CONDENSED LIST. AP- means Within 2%. Physical Quantities, Relations, Dimensional Formulas, etc., p. 18-26 lengths: complete tables, p. 30-34 1 mil = 0.025 40 mm. Ap. *4 1 mm. =39.37 mils. Ap 40 1 cm. =0.393 7 inch. Ap. Vi 1 inch = 2. 540 centimeters. Ap. 10 /i 1 foot =0.304 8 meter. Ap. 3 /l 1 yard = 0.914 4 meter. Ap^Hoor^i 1 meter = 39. 37 inches. Ap. 40 = 3.281 feet. Ap. 1% " =1.094 yards. Ap. i y w 1 kilometer = 3 281. feet. Ap. MX 10000 " =1 094. yards. Ap. 1 100 " =0.621 4 mile. Ap. ^ 1 mile =5 280. feet. Ap. 5 300 " =1 760. yards. Ap. % X 1 000 " =1.609 km. Ap. add t short ton =0.907 2 met. ton. Ap. subtr. Vi lo. = 0.892 9 long ton. Ap. subtr. Mo metric ton = 1.102 short tons. Ap. add Vio lo. =0.984 2 long ton. Ap. 1 long ton = 1.12 short tons. Ap. *% = 1.016 met. tons. Ap. 1 3igit conversion tables, p. 59 ( Weights and Lengths; Wt. of Bars: complete table, p. 62-63 Ib. per mile = 0.281 8 kg. per km. Ap. % m I kg. per kilometer = 3. 548 Ibs. per mile. Ap. % 1 Ib. per yard = 0.496 1 kg. per meter. Ap. H 1 kg. per meter = 2. 016 Ibs. per yard. Ap. 2 do. =0.672 Ib. per ft. Ap. ? llb.perft. = 1.488 kg. perm. Ap. % Pressures: complete table, p. 64-67 1 Ib. per sq. ft. = 4. 882 kg. per sq. meter. Ap. 4 % 1 ft. water column = 62. 43 Ibs. per sq. foot. Ap. 60% do. =0.029 50 atm. Ap. 8 /iro 1 Ib. persq. in. =0.070 31 kg. per sq. cm. Ap. "vioo do.= 0.068 04 atmosphere. Ap.% 1 kg. per sq. cm. = 14. 22 Ibs. per sq. in. Ap. 10 % do. =0.967 8 atmosphere. Ap. subtr. Vao 1 atm. = 14.70 Ibs. per sq. in. Ap. 4 % " =1.033 kg. per sq. cm. Ap. add Vac Digit conversion tables, p. 67 Weights and Volumes: complete table, p. 68-69 1 Ib. per cb. yd. =0.593 3 kg. per cubic meter. Ap. %o kg. per cb. meter= 1.686 Ibs. per cb. yd. Ap. 10 / 6 do. =0.062 43 Ib. per cb. ft. Ap. % ,1 Ib. per cb. ft. = 16. 02 kg. per cb. meter. Ap. 8 % Weights of Water: complete table, p. 70 1 cb. cm. =0.035 27 oz. Ap. %oo 1 cb. inch =16. 39 grams. Ap. 10 % =0.578 ounce. Ap. H 1 pint = 1.043 pounds. Ap. add ^20 CONDENSED LIST. 1 liter = 2. 205 pounds. Ap. 22 /i . Icb. ft. = 62.43 pounds. Ap.%XlOQ " =28.32 kilograms. Ap. 20 % Icb. yd.=1686.1bs. Ap MX 10000 = 764.6 kg. Ap. MX 1 000 = 0.752 5 ton (long). Ap.M lcb.meter = 2 205. pounds = 0.984 2 ton ( long). Ap 1 Volumes of Water: complete table, p. 71 1 gram = 0.061 02 cb. in. Ap. 9ioo 1 ounce = 28.35 cb. cm. Ap. 2 / 7 X 100 1 Ib. = 27.68 cb. in. Ap. 1J ^iX10 " =0.453 6 liter. Ap. Vn " =0.016 02 cb. foot. Ap.%-^100 1 kilogram =61.02 cb. in. Ap. 60 " =0.035 31 cb.ft. Ap.y 2 -3- 100 Energy; Work; Heat: complete table, p. 74-77 1 joule = 0.737 6 ft.-lb. Ap. M = 0.238 9 small calorie. = 0.1020 kg. -met. Ap. Ho 1 ft.-lb. = 1.356 joules. Ap. ^ = 0.3239 small cal. Ap. i%o = 0.138 3 kg. -met. Ap. %o " =0.001 285 thermal unit. Ap. % -5- 1000 1 kg.-m.=9.806 joules. Ap. 10 = 7.233 ft.-lbs. Ap. so/ii = 2.342 g. -calories. Ap. % 1 ther. unit = l 055. joules. Ap. 210 % " =778.1 ft.-lb. Ap. 700 % 4 =107.6kg.-met. Ap. 108 " =0.252 kg.-cal. Ap. \i 1 watt-hour = 2 655. ft.-lbs. Ap. 00 % = 367.1 kg.-met. Ap. 110 % = 0.860 Okg. -gal. Ap.% 1 cal. (kg.) = 4 186. joules. Ap. 4 200 = 3088. ft.-lbs. Ap. 3 100 = 426.9 kg.-met. Ap. 300 ^ = 3. 968 ther. units. Ap.4 = 1.163 watt-hrs. Ap. % 1 met. hp.-hr. = l 952 910. ft.-lbs. = 270000. kg.-met. 1 hp.-hr. = l 980 000. foot-pounds = 273 7 45. kilogram-meters 1 kw.-hr. = 2 655 403. foot-pounds ' =367 123. kilogram-meters Digit conversion tables, p. 77 Relations between Torque and Energy, p. 78 Traction Energy, p. 78 1 ton (met.)-km. =0.684 9 ton (sh.)- mile. Ap. 9 /i 3 1 ton ( sh.)-mile = 1.460 ton (met. )- kilometer. Ap. l % Tractive Force, p. 78 1 Ib. per ton (sh.) =0.500 kilogram per ton (met.) 1 kg. per ton (met.) = 2. 000 pounds per ton (short) Power: complete table, p. 80-82 1 watt =44.26 ft.-lbs. per minute. Ap. %X100 do. = 14. 33 gram -cal. per minute. Ap. Vr X 100 do. =6.119 kg.-met. per minute. Ap. 6 1 met . hp . = 7 5 . 00 kilogram -m eters per sec. Or MX 100 do. =0.986 3 hp. Ap. 1 do.=0.7354 kw. Ap. 22 / 3 -^10 1 hp. = 33 000. ft.-lbs. per min. Ap. MX 100 000 " =1.014 metric hp. Ap. 1 " =0.7457 kilowatt. Ap. M 1 kw. = 1.360 m. hp. Ap. add $i 4 =1.341 hp. Ap. add H Digit conversion tables, p. 82 Forces: complete table, p. 83 1 dyne = 1 .020 milligrams. Ap. 1 1 gram = 980. 6 dynes. Ap. 1000 1 pound =444 791. dynes Moments of Inertia and of Momentum, p. 84 Linear Velocities: complete table, p. 85-86 1 km. per hr.'=16.67 met. per min. Ap. MX 100 1 ft. per sec. =0.681 8 mile per hour. Ap. 68 -MOO 1 mile per hour = 1 .467 ft. per second. Ap. 1% 1 meter per sec. = 3. 6 km. per hour Angular Velocities, p. 86 Frequency, p. 86 Linear Accelerations: complete table, p. 88 Gravity = 32.17 ft. per sec. per sec. Ap. 32 Gravity ==9.806 m. per s. per s. Ap.10 Angular Accelerations, p. 88 Angles: complete table, p. 89 1 deg. =0.017 45 radian. Ap. %-*- 100 1 radian = 57 .30 degrees 1 right angle = 1.571 radians. Ap. ^ Solid angles, p. 89 Grades: complete table, p. 90-91 1 foot per mile = 0.018 94% 1% =52.80 feet per mile Conversion table 1 to 15%, p. 92 Time, p. 94 Discharges; Flow of Water; Irrigation Units, p. 95 1 miner's inch = 1.5 cb. ft. per min. 1 acre-foot = 325 851. gallons Electric and Magnetic Units: Mean, effec. and max. values, p. 97 Resistance, Impedance, React- ance, p. 99 1 Brit. Assn. unit =0.986 7 ohm. Ap. subtr. 1% lohm= 1.014 Brit. Assn. units. Ap. add 1% Resistance and Length, p. 101 Resistance and Cross-section, p. 10) Resistivity; Specific Resist aiice, complete table, p. 103-4 1 ohm, circ. mil, ft., unit =0.001 662 ohm, sq.mm.,m.,unit. Ap. %oo 1 ohm, sq. mm., m., unit = 601. 5 ohm', circ. mil, foot units Resistivity of pure copper: = 10.03 ohm, circ. mil, foot units = 0.016 67 ohm,sq.mm.,m.,unit CONDENSED LIST. Resistivity of mercury: = 565.9 ohm, circ. mil, foot units = 0.940 7 ohm, sq. mm., meter unit Conductance, Admittance, Sus- ceptance, p. 105 Conductivity, Specific Conduc- tance, complete table, p. 107 1 mercury unit=0.017 72 copperunit 1 copper unit = 55. 7 6 mercury units Electromotive Force, complete table, p. 109-111 1 volt = 0.981 Weston cell at 20 C. Ap. subtr. 2% " = 0.697 4 Clark cell 15. Ap. %o 1 Weston cell at 20 C.: = 1.019 4 volts Ap. add 2% = 0.710 9 Clark cell at 15 C. Ap. % 1 Clark cell 15 = 1.434 volts. Ap. i 18 Mechanical , 18 Magnetic, electromagnetic system 20 1 ' electrostatic system 21 Electric, electromagnetic system 22-23 " electrostatic system 24-25 Electrochemical, electromagnetic system 23 electrostatic system 25 Photometric 25 Thermal 26 TABLES OF CONVERSION FACTORS 27-173 General Remarks 27-29 Lengths 29-40 Text 29 Table of Usual Measures 30-31 Table of Unusual, Special Trade or Obsolete Measures 31-32 Table of Foreign Measures 33-34 Tables of Inches in Fractions, Decimals, Millimeters, and Feet. 35-38 Digit Conversion Tables, 1 to 100 39-40 Surfaces 41-44 Text 41 Table of Usual Measures 41-43 Table of Unusual, Special Trade, or Obsolete Measures 43-44 Table of Foreign Measures Digit Conversion Tables Volumes ; Cubic and Capacity Measures 45-55 Text 45-46 Table of Usual Measures 46-51 Digit Conversion Tables 51 Table of Unusual, Special Trade, or Obsolete Measures 52-54 Table of Foreign Measures 64-55 xi Xll TABLE OF CONTENTS. PAGES Weights or Masses 56-61 Text : 56-57 Table of Usual Measures 57-58 Digit Conversion Tables 59 Table of Unusual, Special Trade, or Obsolete Measures 59-60 Table of Relative Weights (used in Chemistry) 60 Table of Foreign Measures 60-61 Weights or Masses and Lengths; Weight of Wires, Rails, Bars; Forces, and Lengths; Film or Surface Tension; Capillarity. . 62-63 Pressures; 'Pressures of Water, Mercury, and Atmosphere; Stress or Force per Unit Area; Weights or Forces and Surfaces; Weights of Sheets, Deposits, Coatings, etc 63-67 Digit Conversion Tables 67 Weights or Masses and Volumes; Densities; Weights of Materi- als ; Masses per Unit of Volume 67-69 Weights and Volumes of Water 69-71 Table of Weights of Water 70 Table of Volumes of Water 71 Energy; Work; Heat ; Vis-Viva; Torque 72-77 Text 72-74 Tables 74-77 Digit Conversion Tables 77 Relations between Torque and Energy 78 Traction Energy 78 Tractive Force; Tractive Effort; Traction Resistance; Traction Coefficient . . . . : . . 78 Power; Rate of Energy ; Rate of Doing Work ; Momentum 79-82 Text 79 Tables 80-82 Digit Conversion Tables 82 Forces ; Weights Considered as Forces 83 Moments of Inertia. Text .' 84 Moments of Inertia in Terms of the Mass 84 Moments of Inertia in Terms of the Surface 84 Moments of Momentum ; Angular Momentum 84 Linear Velocities ; Speeds 85-86 Angular Velocities, Rotary Speeds 86 Frequency; Periodicity; Period; Alternations. 86-87 Linear Accelerations; Rate of Increase in Velocities; Gravity. . . . 87-88 Angular Accelerations; Rate of Increase in Angular Velocities. ... 88 Angles (plane) ; Circular Measures 89 Solid Angles 89 Grades; Slopes; Inclines 90-92 Conversion Table, 1 to 15% 92 Time i 93-95 Text 93 Tabl-s 94-95 Discharges; Flow of Water; Irrigation Units; Volume and Tune. 95 Electric and Magnetic Units 96-144 General Remarks 96 Mean , Effective , and Maximum Values 97 Resistance; Impedance; Reactance 97-100 Resistance and Length for the Same Cross-section 101 Resistance and Cross-section for the Same Length 101 Resistivity; Specific Resistance 101-104 Conductance; Admittance; Susceptance 104-105 Conductivity; Specific Conductance ; 106-107 Electromotive Force; Potential; Difference or Fall of Poten- tial; Stress; Electrical Pressure; Voltage 108-111 E. M. F. of Clark and Weston Cells at Different Tempera- tures Ill Electric Current ; Current Strength or Intensity 112-114 Current Density 114-115 Electrji al Quantity; Charge 115-116 Electrical Capacity 117 'uductance; Coefficient of Self- or Mutual Induction 118-120 TABLE OF CONTENTS. Xlll PAGES Time Constant (of Inductive Circuits) 120 Frequency; Periodicity; Period; Alternations 121 Kinetic Energy of a Current in a Circuit 122 Electrical Energy or Work 1 22-123 Electrical Power 124-125 Electrochemical Equivalents and Derivatives. 125-126 Electrolytic Deposits 126-127 Electrochemical EnePgy 128-129 Magnetic Reluctance; Magnetic Resistance 129-130 Magnetic Reluctivity: Specific Magnetic Reluctance; Mag- netic Resistivity; Specific Magnetic Resistance 130 Magnetic Permeance; Magnetic Conductance; Magnetic Capacity 130-131 Magnetic Permeability; Specific Permeance; Magnetic Con- ductivity 131-132 Magnetic Susceptibility , 132 Magnetomotive Force; Ampere-turns; Magnetic Potential; Difference of Magnetic Potential; Magnetic Pressure. . . * 132-134 Magnetizing Force; Magnetomotive Force per Centimeter; Magnetic Force; Field Intensity; Magnetic Calculations. 