, 
 
 , jo^ A° — z+di+d2+d3+d4:+d5+p 
 
 As A°, z, Id, and p are known, 1° becomes known. 
 
 If z and a were above and 100 on the scale, and 
 if any value for d represents a fraction below 20, 40, 
 60, 80 or 100, then their signs in the equation for I and 
 in the following equations must be changed. 
 
 The temperature for the division on the scale cor- 
 responding to the subscript of t is as follows : 
 
46 HEAT 
 
 ■ to =+z°, 
 
 feo =z-\-l — di, 
 
 Uo =z+2r-di-d2, 
 
 teo =z+3r — di — d2 — d3, 
 
 tso =z+U° — di—d2 — d3 — dA, 
 
 tioo = z+5r — di—d2 — d3 — d4: — d5, 
 
 tioo — A°+p (check). 
 
 h =z+~^ = z+k, ^ 
 
 fe =2 + 2/0, ■ 
 
 tz =z+Zk, 
 feo =2+20/c, 
 
 <21 =i20H 20 =t20-\-li', 
 
 t22 =t20 + 2k'. 
 
 etc. 
 
 Record the number of the thermometer used and 
 prepare a table ranging from to 100. The thermom- 
 eter cahbrated is to be used in the heat experiments 
 that follow. 
 
 EXPERIMENT 30 
 
 CHARLES' LAW— THE PRESSURE OF A GAS VARIES 
 DIRECTLY AS ITS ABSOLUTE TEMPERATURE— P=KT 
 
 Apparatus. — The gas is enclosed in a glass tube, 
 as shown at A in Fig. 30. Inside this glass tube and 
 projecting up through the mercury is a steel rod. This 
 rod throughout its length has threads of one millimeter 
 pitch, is finely pointed at its upper end, and has a microm- 
 eter head of one hundred divisions at the lower end. The 
 mercury in contact with the gas in A communicates 
 through a rubber tube with the mercury contained in a 
 parallel glass tube B. The glass tube A is surrounded by 
 
HEAT 
 
 47 
 
 a glass jacket filled with water which may be heated by 
 passing an electric current through a coil of five turns of 
 No. 24 lala wire wound 
 on the stirrer E. The 
 current may be obtained 
 from the house circuit, 
 using a bank of incan- 
 descent lamps L con- 
 nected in series with the 
 coil of the stirrer. As the 
 lamps in the bank are in 
 parallel, the current may 
 be altered by connect- 
 ing or disconnecting the 
 various lamps. All metal 
 parts coming into contact 
 with mercury are of steel, 
 and those that come in 
 contact with the water in 
 the jacket are of brass. 
 All packings are of 
 leather. The apparatus 
 is supported by two par- 
 allel vertical rods 1.5 
 meters in height and having a tripod base. 
 
 Readings. — The volume of the gas is obtained from the 
 micrometer reading when the point of the rod is even with 
 the mercury surface in A, an adjustment that can be made 
 with great accuracy. The volume V when the microm- 
 eter is at zero on the scale and the volume per unit 
 length (fc) are given. The pressure in centimeters of 
 mercury height is the barometer reading plus the dif- 
 ference in heights of the mercury surface in A and in B. 
 This difference is obtained from the scale C. The 
 slide D carries a vernier and a mirror upon which is an 
 
 Fig. 30. 
 
48 HEAT 
 
 index line which may be placed, free from parallax, even 
 with the mercury surfaces or, in the case of the tube A, 
 even with -the lower end of the rod R. The temperature 
 is obtained from a thermometer suspended in the water 
 surrounding the tube A. 
 
 Method and Manipulation. — The micrometer is kept 
 constant. The apparatus is raised or lowered by means 
 of an auxiliary screw S until the mercury is even with 
 the point. The pressure and temperature are then 
 determined. The temperature is then raised by passing 
 a current. When the desired rise has taken place (say 
 5°), the current is reduced so as to keep the temperature 
 constant. The pressure and temperature are determined 
 as before. Compute the constant K for each set. 
 
 EXPERIMENT.31 
 
 BOYLE'S AND CHARLES' LAWS COMBINED 
 
 Apparatus. — Same as in Exp. 30. 
 
 Method and Manipulation.— When Boyle's and 
 
 PV 
 
 Charles' laws are combined, the relation -=r = R (a 
 
 constant) is obtained, where T is the absolute tem- 
 perature. When the gas in A is at the room temperature 
 determine its pressure, volume, and absolute temperature, 
 as directed in Exp. 30. Raise the temperature about 
 five degrees by passing a current through the coil, being 
 careful to adjust the current so that the temperature 
 is constant while the readings are taken. In order to 
 increase the range of pressure the tube B may be raised 
 arbitrarily. Determine again the pressure, volume, and" 
 absolute temperature of the gas. Continue thus until 
 several sets have been obtained. 
 
 Compute the value of R for each set of observations. 
 
HEAT 
 
 49 
 
 EXPERIMENT 32 
 
 COEFFICIENT OF LINEAR EXPANSION 
 
 The quantity to be determined is the elongation of a 
 unit length of a solid fo'r one degree change in temperature. 
 
 Apparatus. — The rod whose expansion is to be meas- 
 ured is placed as indicated by the dotted hnes in Fig. 
 31. The two tubes en- 
 closing the rod telescope 
 into each other. The 
 tubes rest on a microm- 
 eter screw, and their 
 upper end actuates a 
 lever L. When the rod 
 expands the lever L, is 
 displaced. A telescope is 
 focused so that the image 
 of a scale placed above it 
 is seen in a mirror at- 
 tached to one end of the 
 lever. Any change in the 
 length of the rod is, by 
 this means, detected. The 
 upper end of the tube 
 enclosing the rod has a 
 small hole which admits -pj^ g^ 
 
 the thermometer neces- 
 sary for- recording the initial and final temperatures of 
 the rod. 
 
 Method and Manipulation. — Measure the length of 
 the rod, and place it in the apparatus as indicated in Fig. 
 31. Fill the cups with water. Adjust the telescope 
 and mirror until the cross-line on the vertical scale is 
 seen clearly and until it and the cross-hair in the tele- 
 
50 HEAT 
 
 scope are superposed. Place the thermometer so that 
 the bulb is about half-way down the tube. Record the 
 initial temperature of the rod and the reading of the 
 micrometer. Pass steam through the tube, and, when 
 the cross-line moves away from the cross-hair, lower 
 the rod by turning the micrometer. When the rod has 
 ceased to expand, again superpose the two lines and 
 record the micrometer reading. The difference between 
 the two micrometer readings gives the total expansion. 
 When the thermometer becomes stationary record 
 the temperature of the rod. 
 
 If Z = the length of the rod, 
 
 t = the initial temperature of the rod, 
 ts = the final temperature of the rod, 
 r=the first micrometer reading, 
 r' = the second micrometer reading, 
 then the coefficient of expansion, 
 
 r'—r 
 
 i(ts-ty 
 
 EXPERIMENT 33 
 THE RADIATING POWER OF DIFFERENT SURFACES 
 
 The quantity to be determined is the number of calories 
 lost per second, through radiation, by a unit area of a 
 radiating surface when its temperature is one degree above 
 the temperature of the surroundings. 
 
 Apparatus. — The radiating body consists of a bright or 
 black vessel C filled with water. This vessel is suspended 
 so as to be completely surrounded by a water jacket 
 as shown in Fig. 32. The temperature of the radiating 
 body and that of the surrounding surface are obtained 
 by means of two thermometers placed as shown in figure. 
 
HEAT 
 
 51 
 
 Method and Manipulation. — Determine the mass and 
 area of the vessel C and fill it with water at about 60° C. 
 Observe the temperature 
 of the water in the vessel 
 C and also of the water in 
 the surrounding jacket, at 
 the end of every five 
 minutes during a period 
 of thirty minutes. Obtain 
 the radiation curve by 
 plotting the temperatures 
 as ordinates and times as 
 abscissas. Likewise on the 
 same. sheet and with the 
 same units obtain the 
 curve showing the change 
 in the temperature of the ^iq 32 
 
 surrounding jacket with 
 
 time. Compute the coefficient of radiation at the 
 instants 5, 15, 25 minutes in the following manner. 
 Subtract the temperature t' at 7.5 minutes from the 
 temperature t" at 2.5 minutes as obtained from the 
 radiation curve. Using this value for {t" — t'), compute 
 the radiating power k from the relation 
 
 
 
 
 
 
 e"^^-. 
 
 ^^:^-^ r^r=F^ 
 
 
 - ^ z 
 
 
 Lj u 
 
 
 v:: 
 
 -_-_' 
 
 
 
 
 :-0>: 
 
 ~.zr. 
 
 
 THir^ 
 
 c 
 
 - ■ 
 
 z-'s 
 
 
 7_-_- 
 
 . 
 
 
 
 
 
 k = 
 
 iM+c+e)(t"-n 
 
 TSta 
 
 where M = the mass of water in radiating vessel, 
 c = the thermal capacity of the vessel, 
 e = the thermal capacity of the thermometer, 
 ta = the average temperature less the tempera- 
 ture of the surrounding jacket at this instant, 
 »Sf = the area in sq.cm. of the radiating vessel, 
 r=the time in seconds, i.e., 300. 
 
52 HEAT 
 
 Show the variation of these coefficients by plotting 
 them as ordinates, using the corresponding values of 
 ta as abscissas. 
 
 Make the above determinations in the case of a bright 
 and a black radiating vessel. 
 
 EXPERIMENT 34 
 THE ABSORBING POWER OF DIFFERENT SURFACES 
 
 The quantity to be determined is the number of calories 
 gained per second through radiation by a unit area of an 
 absorbing surface when its temperature is one degree below 
 the temperature of the surroundings. 
 
 Apparatus. — Same as in Exp. 33. 
 
 Manipulation. — Fill the radiating vessel C with water 
 at nearly 0° C, and proceed in the same manner as in 
 Exp. 33. 
 
 EXPERIMENT 35 
 SPECIFIC HEAT 
 
 The quantity to be determined, is the number of calories 
 necessary to raise the temperature of one gram of a given 
 solid one degree. 
 
 Apparatus. — The calorimeter C surrounded by a 
 jacket B packed with felt; a steam jacket for heating the 
 solid; two thermometers; a barometer; a watch. 
 
 Method and Manipulation. — Obtain the mass of the 
 solid to be used, and place it in the inner chamber of 
 the steam jacket. Pass steam through the outer cham- 
 ber until the solid assumes the temperature of the steam. 
 Obtain the mass of the calorimeter and fill it with water 
 at room temperature to a point about two centimeters 
 
HEAT 
 
 53 
 
 from the top. Obtain the mass of both water and calo- 
 rimeter. 
 
 Stir the water constantly, and. observe the temperature 
 of the water in the calorimeter at the end of every minute 
 for a period of about eight minutes. Immediately 
 after the last reading, pass the metal quickly into the 
 calorimeter. Stir, and record the temperature at the 
 
 Fig. 33. 
 
 end of every thirty seconds until the temperature has 
 reached a maximum, after that continue recording the 
 temperature at the end of every minute for an interval 
 of about ten minutes. Record the temperature ts 
 inside the steam jacket. 
 
 To obtain the rise in the temperature of the water, 
 plot the temperature readings on coordinate paper 
 using time as abscissas. Draw a straight hne that will 
 represent the average slope of the rising part of the curve. 
 
54 HEAT 
 
 At the intersections of this line with the lower and upper 
 parts of the curve produced, erect perpendiculars. 
 
 The distance between, the produced curves midway 
 between the two perpendiculars may be taken as the 
 rise (<" — t') in the temperature of the water which would 
 have taken place had there been no loss by radiation, etc. 
 
 If if = the mass of the water, 
 m = the mass of the solid, 
 t' =the initial temperature of the water, 
 <"=the final temperature of the water, 
 ts =the temperature of the steam or solid, 
 c =the thermal capacity of the calorimeter, 
 e =the thermal capacity of the thermometer, 
 
 then the specific heat 
 
 XM+c+e){t"-t') 
 
 S = - 
 
 m{U-t") 
 
 From the values obtained for these quanities, compute 
 the specific heat of the solid. 
 
 EXPERIMENT 36 
 LATENT HEAT OF FUSION 
 
 The quantity to he determined is the number of calories 
 necessary to change one gram of ice to water. 
 
 Apparatus. — A calorimeter surrounded by a felt 
 jacket; one thermometer; a watch. 
 
 Method and Manipulation. — Obtain the mass of the 
 calorimeter and fill it with water at room temperature 
 to a point about four centimeters from the top. Obtain 
 the mass of the water and calorimeter, stir constantly, 
 and record the temperature of the water at the end of 
 every minute for a period of about five minutes. Then, 
 
HEAT 55 
 
 quickly, after the last reading, drop a piece of dry ice of 
 about 100 c.c. volume into the calorimeter so that it will 
 be beneath the gauze stirrer. Record the temperature 
 at the end of every thirty seconds until the ice has melted, 
 then continue taking the temperature at end of every 
 minute for an interval of eight minutes. Plot these 
 readings and from the curve determine the fall in tem- 
 perature of the water by the construction explained 
 under Exp. 35. Weigh the calorimeter and water. The 
 mass of the ice introduced is the difference between this 
 and the mass previously obtained. 
 
 If M' = the mass of the water, ' 
 m = the mass of the ice, 
 t =the initial temperature of the water, 
 t' =the final temperature of the water, 
 e =the thermal capacity of the thermometer, 
 c =the thermal capacity of the calorimeter, 
 
 then the latent heat of fusion 
 
 J. iM+c+e){t-t')-mt' 
 
 m 
 
 EXPERIMENT 37 
 LATENT HEAT OF VAPORIZATION 
 
 The quantity to he determined is the number of calories 
 necessary to vaporize one gram of water. 
 
 Apparatus. — The apparatus is arranged as shown in 
 Fig. 34. Steam is passed from the boiler into the trap 
 and from there down into the immersed tank where it is 
 condensed, and imparts its heat to the tank and sur- 
 rounding water. 
 
 Method and Manipulation. — Obtain the mass of the 
 calorimeter and of the coil. Fill the calorimeter with 
 
56 
 
 HEAT 
 
 water sufficient to cover the coil, and add ice sufficient 
 to cool it 10 or 12° below the temperature of the room. 
 Obtain the mass of the water, and connect the coil and 
 trap as shown in figure. Con- 
 nection with the boiler is not 
 made until the instant the steam 
 is to be admitted. Stir the water, 
 and record its temperature and 
 the temperature of the room at 
 the end of every minute during 
 a period of about eight minutes. 
 Immediately after the last read- 
 ing allow the steam to enter. 
 Stir, and record the temperature 
 at the end of every thirty seconds, 
 and when the temperature is as 
 much above tlie room temperature 
 as it was below at the beginning, 
 disconnect the boiler. Continue 
 recording the temperature at the end of every minute 
 for a period of about eight minutes. Wipe the coil dry, 
 and weigh, and also obtain the temperature of the steam. 
 To obtain the rise in temperature of the water, plot 
 the temperature readings as ordinates and the corre- 
 sponding time as abscissas. Draw a straight vertical 
 line through the point where the temperature curve 
 intersects the room temperature curve produced. The 
 distance between the intersections of this vertical by the 
 lower and upper parts of the temperature curve produced 
 may be taken as the rise (t' — f) in temperature of the 
 water. 
 
 Fig. 34. 
 
 If M = the mass of the water, 
 
 Ml = the mass of the calorimeter, 
 m =the mass of the coil and tank, 
 
HEAT 57 
 
 m' = the mass of the steam condensed, 
 < = the initial temperature of the water, 
 
 t' — the final temperature of the water, 
 
 ts = the temperature of the steam, 
 s = the specific heat of brass, 
 e = the thermal capacity of the thermometer, 
 
 then the latent heat of vaporization 
 
 iM+Mis+ms+e){t' -i)-m'it,-i') 
 
 L = - 
 
 m 
 
 Precautions. — The coil and tank must be dried be- 
 fore using by passing a current of warm air through 
 them. Since the temperature of the water continues 
 to rise after the steam has been cut off, disconnect the 
 tube when the temperature is about one degree below 
 the desired temperature. 
 
 EXPERIMENT 38 
 
 TO DETERMINE THE VAPOR PRESSURE OF WATER AT 
 DIFFERENT TEMPERATURES 
 
 Apparatus. — An air tight boiler provided with a con- 
 denser. A tube leads from the condenser to a large 
 vessel Y containing air, the purpose of which is to ren- 
 der the pressure steady. The vessel V is connected to a 
 manometer for determining the pressure. Pressures 
 less than one atmosphere are obtained by exhausting the 
 air from the boiler and vessel V by means of a water 
 pump connected to the tube F. The bulb of the ther- 
 mometer is placed in oil contained in a closed tube extend- 
 ing into the boiler. 
 
58 
 
 HEAT 
 
 Pump 
 
 'Wafer 
 ^Ouflet 
 
 Fig. 35. 
 
 Method and Manipulation. — Obtain the temperature 
 of boiling when the pressure is one atmosphere, then 
 
 start the exhaust pump and 
 reduce the pressure about 10 
 cm. Turn the stopcock C so 
 as to close the tube leading to 
 the boiler. When the mercury 
 in the thermometer becomes 
 stationary, read the tempera- 
 ture and pressure. Then open 
 C, and again reduce the 
 pressure 10 cm. Close C, and 
 when the thermometer be- 
 comes constant obtain the 
 temperature and pressure. 
 Continue thus until the pres- 
 sure is as small as possible. 
 Obtain the vapor pressure curve by plotting the pres- 
 sures and the corresponding temperatures on coordinate 
 paper, the temperatures being laid off along the axis of 
 abscissas. 
 
 EXPERIMENT 39 
 
 TO DETERMINE THE RELATIVE HUMIDITY OF THE 
 ATMOSPHERE 
 
 The quantity to be determined is the ratio of the mass 
 of the water vapor contained in a unit volume of air to 
 the mass of water vapor it would contain if the air were 
 saturated. 
 
 1. The dew-point method 
 
 Apparatus. — The dew point, or Daniell's hygrometer, 
 consists of two glass bulbs joined by a glass tube as 
 shown in figure. The lower bulb has a bright metallic 
 surface which enables the formation of dew to be more 
 
HEAT 
 
 59 
 
 Fig. 36. 
 
 readily detected. This bulb also contains a thermometer. 
 A thermometer for determining the room temperature 
 is attached to the supporting stand. The instrument 
 contains a quantity of ether sufficient nearly to fill one 
 of the bulbs, no air being present. 
 
 Method and Manipulation. — In a vessel that is well 
 surrounded by felt, place some water and finely divided 
 ice, and in this immerse the upper bulb 
 after the ether has been collected into the 
 lower bulb. Record the air temperature 
 ta, and, when dew begins to form on the 
 bright surface of the lower bulb, record 
 the temperature ts of the metallic surface 
 as indicated by the enclosed thermometer. 
 Remove the ice and water, and note the 
 temperature when the dew disappears. 
 Repeat several times, and obtain the 
 average temperature ts. This gives the temperature at 
 which the amount of vapor present in the air is sufficient 
 to saturate it. 
 
 Since the vapor present is open to the free air, its 
 pressure does not appreciably change when its tem- 
 perature is lowered. The pressure pa of the vapor present 
 in the air is, therefore, the same as the vapor pressure 
 of water at the dew point temperature and may be 
 determined from the table of vapor pressures given on 
 page 231. The saturation pressure ps corresponding 
 to the room temperature is obtained from the same table. 
 From the gas law pv = RmT we have the law that the 
 pressures are to each other as the masses. Therefore 
 the relative humidity is 
 
 rris Ps 
 
 From the values of pa and ps calculate H. 
 
60 HEAT 
 
 2. The wet and dry bulb or the 
 psychrometer method 
 
 Apparatus.— Two thermometers are mounted side 
 by side. The bulb of one is covered with a cloth which 
 is kept saturated with water. 
 
 Manipulation. — Pass a current of air over the two 
 bulbs of the thermometers when they are dry, and obtain 
 
 the readings. Their difference gives the ther- 
 mometer correction. Then saturate the cloth 
 with water and again pass a current of air over 
 the bulbs. Obtain the thermometer readings when they 
 become steady and the barometer reading B. Correct 
 the readings, and from the difference in the two ther- 
 mometer readings, the reading h of the wet-bulb ther- 
 mometer, and the barometric pressure, obtain the vapor 
 pressure pa from Table 25. 
 
 From Table 24 obtain the vapor pressure ps corre- 
 sponding to the reading of the dry-bulb thermometer. 
 From the values pa and ps obtain the humidity H as 
 above. For an approximate value the barometric pres- 
 sure may be taken equal to 76 cm. 
 
HEAT 
 
 61 
 
 EXPERIMENT 40 
 THE MECHANICAL EQUIVALENT OF HEAT 
 
 The quantity to be determined is the number of ergs 
 equivalent to the energy of one calorie. 
 
 Apparatus. — A conical cup a, Fig. 38, is placed on 
 ebonite supports inside a larger cup that is mounted 
 
 Fig. 38. 
 
 on a vertical axis. A friction cone b fits the cup a 
 and supports a flanged wheel c, above which is a 
 heavy metal ring d that serves as a weight. A cord is 
 wound about c, and, passing over a pulley, supports a 
 weight m. 
 
 The friction cone b ia hollow and contains water, a 
 thermometer, and a stirrer. A revolution counter is 
 actuated by the vertical axis. 
 
62 HEAT 
 
 Method and Manipulation. — Fill the cup b with a 
 known mass of water, and let its temperature be about 
 six degrees below the temperature of the room. The 
 purpose of this is to offset the radiation error by allow- 
 ing the temperature of the water to rise during the experi- 
 ment to the same number of degrees above the tem- 
 perature of the room. 
 
 Record the temperature of the water and the read- 
 ing of the counter. Then rotate the cup a with such 
 speed as will keep the weight m suspended. If the 
 counter does not read higher than one hundred, each 
 hundred revolutions must be recorded". The energy 
 expended in overcoming friction between a and 6 is 
 transformed into heat, which raises the temperature of 
 the water. 
 
 Stir the water, and when its temperature is about 
 half a degree below the temperature desired, discon- 
 tinue rotating. 
 
 Record the maximum temperature of the water and 
 the reading of the counter. 
 
 If M' = the mass of (a), (&), and stirrer, 
 Ml =the mass of the water, 
 'm = the mass supported by the cord, 
 s=the specific heat of the metal in a, b, and 
 
 stirrer, 
 w=the number of revolutions, 
 t = the initial temperature of the water, 
 t' = the final temperature of the water, 
 e = the thermal capacity of the thermometer, 
 r=the radius of the wheel c, 
 
 then the work done against friction is 
 
 2i:rmng ergs, 
 
HEAT 63 
 
 and the number of calories into which this work has 
 been transformed is 
 
 (Ml +Ms+e) (<'-<)• 
 Therefore the mechanical equivalent 
 
 2i:rnmg 
 
 J = 
 
 {Mi+Ms+e){t'-ty 
 
ELECTRICITY AND MAGNETISM 
 
 EXPERIMENT 41 
 
 ELECTROSTATICS 
 
 Apparatus. — An electroscope; an ebonite rod; a 
 glass rod; a large and a small piece of woolen cloth; 
 a silk cloth; an insulated brass rod; 
 a metal can; a proof plane. 
 
 Precaution. — Special care must be 
 exercised not to bring an excessive 
 charge near the electroscope as it is 
 liable to tear the leaves. Charge 
 the rod at a distance, bring it slowly 
 toward the electroscope, and stop 
 when the leaves diverge the desired 
 amount. 
 
 Manipulation. — A. 1. Charge the 
 ebonite rod by rubbing it with a 
 woolen cloth. Bring the rod near 
 the electroscope, note effect and ex- 
 plain. Remove the rod, note effect 
 and explain. 
 
 2. Repeat by using a glass rod 
 rubbed with silk. Warm the rod to 
 remove moisture, if it be necessary. 
 
 3. Rub the ebonite rod with wool 
 and gather part of the charge on a 
 
 proof plane. Bring the charged plane into contact with 
 the plate of the electroscope. Note effect and explain. 
 
 64 
 
 Fig. 39. 
 
ELECTRICITY AND MAGNETISM 65 
 
 B. 1. Again charge the proof plane from the rod, 
 and hold it near the charged electroscope. Explain. 
 
 2. Then rub a glass rod with silk, and hold it near 
 the plate of the electroscope. Note effect and explain. 
 
 3. Discharge the electroscope by touching it with the 
 hand. Rub the ebonite rod with wool and hold it near 
 the electroscope. Touch the plate of the electroscope 
 with the hand. Remove the rod first. Explain. 
 
 4. Repeat, and remove the hand first. Explain. 
 
 5. Now bring the charged glass rod near the plate of 
 the charged electroscope. Explain. 
 
 6. Bring a charged ebonite rod near the plate. Explain. 
 
 7. Test the electrification on an insulated brass rod 
 rubbed with wool. 
 
 C. Place a metal can on the plate of the electroscope. 
 
 1. Tie the small piece of woolen cloth to one end of the 
 glass rod. Hold the cloth-end of this rod and the ebonite 
 rod in succession within the can. If either proves to 
 have an electric charge, remove the charge by passing 
 the rod rapidly through a Bunsen flame. 
 
 Hold the ends of the two rods within the can, and rub 
 them together. Then remove the ebonite rod, and 
 observe the leaves of the electroscope. Replace the 
 ebonite rod, and withdraw the cloth. Observe the elec- 
 troscope during the operation and explain. 
 
 2. Collect a negative charge on the proof plane and 
 hold it inside the can without touching it, and move it 
 about. Explain why the charge on the electroscope 
 does not vary with the position of the proof plane. 
 
 3. Again hold the charge on the inside of the can 
 and touch the can with the hand. Explain. Remove 
 the proof plane and then touch the can with it. Does 
 this prove that the induced and inducing charges are 
 equal? What is Faraday's ice-pail experiment? 
 
66 ELECTKICITY AND MAGNETISM 
 
 D. Remove the can, and charge the electroscope. 
 
 1. Hold the hand or a sheet of metal near the plate of 
 the charged electroscope. Explain. 
 
 2. Clamp a sheet of metal close to the plate of the 
 electroscope and connect it to earth by touching the 
 metal support. Charge the electroscope and then note 
 the effect of introducing between the plates sheets of 
 various insulating materials. 
 
 Why must the metal sheet be connected to earth? 
 What is an electric condenser? What is meant by specific 
 inductive capacity? 
 
 3. Touch the plate of the electroscope with a proof 
 plane. Observe the change on the separation of the 
 leaves on removing the proof plane. Explain. 
 
 EXPERIMENT 42 
 TO PLOT A MAGNETIC FIELD 
 
 Apparatus.— Two magnets; a small magnetic needle 
 mounted on a pivot; a large sheet of paper. 
 
 Manipulation. — Place the two magnets on the sheet 
 of paper 10 cms. apart, either parallel or perpendicular 
 to each other, and so that a pole of one is equidistant 
 from the two poles of the other. Mark dots on the paper 
 about one centimeter apart around the edge of one of 
 the magnets. Place the magnetic needle at that end of 
 the magnet from which the north-seeking or -|-pole is 
 repelled and so that the center of the attracted or —end 
 is even with one of the dots and mark with a dot the 
 point on the paper that is even with the center of the 
 repelled end. Then move the needle so that the center 
 of the attracted end is even with this second dot, and 
 continue as before until the second magnet, the edge 
 of the paper, or the other end of the same magnet is 
 reached. Draw a line through this succession of dots. 
 This gives the direction in which the magnetic force 
 
ELECTEICITY AND MAGNETISM 67 
 
 moves the + pole of the needle and hence the direc- 
 tion of a magnetic line of force. Indicate this direction 
 by means of an arrow. In a similar manner plot the 
 magnetic lines of force from all the dots first placed 
 around the magnet. 
 
 If any large portion of the second magnet is not reached 
 by the lines of force coming from the first magnet, place 
 dots a centimeter apart in this region, and plot as before. 
 
 Observe the direction of the magnetic lines of force 
 in regions between poles that are attracting and that are 
 repelling each other. 
 
 Trace the outline of the magnets, and mark each pole 
 with its proper sign. 
 
 EXPERIMENT 43 
 ADJUSTMENT OF MAGNETOMETER 
 
 Apparatus. — A magnetometer; a triangular piece 
 of board having a right angle and one side of which is 
 25 cms. in length. 
 
 Adjustment. — 1. Place the magnetometer on the shelf 
 in such a position that the glass tube containing the 
 suspension fiber is directly in front of the vertical black 
 line drawn on the wall, and that the side of the tube 
 nearest the wall is 25 cms. from the wall. Use the 
 triangular board for locating this position. The value of 
 the horizontal component of the earth's magnetic field 
 varies considerably in different parts of the room due to 
 the presence of steel beams and iron radiators. It is for 
 this reason that the magnetometer is placed at some 
 definite point. 
 
 2. Turn the magnetometer until the horizontal scale 
 is in the magnetic east and west, i.e., perpendicular 
 to the magnetic meridian. The plane surfaces of the 
 cylindrical copper damping block in which the annular 
 
68 
 
 ELECTRICITY AND MAGNETISM 
 
 magnetic needle hangs are perpendicular to the scale. 
 When the plane of the needle is parallel to a plane of this 
 surface or to the plane separating the two halves of the 
 copper block, the scale is perpendicular to the magnetic 
 meridian. In laboratories where there are no appreciable 
 variations in the direction of the earth's field as modified 
 by the magnetic materials of the building, the adjustment 
 
 Fig. 40. 
 
 can be more accurately made by means of a compass 
 attached to a brass bar. 
 
 3. Raise the needle by turning the upper attachment 
 of the suspension fiber a fraction of a turn while watch- 
 ing the needle, and adjust the level of the magnetometer 
 until the needle is free. 
 
 EXPERIMENT 44 
 
 ADJUSTMENT OF TELESCOPE FOR THE TELESCOPE- 
 MIRROR-SCALE METHOD OF MEASUREMENT 
 
 1. Position of Telescope. — Place the telescope stand 
 at a distance of 100 to 150 cms. in front of the magnetom- 
 eter mirror, and .raise the stand until a perpendicular 
 to the mirror passes through a point half-way between 
 
ELECTRICITY AND MAGNETISM 
 
 69 
 
 the axis of the telescope and the reading scale. When 
 the observer sees the image of his eye in the mirror, his 
 eye is in the perpendicular to the mirror. This adjust- 
 ment is often most readily made by holding the eye on 
 the level of the telescope and raising or lowering the 
 stand until the image of the scale is seen in the mirror. 
 
 2. Focusing of Cross-wires. 
 — Adjust eye-piece of the tele- 
 scope until the cross-hairs are 
 seen most distinctly. 
 
 3. Final Adjustment of 
 Height of Telescope Stand. — 
 Focus the telescope upon the 
 image of the scale in the mir- 
 ror. If the image of the scale 
 cannot be found, the telescope 
 stand is usually too high, or 
 too low, or the telescope is not 
 pointed properly. Adjust the 
 height of the stand until the 
 scale divisions appear in the 
 center of the field. 
 
 4. Position of Reading Scale, 
 placed perpendicular to the telescope. 
 
 5. Final Adjustment for Horizontal Position of Tele- 
 scope. — Move telescope stand to the right or to the left 
 until the point on the scale directly above the axis of the 
 telescope appears on the vertical cross-hair of the tele- 
 scope. This point on the scale is called the null point 
 or the zero point. 
 
 6. Parallax. — When the eye-piece is adjusted so that 
 the cross-hairs are seen most distinctly, and the 
 telescope is adjusted until the image of the scale is seen 
 most clearly, there should be no parallax, i.e., the image 
 jn the telescope and the cross-hairs should be in the same 
 
 Fig. 41. 
 
 -The scale must be 
 
70 
 
 ELECTRICITY AND MAGNETISM 
 
 plane, so that when the eye is moved from side to side 
 of the eye-piece, the cross-hair does not change its posi- 
 tion with respect to the scale. If the reading changes 
 as much as 0.1 of a mm. with the motion of the eye, 
 change adjustment of the telescope or of the eye-piece 
 for the cross-hair, or of both, until there is no appreciable 
 parallax.* 
 
 EXPERIMENT 45 
 
 MEASUREMENT OF ANGLES OF DEFLECTION BY THE 
 TELESCOPE-MIRROR-SCALE METHOD 
 
 When the null point of the scale is seen on the cross- 
 
 FiG. 42. 
 
 hair of the telescope, a ray of light coming from the null 
 point of the scale, in Fig. 43, is reflected from the mirror 
 
 * The Telescope-Mirror-Scale Method, Adjustments and Tests, by 
 S. W. Hohnan, 1898. 
 
ELECTKICITY AND MAGNETISM 
 
 71 
 
 at the point P. The incident and reflected rays are 
 in the same vertical plane perpendicular to the mirror. 
 When the mirror is turned through the angle 6, this 
 vertical plane moves from 
 OP, Fig. 44, to Pd, so that 
 a ray of light coming from 
 S in a vertical plane SP, 
 will be reflected in the ver- 
 tical plane OP making the 
 angle SP0 = 2Q. ^ 
 
 If L be the perpendicular 
 distance between the plane 
 of the mirror and the plane 
 of the scale, then 
 
 tan 29 = 
 
 D 
 L 
 
 Fig. 43. 
 
 The value of 29 is obtained from a table of tangents, 
 and 9 from the determined value of 29. 
 
 EXPERIMENT 46 
 TO REMOVE TORSION FROM MAGNETOMETER FIBER 
 
 Apparatus. — Magnetometer of Exp. 43 and telescope of 
 Exp. 44. 
 
 Manipulation. — By means of a pencil carefully turn 
 the magnetic needle 360° in one direction and after 
 the needle comes to rest note the deflection. Turn the 
 needle back to its original null position, take reading 
 and then turn it 360° in the opposite direction, take 
 deflection reading and again the null reading. If the two 
 deflections are equal there is no torsion in the fiber when 
 the needle is in its original position. If the two deflec- 
 tions are not equal, use that position in which it gave 
 
72 ELECTRICITY AND MAGNETISM 
 
 the greater deflection * as a new original position, adjust 
 the telescope again to give the reading of the null 
 point, and then find the two deflections as above. Con- 
 tinue in this manner until the two deflections are equal. 
 If it is necessary to remove torsion of less than one turn, 
 the tube which holds the upper support of the fiber 
 must be turned a fraction of a revolution at a time, and 
 tests for torsion made until it is removed. All readings 
 must be taken as soon as possible after the needle comes 
 to rest, for the large twist given the fiber gradually causes 
 it to obtain a set. To partly correct for the set in the 
 fiber take a reading at the null position of the needle 
 after each observation, and use this in computing that 
 deflection. 
 
 EXPERIMENT 47 
 RATIO OF TORSION OF A MAGNETOMETER FIBER 
 
 Apparatus. — A magnetometer adjusted as in Exps. 
 43, 44, and 46. 
 
 Method. — The magnetometer needle has the direc- 
 tion of the magnetic meridian when its supporting 
 fiber hangs without torsion. If it is deflected from 
 this position, the fiber becomes twisted and exerts a 
 force tending to bring the needle back into the meridian. 
 Both the earth's field and the torsion of the fiber thus 
 tend to move the needle into its original position. The 
 moment of the magnetic force, HmX sin 0, varies as the 
 sine of the angle of deflection, and the torsional moment, 
 T6, as the angle itself. Since the angle of deflection 
 used is small, its sine may be considered equal to the 
 
 _* This position is used in case a cocoon fiber is employed. For 
 a perfectly elastic fiber the position giving the smaller deflection 
 would be taken. 
 
ELECTRICITY AND MAGNETISM 73 
 
 angle; thus as the angle increases the moment of the 
 torsional couple increases at the same rate as that of the 
 magnetic couple. The value of the former may than be 
 expressed as a fraction of the latter. This fraction is 
 called the ratio of torsion of the fiber and is represented 
 by the letter 9. 
 
 The total moment of the forces tending to restore 
 the needle into its original position = i?m>w sin 9 + ^6 
 
 = Hm\ sin 9(1+ „ f^ } \ =Hm\{l-{-Q) sin 9, 
 \ Hmk sm 9/ 
 
 where H = horizontal intensity of the earth's field, 
 m = pole strength of the needle, 
 >, = distance between the poles of the needle, 
 = ratio of torsion, 
 9 = angle of deflection, 
 !r = moment of torsion at unit angle. 
 
 To measure 6 turn the needle through 360°. The 
 fiber is then twisted and turns the needle back a frac- 
 tion of a revolution until the moment of its torsional 
 couple equals the moment of the magnetic couple. 
 This fraction of a revolu- 
 tion is the angle i? of Fig. ^^^-^^^ 
 
 45. The fiber is twisted r'---- -^^T7t:t-^^ 
 
 through (360° -(!)); hence ^^ ' ' " 
 the moment of the tor- -p^^ ^^ 
 
 sional couple = (2x — c])) 7". 
 
 The moment of the magnetic couple at the same 
 time = Z?TOX sin <j). Since the needle is at rest and <j) 
 is small, 
 
 {2x-^)T = Hmk sin ^<pHm\ <^. 
 
 .: ^ ^ ^ . 
 ifmX^2x — (J)' 
 
74 ELECTRICITY AND MAGNETISM 
 
 re T 
 
 But e= 
 
 Hmk sin %^ Hnik' 
 
 e=^t_,=^ 
 
 2-1: -(j) 360°-cl)°' 
 
 Observations. — After torsion is removed from the 
 fiber, a set of scale readings is taken from which the 
 ratio of torsion is to be computed. One method for 
 recording the observations is here given as an illustration: 
 
 ■T3_M 
 
 3 
 
 
 25.00\ 
 
 >25.94 
 25.04/ 
 
 24.12< 
 
 ^24.99\ 
 
 >25.92 
 
 /25.02/ 
 24.09< 
 
 ^24.98 
 
 The perpendicular distance from the mirror to the 
 plane of the scale is measured. 
 
 Calculations. — The null point taken after a deflection 
 is the one employed in computing the magnitude of 
 each deflection. 
 
 The average of the several approximately equal de- 
 flections is taken for computing the value of ^ which is 
 obtained in the manner explained in Exp. 45. 
 