134-137 Magnetic Flux; Lines of Force; Flux of Force; Amount of Magnetic Field; Pole Strength 137-139 Magnetic Flux Density; Magnetic Induction; Lines of Force per Unit Cross-section; Earth's Field 140-142 Magnetic Moment 142 Intensity of Magnetization; Moment per Unit Volume; Pole Strength per Unit Cross-section Magnetic Work or Energy Magnetic Power ' 144 Photometric Units 144-149 Intensity of Light; Candle Power. . . 144-146 Flux of Light ; Spherical or Hemispherical Candle Power 147 Illumination 148 Brightness of Source Quantity of Light Light Efficiency; Power per Candle Power 149 Thermometer Scales 150-163 Reduction Factors for One Degree 150 Reduction Factors for Readings of a Temperature in Degrees. 150 Tables of Values from Absolute Zero to 6000 C . 151-163 Money * 164-166 Fluctuating Currencies 166 Money and Length 166 Money and Weight 167 Scales of Maps and Drawings Paper Measure Miscellaneous Measures 168 Useful Functions of TT 169-WO Useful Numbers 170 Systems of Logarithms 171 Acceleration of G ravity 171 Mechanical Equivalent of Heat. . . 17 1 Specific Heat of Water 171 Miscellaneous Foreign Measures 172-173 Index 175 SYMBOLS AND ABBEEVIATIONS USED IN THE TABLES AND TEXT. The page-numbers refer to the pages where the explanation or most impor- tant use is given. a acceleration, 19 A acre, 43 A, a ampere, 15, 113 a are, 43 ah ampere-hour, 116 ap apothecary measures, 46, 57 Aprx approximate within 2%, 28, 96 atm atmosphere, 66 av avoirdupois weights, 57 a-t ampere-turns, 132 B magnetic induction, 140 B, b, B, b susceptance, 22, 104 B.A.U. British Association unit, 99 bbl barrel, 53 Brit. British or English, 30 BTU Board of Trade unit (kilo- watt-hour), 77 BTU British thermal unit, 74, 75 bu bushel, 50 C,c, C,c capacity, electric, 23, 117 C, c coulomb, 15, 116 C Centigrade degrees, 150 c cord, 54 Cal caloric, large, 76 cal calorie, small, 75 Cal/min calorie (large) per min- ute, 81 cal/min calorie (small) per min- ute, 80 cal/s calorie (small) per second, 81 cb cubic. 46 Ccm circular centimeter 42 Cft circular foot, 42 eg centigram, 57 C.G.S. or CGS centimeter-gram- second, 11 ch chain, 32 Gin circular inch, 42 circ. circular 41 cl centiliter, 52 cm centimeter, 30 cm 2 square centimeter, 42 cm 3 cubic centimeter, 46 cm/s centimeter per second, 85 cm/s 2 centimeter per second per second, 88 CM circular mil, 41 Cmm circular millimeter, 41 cp candle power, 144, 146 cwt hundredweight, 58 d diameter, 41 dg decigram, 57 dkg decagram or dekagram , 59 dkl decaliter or dekaliter, 52 dkm decameter or dekameter, 32 dks decastere or dekastere, 54 dl deciliter, 47 dm decimeter, 30 dm 2 square decimeter, 42 dm 3 cubic decimeter, 48 ds decistere, 53 dwt pennyweight, 59 dyne-cm dyne-centimeter (erg), 7 4 dyne/cm dyne per centimeter, 62 dyne/cm 2 dyne per square centi- meter, 64 e base of Naperian logarithms, 171 e, e brightness of source of light, 16, 148 E, e, E, e electromotive force, 22, 108 Et electromotive force at t degrees, 111 E, e ell, 32 E, E illumination, 16, 148 eff . effective value elmg electromagnetic, 96 elst electrostatic. 96 e.m.f. electromotive force, 108 F Fahrenheit degrees, 150 F farad, 15, 117 F, f force, 18 F magnetomotive force, 132 fl fluid measures, 52 ft foot or feet, 30 ft 2 square foot, 42 ft 3 cubic foot, 49 ft-gr foot-grain, 74 ft-gr/s foot-grain per second, 80 ft-lb foot-pound, 74 XVI SYMBOLS AND ABBREVIATIONS. ft-lb/min foot-pound per minute, 80 ft-lb/s foot-pound per second, 80 ft/C foot per hundred, 90 ft /ft foot per foot, 90 ft/M foot per thousand, 90 ft/min foot per minute, 85 ft/ml foot per mile, 90 ft/s foot per second, 85 ft/s 2 foot per second per second, 88 fur furlong, 32 G, g, G, g conductance, 22, 104 g acceleration of gravity, 171 g gram, 58 gal gallon, 49 gi gill, 52 gr grain, 57 g-C gram -Centigrade heat unit, 75 g-cm gram -centimeter, 74 g-cm/s gram -centimeter per sec- ond, 80 g/cm gram per centimeter, 62 g/cm 2 gram per square centimeter, 64 g/cm 3 gram per cubic centimeter, 69 g/dm 2 gram per square decimeter, 64 g/h gram per hour, 127 g/m gram per meter, 62 g/min gram per minute, 127 gr/in grain per inch, 62 gr/in 3 grain per cubic inch, 68 H heat, 26 H henry, 119 H hydrogen, 126 H magnetic flux density, 140 H magnetizing force, 134 h hour, duration, 94 h hour, time of day, 93 ha hectare, 43 hg hectogram, 60 Hg mercury hhd hogshead, 53 hks hektostere, 54 hi hectoliter, 50 hp horse-power, 81 hp-h horse-power hour, 77 hp-m horse-power-minute, 76 hp-s horse-power-second, 7 ; 5 I,i,7, i current, (elec.) intensity of, 22, 112 1,7 intensity of light, 16, 144 I intensity of magnetization, 142 in inch, 30 in 2 square inch, 42 in 3 cubic inch, 47 in Hg inch of mercury column, 65 in/m inch per mile, 90 int. international J, j joule, 15,74, 123, 143 J mechanical equivalent of heat ,26 k electric inductive capacity, 14, 18, 23, 24 k dielectric constant, 23 K moment of inertia, 19 kg kilogram, 58 kg cal kilogram calorie, 171 kg-C kilogram Centigrade heat unit, 76 , kg-km kilogram -kilometer, 76 kg-km/min kilogram -kilometer per minute, 81 kg-m kilogram -meter, 75 kg-m/min kilogram -meter per min- ute, 80 kg-m/s kilogram -meter per second, 81 kg/cm 2 kilogram per equare centi- meter, 66 kg/cm 3 kilogram per cubic centi- meter, 69 kg/day kilogram per day, 127 kg/h kilogram per hour, 127 kg/hi kilogram per hectf liter, 68 kg/km kilogram per kilometer, 62 kg/1 kilogram per liter, 69 kg/m ki'ogram per meter, 63 kg/m 2 kilogram per square meter, 64 kg/m 3 kilogram per cubic meter, 68 kg/mm 2 kilogram per square milli- meter, 66 kg/t kilogram per ton (metric), 78 kg/yr kilogram per year, 126 kl kilpUter, 54 km kilometer, 30 km 2 square kilometer. 43 kw-h kilowatt-hour, 77 km/hr kilometer per hour. 85 km/hr/min kilometer per hour per minute, 88 km/hr/s ki'ometer per hour per second, 88 km/min kilogram -meter per min- ute, 86 kw kilowatt, 82 kw-m kilowatt-minute, 76 kw-s kilowatt-second, 75 L , 1 , L , I inductance , self-induction , 23, 118 L, I length, 18 1 liter, 48 Ib pound, 58 Ib-C pound Centigrade heat unit, 75 Ib-C/min pound-Centigrade heat unit per miute, 81 Ib-F pound-Fahrenheit heat unit t 75 Ib-F/min do. per minute, 81 Ib/bu pound per bushel, 68 Ib/day pound per day, 127 Ib/ft pound per foot, 63 lb/ft 2 pound per square foot, 64 Ib/ft 3 pound per cubic foot, 68 Ib/gal pound per gallon, 68 Ib/h pound per hour, 127 Ib/in pound per inch, 63 Ib/in 2 pound per square inch, 65 Ib/in 3 pound per cubic inch, 69 Ib/ml pound per mile, 62 Ib/qt pound per qt, 68 Ib/tn pound per ton (av), 78 Ib/yd pound per yard, 62 lb/yd 3 pound per cubic yard, 68 SYMBOLS AND ABBREVIATIONS. XV11 Ib/yr pound per year, 126 li link, 31 log logarithm, 171 log jo common logarithm, 171 loKe Naperian logarithm. 171 m magnet pole, strength, 20 M mass, 18 M mechanical equivalent of heat, 171 m meter, 30 m minute, duration, 94 m minute, time of day, 93 m 2 square meter, 43 m 3 cubic meter. 50 m/min meter per minute, 85 m/s meter per second, 85 m/sec 2 meter per second per sec- ond, 88 mg milligram, 57 mg/mm milligram per millimeter, 62 mg/s milligram per second, 127 mil 2 square mil, 41 min minute, 94 ml milliliter, 46 ml mile, 31 ml 2 square mile, 43 ml-lb m ile-pound , 7 6 ml-lb/min mib-pound per minute, 81 ml/hr mile per hour, 85 ml/hr/min mile per hour per min- ute, 88 ml/hr/sec mile per hour per sec- ond, 88 ml/min mile per minute, 86 mm millimeter, 30 mm 2 square millimeter, 41 mm 3 cubic millimeter, 52 mm Hg millimeter of mercury col- umn 65 mm/m millimeter per meter, 90 m.m.f. magnetomotive force, 132 mo month, 94 moJ gram molecule, 60 N number of turns, 20 n, n any number n frequency, 23, 87, 121 na nail, 31 O ohm, 15 O oxygen, 126 oz ounce, 58 oz/h ounce per hour, 127 oz/min ounce per minute, 127 P page p pole (length), perch, 32 p pressure, 19 P, P power, activity (mechanical, electric, magnetic, etc.), 19 20, 23, 124, 144 p.d. difference of potential, 108 "per" between two units means first divided by second, 4 phys. physical or physics pk peck, 49 pt pint, 47 Q. q. Q> a quantity of electricity, 22, 115 Q, Q quantity of light, 16, 149 qr quarter, 31, 53, 60 qt quart, 48 R, r, R, r resistance (electric), 22, 97 R Reaumur degrees, 150 R reluctance, 129 R rood, 44 r rod, 32 rev revolution, 89 rev/h revolution per hour, 86 rev/min revolution per minute, 86 rev/min/min revolution per min- ute, per minute, 88 rev/min/s revolution per minute per second, 88 rev/s revolution per second, 86 rev/s/s revolution per second per second, 88 rhp revolution per hour, 86 rpm revolution per minute, 86 rps revolution per second, 86 s second, duration, 94 second, time of day, 93 s shilling, 165 s stere, 51 S, s surface, 18 sec second sp specific sp. gr. specific gravity, 18 sq square, 41 S.U. Siemens unit, 99 subtr. subtract T, t time, 18 t temperature degrees, 111 t ton, metric, 58 t-km ton-kilometer, 78 t/cm 2 ton (metric) per square cen- timeter, 67 t/m 2 ton (metric) per square meter, 65 t/m 3 ton (metric) per cubic meter, 69 t/yr ton (metric) per year, 127 tn ton (avoirdupois), 58 tn-ml ton-mile, 78 tn/ft 2 ton (av) per square foot, 66 tn/in 2 ton (av) per square inch, 66 tn/yd 3 ton (av) per cubic yard, 69 tn/yr ton (ay) per year, 127 U, u, U, u difference of potential, 22, 108 U.S. United States v velocity, 19 v, v velocity of light, 14, 21 , 24, 96 V, V volume, 18, 69 V, v volt, 15, 110 vol. volume TV.w watt, 15. 80, 125 W weights, 69 W, W work, energy (mechanical, electric, magnetic, etc.), vis-viva, impact, 19, 20, 23, 122, 143 w-h watt-hour, 76 w-s watt-second, 74 X, x, X, x reactance, 22, 97 x e capacity reactance, 22 XV111 SYMBOLS AND ABBREVIATIONS. y Z Xp magnetic reactance, 22 Y, y, F, y admittance, 22, 104 yd yard, 30 yd 2 square yard, 42 yd 3 cubic yard, 50 r year, 94 , z, Z, z, impedance, 22, 97 a. (alpha) angle, 18 ft (beta) angle, 18 r (gamma) conductivity (electric), 22, 105 r 0.001 milligram, 59 d (delta) density, 18 (theta) temperature, 13, 18, 26 K (kappa) susceptibility, 20, 132 fj. (mu) magnetic permeability, 14, 18, 20, 131 n micron or micro-meter, 31 fifji milli-micron, 31 ^ (nu) reluctivity, 20, 130 n (pi) angle of 180, 89 n ratio of circumference to diam- eter, 169 P (rho) resistivity, 22, 101 (phi) angle of phase difference, 124 0, (phi) magnetic flux, 20, 137 flux of light, 16, 147 a) (omega) angular velocity, 19, 23, 86, 121 (B flux density, magnetic induc- tion, 20, 140 magnetic capacity, 20 JF magnet9motive force, 20, 132 3C magnetizing force, field inten- sity, 20, 134 . 3C flux density, 140 3 intensity of magnetization, 20, 142 3TC magnetic moment, 20, 142 (R reluctance, 20, 129 X multiplication * (hyphen) between two units means their product, 4 * division, first quantity divided by the second % per cent or per hundred, 7 , 90 Voo per mil or per thousand, 8, 90 y square root $/ cube root ~" 1 reciprocal J square, 41 * square root * cube or cubic. 