ELECTRICITY AND MAGNETISM 
 
 75 
 
 EXPERIMENT 48 
 
 THE HORIZONTAL INTENSITY OF THE EARTH'S 
 MAGNETIC FIELD— METHOD OF GAUSS 
 
 This absolute method for measuring the horizontal 
 intensity of the earth's magnetic field consists of two 
 
 
 Fig. 45. 
 
 \ 
 I 
 V / 
 
 / 
 
 
 M 
 independent determinations. The value of -77 is ob- 
 tained from the deflection produced on a short mag- 
 netic needle by a bar magnet, and the value of MH 
 from the time of vibration of this same bar magnet 
 
 in the earth's magnetic field. 
 
 MH=B, 
 
 [B 
 /A' 
 
 M 
 Thus if 7F = A) and 
 
 H=. 
 
 ilf = magnetic moment of the bar magnet, and H= 
 the horizontal intensity of the earth's magnetic field. 
 
76 ELECTRICITY AND M4GNETISM 
 
 1. 10 DETERMINE fj. 
 
 Apparatus. — The magnetometer with reading telescope 
 arid scale, a bar magnet. 
 
 Method and Manipulation. — The magnetometer is 
 first adjusted as in Exps. 43, 44 and 46, and the ratio 
 of torsion determined as in Exp. 47. The deflection of 
 the magnetometer needle is noted when the bar magnet 
 is in position a, Fig. 46. The magnet is then turned 
 
 1 
 
 O 1/ 
 
 ' ' ' -h 
 
 Fig. 46. 
 
 so as to reverse the position of the poles, and another 
 deflection is taken. The bar magnet is then placed in 
 position h, the same distance from the needle, and a 
 similar set of observations taken. The angle of deflec- 
 tion is determined from the average of the scale 
 deflections for the positions a and h of the magnet. 
 
 The distance r is made large compared with the length 
 of the needle. Thus dor— Z. The force /' with which 
 the nearer pole of the magnet acts on a pole of the 
 
 needle =7 ^kh- The force /" due to the other pole 
 
 (r-ir 
 
 of the magnet = . ,,^2 ^^^ ^^*^ ^"^ ^ direction practically 
 
 opposite to that of / '. 
 
 , „_ mm! mm' _ Mmm' _2Mm'r 
 
 ihus;-; -/ -(^_^)2 (^qr^p- (^_p)2 - (f-i2)2 ■ 
 
 The component of this force tending to produce rotation 
 
 2Mm.'r ,„ , , , 2Mm'r q * ^u ^u 1 
 cos (9+q>) = 7-2 — 12^ cos 6. As the other pole 
 
 UZ — 12\2 ^"'^ y"^^'-^ ljZ_l2\2 
 
ELECTRICITY AND MAGNETISM 77 
 
 of the needle is acted on by an equal force, the moment 
 due to this acting couple o-r-j — ^3^2" *^°^ ^> wli^re X is the 
 
 distance between the two poles of the needle and AI = 2lm. 
 
 The horizontal component of the earth's magnetic 
 field acts on each pole of the needle with a force of m' H 
 dynes. The moment of this couple = m'i£X sin 9. 
 
 The suspending fiber is twisted with the deflection 
 of the needle and, tending to untwist, acts on the needle 
 in the same direction as does the earth's field. The 
 
 -SP- 
 
 U , 
 
 Fig. 47. 
 
 total moment of these two forces = m'i?X(l +9) sin 6, 
 as shown in Exp. 47. This moment is in a direction 
 opposite to that due to the action of the magnet and is 
 equal to it because the needle is at rest. 
 
 Thus 
 
 , , _ „, 2 cos 6 = m iiA(l+ej sm 9, 
 whence 
 
 Manipulation. — It is best to use a magnet with a 
 large magnetic moment and to take a large value 
 for r, but in an elementary laboratory where several 
 instruments are in use at the same time this con- 
 dition must be modified. It is necessary, however, 
 that r be at least five times the length of the mag- 
 
78 ELECTEICITY AITD MAGNETISM 
 
 net and twenty times the length of the magnetic 
 
 needle. 
 
 The distance 21 between the poles of the magnet 
 
 is obtained approximately by assuming each pole to 
 
 be one-twelfth the length of the bar from the end of 
 
 the magnet. A considerable error in I produces but a 
 
 M 
 small error in the value of -jy. The distance r is meas- 
 
 xi 
 
 ured on the attached scale and the angle 6 * from the 
 
 deflections observed with the telescope. 
 
 2. To Determine MH. 
 
 Apparatus. — The magnet used in the determination 
 
 M 
 oi jj-; a silk fiber attached to an aluminum saddle for 
 
 suspending the magnet; a box for excluding the air 
 currents from the vibrating magnet; a circular scale; a 
 watch. 
 
 Method. — The general equation for the time of vibra- 
 tion of any body vibrating about an axis with simple 
 
 harmonic motion is < = 2'k-v/— , (Exp. 10, p. 19), where x 
 
 is the linear displacement and a the linear acceleration 
 for that displacement. In a magnetic pendulum sus- 
 pended by a fiber with a negligibly small ratio of torsion 
 the moment of the restoring couple 
 
 Fr = Ioa = niHl sin Q = MH sin 9; 
 
 MH sin 6 
 .% a = J , 
 
 * For limits of observational error in case of functions of angles 
 see p. ?. 
 
ELECTRICITY AND MAGNETISM 79 
 
 whence. 
 
 x_ e_ 7o6 _ 7o 
 a~ a~ MH sin Q'^ MH' 
 
 For small angles, 
 
 and 
 
 ^^^Wm^' 
 
 MF=^, 
 
 where Jo is the moment of inertia of the magnet and 
 t the time of a double vibration. 
 
 Manipulation. — The magnet is suspended at its cen- 
 ter of gravity in the location occupied by the needle 
 
 M 
 of the magnetometer in the determination of -^. The 
 
 £1 
 
 magnet is made to vibrate in an arc of about 30° and 
 care is taken to have the magnet free from any pendu- 
 lar motion. The time of vibration is determined in 
 the ordinary manner. When H is to be determined to 
 within 0.1%, the time of the vibration should be deter- 
 mined to within 0.05%. The moment of inertia of the 
 magnet can be determined experimentally or calculated. 
 The magnet used is usually a cylinder, the moment of 
 inertia of which, when rotating on an axis perpendicular 
 
 to its center, is M 
 
 Ll2"^4j' 
 
 where I is the length of the 
 
 cylinder, r its radius, and M its mass. 
 
 Since the expression for the time of vibration is cor- 
 rect only for very small angles, the comparatively 
 large angle used necessitates a small correction to the 
 
 * Duff, p. 69. 
 
80 ELECTRICITY AND MAGNETISM 
 
 observed time. When t is the observed time, and 6 the 
 average complete arc of vibration, the corrected time 
 
 h = t{l-\&m^\^. 
 
 Note. — Earth currents due to electric street cars, etc., often 
 cause the magnetic needle to vibrate. The position of the needle 
 for any observation must, in such a case, be determined by taking 
 the mean of the extreme positions, the large accidental variations 
 not being taken into consideration. 
 
 EXPERIMENT 49 
 
 THE MAGNETIC MOMENT AND THE POLE STRENGTH OF 
 THE MAGNET USED IN EXP. 48 
 
 M 
 Method. — From Exp. 48 the values of -jj and MH are 
 
 known. 
 
 M 
 
 „=A, MH = B, then M = VaB. 
 tt 
 
 But M=2lm. 
 
 Determine the horizontal force with which each 
 pole of the magnet was acted on in the location where 
 H was determined by the method of Gauss. 
 
 * Ziwet, Part I, p. 121. 
 
ELECTRICITY AND MAGNETISM 
 
 81 
 
 EXPERIMENT 50 
 
 TO DETERMINE H BY COMPARISON 
 
 Method. — The time of vibration of a magnet is found 
 in a place where H is known, and then where H is to 
 be determined. 
 
 - F 
 
 and 
 
 
 EXPERIMENT 51 
 
 THE INCLINATION AND THE INTENSITY OF THE EARTH'S 
 MAGNETIC FIELD* 
 
 Method and Manipulation. — The inclination compass, 
 Fig. 48, is used to measure 
 the inclination or angle of dip. 
 It consists of a magnetic 
 needle having a pivot passing 
 through its center of gravity 
 and attached at the center of 
 a graduated circle. This cir- 
 cle can be rotated about a 
 horizontal axis, and its sup- 
 port about a vertical axis. 
 The circle is placed in a hori- 
 zontal plane so that the needle 
 takes the direction of the mag- 
 netic meridian. The compass 
 is then turned until the needle 
 takes the direction of the axis 
 of rotation of the graduated 
 circle. The circle is now ^^' 
 
 * Henderson, p. 268. Watson, p. 619. Stewart & Gee, II, p. 275. 
 
82 ELECTRICITY AND MAGNETISM 
 
 turned into a vertical position so that its plane is in the 
 magnetic meridian, and the magnetic needle takes the 
 direction of the magnetic field. Readings are taken at 
 both ends of the needle and the inclination thus deter- 
 mined. 
 
 Knowing the horizontal intensity from Exp. 48 and 
 the angle of inclination, the intensity of the earth's 
 
 field can be calculated from F= k- (See Fig. 105.) 
 
 cos 9 
 
 Sources of Error. — 1. The axis of suspension of the 
 needle may not pass exactly through its center of gravity. 
 
 2. The geometrical and magnetic axes of the needle 
 may not coincide. 
 
 EXPERIMENT 52 
 
 TO ADJUST A TANGENT GALVANOMETER AND TO 
 DETERMINE THE RATIO OF TORSION OF ITS FIBER 
 
 Apparatus. — Tangent galvanometer; battery of con- 
 stant electromotive force; commutator; adjustable 
 resistance. 
 
 The tangent galvanometer consists of a vertical coil 
 of wire having a short magnetic needle at its center. 
 If the length of the needle is less than one-twelfth the 
 diameter of the coil that part of the field in which the 
 needle swings may be considered uniform and of the same 
 intensity as the field at the center of the coil.* 
 
 Adjustment. — 1. After placing the galvanometer in 
 the location where the horizontal component of the 
 earth's magnetic field has been determined, free the 
 needle in the same manner as in the case of the magnetom- 
 eter, Exp. 43. 
 
 *Nipher, pp. 162-166. 
 
ELECTKICITY AND MAGNETISM 
 
 83 
 
 2. Turn the galvanometer until the plane of its coil is 
 in the magnetic meridian as nearly as it is possible by- 
 direct observation of the direction of the free mag- 
 netic needle and the parts of the body of the galvano- 
 meter. 
 
 3. Remove torsion from the fiber, and determine 
 the ratio of torsion as in Exps. 46 and 47. 
 
 4. Repeat adjustment 2. 
 
 Fig. 49. 
 
 Fig. 50. 
 
 5. Where the galvanometer needle is not disturbed 
 by earth currents it is possible to make a more accurate 
 adjustment of the plane of the galvanometer coil than 
 the approximate adjustment given above. The galva- 
 nometer is connected to the accessory apparatus as 
 shown in Fig. 50. 
 
 The electric current is regulated by means of the 
 adjustable resistance R until it gives a large deflection 
 
84 ELECTRICITY AND MAGNETISM 
 
 on the scale. It is then reversed by means of the 
 commutator C and the deflection on the other side of 
 the null point is observed. If the two deflections are 
 equal, the coil is parallel to the needle, and is in the 
 magnetic meridian. If the deflections are not equal, 
 the galvanometer is turned so that the plane of its 
 coil approaches the direction of the needle when in the 
 position of its smaller deflection. Continue in this 
 manner until the deflections are equal. If the current 
 changes with time the two deflections must be taken in 
 rapid succession. 
 
 EXPERIMENT 53 
 
 TO DETERMINE THE CONSTANT OF A TANGENT GALVA- 
 NOMETER FROM THE DIMENSIONS OF ITS COIL 
 
 Apparatus. — A tangent galvanometer. 
 
 Magnetic Field About a Wire Carrying an Electric 
 Current. — The intensity of the magnetic field is propor- 
 tional to the current, and, for each element of the cur- 
 rent, varies inversely as the square of the distance from 
 the wire. 
 
 Electromagnetic Unit of Current. — If a conductor be 
 bent into a loop of one centimeter radius, the current 
 through it has unit strength when for each centimeter 
 of the conductor, a force of one dyne is exerted on a 
 unit magnetic pole at the center. Whence at the center 
 of a circular loop consisting of n turns of wire the intensity 
 
 of the field F=-^ — =— — ■, where F is the strength 
 
 of the current in electromagnetic units. 
 The force acting on a pole of m units at the center is 
 
ELECTRICITY AND MAGNETISM 
 
 85 
 
 Practical Unit of Current. — The ampere is 0.1 of the 
 electromagnetic unit of current. 
 
 Method. — Fig. 51 represents a horizontal section of 
 the galvanometer. The earth's field acts on each pole 
 of the needle with a force of mH dynes. The field due 
 
 to the current acts with a force of dynes, and in 
 
 a direction perpendicular to the former force. When 
 the needle is deflected, the torsional couple of the fiber 
 
 U^s^ 
 
 Fig. 51. 
 
 also exerts a force on the needle and tends to move it 
 in the same direction as does the earth's field. 
 Thus when the needle comes to rest 
 
 2%nmXI' 
 
 cos % = mH\ sin %-\-T% = mH'k{l-\-Q) sin 9, 
 
 2%n 
 
86 ELECTRICITY AND MAGNETISM 
 
 %n 
 where 
 
 ^ 5Hr(l+e) , , 
 
 K = ^^ -, a constant. 
 
 Manipulation. — H and 6 are determined as in previous 
 experiments; r is obtained by measuring the diameter 
 of the coil of wire. Calculate the constant of the 
 galvanometer for 1, 2, 3, 5 and 6 turns of the wire. 
 
 EXPERIMENT 54 
 
 TO CALIBRATE AN AMMETER BY MEANS OF A TANGENT 
 GALVANOMETER 
 
 Apparatus. — ^An ammeter; a battery; a tangent gal- 
 vanometer; a resistance rack and a commutator. 
 
 The essential part of an ammeter of the Weston 
 type consists of a coil of wire held by two spiral springs 
 and pivoted on an axis in the magnetic field between 
 the poles of a permanent magnet, as shown in Fig. 53. 
 The deflection of this coil, which is indicated by an 
 attached pointer on a graduated scale, as shown in 
 Fig. 52, depends on the strength of the current flowing 
 through it. The scale is originally graduated by passing 
 known currents through the ammeter and marking 
 the points to which the pointer is deflected. The 
 strength of the magnet changes with time and usage and 
 thus, for accurate work, the ammeter must occasionally 
 be calibrated. 
 
ELECTKICITY AND MAGNETISM 
 
 87 
 
 Great care must he exercised always to connect the + 
 'pole of the battery to the binding post similarly marked and 
 never to send a current through the instrument that produces 
 a deflection greater than the full reading of the scale. Be- 
 fore connections are made to the instrument, put in 
 the circuit all of the resistance on the rack provided 
 and diminish this gradually until the desired current is 
 obtained. 
 
 Method and Manipulation. — Pass a current through 
 the ammeter, tangent galvanometer and resistance rack 
 
 •Fig. 52 
 
 Fig. 53. 
 
 in series. In connecting the apparatus, care must be 
 taken to have the commutator reverse the current 
 through the tangent galvanometer only (see Fig. 50), for 
 the current through the ammeter must always be in the 
 same direction. Measure the current by means of the 
 tangent galvanometer and record the corresponding read- 
 ing of the ammeter. Tabulate the results for several 
 readings which extend at approximately regular intervals 
 over the full range of the scale. 
 
 In reading the ammeter, care must be taken to avoid 
 parallax and to take the " zero " reading. The value 
 
88 ELECTKICITY AND MAGNETISM 
 
 of the current as read on the scale of the ammeter must 
 be corrected by + or — the value of the zero reading. 
 Always record the zero reading. 
 
 The deflections on the tangent galvanometer must 
 be large so that the observational error in the 
 determination of the value of the current is less than 
 the observational error in the corresponding ammeter 
 reading. If, as is nearly always the case, the percentage 
 of error in the readings of the ammeter is practically 
 the same for all the readings, this percentage with its 
 proper sign, + or — , is called the correction factor of the 
 ammeter. Determine this correction factor for the 
 instrument caUbrated. The correction factor is + 
 if it is to be added to the ammeter reading. 
 
 In laboratories where stray earth currents affect the 
 readings of the tangent galvanometer only a rough 
 calibration can be made. Results, however, should be 
 obtained having as high a degree of accuracy as is pos- 
 sible under the existing conditions. 
 
 EXPERIMENT 55 
 
 THE CORRECTION FACTOR OF AN AMMETER— SILVER 
 VOLTAMETER METHOD 
 
 Apparatus. — Silver voltameter; silver nitrate solu- 
 tion; ammeter; resistance rack; battery. 
 
 Method. — Accurate determinations have been made 
 of the mass of silver deposited by a known current from 
 a solution of silver nitrate.* It has been found that the 
 deposit for any given time is proportional to the cur- 
 
 *See definition of international ampere in Appendix. 
 
ELECTRICITY AND MAGNETISM 
 
 89 
 
 rent and that 0.001118 gram is deposited by one ampere 
 in a second. Hence, the 
 average strength of 
 current is 
 
 any 
 
 m 
 
 O.OOlllSi' 
 
 Fig. 54. 
 
 where m is the mass of 
 silver deposited and t the 
 time. 
 
 Manipulation. — The 
 voltameter V and am- 
 meter A are connected to 
 the battery as shown in 
 Fig. 54. The cup of the 
 voltameter is the cathode 
 and is connected to the 
 
 negative pole of the battery. The voltameter, before 
 filling with AgNO.3, is short-circuited and the resist- 
 ance is adjusted until a conveniently large deflection 
 is obtained on the ammeter. The short-circuit is then 
 removed and the circuit is opened. The anode is 
 placed within a porous cup to prevent particles dropping 
 from it into the cathode cup, and is so adjusted as to be 
 at the approximate center of the latter. The cups are 
 then filled with the silver nitrate solution and the cir- 
 cuit is closed, the time of closing being, accurately noted. 
 The current is allowed to flow about an hour and is kept 
 constant by occasional slight changes in the adjustable 
 resistance. Finally, when the circuit is opened, the 
 time is again noted. 
 
 The silver nitrate solution is poured into the stock 
 bottle through a filter paper to save any crystals of 
 silver that may break off. A 5% silver nitrate solution 
 gives the best deposit for currents of less than 0.5 
 
90 ELECTRICITY AND MAGNETISM 
 
 ampere, but any solution up to 30% may be used. 
 The area of the cathode should be between 50 and 500 
 square centimeters per ampere. 
 
 To clean the silver cathode: 
 
 If the cathode cup contains deposited silver crystals, it 
 is usually clean, and needs drying only. If the crystals 
 are not clean, the cup must be polished inside and then 
 cleaned as directed: 
 
 1. Polish with clean cloth, water and fine sand. 
 
 2. Wash to remove sand. 
 
 3. Wash with soap and water, and then with water 
 alone. 
 
 4. Wash with a warm solution of potassium cyanide 
 (1%). Wash in distilled water. 
 
 5. Dry in hot air bath at a temperature between 100° 
 and 120° C, and leave to cool in a desiccator before 
 weighing. 
 
 To wash cathode after deposit: 
 
 1. Rinse four times with hot distilled water pouring 
 the water each time through a filter to save any loose 
 crystals whose weight must be added to that of the 
 cathode. 
 
 2. Fill with hot distilled water and let stand about 
 thirty minutes. The final washing should give no re- 
 action with HCl. 
 
 4. Dry in air bath as before and cool, before weighing. 
 If a large deposit of crystals is present, keep cathode 
 in air bath at least twenty minutes. 
 
 Note. — Use great care not to touch the inner surface of the 
 cathode cup with the fingers during and after washing. It is best 
 to place a large resistance in the circuit so that any variations in 
 the resistance of the voltameter can cause but little change in the 
 strength of the current. 
 
 Copper, water and zinc voltameters are sometimes used but 
 are not as accurate for small currents as the silver voltameter. 
 
ELECTEICITY AND MAGNETISM 
 
 91 
 
 References. — Stewart & Gee, II. p. 253. Henderson, p. 151. 
 K. E. Guthe, Bulletin Bureau of Standards, Vol. 1, pp. 21 and 349; 
 1904-5. 
 
 THE AYRTON SHUNT 
 
 e 
 
 WVWVWVWVW\^VWWWAA'v^ 
 
 <-iS— . 
 
 The Ayrton shunt is constructed in such a manner 
 that any galvanometer to 
 which it is connected, 
 whatever may be its re- 
 sistance, receives the same 
 fraction of the whole cur- 
 rent when the switch is 
 on the same contact pin. 
 This makes the shunt 
 universal. It consists of Fig. 55. 
 
 a series of resistances 
 
 represented by »S in Fig. 55. The galvanometer is connected 
 to the extremities of this series. The source of electricity 
 is connected to A and P, the latter being a switch which 
 can be moved to include different fractions of the whole 
 resistance S. Let I'l be the current flowing through the 
 galvanometer, I the whole current, and G the resistance 
 of the galvanometer. Then if P is connected to B so 
 as to include the whole of the resistance S between 
 A and P, 
 
 from which 
 
 ii 
 
 S 
 
 I-ii G' 
 
 11 = 
 
 G+S 
 
 (I) 
 
92 ELECTRICITY AND MAGNETISM 
 
 When the switch includes — th of the whole resistance, 
 
 n 
 
 between A and P, 
 
 ii n 
 
 I-ii' 
 
 G+ 
 
 H^) 
 
 from which 
 
 I^'-ifS^=:^-G+sI (II) 
 
 From equations (I) and (II) 
 
 ix' I 
 n 
 
 when the whole current is the same in both cases. The 
 ratio of the two currents is independent of the resistance 
 of the galvanometer, and depends only on the ratio the 
 whole resistance S bears to the fraction of the resistance 
 included between A and P- 
 
 The resistances of the shunt have usually the ratios 
 1 : 9 : 90 : 900, so the resistances that may be included 
 between A and P bear to each other the ratio of 1 : 
 10 : 100 : 1000. The corresponding ratios of the cur- 
 rents flowing through the galvanometer are 0.001 : 0.01 : - 
 
ELECTEICITY AND MAGNETISM 
 
 93 
 
 0.1 : 1. These latter ratios are usually stamped on the 
 shunt as shown in 
 Fig. 56. When the 
 switch is on the pin 
 marked 0, the source 
 of electricity is short- 
 circuited, and when it 
 is on the pin marked 
 Inf., the circuit is 
 open. 
 
 The same ratios may 
 be employed in com- 
 paring quantities of 
 
 electricity. The inductance in the galvanometer can be 
 shown to have no effect upon them. 
 
 Inf. 
 O 
 
 Fig. 56. 
 
 EXPERIMENT 56 
 
 THE RELATION BETWEEN THE CURRENT AND ITS 
 DEFLECTION ON A MOVING COIL GALVANOMETER 
 
 Apparatus. — For most electrical measurements the 
 tangent galvanometer is not sufficiently sensitive. There 
 are two types of sensitive galvanometers; in one, the 
 coil is fixed, and the magnet is movable; in the other 
 the magnet is fixed, and the coil is movable. The first 
 type is illustrated in Fig. 57. This differs from the tan- 
 gent galvanometer in having a coil of small diameter 
 and a comparativeh' large number of turns of wire. The 
 sensitiveness of the instrument is varied by sliding the 
 coil nearer to or farther from the magnetic needle. The 
 second type is known as the moving-coil or d'Arsonval 
 galvanometer and is illustrated in Fig. 58. The per- 
 
94 
 
 ELECTRICITY AND MAGNETISM 
 
 manent magnet serves as the frame of the instrument, 
 and the coil is suspended in the magnetic field be- 
 tween the poles of 
 ^ this magnet. The 
 
 sensitiveness is varied 
 by means of an ex- 
 ternal shunt which 
 allows Only a part of 
 the current to flow 
 through the galva- 
 nometer coil. This 
 type of galvanometer 
 is not appreciably 
 affected by external 
 magnetic distur- 
 bances, and is, there- 
 fore, the type now 
 most commonly em- 
 ployed. 
 
 Method and Ma- 
 nipulation. — ^A milU- 
 ammeter A and a 
 shunted moving-coil 
 galvanometer G are 
 connected in series with a resistance box R and a 
 battery B. The galvanometer is also provided with 
 a short-circuiting key K. 
 
 The sensitiveness of the galvanometer is adjusted 
 until both instruments give about a full scale deflection. 
 After adjustment, the sensitiveness of the galvanoftieter 
 must not be changed during the experiment. Deflec- 
 tions are taken with about ten different currents, de- 
 creasing in value uniformly from the highest to the 
 lowest. 
 The deflections of the galvanometer should preferably 
 
 Fig. 57. 
 
ELECTEICITY AND MAGNETISM 
 
 95 
 
 be to the right as viewed through the reading telescope. 
 The magnitude of a deflection must be reckoned from the 
 zero reading taken immediately after the deflection. 
 If the galvanometer is not aperiodic, i.e., if on returning 
 to the zero point the coil swings beyond it, the coil 
 must be made aperiodic while it is returning, by 
 
 Fig. 58. 
 
 short-circuiting the galvanometer through key shown 
 in Fig. 59. In case this does not produce a sufficient 
 amount of damping, the motion of the galvanometer 
 coil may be checked by induced currents produced by 
 moving a magnet within a coil C, placed for that purpose 
 in the short-circuiting branch. This operation is made 
 necessary by the presence of magnetic impurities * 
 within the coil which have their direction of magnetiza- 
 tion changed as the coil passes into different parts of the 
 magnetic field. This changes the zero point, and affectsf 
 
 * Physical Review, Vol. XXXII, p. 297, 1911. 
 t Physical Review, Vol. XXIII, p. 400, 1906. 
 
96 
 
 ELECTRICITY AND MAGNETISM 
 
 somewhat the sensibihty of the instrument. Even with 
 this precaution, the zero point will usually change after 
 successive deflections of increasing magnitude. In such 
 cases it is desirable to take the largest deflections first. 
 The temperature coefiicient * of a moving coil gal- 
 vanometer for continuous currents is about 0.00018 
 when the upper suspension is of phosphor-bronze, and 
 0.00005 when it is of steel. 
 
 Fig. 59. 
 
 Calculations . — The strength of the current in the 
 main circuit is given by the ammeter. This same cur- 
 rent, or a fractional part of it, passes through the sensitive 
 galvanometer and produces there the observed deflection. 
 Plot a curve on coordinate paper, showing the relation 
 between the current and the deflection produced on the 
 sensitive galvanometer. The scale used in plotting must 
 be large enough to show the relation to the limits of ob- 
 servational error. A record of the distance from the 
 galvanometer mirror to the scale and a statement of the 
 
 * Physical Review, Vol. XXVIII. p. 277, 1909. 
 
ELECTRICITY AND MAGNETISM 
 
 97 
 
 direction of the deflections should be made on the same 
 coordinate paper. 
 
 Note. — In many electrical measurements it is only necessary 
 to determine the ratio of the electric currents flowing through 
 the galvanometer. The cm-ve here obtained is used for determin- 
 ing this ratio. Usually the currents are proportional to the deflec- 
 tions for some distance so that for deflections within this range 
 
 i _d 
 i' d'' 
 
 EXPERIMENT 57 
 
 to make an ammeter from a moving coil 
 galvanomete;r 
 
 Apparatus. — Galvanometer; ammeter; battery; key; 
 small wire resistance; two resistance boxes. 
 
 Method and Manipulation. — The galvanometer with a 
 resistance box in its circuit is connected to the extrem- 
 
 B- 
 
 K 
 
 r— I 
 -O O- 
 
 FiG. 60. 
 
 ities of a small wire resistance S. This combination 
 makes an ammeter, and, by adjusting the resistance r, 
 a convenient whole number of divisions is made to 
 represent a current given by somp whole number of 
 di-^dsions on the ammeter A, Fig. 60. 
 
98 ELECTRICITY AND MAGNETISM 
 
 After the adjustment is made for the highest current 
 employed, decrease the strength of the current by con- 
 venient step's. Plot the values of the currents given 
 by the anuneter as abscissas and the amounts, + or — , 
 by which the values given by the galvanometer differ 
 from them, as ordinates. 
 
 This is the correction curve of the galvanometer when 
 used as an ammeter with the particular resistance and 
 shunt. If the galvanometer gives higher values than 
 the ammeter, the differences are plotted above the line 
 of reference. The curve must be properly labeled. 
 
 The best arrangement of resistances is such as, with 
 the aid of the damping rectangle, make the galvanometer 
 just aperiodic. 
 
 EXPERIMENT 58 
 
 DETERMINATION OF THE DIFFERENCE OF POTENTUL 
 BETWEEN TWO POINTS ON A CONDUCTOR CARRY- 
 ING A KNOWN ELECTRIC CURRENT— CALORIMETER 
 METHOD 
 
 Apparatus. — Electric calorimeter; ammeter; battery; 
 resistance rack; key; watch. 
 
 An electric calorimeter differs from an ordinary cal- 
 orimeter in having a resistance coil immersed in the 
 liquid. It is used for measuring the heating effect of a 
 current sent through this coil. 
 
 Method. — The difference of potential between two 
 points in absolute units is measured by the work done 
 in moving a unit quantity of electricity from one point 
 to the other. All the work is transformed into heat 
 within the conductor. The amount of this heat H, 
 in calories, can be measured by means of the calorim- 
 
ELECTRICITY AND MAGNETISM 99 
 
 eter. Whence from the definition, the difference of 
 potential 
 
 „, work done _JH ^ J(MS+M'S'+K)(t-t') 
 Q' FT FT 
 
 _ 10J{MS+M'S'+K)it-t') 
 IT 
 
 where J is the mechanical equivalent of heat. 
 
 As the specific heat of water varies with the tem- 
 perature, the amount of heat required to raise a gram 
 of water 1° C. varies: 
 
 atl5°C. J = 4.189X10''ergs, 
 
 20° C. J = 4.179X107 " , 
 
 25° C. J = 4.173X107 " , 
 
 30° C. / = 4.171X107 ". 
 
 The practical unit of difference of potential is the 
 volt and is equal to 10^ electromagnetic units. Thus 
 
 Manipulation. — The current is sent through the calor- 
 imeter, ammeter and resistance rack in series, as shown 
 in Fig. 61. Place the coil in a beaker of water and 
 adjust the resistance rack until the current has the desired 
 value, usually from four to ten amperes. Never send a 
 current through the coil without having first immersed it in 
 water. The calorimeter is weighed and then filled to 
 within about 2 cm. from the top with distilled water at 
 room temperature and again weighed. Wipe the coil 
 dry, and place it in the calorimeter. Before turning on 
 the current, take readings of the temperature of the 
 water at intervals of a minute for about five minutes. 
 
100 
 
 ELECTRICITY AND MAGNETISM 
 
 Then turn on the current, noting the time at that 
 instant. Read the thermometer eveiy thirty seconds 
 until the temperature has increased about eight or ten 
 degrees. Shut off the current, again noting the time 
 at the instant of breaking the circuit. Continue taking 
 temperature readings at intervals of one minute for a 
 period of eight minutes. Plot the readings taken, and 
 
 AAAAAAAAA/ 
 R 
 
 Fig. 61. 
 
 from the curve determine the rise in temperature by the 
 construction explained in Exp. 35. 
 
 Calculation. — From the data obtained calculate the 
 difference of potential in electromagnetic and in prac- 
 tical units. 
 
ELECTKICITY AND MAGNETISM 101 
 
 EXPERIMENT 59 
 
 TO SHOW THAT BETWEEN ANY TWO POINTS ON A 
 CONDUCTOR THE DIFFERENCE OF POTENTIAL AND 
 THE CURRENT ARE PROPORTIONAL— TO MEASURE 
 A RESISTANCE BY THE CALORIMETER METHOD 
 
 Apparatus. — Same as in Exp. 58. 
 
 Method. — The difference of potential between two 
 points on a conductor is determined for at least three 
 different currents by the calorimeter method as in Exp. 
 58. A convenient ratio for the values of the lowest 
 and highest currents used is 1 : 2. 
 
 A. To show that the difference of potential and the 
 
 E 
 current vary at the same rate, i.e., -y = a constant, 
 
 plot on coordinate paper the values of E and / obtained 
 in the experiment. The value of the constant depends 
 on the length, thickness, and material of the con- 
 ductor, and is called the resistance of that conductor. 
 
 It is usually represented by the letter R. 
 w J? 
 
 The relation -y = R, or 7 = ^ is known as Ohm's Law. 
 
 B. Obtain the constant R from the above plotted 
 curve, thus determining the resistance of the conductor. 
 
 EXPERIMENT 60 
 TO TEST A VOLTMETER BY THE CALORIMETER METHOD 
 
 Apparatus. — Same as in Exp. 58; a voltmeter. 
 
 Manipulation. — Attach the voltmeter to the extremities 
 of the calorimeter coil used in Exp. 58. Determine the 
 fall of potential between the two points from the calorim- 
 eter observations, and note the voltmeter readings. 
 
102 
 
 ELECTKICITY AND MAGNETISM 
 
 Calculate the correction factor of the voltmeter. 
 State why it is necessary for a voltmeter to have a high 
 resistance. 
 
 EXPERIMENT 61 
 
 TO MAKE A VOLTMETER FROM A MOVING COIL 
 GALVANOMETER 
 
 The voltmeter reads the difference of potential be- 
 tween the extremities directly in volts. The moving 
 part of the galvanometer corresponds to the moving coil 
 of the voltmeter, and the only requirement necessary to 
 
 change the galvanometer into 
 a direct reading voltmeter is 
 the introduction of the proper 
 high resistance into its circuit. 
 Apparatus. — Galvanometer ; 
 voltmeter; resistance rack; 
 two resistance boxes; battery; 
 key. 
 
 Manipulation. — Connect the 
 galvanometer and the volt- 
 meter to the extremities of 
 the same resistance as shown 
 in Fig. 62. The high resist- 
 ance r in the galvanometer 
 circuit is adjusted until some 
 convenient whole number of 
 large divisions on the scale 
 represents one volt. If a box with a sufficiently high 
 resistance is not available, a shunt may be placed next to 
 the galvanometer to reduce the sensitiveness. This 
 however should be avoided. 
 
 Fig. 62. 
 
ELECTRICITY AND MAGNETISM 103 
 
 After the above adjustment has been made, vary the 
 difference of potential between the extremities of U, and 
 plot the various observed readings of the voltmeter as 
 abscissas and the amounts, + or — , by which the values 
 given by the galvanometer differ from them, as ordinates. 
 
 This is the correction curve for the galvanometer when 
 used as a voltmeter. If the galvanometer gives higher 
 values than the voltmeter, the differences are plotted 
 above the line of reference. 
 
 EXPERIMENT 62 
 
 ELECTROMOTIVE FORCE OF CELLS IN SERIES AND IN 
 PARALLEL 
 
 Apparatus. — Same as in Exp. 61. 
 
 Manipulation. — Determine the electromotive force of 
 each of two cells, and then of the two in series and in 
 parallel. Make the measurements with the galvanometer 
 adjusted in the previous experiment and also with the 
 voltmeter. 
 
 The electromotive forces may be compared with an 
 unadjusted galvanometer having a sifiiciently high re- 
 sistance. The deflections are approximately proportional 
 to the electromotive forces. 
 
 In any case, the galvanometer or the voltmeter 
 measures only the difference of potential between the 
 binding posts to which it is connected. The fall of poten- 
 tial within the battery is not measured, and therefore the 
 readings are too small by that amount. If the resistance 
 of the galvanometer is large, this error is negligibly small. 
 
104 ELECTRICITY AND MAGNETISM 
 
 EXPERIMENT 63 
 ELECTRO-CHEMICAL SERIES 
 
 The electromotive force produced by two substances 
 in any given liquid depends on the nature of the sub- 
 stances. In a list arranged so that each substance is 
 electro-positive to any one lower on the list, the electro- 
 motive force between any two substances is equal to the 
 sum of the electromotive forces between all of the intervening 
 substances taken consecutively. 
 
 Apparatus. — Dilute hydrochloric acid in a tumbler; 
 clamps for attachment of substances tested; pieces of 
 iron, zinc, copper, carbon and lead; calibrated high 
 resistance galvanometer; battery with marked electrodes. 
 
 Manipulation. — By noting the direction of the de- 
 flections produced by the battery and the different pairs 
 of substances when placed in dilute hydrochloric acid, 
 make a list of these substances so that each is electro- 
 positive to the one below it. Note the galvanometer 
 deflections produced by each of the substances with the 
 one next to it on the list; also by the first and last sub- 
 stance on the list. The readings are taken as quickly 
 as possible to reduce errors due to polarization. The 
 plates are washed before each repetition of the observa- 
 tions. 
 
 Calculations. — Correct the readings for the electro- 
 motive forces by use of correction curve of Exp. 61 
 and show that the E.M.F. of the first with the last 
 substance on the list is equal to the sum of the electro- 
 motive forces of all the intervening couples. 
 
ELECTKICITY AND MAGNETISM 
 
 105 
 
 EXPERIMENT 64 
 
 RESISTANCE— SUBSTITUTION METHOD 
 
 Apparatus. — A sensitive galvanometer; a battery; 
 a commutator; a resistance rack; a resistance box; the 
 resistance to be measured. 
 
 The resistance box contains a number of coils of 
 wire of known resistance. These have been accurately 
 measured by the manufacturer and properly marked. 
 It is used for introducing a known resistance into an 
 electric circuit, as is necessary in many electrical deter- 
 minations. One of the uses is the comparison of an 
 unknown resistance with the known resistances of the 
 box, in a manner similar to the comparison of an unknown 
 mass with the known masses of the balance weights. The 
 coils are connected in series, as shown in Fig. 63. When a 
 
 <Z)^ 
 
 lUB 
 
 — ^1 /WVWv 
 
 Fig. 63. 
 