46 cube root 10 n for condensed numbers, 8 > degree, 89 ' foot ' minute, 89 ' inch " second, 89 '" line (1/12 inch) pound sterling, 165 $ dollar in the United States, 166 3 dram, apothecary, 59 ^ ounce, apothecary, 59 9 scruple, 59 CO frequency, 87, 121 INTRODUCTION. INTER-RELATION OF UNITS. In order to establish a stable, systematic set of relations between the various units in use, the first requisite is to find which the proper funda- mental relations are, otherwise the secondary relations may have different values depending upon how they have been calculated, that is, they will form an unstable system. Such simple relations as those between pounds, ounces, grams, kilograms etc., are definitely established by law and are well known. The relations between units of length and units of capacity (volume) are slightly less simple but are also, or at least should be, estab- lished by law. The relation between what are commonly called weights (more correctly masses) and lengths, becomes somewhat more complicated, but by means of the mass of water these two kinds of units become defi- nitely connected with each other, at least in the metric system, and from that to all others; this relation is also defined by law. But with the more widely differing units the relations become more complicated. It may even seem at first thought as though there was no relation between such units as a foot and a horse-power or a watt; or between a pound and a degree of a thermometer, or between a foot and an ampere of electric current, etc. Yet all such units can be shown to be con- nected with each other, often more simply than might at first appear. In a set of values of different units in terms of the others it is therefore neces- sary to find and establish all the necessary fundamental relations, and no more, or else the system is not a stable one, and the derived values become different, depending upon how they are calculated. To find the connecting links between such widely different units, it is necessary to reduce them to some unit or quantity common to all, as a direct numerical equivalent can be given only between units of the same kind. This common unit, or quantity, and the only one, is energy. Energy can be expressed in terms of combinations of all the different units com- monly used in practice, and by expressing the same amount of energy in terms of these different combinations of units their relations to each other may be established. An analysis shows that there are three typically different sets of well- established units in common use, in terms of which energy is expressed or measured, and these three sets or groups include all the units generally used. The first and most stable or fundamental are the absolute units and all those having a known and invariable relation to them; these include the electrical units, which are the newest, and were wisely based on the absolute ones. They are the same throughout the universe; they are invariable and are independent of any constant of nature, except perhaps the unit of time, which depends on the revolution of the earth around the sun, but as this has been so accurately determined it may safely be considered here as an absolute quantity; it is at least invariable throughout the universe. The next group of units is the one involving the constant of nature called the attraction of gravitation of our earth, and they are therefore often called gravitation units; they are. therefore, purely terrestrial units, and would be quite different on other celestial bodies, and, in fact, are, strictly speaking, different even on different parts of this earth, although for most purposes they may be considered to be the same. They include such units as the pound or kilogram considered as weights, the horsa- 2 INTRODUCTION. power ay vfsually defined, etc. It will be found upon investigation that this whole group, is linked "with 'the first group, namely, the absolute and elec- trical units, 'through the value 01 the acceleration of gravity, a terrestrial constant of nature, and unfortunately one which is variable and has no universally .accepted normal value. If some standard normal value of this constant ^vcre cn*yersaily established and were fixed definitely, this group of 'units would be connected with the absolute units by a fixed relation, and the former would then all have definite and invariable values in terms of the absolute units. The acceleration of gravity is therefore the connect- ing link, and the only one, between these two groups. Throughout the tables in this book the standard value of the acceleration of gravity has been taken as 980.596 6 cm (the authority is given under units of Accel- eration). The third and last group of units is the one involving a property of water, which is also a constant of nature. This group includes such units as the calorie, the thermal unit, the degrees of thermometer scales, etc. It will be found upon investigation that this whole group of units is linked with the first group, namely, the absolute and electrical units, through the value of the specific heat of water; and moreover this is the only con- necting link. This constant differs from the acceleration of gravity, which is the connecting link between the first and second groups, in that it has not a variable value like gravity; but, on the other hand, its value has not yet been determined with such accuracy. It is like gravity in that it is a constant of nature whose value must be determined by experiment and definitely established before fixed and stable relations between this group of units and the absolute units can be established. The relations between the various units in any one group are, as a rule, fixed by definition or by law; the horse-power, for instance, is defined in terms of the foot-pound and the minute; or the calorie is defined in terms of the thermometer scale and a quantity of water. Such relations are there- fore established and require no experimentally determined constant in order to express their relations to each other; no matter what the value of the specific heat of water is, the relation between the calorie and the thermometer scale is fixed. But when we wish to express the values of any of the units of one group in terms of units of the other, then the numer- ical values of these two connecting links, namely, the acceleration of gravity and the specific heat of water, must be known. If, for instance, horse- powers are to be expressed in absolute or electrical units like watts, the value of gravity must be known ; or if calories are to be expressed in abso- lute or electrical units like joules, the value of the specific heat of water must be. known; or if calories are to be expressed in foot-pounds, then both these constants of nature must be known. Moreover, the values of gravity and the specific heat of water are the only two constants that must be known in order to reduce any of these different units of energy to any of the others. By accepting or fixing a definite value for each of these two constants the relations between all the units of these three groups become linked together into one single stable system in which all the relations between any of the units remain absolutely the same no matter how they have been calculated. As no standard values of either of these two constants have as yet been universally accepted or agreed upon, which is very unfortunate, the writer has, in the preparation of these tables, selected certain values as the ones which seem to be the best that exist at the present time, and which at least have a semi-official en- dorsement. As the establishment of fixed values for these two constants forms an inflexible stable system of units, there must exist some relation between these two constants themselves. It can readily be shown that the specific heat of water divided by the acceleration of gravity gives the mechanical equivalent of heat, when all are reduced to the same terms. The value used throughout this book is 426.9 kilogram-meters per kilogram-Centigrade heat unit (see authority under units of Energy). When thus defined this constant becomes a secondary or derived one.t which is as it should be, t While this is, strictly speaking, the more rational way of deducing those constants, yet the author has preferred to accept the simpler value above given for the mechanical equivalent and has made the specific heat the in- commensurable derived unit for reasons explained under units of Energy. INTER-RELATION OF UNITS. because it does not involve the absolute units, but is the relation between the second and third groups, both of which are separated from the absolute units by one of these two empirical and therefore less stable constants. If the heat units and what are called the gravitational units, like the pound (weight), the horse-power, etc., had originally been denned in terms of the absolute units, as was wisely done with the electrical units, there would have been no necessity of knowing the values of gravity, the specific heat of water, or the mechanical equivalent of heat, in order to establish fixed and invariable relations between all the units in use. Such fixed rela- tions would then be established definitely by definition. Concrete stand- ards of measurement complying as closely as possible with the defined values would then require the experimental determinations of these con- stants, but the exact theoretical relations would not. Such is the case with the electrical units which are defined with absolute accuracy in terms of the absolute units and require no experimentally determined constant to connect them with the absolute units; in the electrical system of units the problem therefore is not to find such an empirical relation, but to deter- mine concrete standards which comply with the defined relations as closely as possible. If those on whom the duty will fall to establish a system of units of light would follow the good example set by those who established the electrical units, and base them on the absolute system of units, instead of on another different constant of nature, like the candle or incandescent platinum, they would avoid creating a fourth group of units whose empirical relation to the absolute, electrical, and to the other two groups of units would have to be determined, and would be uncertain until it was. What is called radiated light, usually measured in spherical candle-power, is a true power or a rate of work, and the absolute unit would therefore