 Fig. 64. 
 
 plug is removed, as shown at A, the whole current must 
 pass through the coil of wire, and when the plug is in, as 
 shown at B, practically the whole current passes through 
 the plug. In order that the resistance of the plug may 
 be negligible, it must be clean, and make a good contact. 
 When putting a plug into ihe box apply a slight pressure 
 &nd turn to the right until a good contact is made. 
 
106 ELECTKICITY AND MAGNETISM 
 
 When removing it, turn to the left. Use the box only 
 when absolutely necessary and then for as short a time as 
 possible. Heating of the coils by a current may per- 
 manently change their resistance, and, for this reason, 
 too large a current must never be passed through them. 
 
 Method and Manipulation. — Place the unknown re- 
 sistance in the circuit with the sensitive galvanometer 
 and adjust the current and sensitiveness until a large 
 deflection is obtained. Substitute the resistance box 
 for the unknown resistance, and introduce such a resistance 
 as will give the same deflection on the galvanometer as 
 was obtained when the unknown resistance was in the 
 circuit. The known resistance in the box must then 
 equal the unknown resistance. If no set of resistances 
 in the box gives exactly the same deflection the value 
 of the unknown resistance is obtained by interpolation. 
 
 Sources of Error. — The resistance of the circuit may 
 change during the experiment on account of variations 
 in the battery, heating of the wire by the current, or 
 changes in the room temperature. An error may also 
 be introduced by the commutator if the resistances in 
 its contacts or connections on the two sides are not 
 exactly equal. 
 
 These errors are avoided by taking the final readings 
 in rapid succession and by interchanging the positions 
 of the compared resistances. 
 
 EXPERIMENT 65 
 RESISTANCE— FALL OF POTENTIAL METHOD 
 
 Apparatus. — As shown in figure. 
 
 Method. — A current is passed through a known re- 
 sistance R, and an unknown resistance X placed in 
 series. A high-resistance galvanometer can be con7 
 
ELECTRICITY AND MAGNETISM 
 
 107 
 
 nected to the extremities of either of these resistances by 
 means of a commutator, as shown in Fig. 65. The 
 resistance R is adjusted so that when the galvanometer 
 is connected to R the deflec- 
 tion is nearly the same as 
 when connected to X. The 
 galvanometer, when connected 
 to either R or X, must have 
 a current flowing through it 
 proportional to the fall of 
 potential across these resistan- 
 ces, for in both cases the 
 resistance of the galvanometer 
 is the same. If the resistance 
 of the galvanometer circuit is 
 sufficiently high, its introduc- 
 tion as a shunt to either R or 
 X does not appreciably change 
 
 the fall of potential between the points to which it is 
 attached. The ratio of the currents flowing through the 
 galvanometer in the two cases is 
 
 OAVO— 1 
 
 K 
 
 I — I 
 -o o— 
 
 FiQ. 65. 
 
 and 
 
 I_^E_^X 
 r'~E'~R' 
 
 ^=T^^d'^- 
 
 Manipulation. — Some suitable resistance is ■"" placed 
 in R, and deflections are taken with the galvanometer 
 connected in succession to the extremities of the two 
 resistances. If, for the two cases, the deflections are not 
 equal, a rough calculation of X is made, and such a 
 resistance is introduced in R as is most nearly equal to 
 
108 ELECTRICITY AND MAGNETISM 
 
 this calculated value. A set of deflections is then taken 
 for the final readings. 
 
 Sources of Error. — The observational error is reduced 
 when the resistances, and therefore the deflections, 'are 
 nearly equal. If R and X are not small compared with 
 the resistance of the galvanometer, the introduction of 
 the galvanometer changes appreciably the difference 
 of potential between the ends of the resistance. But 
 if ' R and X, although large, are made nearly equal, 
 the introduction of the galvanometer lowers each dif- 
 ference of potential nearly the same fraction of itself 
 so that the ratio of the differences of potential as they 
 exist with the galvanometer connected is almost the 
 same as their former ratio. 
 
 RG 
 The resistance of wires in parallel =---—-. If (? is large com 
 
 R+G 
 
 •art 
 
 pared with R, — — —^R. In such a case the placing of (? in parallel 
 R-\-G 
 
 with R does not appreciably change the resistance of that part 
 
 of the circuit between the ends of R and therefore does not change 
 
 the difference of potential. For this reason the introduction of 
 
 the high resistance galvanometer does not change the difference of 
 
 potential between the extremities of the resistances B or X 
 
 experiment:66 
 
 THE RESISTANCE OF A UNIFORM WIRE IS PROPORTIONAL 
 TO ITS LENGTH 
 
 Apparatus. — Galvanometer; key; battery; resistance 
 rack; wire with sliding contact. 
 
 Method and Manipulation. — ^A constant current is 
 passed through a long uniform high-resistance wire. 
 A high-resistance galvanometer is attached to one end 
 of the wire and to a sliding contact which is first moved 
 
ELECTRICITY AND MAGNETISM 
 
 109 
 
 to point 1, Fig. 66, and a deflection is taken. The 
 sliding contact is then moved to point 2, and a deflection 
 is again taken. Con- 
 tinue in this manner in (~^x 
 
 rapid succession until ^— ^ 
 
 deflections for several 
 equal steps have been 
 taken. The small por- 
 tions of the main cur- 
 rent that flow through 
 the galvanometer are 
 to each other as the 
 differences of potential 
 
 ^AAAWV^A. 
 
 R 
 
 Fig. 66 
 
 producing them, 
 of the currents is known from the deflections : 
 
 The ratio 
 d ^i _ e 
 
 Use curve of Exp. 56 if the deflections are large. 
 
 Plot a curve showing the relation between the dif- 
 ferences of potential and the lengths of the wire included 
 between the galvanometer terminals. 
 
 In Exp. 65 it was shown that E <xi R, i.e., 
 
 If the curve shows that 
 
 then 
 
 t = i 
 
 !L = l 
 
 r~l' 
 
110 
 
 ELECTRICITY AND MAGNETISM 
 
 EXPERIMENT 67 
 
 TO DETERMINE A RESISTANCE BY MEANS OF A 
 WHEATSTONE WIRE BRIDGE 
 
 Apparatus. — ^A Wheatstone wire bridge and acces- 
 sories; the resistance to be measured. 
 
 Method. — The connections are made so that the 
 
 known resistance R, and the resistance to be measured, 
 
 X, form two arms of the Wheatstone bridge, while the 
 
 uniform wire LL' forms the other two arms as shown 
 
 in Fig. 67. The slide S is moved along the wire until 
 
 the potential at S is the same as that at P so that no 
 
 current flows through the galvanometer. The bridge 
 
 is then said to be " balanced," and the fall of potential 
 
 e = ei, 
 and 
 
 e' = ei'. 
 But 
 
 fl_e^_ei _L 
 
 X~e'~7l~L" 
 
 ■■■ ^4'«- 
 
ELECTRICITY AND MAGNETISM 
 
 111 
 
 To avoid an error due to the connecting wires and to 
 the contacts at the ends of the bridge wire, a balance 
 of the bridge may be obtained with the galvanometer- 
 connected successively to the three places indicated in 
 Fig. 68. It may be necessary to introduce small re- 
 sistances Y and Z to bring the points A and C a short 
 distance from the extremities of the bridge wire. L 
 and U are now the distances AB and BC* 
 
 Fig. 68. 
 
 Manipulation. — The current is allowed to flow through 
 the circuit only while an observation is being taken. 
 The main circuit is closed an instant sooner than the 
 galvanometer circuit to avoid a deflection by the in- 
 stantaneous currents that may be induced in any part 
 of the circuit at the time of closing. The galvanometer 
 is first greatly decreased in sensitiveness by means of a 
 shunt, and after a balance is roughly obtained the sen- 
 sitiveness is increased again until a movement of a milli- 
 meter of the slide on the bridge wire causes a deflection 
 of at least a millimeter on the galvanometer scale. The 
 
 * For high accuracy the bridge wire must be caUbrated. See 
 Carhart and Patterson, p. 78. 
 
112 
 
 ELECTRICITY AND MAGNETISM 
 
 slide is pressed against the guide holding the scale, and 
 when a final balance is obtained a reading is taken 
 opposite the point of contact. 
 
 An approximate value of X is obtained by a pre- 
 liminary trial, and then the resistance in the box R is 
 made nearly equal to this, so that in the final observa- 
 tions L will be nearly equal to L'. 
 
 Determine the resistance of a coil of wire by the two 
 methods of Fig. 67 and Fig. 68. 
 
 EXPERIMENT 68 
 
 TO DETERMINE A RESISTANCE BY MEANS OF THE 
 WHEATSTONE BOX BRIDGE 
 
 The Wheatstone box bridge is a resistance box with 
 keys and binding posts so placed as to make a convenient 
 arrangement for use as a Wheatstone bridge. A diagram 
 of one form of the bridge is shown in Fig. 69. The two 
 
 @ A B ® 
 
 ^ 000080000-^ 
 
 )00000000 Line® 
 
 QQQOQOOOO 
 
 @ 
 
 Bq 6q 
 
 f 
 
 m 
 
 Fig. 69. 
 
 sets of coils, A and B, are called the " ratio arms " or 
 " bridge arms," while the coils of the set C are usually 
 designated as the " rheostat " coils. The resistance to 
 be determined is connected to the binding posts e and /. 
 A balance of the bridge is obtained in the same manner 
 
ELECTRICITY AND MAGNETISM 113 
 
 as in Exp. 67, except that instead of changing the ratio 
 of A to S the resistance in C is varied. 
 
 Since 
 
 A 
 
 C 
 
 B' 
 
 X' 
 
 X = 
 
 -i- 
 
 If the value of the resistance to be measured is known 
 approximately, the ratio of the resistance in the ratio 
 arms is made such as to require the highest possible 
 resistance in the rheostat arm to produce a balance of 
 the bridge, but if it is not known, its approximate value 
 may first be determined by using a ratio of unity, pref- 
 erably 100 : 100. When the final balance is obtained, 
 the resistance in the rheostat branch should be at least 
 1000 ohms, whenever possible, so that if the adjustment 
 is made to within the nearest ohm, the resistance is 
 measured to within 0.1%, the usual reliable limit of 
 accuracy of an ordinary resistance box. 
 
 It is preferable to use only one coil in each ratio arm, 
 and to avoid as far as possible the use of the one ohm 
 ratio coils. 
 
 The galvanometer circuit should be closed first to test 
 for thermo-electric currents in the circuit. If any de- 
 flection of the galvanometer is observed, the bridge 
 must be balanced to this deflected position and not 
 to the null point. 
 
 If the circuit contains self-induction, first the battery 
 circuit and then the galvanometer circuit must be closed. 
 This is the usual procedure after the test for thermo- 
 electric currents is made. 
 
 The circuit should be closed for a short time only, to 
 avoid heating the coils and changing their resistance. 
 
114 electricity] 4.ND MAGNETISM 
 
 The current-carrying capacity of the coils in an ordinary 
 resistance box is about 0.25 watt. When necessary, 
 0.5 watt may be used for a short time only. 
 
 In the measurement of small resistances, the resistance 
 of the connecting wires may not be negligible. It must 
 then be determined separately. 
 
 Measure the resistance of the coil of wire used in Exps. 
 65 and 67 and compare results; also measure the resistance 
 of two coils separately, in series, and in parallel. Com- 
 pare the last two results with values obtained by calcula- 
 tion from the determined resistances of the individual 
 coils. 
 
 EXPERIMENT 69 
 THE TEMPERATURE COEFFICIENT OF COPPER WIRE 
 
 Method. — Determine the resistance of a coil of wire 
 at the temperature of melting ice, and then at the tem- 
 perature of steam. The student is to select the method 
 considered the most suitable. 
 
 Let Ro = the resistance at 0° C, and R' at the tem- 
 perature t. 
 
 Then 
 
 R — Ro 
 
 R' = Ro(.l+at), and a- 
 
 Rot 
 
 Apparatus. — A metal jacket for enclosing a coil of 
 wire when placed into ice or steam; a thermometer. 
 
 Manipulation.- — The coil must be surrounded by 
 steam or ice for at least fifteen minutes before observa- 
 tions are taken to insure the temperature of the wire 
 being the same as that of the ice or steam. The tem- 
 perature has reached its final value when the resistance 
 does not change with time. 
 
ELECTRICITY AND MAGNETISM 
 
 115 
 
 EXPERIMENT 70 
 MEASUREMENT OF VARIOUS RESISTANCES 
 
 Determine with the Wheatstone box bridge: 
 
 1. The resistance of a coil of copper wire at 20° C. 
 
 2. The resistance per foot of a copper wire of known 
 gauge, at 20° C. Calculate from the result the specific 
 resistance of copper. 
 
 3. The resistance of a high resistance coil at 20° C. 
 
 4. The resistance of a carbon megohm. 
 
 5. The resistance of the human body with dry and 
 then with moist contacts. For moist contact insert 
 each thumb into a tube containing salt water. 
 
 6. A resistance containing a small electromotive force. 
 
 7. A resistance with self-inductance. 
 
 EXPERIMENT 71 
 TO ADJUST A RESISTANCE 
 
 Apparatus. — A coil of wire connected by means of 
 clamps to two binding posts attached to an insulating 
 base, as shown in Fig. 70; 
 a standard resistance; a 
 Wheatstone box bridge. 
 
 Method and Manipula- 
 tion. — Balance the bridge 
 with the standard resistance 
 placed in one arm as though 
 it were an unknown resist- 
 ance. Use such a ratio in 
 the bridge arms as will 
 bring the greatest possible 
 resistance into the rheostat 
 branch. Replace the standard resistance with the coil 
 
 
 
 /wv\ 
 
 
 
 o 
 o ■ 
 
 
 o 
 o 
 
 
 @ 
 
 
 @ 
 
 Fig. 70. 
 
116 ELECTEICITY AND MAGNETISM 
 
 that is to be adjusted. If the bridge retains its 
 balance the standard resistance and the coil are 
 equal. The Umits of observational error can be 
 determined by introducing the smallest available re- 
 sistance in the rheostat branch and noting the amount 
 of deflection from the balance position. If on exchanging 
 the coils the balance is disturbed, loosen one of the clamps 
 and shorten or lengthen the wire until a balance is ob- 
 tained. In order to avoid thermo-electric currents the 
 final observations should not be taken too soon after the 
 coil has been handled. 
 
 EXPERIMENT 72 
 
 MEASUREMENT OF HIGH RESISTANCE— DIRECT 
 DEFLECTION METHOD 
 
 Apparatus. — Galvanometer; megohm; high imknown 
 resistance; two cells. 
 
 Method and Maniptilation. — The Wheatstone bridge 
 is not suitable for measuring resistances of many million 
 ohms. The direct deflection method is employed when 
 the resistances to be measured are not too high. The 
 galvanometer is first connected in series with a megohm 
 (million ohms) and the deflection is observed. It is then 
 connected with the unknown high resistance, and a 
 battery of such electromotive force is used as will produce 
 nearly the same deflection given in the previous case. 
 If the same cell is not used, both electromotive forces 
 must be measured. The resistance of the battery being 
 assumed neghgible, 
 
 . E 
 ^ X 
 
ELECTRICITY AND MAGNETISM 117 
 
 but 
 
 therefore, 
 
 and 
 
 i 
 
 E 
 
 X 
 
 T 
 
 ^E' 
 
 'R' 
 
 i 
 
 T 
 
 d 
 
 "d" 
 
 
 d 
 
 E 
 
 X 
 
 d'' 
 
 E~' 
 
 R' 
 
 X 
 
 d 
 d' 
 
 
 EXPERIMENT 73 
 SPECIFIC RESISTANCE OF AN ELECTROLYTE 
 
 Apparatus. — Induction coil and battery; telephone 
 receiver; Wheatstone box bridge; electrolyte in vessel 
 with suitable electrodes. 
 
 On account of polarization, the resistance of an elec- 
 trolyte cannot be measured by any direct current, method 
 with accuracy. The polarization is avoided by using 
 a rapidly alternating current from the secondary of an 
 induction coil in place of the steady current from a bat- 
 tery. The time that the current continues in one direc- 
 tion is so short that no appreciable accumulation can 
 form at the electrodes to produce an opposing difference 
 of potential. The telephone is a very delicate detector 
 of alternating currents, and must be substituted for the 
 galvanometer. 
 
 Manipulation. — ;Measure the resistance of an electro- 
 lyte of known density at a known temperature in a 
 tube of known diameter, using plate electrodes. Change 
 the separation of the electrodes by a known amount. 
 
118 
 
 ELECTKICITY AND MAGNETISM 
 
 and again measure the resistance. The difference in the 
 resistances gives the resistance of a known column of the 
 electrolyte. This eliminates uncertainty as to any re- 
 maining polarization at the electrodes and as to the 
 exact location of the ends of the column. 
 
 From the resistance and the dimensions of the column, 
 calculate the specific resistance. 
 
 EXPERIMENT 74 
 ELECTROMOTIVE FORCE OF A BATTERY— OHM'S METHOD 
 
 Apparatus and Method. — Connect the battery in 
 series with an ammeter, a resistance rack and a resistance 
 box. Measure the current with no resistance in the box, 
 and then with a resistance R. Let r be the resistance of 
 the circuit. Then for the first case 
 
 1= 
 
 E 
 
ELECTRICITY AND MAGNETISM 
 and for the second case 
 
 119 
 
 7' = -^ 
 
 whence 
 
 and 
 
 r+R' 
 I _^r+R 
 
 r r ' 
 I'R ' 
 
 r = 
 
 The resistance of the circuit then is known for each of 
 the two known currents and E=Ir = I'{r-\-R). 
 
 EXPERIMENT 75 
 
 ELECTROMOTIVE FORCE OF A BATTERY— 
 POGGENDORFF'S METHOD 
 
 Apparatus. — As shown 
 in figure. 
 
 Method.— Connect 
 the cell E, whose E.M.F. 
 is to be determined, as 
 indicated in Fig. 72. 
 The battery B must 
 have a greater E.M.F. 
 than E. A definite re- 
 sistance is placed in R, 
 and the resistances r 
 and S are varied until 
 the sensitive galvan- 
 ometer shows no de- 
 flection when the key K is closed, 
 potential E' between the ends of 
 
 ^ 
 
 R 
 
 
 
 K^l/^ 
 
 I) 
 
 3 ^ 
 
 ? \ 
 
 / 
 
 ' s „ 
 
 Fig. 72, 
 
 The difference of 
 the resistance R, 
 
120 ELECTRICITY AND .MAGNETISM 
 
 produced by the battery B, must then be equal to the 
 E.M.F. of the cell E, because the two acting in opposite 
 directions through the galvanometer circuit produce 
 no current. Then 
 
 E=E'^IR. 
 
 The current I is measured by means of an ammeter 
 and R is known. 
 
 The proper value of the resistance to be placed in R 
 is calculated from a knowledge of the approximate 
 value of E and of a current that will give nearly a full 
 scale deflection on the ammeter employed. This will 
 insure a large deflection on the ammeter when the balance 
 is obtained. 
 
 EXPERIMENT 76 
 
 ELECTROMOTIVE FORCE OF A BATTERY- 
 POTENTIOMETER METHOD 
 
 Apparatus. — A sensitive galvanometer; a battery 
 whose electromotive force is to be determined; a stand- 
 ard cadmium cell; . a battery of constant E.M.F., pref- 
 erably a storage cell; resistance boxes, etc. The crystal 
 type of the cadmium cell is shown in Fig. 73. Its E.M.F. 
 is 1.0183 volts and its temperature coefficient is negUgible. 
 
 A high resistance must be placed in the circuit of the 
 standard cell to protect it from excessive currents. The 
 electromotive force of a standard cell was determined 
 by the method of Exp. 75, and since, with proper usage, 
 it remains constant, it is used as a standard for the deter- 
 mination by comparison of unknown electromotive forces. 
 
 The standard cell is never used as a source for the 
 production of electric currents. 
 
ELECTKICITY AND MAGNETISM 
 
 121 
 
 Method. — The potentiometer is an apparatus for com- 
 paring falls of potential by a modification of Poggen- 
 dorff's method. For comparing electromotive forces of 
 batteries, the apparatus is arranged as shown in Fig. 74. 
 5 is a battery whose E.M.F. is greater than the E.M.F. 
 to be determined. Es is a standard cell whose E.M.F. 
 is known. The resistance of the galvanometer circuit 
 
 'HM 
 
 Fig. 73. 
 
 Fig. 74. 
 
 should be at least 10,000 ohms to protect the standard 
 cell. The cells are connected in opposition ; a large resist- 
 ance is placed in R, and then Ri is adjusted until there 
 is no deflection on the galvanometer. The fall of potential 
 in the main circuit between the ends of the box must 
 
 then equal the E.M.F. of Es, and is equal to ^ per ohm, 
 
 R' being the resistance in the box R. The standard 
 cell is then replaced by the battery whose E.M.F. is 
 to be determined. The resistances in R and Ri are 
 then adjusted, their sum being maintained constant, 
 
122 ELECTRICITY AND MAGNETISM 
 
 until the fall of potential between the extremities of 
 B is equal to the E.M.F. of the cell. The fall of potential 
 
 along the line of the main circuit then is ~ per ohm, 
 
 where R" is the resistance in R. But since R+Ri is 
 constant, the fall of potential per ohm is constant and 
 
 R'~R 
 
 s -^x 
 
 and 
 
 Ex=-^Es. 
 
 Manipulation. — If the resistance R' originally placed 
 
 in R is numerically 1000 times as large as the E.M.F. 
 
 7?" 
 of the standard cell, the E.M.F. of the battery Ex=-^^. 
 
 This is usually done as it simplifies the computations. 
 
 A commercial form of the potentiometer is shown in 
 Fig. 78. This accomplishes somewhat more conven- 
 iently the same object as the resistance boxes of Fig. 74. 
 
 Refebences. — Cadmium cell, Phil. Mag. V, 48, 1899, p. 152; 
 Bulletin of the Bureau of Standards, 1907, Vol. IV, p. 1. 
 
 EXPERIMENT 77 
 
 TO CALIBRATE A VOLTMETER BY THE POTENTIOMETER 
 METHOD 
 
 Apparatus. — As shown in Fig. 75. 
 
 Method and Manipulation. — The standard cell is 
 placed at E^, and the resistance R is made numerically 
 equal to 1000 times the E.M.F. of the standard cell. 
 
ELECTEICITY AND MAGNETISM 
 
 123 
 
 The resistance Ri is made such that R-}-Ri is numer- 
 ically equal to 1000 times the voltage reading to be 
 tested. For example, in testing the 1 .500 volt point the 
 resistance R-\-Ri is 
 made equal to 1500 
 ohms. A resistance is 
 placed in R2. The 
 circuit of the battery B 
 is then closed, and the 
 resistance R2 is adjust- 
 ed until the voltmeter 
 indicates the proper 
 voltage. The stand- 
 ard cell circuit is then 
 closed for the ^rs< time. 
 If no deflection is pro- 
 duced on the sensitive 
 galvanometer, the po- 
 tentiometer is bal- 
 anced, and the fall 
 of potential between 
 
 X and Y equals the electromotive force of the standard cell. 
 Then the fall of potential in R is one volt for every 1000 
 
 ohms, and between X and Z^ 
 
 
 z> 
 
 J 
 
 
 
 
 
 m^ 
 
 ^i 
 
 
 
 1 r 
 
 J 
 
 
 X Y 
 
 — 
 
 06 
 z 
 
 
 -0 0- 
 
 
 
 
 
 R 
 
 
 R, 
 
 
 Rt 
 
 
 B 
 
 1 1 " 
 
 K 
 
 — 1 
 0- 
 
 
 
 
 1 1 
 
 
 
 
 
 Fig. 75. 
 
 1000 
 
 li R+Ri = \500 
 
 ohms, the fall of potential between X and Z=—-rj^= 1.500 
 
 volts. The current through the voltmeter is caused by 
 the difference of potential between the points X and Z, 
 to which the voltmeter is connected. If the voltmeter 
 indicates the known value of the difference of potential, 
 it reads correctly. If a deflection is produced on the 
 galvanometer when the standard cell circuit is closed, 
 R2 must be adjusted until there is no deflection. The 
 difference of potential between X and Z will then be 
 
124 ELECTRICITY AND MAGNETISM 
 
 1.500 volts as before, but the voltmeter may indicate 
 some other nimiber. The difference between the two is 
 the error of the voltmeter reading at that point. To 
 test whether the galvanometer has sufficient sensitive- 
 ness, change R2 by an amount that will cause the volt- 
 meter to change its reading by one-tenth of its smallest 
 division. If the galvanometer then gives an appreciable 
 deflection it is sufficiently sensitive. 
 
 The corrections for the other points on the voltmeter 
 are determined in a similar manner by first placing the 
 proper resistance in Ri and then adjusting by means 
 of R2. If a scale giving a maximum reading of less than 
 two volts is calibrated, only one storage cell is used. 
 A five volt scale requires three storage cells. This 
 method is usually modified for voltages higher than this, 
 and for those less than 1.1 volts. 
 
 If R2 does not have a sufficient range of adjustment 
 the resistance of the circuit can be changed by use of 
 a sliding resistance. An adjustment to the nearest 
 one-tenth of an ohm is usually sufficient. 
 
 Calibrate a 1.5 volt scale using the points 1.1, 1.2, 
 1.3, 1.4 and 1.5. Tabulate results, showing the per- 
 centage of error in each case. Use the sign -|- or — to 
 show whether the correction is to be added to or sub- 
 tracted from the observed reading. If the per cent, of 
 error is the same in each case within the limits of obser- 
 vational error, this per cent, is called the correction factor 
 of the voltmeter, and by its use readings can be correcetd 
 without referring to the calibration table. Draw a 
 correction curve as in Exp. 61. 
 
 Is the fall of potential per ohm in R2 the same as in RI 
 
 Should the voltmeter branch be provided with a key? 
 
 How does the strength of the current in the voltmeter 
 compare with that in RI 
 
 Explain how it is possible to calibrate a voltmeter 
 
ELECTRICITY AND MAGNETISM 
 
 125 
 
 when it is connected to points between which the resistance 
 is high. 
 
 Why is it then that the voRmeter cannot be used to 
 measure the voltage between points between which the 
 resistance is high? Does it measure the voltage as it is 
 after the voltmeter is connected? 
 
 What is an electrostatic voltmeter? 
 
 EXPERIMENT 78 
 
 TO CALIBRATE A MILLIAMMETER BY THE 
 POTENTIOMETER METHOD 
 
 Apparatus. — The diagram shown in Fig. 76 is a mod- 
 ification of that used in Exp. 76. 
 
 'WWWW^Ar 
 
 Fig. 76. 
 
 Manipulation. — The potentiometer is balanced in the 
 same manner as in Exp. 76. The difference of potential 
 between the ends of the known resistance is equal to 
 the E.M.F. of the standard cell and thus the value of 
 
 the current in the circuit is 
 
 R 
 
 For other values of the 
 
 current R is changed, and Ri adjusted, until the balance 
 is again obtained. The results are tabulated, and the 
 correction factor obtained as in Exp. 77, 
 
126 
 
 ELECTRICITY AND MAGNETISM 
 
 EXPERIMENT 79 
 
 TO CALIBRATE AN AMMETER BY THE POTENTIOMETER 
 METHOD 
 
 Apparatus. — The apparatus may be arranged as shown 
 in either Fig. 77 or Fig. 79. 
 
 Method and Manipulation.— I. In Fig. 77 the current 
 from the storage battery B' is passed through the ammeter 
 
 Fig. 77. 
 
 to be calibrated, a resistance rack, and a standard low 
 resistance S made specially for such purposes to withstand 
 large currents. The fall of potential is determined by 
 comparison with a standard cell using the potentiometer 
 arrangement of Exp. 76. By means of a commutator, 
 the standard cell S.C. is first connected to the resistance 
 R and a balance is obtained. The commutator bar is 
 then turned and the fall of potential E at the extremities 
 
ELECTEICITY AND MAGNETISM 
 
 127 
 
 of S is determined. 
 
 E 
 
 The current through the ammeter is 
 
 then equal to 
 
 S' 
 
 The approximate values of the resistances required to 
 produce a balance are always first calculated from the 
 approximately known value of the current. 
 
 Test at least five points on the ammeter properly 
 distributed over the full range of the scale, and plot 
 the correction curve as in Exp. 61. 
 
 Fig. 78. 
 
 Commercial instruments called " potentiometers," 
 shown in Fig. 78, operate essentially in the above manner. 
 All the keys and all the resistances except P and S, 
 are conveniently placed on and in one box. These 
 instruments may be used to calibrate either voltmeters 
 or ammeters or to measure any electromotive force 
 within their range. 
 
 II. The other method is shown in Fig. 79. This has an 
 advantage over the first method in that only ong storage 
 battery is required. The current in the main circuit 
 
128 
 
 ELECTRICITY AND MAGNETISM 
 
 is regulated until the ammeter gives the reading to be 
 tested. The resistance in R+R' must be at least 2000 
 
 times that in S. The 
 current flowing through 
 R+R' is then a negli- 
 gibly small portion of 
 the whole current, and 
 that flowing through S 
 may be assumed to be 
 the whole current flowing 
 through the ammeter. 
 
 The value of the cur- 
 rent flowing through the 
 resistance *S is deter- 
 mined from the differ- 
 ence of potential E be- 
 tween its extremities. 
 
 The standard cell is 
 balanced against the 
 difference of potential 
 in the resistance R', 
 the approximate value 
 of the resistance required for this being flrst calculated. 
 The fall of potential in R' included in the standard cell 
 circuit is then equal to the electromotive force of the 
 standard cell, and the fall of potential, E, between the 
 extremities of R+R' and of S can be calculated. 
 
 The current flowing through S and the ammeter is 
 
 E 
 obtained from /= „. This is repeated for various values 
 
 of the current. 
 
 If the current through R is not a negligible portion 
 
 of the whole, its value is calculated from the known 
 
 E 
 difference of potential and resistance. Thus t" = ^ . „„ 
 
 K+K 
 
 Fig. 79. 
 
ELECTRICITY AND MAGNETISM 129 
 
 and the total current through the ammeter is 
 
 E , E 
 
 I+i = 
 
 S^ R+R'' 
 
 EXPERIMENT 80 
 CALIBRATION OF AN AMMETER— VOLTMETER METHOD 
 
 Apparatus. — Ammeter; voltmeter; standard resistance; 
 battery; resistance rack; key. 
 
 Method. — The current is passed through the ammeter 
 in series with a resistance rack and a standard resistance. 
 A voltmeter that has been carefully calibrated is at- 
 tached to the posts of the standard resistance. The 
 current flowing through the ammeter is determined 
 from the fall of potential between the ends of the known 
 resistance R. 
 
 If the current flowing through the voltmeter is not 
 
 a negligible portion of the whole current, its value is 
 
 E 
 determined from 1 = 77, where G is the resistance of the 
 
 voltmeter. The total current flowing through the 
 
 ammeter is Z+i= ^- + 77. 
 
 The errors in the readings are tabulated, and the 
 correction curve is plotted as in the previous experiments. 
 
 Precaution. — The voltmeter should have such a sen- 
 sibility that with the standard resistance used the 
 deflections will be nearly equal to those of the ammeter. 
 
 Note. — The methods for cahbratmg ammeters are methods for 
 measuring an electric current without the use of an ammeter. The 
 current is measured in terms of difference of potential and resistance. 
 
130 ELECTRICITY AND MAGNETISM 
 
 THEORY OF THE BALLISTIC GALVANOMETER 
 
 A ballistic galvanometer is used for measuring the 
 quantity of electricity that passes through the instrxmient 
 during an instantaneous discharge such as is obtained 
 from a condenser or from an induced current. Any 
 galvanometer may be employed for this purpose and 
 when so employed is called a " ballistic galvanometer." 
 In place of a continuous deflection which an electric 
 current produces, the quantity of electricity sets the 
 moving part (magnet or coil) into motion and causes 
 it to swing a definite distance depending on the quantity 
 of discharged electricity. The distance of the swing, 
 as represented on the reading scale, is called the " throw " 
 of the galvanometer. 
 
 The galvanometer is more sensitive if the moving 
 part is undamped, and the throws are more easily read 
 if the period of vibration is long. If the coil is undamped, 
 the throws are also more nearly proportional to the 
 quantities of electricity; and if the period of vibration 
 is long, the whole of the instantaneous current passes 
 through the coil before the moving part is appreciably 
 deflected, a condition required by the theory of the 
 instrument. .For these reasons a baUistic galvanometer 
 has its moving part imdamped, and its period of vibra- 
 tion long. 
 
 If the moving part be damped so as to be aperiodic 
 or nearly so, the constant of the galvanometer cannot 
 be calculated, but can be determined by the indirect 
 methods of Exps. 84 and 101, which are the most con- 
 venient methods to employ in any case. 
 
 The relation existing between the throw and the quan- 
 tity of electricity is here derived for the moving-coil type 
 of the galvanometer. 
 
ELECTRICITY AND MAGNETISM 
 
 131 
 
 A. Moment of the force tending to rotate the coil. 
 When a current I' flows through the coil in the direc- 
 tion indicated in Fig. 80, the coil, when viewed from the 
 
 Fig. 80. 
 
 top, is deflected in a direction contrary to the hands of a 
 watch. The moment of the force tending to produce 
 rotation is all due to the current in the vertical parts 
 of the coil. 
 
 Fig. 81. 
 
 If F be the strength of the magnetic field and n the 
 number of turns on the coil, the force with which each 
 vertical side of the coil is acted on, is FLnl' dynes. 
 When the plane of the coil is parallel to the magnetic 
 lines of force the moment of this couple is FLnDF; 
 
132 ELECTKICITY AND MAGNETISM 
 
 and when it makes the angle cj) with that direction, the 
 moment is FLnDI' cos <^. [See Fig. 81.] In case of 
 a radial field the plane of the coil, upon deflection, re- 
 mains approximately parallel to the lines of force and 
 cos <i) = l. 
 
 B. The constant K' of the galvanometer for continuous 
 currents. 
 
 The moment of the force producing rotation must 
 equal the moment T^, of the torsional couple, when 
 the coil comes to rest. Thus T^ = FLnDI' cos ^ 
 
 FLnD cos <j) cos ^ 
 .: K'=I^, (I) 
 
 r„, loT _-- _ lori ,„, 
 
 [^=FL^- PLnD = -^\. . . . (II) 
 
 C. The constant K giving the relation between the 
 quantity of electricity and the throU) of the coil. 
 
 Let I" be the mean value of the transient current. 
 
 During the passage of the current a force is exerted 
 
 upon the coil, the moment of which is fr = FLnDI" and 
 
 fr 
 the angular acceleration which it produces is a — -^, 
 
 Iq being the moment of inertia of the coil. The angular 
 velocity of the coil developed while the current is flowing 
 for the time t is 
 
 , frt FLnDF't FLnDQ' 
 
 (^ = (Xt = -j^ = J = f . 
 
 -/o io io 
 
 « = 1«Q' = ^f'- • • • (III) 
 
ELECTKICITY AND MAGNETISM 133 
 
 Substituting equation (II) in (III), 
 
 ^ _ IO/qg) _ loK^i^ / jTTN 
 
 ^""ToF' rp ■ ■ • ■ • ■ uv; 
 
 K' 
 
 The kinetic energy acquired by the coil = |/ow^, which, 
 
 neglecting damping, is all changed into potential energy 
 
 when the coil reaches the end of the throw. The average 
 
 torsional moment that the moving coil acts against in 
 
 0-i- T'fl 
 its motion to the end of the angle of throw 8 is — =•— = IT'S. 
 
 The work done by the moving coil in twisting the sus- 
 pensions = the average torsional moment X the angular 
 distance = |re-6 = |r62. When all the kinetic energy 
 has been converted into potential energy at the end 
 of the throw, 
 
 "=V: 
 
 T 
 
 Substituting w in equation (IV), 
 
 Q=^^-6=^-x'e. . . . (V) 
 
 The complete period of vibration of the galvanometer 
 
 coil t = 2x-i/-^, from which^/T^f = ;7-. Substituting this 
 \ 1 \ 1 Aiz 
 
 value in equation (V), 
 
 Q = ^9, (VI) 
 
 n which all the quantities can be easily determined. 
 
134 ELECTRICITY AND MAGNETISM 
 
 D. Damping correction. — Equation (VI) is developed 
 on the assumption that all the kinetic energy imparted 
 to the coil by the electric discharge is changed into 
 potential energy when the coil has reached the end of 
 the throw. This, however, is not the case. Some of 
 the energy is consumed in the production of induced 
 currents, in overcoming the resistance of the air, and in 
 the internal resistance of the suspending fiber. For 
 this reason a correction must be applied to equa- 
 tion (VI). 
 
 The first throw of the coil is smaller than it would be if 
 there were no damping. Any observed throw must be 
 multiplied, therefore, by some factor that will increase 
 its value to what it would be if undamped. 
 
 While the coil is Aabrating, the successive angles of 
 vibration bear to each other a practically constant ratio. 
 
 Thus 
 
 61 62 03 6(n-l) , 
 
 ei = p02 = P% = p^64 = p<":'^6, 
 
 '»• 
 
 For example, to find what the angle 63 would have 
 been had there been no damping in the last two vibra- 
 tions, 63 is multiphed by p^, i.e., by p with an exponent 
 numerically equal to the number of intervening vibrations. 
 
 When an actual throw is taken, the coil starts from the 
 center of its natural arc of vibration and has moved, 
 therefore, through but one-half of a single vibration. 
 To find what the angle of throw would have been without 
 damping, multiply the observed angle 61 by p with an 
 exponent numerically equal to the number of intervening 
 vibrations, i.e., ^. 
 
 Then 
 
 e=p*ei, 
 
ELECTRICITY AND MAGNETISM 135 
 
 and 
 
 Q=-^^h=Kh (VII) 
 
 . K'tVV 
 
 For open circuit work p^ is nearly constant, but 
 for throws in closed circuit work the amount of damping 
 is dependent on the resistance of the apparatus. The 
 factor p^ must be determined for each circuit employed. 
 In moving coil galvanometers of low resistance, the amount 
 of damping on closed circuit is usually so great as to 
 make the instrument aperiodic. In such a case, the 
 constant of the galvanometer cannot be determined 
 by this method. 
 