be a dyne-centimeter per second. If the practical unit could be defined as a decimal multiple or fraction of this, like the watt is, it would at once become a definitely known unit. It would then become necessary, as it was with the three funda- mental electrical units, to establish a concrete standard which would com- ply as definitely as possible with the defined unit. If, however, the prac- tical unit is defined on some independent basis, like the present unsatis- factory and uncertain temporary light units, it will become necessary to determine experimentally the mechanical equivalent of light, before any of the much-needed relations could be established with the other existing units, like the electrical, gravitational, or heat units. In the case of light units the matter becomes complicated on account of the different wave- lengths and combinations of wave-lengths which have different effects on the eye in perceiving light. Owing to the absence of any known relations between light energy and energy stated in other units, no relations ^between those groups of units could be given in this book. The relations above described, which exist between the different groups of units, maybe represented graphically as shown in the following diagram: GRAVITATIONAL, UNITS and their derivatives; gram or pound as weights, foot-pounds, horse- powers, etc. HEAT UNITS and their derivatives; calorie, thermal unit, thermometer scale, etc. LIGHT UNITS and their derivatives; candle power, lux, etc. ABSOLUTE or ELECTRICAL UNITS and their derivatives; gram or pound as masses, dyne, centimeter, foot, second, watts, joules, etc. 4 INTRODUCTION. The most fundamental, namely, the absolute or electrical units, are shown at the bottom. The gravitational units to the left are .shown linked to these through the value of the acceleration of gravity. The heat units to the right are shown linked to the absolute and electrical units through the value of the specific heat of water. Finally, the gravitational units are linked to the heat units through the mechanical equivalent of heat, which is the quotient of the other two linking values. The relation of any one of the three to either one of the others can be found either by means of the direct link or indirectly by means of the two other links; the result must be the same if the values expressed by these links are definitely established. The group of units of light at present in temporary use are shown as a fourth group, the connecting links of which are unfortunately not yet known, for which reason no relations to other units can be given, although such definite relations it seems ought of necessity to exist at least when light is considered as a radiation of some definite wave-length. COMPOUND NAMES OF UNITS. When in compound names of units of measurement the names of the sim- ple units are joined by a hyphen, as in foot-pounds, horsepower-hours, etc., it signifies the product of the simple units; that is, an amount of energy stated in foot-pounds means that the number of feet is multiplied by the number of pounds, to give the foot-pounds; in other words, the compound quantity varies directly with each of the others. If, however, the simple units are joined by the word "per" as in feet per second, pounds per mile, etc., it means that the first is divided by the second; that is, a velocity in feet per second means that the number of feet is divided by the number of seconds. In other words, the compound quantity varies directly with the first and in- versely with the second. An acceleration in "miles per second per second " means that the velocity in miles per second is divided by the number of seconds during which this velocity was acquired. The above is the correct practice, but it is unfortunately not followed by all writers. For instance, "horsepower per hour" is wrong; it should be horsepower-hour, as the number of hours is a multiplier and not a divisor. DISTINCTION BETWEEN UNITS AND QUANTITIES MEASURED IN THOSE UNITS. Serious errors are not infrequently made by failing to distinguish between the calculations of the values of the units themselves and the calculation of quantities measured in terms of those units. By using the reduction factors in the way they are given in the tables in this book, no mistakes can be made as the units have all been calculated, and the value of each unit is given in terms of each of the others; they are just like the selling prices of one apple in terms of different moneys, the only remaining calculation is to multiply the quantity of these units by the reduction factor given in the table; thus, if 1 foot-pound = 0.14 kilogram-meter, then 3 of these foot- pounds = 3X0.14 kilogram-meters. But in determining the reduction fac- tors themselves, or in similar calculations, which are not infrequent, errors are very apt to be made by multiplying where one should divide. A fre- quent case is that in which a formula reads in, say, distances in meters and weights in kilograms and it is desired to change it to read in feet and pounds; this will be described in the next section; or, in dealing with compound units, such as foot-pounds (ft X Ibs) and feet per pound (ft *- Ibs), which were described above. A clear distinction should be made between the units themselves and the quantities measured or expressed in terms of those units. Thus, 1 foot = 12 inches is the value of one unit in terms of another, but it is not correct to interpret this by saying: a length in feet = 12 X its length in inches, as that would be 144 times too great. It should be interpreted as follows: If 1 foot equals 12 inches, then a length of any other number of feet must be mul- tiplied by 12 to reduce that length to inches, or to express it in terms of inches. Simple as this may seem in this almost self-evident illustration mistakes are easily and frequently made in more obscure relations, especially DISTINCTION BETWEEN UNITS AND QUANTITIES. 5 in substituting one kind of a unit for another in a formula, which will be described in the next section. The larger the unit the smaller will be the number of those units which are contained in a given quantity; that is, the smaller will be the number ex- pressing the size of that quantity in terms of those units. For instance, a meter is greater than a yard, hence the number of meters in a given distance is less than the number of yards. This self-evident rule is often of value as a check to avoid confusion between the calculation of the units them- selves and the quantities in terms of those units; errors of this kind are especially liable to occur when the two units have nearly the same values. This rule should not be confounded with another quite different case in which the values of two different units are given in terms of the same third unit; for instance, 1 kilogram -meter = 2. 3 heat units and 1 foot-pound = 0.32 of the same heat units; here the foot-pound is a smaller unit than the kilo- gram-meter, hence its value in terms of a third unit will of course also be smaller. But 1 heat unit = 3.1 foot-pounds and 1 heat unit = 0.43 kilogram- meter; here the same quantity, 1 heat unit, is expressed first in small units, namely, foot-pounds, then in large units, kilogram-meters; hence 3.1, the number of the smaller units, is of course greater than 0.43, which is the number of the larger units. In calculating one unit from another unit, great care must be taken to avoid multiplying when one should divide, or the reverse. Thus if the unit 1 kilogram-meter equals 2.3 heat units, it would not be correct to say that because the unit 1 kilogram = 2. 2 Ibs and 1 meter = 3. 3 feet, therefore the unit 1 foot-pound = 2. 3 X (2. 2X3. 3) = 16. 8 of those heat units; it should be 1 foot-pound = 2. 3 -4- (2.2X3. 3) = 0.31 heat units. The safest way to avoid such errors is to remember that the statement 1 kilogram-met er= 2.3 heat units really is a true equation between units and means 1 kilogram X 1 meter = 2. 3 heat units; one should then substitute for 1 kilogram its equal in terms of the other unit, namely 2.2X 1 pound, and for 1 meter its equal, namely 3.3X1 foot; then 2.2X1 pound X 3.3 XI foot = 2. 3 heat units, which when reduced gives the unit 1 foot-pound = 2. 3 -=-(2. 2X3. 3) or 0.31 heat unit. These terms here always refer to units and not to quantities meas- ured in terms of those units, which latter is the case in formulas. Expressing this relation in algebraical terms, it means A X B = 2. 3C, in which A is one kilogram and not a weight denoted in kilograms, B is one meter and not a distance expressed in meters, and C is one heat univ ; they should here be considered as concrete things like a certain piece of brass, a certain stick, and a certain lump of coal, as distinguished from abstract units or as distinguished from a mere number representing a measurement in terms of those units. Now if in the same way a is one pound or a differ- ent piece of brass, and b is one foot or a different stick, then as 1 kilogram = 2.2 pounds, it follows that ^4=2.2a and similarly /? = 3.36. Substituting these in the first equation gives 2.2aX3.3b = 2.3C, which when reduced gives aX6 = 2.3-K2.2X3.3)C = 0.31C. In formulas, however, the letters do not stand for the units themselves, but for quantities measured in terms of the units, and hence one must multi- ply instead of divide, or the reverse, PS will be explained in the next section. A similar error is very apt to arise in calculating units containing the word "per," which word signifies a division, as was explained above under compound units. This is the case, for instance, in finding the value of 1 kilogram per kilometer from the value of 1 pound per foot, or a velocity of 1 mile per minute from that of 1 foot per second. It must be remembered that the unit following the word "per" is a divisor. The safest way always is, as above described, to write out the whole expression carefully, then substitute equivalent values and reduce. Thus: 1 mile per minute = 26. 