 Magnetic impurities within the coil influence * some- 
 what the magnitude of the throw. Equation (VII) 
 makes no correction for this effect which is usually 
 not very large. 
 
 The temperature coefficient of this type of galvanometer 
 for ballistic throws is about —0.00017. 
 
 EXPERIMENT 81 
 
 DETERMINATION OF THE CONSTANT OF A BALLISTIC 
 GALVANOMETER ON OPEN dRCUIT 
 
 The constant of a ballistic galvanometer, K = , 
 
 is obtained by determining K', t, and Vp- 
 
 Apparatus. — As shown in Fig. 82. 
 
 Method. — A. Determination of K'. — The apparatus 
 may be connected as shown in the figure. The resist- 
 
 * Physical Review, Vol. XXIII, p. 400, 1906; Vol, XXXII, p. 
 297, 1911. 
 
136 
 
 ELECTKICITY AND MAGNETISM 
 
 m 
 
 ance Ri should be small, but the resistance R2 has 
 a value of several hundred ohms. The voltmeter 
 
 V gives the difference of po- 
 tential between the extremities 
 of R1+R2, from which the 
 difference of potential e at the 
 extremities of Ri can be calcu- 
 lated. The resistance R3 should 
 have a value of about 10,000 
 ohms. If G is the resistance of 
 the galvanometer, the value of 
 the current flowing through it is 
 
 
 ,B K 
 
 / = 
 
 Rs+G- 
 
 Fig. 82. 
 
 If the angle of deflection 
 produced by this current is (J* 
 radians, 
 
 K'=I 
 
 cose]) 
 
 where cos. <J)=c=l in galvanometers where the magnetic 
 field is approximately radial, so that 
 
 H 
 
 B. Determination of t. — Cause the coil to vibrate by 
 deflecting it by means of a small current and then 
 opening the circuit. Determine t by the usual method. 
 The passages of the zero mark on the scale are 
 observed in the telescope. A wire may be hung over 
 the zero mark, if necessary, to make the passages visible. 
 A suflBciently large current may often be obtained for 
 
ELECTRICITY AND MAGNETISM 137 
 
 deflecting the coil by holding in each hand one of the 
 wires leading to the galvanometer, the fingers of one 
 hand being moistened. 
 
 C. Determination of Vp- — Cause the coil to vibrate 
 and observe the scale readings at the extremities of the 
 vibrations. When a circular scale is used, 
 
 '1 n 
 
 dn+l 6b+1 
 
 where the terms d represent the observed lengths of 
 any two arcs. Determine several values of p" and from 
 their average obtain p^. 
 
 Manipulation. — A thermo-electric current may exist 
 in the galvanometer branch of the apparatus. This 
 affects the zero reading which must be taken with the 
 thermo-electric current present and should be read 
 immediately after the observed deflection. While the 
 coil is returning to the null point, it must be damped by 
 means of a short-circuiting key as described in Exp. 56. 
 
 The magnetic field in most galvanometers is not 
 uniform, and therefore the same current produces dif- 
 ferent deflections on the two sides of the zero point. 
 For the purpose of determining the constant of a ballistic 
 galvanometer, the value of K' is desired for the field 
 as it is at the zero point, and not as it may be in any 
 deflected position of the coil. This can be obtained 
 approximately from the average of the deflections on 
 the two sides of the null point and by having these 
 deflections as small as is consistant with the desired 
 degree of observational accuracy. A deflection of 5 cm. 
 usually is used. 
 
 In a moving coil galvanometer p* varies somewhat 
 with the amplitude of vibration. It is therefore desir- 
 able to obtain its value for nearly the same amplitude as 
 
138 
 
 ELECTRICITY AND MAGNETISM 
 
 that of the throw in any particular case. This is ob- 
 tained by taking only a few vibrations immediately 
 following a throw. 
 
 The Vp must be determined for each particular 
 circuit when the galvanometer is employed on closed 
 circuit. For open circuit work it has a definite value 
 smaller than that on any closed circuit. 
 
 Obtain the value of Vp on open circuit. 
 
 EXPERIMENT 82. 
 
 QUANTITY OF ELECTRICITY IN A CHARGED CONDENSER 
 
 K 
 
 Apparatus. — Condenser; galvanometer; standard cell; 
 high resistance; commutator; key. 
 
 Method. — The condenser is charged by connecting 
 
 A and B, Fig. 83 to a 
 secondary standard cell 
 in series with a high 
 resistance. 
 
 It is then discharged 
 by disconnecting B from 
 A and immediately con- 
 necting to D. This is 
 done by means of a key 
 or a commutator. 
 
 Q=2!l0i coulombs, 
 
 Q' = xV Q electromagnet- 
 ic units, 
 
 Q"= 3x1010 Q' electro- 
 static units. 
 
 6D 
 oB 
 9A 
 
 ' www \\- 
 
 FiG. 83. 
 
 Determine the quantity in each of the three units. 
 
ELECTEICITY AND MAGNETISM 139 
 
 Manipulation. — To bring the galvanometer coil to 
 rest, siiort-circuit the galvanometer by means of the 
 key K. If on opening the short-circuit the coil does 
 not remain at rest, a thermo-electric current existed 
 during the short-circuit. If this is the case, estimate 
 the position of the null point from the vibrations and 
 again short-circuit the galvanometer. Then open the 
 short-circuiting key, and as the coil passes the known 
 position of the null point, again short-circuit the galva- 
 nometer for an instant by tapping the key. With a 
 little practice, the coil can quickly be brought to rest 
 in this manner. If the vibrations are not completely 
 removed when the throw is to be taken, the condenser 
 is discharged when the coil is at either of its elonga- 
 tions; the throw, however, is computed from the null 
 point and not from the elongation. 
 
 A high resistance galvanometer is not made ape- 
 riodic on short-circuit, and cannot be brought to rest 
 in this manner. If, however, a coil of wire is introduced 
 into the short-circuiting branch, a proper motion of a 
 magnet within this coil will produce the desired effect. 
 When the galvanometer coil is at rest at the null point 
 the short-circuiting key is opened. 
 
 In galvanometers that are aperiodic on closed cir- 
 cuit, the effect of the thermo-electric currents may also 
 be eliminated by either one of the two following methods : 
 
 1. A thermo-electric couple is made from a piece of 
 spring brass wire by annealing the central portion of 
 it by heating. This is introduced into the short-cir- 
 cuiting branch of the galvanometer, and the coil is 
 brought to the null point by warming with the fingers 
 the proper part of the couple. 
 
 2. A small part of the galvanometer circuit is made 
 a part of a secondary circuit containing a large resistance. 
 The fall of potential in this part is made to oppose the 
 
140 
 
 ELECTRICITY AND MAGNETISM 
 
 thermo-electromotive force in the galvanometer and is 
 adjusted by altering the large resistance until it is equal 
 to it. 
 
 EXPERIMENT 83. 
 
 CAPACITY OF MICA CONDENSER— ABSOLUTE METHOD 
 
 Apparatus. — The condenser, galvanometer, and 
 " working " or secondary standard cell are connected 
 
 as shown either in Fig. 84 
 or Fig. 85. 
 
 Method and Manipu- 
 lation. — The obiect to be 
 accomplished in either 
 case is to charge the 
 condenser by means of a 
 standard cell and then by 
 means of one tap on the 
 key or by the reversal 
 of the commutator rocker, 
 to open the battery circuit, 
 connect the condenser to the galvanometer, and again 
 open the circuit. 
 
 In Fig. 84 a key with three metallic springs is em- 
 ployed. In the position shown, the condenser C is being 
 charged by means of the cell E. When the key is de- 
 pressed, the springs K, L and M are brought into metallic 
 connection in succession and accomplish the above 
 mentioned results. The key must be kept down until 
 the full throw is obtained. Fig. 86 shows a form of the 
 key combined with a key for short-circuiting the gal- 
 vanometer. 
 
 In Fig. 85 an ordinary Pohl commutator without 
 cross-bars is employed. The height of the mercury in 
 
 Fig. 84. 
 
ELECTRICITY AND MAGNETISM 
 
 141 
 
 the cups is adjusted so that when the rocker is turned 
 from the charging position, shown in the figure, the cell 
 circuit is opened first at cup 1. The rocker then makes 
 a contact in cup 2 before leaving cup 3, discharging the 
 
 Fig. 85. 
 
 condenser during the time the two cups are in connec- 
 tion. The period of discharge can be varied by changing 
 the height of the mercury in the cup 3. 
 
 Fig. 86. 
 
 The capacity determined in this manner is tem- 
 porarily called the " free charge " capacity to distinguish 
 it from the " ordinary " capacity value as obtained 
 
142 ELECTRICITY AND MAGNETISM 
 
 when the condenser remains in circuit with the galva- 
 nometer during the whole period of its throw. 
 
 When a condenser discharges, the free charge follows 
 the law developed in Exp. 89. In ordinary circuits the 
 time required to discharge the whole measurable part 
 of the electricity is less than .01 second. The absorbed 
 charge, however, continues to flow for an indefinite 
 period. In the " free charge " capacity value * only a 
 negligible portion of the absorbed charge is included, 
 while in the " ordinary " capacity value the absorbed 
 charge plays a considerable part, the magnitude of which 
 depends on the period of the galvanometer. The latter 
 is a variable quantity and has but httle value. 
 
 Determine both the " free charge " and the " ordinary " 
 capacity values. 
 
 C = -^ farads = „ microfarads 
 
 EXPERIMENT 84 
 
 CONSTANT AND FIGURE OF MERIT OF A BALLISTIC 
 GALVANOMETER— STANDARD CONDENSER METHOD 
 
 Apparatus. — As in Fig. 84. 
 
 Method and Maniptxlation. — Discharge the free charge 
 of a condenser of known capacity, through the galvano- 
 meter as in Exp. 83. 
 
 Q = CE = K^ = Fd. 
 
 „_CE T<_CE 
 
 * Physical Review, Vol. xxii, p. 65, 1906; Vol. xxvii, p. 65, 
 1908. 
 
ELECTRICITY AND MAGNETISM 143 
 
 i^ = figure of merit, i.e., the quantity of electricity 
 that produces a throw represented by one centimeter 
 on the scale; d = throw in centimeters. 
 
 This is an indirect method, and is the one most com- 
 monly employed when the galvanometer is to be used on 
 open circuit. The figure of merit replaces the constant 
 especially when a circular reading scale is used. The 
 millimeter is often employed in place of the centimeter 
 as the unit of throw. 
 
 EXPERIMENT 85 
 CAPACITY OF A LEYDEN JAR 
 
 Apparatus. — As in Fig. 84; Leyden jars. 
 
 Method and Manipulation. — Charge several similar 
 Leyden jars, placed in parallel, with a known electro- 
 motive force of about 100 volts, and immediately dis- 
 charge through the ballistic galvanometer. 
 
 ^ Q Fd. , lO^Fd . . , 
 C = ^ir = -pr larads = — ^j— microfarads. 
 E E E 
 
 The capacity of one jar 
 
 C 
 
 The capacity in electrostatic units (centimeters) 
 
 ^9X10^ 
 E 
 
 Determine the capacity in microfarads and in electro- 
 static units. 
 
 Place a resistance of at least 10,000 ohms in series 
 with the battery to protect the galvanometer against 
 accident. 
 
144 ELECTRICITY AND MAGNETISM 
 
 EXPERIMENT 86 
 
 CAPACITY OF A PARAFFIN PAPER CONDENSER- 
 COMPARISON METHOD 
 
 Apparatus.— As in Fig. 84; paraffin paper condenser. 
 
 Method and Manipulation.— Charge a standard mica 
 
 condenser with a battery, and then discharge through 
 
 a baUistic galvanometer. In the same manner charge 
 
 and discharge the paraffin condenser. 
 
 Then 
 
 Q = CE, 
 and 
 
 Qi = CiE; 
 
 C Q^ d- 
 
 .: c,=fe. 
 
 This method should be employed whenever only a few 
 measurements are to be taken. The figure of merit 
 saves time in computation whenever a large number of 
 measurements are made. In the measurement of capac- 
 ities, the figure of merit is often used to represent the 
 capacity in place of the quantity of electricity for each 
 centimeter (sometimes millimeter) of throw. This can 
 be employed only when the charging is made always 
 by the same electromotive force. 
 
 C^^d = F'd, 
 
 where F' represents the capacity in microfarads for each 
 centimeter of throw. 
 
 Determine both the " free charge " and the " ordinary " 
 capacity of each of two paper condensers manufactvu-ed 
 by different firms. 
 
ELECTRICITY AND MAGNETISM 145 
 
 EXPERIMENT 87 
 RESIDUAL CHARGES IN A CONDENSER 
 
 Apparatus. — Same as for Exp. 86. 
 
 Method and Manipulation.^The residual charge of a 
 condenser is the charge acquired by the Uberation of the 
 absorbed electricity. After a condenser is discharged, it 
 gradually acquires an appreciable charge. This effect is 
 usually larger in paper than in mica condensers. 
 
 Connect a paper condenser to a special discharge key 
 in the same manner as in Fig. 84, Exp. 83, but into the 
 battery branch place a key for opening the battery 
 circuit. 
 
 Charge the condenser for five minutes, and then per- 
 manently open the battery circuit by means of the 
 additional key. Discharge the condenser at a known 
 time, and keep the discharge key down. Bring the 
 galvanometer coil to rest, and at the end of two minutes 
 release the key. The electricity liberated during the 
 two minutes will discharge through the galvanometer. 
 At the end of another two minutes press down the key 
 and obtain the second residual discharge. Continue 
 in this manner until several discharges have been obtained. 
 
 Plot the throws obtained on coordinate paper. 
 
 EXPERIMENT 88 
 
 ELECTROMOTIVE FORCE OF A CELL— CONDENSER 
 METHOD 
 
 Apparatus. — As in Fig. 84; cell. 
 
 Method and Manipulation. — Charge a mica condenser 
 with the cell whose electromotive force is to be deter- 
 mined, and discharge through a ballistic galvanometer. 
 
and 
 
 146 ELECTRICITY AND MAGNETISM 
 
 Then charge the condenser with a standard cell, and 
 discharge. 
 
 Q = CE, 
 and 
 
 Qi = CEi, 
 
 ■ ^=^ = 1 
 Ei Qi 61' 
 
 E = -r-Ei=^-r El. 
 01 ai 
 
 Any difference of potential may be determined in this 
 manner. It is, therefore, possible to measure a current 
 by this method in terms of resistance and electromo- 
 tive force. 
 
 EXPERIMENT 89 
 
 INSULATION RESISTANCE— METHOD OF LEAKAGE 
 
 Apparatus. — As in Fig. 87. 
 
 Method. — Discharge a condenser through a ballistic 
 galvanometer and determine Qi, the quantity of elec- 
 tricity discharged. Recharge the condenser, and allow it 
 to discharge through the high resistance for a known 
 • time. Then discharge through the galvanometer and 
 determine Q2, the quantity remaining in the condenser. 
 
 —-^=7, the current which flows through the high 
 
 resistance R. 
 
 
 But 
 
 
 
 rE Q 
 
 R CR- 
 
 •' 
 
 , Q dQ 
 ' CR df 
 
ELECTRICITY AND MAGNETISM 
 
 147 
 
 1 r*' 
 
 C ^, 
 
 R- 
 
 dQ 
 
 Q: Q' 
 t 
 
 Qi 
 
 Qi 
 
 Clogerr 2.3026Clog„ 
 
 If the condenser loses an appreciable part of its charge 
 through leakage in itself, the resistance R, as deter- 
 mined above is the resistance of the high resistance and 
 condenser in parallel. To obtain Ri, the value of the 
 high resistance, determine Re, the resistance of the con- 
 denser alone in the same manner as R was determined. 
 Then 
 
 RxRc 
 
 and 
 
 R- 
 
 Rx — 
 
 Rx-\-Rc 
 
 RRc 
 Re — R 
 
 Manipulation. — The apparatus may be connected as 
 shown in Fig. 87. The 
 commutators A and B 
 must have high insula- 
 tion. When the com- 
 mutator rockers make 
 connections at a ^and at 
 n, the condenser is being 
 charged. When the rock- 
 er of commutator B is 
 turned from n to m the 
 condenser discharges 
 through the high resist- 
 ance R, and when the 
 rocker of commutator A 
 
 is turned from a to b, the condenser discharges through 
 the galvanometer. 
 
 Fig. 87. 
 
148 ELECTRICITY AND MAGNETISM 
 
 The condenser should be charged at least five minutes 
 on account of the absorbed charge as this method is 
 subject to considerable error due to the absorption ot 
 the charge after the charging battery is disconnected. 
 The condenser should be allowed to discharge through 
 the high resistance until only one-half its charge remains, 
 but for all ordinary purposes five minutes is a sufficient 
 length of time. A high degree of accuracy is usually 
 not required in the measurement of such high resistances. 
 
 1. Measure the resistance of a mica condenser. 
 
 2. Measure the resistance of some high resistance. 
 
 3. Calculate the time it takes the condenser to dis- 
 charge completely through the galvanometer. 
 
 4. Calculate the time it requires the condenser to 
 discharge all but '0.1 per cent of its charge through the 
 galvanometer. 
 
 EXPERIMENT 90 
 
 INSULATION RESISTANCE— DIRECT DEFLECTION 
 METHOD 
 
 Let 72 be a known resistance of at least 1,000,000 ohms. 
 Connect this in series with a known electromotive force 
 and a sensitive galvanometer. 
 
 Then 
 
 . E 
 
 the resistance of the galvanometer and battery being 
 neglected. Then in place of R, introduce the insulator 
 X, and if necessary to produce a proper deflection increase 
 the electromotive force of the battery. 
 
ELECTRICITY AND MAGNETISM 149 
 
 ., E' 
 
 
 I -■ 
 
 X' 
 
 d 
 d' 
 
 i 
 
 X E 
 R'E" 
 
 
 X- 
 
 E' d 
 
 Measure the resistance of the same insulator as in Exp. 
 89 and compare results. 
 
 A high resistance galvanometer is the most sensitive 
 for this purpose. However, use the galvanometer of 
 Exp. 89. In place of damping the coil by means of a 
 metal rectangle, make the galvanometer just aperiodic 
 by means of a proper resistance in a shunt. Although 
 this reduces the sensitiveness, the greater convenience 
 often makes this a desirable method of damping. 
 
 It will be observed that this experiment is the same 
 as Exp. 72, the only variation being the use of a gal- 
 vanometer without a damping rectangle. 
 
 EXPERIMENT 91 
 
 WORK DONE m CUTTING LINES OF FORCE— MAGNITUDE 
 OF INDUCED ELECTROMOTIVE FORCE 
 
 From the definition of an electro-magnetic unit of 
 current it follows that a wire 1 cm. in length placed per- 
 pendicular to the lines of force in a unit magnetic field, will 
 be acted on with a force of one dyne when a unit current 
 is flowing through the wire. 
 
 In Fig. 88, as soon as the •ware I is moved, it cuts lines 
 of force, and an E.M.F. is induced in the circuit, which 
 causes a current to flow. This current causes the field 
 
150 
 
 ELECTRICITY AND MAGNETISM 
 
 to act upon the wire, from the above consideration, 
 with a force of HII' dynes, in a direction opposed to the 
 motion of the wire. The mechanical work done in 
 moving this wire through the distance d, is 
 
 w=Hm'=Nr. 
 
 U 1 A 
 
 I 
 ( 
 
 i. 
 
 B 
 
 Fig. 88. 
 
 All this mechanical work is expended in supplying the 
 energy for the electric current, which is represented by 
 
 and 
 
 W = Q'E' = rtE', 
 :. rtE'=Nr, 
 
 p,_N_dn p_E' _ I dn 
 t~df 10^~10^dt' 
 
 Calculate the electromotive force induced when a 
 coil of 100 loops is cutting magnetic lines of force at the 
 rate of 100,000 per second. How much work is done 
 in a second in cutting the lines of force if the resistance 
 in the circuit is 10 ohms? Get the work in ergs, joules, 
 and foot-pounds. 
 
 Care must be taken to interpret properly the term 
 " cutting lines of force." Would any current be induced 
 in a loop of wire moving in a uniform field? 
 
ELECTEICITY AND MAGNETISM 151 
 
 EXPERIMENT 92 
 
 THE NUMBER OF LINES OF INDUCTION (MAGNETIC 
 FLUX) WITHIN A SOLENOID— THE CONSTANT OF A 
 SOLENOID 
 
 The strength of the field within a solenoid is numerically- 
 equal to the work required to move a unit magnetic 
 pole in it through a distance of 1 cm. A unit pole has 
 
 Fig. 89. 
 
 radiating from it 4tc lines of force. If the solenoid is of 
 sufficient length that all these lines can be assumed to 
 penetrate its loops of wire, the work done in moving the 
 pole a distance equal to the thickness of the wire is 
 
 TF=iV/'= 4x7' ergs, 
 
 for each line of force will have cut the wire once. In 
 moving the distance of one centimeter, each line cuts 
 
 Y wires, so that the work done is 
 
 If the number of lines passing through the ends of 
 the solenoid is not negligible, but still is small, a correc- 
 tion can be made in the following manner : 
 
152 
 
 ELECTRICITY AND MAGNETISM 
 
 The surface area of a sphere circumscribed about a 
 solenoid is 4%R^, and the area of the two segments cut 
 off by the ends of the solenoid is approximately 2xr2. 
 
 Fig. 90. 
 
 The number of lines from the unit pole that penetrate 
 the loops of wire is 
 
 The strength of the field near the center of the solenoid 
 therefore is 
 
 and the total number of lines is 
 
 -^C-.-(^) (I) 
 
ELECTEICITY AND MAGNETISM 153 
 
 The constant K represents the number of hnes threading 
 through the center portion of the solenoid when the 
 current flowing is one ampere, and A is the area of the 
 cross-section, the diameter being measured from center 
 to center of wire. 
 
 An equation* for the intensity of the magnetic field 
 at any point on the axis of a solenoid is 
 
 /„ 2T:n/, 
 
 ■ u = (cos a+cos a ), 
 
 Fig. 91. 
 
 where the angles are those indicated in Fig. 91. 
 
 The intensity of the field at the center of the solenoid 
 
 from this equation is 
 
 rr 2%nl 
 
 tL = -. COS a, 
 
 and 
 
 A^ = ff A = — y— cos a = if/, 
 
 „ 2TznA 
 
 K = —-^cosix = „ . .(11) 
 
 Both of these formula; are only close approximations. 
 When the coil has more than one layer of wire, the values 
 of E^ or iV are the sum of the values for the separate layers. 
 
 * Stewart and Gee, Vol. 2, p. 329. 
 
154 
 
 ELECTRICITY Ar?D MAGNETISM 
 
 The intensity of tiie magnetic field due to one ampere 
 within U. of M. solenoid No. 5, at different points on 
 the axis, extending from one end to the center, is shown 
 in Fig. 92. 
 
 lb 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 , 
 
 
 
 
 
 
 
 
 
 1 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 Vi 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 5: 
 
 
 
 
 
 
 
 
 
 
 
 
 
 B ]Z 16 
 
 Qeniimeters From EndoF6olenoid 
 
 Fig. 92. 
 
 20 
 
 24 
 
 Calculate K for some solenoid. 
 
 EXPERIMENT 93 
 
 CALCULATION OF THE COEFFICIENT OF MUTUAL 
 INDUCTION OF A CURRENT INDUCTOR 
 
 The coefficient of mutual induction is the ratio 
 between the induced electromotive force in the secondary 
 and the rate of change of the current in the primary. 
 
 E 
 di' 
 Jt 
 
ELEOTKICITY AND MAGNETISM 
 
 155 
 
 Imagine the current in the primary to be changing 
 at the rate of one ampere per second. Then the coef- 
 ficient of mutual induction is numerically equal to the 
 
 
 
 fi S 
 
 
 
 s 
 
 
 
 ^ 
 
 Ul=J U=M 
 
 Fig. 93. 
 
 induced electromotive force. The magnetic lines of 
 force due to one ampere close up, in one second, if the 
 current is diminishing or pass outward if the current 
 is increasing; in either case all these lines, if the secondary 
 is wound over the central portion of a long solenoid, 
 cut the loops of the secondary coil. The number of 
 these lines K, produced by each ampere, can be calculated 
 by means of one of the equations in Exp. 92. The in- 
 duced electromotive force Ei', in each loop of the 
 secondary is numerically equal to the number of lines 
 that cut it per second, which in this case is K. The 
 electromotive force in volts 
 
 and 
 
 '' 108 108' 
 
 M = n2Ei = 
 
 108' 
 
 where n2 is the number of turns in the secondary coil. 
 Calculate the coefficient of mutual induction of a 
 current inductor. 
 
156 
 
 ELECTEICITY AND MAGNETISM 
 
 EXPERIMENT 94 
 
 RELATION BETWEEN THE QUANTITY OF INDUCED ELEC- 
 TRICITY IN THE SECONDARY CIRCUIT AND THE 
 CURRENT IN THE PRIMARY 
 
 Arrange the apparatus as shown in Fig. 94. The 
 galvanometer circuit containing the secondary of the 
 mutual induction coil or current inductor M is opened 
 
 h 
 
 r' 
 
 L© 
 
 4:^ 
 
 s 
 
 Fig. 94. 
 
 immediately after the opening of the primary circuit. 
 Read the ammeter, and observe the throw on the ballistic 
 galvanometer when the circuits are opened. Repeat 
 for several different currents. Determine the relation 
 between Q and / by plotting on coordinate paper. The 
 throws may be plotted in place of the quantities. 
 
 Whenever it is possible, the resistance in the galva- 
 nometer circuit should not be less than that which makes 
 the coil just aperiodic. 
 
 The galvanometer circuit, when closed, may contain 
 a thermo-electric current so that the coil is not at rest 
 at the true null point. 
 
ELECTKICITY AND MAGNETISM 157 
 
 The magnitude of the throw must be reckoned from 
 the null point as it is on open circuit. This is similar 
 to the case in Exp. 82 where the throws are taken from 
 the elongation of a vibrating coil. 
 
 EXPERIMENT 95 
 
 RELATION BETWEEN THE QUANTITY OF IITDUCED ELEC- 
 TRICITY AND THE RESISTANCE IN THE SECONDARY 
 CIRCUIT 
 
 Use the same apparatus as in Exp. 94. The current 
 in the primary circuit is kept constant while the resistance 
 in the secondary is varied. The throw on the ballistic 
 galvanometer is observed for each of several resistances 
 when the circuits are opened. The total resistance Rs 
 in the secondary circuit must be known for each of the 
 observations. Determine the relation between Qs and 
 
 p- by plotting. 
 
 The resistance Rs includes the resistance of the galva- 
 nometer and the secondary coil. Its largest value should 
 be at least four times that of the smallest. 
 
 EXPERIMENT 96 
 COEFFICIENT OF MUTUAL INDUCTION 
 
 From the definition of the coefficient of mutual induc- 
 tion 
 
 y[»- _ ^ _ ;Es _ IstRs _ QsRs 
 
 Cll 1 p 1 p 1 p 
 
 It T 
 
 where 72s is the total resistance in the secondary circuit, 
 and Qs the quantity of induced electricity in the secondary 
 
158 ELECTRICITY AND MAGNETISM 
 
 when the primary circuit containing the current Ij, is 
 either closed or opened. From this equation 
 
 which shows the relations proven experimentally in 
 Exps. 94 and 95. 
 
 Measure the coefficient of mutual induction of the 
 current inductor of Exp. 93, and compare with the 
 calculated results. 
 
 Determine the value of M, also, from the observations 
 of Exp. 95. 
 
 EXPERIMENT 97. 
 CALCULATION OF THE COEFFICIENT OF SELF-INDUCTION 
 
 The coefficient of self-induction is the ratio of the 
 electromotive force induced in a coil to the rate at which 
 the current in it is changing. 
 
 di' 
 di 
 
 This is similar to the definition of the coefficient of 
 mutual induction, except that in this case the magnetic 
 lines of force cut the loops of the coil that produces 
 them. The coefficient of self-induction is numerically 
 equal to the induced electromotive force when the cur- 
 rent is changing at the rate of one ampere per second. 
 
 If all the lines that pass through the central section 
 of a solenoid passed out of the ends, then when the 
 
ELECTRICITY AND MAGNETISM 159 
 
 current is changing at the rate of one ampere per second 
 K lines of force would cut each loop of the coil, so that 
 the induced electromotive force 
 
 
 where ni is the number of loops and K the constant of 
 the solenoid. 
 
 Therefore, 
 
 ^ = ^a=^| (I) 
 
 Since about one-half of the hues pass out of the 
 solenoid before reaching the ends, as was shown in Fig. 
 92, Exp. 92, this equation gives only a rough approxima- 
 tion. The following are more exact formulae: 
 
 J _ 4t%^vP 
 
 2a^-\-aH^ Sa^' 
 
 10« lV4aq:]2 3^ J'* • • ^"^ 
 
 -0.0156 (y)'].t . (Ill) 
 
 in which a = radius, D = diameter, ? = length of coil, and 
 n = number of turns of wire per centimeter. 
 
 * Bulletin Bureau of Standards, Vol. Ill, p. 303, 1907; Vol. IV, p. 
 385, 1907-8. 
 
 t Fleming, Manual of Radiotelegraphy, p. 286. 
 
160 
 
 ELECTRICITY AND MAGNETISM 
 
 EXPERIMENT 98 
 
 COEFFICIENT OF SELF NDtJCTION— ZELENY'S MODI- 
 FICATION OF MAXWELL'S METHOD * 
 
 The coil Ri, whose coefficient of self-induction is 
 to be measured, is connected to three non-inductive 
 resistances Mdth which it forms a Wheatstone bridge as 
 shown in Fig. 95. When ordinary resistance boxes are 
 
 Fig. 95. 
 
 used for the non-inductive resistances a sliding resistance s, 
 is added into one arm of the bridge so as to make possible 
 an exact balance of the resistances in the four arms. 
 
 The current / in the main circuit is measured after 
 the balance of the resistances is obtained, and from this 
 the current /i flowing through Ri is calculated. 
 
 (Rs+Ri)! 
 
 Ri-\-R2-^R3-\-R4: 
 * Physical Review, Vol. XXIV, p. 260, 1907. 
 
 (I) 
 
ELECTRICITY AND MAGNETISM 161 
 
 The circuit is then opened, and the induced electro- 
 motive force in Ri causes an instantaneous current to 
 flow through the resistances and the balhstic galvanom- 
 eter. The whole quantity Q of the induced electricity 
 passes through Rj and R3, but it divides into two parts 
 in passing through the galvanometer and the resistance 
 R2-I-R4. The part passing through the galvanometer 
 
 ^ {R2+Ri)Q 
 ^ R2+Ri+G' 
 
 so that 
 
 QJB2+S4+G ,__, 
 
 = R2+R. g- • • • • • (II) 
 
 The quantity q is obtained from the throw of the 
 galvanometer coil. 
 
 The induced electromotive force is situated in Ri, 
 so that the resistance of the Wheatstone net for the 
 induced electricity is 
 
 «=«'+*+lwl- • ■ ■ ("n 
 
 J. e IjR litR QR /Txr\ 
 
 ^-^-jT^^^h- • • • • (IV) 
 
 dt t 
 
 The values of Q, R and 7i can be calculated from 
 Equations (I), (II) and (III). 
 
 It is often convenient to make jR2 = jB4 = 100 ohms, 
 and to introduce a resistance box with tenths of ohms 
 into the -inductance branch, so that R^ may be made 
 some multiple of ten. In the equations, Ri is the sum 
 
162 ELECTRICITY AND MAGNETISM 
 
 of all resistances in that branch. The resistance G 
 represents the sum of all the resistances in the galvanom- 
 eter branch. 
 
 While obtaining a balance, employ a shunt with 
 the galvanometer. For the final observations the shunt 
 circuit usually shoiild be opened. If this is not the 
 case, care must be taken to obtain the figure of merit 
 of the' galvanometer with the same resistance in the 
 shunt. 
 
 When the throw D is obtained from a standard 
 condenser C charged with the electromotive force E, 
 and when d represents the throw due to the induced 
 electricity, while I is the current flowing through the 
 ammeter, and when R2=Ri, 
 
 which is obtained from equation (IV) by substituting 
 equations (I), (II) and (III), and simplifying. 
 
 The self-induction in the galvanometer coil does 
 not affect this equation, for when the current in the 
 main circuit is broken the induced current caused to 
 flow through the galvanometer increases from zero to 
 a maximum and then decreases to zero. The quantity 
 of electricity induced in the galvanometer coil while 
 the transient current in it is increasing, is equal to the 
 quantity induced while the current is decreasing. The 
 two quantities are, however, sent in opposite directions 
 and their algebraic sum being zero they do not affect 
 the throw of the galvanometer coil. 
 
 It is often most convenient to use the three resistances 
 of a Wheatstone box bridge. The connections are made 
 as shown in Fig. 96, the keys on the box bridge not being 
 used. 
 
ELECTRICITY AND MAGNETISM 
 
 163 
 
 The heating of the copper coil often alters the balance 
 of the bridge quite rapidly making it necessary to move 
 
 Fi(5. 06. 
 
 the adjusting sHde continualljr. The throws are meas- 
 ured from the zero on open circuit. 
 
 EXPERIMENT 98a 
 
 COEFFICIENT OF SELF-INDUCTION— ANDERSON'S 
 METHOD* 
 
 This method of measuring self -inductance has an advant- 
 age over the method of the previous experiment in that 
 it does not require the measurement of the current and of 
 the quantity of electricity. The coil whose inductance 
 L is to be measured is placed in one arm of the Wheat- 
 stone bridge together with a sliding resistance. The 
 i-esistance r is placed in the galvanometer branch, and 
 the condenser C is connected as shown in Fig. 97. 
 
 * Philosophical Magazine, Vol. XXXI, p. 329, 1891. Physical 
 Review, Vol. XXIV, p. 265, 1907. 
 
164 
 
 ELECTRICITY AND MAGNETISM 
 
 As in the previous method, it is often convenient to 
 employ the resistances of a box bridge. The connec- 
 tions can be made as shown in Fig. 98. It is then desir- 
 able to introduce a box having tenths of an ohm into the 
 inductance branch. 
 
 The resistance R may be made equal to P The 
 bridge is balanced with a continuous current as in Exp. 
 
 Fig. 97. 
 
 98. The resistance r or the capacity C, or both, are 
 then adjusted until no throw is produced on the gal- 
 vanometer when the key for opening the circuits is de- 
 pressed. The induced electromotive force in Q and the 
 discharge from the condenser act in opposite directions 
 through the galvanometer, and with the proper value 
 of r the two effects are equal so that there is no difference 
 of potential between the extremities M and N of the 
 galvanometer. The condition is the same at both the 
 
ELECTKICITY AND MAGNETISM 
 
 165 
 
 opening and the closing of the circuits. The bridge is 
 then balanced for both continuous and intermittent 
 currents. 
 
 The condition for a balance with continuous current is 
 that 
 
 R S 
 
 (I) 
 
 Fig. 98. 
 
 The additional condition for a balance with intermittent 
 currents is that there shall be no difference of potential 
 between il and N at the opening or closing of the cir- 
 cuits. Let X be the quantity of electricity that has 
 flowed into the condenser during the time t after the 
 closing of the circuits, and z the quantity that has passed 
 through R. Then x-\-z has passed through P Also 
 let y be the quantity that has passed through Q and S 
 during the same time. Since M and N have the same 
 potential, the difference of potential between the ends 
 of the condenser must equal the difference of potential 
 between the extremities of S, so that 
 
 x__„dy 
 C dt' 
 
 (11) 
 
 <^y ^„ 
 
 where ■— is the current flowing through S. 
 
166 ELECTRICITY AND MAGNETISM 
 
 The fall of potential from 5 to D must be the same by 
 both paths, so that 
 
 4r^^t ("I) 
 
 The fall of potential from A through B to N must 
 be equal to that from A to M. The latter contains 
 the inductance coil, the induced electromotive force in 
 
 , • , .' 1 , -rdidy) ^dP"u 
 
 which IS equal to L— jT— =iv-p^. 
 
 Then 
 
 dx (dx dz\_dy ^ 
 
 "^dt^^Kdt^dtJ'^dt^^- ' ■ ^^^ 
 
 Substituting from equations (II) and (III) in (IV), 
 
 , , ^.dx .Plx.dxX Q X , L dx „„ 
 
 ^'+P^dt+R(c+'dt)^S-C+SCdf- • ^^) 
 
 . .dx Prdx x/P Q\ L dx , , 
 
 This equation represents the conditions necessary for a 
 balance with intermittent or variable currents. The 
 Wheatstone net is at the same time balanced also for 
 continuous currents so that from equation (I), 
 
 R S 
 Then equation (VI) becomes 
 
 from which since PS = QR, 
 
 L = C{r(,Q+S)+PS] (VII) 
 
ELECTRICITY AND MAGNETISM 
 
 167 
 
 For sensitiveness of the final adjustment it is desirable 
 to make R and S large, and r small. Since Q is usually 
 small, P will also be small. When the maximum sen- 
 sitiveness is not required, it is often advantageous to 
 make R = P = 10 or 100. This makes r larger which is a 
 condition convenient for adjustment. 
 
 If thermo-electric currents are present in the gal- 
 vanometer circuit they must either be eliminated by a 
 potentiometer method or proper allowance must be 
 made for them. 
 
 For the measurement of very small self-inductances, 
 a double commutator is usually employed. It consists 
 of two two-part commutators 
 mounted on a common shaft, 
 usually on opposite ends of the 
 shaft of a small motor. The 
 battery is connected through 
 one commutator while the 
 galvanometer is connected 
 through the other, as shown 
 in Fig. 99. The commutators 
 are so placed that when one 
 causes an alternating current 
 
 to flow through the arms of a bridge, the other so com- 
 mutates the alternating current in the galvanometer 
 branch as to send a succession of unidirectional pulses 
 through the galvanometer. This can be made to in- 
 crease the sensitiveness of the method about one hun- 
 dred times. A rapid succession of pulses acts upon the 
 galvanometer in place of one. 
 
 mih 
 
 Fig. 99 
 
168 
 
 ELECTRICITY AND MAGNETISM 
 
 EXPERIMENT 99 
 
 COMPARISON OF TWO COEFFICIENTS OF SELF- 
 INDUCTION 
 
 The apparatus is connected as shown diagramatically 
 in Fig. 100. The battery and galvanometer are usually 
 connected through a double key and preferably through 
 a double commutator. 
 