8 meters per second, may be written 1 mile -*- 1 minute = 26.8 meters per second ; then to find the value of 1 foot per second from it, substitute 5 280 feet for 1 mile, and 60 seconds for 1 minute, thus (5 280 X 1 foot) -J- (60 X 1 second) = 26.8 meters per second; then reduce by dividing both sides by 5 280 and multiplying both by 60, giving 1 foot-f-1 second, or 1 foot per second = 26.8 -s-5 280X60 = 0.30 meter per second. When the unit following the word "per" is not to be changed, then the division just referred to does not enter into the calculation. Thus to reduce some value of 1 foot per minute to that of 1 mile per minute or to that of 1 meter per minute, is merely a reduction of feet to miles, or to meters. INTRODUCTION. REDUCING FORMULAS FROM ONE KIND OF UNITS TO ANOTHER. It often occurs in practice that a formula is given for one set of units, such as meters, kilograms, seconds, etc., and it is desired to use it for other units such as .feet, pounds, minutes, etc. While this is generally a simple calculation, it is very apt to be made incorrectly, giving an entirely wrong result. The error arises from the fact that one is apt to forget the distinc- tion between a unit and a quantity measured in terms of that unit (see the preceding section) which distinction involves the difference between whether one should multiply or divide. For instance, a foot is larger than an inch, but the number of feet expressing a certain distance is smaller than the num- ber of inches expressing that same distance. The best way to avoid mistakes is to remember that in the usual formulas the letters represent quantities measured in terms of certain units; they do not represent the units themselves. To change a formula from one kind of units to another, it is therefore necessary to replace each letter of the original formula by another letter combined with such a reduction factor as will reduce the measurement made in terms of the new unit, to that made in terms of the original one. Thus if a formula contains the letter L as repre- senting length expressed in meters, and it is desired to change this to I expressed in feet, then substitute for L the equivalent Z-j-3.28 or ZX 0.305, because any length I measured in feet when divided by 3.28 (or multiplied by 0.305) will be that same length expiessed in meters; and as the original formula is correct for meters, it will be correct to substitute IX 0.305 for L because the number thus obtained will be the same as the number which expresses L in meters. Or, if the original formula contains the letter W, representing weight expressed in kilograms, and it is desired to change it to w expressed in pounds, then substitute for W the equivalent wX 0.454, because any weight in pounds when multiplied by 0.454 will give the same number as when expressed in kilograms, which latter number is what the original formula called for. After having thus replaced each letter by a new one and a constant, all these constants may be combined into one if desired. It must not be forgotten that the quantity which the whole formula rep- resents, that is, the quantity which is to be calculated by means of the formula, is generally also in terms of some unit (unless it is a mere ratio, percentage, or number), and care must therefore be taken to also substi- tute a new letter and reduction factor for it, if it is desired to change it also. For instance, if a formula for determining the metric horse-power from data given in meters and kilograms, is to be changed to read in feet and pounds, as just described, the value obtained in applying the new formula would still be in metric horse-power, notwithstanding that the formula has been reduced to feet and pounds; if the result is to be in English horse-powers, then the letter representing the metric horse-power must likewise be changed to a new one with its reduction factor, just as was clone with the others. This will be illustrated below by an example. The safest rule of thumb for the changing of the units of a formula is therefore as follows: Substitute for each letter of the old formula a new letter multiplied by the value of the NEW unit in terms of the OLD one, as given in the tables in this book. Then the new letters will represent quantities meas- ured in terms of the new units. Thus for L (meaning meters) substitute I (meaning feet) X 0.305, this number being the value of one foot (new unit) in terms of meters (old unit), as obtained from the tables. For if 1 foot = 0.305 meters, any other number of feet (namely Z) must be multiplied by 0.305 to reduce it to the number of meters (L) called for by the original formula; or, in other words, L and IX 0.305 arc equivalents, and either may be substituted for the other. Or put into algebraical terms: if according to the table 1 foot = 0.305 meter, then I feet are equal to IX 0.305 meters; or a length represented by I in feet is represented by IX 0.305 in meters; but the latter quantity is numerically equal to I/, hence L = lX 0.305, which is a true equation in which the letters 'represent the same length measured first in meters and then in feet; and (IX 0.305) may therefore be substituted for L in any formula. . RATIOS. PERCENTAGE. 7 The above rules are illustrated in the following example: Let in which P represents the power in metric horse-power which is required to operate a windlass or elevator to raise W metric tons L meters in height in T seconds. The constant 16 includes the friction loss, and the reduction factors. It is required to change this to English horse-powers (p), short tons (w), feet (Z), and minutes (/). From the tables, one English horse-power (the new unit) is equal to 1.01 metric horse-powers (the old unit), hence P = pXl.01. One short ton of 2000 Ibs. is equal to 0.907 metric ton, hence JF = u?X0.907. One foot equals 0.305 meter, hence L = lX 0.305, and one minute equals 60 seconds, hence T = oo%- For five places H>oo% to J4ooo%t etc. The possible error is also less as the left-hand digit is greater. Thus the abbreviated number 10.1 may mean anything from 10.05 to nearly 10.15, and the greatest error may therefore be 5 in the next (2d) place of deci- mals, or numerically this would be 5 times the left-hand digit 1. But in the abbreviated number 99.9, which may mean anything from 99.85 to nearly 99.95, the greatest error, namely 5, is only about 5 /io times the left-hand digit 9; that is, in percentage only about Vio as great an error as it was for 10.1. It follows therefore that values beginning (at the left hand) with the low digits 1, 2, 3, etc., should be stated to one place more than those beginning with the high digits 9, 8, 7, etc., if the accuracies of the 10 INTRODUCTION. abbreviated values are to be more nearly the same for all. This can be made use of to advantage in tables of such approximate or abbreviated values. If, for instance, such a table is intended to be limited to three places of figures, the greatest error will be 0.^% for the 1 w numbers (100) and 0.05% for the high ones (999); but by giving f ; ur places of figures for all numbers between 100 and 499, and three places for those from 500 to 999, making an increase of only 1 figure for every 6, or about 16%, the greatest error will be reduced fivefold, namely to 0.1%. The table will then take a mean position (in accuracy) between a throe-pk.ce and a four-pi: ,ce table; the variations of the greatest errors in diiierent parts of the table will, however, be the same in all taree. The location of the .ecimal point Ices not, of course, affect the percentage accuracy; thus 101., when it represents an approximate or abbreviated value, is just as accurately stated as 0.000 1:1 is. A zero (0) at tae right-hand end of a whole number may mean either a definite known quantity like a digit or ar. indefinite unknown one, merely filling a vacant pb.ce of figures; there is unfortunately no way of distin- guishing between them. Thus in the value 5280. feet to the mile, the zero might either mean that the last place of figures is exactly correct (which is the case in this particular value) or that only three places are correct and the fourth unknown. In a decimal fraction, however, a zero in the last place is always, or should always be, understood to be a definite known quantity; thus 0.150 is understood to be known or correct to three places, while 0.15 is known or correct to only two places. The following table gives the errors in abbreviated numbers in a more concise form: Abbreviated numbers. Greatest error. Mean probable error. 10 5.% 2.5% 50 1.% 0.5% 100 0.5% 0.25% 500 0.1% 0.05% 1 000 0.05% 25 parts in 100 000 5 000 0.01% 5 parts in 100 000 10 000 5 parts in 100 000 25 parts in 1 000 000 50 000 1 part in 100 000 5 parts in 1 000 000 100 000 5 parts in 1 000 000 25 parts in 10 000 000 500 000 1 part in 1 000 000 5 parts in 10 000 000 It will thus be seen that by using only three places of figures to represent any data or results (provided the last right-hand figure has been raised by unity when the next is 5 or over) the greatest possible error is only half a percent, and the probable mean error is about Vio to %<>% Three places of figures therefore suffice for most engineering data. In the more usual simple calculations with three-place values, the mean probable error in the result is in general no greater than this, but the greatest possible error becomes larger and may affect the third place by one unit. Successive multiplications like cubing increase the error. Hence if the result is to be quite correct to three places, then four places must be used to start with. ACCURACY OF LOGARITHMS. In general, the number of (decimal) places in the logarithms themselves should be one greater than the number of places of figures, in order to be as accurate (in percent) as the arithmetical calculations would be. The errors are therefore easily obtained from the above table for numbers. Four- place tables are therefore sufficiently accurate for three-place figures. ABSOLUTE SYSTEM. C. G. S. SYSTEM. 