 Fig. 100. 
 
 Let Ri and i?2 have the coefficients of self-induction 
 Li and L2, and let R3 and /24 be inductionless resistances. 
 One of the inductances L2 is a variable standard. 
 
 The resistances Rz and R^ are adjusted to give a balance 
 with steady current. The variable standard is then 
 adjusted until a balance is obtained for intermittent 
 
ELEOTEICITY AND MAGNETISM 169 
 
 currents also. This adjustment does not alter the 
 resistances in the bridge. 
 
 Let I'l be the current through Ri and R2, and 13 that 
 through i?3 and Ri, at the same instant t after closing 
 the circuit. Since no current flows through the gal- 
 vanometer the potential at C must be the same as that 
 at D. Hence 
 
 Riii+Li-^=Rzi3, ...... (I) 
 
 R2ii-\-L2-T7 = Rdz. ...... (II) 
 
 From which 
 
 RiRd\iz+LiRi-^z = R2Rziiiz+L2Rz--^z. (Ill) 
 
 From the condition of a balance with steady current, 
 RiRi = R2R3- This eliminates two of the terms in 
 equation (III). The remaining terms form the equation 
 
 LiRi^LzRz) 
 and 
 
 L,=|l.. .... (IV) 
 
 The variable standard of self-induction contains two 
 coils without iron joined in series, one of them fixed and 
 the other movable about a vertical axis. The self- 
 induction of the two depends upon their relative position. 
 A graduated scale indicates the inductance in milli- 
 henries for any position of the movable coil. 
 
170 
 
 ELECTRICITY AND MAGNETISM 
 
 EXPERIMENT 100 
 
 ABSOLUTE DETERMINATION OF RESISTANCE— STANDARD 
 SOLENOID OR CURRENT INDUCTOR METHOD 
 
 Connect the apparatus as shown in Fig. 101 so that 
 the primary circuit opens an instant before the secondary. 
 
 j^mi 
 
 Fig. 101. 
 
 The resistance x, to be measured, can be placed into 
 or taken out of the galvanometer circuit.. Throws are 
 obtained with the resistance in and out of the circuit. 
 
 R = 
 
 Nnz _KIin2 
 
 In a similar manner when x is in the galvanometer 
 circuit, 
 
 Nn2 _'K l2n2 
 10»Q2~l(fiQ2' 
 
 R+X = 
 
ELECTEICITY AND MAGNETISM 171 
 
 Then 
 
 y ^ Kl2n2 Khn2 ^ Kn2 /h h\ ^Kn2 (h h\ 
 IO8Q2 lO^Qi ~ 108 [q^ qJ - iQSi? \d2 dil ' 
 
 where K is the constant of the solenoid, A^2 the 
 number of turns in the secondary coil, and F the figure 
 of merit of the galvanometer. 
 
 The resistance of the circuit should not be less than 
 that which makes the coil just aperiodic. The currents 
 Ii and I2 should have such magnitudes as to make the 
 throws di and ^2 nearly equal. 
 
 The throws must be reckoned from the zero on open 
 circuit. 
 
 Measure the resistance of the 200 ohm coil in a 
 resistance box. 
 
 EXPERIMENT 101 
 
 FIGURE OF MERIT— BY THREE METHODS 
 
 The figure of merit of a galvanometer is usually 
 determined by one of the following three methods: 
 
 1. Condenser Method. 
 
 F = 9: = 91 
 
 d d' 
 
 This is given in Exp. 84 and is the method commonly 
 employed for determining the figure of merit on open 
 circuit. Only the free charges of the condenser can be 
 used as explained in Exp. 83. 
 
 This method is not commonly used for determining 
 the figure of merit on closed circuit. It may, however, 
 be employed if proper precautions are taken. The con- 
 denser is discharged through the galvanometer which, 
 before the coil moves an appreciable distance, is placed 
 
172 
 
 ELECTKICITY AND MAGNETISM 
 
 on closed circuit with the apparatus with which it is to 
 be used. This can be approximately accomplished 
 by means of one tap on the key shown in Fig. 102. The 
 key is held down until the reading of the throw is ob- 
 tained. An error is often produced by the coil moving 
 an appreciable distance before the circuit is closed. 
 To reduce this error the circuit should have a resistance 
 
 Fig. 102. 
 
 not less than that which makes the galvanometer just 
 aperiodic, and the tap on the key should be rapid. The 
 greater the period of the galvanometer, the less is the 
 danger of error from this cause. 
 
 A thermo-electric current in the circuit does not 
 produce its full deflection during the period of the throw. 
 Only a fraction* of the full thermo-electric deflection 
 is effective, and this fraction depends on the damping 
 of the galvanometer and can be obtained from the curve 
 
 * Physical Review, Vol. XXIII, p. 414, 1906. 
 
ELECTKICITY AND MAGNETISM 
 
 173 
 
 of Fig. 103. After an observed throw, the coil is allowed 
 to swing beyond the zero point, on the closed circuit. 
 The distance it moves beyond the zero gives the neces- 
 
 60 
 
 ||40 
 
 •I 
 
 s 
 
 eJ«5_ 1 
 
 10 
 
 30 40 50 60 70 80 
 
 Correcfion, in%oFfhermo-elecfricdenecfion, 
 fobeaddec/foorsubfnTcfedfromihrow 
 
 Fig. 103. 
 
 sary datum for obtaining the correction from the curve 
 and the observed thermo-electric deflection. 
 
 2. Standard Mutual Inductance Method. — This 
 method is particularly adapted for determining the figure 
 of merit on closed circuit. The secondary coil is made 
 
174 ELECTRICITY AND MAGNETISM 
 
 a part of the galvanometer circuit. From Exp. 96 it 
 follows that 
 
 d dRa ' 
 
 The figure of merit on open circuit may be determined 
 by this method by opening the secondary circuit imme- 
 diately after the opening of the primary circuit. This 
 is best accomplished by means of a key as shown in 
 Fig. 84. The resistance of the galvanometer circuit 
 should not be less than that which makes the coil just 
 aperiodic. The standard mutual induction coil is a set 
 of two coils whose coefficient of mutual induction is 
 known, being given by the manufacturer or having been 
 previously determined. 
 
 3. Current Inductor or Standard Solenoid Method. — 
 This method is the same as the standard mutual 
 induction coil method except that the primary coil is a 
 standard solenoid whose constant can be calculated. 
 
 Q^MIv^n2KI^ 
 d dRs lO^dR/ 
 
 Determine the figure of merit of a galvanometer on 
 open circuit by each of the three methods. 
 
 EXPERIMENT 102 
 
 NUMBER OF LINES IN A MAGNETIC FIELD BY 
 COMPARISON METHOD 
 
 Instead of obtaining a value for the figure of merit, 
 it is usually more convenient to make a direct comparison 
 of the unknown number of lines cut with the known 
 number in a standard coil. In this case the secondary 
 
ELECTRICITY AND MAGNETISM 175 
 
 coil of the standard and the coil cutting the field to be 
 measured are placed in series with a galvanometer so 
 that the resistance of the circuit Rs, is common to both. 
 
 „__]_dw_ 1 Nn2 
 E=IRs. 
 
 Then 
 
 and 
 
 1 Nn2_jjy 
 108 i ^«" 
 
 Nn2 = 10^QRs.. 
 
 where the term Nn2 is called the " number of lines 
 cut," and is the niunber of lines in the magnetic field 
 multiplied by the number of turns in the coil that cut 
 these lines. 
 
 This is true for any coil cutting or being cut by 
 magnetic lines of force. In the standard coil, M is 
 calculated or previously determined. In it 
 
 Q^MIp Nn2 
 
 Rs lO^Rs' 
 Nn2 = l(^QsRs = lO^MI^. 
 
 Let Nx be the number of lines cut in the field to be 
 measured and Ux the number of turns on the coil cutting 
 them, then 
 
 NxUx^lO^QxRs. 
 
 But in the standard coil, 
 
 Nn2 = lO«Mh = l(fiQsRs. 
 
176 ELECTRICITY AND MAGNETISM 
 
 Then, 
 
 NxUx _ Nxnx _ IQ^QxRs _Qx _dx 
 Nm ~ WMI^ " lO^QA ~ Q." dj 
 
 from which 
 
 WM dx ■ 
 rix ds 
 
 Nx = ^^^-I^^ 
 
 The value of I-p should be such as to make ds nearly- 
 equal to dx. 
 
 1. Determine the number of lines about the pole of 
 a magnet by cutting them with a coil of wire, and then 
 calculate the pole strength. 
 
 2. In a similar manner determine the number of 
 lines between the poles of an electro-magnet with different 
 current strengths in the field coil. Show the relation 
 between the current and the number of lines by plotting. 
 
 The coil should be moved through the field as rapidly 
 as possible. As in all similar cases the resistance in the 
 circuit should not be less than that which makes the 
 galvanometer coil just aperiodic. Such determinations 
 should be made on closed circuit. 
 
 EXPERIMENT 103 
 
 HORIZONTAL COMPONENT OF THE EARTH'S MAGNETIC 
 FIELD— EARTH INDUCTOR METHOD 
 
 The earth inductor consists of a coil of wire that can 
 be rotated in the earth's field. In Fig. 104 the coil 
 is wound on the circular frame. The square frame can 
 be placed in any position and clamped. The circular 
 frame rotates on an axis perpendicular to the axis of the 
 square one. 
 
ELECTRICITY AND MAGNETISM 
 
 177 
 
 The square frame is clamped in a vertical position 
 and by the aid of a magnetic needle its plane is placed 
 perpendicular to the magnetic meridian. The coil 
 is then connected in series with a ballistic galvanometer, 
 a standard mutual inductance coil and sufficient resistance 
 to make the galvanometer approximately just aperiodic. 
 
 Fig. 104. 
 
 The frame of the coil is placed into the plane of the 
 square frame, and then is rotated quickly through half 
 a revolution. The quantity of induced electricity is 
 obtained from the throw by comparison with the throw 
 obtained from the standard mutual inductance coil. 
 
 The number of lines of force passing thorugh the 
 coil in its initial position is AH, where A is the average 
 area of the loops in the coil. In turning through 180° 
 these lines are cut twice. The electromotive force gen- 
 erated by a conductor cutting fines of magnetic force 
 is the rate at which they are cut. 
 
178 ELECTRICITY AND MAGNETISM 
 
 In the n loops, 
 
 „ 1 dn 2AHn ,.„ 
 
 •• ■"" 2An • 
 
 ■ EXPERIMENT 104 
 
 INCLINATION OF THE EARTH'S MAGNETIC FIELD- 
 EARTH INDUCTOR METHOD 
 
 The earth inductor is adjusted in a vertical plane as 
 in Exp. 103. The coil is tiu-ned quickly through 180°, 
 and the throw on the galvanometer is noted. The square 
 frame is then clamped in a horizontal position, and the 
 throw is again taken for half a turn of the coil. 
 
 In the first case 
 
 2AnH 
 
 Qn = 
 
 WR 
 
 In the second case the vertical component V of the 
 earth's field is effective. Then 
 
 2An7 
 ^"~ 108/2 • 
 
 from which the angle of inclination is calculated. It 
 is not necessary to use the standard inductance coil. 
 
ELECTRICITY AND MAGNETISM 
 
 179 
 
 EXPERIMENT 105 
 
 INTENSITY OF THE EARTH'S MAGNETIC FIELD 
 
 H 
 
 When the ^inclination and H are 
 known, the intensity of the earth's 
 field 
 
 H 
 
 cos 6' 
 
 Compute from the observations of 
 Exps. 104 and 105. 
 
 Fig 105. 
 
 EXPERIMENT 106 
 
 MAGNETIC PROPERTIES OF IRON— I, B, (J,, R AND 
 HYSTERESIS 
 
 Apparatus. — In Fig. 106 >S is a standard mutual 
 inductance coil for obtaining the constant of the gal- 
 vanometer on closed circuit, T is a solenoid placed perpen- 
 dicular to the magnetic meridian, which has the number 
 of turns of wire known for both its coils, f/ is a set of two 
 coils one sliding over the other for the purpose of vary- 
 ing the inductance, P and P' are Pohl commutators, and 
 K is another Pohl commutator with its cross-bars removed. 
 
 The piece of iron to be tested should be in the form 
 of a wire or several fine wires somewhat shorter than 
 the primary coil of T. 
 
 Manipulation. — Adjust the coils of V until the in- 
 ductance is equal and opposed to the inductance in T, 
 and with the largest current to be employed no throw 
 
180 
 
 ELECTEICITY AND MAGNETISM 
 
 is produced on the galvanometer on closing or opening 
 the battery circuit. 
 
 Place the iron into the primary coil of T when the 
 battery circuit is open. By means of a copper wire or a 
 
 string attached to the 
 iron quickly withdraw 
 the iron from the coil. 
 This should produce no 
 throw on the galvanom- 
 eter. If it does, try to 
 remove the magnetism 
 from the iron, and if 
 any finally remains, take 
 the initial magnetism 
 into consideration as a 
 part of the total mag- 
 netism that will be in- 
 duced in the iron. Care 
 must be taken to make 
 the correction in the 
 proper sense. 
 
 With the iron in T, 
 increase the current by 
 equal steps until finally 
 a considerable increase 
 Fig. 106. produces but little effect 
 
 upon the galvanometer, 
 then, in the same manner, decrease the current to zero. 
 Then reverse the commutators P and P' and repeat 
 the process, and then again reverse and increase the 
 current to the maximum. The ballistic throw and the 
 ammeter reading are taken at each increase. These give 
 the data for calculating the increases in the magnetizing 
 force and the induction. 
 
 Obtain throws from the standard mutual inductance 
 
ELECTRICITY AND MAGNETISM 181 
 
 coil. Large throws should be compared with large, 
 and small with small. 
 H. — Magnetizing force. 
 
 
 1 — 
 
 2 
 
 P 
 
 + T^) 
 
 Nx. — N' umber of lines induced at each increase of the 
 magnetizing force.' 
 
 IQ^QR 
 
 N, = N"-N'-- 
 
 nz 
 
 This number, {N"-N'), is the number Nx of Exp. 102, 
 and is determined in the same manner. 
 /. — Intensity of magnetizatioyi. 
 
 J _M _ml _4:T:m _ N 
 ~"F ~ si ~ 4xs ~4x5' 
 
 where A'' is the total number of lines in the iron, and s 
 the cross-section of the iron in square centimeters. 
 The number of induced lines per square centimeter, 
 
 ^ A T 
 
 — = iizl. 
 s 
 
 B. — Magnetic induction refers to the number of lines 
 of force per square centimeter through the iron. It 
 includes the lines due to the magnetizing force and the 
 induced lines. 
 
 B = H+ii:I. 
 
 K. — Magnetic susceptibility is the ratio of the intensity 
 of magnetization to the magnetizing force. 
 
182 ELECTKICITY AND MAGNETISM 
 
 Magnetic permeability is the ratio of the magnetic 
 induction to the magnetizing force. 
 
 B 
 
 Magnetic reluctance R= — . 
 
 » [J.S 
 
 1. Tabulate the values of H and B for all the observed 
 points, and plot the values of H as abscissas and those 
 of B as ordinates. The first branch is called the magnetiza- 
 tion curve, and the closed part, the hysteresis curve. 
 
 Since H is proportional to the magnetizing current 
 and B to the algebraic sum of the throws, the nature 
 of the hysteresis curve can be obtained by plotting the 
 currents with the sums of the throws. When the rocker 
 of the conamutator leading to the galvanometer is re- 
 versed, the sign of the throws is reversed. 
 
 2. Tabulate the values of H, N, I, B, ^, and R for 
 a few important points on the hysteresis curve. There 
 are two values for these quantities for each value of H. 
 The direction of the quantities is designated by -|- and 
 — signs. 
 
 These values are not exactly the same as would be 
 obtained with the iron made in the form of a ring. The 
 reluctance is that of the iron and the air through which 
 the lines of force pass. This influences all the other 
 quantities. When the wire is long compared to its 
 diameter, these values are nearly the same as those 
 obtained with a ring. 
 
SOUND 
 
 EXPERIMENT 107 
 
 VELOCITY OF A TRANSVERSE WAVE ALONG A CORD 
 
 Apparatus. — A cord about 20 meters long, and of 
 about 4 mm. diameter, has one end attached to a rigid 
 support. At the other end, the cord passes over a pulley 
 and supports a weight that may be varied from one to 
 four kilograms. The pulley should have ball bearings so 
 that the friction may be as small as possible. 
 
 Fig. 107. 
 
 D 
 
 Method and Manipulation. — Attach a weight of one 
 kilogram to the end of the cord. Start the wave by 
 giving the cord a quick transverse blow near one end 
 and at the same instant start the stop watch. Count 
 the number of returns of the wave that are clearly 
 perceptible, and stop the watch at the last return counted. 
 The number of returns times twice the distance between 
 the supports divided by the time in seconds gives the 
 velocity. 
 
 The velocity of the wave is d = ^/— , where T is the 
 
 183 
 
184 SOUND 
 
 tension of the cord in dynes and m the mass per unit 
 length of the cord. Determine the mass m by weighing 
 and measuring the entire length of the cord. For T 
 the weight supported by the cord multipUed by the 
 acceleration of gravity may be used. From these values 
 compute V the velocity. This should check with the 
 observed value obtained as directed above. Repeat by 
 using a weight of four kilograms. 
 
 EXPERIMENT 108 
 TO TEST EXPERIMENTALLY THE EQUATION n=— -J^ 
 
 Apparatus. — ^A wire that is attached to the table at 
 one end passes over two supports and carries a weight 
 at the other end. At the middle point of the wire is a 
 mechanism for determining the number of vibrations. 
 As the wire vibrates it makes and breaks the current 
 through an electromagnet which keeps it vibrating, and 
 also actuates a weighted escapement that registers the 
 number of vibrations. 
 
 Method and Manipulation. — A. Determine m the 
 mass of the wire per unit length, I the length of the wire 
 between the two supports, and T its tension in dynes. 
 Substitute these values in the eqviation, and obtain n, 
 the number of vibrations of the wire per second. 
 
 B. Cause the wire to vibrate, and observe the num- 
 ber of revolutions made by the escapement wheel in a 
 given time. From this determine n, the number of 
 vibrations made by the wire in a second and compare 
 this value with that obtained from the equation. 
 
 C. Repeat, using a different tension. 
 Reference. — Sound, by Poynting & Thompson, p. 82. 
 
SOUND 
 
 185 
 
 [/ 
 
 EXPERIMENT 109 
 VELOCITY OF SOUND IN AIR 
 
 Apparatus. — A large vertical glass tube has at its 
 upper end an ear tube e and a tuning fork as shown 
 in Fig. 108. The lower end is connected to the vessels 
 a and b by means of tubes as shown in figure, c and d are 
 stop cocks. When d is closed and 
 c is open the water in the resonance 
 tube rises. When c is closed and d 
 is open the water surface is lowered. 
 
 Method and Manipulation. — Al- 
 low the water to enter the tube. 
 As the water rises, the reenforce- 
 ment of the sound is detected 
 through the ear tube. Place a 
 rubber band at the position of the 
 water surface when the reenforce- 
 ment is a maxirhum. Repeat 
 several times by turning the stop 
 cocks c and d. In this way locate 
 the position of the water surface 
 for each point where the reenforce- 
 ment of the sound is a maximum. 
 The distance between two reenf orce- 
 ments is equal to a half wave 
 length of the sound. Place bands 
 at the highest and lowest positions 
 noted, and measure the distance d between them. Repeat 
 several times. 
 
 Let X be the wave length and i the number of inter- 
 nodes in the distance d, then 
 
 Fig. 108. 
 
 X = 
 
 2d 
 
186 SOUND 
 
 The velocity Vt of the sound in the tube at the tem- 
 perature t° of the room=Xn, where n is the frequency 
 of the fork. The velocity at 0° C. 
 
 70= ^' 
 
 Vl+at 
 
 The velocity Va in ffee air is larger than in the tube 
 and is obtained from the equation 
 
 Va = 7o(^l +2:507989\ ^^^^.^^ ^^^ g^^^^ ^^^^^ ^ 254.) 
 \ 2rV%n I 
 
 EXPERIMENT 110 
 VELOCITY OF SOUND IN A SOLID— KUNDT'S METHOD 
 
 Apparatus. — The solid in which the velocity is to be 
 determined must be in the form of a long rod or tube. 
 This is firmly clamped at its central point. A cork 
 disc is firmly attached to one end which is placed into 
 the end of a horizontal glass tube. The other end of 
 the glass tube contains a movable piston. 
 
 Method and Manipulation. — Distribute cork dust in- 
 side the tube so as to form a thin layer throughout its 
 length. Stroke the free half of the rod with a resined 
 cloth, thus producing in it longitudinal vibrations. The 
 cork disc transmits tbese vibrations to the air in the 
 tube. By means of the piston at the other end, the 
 length of the vibrating air column is adjusted until 
 stationary waves are formed. Since compression and 
 rarefaction take place only at the nodes, while at the 
 internodes the air sweeps to and fro, the dust at the for- 
 mer remains undisturbed. This affords a means of 
 locating the nodes and determining the wave-length in 
 air of the sound emitted by the rod. 
 
SOUND 137 
 
 When a rod is clamped at its middle the wave-length, 
 in its own substance, of the sound it emits is twice the 
 length of the rod 
 
 Measure the distance d between the extreme nodes 
 in the glass tube. 
 
 Let i = number of internodes in length d, 
 I = length of rod, 
 
 Fj = velocity in air at temperature of room, 
 Vs = velocity in solid, 
 Xo = wave-length in air, 
 Xs = wave-length in solid. 
 
 Since the period of the waves in the rod and in the 
 air is the same 
 
 where 
 
 and 
 
 
 __2d 
 
 I 
 
 Is =21. 
 
 Use 332 meters per second as the velocity in air at 
 zero degrees and from this determine Vt, as in Exp. 109. 
 
 Make several determinations of la- 
 
 Substitute the values obtained and solve for Vs. 
 
 The difference between the velocity in free air and 
 in a closed tube is disregarded in the above. 
 
188 
 
 SOUND 
 
 EXPERIMENT 111 
 
 FREQUENCY OF A FORK BY MEANS OF STATIONARY 
 WAVES PRODUCED BY IT IN A CORD 
 
 Apparatus. — A braided silk line of about 130 cm. 
 length and | mm. diameter is attached to an electric 
 fork and supports a weight-pan as shown in Fig. 109. 
 
 Manipulation.— Adjust the weights until the number 
 of loops is as large as possible. Measure the distance 
 from pulley to the first node a from the fork. From 
 this length obtain the value of the wave-length X. Ob- 
 
 FiG. 109. 
 
 tain also the mass per unit length by weighing accurately 
 a longer length of the same line. The tension T is the 
 weight supported by the line times the acceleration of 
 gravity. Compute the frequency n from the relation 
 
 _i It 
 
 '-xVm- 
 
 Then increase the weight until the number of loops 
 is one less, and again determine n. Continue thus until 
 the number of loops becomes as small as possible. From 
 these results determine the average value of n. 
 
SOUND 
 
 189 
 
 EXPERIMENT 112 
 
 FREQUENCY OF A TUNING FORK BY THE GRAPHICAL 
 METHOD 
 
 Apparatus. — The tuning fork is adjusted so that it 
 will trace its vibrations by means of a stylus on the 
 smoked surface of a rotating drum. A time-marker 
 giving seconds is adjusted so that its tracing is parallel 
 and close to the tracing of the fork. 
 
 Manipulation. — Cover the surface of the drum with 
 
 Fia. 110. 
 
 glazed paper, and smoke the surface by holding a lu- 
 minous flame against it while the drum is rotating. 
 
 Let the fork and marker make their tracings on the 
 surface as directed above. 
 
 Determine the frequency by counting the total 
 number of vibrations made by the fork during the total 
 number of seconds recorded by the marker. 
 
LIGHT 
 
 EXPERIMENT 113 
 
 SPHERICAL MIRRORS 
 
 Apparatus. — The spherical mirror M is mounted so 
 that it may be rotated about a vertical axis, also so 
 that it may be moved along the metric scale that sup- 
 ports it. A vertical rod o serves as the object. Back 
 
 of the rod o and mounted on the same slide is a plane 
 mirror N placed vertically and at an angle of forty-five 
 degrees with the supporting scale. The purpose of this 
 mirror is to produce a bright background against which 
 the image may be clearly seen. The image I is located 
 by adjusting a vertical rod c until there is no parallax 
 between it and the image. The position of the mirror 
 is determined by placing the end of the rod I against 
 the surface of the mirror and obtaining the reading on 
 the scale opposite the vertical edge attached to I and 
 
 190 
 
LIGHT 191 
 
 adding to this reading the length d obtained by means 
 of a vernier caliper. The readings a and h for the posi- 
 tion of the object and image on the scale are obtained 
 by means of the square s. 
 
 A — Real Image 
 Method. — In the general equation 
 
 1,112 , , 
 
 — H — = -r=n = a constant, 
 u V f R 
 
 u is the distance of the object from the mirror, v the 
 distance of the image, / the focal length, and R the 
 radius of curvature of the mirror. Determine v for each 
 of several values of u and substitute in succession the 
 values of each set in the general equation. Solve for / 
 and average. The equation is verified if the several 
 values obtained are constant. 
 
 B — Virtual Image 
 
 Apparatus. — Same as in A except that the concave or 
 convex mirror has a small hole at center or has a small 
 area from which the silvering has been removed. ' 
 
 Method. — To obtain the distance of the virtual image 
 place the rod c behind the mirror, and adjust until 
 there is no parallax between the image and the rod as 
 seen through the hole or unsilvered area. 
 
 Verify the equation for each of the two mirrors, as 
 in case A. 
 
192 LIGHT 
 
 EXPERIMENT 114 
 
 TO MEASURE THE FOCAL LENGTHS AND RADH OF 
 CURVATURE OF SPHERICAL MIRRORS 
 
 Method. — Find the distance of the image from each 
 of the two kinds of mirrors used in Exp. 113, when the 
 rays coming from the object are nearly parallel. For 
 this purpose use an object situated at a long distance. 
 Then in> each case /=«, and R = 2f. 
 
 Find R by moving the mirror towards the object 
 until u = v. Then u = v = R. 
 
 The values thus obtained for / and R should check 
 with the values obtained in A and B of the previous 
 experiment. 
 
 EXPERIMENT 115 
 LENSES 
 
 Apparatus. — A convex lens; a half of a convex lens; 
 a half of a concave lens. Apparatus as shown in Fig. 112. 
 
 In this case the reading on the scale for the midway 
 point of the thickness of the lens must be determined. 
 This is done by placing the end of the rod I against the 
 surface of the lens and obtaining the reading on the 
 scale for the vertical edge, adding the distance d and one- 
 half the thickness of the lens as determined by means 
 of a vernier caliper. 
 
LIGHT 193 
 
 A. — Real Image 
 In the general equation 
 
 = -r = a constant, 
 
 V u f 
 
 u is the distance of the object from the center of the 
 lens, V the distance of the image, and / the focal length 
 of the lens. Distances on the same side of the lens as 
 the object are positive. Determine v, by parallax method, 
 for each of several values of u and solve for/ and average. 
 
 B. — Virtual Image 
 
 a. Place the object in front of a half of a double con- 
 vex lens in a position where a virtual image is obtained. 
 Move the rod c behind the lens and observe it over the 
 cut end of the lens, so that the rays from it do not pass 
 through the lens. The place where no parallax exists 
 between the image of the object and the rod c is the 
 location of the virtual image. Verify equation as in A. 
 
 b. Use a half of a double concave lens, locate the 
 position of the virtual image by the parallax method, 
 and verify the general equation as in A above. 
 
 EXPERIMENT 116 
 
 TO MEASURE THE FOCAL LENGTH OF LENSES 
 
 Method. — Find by the parallax method the distance 
 of the image of a distant object for each of the two kinds 
 of lenses used in Exp. 115. This gives directly the focal 
 lengths of the lenses, and the values obtained should 
 check with those obtained in Exp. 115. 
 
194 LIGHT 
 
 EXPERIMENT 117 
 THE MAGNIFYING POWER OF OPTICAL INSTRUMENTS 
 
 From a point upon an object, the eye receives a cone 
 of rays the angle of which becomes greater as the distance 
 between the eye and object becomes less. The point 
 is clearly seen when the eye is able to bring this cone 
 of rays to a focus upon the retina. The distance at 
 which the point is seen most clearly is in the case of most 
 eyes about 25 cm. If the object is within this distance 
 it can still be seen distinctly by the aid of a lens which 
 converges the rays so as to make them appear to come 
 from a greater distance which by proper adjustment 
 of the lens can be made that of the most distinct vision 
 (25 cm.). This lens, or simple microscope is then said to 
 be focused for the object. 
 
 The angle between the axes of the two cones of rays 
 received by the eye from two extreme points of the 
 object or image determines the apparent size of the 
 object or image. 
 
 The magnification in a simple microscope is defined 
 as the ratio of the apparent size of the image to the 
 apparent size of the object when both are at the dis- 
 tance of most distinct vision, or in other words, the 
 ratio of the angle subtended at the eye by the image 
 to the angle subtended by the object when each is placed 
 at the distance of most distinct vision. 
 
 In any optical instrument where the object is farther 
 away than the distance of most distinct vision the 
 magnifying power is the ratio between the angles sub- 
 tended at the eye by the image and the object respect- 
 ively, the image being at the distance of most distinct 
 vision. 
 
LIGHT 195 
 
 A. Magnifying power of a convex lens when the object 
 viewed is at a great distance. The image in this case is 
 formed at practically the principal focus of the lens, and 
 the distance of the object from the lens, D in Fig. 113, can 
 be considered as practically the distance of the object from 
 the eye. In observing the image, the eye is placed 25 cm. 
 from it, i.e., at the distance of most distinct vision. The 
 
 
 zscm. 
 
 \ 
 
 Fig. 113. 
 
 angle formed at the eye by the image /, is approximately 
 equal to ^ and the angle formed by the object itself is 
 
 A A I 
 
 D+F+25 D F' 
 
 J J w 
 
 Therefore the magnifying power M = -^-^r=, = —^. 
 
 Manipulation. — Place a scale horizontally at a distance 
 of 25 cm. from the eye. Adjust the lens so that there 
 is no parallax between this scale and the image / of the 
 distant object. 
 
 1. Determine the distance F and compute the magnify- 
 ing power M. 
 
 2. With the eye 25 cm. from I, measure the distance 
 between two points on the image by means of the scale 
 at /. Remove the lens, and with the eye and scale in 
 the same position, measure the distance between the 
 
196 LIGHT 
 
 same two points on the object directly. The ratio of these 
 distances should check with the computed value of M. 
 B. Magnifying power of a convex lens when the object 
 viewed is within the principal focus. (Simple Micro- 
 scope.) — The image produced is a virtual image and the mi- 
 croscope is said to be in focus for the object when the image 
 
 is seen most distinctly, 
 
 '"^"---v^----,. i-e., when it is at the 
 
 ~~""~-V,-_---^/yg^, distance of most dis- 
 
 ^~^^^^^^ tinct vision or about 
 
 ,,--,'r-- ""Y 25 cm. from the eye. 
 
 ,,.-x'-'''"' \—s-- -pjjg object observed is 
 
 :'''.' --^s J nearer the eye than the 
 
 p ^^. image and for com- 
 
 parison of magnifica- 
 tion must be assumed to be placed 25 cm. from the 
 eye or at the same distance as the image. The eye is 
 assumed to be placed very near the, lens so that the 
 angle formed at the center of the lens is nearly the same 
 as that formed at the eye. 
 Then 
 
 25 ■ 2 AS' 
 
 Manipulation. — Focus the lens on a vertical graduated 
 scale A, and place another similar scale 25 cm. from the 
 lens. With one eye, observe the image of the first scale, 
 and with the other, view the second scale directly. Adjust 
 the distance ;S by moving A until the image of the first 
 scale appears beside the second scale and has n6 parallax 
 with it. 
 
 1. Determine the value of S and calculate the magnify- 
 ing power of the lens from the equation 
 
 S 
 
LIGHT 
 
 197 
 
 2. With one eye observe the image of the first scale 
 and with the other eye view the second scale directly. 
 Determine the number of scale divisions on one that 
 corresponds to a given number on the other. The ratio 
 of these is the magnification and should check with the 
 computed value. 
 
 C. Magnifying power of a compound microscope. — 
 The object A is a small distance beyond the focus 
 of the objective N, so as to form a real image 7i at 
 
 some distance from the objective. This image is within 
 the principal focus of the eye-piece which acts as a 
 simple microscope. 
 
 The real image is larger than the object, its magnifi- 
 cation being 
 
 *=i4 
 
 The magnification of this image by the eye-piece is 
 25 
 
 M2 = 
 
 S' 
 
198 
 
 LIGHT 
 
 The total magnification of the object is 
 
 25L 
 
 M=MiM2 = 
 
 RS- 
 
 This assumes the eye to be very near the lens. 
 
 Manipulation. — Determine the distance S for the 
 lens X, to be used as the eye-piece, as directed under 
 manipulation in case B above. Having done this leave 
 the scale at 7i in position and remove the lens X. Place 
 a vertical scale A and the lens N, which is the objective, 
 so that a real image of A is formed. Then adjust A and 
 N until there is no parallax between the real image of A 
 and the scale at 7i. Replace the lens X. The final image 
 I2 of A is then seen at a distance of 25 cm. from X, the 
 eye being close to the lens X. 
 
 1. Measure the distance R, L and S, and compute 
 the magnification M. 
 
 2. With one eye, observe the image I2 through the 
 lens, and with the other eye, observe directly the scale 
 placed at A. The ratio of the number of divisions on 
 I2 that correspond to a given number on the scale A is 
 the magnification, and should check with the computed 
 value of M, 
 
 Fig. 116. 
 
 D. Magnifying power of an astronomical telescope. — 
 
 The object A, Fig. 116, is assumed to be at a con- 
 
LIGHT 
 
 199 
 
 siderable distance from the lens iV. The magnifying 
 
 F 
 
 power of the objective as shown in section A is Mi=-^. 
 
 When the image falls within the principal focus of the 
 eye-piece X, the magnification of this image by the 
 
 25 
 
 eye-piece is M2 = -a, as shown in section B. 
 
 F 
 
 The total magnifying power M = M1M2 = 
 
 s- 
 
 Manipulation. — The procedure is the same as under 
 manipulation in section C, except that the lenses X and 
 N are interchanged, and a distant object is used instead 
 of the scale at A. 
 
 1. Determine the distances F and ;S and compute 
 the magnification M. 
 
 2. Hold a scale at a distance of 25 cm. from the eye- 
 piece. On this scale measure the distance between two 
 points on the image by viewing the image with one eye 
 and the scale with the other. Then with the eye and 
 scale in the same position, measure on the scale directly 
 the distance between the same two points on the object. 
 The ratio should check with the computed value for M. 
 
 Fig. 117-. 
 
 E. Terrestrial telescope. — Place the reversing lenses 
 between the objective and the eye lens and at a distance 
 from the real image equal to the focal length of the first 
 reversing lens. Since the two lenses are alike, the second 
 image will be formed at a distance equal to the same 
 focal length. Place the eye lens at a distance S from the 
 second image. Observe the erect image. 
 
200 
 
 LIGHT 
 
 EXPERIMENT 118 
 ADJUSTMENT OF THE SPECTROMETER 
 
 The spectrometer consists of a circular graduated 
 disc mounted horizontally on a vertical axis supported 
 by a leveling base. Two arms extend horizontally 
 from the axis. One supports a telescope and the other 
 a collimator. The arm supporting the telescope may 
 be undamped from the vertical axis and rotated so as 
 
 Fig. 118. 
 
 to make any desired angle with the collimator. Beneath 
 the telescope is a tangent screw for moving the tele- 
 scope slowly. Surrounding the graduated disc is a rim 
 which moves with the telescope and on which are two 
 verniers placed 180° apart. 
 
 The telescope is made of three tubes. The first is 
 the eye-piece and contains a thin glass plate set at an 
 angle of 45° with the axis of the telescope and directly 
 opposite a round opening in the side of the eye-piece. 
 The second contains the cross-hairs, and is moved in 
 the third by means of a rack and pinion. The third 
 contains the objective, and is fixed to the supporting 
 
LIGHT 201 
 
 arm. The collimator consists of two tubes. One is fixed 
 to the arm, and contains a lens similar to the objective 
 of the telescope. The other slides in the first and con- 
 tains an adjustable slit. 
 
 1. Focus the Eye-piece on the Cross-hairs. — This is 
 done by moving the eye-piece in or out until the cross- 
 hairs are seen distinctly. 
 
 2. Focus the Telescope for Parallel Rays. — In the 
 experiments to follow it is necessarj^ that rays coming 
 from the collimator be parallel and henCe the telescope 
 must be adjusted for parallel rays. This may be done 
 by focusing the telescope on a distant object. The fol- 
 lowing is a more exact method and is the one to be used. 
 
 Place a prism at the center of the spectrometer disc 
 and adjust by means of the screws underneath until 
 one face is about perpendicular to the axis of the telescope. 
 Place a light at the side of the eye-piece so that the rays 
 enter the opening as shown in Fig. 118. The glass plate 
 reflects a portion of the rays down the barrel of the 
 telescope. In passing the cross-hairs some of the rays 
 are cut off. The remaining rays pass through the object- 
 ive to the face of the prism where a portion is again 
 reflected. Since the face of the prism was placed nearly 
 perpendicular to the axis of the telescope a portion of 
 the reflected rays will again enter the telescope. If the 
 cross-hairs are in the pjincipal focus, rays proceeding 
 from them are parallel while passing from the objective 
 to the prism. In this case, the rays reflected by the 
 prism will be parallel, and on entering the telescope, 
 will be brought to a focus in the plane of the cross-hairs 
 forming an image of them which may be seen through 
 the eye-piece. Therefore, when the telescope is adjusted 
 for parallel rays the cross-hairs and their image are in 
 the same plane and no parallax is observed when the eye 
 is moved in front of the eye-piece. 
 
202 LIGHT 
 
 3. Adjust the Collimator for Parallel Rays.— This is 
 
 done by turning the adjusted telescope into hne with the 
 collimator and moving the colUmator slit in or out until 
 it is seen through the telescope clearly and without 
 parallax. 
 
 4. Alignment.— Align the Telescope and Collimator 
 with the center of the spectrometer disc. This is done 
 by unclamping the telescope or the collimator from its 
 supporting arm, sighting along its barrel and turning 
 until it points to the center of the disc- 
 s' Level the Disc of the Spectrometer. — Place a level 
 
 on the disc so that it is parallel to a line through two of 
 the screws in the base, and level. Then place the level 
 at right-angles to its first position, and level by turning 
 the third screw. 
 
 6. Level the Telescope and the Collimator. — Place 
 the level on top of the barrel of the telescope, and ad- 
 just by turning the vertical screw underneath. In a 
 similar manner level the collimator. 
 
 7. To Adjust the Face of a Prism so that it is Per- 
 pendicular to the Axis of the Telescope. — ^When the 
 face is perpendicular, rays coming from the objective to 
 the prism parallel to the axis of the telescope will be 
 reflected back in their paths. In this case, since 
 the intersection of the cross-hairs is in the principal 
 focus, rays coming from the intersection will retrace their 
 paths and hence come to a focus at the intersection. The 
 superposition of the cross-hairs and their image is there- 
 fore the test for perpendicularity. 
 
 Eccentricity.' — If the rim that carries the verniers does 
 not rotate about the center of the graduated disc, the 
 zero line on the vernier will, for the same degree of rota- 
 tion, move over an unequal number of divisions on the 
 disc for different portions of the circumference and hence 
 igive rise to an error in angular measurements. This 
 
LIGHT 
 
 203 
 
 error is partly eliminated by measuring the angle with 
 each of the two attached verniers placed 180° apart and 
 taking the average. 
 
 EXPERIMENT 119 
 
 ANGLE OF INCIDENCE IS EQUAL TO THE ANGLE OF 
 REFLECTION 
 
 Apparatus. — A spectrometer and a glass prism. 
 
 Method and Manipulation. — Adjust the spectrometer 
 according to the directions given in Exp. 118. Place the 
 prism as indicated in Fig. 119. Move the telescope into 
 
 S 
 
 Fig. 119. 
 
 the position N and adjust the face of the prism so 
 that it is perpendicular to the axis of the telescope. Ob- 
 tain the reading N for this position. Then turn the tele- 
 scope into the position R so that the intersection of 
 the cross-hairs coincides with the center of the reflected 
 slit. Obtain the reading R for this position. Then 
 remove the prism, and turn the telescope into line with 
 the coUimator, and obtain the reading S for this position. 
 
 i = 180-iS-N), 
 
 and 
 
 r = R-N, 
 
 then if 
 
 i = r, 
 R-N = 180-{S-N). 
 
204 
 
 LIGHT 
 
 EXPERIMENT 120 
 
 ANGLE OF A PRISM 
 
 Apparatus. — A spectrometer and a prism. 
 Method and Manipulation. — -Adjust the spectrometer. 
 Since the angle between normals to two faces of the prism 
 has to be measured, it is necessary that the prism be 
 perpendicular so that the cross-hairs and their reflected 
 image will coincide for each face. Adjust one face 
 so that it is perpendicular, then bring the telescope 
 .c to the other and adjust so 
 
 that the cross-hairs and 
 their image coincide. It is 
 here essential not to disturb 
 the perpendicularity of the 
 face previously adjusted. 
 This is avoided by turning 
 the proper screw. The 
 prism is placed on the plate 
 that supports it in such a 
 way that each face is perpendicular to a line through two 
 of the supporting screws. (See Fig. 120.) If the face de 
 has been made perpendicular, the face dj may be adjusted 
 without disturbing de by turning the screw a as it will 
 be moved only in its own plane. After the second face 
 has been adjusted, turn the telescope back to the first, 
 and determine if its adjustment has been disturbed. If 
 so, correct, and then again test the second. 
 
 When the prism has been made perpendicular super- 
 pose the cross-hairs and their image and record the 
 reading for the normal R. Then turn the telescope 
 to the other face and record the reading for the normal 
 N. The angle of the prism is 
 
 A = 180-(i?-iV). 
 
 Fig. 120. 
 
LIGHT 205 
 
 A second method for obtaining tlie angle of the prism 
 is to place the collimator at M, Fig. 120, and measure 
 the angle between the reflected rays g and h. This 
 angle is twice the angle of the prism. 
 
 EXPERIMENT 121 
 REFRACTIVE INDEX OF GLASS FOR SODIUM LIGHT 
 
 Apparatus. — A spectrometer and the glass prism 
 whose angle was determined in Exp. 120, a Bunsen burner 
 and a bead of NaCl2 on a wire. 
 
 Method and Manipulation. — Adjust the spectrometer 
 as before, and then adjust the prism so that its reflecting 
 faces are perpendicular to the disc of the spectrometer. 
 To measure the angle of 
 minimum deviation, turn the 
 prism into the position indi- 
 cated in Fig. 121 and allow 
 sodium light to pass from -p .^i ■ 
 
 the collimator through the 
 
 prism and into the telescope on the other side. Turn 
 the prism and telescope until the refracted slit is seen. 
 It will be found on turning the prism in one direction 
 that the image of the slit moves toward the axis of the 
 collimator produced, stops, and then returns. The 
 turning point, therefore, marks the direction of the 
 beam when it is deviated the least and the angle between 
 this and the original direction is the angle of minimum 
 deviation to be determind. Place the intersection of 
 the cross-hairs on the image of the slit when the latter 
 is at the turning point. Record the reading B for this 
 position. Then remove the prism, and turn, the tele- 
 scope into line with the coUimator and record the read- 
 ing S^. 
 
206 
 
 LIGHT 
 
 The angle of minimum deviation D = S—B, and the 
 refractive index 
 
 sin|(A+Z)) 
 ^ sin §A ' 
 
 where A is the angle of the prism. 
 
 EXPERIMENT 122 
 
 WAVE LENGTH OF LIGHT— FIRST DIFFRACTION 
 GRATING METHOD 
 
 Apparatus. — A diffraction grating; a scale with a slit 
 (a) Fig. 122; a Bunsen flame; a bead of NaCb for the 
 
 Fig. 122. 
 
 production of the sodium light; Li CI; etc. The apparatus 
 is arranged as shown in Fig. 122. 
 
 Method. — The diffracted images of the slit are ob- 
 served through the diffraction grating G placed at a 
 distance of about a meter. Determine the distance from 
 the image of the highest order on one side to the image 
 of corresponding order on the other side. One half of 
 this distance is S the distance to the slit. If the grating 
 used is large, care must be taken to have the eye near 
 the grating and at a point directly above the scale M. 
 
LIGHT 
 
 207 
 
 The rays coming from the sht can be assumed to be 
 nearly parallel, and, if the first diffracted image is ob- 
 served, 
 
 l = dsmQ, (Fig. 123), 
 
 but 
 
 sm = 
 
 R' 
 
 1 = 4. 
 
 Fig. 123. 
 
 In case the nth image is observed, 
 nk = d sin 6, 
 
 and 
 
 
 Find the wave lengths for the sodium and the lithium 
 lights, using various values of L. 
 
208 
 
 LIGHT 
 
 EXPERIMENT 123 
 
 WAVE LENGTH OF LIGHT— SECOND DIFFRACTION 
 GRATING METHOD 
 
 Apparatus. — A spectrometer and a diffraction grating. 
 Method and Manipulation. — Adjust the spectrometer. 
 Then place the grating perpendicular to the collimator. 
 This may be done in the following manner: Turn the 
 telescope into line with the collimator; and place the 
 cross-hairs on the image of the slit; place the grating 
 on the stand between the telescope and the collimator, 
 
 and adjust -by rotating and 
 tilting the grating until it is 
 perpendicular to the axis of 
 the telescope. Then as the 
 telescope and the collimator 
 are in line, the grating is 
 also perpendicular to the col- 
 limator. Allow sodium light 
 to pass through the slit. Turn the telescope to one side 
 until the first diffracted image of the slit enters the field 
 of the telescope. Place the cross-hairs on the image, 
 and record the reading N. Turn the telescope to the 
 first diffracted image on the other side, and obtain the 
 reading M. 
 
 If n=the number of lines per millimeter on the 
 grating, 
 6 = the angle of diffraction, 
 X = the wave length. 
 d = the distance between two adjacent lines, 
 
 a*i 
 
 Fig. 124. 
 
 then 
 
 N-M 
 
 and 
 
 X = d sin 6. 
 
LIGHT 
 
 209 
 
 EXPERIMENT 124 
 
 POLARIZED LIGHT 
 
 Apparatus. — A polariscope; a uniaxial crystal cut 
 parallel to its optic axis (selenite or mica); a uniaxial 
 crystal cut perpendicular to its optic axis (Iceland spar) ; 
 a biaxial crystal cut perpendicular to a line bisecting the 
 angle between the axes (niter crystal); a quartz crystal 
 cut perpendicularly to its optic axis. 
 . The polariscope consists of a polarizer made of a num- 
 
 FiG. 125. 
 
 ber of glass plates and placed in a horizontal position. 
 At one end of the polarizer is a vertical screen of ground 
 glass for diffusing the light passed through it to the 
 polarizer. From the polarizer, the light passes to a 
 large lens serving as a condenser near the focus of which 
 is placed a smaller lens which renders the rays parallel. 
 The light then enters the analyzer consisting of a Nicol's 
 prism. 
 
 Manipulation. — Assume the light waves coming from 
 the polarizer to be vibrating in a plane perpendicular 
 to the plane containing the incident aid reflected rays. 
 
 The principal section of the Nicol's prism coincides 
 with the plane of vibration of the light that it transmits. 
 
210 LIGHT 
 
 Determine its direction with reference to the angles in 
 the end view of the prism. 
 Uniaxial Crystal Cut Parallel to the Optic Axis. 
 
 1. Turn the analyzer until its principal section is 
 horizontal and place a one-half wave plate of selenite 
 or mica crystal in the parallel rays, i.e., position A 
 Fig. 125. Since the crystal is cut parallel to the optic 
 axis it has two planes of vibration that transmit polarized 
 white light. Rotate the crystal, and determine their 
 direction with reference to the frame of the crystal. 
 
 2. Use sodium light instead of white, and rotate the 
 crystal until its planes of vibration make an angle of 
 45° with the vertical. Rotate the analyzer, and observe 
 the positions for which light and darkness are obtained. 
 Explain 
 
 3. Replace the white light and rotate the crystal 
 until its planes of vibration make an angle of 45° with 
 the vertical. Rotate the analyzer, and observe if two 
 colors are obtained and if they are complementary. 
 Explain. 
 
 4. Rotate the crystal, and observe if more than one 
 color is obtained. Explain. 
 
 5. Remove the one-half wave plate, and insert in 
 its place a quarter wave plate. Rotate the analyzer and 
 observe if any change takes place when sodium light is 
 used. Explain. 
 
 6. Repeat using white light. Explain. 
 
 Uniaxial Crystals Cut Perpendicularly to the Optic 
 Axis. 
 
 1. Place the principal section of the analyzer vertical 
 and insert the crystal of Iceland spar in the converging 
 rays of white light, i.e., position B. Observe the black 
 cross. Explain. 
 
 2. Between the arms of the cross observe the colored 
 rings. Explain. 
 
LIGHT 211 
 
 3. Rotate the crystal and observe if any change 
 takes place. Explain. 
 
 4. Rotate the analyzes 90°. Observe the white 
 cross and at any point between the arms observe that 
 the color is complementary to that seen in position 2. 
 Explain. 
 
 Biaxial Crystal Cut Perpendicularly to the Bisector 
 of the Angle Between the Optic Axes. 
 
 1. Place the principal section of the analyzer ver- 
 tical. Place the biaxial (niter) crystal in position B. 
 Rotate the crystal until a line joining the centers of the 
 two observed circles makes an angle of 45° with the 
 vertical. Observe the hyperbola and the colored bands 
 between its arms. Explain. 
 
 2. Rotate the analyzer 90°. Observe the white hy- 
 perbola and at intervening points note that the color is 
 complementary to that observed in position 2. Explain. 
 
 '3. Rotate the crystal 45° and observe the white 
 cross. Then rotate the analyzer 90° and observe the 
 black cross. Explain. 
 
 Quartz Crystal Cut Perpendicularly to the Optic 
 Axis. 
 
 1. Rotate the analyzer until its principal section is 
 horizontal. Place the quartz crystal in position A and 
 use a sodium flame as the source of light. Rotate the 
 analyzer until the light becomes a maximum. Observe 
 the angle of rotation. 
 
 2. Replace the sodium flame by lights of different 
 colors, and measure in each case the angle of rotation. 
 
 3. Use white light, and observe the different colors 
 obtained by rotating the analyzer. Explain. 
 
 4. Place the crystal at B, and observe the rings. 
 Explain. 
 
APPENDIX 
 
 THE THEORY OF ERRORS 
 
 The theory of errors is based upon a consideration 
 of probability and therefore takes into account only 
 the accidental errors. 
 
 Probability, expressed in mathematical terms, is a 
 common fraction of which the numerator is the number 
 of ways in which a predicted event may occur, and the 
 denominator, the total number of ways in which it can 
 occur, e.g., If a rifleman can hit the center of a target 
 five times out of twelve shots the probability that he 
 will hit the center in any future shot is five in iJwelve, 
 or T^^, and of missing, t's. In general, if an event can 
 happen in a ways and fail in b ways, the probability of 
 
 the event happening = -^7^ and of failing = -^-t. One 
 
 is certainty and zero is impossibility. 
 
 In' dealing with errors it becomes necessary to for- 
 mulate some law whereby it becomes possible to judge 
 as to the probable accuracy of an observation. Such a 
 law is derived in the Theory of Least Squares, from the 
 following three axioms arising from experience: 
 
 I. Positive and negative errors are equally probable. 
 II. The smaller the error, the greater the probability 
 
 of its occurrence. 
 III. Very large or very small errors cannot happen at all. 
 
 213 
 
214 
 
 APPENDIX 
 
 This law is expressed by the equation, 
 
 y = ke' 
 
 -h'xf 
 
 where x represents the magnitude of the error and y the 
 probability of its occurrence, h and k are constants. 
 This equation is represented by the following curve : 
 
 
 
 
 
 
 
 
 
 Y 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 ^ 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 \ 
 
 \s 
 
 
 
 
 
 
 
 
 
 
 y 
 
 
 
 
 
 
 1 " 
 
 \ 
 
 
 
 
 
 
 
 ^ 
 
 ^' 
 
 
 
 
 
 
 
 1 
 
 1 
 1,. 
 
 
 r\ 
 
 
 
 
 A comparison of the curve and equation with the 
 axioms shows complete conformity; the curve is sym- 
 metrical with respect to the Y axis, showing equal proba- 
 bility of positive and negative errors; for small values of 
 X, y is 'large showing agreement with the second axiom; 
 and for large values of x, y is practically zero, showing 
 that large errors, positive or negative, do not occur. 
 
 The Arithmetical Mean. — If a certain quantity is 
 measured a number of times under the same conditions, 
 the readings will differ among themselves due to acci- 
 dental errors. By means of the axioms upon which the 
 probability equation is based. Least Squares shows that 
 the most probable value of the quantity measured is 
 the arithmetical mean of the readings taken. 
 
 Appljdng the equation of probability given above to 
 the arithmetical mean gives for the probable error in 
 the mean: 
 
 \n(n—l) 
 
APPENDIX 215 
 
 where n is the number of readings and Sw^, the sum o'f 
 the squares of the differences between the arithmetical 
 mean and the readings. Table 15 on page 226 gives the 
 values of K for various values of n. The labor involved 
 in squaring the differences may be avoided by using the 
 following more approximate form: 
 
 E= ±0.8453 ^.^ifL = Ki 2(+y) 
 nVn— 1 
 
 where Sw is the arithmetical sum of the differences. 
 
 Table 16 on page 226 gives the values of Ki for various 
 values of n. 
 
 Suppose the following to be the results of a repeated 
 measurement : 
 
 2.53' .008 
 
 2.55 .012 
 
 2.54 .002 
 
 2.54 .002 
 
 2.53 .008 
 
 The mean = 2.538, and 2y=.032 
 
 From the table K\ for n = 5 is .0845. Therefore, 
 £ = ±.0845X. 032 = . 0027. The result is written 
 2.5380±27, meaning that the probabilty is that the true 
 result may be the average +.0027 or -.0027. The 
 probable numerical error in this case is .0027 and the 
 
 , ,, • . • -0027 27 
 
 probable error m present is 2~5S8^ 2'i'^S0 ~ P^"^ cqtA. 
 
 Error in a Single Reading. — It frequently happens that 
 only one reading for a quantity is obtained. In this 
 case the possible error may be used which may be taken 
 equal to one of the estimated parts; e.g., if in the reading 
 2.56 the last figure was obtained by estimating, the 
 
216 APPENDIX 
 
 reading is liable to contain an error as large as 0.01. This 
 fact may be expressed by writing the result 2.56 ±1 
 meaning that the true value probably lies between 
 2.56+.01 and 2.56-. 01. 
 
 Propagation of Error. — When an unknown quantity 
 is to be determined from a set of observations by means 
 of a computation, the question presents itself: How do 
 the observational errors propagate themselves through 
 the computation and what will be the probable error in 
 the final result? The answer is contained in the follow- 
 ing general equation for the propagation of error: 
 
 >/©''"+©''''+-. 
 
 R = ± 
 
 where R is the probable error' of the result. 
 
 Z is any function of the observed quantities, i.e., 
 
 ri, /•2, etc., are the probable errors of zi, Z2, . . . , 
 r:— , -r— , . . . , are the partial derivatives of the functions 
 
 OZl 0Z2 
 
 with respect to zi, Z2, ■ ■ ■ 
 
 The application of this equation in the cases which 
 in general, arise, may be illustrated as follows: 
 
 Addition and subtraction: 
 
 If Z = Zl±Z2±Z3±. . . . 
 
 .-. R = ±Vri^+r2^+r3^+. . . 
 
 The probable error in a sum or difference is therefore the 
 square root of the sum of the squares of all the individual 
 errors. 
 
APPENDIX 217 
 
 The calculation of R is facilitated by the use of the 
 table of squares on page 239. 
 Multiplication: 
 
 In this case Z=KziZ2Z3 . . ., 
 
 IZ „ 
 
 ^ = AZ223 • • •, 
 CZl 
 
 IZ „ 
 
 ■^— = KZiZz . . ., 
 
 002 
 
 :^ = KziZ2 . . ■ etc., 
 
 003 
 .-. S= ±ZV(2223 . . .)W+{ZlZ3 . . .)W + (3l22...)W+ • .. 
 
 To compute the probable limit of error R by means 
 of this equation becomes quite laborious, especially where 
 Z is the product of several observations. 
 
 Instead of the probable error the possible error may 
 
 be used which is the sum of the per cents of error in each 
 
 of the observations that enter into the product; e.g., 
 
 if zi±ri, Z2±r2, Zsi^s, are the observations with their 
 
 individual numerical errors ri, r2, and rz, then since the 
 
 , J, ri r2 , n 
 
 per cents of error are — , — , and — . 
 
 2l 22 23 
 
 '^— K)K)('-S). 
 
 :. Z = 2i2223 U± (-+-+-) 
 ( \2i 22 23/ 
 
 if the small quantities resulting from the products of 
 the per cents are neglected. 
 
 In a product therefore the possible arror in per cent may 
 be taken equal to the sum of the per cent errors in each of 
 the observations. 
 
218 APPENDIX 
 
 Division: 
 
 Suppose in this case Z = K—. 
 
 dZl Z2 
 
 S22 22^' 
 
 
 ii;=±-x/ri2+?lr22 
 
 i_2 
 
 22^ 
 
 In this case as in multipHcation the possible error may 
 be used. Thus: 
 
 -77^ = 11^^(2+2) i- See table 37, page 244. 
 
 7n a quotient therefore the possible error in per cent may 
 be taken equal to the sum of the per cent errors in numerator 
 and denominator. 
 
 The Case of a Quantity Raised to a Power n: 
 
 Suppose Z = Kz", 
 
 Iz ^""^ ' 
 :. R = ± VKSn^gf^-DV, 
 
 = ±Zn-. 
 z 
 
 In this case therefore the probable error in per cent is the 
 per cent of ert-or in the observation multiplied by the exponent. 
 
APPENDIX 
 
 219 
 
 1. LENGTH 
 
 1 centimeter 
 1 meter 
 1 kilometer 
 Imil 
 
 .39370 inch 
 3.2809 feet 
 0.62137 mile 
 
 .001 in. 
 
 1 inch 
 1 foot 
 1 m le 
 
 1 micron = 
 1 Angstrccm 
 unit = 10 
 
 = 2.54 cm. 
 = 30.48 cm. 
 = 1.G09 km. 
 = .001 mm. 
 
 cm. 
 
 2. AREA 
 
 1 sq.cm. = 0.155 sq.in. 
 
 1sq.m. . =10.764 sq.ft. 
 
 1 circular cm. = 0.1217 sq.in. 
 
 .7S54d! 
 
 area of circle = (— rf') 
 
 1 circular mill = 0.000001 sq.in. 
 
 1 sq.in. = 6.4510 sq.cm. 
 
 1 sq.ft. =929.03 sq.cm. 
 
 1 sq. mile = 2.59 sq.km. 
 1 circular in. = 5.0671 Kq.cm. 
 1 circular mil = 0.000.50671 
 sq.mm. 
 
 
 3. VOLUME 
 
 
 1 cu.cm. 
 1 cu.m. 
 1 liter 
 1 liter 
 
 = 0.061 cu.in. 
 = 35.315 cu.ft. 
 = 61.023 cu.in. 
 = 2.11336 pints 
 (hquid) 
 
 1 cu.in. 
 
 1 cu.ft. 
 
 1 cu.foot 
 
 1 pint (hquid) 
 
 = 16. .387 cu.cm. 
 = 0.028317 cu.m. 
 = 28.317 liter,s 
 = 0.47318 liter 
 
 
 4. MASS 
 
 
 1 gram 
 1 kilogram 
 
 = 15.432 grains 
 = 2.205 pounds 
 
 1 ounce (av.) 
 1 pound (av.) 
 
 = 28.35 grams 
 = 453.59 grams 
 
 
 5. CIRCULAR MEASURE 
 
 
 1 radian 
 
 ■K 
 
 = 57.296 degrees 
 = 3.1416 
 
 1 degree = 
 Logx 
 
 = 0.017453 radian 
 = .49714987 
 
 1 
 
 = 0.31831 
 
 
 
 
 = 9.8696 
 = 1.77245 
 
 
 
 
 6. FORCE 
 
 
 1 dyne 
 
 1 dyne 
 1 djme 
 
 1 megadyne 
 
 = 0.00007233 
 
 poundal 
 = 0.00102 gi-am 
 = 22.48X10-' 
 
 pounds 
 = 1000000 dynes 
 
 1 poundal = 
 1 poundal = 
 1 poundal = 
 
 = 13825 dynes 
 = 0.03108 pound 
 = 14.10 gi'ams 
 
22G 
 
 APPENDIX 
 
 
 7. PRESSURE 
 
 
 1 atmosphere 
 
 
 1 atmosphere 
 
 
 (standard) 
 
 = 76.0 cm. 
 
 (76 cm.) 
 
 = 29.921 in. 
 
 1 atmosphere 
 
 = 1.0333 kg. per 
 
 1 atmosphere 
 
 = 14.697 lb. per 
 
 
 sq.cm. 
 
 
 sq.in. 
 
 1 atmosphere 
 
 = 1.0133 mega/- 
 
 1 atmosphere 
 
 = 33.90 ft. of 
 
 
 dynes per sq. 
 
 
 water 
 
 
 cm. 
 
 1 poundal per 
 
 
 1 dyne 
 
 
 sq.in 
 
 =2142.95 dynes 
 
 per sq. cm 
 
 =0.0004666 
 
 
 per sq.cm. 
 
 
 poundal per 
 
 1 pound per 
 
 
 
 sq.in. 
 
 sq.in. 
 
 = 70.31 grams 
 
 1 gram wt. per 
 
 
 
 per sq.cm. 
 
 sq.cm. 
 
 =0.014223 lb. per 
 
 1 in. of mer- 
 
 
 
 sq.in. 
 
 cury (0° C.) 
 
 = 34.533 g.wt. per 
 
 1 cm. of mer- 
 
 
 
 sq.cm. 
 
 cury (0° C.) 
 
 = 13.596 g.wt. per 
 
 1 in. of mer- 
 
 
 
 sq.cm. 
 
 cury (0° C.) 
 
 =0.4912 lb. wt. 
 
 1 cm. of mer- 
 
 
 
 per sq.in. 
 
 cm-y (0° C.) 
 
 = 0.19338 lb. wt. 
 per sq.in. 
 
 
 
 1 megabarie 
 
 = 1 megadyne 
 per sq.cm. 
 
 
 
 1 megabarie 
 
 = 75.0068 cm. 
 of Hg. 
 
 
 
 
 8. EN 
 
 ERGY 
 
 
 1 erg = 2.373 X 10 - 6 ft.-poundal 
 
 1 ft.-poundal 
 
 =421,390 ergs 
 
 1 erg =7.376X10-' ft.-pound 
 
 1 ft.-pound 
 
 = 1.35573 joules 
 
 1 g.cm.=7.233X10-'ft.-pomids 
 
 1 ft.-pound 
 
 = 13825.5 g.cm. 
 
 1 joule = 10' ergs 
 
 
 
 
 9. P( 
 
 )WER 
 
 
 1 watt 
 
 = 10' ergs per sec. 
 
 1 ft.-poundal 
 
 
 1 watt 
 
 = 23.731 ft.-poun- 
 
 per sec. 
 
 = 421390 ergs 
 
 
 dals per sec. 
 
 
 per sec. 
 
 1 watt 
 
 =0.7376 ft.-lb. 
 
 1 ft.-pound per 
 
 
 
 per sec. 
 
 sec. 
 
 = 1.35573watt3 
 
 1 watt 
 
 =0.001341 horse- 
 
 1 horse-power sec. =745.65 watts 
 
 
 power-sec. 
 
 1 horse-power sec. =550 ft.- 
 
 1 kilowatt-hr. 
 
 = 2,655,403 ft.- 
 
 
 pounds. 
 
 
 pounds. 
 
 1 horse-power sec. = 178.122 
 
 1 kilowatt-hr. 
 
 = 1,3411 horse- 
 power-hour 
 
 
 calories 
 
 1 kilowatt-hr. 
 
 = 859,975 calories 
 
 
 
APPENDIX 
 
 221 
 
 10. HEAT 
 
 1 calorie (g.c.) = . 0039683 B.T.U. 
 1 calorie " =4.1862 joules 
 1 calorie " =3.088 ft.-lbs. 
 1 calorie " =0.005614 horse- 
 power-sec. 
 
 1 B.T.U. =252.00 calories 
 1 B.T.U. = 1055 joules 
 1 B.T.U. =778.1 ft.-lbs. 
 1 B.T.U. = 1.4147 horse-power- 
 sec. 
 
 11. TIME 
 
 1 sidereal sec. — .99727 sec. (mean solar). 
 
 1 sec. (mean solar) = 1.002738 sidereal sec. 
 
 Length of seconds pendulum latitude 45° = 99.3555 cm. =39.1163 in. 
 
 logic N 
 e 
 
 12. LOGARITHM 
 
 = .43429 logeiV 
 = 2.7183 
 
 loge AT =2.3026 logioAT. 
 
 13. DATA WITH REGARD TO EARTH 
 
 Force of gravity (Minneapolis (j) = 
 
 44° 58'.7, X=93° 13'.9, /t = 256 meters) =980.596 dynes 
 
 Gravitation constant =666.07X10~i<' 
 
 Mean density of the earth =5.52470 
 
 Half diameter of earth (equatorial) =6378.2 km. 
 
 Half diameter of earth (polar) =6356.5 km. 
 
 Half diameter of earth (average) =6367.4 km. 
 
 Half diameter of earth (average) =3956.5 miles 
 
 Mean distance from earth to sun =9.29X10' miles 
 
 Me^n distance from earth to moon =2.39X10^ miles 
 
 1° latitude at 40° latitude =69.0 miles 
 
 Velocity of hght per sec. 
 
 14. VELOCITY 
 
 = 186330 miles 
 
 Velocity of sound per sec. (air 0° C.) = 1089 ft. 
 Velocity of sould per sec. (brass) =11480 ft. 
 Velocity of sound per sec. (cast steel) = 16360 ft. 
 Velocity of sound per sec. (water) =4728 ft. 
 
 = 299870 kilo- 
 meters 
 = 331.92 meters 
 = 3500 meters 
 =4990 meters 
 = 1441 meters 
 
222 
 
 APPENDIX 
 
 ■Qo inioj Saijioa 
 
 208 
 1800 
 
 1440 
 
 -186. 
 
 1430 
 
 61. 
 770 
 
 3600 
 
 - 33.6 
 
 2200 
 
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 ■3 S a C H 3 3 
 
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APPENDIX 
 
 223 
 
 
 
 
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 O O O O ^ O ~ f.'O :S O O S S « « ^ » "i 53 .Si .S .15 
 
224 
 
 APPENDIX 
 
 'Oo »nTOd; Sajiioa 
 
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 t^ ■ O iO 
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APPENDIX 
 
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 M CB H H H H H H t" p ;> tsj 
 
226 
 
 APPENDIX 
 
 o 
 
 H 
 
 o 
 
 a 
 
 H 
 O 
 
 I 
 
 P 
 
 
 
 
 lO o t~ 
 
 
 
 (33 CO CO 
 
 
 OJ 
 
 1> CO IN 
 
 
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 W Tl( 
 
 
 O 
 
 
 d 
 
 ;: 
 
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 05 
 
 0.0332 
 .0105 
 .0055 
 
 00 
 
 0.0399 
 .0114 
 .0058 
 
 I> 
 
 0.0493 
 .0124 
 .0061 
 
 CD 
 
 0.0630 
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 e 
 
 > 
 
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 0.0845 
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 .01220 
 .0167 
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 0.1993 
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APPENDIX 
 
 227 
 
 18. ALLOYS 
 
 Den- 
 sity. 
 
 Speci- 
 fic 
 I-Ieat. 
 
 CoefEcienb 
 
 of 
 
 Thermal 
 
 Expansion. 
 
 Specific 
 Resist- 
 ance. 
 Microms 
 per c.c 
 
 Temp. 
 Coefficient 
 
 % 
 
 Brass(70Cu.+30Zn)., 
 
 German silver 
 
 Platinum and Iridium. 
 
 Constantine 
 
 Manganin 
 
 lala (hard) 
 
 lala (soft) 
 
 8.44 
 
 S.3 
 
 21.62 
 
 8.97j 
 8.6 
 
 ,0883 
 ,0946 
 
 .1859X10-^ 
 ,1836 
 0884 
 
 8.0 
 31.5 
 ■29.90' 
 48 . 30 
 39.14 
 50.2 
 47.1 
 
 .40 
 .025 
 .087 
 .0014 
 .0017 
 - .0011 
 -f 0.0005 
 
 19. GASES 
 
 Gas. 
 
 Air 
 
 Ammonia 
 
 Carbon dioxide. . . 
 
 Carbon monoxide 
 
 Chlorine 
 
 _, \ from . . 
 
 Coal gas > ^ 
 
 J to 
 
 Hydrogen 
 
 Nitrogen 
 
 Oxygen 
 
 Steam at 100°. . . . 
 
 tn >> 
 
 
 
 
 Grams 
 
 Pounds 
 
 coo 
 
 per 
 Liter. 
 
 Cu.ft. 
 
 *71 
 
 71 
 
 71 
 
 1.000 
 
 1.2928 
 
 .08071 
 
 0.597 
 
 0.7621 
 
 .04758 
 
 1.5291 
 
 1.9652 
 
 . 12269 
 
 0.9672 
 
 1.2506 
 
 .07807 
 
 2.491 
 
 3.1666 
 
 . 19769 
 
 0.320 
 
 414 
 
 .02583 
 
 0.740 
 
 0.957 
 
 .05973 
 
 0.0696 
 
 0.09004 
 
 .005621 
 
 0.9673 
 
 1.2542 
 
 .07829 
 
 1.0530 
 
 1.4292 
 
 .08922 
 
 0.4690 
 
 0.5810 
 
 .0363 
 
 Specific Heat 
 
 at Constant 
 
 Pressure. 
 
 232 
 (0-100°) 
 (23-100) 
 (15-100) 
 (23-99) 
 (13-202) 
 
 0.2374 
 0.5202 
 .2025 
 0.2425 
 0.1241 
 
 (21-100) 
 ( 0-200) 
 (13-207) 
 100° 
 
 3.410 
 0.2438 
 0.2175 
 0.421 
 
 Exp. CoefF. 76. 
 
 Con- 
 stant 
 Vol- 
 ume. 
 
 xio- 
 
 Con- 
 stant 
 Pres. 
 XlO-2 
 
 223 
 .36650 .3671 
 
 .36856 
 .36667 
 
 .36504 
 . 36682 
 .36681 
 
 .3710 
 .3669 
 
 .4187 
 
 * No. of Smithsonian Table, 
 
228 
 
 APPENDIX 
 
 xn 
 
 Pi 
 
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 d d d d 
 
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 0"-l(NCO-*>OtOt~00050-H(MCO-*10CDI>0005 
 (NIM(N(M(N(NIM<N(N(NMMCOCOCOCOeoroci3CO 
 
 1 
 
 P 
 
 mii>coi-i ec t^ (N 05 1> t^ 00 o CO t~ M o 00 1> 
 >-ioooO'-ioo>-i-H(Nc<3-*coi^ooo<Neo'ra 
 
 OOOOOOOOOOOOOOOO'-l.-HiMr-i 
 1-H 1-H i-H t-( r-i 
 
 
 t>rar^05 05 1~ ro 00 1-1 CO M (M o l> CO b- o (N CO 
 
 000505a5OO5 05O50000t~CDl2Tt<(Ni-(0500lC->l< 
 05050505005050505050505050505 05 00 00 OO OO 
 
 05 O 05 05 05 
 
 d ^'d d d 
 
 d 
 
 Oi-tlMCO'^lC^t^00050i-l(NCO'«#iOCOt^0005 
 
APPENDIX 
 
 229 
 
 21. BOILING POINT OF WATER UNDER DIFFERENT 
 BAROMETRIC PRESSURES 
 
 Bar. 
 Pres- 
 
 .0 
 
 
 I 
 
 .2 
 
 .3 
 
 .4 
 
 .5 
 
 *? 
 
 .7 
 
 .8 
 
 .9 
 
 sure. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 72. 
 
 98 
 
 50 
 
 98 
 
 34 
 
 98 
 
 57 
 
 98 
 
 61 
 
 98 
 
 65 
 
 98 
 
 69 
 
 98 
 
 73 
 
 98 
 
 77 
 
 98 
 
 80 
 
 98 
 
 84 
 
 73. 
 
 98 
 
 88 
 
 98 
 
 92 
 
 98 
 
 96 
 
 98 
 
 99 
 
 99 
 
 03 
 
 99 
 
 07 
 
 99 
 
 11 
 
 99 
 
 14 
 
 99 
 
 18 
 
 99 
 
 22 
 
 74. 
 
 99 
 
 26 
 
 99 
 
 30 
 
 99 
 
 33 
 
 99 
 
 37 
 
 99 
 
 41 
 
 99 
 
 44 
 
 99 
 
 48 
 
 99 
 
 62 
 
 99 
 
 56 
 
 99 
 
 59 
 
 75. 
 
 99 
 
 63 
 
 99 
 
 67 
 
 99 
 
 71 
 
 99 
 
 74 
 
 99 
 
 78 
 
 99 
 
 82 
 
 99 
 
 85 
 
 99 
 
 89 
 
 99 
 
 93 
 
 99 
 
 96 
 
 76. 
 
 100 
 
 00 
 
 100 
 
 04 
 
 100 
 
 07 
 
 100 
 
 11 
 
 100 
 
 15 
 
 100 
 
 18 
 
 100 
 
 22 
 
 100 
 
 26 
 
 100 
 
 29 
 
 100 
 
 33 
 
 77. 
 
 100 
 
 36 
 
 100 
 
 40 
 
 100 
 
 44 
 
 100 
 
 47 
 
 100 
 
 51 
 
 100 
 
 55 
 
 100 
 
 58 
 
 100 
 
 62 
 
 100 
 
 65 
 
 100 
 
 69 
 
 22. BAROMETER CORRECTION AT DIFFERENT TEMPER- 
 ATURES, THE BAROMETER SCALE BEING CORRECT 
 AT 62° F. (BRASS SCALE) 
 
 The correction is to be subtracted from the observed height. 
 
 '^I.'J'P- 
 
 Pressure in Inches. 
 
 Temp. 
 
 Pressu 
 
 re in Inches. 
 
 F°. 
 
 
 
 
 
 
 
 
 
 28 
 
 29 
 
 30 
 
 
 28 
 
 29 
 
 30 
 
 64 
 
 -.090 
 
 .093 
 
 .096 
 
 74 
 
 -.115 
 
 .119 
 
 .123 
 
 65 
 
 .092 
 
 .095 
 
 
 099 
 
 75 
 
 .117 
 
 .122 
 
 .126 
 
 66 
 
 .095 
 
 .096 
 
 
 101 
 
 76 
 
 .120 
 
 .124 
 
 .128 
 
 67 
 
 .097 
 
 101 
 
 
 104 
 
 77 
 
 .122 
 
 .127 
 
 .131 
 
 68 
 
 .100 
 
 .103 
 
 
 107 
 
 78 
 
 .125 
 
 .129 
 
 .134 
 
 69 
 
 .102 
 
 .106 
 
 
 110 
 
 79 
 
 .127 
 
 .132 
 
 .137 
 
 70 
 
 .105 
 
 .109 
 
 
 112 
 
 80 
 
 .130 
 
 .135 
 
 .139 
 
 71 
 
 .107 
 
 .111 
 
 
 115 
 
 81 
 
 .132 
 
 .137 
 
 .142 
 
 72 
 
 .110 
 
 .114 
 
 
 118 
 
 82 
 
 .136 
 
 .140 
 
 .145 
 
 73 
 
 .112 
 
 .116 
 
 
 120 
 
 83 
 
 .137 
 
 .142 
 
 .147 
 
230 
 
 APPENDIX 
 
 22a. BAROMETER CORRECTION. BRASS SCALE CORRECT 
 AT 0°C, METRIC MEASURE 
 
 The correction is to be subtracted from the observed height. 
 