11 ABSOLUTE SYSTEM of UNITS. The O. G. S. SYSTEM. Among the physical quantities there are some that are fixed, definite, independent of each other, and invariable all over the universe, while others are variable, indefinite, dependent, arbitrary, or involve empirical con- stants. A length, for instance, belongs to the former class and is invariable throughout the universe, while a weight considered as a force, is different on different parts of the earth or on different planets. The former are for the sake of a distinction, called absolute quantities. Most physical quantities are dependent on others or are derived from others; for instance, a surface is the product of two lengths, and a velocity is a length passed through in a certain time. But there are a few that are independent of any others and may, therefore, be considered to be funda- mental; a length, for instance, cannot be derived from anything else. By selecting three of the latter from among the absolute quantities most and perhaps all of the other physical quantities may be derived from them, or defined in terms of them, thus making a single uniform system in which all quantities can be expressed in terms of some function or combination of these three fundamental quantities. When a definite amount of each of these fundamental quantities is taken as a unit, such a system is called an absolute system of units. Various systems of this kind have been devised differing in the three quantities which have been selected as the fundamental ones. The most important one and the only one which is in general use, is based on the three quantities, length, mass* and time, usually represented by the letters L, M , and T respectively, or I, m, and t, and when the unit amounts are taken as the centimeter the gram (mass), and the second (mean solar), the system is called the centimeter-gram-second system, usually denoted as the C. G. S. system. In this system the units of each of the derived quantities are the amounts which correspond to the unit amounts of those of the fundamental quanti- ties which are involved. Thus the unit of velocity in this system is one centimeter per second; the unit of force, called a dyne, is that whi-;h act- ing on a mass of one gram at rest produces in one second a velocity of one centimeter per second; the unit of energy, called an erg, is one dyne of force acting through one centimeter; the unit of electricity or magnetic pole is that which attracts another equal amount at one centimeter dis- tance, with a force of one dyne; etc., etc. Sometimes the units of the derived quantities depend upon how they are defined, in which case there may be several different units. The most important case of this kind is that of the electrical units; in the one system, called the electrostatic system, a whole set of units is based on the definition that a unit of electricity is that which attracts an equal amount one centi- meter distant, with a force of one dyne, while in the other, called the electro- magnetic system, the unit current is defined as that which, flowing through an arc of one centimeter, curved to one centimeter radius, generates a unit magnetic pole at the center; this definition thus connects the unit of elec- tricity with the unit of magnetism ; on it the electrical units in common use are based. The relation between the two units of each electric quantity thus defined is found to be the velocity of light in the C. G. S. system, or some power of it. * Mass represents amounts of matter and must be clearly distinguished from weight. A given mass of iron, for instance, is the same whether it be solid, liquid, volatilized, oxidized, dissolved, etc., and is the same all over the universe, while its weight is enormously greater on the sun than -on the earth. Two given masses always have the same attraction of gravi- tation to each other at the same distance. 12 INTRODUCTION. Sometimes the units thus defined are inconveniently large or small for practical purposes, and therefore certain practical units have been chosen which are made some multiple of 10 times as small or as large as the C. G. S. unit. Thus the ampere is 1/10 and the volt is one hundred million times the C. G. S. unit (electromagnetic) of current arid electromotive force respectively. The definitions of all these units and the relations between them are given in their respective places in the table of conversion factors. While most physical quantities can thus be deduced from, or denned in terms of, the centimeter, the gram, and the second, yet it has not yet been possible to do this with all. Among the important exceptions are tem- perature, magnetic permeability, and electric inductive capacity; they should therefore be added to the fundamental quantities in defining or deducing some of the quantities. For most purposes the two latter may be, and in fact are, eliminated from consideration by making them equal to the numeral 1 or unity in the definitions. They are therefore called "suppressed" fundamental quantities. There are in general two ways of establishing units of various quantities. One way is to define an absolute unit in accordance with the C. G. S. system, and then establish a concrete unit in terms of the absolute one, which will represent the latter or some simple decimal multiple of it, as closely as possible; this seems to be the more rational way, as it maintains the whole system uniform, although it is sometimes difficult to establish the concrete unit correctly; this is the method adopted in establishing the electrical and magnetic units. The other way is to arbitrarily establish the concrete unit first, by selecting some convenient standard, and then determining experimentally what its relation is to the natural, absolute unit; this is the method which was adopted in establishing the units of weight (con- sidered as a force and not as a mass), heat, temperature, light, etc.; it has the disadvantage that the relations to other units then always involve an empirical constant which is necessarily incommensurate, like the accelera- tion of gravity, the mechanical equivalents of heat and of light ; but it has the advantage that the concrete unit is established definitely at the start, and is not subject to occasional readjustment like the concrete electrical units. DIMENSIONAL FORMULAS. When any physical quantity is represented algebraically in terms of the fundamental quantities, the expression is called its dimensional formula. Thus, in the length, mass, and time or L, M, and T system, a surface which is the product of two lengths is represented by L X L or L 2 , a volume by L 3 ; these are the dimensional formulas of a surface and a volume, and the exponent of the letter is called its dimension. It frequently happens in the more complex formulas, that some of these letters occur as divisors, that is, as the denominator of a fraction, and to avoid stating the formula in terms of a fraction, the exponents are made negative in such cases. Thus a velocity which is length divided by time, or L/T, is generally written LT~ *; an acceleration is a velocity divided by time or LT~ l /T, which is written LT~ 2 ; a force is this multiplied by mass or L M T~ 2 and energy is a force multiplied by a length or L 2 M T~ 2 , etc. Fractional exponents denote roots, thus La means the square root of L; or L~~i denotes the cube of the square root of L in the denominator, etc. In some cases the letters cancel each other, leaving the exponent 0; the dimensional formula is then simply 1, sometimes called a number; an angle, for instance, is defined as an arc divided by the radius, that is Z/-5-Z/ = Z/ = 1; DIMENSIONAL FORMULAS. 13 the same is true of an efficiency which is energy divided by energy, or power divided by power, etc. When intelligently interpreted and applied, such dimensional formulas are often very useful. They frequently give an idea of the physical nature of a quantity, and more particularly of its relation to other quantities; they sometimes show that several quantities which have been defined in entirely different ways, and which originated differently, are really the same; they sometimes point out the existence of some law; they aid one in determining what the rational unit is, of a quantity for which no unit has been chosen; they are useful for finding whether a physical formula is correct and complete; these are only a few of the uses of such formulas. The quotient of two such formulas often shows that the relation between the quantities is a third simple, well-known quantity. Thus the relation between energy and pressure is a volume, or the relations between the formulas for the electrical units defined electrostatically, and those defined electromagnetically, is in this way shown to be the velocity of light or its square, which is one of the foundations of Maxwell's electromagnetic theory of light. It must not be forgotten that the dimensional formulas usually used are based on the fundamental units, length, mass, and time, and that they would be quite different if other fundamental quantities are used ; they are therefore only relative and not absolute, and their chief use is therefore to show the relations between different quantities rather than the physical nature of any one considered by itself. Although dimensional formulas are very useful, great caution should be exercised in applying them, as inconsistencies and absurdities may result if they are treated as mere algebraic formulas. Energy, for instance, is force X length, which gives the formula L 2 M T~ 2 , but torque is also force X length, and therefore has the same formula, although it is physically an entirely different quantity. When torque acts through an angle it becomes energy, hence torque X angle = energy, but as the dimensional formula of an angle is 1 the formula for torque is not changed by multiplying it by an angle. The length involved in torque is perpendicular to the force, while in energy it is in the same direction, therefore the two lengths have a differ- ent physical meaning in the two cases, as they are at right angles to each other; as this cannot be indicated in the formula, it shows that the system is defective. The "angle" in such a case has been termed a "suppressed quantity" in the formula, and it is due to such quantities that errors may arise in using dimensional formulas. Its real existence in the formula for torque should at least be indicated by some auxiliary letter like a, for in- stance, calling attention to its suppression in the rest of the formula. It seems to have been impossible so far to find the true dimensional formula of temperature, or even to determine whether one exists; it is therefore safest at present to consider it an auxiliary fundamental quantity, usually represented by 6, and add it to the formula. Further remarks on this sub- ject will be found in a footnote under the thermal units in the table of physical quantities given below. In the formulas for the electrical and magnetic units there are also some "suppressed" quantities similar in some respects to the angle above men- tioned; like in the case of temperature, their dimensional formulas are not known, they are generally omitted in the formulas for other derived quantities, but without them these formulas are not complete and may mislead. Ordinarily, they are considered to be unity and therefore are algebraically eliminated from the formulas, but as was shown above con- cerning the quantity "angle," this may lead to misconceptions. Until their dimensional formulas are known it is recommended to consider them as auxiliary fundamental quantities and to add to the formulas a symbol which represents them, in order to call attention to the fact that they really exist but are suppressed in the particular system used.