 Height in 
 
 a 
 in mm. for 
 
 Height in 
 
 a 
 in mm. for 
 
 Height in 
 
 a 
 in mm. for 
 
 mm. 
 
 each °C. 
 
 mm. 
 
 each -C. 
 
 mm. 
 
 each °C. 
 
 500 
 
 0.0813 
 
 600 
 
 0.0975 
 
 700 
 
 0.1137 
 
 510 
 
 .0830 
 
 610 
 
 .0992 
 
 710 
 
 .1154 
 
 520 
 
 .0846 
 
 620 
 
 .1008 
 
 720 
 
 .1170 
 
 530 
 
 .0862 
 
 630 
 
 .1024 
 
 730 
 
 .1186 
 
 540 
 
 .0878 
 
 640 
 
 .1040 
 
 740 
 
 .1202 
 
 550 
 
 .0894 
 
 650 
 
 .1056 
 
 750 
 
 .1218 
 
 560 
 
 .0911 
 
 660 
 
 .1073 
 
 760 
 
 .1235 
 
 570 
 
 ,0927 
 
 670 
 
 .1089 
 
 770 
 
 .1251 
 
 580 
 
 .0943 
 
 680 
 
 .1105 
 
 780 
 
 .1267 
 
 590 
 
 .0959 
 
 690 
 
 .1121 
 
 790 
 800 
 
 .1283 
 .1290 
 
 3. PRESSURE OF SATURATED MERCURY VAPOR IN MM. 
 OF MERCURY 
 
 t°C. 
 
 Prea- 
 sure. 
 
 ("C. 
 
 Prea- 
 aure. 
 
 f.C. 
 
 Prea- 
 aure. 
 
 f.C. 
 
 Prea- 
 sure. 
 
 r.C. 
 
 Prea- 
 siire. 
 
 
 
 0.00047 
 
 10 
 
 0.00080 
 
 20 
 
 0.00133 
 
 70 
 
 0.050 
 
 120 
 
 0.t79 
 
 2 
 
 .00052 
 
 12 
 
 .00089 
 
 30 
 
 .0029 
 
 80 
 
 .093 
 
 130 
 
 1.240 
 
 4 
 
 .00058 
 
 14 
 
 .00099 
 
 40 
 
 .0063 
 
 90 
 
 .165 
 
 140 
 
 1.930 
 
 6 
 
 .00064 
 
 16 
 
 .00109 
 
 50 
 
 .0130 
 
 100 
 
 .285 
 
 150 
 
 2.930 
 
 8 
 
 .00072 
 
 18 
 
 .00121 
 
 60 
 
 .0260 
 
 110 
 
 .478 
 
 160 
 
 4.380 
 
APPENDIX 
 
 231 
 
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 i-HC^CO'^iO<:DI>OOOiOi-l(MCO'^iO':DI:^OOCsO 
 
 
 
 l>-C0CDiO{NCOt>l>TH'-it~*(Mb-(N00cC'»OCDOt^ 
 
 T**i:DoOrHioo:)ThOt>iocococoioi>»-Hco(NOoo 
 
 OOa30(NCO-^CD0005i-:(XiiOI>0)rH^cDOi(M-^ 
 
 d 
 
 i-((MCO'^lOc01>-OOOiOT-(C<ICO'^lQ<:Ot^OOO>0 
 
 1 
 
 S S CD o-io O3=^oi'0^i>rt^i-Hooa:iio^cococo 
 
 ■<^10":ii:C'<X)':DI>I>00OiOjOi— lrHtNC0THlO'X'l> 
 
 d 
 
 o 
 
 i-l(MCO"^»OCDt^00050'-<<NCO-^>001^000iO 
 
232 
 
 APPENDIX 
 
 26. DRY AND WET BULB HYGROMETER 
 
 This table gives the vapor pressure corresponding to various 
 values of the differences t—h between the readings of the dry and 
 wet bulb thermometers and the temperature h of the wet bulb 
 thermometer. The table was calculated for a barometric pressure 
 B equal to 76 centimeters and a correction is given for each centi- 
 meter at the top of the columns. For pressures less than 76 the 
 correction is to be added. 
 
 ti°C. 
 
 t-h 
 =0 
 
 2 
 
 34 
 
 6 
 
 8 
 
 10 
 
 12 
 
 14 
 
 16 
 
 
 Correction 
 for B per cm. 
 
 .013 
 
 .026 
 
 .040 
 
 .053 
 
 .066 
 
 .079 
 
 .092 
 
 .106 
 
 IBS 1 
 
 
 mm. 
 
 
 
 
 
 
 Example:* —h= 7.2 
 <i=10.0 
 
 
 
 
 4.60 
 
 3.60 
 
 2.59 
 
 1.59 
 
 0.59 
 
 
 B=7i.5 
 TabuIarNo. =6.12- 
 
 0.100 
 
 1 
 
 4.94 
 
 3.93 
 
 2.92 
 
 1.92 
 
 0.92 
 
 
 6X.101=5.Sl 
 
 0.100 
 
 2 
 
 5.30 
 
 4.29 
 
 3.29 
 
 2.'S8 
 
 1.28 
 
 0.27 
 
 Correction for 
 B = 1.5X.048 = .07 
 
 0.100 
 
 3 ■ 
 
 5.69 
 
 4.68 
 
 3.68 
 
 2.67 
 
 1.66 
 
 0.66 
 
 Hence, p =5.58 
 
 0.101 
 
 4 
 
 6.10 
 
 5.09 
 
 4.09 
 
 3.08 
 
 2.07 
 
 1.06 
 
 0.05 
 
 
 
 0.101 
 
 6 
 
 6.63 
 
 5.52 
 
 4.51 
 
 3.50 
 
 2.49 
 
 1.48 
 
 0.48 
 
 
 
 0.101 
 
 6 
 
 7.00 
 
 5.99 
 
 4.98 
 
 3.97 
 
 2.96 
 
 1.95 
 
 0.94 
 
 
 
 O.lQl 
 
 7 
 
 7.49 
 
 6.48 
 
 5.47 
 
 4.45 
 
 3.44 
 
 2.43 
 
 1.42 
 
 0.41 
 
 
 0.101 
 
 8 
 
 8.02 
 
 7.01 
 
 5.99 
 
 4.98 
 
 3.97 
 
 2.96 
 
 1.94 
 
 0.93 
 
 
 0.101 
 
 9 
 
 8.57 
 
 7.56 
 
 6.54 
 
 5.53 
 
 4.51 
 
 3.60 
 
 2.49 
 
 1.48 
 
 0.46 
 
 0.101 
 
 10 
 
 9.17 
 
 8.16 
 
 7.14 
 
 6.12 
 
 5.11 
 
 4.09 
 
 3.08 
 
 2.07 
 
 1.06 
 
 0.101 
 
 11 
 
 9.79 
 
 8.77 
 
 7.76 
 
 6.74 
 
 5.73 
 
 4.71 
 
 3.69 
 
 2.68 
 
 1.66 
 
 0.102 
 
 12 
 
 10.46 
 
 9.44 
 
 8.43 
 
 7.41 
 
 6.39 
 
 5.37 
 
 4.36 
 
 3.34 
 
 2.32 
 
 0.102 
 
 13 
 
 11.16 
 
 10.14 
 
 9.12 
 
 8.10 
 
 7.09 
 
 6.07 
 
 5.05 
 
 4.03 
 
 3.01 
 
 0.102 
 
 14 
 
 11.91 
 
 10.89 
 
 9.87 
 
 8.85 
 
 7.83 
 
 6.81 
 
 5.79 
 
 4.77 
 
 3.71 
 
 0.102 
 
 16 
 
 12.70 
 
 11.68 
 
 10.66 
 
 9.64 
 
 8.62 
 
 7.60 
 
 6.58 
 
 5.66 
 
 4.54 
 
 0.102 
 
 16 
 
 13.54 
 
 12.52 
 
 11.50 
 
 10.47 
 
 9.45 
 
 8.43 
 
 7.41 
 
 6.39 
 
 5.37 
 
 0.102 
 
 17 
 
 14.42 
 
 13.40 
 
 12.37 
 
 11.35 
 
 10.33 
 
 9.31 
 
 8.28 
 
 7.26 
 
 6.24 
 
 0.102 
 
 18 
 
 15.36 
 
 14.34 
 
 13.31 
 
 12.29 
 
 11.26 
 
 10.24 
 
 9.21 
 
 8.19 
 
 7.17 
 
 0.102 
 
 19 
 
 16.35 
 
 15.33 
 
 14.30 
 
 13.27 
 
 12.25 
 
 11.22 
 
 10.20 
 
 9.17 
 
 8.15 
 
 0.102 
 
 20 
 
 17.39 
 
 16.37 
 
 15.34 
 
 14.31 
 
 13.28 
 
 12.26 
 
 11.23 
 
 10.21 
 
 9.18 
 
 0.103 
 
APPENDIX 233 
 
 26. WAVE-LENGTHS IN ANGSTRCEM UNITS (lo-s CM.) 
 
 Smithsonian Table No. 151, 
 
 Barium 
 
 5535.7 
 
 Caesium 
 
 4555.4 
 
 Caesium 
 
 4593.3 
 
 Calcium 
 
 5589.0 
 
 Cadmium. . . 
 
 4799.9 
 
 Cadmium. . . 
 
 5085.8 
 
 Cadmium. . . 
 
 6438.5 
 
 Helium 
 
 5876.8 
 
 Helium 
 
 5876.2 
 
 Hydrogen. . . 
 Hydrogen.. . 
 Hydrogen. . . 
 Hydrogen.. . 
 Lithium . . . . 
 Magnesium . 
 Magnesium . 
 Magnesium . 
 Mercury.. . . 
 
 4101.8 
 
 4340.7 
 
 4861.6 
 
 6563.0 
 
 6708.2 
 
 5167.6 
 
 5172.9 
 
 5183.8 
 
 5461.0 
 
 Potassium . 
 Potassium . 
 Rubidium. . 
 Sodium. . . . 
 Sodium. . . . 
 Strontium. . 
 Strontium. . 
 Strontium. . 
 Thallium. . 
 
 7668.5 
 7701.9 
 6298.7 
 5890.2 
 5896.2 
 4607.5 
 5481.2 
 6408.6 
 5350.6 
 
 27. INDEX OF REFRACTION 
 
 
 D Line. 
 
 
 D Line. 
 
 Water 
 
 1.3335 
 1.3624 
 1.6293 
 1.6280 
 1.6604 
 1.6410 
 1.7102 
 1.5442 
 1.5533 
 1.6585 
 1.4864 
 
 Bpnzol 
 
 1 5014 
 
 Alcohol 
 
 Ether, ethyl 
 
 Chloroform 
 
 Glycerine 
 
 Aluminum 
 
 Conner 
 
 1 3560 
 
 Carbon bisulphide 
 
 1 4490 
 
 Glass, hght crown (S=2.50) . 
 Glass, heavy crown (5 = 3.00) 
 Glass, hght flint (S =2.87) . . . 
 
 1.4729 
 1.44 
 641 
 
 Glass, heavy flint (S =4.22) . . 
 
 Gold 
 
 0.366 
 
 Quartz, crystal H-ord 
 
 Iron. 
 
 2 36 
 
 Quartz, crystal+ext 
 
 Silver 
 
 181 
 
 Iceland spar, ord 
 
 Antimony 
 
 Air, 0° 76 cm 
 
 3 04 
 
 Iceland spar, ext 
 
 1.000293 
 
 28. ELASTIC CONSTANTS 
 Dynes per sq.cm. 
 
 
 Young's 
 Modulus. 
 
 F.igidity. 
 
 Elastic 
 Limit. 
 
 Velocity 
 
 of Sound. 
 
 Cm. per 
 
 Sec. 
 
 
 6.5X10" 
 7.7- 9.3 
 8.5-11.6 
 6.9-10.8 
 19.3-20.9 
 20.1-21.6 
 15 
 7 
 0.7- 1.5 
 
 2.4-3.3X10" 
 3.1-3.6 
 3.2-4.S 
 2.6-4.2 
 7.7-8.5 
 8.0-8.8 
 6.1-6.5 
 2.6 
 0.07-0.12 
 
 
 5.10X10° 
 
 Braaa ... 
 
 4.5-11 XIO^ 
 
 3- 7 
 
 7 
 
 20 
 
 33-40 
 
 14-26 
 
 3-11 
 
 1.5-2.4 
 
 3.56 
 
 
 3.74 
 
 
 4.32 
 
 Iron, wrouglit 
 
 5.06 
 5.22 
 
 
 2.69 
 
 Silver 
 
 2.61 
 
 Wood 
 
 3.0 
 
 
 
234 
 
 APPENDIX 
 
 29. COMPRESSIBILITy 
 
 
 Compression Per 
 Megabarye. 
 
 Elasticity of 
 Volume. 
 
 Water 
 
 6.0X10-6 
 3.9X10-" 
 2.6XlO-» 
 
 2.03X101" 
 
 Mercury. 
 
 Glass 
 
 2.60X10" 
 3.90X10" 
 
 
 
 30. FUNDAMENTAL ELECTRICAL UNITS 
 
 Magnetic Units 
 
 Unit magnetic pole is a pole of such strength that it 
 repels a like pole of equal strength at a distance of one 
 centimeter, in air, with a force of one dyne. 
 
 Magnetic field of imit intensity is a field that acts on 
 a unit magnetic pole placed in it with a force of one dyne. 
 
 Electrostatic Units 
 
 Electrostatic unit of quantity of electricity is such a 
 quantity which acts on an equal quantity at a distance 
 of one centimeter, in air, with a force of one dyne. 
 
 Electrostatic unit of current exists when an electro- 
 static unit of quantity of electricity flows per second past 
 any point. 
 
 Electrostatic unit of -potential exists at any point when 
 it requires an erg of work to carry to it an electrostatic 
 unit of electricity from an infinite distance or from a 
 point of zero potential. 
 
 Difference of potential of one electrostatic unit exists 
 between two points when it requires an erg of work to 
 carry an electrostatic unit of quantity of electricity from 
 one point to the other. 
 
APPENDIX 235 
 
 Electromagnetic Units 
 
 Electromagnetic unit of current is that current which 
 flowing in the circumference of a circle of one centimeter 
 radius, will, for every centimeter length, exert in air a 
 force of one dyne on a unit magnetic pole placed at the 
 center. 
 
 Electromagnetic unit of quantity is that quantity of 
 electricity which passes any point in a circuit when one 
 electromagnetic unit of current is flowing. [This has 
 been found by experiment to be SXIO^*^ times greater 
 than the electrostatic unit.] 
 
 Electromagnetic unit of difference of -potential is the 
 difference of potential that exists between two points 
 when it requires an erg of work to carry an electromag- 
 netic unit of quantity of electricity from one point to the 
 other., [The unit is therefore smaller than the electro- 
 static unit by the factor 3X101°.] 
 
 Electromagnetic unit of resistance is a resistance through 
 which one electromagnetic unit of difference of potential 
 causes one electromagnetic unit of current to flow. 
 
 Practical Units 
 
 Ampere is one-tenth of an electromagnetic unit of 
 current. 
 
 Coulomb is that quantity of electricity which passes 
 any point in one second when the current is one ampere. 
 
 Volt is equal to 10® electromagnetic units of difference 
 of potential. [300 volts therefore equal one electro- 
 static unit of difference of potential.] 
 
 Ohm is a resistance through which one volt causes 
 one ampere to flow. [One ohm therefore equals 10^ 
 electromagnetic units of resistance.] 
 
236 APPENDIX 
 
 Joule is the work done by a coulomb of electricity in 
 moving through a difference of potential of one volt. 
 [It is 10'^ ergs, and the work in joule§.= ElL] 
 
 Watt is the electrical unit of power and represents 
 the work done at the rate of one joule per second. [Watts 
 =EI.] A kilowatt = 1000 watts. 
 
 Farad is such a capacity where a difference of potential 
 of one volt is produced by a coulomb of electricity. 
 [The part of charge absorbed by the dielectric is, not 
 considered as a part of the quantity.] A microfarad = 
 0.000001 farad. 
 
 Henry is such an inductance in which one volt is 
 induced when the inducing current varies at the rate of 
 one ampere per second. This applies to both self- 
 induction and mutual induction. A milhhenry = 0.001 
 henry. 
 
 Maxwell is the practical unit of magnetic flux. It is 
 a single line of force. The flux in a field is n maxweUs 
 when it contains n magnetic lines of force. 
 
 Gauss is the practical unit of the density of magnetic 
 flux (magnetic induction B). The magnetic induction 
 is one gauss when there is one maxwell (or line of force) 
 per square centimeter. 
 
 31. DEFINITIONS OF THE FUNDAMENTAL ELECTRICAL 
 UNITS BY THE LONDON CONFERENCE (1908) 
 
 These fundamental units are: 
 
 1. The Ohm, the unit of electric resistance, which has 
 the value of 1,000,000,000 (lO^) in terms of the centimeter 
 and the second; 
 
 2. The Ampere, the unit of electric current, which 
 has the value of one-tenth (0.1) in terms of the centi- 
 meter, gram, and second; " 
 
 3. The Volt, the unit of electromotive force, which 
 
APPENDIX 237 
 
 has the value of 100,000,000 (lO^) in terms of the cen- 
 timeter, gram, and second; ,i 
 
 4. The Watt, the unit of power, which has the value 
 10,000,000 (10^) in terms of the centimeter, the gram, 
 and the second. i 
 
 As a system of units representing the above and suf- 
 ficiently near for the purpose of electrical measurements, 
 and as a basis for legislation, the conference recommended 
 the adoption of the international ohm, the international 
 ampere, the international volt, and the international 
 watt, defined as follows : 
 
 1. The International Ohm is the resistance offered to an 
 unvarying electric current by a column of mercury at 
 the temperature of melting ice, 14.4521 grams in mass, 
 of a constant cross-sectional area and of a length of 
 106.300 centimeters. 
 
 2. The International Ampere is the unvarying electric 
 current which, when passed through a solution of nitrate 
 of silver in water, deposits silver at the rate of 0.00111800 
 of a gram per second. 
 
 3. The International Volt . is the electrical pressure 
 which, when steadily applied to a conductor the resistance 
 of which is one international ohm will produce a current 
 of one international ampere. 
 
 4. The International Watt is the energy expended per 
 second by an unvarying electric current of one inter- 
 national ampere under an electric pressure of one inter- 
 national volt. 
 
238 A.PPENDIX 
 
 32. THE NEW VALUE OF THE WESTON NORMAL CELL 
 
 The electromotive force of the Weston normal cell 
 derived from the international ohm and the international 
 ampere according to the resolutions of the London 
 Conference is, within one part in 10,000, 
 
 i; = 1.01830 international volts at 20° C. 
 
 The formula for the temperature coefficient of the 
 Weston normal cell adopted by the London Conference, 
 based on the investigations of the Bureau of Standards, 
 is as follows: 
 
 Et = E20 - 0.0000406 (t - 20°) - 0.00000095 (t - 20)2 
 +0.00000001 (<-20)3. 
 
 33. RELATION BETWEEN ELECTRICAL UNITS 
 
 1 amp. =0.1 electromagnetic units =3x10^ electrostatic units. 
 
 1 volt =108 
 
 1 ohm =10^ 
 
 300 
 1 
 
 "9X1011 
 
 1 farad =-^5 '' =9x10" 
 
 10" 
 
 1 henry =10^ il) 
 
APPENDIX 
 
 239 
 
 34. SQUARES, CUBES AND RECIPROCALS 
 
 Nos. 
 
 Squares. 
 
 Cubes. 
 
 Reciprocals. 
 
 Nos. 
 
 Squares. 
 
 Cubes. 
 
 Reciprocals. 
 
 1 
 
 1 
 
 1 
 
 1.000000000 
 
 51 
 
 26 01 
 
 132 631 
 
 .019607843 
 
 2 
 
 4 
 
 8 
 
 . 500000000 
 
 52 
 
 27 04 
 
 140 608 
 
 .019230769 
 
 3 
 
 9 
 
 27 
 
 .333333333 
 
 53 
 
 28 09 
 
 148 877 
 
 .018867925 
 
 4 
 
 16 
 
 64 
 
 .250000000 
 
 54 
 
 29 16 
 
 137 464 
 
 .018518519 
 
 5 
 
 25 
 
 125 
 
 .200000000 
 
 55 
 
 30 25 
 
 166 373 
 
 .018181818 
 
 6 
 
 36 
 
 216 
 
 . 166666667 
 
 56 
 
 3136 
 
 173 616 
 
 .0178,57143 
 
 7 
 
 49 
 
 343 
 
 .142857143 
 
 57. 
 
 32 49 
 
 185 193 
 
 .017.543860 
 
 8 
 
 64 
 
 512 
 
 . 125000000 
 
 58 
 
 33 64 
 
 195 112 
 
 .017241379 
 
 9 
 
 81 
 
 729 
 
 .111111111 
 
 59 
 
 34 81 
 
 203 379 
 
 .016949133 
 
 10 
 
 100 
 
 1000 
 
 . 100000000 
 
 60 
 
 36 00 
 
 216 000 
 
 .016666667 
 
 11 
 
 121 
 
 1331 
 
 .090909091 
 
 61 
 
 37 21 
 
 226 981 
 
 .016393443 
 
 12 
 
 144 
 
 1728 
 
 .083333333 
 
 62 
 
 38 44 
 
 238 328 
 
 .016129032 
 
 13 
 
 169 
 
 2 197 
 
 .076923077 
 
 63 
 
 39 69 
 
 230 047 
 
 .015873016 
 
 14 
 
 196 
 
 2 744 
 
 .071428571 
 
 64 
 
 40 96 
 
 262 144 
 
 .015625000 
 
 15 
 
 2 25 
 
 3 375 
 
 .066666667 
 
 65 
 
 42 25 
 
 274 623 
 
 .015384615 
 
 16 
 
 2 56 
 
 4 096 
 
 .062500000 
 
 66 
 
 43 56 
 
 287 496 
 
 .015151515 
 
 17 
 
 2 89 
 
 4 913 
 
 .058823529 
 
 67 
 
 44 89 
 
 300 763 
 
 .014925373 
 
 IS 
 
 3 24 
 
 5 832 
 
 .055.555556 
 
 68 
 
 46 24 
 
 314 432 
 
 .014705882 
 
 19 
 
 3 61 
 
 6 859 
 
 .052631579 
 
 69 
 
 47 61 
 
 328 509 
 
 .014492754 
 
 20 
 
 4 00 
 
 8 000 
 
 .050000000 
 
 70 
 
 49 00 
 
 343 000 
 
 .014285714 
 
 21 
 
 4 41 
 
 9 261 
 
 .047619048 
 
 71 
 
 50 41 
 
 357 911 
 
 .014084307 
 
 22 
 
 4 84 
 
 10 648 
 
 .045454545 
 
 72 
 
 5184 
 
 373 248 
 
 .013888889 
 
 23 
 
 5 29 
 
 12 167 
 
 .043478260 
 
 73 
 
 53 29 
 
 389 017 
 
 .013698630 
 
 24 
 
 5 76 
 
 13 824 
 
 .041666667 
 
 74 
 
 54 76 
 
 405 224 
 
 .013313314 
 
 25 
 
 6 25 
 
 L5 625 
 
 .040000000 
 
 75 
 
 56 25 
 
 421 875 
 
 .013333333 
 
 26 
 
 6 76 
 
 17 576 
 
 .038461538 
 
 76 
 
 57 76 
 
 438 976 
 
 .013157895 
 
 27 
 
 7 29 
 
 19 683 
 
 .037037037 
 
 77 
 
 59 29 
 
 456 533 
 
 .012987013 
 
 28 
 
 7 84 
 
 21 952 
 
 .035714286 
 
 78 
 
 60 84 
 
 474 552 
 
 .012820513 
 
 29 
 
 8 41 
 
 24 389 
 
 .034482759 
 
 79 
 
 62 41 
 
 493 039 
 
 .012658228 
 
 30 
 
 9 00 
 
 27 000 
 
 .033333333 
 
 80 
 
 64 00 
 
 312 000 
 
 .012500000 
 
 31 
 
 9 61 
 
 29 791 
 
 .0322.58065 
 
 81 
 
 65 61 
 
 331 441 
 
 .012345679 
 
 32 
 
 10 24 
 
 32 768 
 
 .031250000 
 
 82 
 
 67 24 
 
 331 368 
 
 .012195122 
 
 33 
 
 10 89 
 
 35 937 
 
 .030303030 
 
 83 
 
 68 89 
 
 371 787 
 
 .012048193 
 
 34 
 
 1156 
 
 39 304 
 
 .029411765 
 
 84 
 
 70 56 
 
 392 704 
 
 .011904762 
 
 35 
 
 12 25 
 
 42 875 
 
 .028571429 
 
 83 
 
 72 25 
 
 614 125 
 
 .011764706 
 
 36 
 
 12 96 
 
 46 656 
 
 .027777778 
 
 86 
 
 73 96 
 
 636 056 
 
 011627907 
 
 37 
 
 13 69 
 
 50 653 
 
 027027027 
 
 87 
 
 75 69 
 
 658 503 
 
 .011494253 
 
 38 
 
 14 44 
 
 54 872 
 
 .026315789 
 
 88 
 
 77 44 
 
 681 472 
 
 .011363636 
 
 39 
 
 15 21 
 
 59 319 
 
 .025641026 
 
 89 
 
 79 21 
 
 704 969 
 
 .011235955 
 
 40 
 
 16 00 
 
 64 000 
 
 .026000000 
 
 90 
 
 8100 
 
 729 000 
 
 .011111111 
 
 41 
 
 16 81 
 
 68 921 
 
 .024390244 
 
 91 
 
 82 81 
 
 7.53 371 
 
 .010989011 
 
 42 
 
 17 64 
 
 74 088 
 
 .023809524 
 
 92 
 
 84 64 
 
 778 688 
 
 .010869565 
 
 43 
 
 18 49 
 
 79 507 
 
 .023255814 
 
 93 
 
 86 49 
 
 804 357 
 
 .0107,52688 
 
 44 
 
 19 36 
 
 85 184 
 
 .022727273 
 
 94 
 
 ' 88 36 
 
 830 584 
 
 .010638298 
 
 45 
 
 20 25 
 
 91 125 
 
 .022222222 
 
 95 
 
 90 25 
 
 857 375 
 
 .010526316 
 
 46 
 
 21 16 
 
 97 336 
 
 .021739130 
 
 96 
 
 92 16 
 
 884 736 
 
 .010416667 
 
 47 
 
 22 09 
 
 103 823 
 
 .021276600 
 
 97 
 
 94 09 
 
 912 673 
 
 .010309278 
 
 48 
 
 23 04 
 
 110 592 
 
 .020833333 
 
 98 
 
 96 04 
 
 941 192 
 
 .910204082 
 
 49 
 
 24 01 
 
 117 649 
 
 .020408163 
 
 99 
 
 98 01 
 
 970 299 
 
 .010101010 
 
 50 
 
 25 00 
 
 125 000 
 
 .020000000 
 
 100 
 
 100 00 
 
 1 000 000 
 
 .010000000 
 
240 
 
 APPENDIX 
 
 35. LOGARITHMS OF TRIGONOMETRIC FUNCTIONS 
 
 Angle. 
 
 Log Sin. 
 
 Log Tan. 
 
 Log Sec. 
 
 Log Csc. 
 
 Log Cot. 
 
 Log Cos. 
 
 
 
 
 —00 
 
 — 00 
 
 0.0000 
 
 00 
 
 00 
 
 0.0000 
 
 90 
 
 1 
 
 8.2419 
 
 8.2419 
 
 0.0001 
 
 1.7581 
 
 1.7581 
 
 9.9999 
 
 89 
 
 3 
 
 8.5428 
 
 8.5431 
 
 0.0003 
 
 1.4572 
 
 1.4569 
 
 9.9997 
 
 88 
 
 3 
 
 8.7188 
 
 8.7194 
 
 0.0006 
 
 1.2812 
 
 1.2806 
 
 9.9994 
 
 87 
 
 4 
 
 8.8436 
 
 8.8446 
 
 0.0011 
 
 1.1564 
 
 1.1554 
 
 9.9989 
 
 86 
 
 5 
 
 8.9403 
 
 8.9420 
 
 0.0017 
 
 1.0597 
 
 1.0580 
 
 9.9983 
 
 85 
 
 6 
 
 9.0192 
 
 9.0216 
 
 0.0024 
 
 0.9808 
 
 0.9784 
 
 9.9976 
 
 84 
 
 7 
 
 9.0859 
 
 9.0891 
 
 0.0032 
 
 0.9141 
 
 0.9109 
 
 9.9968 
 
 83 
 
 8 
 
 9.1436 
 
 9.1478 
 
 0.0042 
 
 0.8564 
 
 0.8522 
 
 9.9958 
 
 82 
 
 9 
 
 9.1943 
 
 9.1997 
 
 0.0054 
 
 0.8057 
 
 0.8003 
 
 9.9946 
 
 81 
 
 10 
 
 9.2397 
 
 9.2463 
 
 0.0066 
 
 0.7603 
 
 0.7537 
 
 9.9934 
 
 80 
 
 11 
 
 9.2806 
 
 9.2887 
 
 0.0081 
 
 0.7194 
 
 0.7113 
 
 9.9919 
 
 79 
 
 12 
 
 9.3179 
 
 9.3275 
 
 0.0096 
 
 0.6821 
 
 0.6725 
 
 9.9904 
 
 78 
 
 13 
 
 9.3521 
 
 9.3634 
 
 0.0113 
 
 0.6479 
 
 0.6366 
 
 9.9887 
 
 77 
 
 14 
 
 9.3837 
 
 9.3968 
 
 0.0131 
 
 0.6163 
 
 0.6032 
 
 9.9869 
 
 76 
 
 15 
 
 9.4130 
 
 9.4281 
 
 0.0151 
 
 0.5870 
 
 0.5719 
 
 9.9849 
 
 76 
 
 16 
 
 9.4403 
 
 9.4575 
 
 0.0172 
 
 0.5597 
 
 0.5425 
 
 9.9828 
 
 74 
 
 17 
 
 9.4659 
 
 9.4853 
 
 0.0194 
 
 0.5341 
 
 0.5147 
 
 9.9806 
 
 73 
 
 18 
 
 9.4900 
 
 9.5118 
 
 0.0218 
 
 0.5100 
 
 0.4882 
 
 9.9782 
 
 72 
 
 19 
 
 9.5126 
 
 9.5370 
 
 0.0243 
 
 0.4874 
 
 0.4630 
 
 9.9757 
 
 71 
 
 20 
 
 9.5341 
 
 9.5611 
 
 0.0270 
 
 0.4659 
 
 0.4389 
 
 9.9730 
 
 70 
 
 21 
 
 9.5543 
 
 9.6842 
 
 0.0298 
 
 0.4457 
 
 0.4158 
 
 9.9702 
 
 69 
 
 22 
 
 9.5736 
 
 9.6064 
 
 0.0328 
 
 0.4264 
 
 0.3936 
 
 9.9672 
 
 68 
 
 23 
 
 9.5919 
 
 9.6279 
 
 0.0360 
 
 0.4081 
 
 0.3721 
 
 9.9640 
 
 67 
 
 24 
 
 9.6093 
 
 9.6486 
 
 0.0393 
 
 0.3907 
 
 0.3514 
 
 9.9607 
 
 66 
 
 35 
 
 9.6259 
 
 9.6687 
 
 0.0427 
 
 0.3741 
 
 0.3313 
 
 9.9573 
 
 65 
 
 26 
 
 9.6418 
 
 9.6882 
 
 0.0463 
 
 0.3582 
 
 0.3118 
 
 9.9537 
 
 64 
 
 27 
 
 9.6570 
 
 9.7072 
 
 0.0501 
 
 0.3430 
 
 0.2928 
 
 9.9499 
 
 63 
 
 28 
 
 9.6716 
 
 9.7257 
 
 0.0541 
 
 0.3284 
 
 0.2743 
 
 9.9459 
 
 62 
 
 29 
 
 9.6856 
 
 9.7438 
 
 0.0582 
 
 0.3144 
 
 0.2562 
 
 9.9418 
 
 61 
 
 30 
 
 9.6990 
 
 9.7614 
 
 0.0625 
 
 0.3010 
 
 0.2386 
 
 9.9375 
 
 60 
 
 31 
 
 9.7118 
 
 9.7788 
 
 0.0669 
 
 0.2882 
 
 0.2212 
 
 9.9331 
 
 59 
 
 33 
 
 9.7242 
 
 9.7958 
 
 0.0716 
 
 0.2758 
 
 0.2042 
 
 9.9284 
 
 58 
 
 33 
 
 9.7361 
 
 9.8125 
 
 0.0764 
 
 0.2639 
 
 0.1875 
 
 9.9236 
 
 57 
 
 34 
 
 9.7476 
 
 9.8290 
 
 0.0814 
 
 0.2524 
 
 0.1710 
 
 9.9186 
 
 56 
 
 35 
 
 9.7586 
 
 9.8452 
 
 0.0866 
 
 0.2414 
 
 0.1548 
 
 9.9134 
 
 55 
 
 36 
 
 9.7692 
 
 9.8613 
 
 0.0920 
 
 0.2308 
 
 0.1387 
 
 9.9080 
 
 54 
 
 37 
 
 9.7795 
 
 9.8771 
 
 0.0977 
 
 0.2205 
 
 ; 0.1229 
 
 9.9023 
 
 53 
 
 as 
 
 9.7893 
 
 9.8928 
 
 0.1035 
 
 0.2107 
 
 0.1072 
 
 9.8965 
 
 52 
 
 39 
 
 9.7989 
 
 9.9084 
 
 0.1095 
 
 0.2011 
 
 0.0916 
 
 9.8905 
 
 61 
 
 40 
 
 9.8081 
 
 9.9238 
 
 0.1157 
 
 0.1919 
 
 0.0762 
 
 9.8843 
 
 50 
 
 41 
 
 9.8169 
 
 9.9392 
 
 0.1222 
 
 0.1831 
 
 0.0608 
 
 9.8778 
 
 49 
 
 42 
 
 9.8255 
 
 9.9544 
 
 0.1289 
 
 0.1745 
 
 0.0456 
 
 9.8711 
 
 48 
 
 43 
 
 9.8338 
 
 9.9697 
 
 0.1359 
 
 0.1662 
 
 0.0303 
 
 9.8641 
 
 47 
 
 44 
 
 9.8418 
 
 9.9848 
 
 0.1431 
 
 0.1582 
 
 0.0152 
 
 9.8569 
 
 46 
 
 45 
 
 9.8495 
 
 0.0000 
 
 0.1505 
 
 0.1505 
 
 0.0000 
 
 9.8495 
 
 45 
 
 
 Log Cos. 
 
 Log Cot. 
 
 Log Csc. 
 
 Log Sec. 
 
 Log Tan. 
 
 Log Sin. 
 
 Angle. 
 
APPENDIX 
 
 241 
 
 TRIGONOMETRIC FUNCTIONS (Continued) 
 
 Arc. 
 
 Angle. 
 
 Sin. 
 
 Tan. 
 
 Sec. 
 
 Cosec. 
 
 Cot. 
 
 Cos. 
 
 
 
 0.0000 
 
 
 
 0.0000 
 
 0.0000 
 
 1 . 0000 
 
 CO 
 
 00 
 
 1.0000 
 
 90 
 
 1.5708 
 
 0.0176 
 
 1 
 
 0.0176 
 
 0.0176 
 
 1.0002 
 
 57.2987 
 
 57.2900 
 
 0.9998 
 
 89 
 
 1.5533 
 
 0.0349 
 
 8 
 
 0.0349 
 
 0.0349 
 
 1.0006 
 
 28.6537 
 
 28.6363 
 
 0.9994 
 
 88 
 
 1.5359 
 
 0.0.524 
 
 3 
 
 0.0623 
 
 0.0624 
 
 1.0014 
 
 19.1073 
 
 19.0811 
 
 0.9986 
 
 87 
 
 1.5184 
 
 0.0698 
 
 4 
 
 0.0698 
 
 0.0699 
 
 1.0024 
 
 14.3356 
 
 14.3007 
 
 0.9976 
 
 86 
 
 1..5010 
 
 0.0873 
 
 5 
 
 0.0872 
 
 0.0875 
 
 1.0038 
 
 11.4737 
 
 11.4301 
 
 0.9962 
 
 85 
 
 1.4835 
 
 0.1047 
 
 6 
 
 0.1046 
 
 0.1051 
 
 1.0055 
 
 9.5668 
 
 9.5144 
 
 0.9946 
 
 84 
 
 1,4661 
 
 0.1222 
 
 7 
 
 0.1219 
 
 0.1228 
 
 1.0075 
 
 8.2055 
 
 8.1443 
 
 0.9925 
 
 83 
 
 1 , 4486 
 
 0.1396 
 
 8 
 
 0.1392 
 
 0.1405 
 
 1.0098 
 
 7.1853 
 
 7.1154 
 
 0.9903 
 
 83 
 
 1.4312 
 
 0.1571 
 
 9 
 
 0.1564 
 
 0.1584 
 
 1.0125 
 
 6.3925 
 
 6.3138 
 
 0,9877 
 
 81 
 
 1.4137 
 
 0.1745 
 
 10 
 
 0.1763 
 
 0.1763 
 
 1.0154 
 
 5.7688 
 
 6.6713 
 
 0.9848 
 
 80 
 
 1.3963 
 
 0.1920 
 
 11 
 
 0.1908 
 
 0.1944 
 
 1.0187 
 
 5.2408 
 
 5.1446 
 
 0.9816 
 
 79' 
 
 1.3788 
 
 0.2094 
 
 13 
 
 0.2079 
 
 0.2126 
 
 1.0223 
 
 4.8097 
 
 4.7046 
 
 0.9781 
 
 78 
 
 1.3614 
 
 0.2269 
 
 13 
 
 0.2250 
 
 0.2309 
 
 1.0263 
 
 4.4454 
 
 4.3316 
 
 0.9744 
 
 77 
 
 1.3439 
 
 0.2443 
 
 14 
 
 0.2419 
 
 0.2493 
 
 1.0306 
 
 4.1336 
 
 4.0108 
 
 0.9703 
 
 76 
 
 1.3264 
 
 0.2618 
 
 15 
 
 0.2688 
 
 0.2679 
 
 1.0363 
 
 3.8637 
 
 3.7321 
 
 0.9659 
 
 75 
 
 1.3090 
 
 0.2793 
 
 16 
 
 0.2756 
 
 0.2867 
 
 1.0403 
 
 3.6280 
 
 3.4874 
 
 0.9613 
 
 74 
 
 1.2916 
 
 0.2967 
 
 17 
 
 0.2924 
 
 0.3067 
 
 1.0467 
 
 3.4203 
 
 3.2709 
 
 0,9563 
 
 73 
 
 1.2741 
 
 0.3142 
 
 18 
 
 0.3090 
 
 0.3249 
 
 1.0615 
 
 3.2361 
 
 3.0777 
 
 0,9511 
 
 72 
 
 1.2566 
 
 0.3316 
 
 39 
 
 0.3256 
 
 0.3443 
 
 1.0676 
 
 3.0716 
 
 2.9042 
 
 0,9455 
 
 71 
 
 1.2392 
 
 0.3491 
 
 20 
 
 0.3420 
 
 0.3640 
 
 1.0642 
 
 2.9238 
 
 2.7475 
 
 0.9397 
 
 70 
 
 1.2217 
 
 0.3665 
 
 21 
 
 0.3584 
 
 0.3839 
 
 1.0711 
 
 2.7904 
 
 2.6061 
 
 0.9336 
 
 69 
 
 1.2043 
 
 0.3840 
 
 22 
 
 0.3740 
 
 0.4040 
 
 1.0785 
 
 2.6695 
 
 2.4761 
 
 0.9272 
 
 68 
 
 1.1868 
 
 0.4014 
 
 23 
 
 0.3907 
 
 0.4245 
 
 1.0864 
 
 2.5593 
 
 2.3669 
 
 0,9205 
 
 67 
 
 1.1694 
 
 0.4189 
 
 24 
 
 0.4067 
 
 0.4452 
 
 1.0946 
 
 2.4586 
 
 2.2460 
 
 0.9135 
 
 66 
 
 1.1519 
 
 0.4363 
 
 25 
 
 0.4226 
 
 0.4663 
 
 1.1034 
 
 2.3662 
 
 2.1446 
 
 0.9063 
 
 65 
 
 1.1345 
 
 0.4538 
 
 26 
 
 0.4384 
 
 0.4877 
 
 1.1126 
 
 2.2812 
 
 2.0603 
 
 0.8988 
 
 64 
 
 1.1170 
 
 0.4712 
 
 37 
 
 0.4640 
 
 0.5095 
 
 1.1223 
 
 2.2027 
 
 1.9626 
 
 0.8910 
 
 63 
 
 1.0996 
 
 0.4887 
 
 38 
 
 0.4695 
 
 0.5317 
 
 1 . 1326 
 
 2.1301 
 
 1.8807 
 
 0.8829 
 
 62 
 
 1.0821 
 
 0,5061 
 
 39 
 
 0.4848 
 
 0.5543 
 
 1.1434 
 
 2.0627 
 
 1.8040 
 
 0,8746 
 
 61 
 
 1.0647 
 
 0.5236 
 
 30 
 
 0.6000 
 
 0.5774 
 
 1 . 1547 
 
 2.0000 
 
 1.7321 
 
 0.8660 
 
 60 
 
 1.0472 
 
 0.5411 
 
 31 
 
 0.5150 
 
 0.6009 
 
 1 . 1666 
 
 1.9416 
 
 1.6643 
 
 0.8572 
 
 59 
 
 1.0297 
 
 0.6585 
 
 32 
 
 0.5299 
 
 0.6249 
 
 1 . 1792 
 
 1.8871 
 
 1.6003 
 
 0.8480 
 
 58 
 
 1.0123 
 
 0.6760 
 
 33 
 
 0.6446 
 
 0.6494 
 
 1.1924 
 
 1.8361 
 
 1.6399 
 
 0.8387 
 
 57 
 
 0.9948 
 
 0.6934 
 
 34 
 
 0.6692 
 
 0.6746 
 
 1.2062 
 
 1.7883 
 
 1.4826 
 
 0.8290 
 
 56 
 
 0.9774 
 
 0.6109 
 
 35 
 
 0.6736 
 
 0.7002 
 
 1.2208 
 
 1.7434 
 
 1.4281 
 
 0.8192 
 
 55 
 
 0.9699 
 
 0.6283 
 
 36 
 
 0.6878 
 
 0.7266 
 
 1.2361 
 
 1.7013 
 
 1.3764 
 
 0.8090 
 
 64 
 
 0.9426 
 
 0.6458 
 
 37 
 
 0.6018 
 
 0.7636 
 
 1.2621 
 
 1.6616 
 
 1.3270,0.7986 
 
 53 
 
 0.9250 
 
 0.6632 
 
 38 
 
 0.6157 
 
 0.7813 
 
 1.2690 
 
 1.6243 
 
 1.2799 0.7880 
 
 63 
 
 0.9076 
 
 0.6807 
 
 39 
 
 0.6293 
 
 0.8098 
 
 1.2868 
 
 1.5890 
 
 1.2349 0.7771 
 
 51 
 
 0.8901 
 
 0.6981 
 
 40 
 
 0.6428 
 
 0.8391 
 
 1.3054 
 
 1.5567 
 
 1.1918 
 
 0.7660 
 
 50 
 
 0.8727 
 
 0.7156 
 
 41 
 
 0.6561 
 
 0.8693 
 
 1.3260 
 
 1 . 6243 
 
 1.1504 
 
 0.7547 
 
 49 
 
 0.8552 
 
 0.7330 
 
 43 
 
 0.6691 
 
 0.9004 
 
 1.3456 
 
 1.4945 
 
 1.1106 
 
 0.7431 
 
 48 
 
 0.8378 
 
 0.7505 
 
 43 
 
 0.6820 
 
 0.9326 
 
 1.3673 
 
 1.4663 
 
 1.0724 
 
 0.7314 
 
 47 
 
 0.8203 
 
 0.7679 
 
 44 
 
 0.6947 
 
 0.9667 
 
 1.3902 
 
 1.4396 
 
 1.0366 
 
 0.7193 
 
 46 
 
 0.8029 
 
 0.7854 
 
 45 
 
 0.7071 
 
 1.0000 
 
 1.4142 
 
 1.4142 
 
 1.0000 
 
 0.7071 
 
 45 
 
 0.7854 
 
 
 
 Cos. 
 
 Cot. 
 
 Cosec. 
 
 Sec. 
 
 Tan. 
 
 Sin. 
 
 Angle. 
 
 Arc. 
 
242 
 
 APPENDIX 
 
 LOGARITHMS OF NUMBERS FROM 1 TO 1000 
 
 
 
 
 1 
 
 3 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 S 
 
 9 
 
 123 
 
 4 5 6 
 
 7 8 9 
 
 10 
 
 11 
 
 13 
 13 
 14 
 
 0000 
 0414 
 0792 
 1139 
 1461 
 
 0043 
 0453 
 0828 
 1173 
 1492 
 
 0086 
 0492 
 0864 
 1206 
 1523 
 
 0128 
 0531 
 0899 
 1239 
 1553 
 
 0170 
 0569 
 0934 
 1271 
 1584 
 
 0212 
 0607 
 0969 
 1303 
 1614 
 
 0253 
 0646 
 1004 
 1335 
 1644 
 
 0294 
 0682 
 1038 
 1367 
 1673 
 
 0334 
 0719 
 1072 
 1399 
 1703 
 
 0374 
 0755 
 1106 
 1430 
 1732 
 
 4 812 
 4 8 11 
 3 7 10 
 3 6 10 
 36 9 
 
 172125 
 15 19 23 
 14 17 21 
 13 16 19 
 12 15 18 
 
 29 33 37 
 26 30 34 
 24 28 31 
 23 26 29 
 2124 27 
 
 15 
 16 
 17 
 18 
 19 
 
 1761 
 2041 
 2304 
 2553 
 2788 
 
 1790 
 2068 
 2330 
 2577 
 2810 
 
 1818 
 2095 
 2355 
 2601 
 2833 
 
 1847 
 2122 
 2380 
 2625 
 2856 
 
 1875 
 2148 
 2405 
 2648 
 2878 
 
 1903 
 2176 
 2430 
 2672 
 2900 
 
 1931 
 2201 
 2455 
 2695 
 2923 
 
 1959 
 2227 
 2480 
 2718 
 2945 
 
 1987 
 2253 
 2504 
 2742 
 2967 
 
 2014 
 2279 
 2529 
 2765 
 2989 
 
 3 6 8 
 3 5 8 
 2 5 7 
 2 5 7 
 
 2 4 7 
 
 11 14 17 
 
 11 13 16 
 
 10 12 15 
 
 912 14 
 
 9 1113 
 
 20 22 25 
 18 2124 
 17 20 22 
 16 19 21 
 16 18 20 
 
 30 
 31 
 33 
 33 
 24 
 
 3010 
 3222 
 3424 
 3617 
 3802 
 
 3979 
 4150 
 4314 
 4472 
 4624 
 
 3032 
 3243 
 3444 
 3636 
 3820 
 
 3054 
 3263 
 3464 
 3655 
 3838 
 
 3075 
 3284 
 3483 
 3674 
 3856 
 
 3096 
 3304 
 3602 
 3692 
 3874 
 
 3118 
 3324 
 3622 
 3711 
 3892 
 
 3139 
 3345 
 3641 
 3729 
 3909 
 
 3160 
 3365 
 3560 
 3747 
 3927 
 
 3181 
 3385 
 3579 
 3766 
 3945 
 
 3201 
 3404 
 3598 
 3784 
 3962 
 
 2 4 6 
 2 4 6 
 2 4 6 
 2 4 6 
 2 4 5 
 
 81113 
 810 12 
 810 12 
 7 9 11 
 7 9 11 
 
 15 17 19 
 14 16 18 
 14 15 17 
 13 15 17 
 12 14 16 
 
 35 
 36 
 27 
 
 28 
 29 
 
 3997 
 4166 
 4330 
 4487 
 4639 
 
 4014 
 4183 
 4346 
 4502 
 4654 
 
 4031 
 4200 
 4362 
 4518 
 4669 
 
 4048 
 4216 
 4378 
 4533 
 4683 
 
 4065 
 4232 
 4393 
 4648 
 4698 
 
 4082 
 4249 
 4409 
 4564 
 4713 
 
 4099 
 4265 
 4425 
 4579 
 4728 
 
 4116 
 4281 
 4440 
 4594 
 4742 
 
 4133 
 4298 
 4456 
 4609 
 4757 
 
 2 3 6 
 2 3 5 
 2 3 5 
 2 3 5 
 1 3 4 
 
 7 9 10 
 7 8 10 
 6 8 9 
 6 8 9 
 6 7 9 
 
 12 14 15 
 11 13 15 
 11 13 14 
 11 12 14 
 10 12 13 
 
 30 
 31 
 32 
 33 
 34 
 
 4771 
 4914 
 5051 
 5185 
 5315 
 
 4786 
 4928 
 5065 
 5198 
 5328 
 
 4800 
 4942 
 5079 
 5211 
 5340 
 
 4814 
 4955 
 5092 
 5224 
 5353 
 
 4829 
 4969 
 5105 
 5237 
 5366 
 
 4843 
 4983 
 5119 
 5250 
 5378 
 
 4857 
 4997 
 5132 
 5263 
 6391 
 
 4871 
 6011 
 5145 
 5276 
 5403 
 
 4886 
 5024 
 5169 
 6289 
 6416 
 
 4900 
 5038 
 6172 
 5302 
 5428 
 
 1 3 4 
 1 3 4 
 1 3 4 
 13 4 
 1 3 4 
 
 6 7 9 
 6 7 8 
 5 7 8 
 5 6 8 
 5 6 8 
 
 10 11 13 
 
 10 11 12 
 
 9 11 12 
 
 910 12 
 
 910 11 
 
 35 
 36 
 37 
 38 
 39 
 
 5441 
 5563 
 5682 
 5798 
 5911 
 
 5453 
 5575 
 5694 
 6809 
 5922 
 
 5465 
 5587 
 5705 
 5821 
 5933 
 
 5478 
 5599 
 5717 
 5832 
 5944 
 
 5490 
 5611 
 5729 
 6843 
 5955 
 
 5502 
 5623 
 5740 
 5856 
 5966 
 
 6614 
 6636 
 6752 
 6866 
 6977 
 
 5527 
 5647 
 5763 
 5877 
 5988 
 
 5539 
 5658 
 5775 
 5888 
 5999 
 
 5551 
 5670 
 5786 
 5899 
 6010 
 
 1 2 4 
 12 4 
 12 3 
 12 3 
 1 2 3 
 
 5 6 7 
 5 6 7 
 5 6 7 
 5 6 7 
 4 5 7 
 
 910 11 
 810 11 
 8 9 10 
 8 9 10 
 8 9 10 
 
 40 
 41 
 43 
 43 
 
 44 
 
 6021 
 6128 
 6232 
 6335 
 6435 
 
 6031 
 6138 
 6243 
 6345 
 6444 
 
 6042 
 6149 
 6263 
 6366 
 6464 
 
 6053 
 6160 
 6263 
 6366 
 6464 
 
 6064 
 6170 
 6274 
 6375 
 6474 
 
 6075 
 6180 
 6284 
 6385 
 6484 
 
 6085 
 6191 
 6294 
 6395 
 6493 
 
 6096 
 6201 
 6304 
 6406 
 6603 
 
 6107 
 6212 
 6314 
 6415 
 6513 
 
 6117 
 6222 
 6325 
 6425 
 6522 
 
 1 2 3 
 12 3 
 12 3 
 12 3 
 12 3 
 
 4 5 6 
 4 5 6 
 4 5 6 
 4 5 6 
 4 5 6 
 
 8 9 10 
 7 8 9 
 7 8 9 
 7 8 9 
 7 8 9 
 
 45 
 46 
 
 47 
 48 
 49 
 
 6532 
 6628 
 6721 
 6812 
 6902 
 
 6542 
 6637 
 6730 
 6821 
 6911 
 
 6661 
 6646 
 6739 
 6830 
 6920 
 
 6661 
 6656 
 6749 
 6839 
 6928 
 
 6671 
 6665 
 6758 
 6848 
 6937 
 
 6580 
 6675 
 6767 
 6857 
 6946 
 
 6590 
 6684 
 6776 
 6866 
 6955 
 
 6599 
 6693 
 6785 
 6875 
 6964 
 
 6609 
 6702 
 6794 
 6884 
 6972 
 
 6618 
 6712 
 6803 
 6893 
 6981 
 
 12 3 
 1 2 3 
 12 3 
 1 2 3 
 12 3 
 
 4 5 6 
 4 5 6 
 4 5 5 
 
 4 4 5 
 4 4 5 
 
 7 8 9 
 7 7 8 
 6 7 8 
 6 7 8 
 6 7 8 
 
 60 
 61 
 53 
 53 
 54 
 
 6990 
 7076 
 7160 
 7243 
 7324 
 
 6998 
 7084 
 7168 
 7251 
 7332 
 
 7007 
 7093 
 7177 
 7259 
 7340 
 
 7016 
 7101 
 7185 
 7267 
 7348 
 
 7024 
 7110 
 7193 
 7275 
 7356 
 
 7033 
 7118 
 7202 
 7284 
 7364 
 
 7042 
 7126 
 7210 
 7292 
 7372 
 
 7050 
 7135 
 7218 
 7300 
 7380 
 
 7059 
 7143 
 7226 
 7308 
 7388 
 
 7067 
 7152 
 7235 
 7316 
 7396 
 
 12 3 
 12 3 
 1 2 2 
 12 2 
 12 2 
 
 3 4 5 
 3 4 5 
 3 4 5 
 3 4 5 
 3 4 5 
 
 6 7 8 
 6 7 8 
 6 7 7 
 6 6 7 
 6 6 7 
 
APPENDIX 
 
 243 
 
 LOGARITHMS OF NUMBERS FROM 1 TO 1000 {Con.) 
 
 
 
 
 1 
 
 3 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 13 3 
 
 4 5 6 
 
 7 8 9 
 
 55 
 56 
 57 
 
 5S 
 59 
 
 7404 
 7482 
 7559 
 7634 
 7709 
 
 7412 
 7490 
 7566 
 7642 
 7716 
 
 7419 
 7497 
 7574 
 7649 
 7723 
 
 7427 
 7505 
 7582 
 7657 
 7731 
 
 7435 
 7513 
 7589 
 7664 
 7738 
 
 7443 
 7520 
 7597 
 7672 
 7745 
 
 7451 
 7528 
 7604 
 7679 
 7752 
 
 7469 7466 
 7536:7543 
 76127619 
 7686 7694 
 7760,7767 
 
 7474 
 7551 
 7627 
 7701 
 7774 
 
 1 2 2 
 12 2 
 12 2 
 112 
 1 1 2 
 
 3 4 5 
 3 4 5 
 3 4 5 
 3 4 4 
 3 4 4 
 
 5 6 7 
 5 6 7 
 5 6 7 
 5 6 7 
 5 6 7 
 
 60 
 61 
 63 
 63 
 64 
 
 7782 
 7853 
 7924 
 7993 
 8062 
 
 7789 
 7860 
 7931 
 8000 
 8069 
 
 7796 
 7868 
 7938 
 8007 
 8075 
 
 7803 
 7875 
 7945 
 8014 
 8082 
 
 7810 
 7882 
 7952 
 8021 
 8089 
 
 7818 
 7889 
 7959 
 8028 
 8096 
 
 7825 
 7896 
 7966 
 8035 
 8102 
 
 7832 
 7903 
 7973 
 8041 
 8109 
 
 7839 
 7910 
 7980 
 8048 
 8116 
 
 7846 
 7917 
 7987 
 8055 
 8122 
 
 112 
 112 
 112 
 112 
 112 
 
 3 4 4 
 3 4 4 
 3 3 4 
 3 3 4 
 3 3 4 
 
 5 6 6 
 5 6 6 
 5 6 6 
 5 5 6 
 5 5 6 
 
 65 
 66 
 67 
 68 
 69 
 
 8129 
 8195 
 8261 
 8325 
 8388 
 
 8136 
 8202 
 8267 
 8331 
 8395 
 
 8142 
 8209 
 8274 
 8338 
 8401 
 
 8149 
 8215 
 8280 
 8344 
 8407 
 
 8156 
 8222 
 8287 
 8351 
 8414 
 
 8162 
 8228 
 8293 
 8357 
 8420 
 
 8169 
 8235 
 8299 
 8363 
 8426 
 
 8176 
 8241 
 8306 
 8370 
 8432 
 
 8182 
 8248 
 8312 
 8376 
 8439 
 
 8189 
 8254 
 8319 
 8382 
 8445 
 
 112 
 1 1 2 
 1 1 2 
 112 
 112 
 
 3 3 4 
 3 3 4 
 3 3 4 
 3 3 4 
 2 3 4 
 
 5 5 6 
 5 S 6 
 5 5 6 
 4 5 6 
 4 5 6 
 
 70 
 71 
 
 72 
 73 
 
 74 
 
 8451 
 8513 
 8573 
 8633 
 8692 
 
 8457 
 8519 
 8579 
 8639 
 8698 
 
 8463 
 8525 
 8585 
 8645 
 8704 
 
 8470 
 8531 
 8591 
 8651 
 8710 
 
 8476 
 8537 
 8597 
 8657 
 8716 
 
 8482 
 8543 
 8603 
 8663 
 8722 
 
 8488 
 8549 
 8609 
 8669 
 8727 
 
 8494 
 8555 
 8615 
 8675 
 8733 
 
 8500 
 8561 
 8621 
 8681 
 8739 
 
 8506 
 8567 
 8627 
 8686 
 8745 
 
 1 1 2 
 1 1 2 
 112 
 1 1 2 
 1 1 2 
 
 2 3 4 
 2 3 4 
 2 3 4 
 2 3 4 
 2 3 4 
 
 4 5 6 
 4 5 5 
 4 5 5 
 4 5 5 
 4 5 5 
 
 75 
 76 
 
 77 
 78 
 79 
 
 8751 
 8808 
 8865 
 8921 
 8976 
 
 8756 
 8814 
 8871 
 8927 
 8982 
 
 8762 
 8820 
 8876 
 8932 
 8987 
 
 8768 
 8825 
 8882 
 8938 
 8993 
 
 8774 
 8831 
 8887 
 8943 
 8998 
 
 8779 
 8837 
 8893 
 8949 
 9004 
 
 8785 
 8842 
 8899 
 8954 
 9009 
 
 8791 
 8848 
 8904 
 8960 
 9015 
 
 8797 
 8854 
 8910 
 8965 
 9020 
 
 8802 
 8859 
 8915 
 8971 
 9025 
 
 1 1 2 
 1 1 2 
 1 1 2 
 1 1 2 
 1 1 2 
 
 2 3 3 
 2 3 3 
 2 3 3 
 2 3 3 
 2 3 3 
 
 4 5 5 
 4 5 5 
 4 4 5 
 4 4 5 
 4 4 5 
 
 80 
 81 
 82 
 S3 
 84 
 
 9031 
 9085 
 9138 
 9191 
 9243 
 
 9036 
 9090 
 9143 
 9196 
 9248 
 
 9042 
 9096 
 9149 
 9201 
 9253 
 
 9047 
 9101 
 9154 
 9206 
 9258 
 
 9053 
 9106 
 9159 
 9212 
 9263 
 
 9058 
 9112 
 9165 
 9217 
 9269 
 
 9063 
 9117 
 9170 
 9222 
 9274 
 
 9069 
 9122 
 9175 
 9227 
 9279 
 
 9074 
 9128 
 9180 
 9232 
 9284 
 
 9079 
 9133 
 9186 
 9238 
 9289 
 
 1 1 2 
 1 1 2 
 1 1 2 
 1 1 2 
 112 
 
 2 3 3 
 2 3 3 
 2 3 3 
 2 3 3 
 2 3 3 
 
 4 4 5 
 4 4 5 
 4 4 5 
 4 4 5 
 4 4 5 
 
 85 
 86 
 87 
 88 
 89 
 
 9294 
 9345 
 9395 
 9445 
 9494 
 
 9299 
 9350 
 9400 
 9450 
 9499 
 
 9304 
 9355 
 9405 
 9455 
 9504 
 
 9309 
 9360 
 9410 
 9460 
 9509 
 
 9315 
 9365 
 9415 
 9465 
 9513 
 
 9320 
 9370 
 9420 
 9469 
 9518 
 
 9325 
 9375 
 9425 
 9474 
 9523 
 
 9330 
 9380 
 9430 
 9479 
 9528 
 
 9335 
 9385 
 9435 
 9484 
 9533 
 
 9340 
 9390 
 9440 
 9489 
 9538 
 
 1 1 2 
 1 1 2 
 1 1 
 Oil 
 1 1 
 
 2 3 3 
 2 3 3 
 2 2 3 
 2 2 3 
 2 2 3 
 
 4 4 5 
 4 4 5 
 3 4 4 
 3 4 4 
 3 4 4 
 
 90 
 91 
 93 
 93 
 94 
 
 9542 
 9590 
 9638 
 9685 
 9731 
 
 9547 
 9595 
 9643 
 9689 
 9736 
 
 9552 
 9600 
 9647 
 9694 
 9741 
 
 9557 
 9605 
 9652 
 9699 
 9745 
 
 9562 
 9609 
 9657 
 9703 
 9750 
 
 9566 
 9614 
 9661 
 9708 
 9754 
 
 9571 
 9619 
 9666 
 9713 
 9759 
 
 9576 
 9624 
 9671 
 9717 
 9763 
 
 9581 
 9628 
 9675 
 9722 
 9768 
 
 9586 
 9633 
 9680 
 9727 
 9773 
 
 1 1 
 Oil 
 1 1 
 1 1 
 Oil 
 
 2 2 3 
 2 2 3 
 2 2 3 
 2 2 3 
 2 2 3 
 
 3 4 4 
 3 4 4 
 3 4 4 
 3 4 4 
 3 4 4 
 
 95 
 96 
 97 
 98 
 99 
 
 9777 
 9823 
 9868 
 9912 
 9956 
 
 9782 
 9827 
 9872 
 9917 
 9961 
 
 9786 
 9832 
 9877 
 9921 
 9965 
 
 97919795 
 9836,9841 
 9881 9883 
 9926 9930 
 9969,9974 
 
 9800 
 9845 
 9890 
 9934 
 9978 
 
 9805 
 9850 
 9894 
 9939 
 9983 
 
 9809 
 9854 
 9899 
 9943 
 9987 
 
 9814 
 9859 
 9903 
 9948 
 9991 
 
 9818 
 9863 
 9908 
 9952 
 9996 
 
 1 1 
 1 1 
 1 1 
 1 1 
 1 1 
 
 2 2 3 
 2 2 3 
 2 2 3 
 2 2 3 
 2 2 3 
 
 3 4 4 
 3 4 4 
 3 4 4 
 3 4 4 
 3 3 4 
 
244 
 
 APPENDIX 
 
 37. APPROXIMATE EXPRESSIONS 
 
 when a, &, . . . and ai, bi, . . are small. 
 
 (l-a)2 = l-2a. 
 
 (l+o)2 = l+2a. 
 1 
 
 1+a 
 
 1 
 (1+a)' 
 
 = l-a. 
 
 = l-2o. 
 
 Vl+o 
 
 (l±a)(:±6)(l±c) 
 
 (l±a)(l±6) ■ ■ . 
 (l±ai)(l±6i) . . 
 
 ^1-0 = 1-1 
 1 
 
 a. 
 
 1-a 
 
 1 
 
 -a)' 
 1 
 
 VI -a 
 
 = l±a±6±c . . 
 
 = l+o. 
 = l+2a. 
 1+ia. 
 
 = l±a±6 
 
 T01T61 
 
 V^ = -^ 
 
 sin 8=0. 
 
 cos a = l. 
 
 .tan a = a. 
 
INDEX 
 
 PAGE 
 
 Absorption, electric 142 
 
 Absorbing power 52 
 
 Acceleration, angular . 27 
 
 of gravity 32-34 
 
 Adjustment of telescope 68 
 
 Alloys 227 
 
 Ammeter 87, 97 
 
 calibration of, 87, 125, 
 129 
 
 Ampere, definition of 86, 235 
 
 Angle of a prism 204 
 
 ' ' of incideuce= angle of 
 
 flection 203 
 
 " of torsion 20 
 
 Angular acceleration 27 
 
 ' ' displacement 21 
 
 ' ' velocity 29 
 
 Aperiodic 95 
 
 Approximation, table of 244 
 
 Areas, table of 219 
 
 Atmospheric pressure 41 
 
 Ayerton shunt 91 
 
 Balance 9, 36 
 
 ' ' ratio of length of arms 12 
 
 Ballistic galvanometer 130 
 
 " " " con- 
 
 stant.. 132, 135, 142 
 
 Barometer 41 
 
 ' ' corrections, table 
 
 of 229,230 
 
 Bending of a rod 15 
 
 Boiling point of water 44 
 
 " table of 229 
 Boyle's and Charles' laws 
 
 combined 48 
 
 Boyle's law 35 
 
 PAGE 
 
 Cadmium standard cell 121 
 
 Calibration of ammeter, 125, 126, 129 
 
 Calibration of hydrometer ... 38 
 
 " " thermometer. . 43 
 
 " " voltmeter 120 
 
 Caliper, micrometer 5 
 
 ' ' vernier 2 
 
 Colorimeter 51, 52, 98 
 
 Capacity — absolute method . . 140 
 ' ' — comparison method 144 
 
 " of Lej'den jar 143 
 
 " " paraffin condenser 144 
 
 Cell, standard 121, 238 
 
 Changing from closed and open 
 
 circuit 156 
 
 Changing from open to closed 
 
 circuit 171 
 
 Charles, law of 46 
 
 Charge, free 141 
 
 residual 145 
 
 Circular measure 219 
 
 Coefficient of linear expansion. 49 
 " " mutual induc- 
 tion 151, 157 
 
 " " selt-induetion, 
 
 158, 160, 163 
 " temperature. ..114, 135 
 Coefficient temperature gal- 
 vanometer 96 
 
 Coefl&cients, temperature table 
 
 of 222 
 
 Coincidence method 32 
 
 Commutator, double 167 
 
 Constant of ballistic galva- 
 nometer. . 132, 135, 142 
 " of tangent galva- 
 nometer 84 
 
 245 
 
246 
 
 INDEX 
 
 PAGE 
 
 Conversion tables 219 
 
 Correction factor 88, 89, 124 
 
 " curve 103 
 
 Cubes, table of 239 
 
 Coulomb 235 
 
 Current inductor 174 
 
 Curvature, radius of S, 191 
 
 Damping of galvanometer . . . 134 
 
 Density of solids 35 
 
 " water, table of 228 
 
 Diameter 6 
 
 Dip, magnetic 81 
 
 Earth, data with regard to. . . 221 
 
 ' ' inductor 175 
 
 Earth's magnetic field, 81, 175, 179 
 
 Electro-chemical series 104 
 
 E.M.F. of battery 120, 145 
 
 ' ' — calorimeter method . 98 
 ' ' — condenser method . . 145 
 ' ' of cells in series and in 
 
 parallel ." 103 
 
 ' ' of standard cells 238 
 
 " — Ohm's method 118 
 
 " — Poggendorff's meth- 
 od 119 
 
 ' ' — Potentiometer meth- 
 od 120 
 
 . . <*™ , -„ 
 
 = * ^^° 
 
 Electrostatics 64 
 
 Electrostatic units 234 
 
 Electrolyte, resistance of 117 
 
 Elements, data with regard to. 222 
 Elongation of a spiral spring . . 18 
 
 " " wire 17 
 
 Energy units, table of 220 
 
 Error in computation ix 
 
 propagation of 216 
 
 Errors, accidental , . . . viii 
 
 ' ' constant viii 
 
 Errors, theory of 213 
 
 Estimating 1 
 
 Farad 236 
 
 Faraday's ice-pail 65 
 
 Figure of merit 142, 171 
 
 Focal lengths of lenses 93 
 
 " " " mirrors 192 
 
 Forces, table of 219 
 
 PAGE 
 
 Frequency of tuning fork . 188, 189 
 
 Fr = al 28 
 
 Fs = iu'1 30 
 
 Fusion, latent heat of 54 
 
 Galvanometer 82 
 
 ' ' , movingcoil, 93, 130 
 
 " , constant of . . . 84 
 
 Gases 227 
 
 Gauss' method 75, 236 
 
 Graphic interpretation of obser- 
 vations ix, 16 
 
 Grating, diffraction 206 
 
 Gravity, acceleration of 32 
 
 Heat, units of 221 
 
 Henry 236 
 
 Hooke's law 16 
 
 Humidity, relative 58 
 
 Hydrometer 38 
 
 Hygrometer 58 
 
 Hysteresis 182 
 
 I=Ia+Md^ 24 
 
 Image, real 191, 193 
 
 virtual 191, 193 
 
 Inclination, magnetic 178 
 
 Induction, lines of 151 
 
 Induction, coefficient of mu- 
 tual 154 
 
 Induction, coefficient of self- 
 
 158, 160, 163 
 
 Induction, magnetic 181 
 
 Inertia, moment of 23 
 
 Insulation, resistance .... 146, 148 
 Intensity of the earth's field, 
 
 75, 175, 179 
 Intensity of magnetization .. . 181 
 Iron, magnetic properties of. . 179 
 
 Joule 236 
 
 Kinetic energy 30 
 
 Latent heat of fusion 54 
 
 " " " vaporization . . 55 
 
 Least count 3 
 
 Length, measurement of 1 
 
 Lengths, table of 219 
 
 Lenses 192 
 
 ' ' focal lengths of 193 
 
 Logarithms 221, 242 
 
INDEX 
 
 247 
 
 PAGE 
 
 Magnetic dip or inclination 81, 178 
 
 fitiid 66 
 
 ' ' hysteresis 182 
 
 ' ' induction 181 
 
 " intensity 181 
 
 " lines, number of .. . 174 
 
 ' ' moment 80 
 
 " permeability 182 
 
 ' ' properties of iron. . 179 
 
 ' ' reluctance 182 
 
 " susceptibility 181 
 
 units 234 
 
 Magnetization curve 182 
 
 Magnetizing force 181 
 
 Magnetometer 67 
 
 Magnifying power of micro- 
 scopes and telescopes . . . 194 
 
 Mass, measurement of 8 
 
 Maxwell 236 
 
 Masses, table of 219 
 
 Mechanical equivalent of heat 61 
 
 Microfarad 236 
 
 Micrometer caliper 5 
 
 Microscpoes, magnifying pow- 
 er of 196, 197 
 
 Mirrors, spherical 190 
 
 Mohr's balance 36 
 
 Moment of inertia 23, 24 
 
 " of a force 29 
 
 ' ' of torsion 21 
 
 Momentum 31 
 
 Mutual induction 154, 157 
 
 Mutual-induction coil, stand- 
 ard 174 
 
 ' ' notation. . . ix 
 
 ^=i-Vi 
 
 184 
 
 Onm, definition of 235 
 
 Ohms laws, proof of 101 
 
 J'arallax 2, 69 
 
 Parallelogram of forces 25 
 
 Pendulum, compound 34 
 
 ' ' simple 32 
 
 " torsional 21 
 
 Permeability, magnetic 182 
 
 Polarized light 209 
 
 Pole strength of a Magnet. ... 80 
 
 Potential difference 98 
 
 FAGB 
 
 Potentiometer 127 
 
 method — cali- 
 bration of ammeter . 125, 126 
 
 Potentiometer method — cali- 
 bration of voltmeter. . . . 122 
 
 Potentiometer method — de- 
 termination of E.M.F . . . 120 
 
 Power units of 220 
 
 Pressures, table of 220 
 
 Quantity of electricity meas- 
 ured 138 
 
 Quantity of induced electric- 
 ity 156, 167 
 
 Radian 219 
 
 Radii of curvature 8, 192 
 
 Radiating power 50 
 
 Reciprocals, table of 239 
 
 Refractive index 205 
 
 " table of 233 
 
 Reluctance, magnetic 182 
 
 Resistance, absloute determi- 
 nation of 170 
 
 Resistance, adjustment of a . . 115 
 
 box 112 
 
 " " carrying ca- 
 pacity of . . 1 14 
 " — calorimeter meth- 
 od 101 
 
 " temperature coefii- 
 
 cient for 114 
 
 " definition of 235 
 
 ' ' — standard solenoid 
 
 method 170 
 
 — fall of potential 
 
 method 106 
 
 high 116, 146, 148 
 
 " — method of leak- 
 age 146 
 
 " of a uniform wire . 108 
 ' ' — substitution meth- 
 od 105 
 
 ' ' -Wheatstone bridge 
 
 method 110, 112 
 
 Resistances, measurement of 
 
 various 115 
 
 Self-induction, coefficient of, 
 
 158, 160, 163 
 Self-induction, "comparison of 
 
 coefficients of 168 
 
248 
 
 INDEX 
 
 FAGE 
 
 Sensibility 10 
 
 Sensitive galvanometer 93 
 
 Sensitiveness of a galvanom- 
 eter 94 
 
 Simple harmonic motion .. . 19, 21 
 
 Shunt, Ayerton 91 
 
 Solenoid, standard 170 
 
 Sound, velocity of 183, 185 
 
 Specific gravity of solids 35 
 
 " liquids .... 36 
 
 heat 52 
 
 heats, table of 222 
 
 Spectrometer, adjustment of . 200 
 
 Spherometer 7 
 
 Spiral spring, elongation of. . . 18 
 " " time of vibra- 
 
 tionof 19 
 
 Squares, table of 239 
 
 Standard cell 100 
 
 Strain 16 
 
 Stress 16 
 
 Surface tension 39 
 
 Susceptibility, magnetic 181 
 
 Tables, reference Appendix 
 
 Telescope and scale 70 
 
 Telescope, terrestrial 199 
 
 Telescopes, magnifying power 
 
 of 198 
 
 Thermo-electric current 139 
 
 Thermometer, calibration of. . 43 
 
 Time of vibration 13, 32 
 
 Time, units of 221 
 
 Torsion ol a rod 20 
 
 PAGE 
 
 Torsion, pendulum 21 
 
 ratio of 71,72,82 
 
 Trigonometric functions 241 
 
 Tuning fork, frequency of, 
 
 188, 189 
 
 Units, electrical 234, 238 
 
 Vaporization, latent heat of . . 55 
 
 Vapor pressure 57 
 
 Vapor pressures, table of, 230, 231 
 
 Velocities, table of 221 
 
 Velocity, angular 29 
 
 ' ' of sound along a cord 183 
 ' ' of sound in a solid. . 186 
 
 " " "air 185 
 
 Vernier calipers 2 
 
 Vibrating coil, throw of 138 
 
 Volt, definition of 235 
 
 Voltameter 89 
 
 Voltmeter 101, 102 
 
 Voltmeter, calibration of 120 
 
 Volume, measurement of 2 
 
 Volumes, table of 219 
 
 Water, table of densities of. . . 228 
 
 ' ' boiling-point of 229 
 
 Watt 236 
 
 Wave length of light 206, 208 
 
 Wave-lengths, table of 233 
 
 Wheatstone bridge 110 
 
 Work=iV/i 149 
 
 Young's modulus 17 
 

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