* In the dimensional formulas given below in the table of physical quantities, they have been * Prof. Ruecker, in the paper referred to below, says: "I think the sym- bols are thus made to express the limits of our knowledge and ignorance on the subject more exactly than if we arbitrarily assume that some one of *,he quantities is an abstract number." 14 INTRODUCTION. added in parentheses; for ordinary purposes they may be considered as being unity. These quantities are the electric inductive capacity repre- formulas should be unity; a definition, for instance, might be based on a unit distance, but it would be quite wrong to therefore leave out of the dimensional formula the L which represents it. Energy being always the same quantity in all systems, never has any of these suppressed factors in its formula; the same is true of power. Although the dimensional formulas of these two suppressed quantities individually are unknown, that of both combined is known in the follow- ing form: in which v is the velocity of light in the C. G. S. system ; hence when the formula of one of them is known, that of the other can be determined. It was thought by Williams (see reference below) that fi may be a density. The dimensional formulas of the photometric quantities have, it seems, never been given. Those in the following table are suggested by the author. A concise discussion of the deduction of the dimensional formulas of many of the usual physical quantities will be found in the Smithsonian Physical Tables, edited by Prof. Thomas Gray, 2d edit. p. xv to 4. The "suppressed" quantities are discussed in a paper by Ruecker in the Phil. Mag., Feb. 1889, vol. 27, p. 104, supplemented by another by Williams, Phil. Mag., 1892, p. 234. The table of physical quantities given below con- tains all the quantities whose formulas are given in these references besides numerous others. DECISIONS OF INTERNATIONAL ELECTRICAL CONGRESSES Concerning Electric, Magnetic, and Photometric Units and Definitions. THE following is a brief summary of these decisions, adoptions, and recommendations.* The official congress of 1881 in Paris adopted the fundamental units centimeter, gram (mass), and second, for the electric measures; the ohm as equal to 10 9 C. O. S. units t; the volt as equal to 10 s C. G. S. units f; the ampere as the current produced by a volt through an ohm; the coulomb as the quantity corresponding to an ampere for one second; the farad as the capacity corresponding to a charge of a coulomb by a volt. It in- structed an international commission to determine what the length of a in order to have a resistance of one ohm as above defined; the commission reported in 1884 that this length was 106 centimeters and called this ohm the legal ohm; it also recommended that this be adopted internationally; it also recommended that the ampere be made equal to 10 -1 C. G. S. (electromagnetic) units, and that the volt be that electromotive force which maintains a current of one ampere (presumably as just defined) through a resistance of one legal ohm; (this volt has since often been re- ferred to as the legal volt, although there seems to be no legal sanction for this name). The congress of 1881 also recommended that an inter- national commission be appointed to define a standard of light; this com- mission reported in 1884 that the unit of each kind of simple light be * For a more detailed summary up to 1900 see Recapitulation des Deci- sions des Conyres Anterieurs, by Hospitalier, in the report entitled "Con- gres International d'Electricite/' 1900, pages 11-22; for the adoptions of the congress of 1900 see pages 369 and 370 of that report. f The electromagnetic system was unquestionably meant, and is under- stood to be meant in all that follows here. DECISIONS OF CONGRESSES. 15 the quantity of light of the same kind emitted perpendicularly from a square centimeter of surface of melted platinum at the temperature of its solidi- fication; and that the practical unit of -white light be the total light thus emitted. (The name violle has since come into use for this unit, although apparently without official adoption.) The official congress of 1889 in Paris adopted the joule as equal to 10 7 C. G. S. units of work (ergs) and defined it also as the energy represented per second by one ampere through one ohm; (the ohm here referred to is presumably that defined in terms of the absolute system, and not the "legal " ohm); the watt as equal to 10 7 C. G. S. units of power (ergs per second), and therefore equal to a joule per second; the kilowatt as the industrial unit of power in place of the horse-power; the bougie dec! male (decimal candle) for the practical unit of light as equal to the twentieth part of the absolute standard of light defined by the commission in 1884 (namely, the Elatinum standard described above and called the violle); the quadrant :>r the practical unit of self-induction (now called inductance) as equal to 10 9 centimeters; the period of an alternating current was defined to be the duration of one complete oscillation, and the frequency the num- ber of periods per second; the mean intensity (of a current) was defined by an algebraic expression which signifies the arithmetical average of all the instantaneous values; the effective intensity (of a current) was defined to be the square root of the mean square of the intensity of the current, and the effective electromotive force, the square root of the mean square of the electromotive force ; the apparent resistance (now called impedance) was defined to be the factor by which the effective in- tensity of the current must be multiplied to give the effective electromo- tive force; the positive plate of an accumulator was defined to be that which is the positive pole during discharge. The unofficial congress of 1891 at Frankfort (Germany) decided * that all units be expressed in Roman type, all physical quantities in italics, and all physical constants and angles in Greek type; also that the quantities ampere, coulomb, farad, joule, ohm, volt, and watt be expressed by their initial letters, A, C, F, J, O, V, and W. (These decisions were not con- firmed by the subsequent official congress, and have not come into general use.) The official congress of 1893 in Chicago f adopted the international ohm, based on the ohm equal to 10 9 C. G. S. units and represented by the resistance of a column of mercury at C., 106.3 centimeters long, weigh- ing 14.452 1 grams, and having a uniform cross-section; (this cross-sec- tion, although not so stated officially, is practically one square millimeter); the international ampere as equal to lO" 1 C. G. S. units, represented for practical purposes by the current which will deposit 0.001 118 gram of silver per second; the international volt as that electromotive force which will maintain one international ampere through one international ohm, represented for practical purposes by 1-4- 1.434 of that of a Clark cell at 15 C. ; the international coulomb as the quantity corresponding to one international ampere in one second; the international farad as corresponding to a charge of one international coulomb by one interna- tional volt; the joule as equal to 10 7 C. G. S. units, and represented in practice by the energy in one second of an international ampere passing through an international ohm ; the watt as equal to 10 7 C. G. S. units and represented in practice by one joule per second; the henry as "the induc- tion in a circuit when the electromotive force induced in this circuit is one international volt, while the inducing current varies at the rate of one ampere per second'-' (presumably meaning the international ampere). These eight units, substantially as defined by this international congress, were made legal in the United States by Act of Congress in 1894. For practical magnetic units, the C. G. S. units were commended by the Chicago congress, but no names were given them; a report of a committee on notation and nomenclature was received and ordered printed as an appen- * See Electrical World, vol. 18, 1891, page 248. t For further details see page 20 of the Proceedings of the International Electrical Congress held at Chicago, 1893, published by the Amer. Inst. Elect. Engineers. INTRODUCTION. dix, no further action being taken on it; (see the table of physical quanti- ties below, in which that report is included). The unofficial congress of 1896 in Geneva adopted the bougie decimale angle; the lux as the unit 01 illumination [E] equal to one lumen per square meter; the bougie per square centimeter as the unit of brightness [e\\ the lumen-liour as the unit of quantity of light [O]. The official congress of 1900 in Paris adopted the name gauss for the C. G. S. unit of intensity of magnetic field or flux density, and the name maxwell for the C. G. S. unit of magnetic flux. For the values of these official units in terms of each other and of other units, see the respective tables of measures. PHYSICAL QUANTITIES AND RELATIONS. 17 TABLES of PHYSICAL QUANTITIES and RELATIONS. The following table gives the physical quantities and relations in use, with their names, symbols, derivation, dimensional formulas in the C. G. S. system, and the units whenever such units have been generally adopted. A similar though much smaller table, limited chiefly to the more important electrical and magnetic quantities, was recommended by the Committee on Notation of the International Electrical Congress of 1893 in Chicago;* this has been included here after revision, correction, and some rearrange- ment. The "suppressed" factors k and have been added, which neces- sitated some changes in the derivational formulas of the Congress table, notably that of magnetic flux which now becomes BS instead of HS. The dimensional formulas then become consistent throughout; moreover, they then represent the conditions of practice better, as iron is used in nearly all forms of magnets. The magnetic quantities based on the electrostatic system, which are riot generally given in text-books, have here been added for the sake of completeness. For the definitions of the units and the quantitative relations between them see the respective tables of measures. * Reprinted with some additions in the "Electrical World and Engineer," vol. 37, January 5, 1901, p. 501. 18 INTRODUCTION. PHYSICAL QUANTITIES AND RELATIONS. Name. Sym- bol. Derivation. Dimensional Formula. 'C. G. S. Unit.* Fundamental. L,l L M T d k n L2 3 number number L-i L-2 L-3M number LMT-* LT-z L*M-iT~* L S T~* MT-z centimeter gram (mass) second (mean solar) 1 sq. centimeter cubic centimeter radian 1 sq. cm at 1 cm radius gram (mass) per cb. cm 1 dyne dyne per gram dyne per centi- meter Mass . . M T,t Time Auxiliary funda- mental quantities. Temperature Electric inductive capacity * k H S,s V a,/? Magnetic permea- bility Geometric. Surface Volume. . . LL LLL arc Angle, plane Angle, solid radius spherical area Curvature or Tor- [.... }- d sp. gr. ("' (.:.. }::; . radius 2 angle Specific curvature of a surface Mechanical. Weight: see Mass and Force. Density. . . length solid angle surface M V density Specific gravity. . . Force. density Ma F M FL* M 2 FL* M F L Intensity of attrac- tion or force at a point. . Gravitation con- stant Force of a center of attraction or strength of a cen- ter Surface tension. . . * For the names and abbreviations of the numerous practical units and for their values in the C. G. S. units, see the various tables of measures. PHYSICAL QUANTITIES AND RELATIONS. 19 PHYSICAL QUANTITIES AND RELATIONS (Continued). Name. Sym- bol. Derivation. Dimensional Formula. C. G. S. Unit. Pressure or inten- sity of stress. . . . P F S L-.MZ- barie* Modulus of elas- ticity IF L 1 MT~ 2 Resilience. . . LS W L~ 1 MT~ 2 Torque, moment, couple V FLor W L 2 MT~ 2 1 angle yne cen ime er Directive force (as \ torque in suspensions). . Fv angle L 2 MT~ 2 Moment of inertia. Inertia K MX radius 2 L 2 M LM gm (mass) cm sq. L Velocity, linear. . . V L T LT~< centimeter per second; kine Velocity, angular 0) angle T-i radian per second Acceleration, lin- ear . . T V LT~ 2 centimeter per T sec. per sec. Acceleration, an- gular. . <0 T -2 radian per sec. Momentum or quantity of mo- tion. . [.... T mass X veloc- ity i LMT-i per sec. Moment of mo- mentum or angu- lar momentum . . \- momentum X length MT* Energy, work. . . . Vis- viva. . . W W FL L 2 MT~ 2 L 2 MT~ 2 erg erg Impact. . w L 2 MT~ 2 erg Power or activity.. P W T erg per second Efficiency power number 1% power * It equals a dyne per sq. centimeter. This name has not yet been defi- nitely adopted ; some use it for the pressure of one atmosphere. t More correctly this should also contain the reciprocal of an angle. 20 INTRODUCTION. PHYSICAL QUANTITIES AND RELATIONS (Continued) Name. Sym- bol. Derivation. Dimensional Formula. C. G. S, and Practical Units. Magnetic. (El'mag. system.) Reluctance ormag- netic resistance. . (R L_ Hr') oersted * Reluctivity ; spe- cific reluctance. V number X(f*~ l ) Permeance 1 J (R Magnetic capacity C 1 (R UA Permeability or specific induc- tive capacity . . . (B number X(/*) Susceptibility. . . . K 3 5C number X( At) Magnetomotive force . ff 4nNn L^M^T~ l (fi~^) gilbert T Magnetic poten- W tial L^M^T~~ l (fj~^) m Magnetizing force. 3C ~L~I L~^M^T~ l (n~^) gilbert 1 per cen- timeter; gauss Field intensity. . . JC F m L~~b M % T~ l ( fT~% ) gauss Flux density (B ~~S L $M$T l (ffc) gauss Magnetic indue- tion. . (B L%M?T- l ([jp) gauss Intensity of mag- arc netization 3 L~^M^T~ 1 (fi^f) ~V Flux or magnetic lines of force .... &s L$M^T~ l (t&) mixwell Strength of pole or ) quantity of mag- \m \/L 2 Ffji L*M%T~ l ([jfc) netism ) Magnetic moment. 9TC ml WMt^-i(^t) Magnetic energy... W 03 Lwr-2 erg Magnetic power. . P Jg| Lwr-a erg per second * Provisionally adopted t N=: number of turns. I" Provisionally adopted unit is the ampere-turn. by the Amer. Inst. of Electrical Engineers. t L = length of coil. n = frequency, by the Amer. Inst. Elec. Engineers. The usual 1 gilbert = 0.795 8 ampere-turn. PHYSICAL QUANTITIES AND RELATIONS. 21 PHYSICAL QUANTITIES AND RELATIONS (Continued). Name. Sym- bol. Derivation. Dimensional Formula. Ratio of Electro- static to Elec'mag- n'tic Units* Magnetic. (Electrostatic system.) Reluctance or magnetic (R & V 2 Reluctivity; specific re- luctance. . . . 1 V 2 Permeance. A* 1 L~ l T 2 (k~ l ) v~ 2 Magnetic capacity Permeability or specific inductive capacity. . . . Susceptibility. ....'.... K (R 1 (R ffl 3C 3 v -2 Magnetomotive force. . . Magnetic potential. . . 3C $(${ L^M^T~ 2 (k^) L%M%T~ 2 (kb) V Magnetizing force. . . 5C l$M%T~ 2 (k^) Field intensity. . . 3C L F Flux density. m (P L-%M%(k-^) Magnetic induction Intensity of magnetiza- tion 3 S ~~S L*M%(k~^) ,-. Flux (or magnetic lines of force) . f V ET /*M>-*> Strength of pole or quan- W T ^ Jl/T^f 1* ^t\ Magnetic moment Magnetic energy. . W magn.poten. ml L 2 MT 2 1 Magnetic power. ... p W L 2 MT ^ I T * vis the velocity of light in the C. G. S. system, namely 3X10 10 centi- 22 INTRODUCTION. PHYSICAL QUANTITIES AND RELATIONS (Continued). Name. Sym- bol. Derivation. Dimensional Formula. Practical Units.* Electric. ( El' mag. system.) p Resistance. . . R. rt LT~*( ) ohm Resistivity or spe- cific resistance. . P RS L L 2 T~ l (n) ohm-centimeter X, x f I hm Magnetic react- 2nnC ance X ohm Capacity react- ance or condens- ance . . I Lr-'(,> ohm Impedance . Z, g\ V^T^ 2 L ohm Conductance G,g1[ 1 r L"~ 1 7 7 (/<~ 1 ) mho Conductivity or specific conduct- ance . . ... \> 1 P *+$ mho per centi- meter Admittance . . Y y\ 1 1 or L -i T( -i) mho Z z Susceptance. .... B.6t X/^? L-iT(^-0 mho Electromotive force E, e 2 L%M^T~ 2 ( 1$ ) volt T Potential W z,itf*r-*(A*) volt Difference of po- Q tential .... U,u RI L^M^T~ 2 (fi^) volt Electromotive E force at a point . ~L L^M^T~ 2 (n^) volt Intensity of elec- F , tric field L%M%T- 2 (fj?} ~Q ET Vector potential . ~J7 L^M^T~ l (fj^} Current M E R L^M^T~ l (fj~^) ampere Current density I L~i M^ T~ l ( u~^ ) ampere per sq. cm S Quantity of elec- tricity; charge. Surface density or Q,Q IT L^M^djT ^) coulomb ; am- pere-hour electric displace- f _*_ L-$Mb(n~b') coulomb per sq. ment S cm * The C. G. S. units have no names. For the relations between the values of these units and the C. G. S. units, see the various tables of measures. * Vector quantities when used should be denoted by capital italics. PHYSICAL QUANTITIES AND RELATIONS. 23 PHYSICAL QUANTITIES AND RELATIONS (Continued*,. Name. Sym- bol. Derivation. Dimensional Formula. Practical Units. Capacity C, c Q L~ l T 2 (/j~ l ) micro-farad Electric inductive capacity, or Di- electric constant, or Specific induc- tive capacity. . . . Inductance or co- efficient of self- induction or elec- tro-kinetic iner- tia || j I MM E C L c.^* I number henry Mutual inductance L(n) henry Inductance factor : see below. Tim* 1 constant L T second ; henry Period J_ T per ohm second Frequency n n periods per sec. Angular velocity.. Electro-kinetic momentum Thermoelectric height or specific heat of electricity Coefficient of Pel- tier effect Ditto \ ' | .... litn, /Xinductance E heat IT energy T-i radians per sec. Electric energy . . . Kinetic energy... . Electric power. . . . W W p IT EQ T* realP number joule; watt-hour joule watt; kilowatt 1 apparent P wattless P Electrochemi- cal. ( El' mag. system.) Ionic charge. . . apparent P Q coulomb per uni- Electrochemical equivalent Electric deposition M M Q M T T'" valent gram ion gram per cou- lomb gram per second * N = number of turns. t = temperature. 24 INTRODUCTION. PHYSICAL QUANTITIES AND RELATIONS (Continued}. Name. Sym- bol. Derivation. Dimensional Formula. Ratio of Electro- static to Elec 'mag- netic Units. Electric. (Electrostatic system.) E Resistance R, r v 2 ~T Resistivity or specific 1 resistance P T(k~*) v -2 r Reactance X,x x jj iffif 1\ v 2 Magnetic reactance. . . . 2nnL L-iTO-i) Capacity reactance or 1 * 1 7-1 -1 -2 condensance \ c Impedance Z, z \&+* *-> -* Conductance. . . . G g 7 LT~ l (k) ^ E Conductivity or specific Q V 2 T STE/L Admittance ... Y,y 1 LT-i(k) V 2 z Susceptance. . . . B, b vA/2 f]2 LT~ l (k} V 2 Electromotive force or w potential E, e - L^M^T~^(k~^} v~ * ~Q Difference of potential U,u ei-e 2 L^M^T~ l (k~^) v~ l Electromotive force at i E 111 a point. . r . . . . Y lj JVl I (K * V Intensity of electric field F Q L-Wr-^ fa ET Vector potential L-%M^(k-%) v~i ~J7 I i Q L$M$T- 2 (k$) V ' T Current density. . . . I L-^M^T~ 2 (k^) V S Quantity of 'electricity, charge Q, Q \/ L 2 F L%M^T- l (k^) V Surface density or elec- I Q tric displacement ) ' 8 L M T l (k ) V Capacity C, c Q L V 2 E Electric induction ca- pacity or Dielectric constant or specific in- u ~L C . C number (k) V* ductive capacity L ' L number 1 PHYSICAL QUANTITIES AND RELATIONS. 25 PHYSICAL QUANTITIES AND RELATIONS (Continued). Name. Sym- bol. Derivation. Dimensional Formula. Ratio of Electro- static to El ec 'mag- netic Units. Inductance or coefficient of self - induction or electrokinetic inertia. . Mutual inductance . },, xm 2nn L-i!T2(jfc-i) L -i T 2( k -i) T T T-i L*M*(fc-*) L*M* tr-2 tr* 1 1 1 v-i r-i v- v-i 1 1 V V 1 L R 1 n Period Frequency . n Electrokinetic momen- tum Thermo-electric height or specific heat of elec- tricity Coefficient of Peltier ef- fect } /Xinductance E e heat IT energy IT EQ W T Q M M Q M T QT Q T Te(k*) L-%M*Te(k-*) L*MlT- l (k-l) L 2 MT~ 2 L 2 MT~* LlM~*T- l (k*) L-%M*T(k-*) MT-i L2MT-2 L 2 MT~* L*MT~ 3 MI 7 " 3 MT~ 3 Ditto Electric energy W P Electric power Electrochemical. (Electrostatic system.) Ionic charge. . . . Electrochemical equiva- lent Electric deposition. . . . Photometric.* Quantity of light Q I e E Unit. lumen hour lumen candle ; hef- ner candle per sq. cm lux Flux of light solid angle 7 S 9 S Illumination * The dimensional formulas for the photometric quantities here given have been deduced by the author on the basis that radiated light is power, or that quantity of light is energy; according to this, the formula for can- dle-power is that for power divided by that for a solid angle; the latter, having the dimensional formula 1 in the C. G. S. system, is a "suppressed" factor which does not appear in the formula. 26 INTRODUCTION. PHYSICAL QUANTITIES AND RELATIONS (Concluded). Name. Symbol. Derivation. Dimensional Formulas. t Dynamical Thermal Thermo- metric. =H+M. Thermal.* Heat . .. H energy L 2 MT~ 2 MB L 3 L 2 MT* Rate of heat production... \ H T L 2 MT~* Temperature. . ft n 01 0J T 2jr 2 Coefficient of expansion. . . (V-V) 0-1 ,-. 0-1 L-*r V(0-0') Entropy. . . H e L2MT-20-1 M L 3 M Latent heat. . . H M L>I- L3M-10 VT-' Conductivity. . HL TL 2 d LMT-0-1 L -i MT -i L 2 T -1 L-WT-i Emissivity or ff immissivity. . M T 3 i L-2MT- 1 LT~ i L~ 2 MT~^ TL 2 d Specific heat. H H number number number number Capacity MXsp. heat M M A/ M Mech ani c a] equivalent.. . J mec. energy number L 2 T 2 0-! M number heat-energy LT 2 * For the names and definitions of heat-units see tables of units of Energy. t The dimensional formulas of the various thermal quantities or relations are still a matter of some conjecture, excepting only that of quantity of heat, which is simply energy, and that of the rate of production or trans- mission of heat or radiant heat, which is simply power; that of tempera- ture is the uncertain factor. For this reason four different systems are here given, based on four different fundamental conceptions. The first is based on the dynamical units, that is, on the formula of energy combined with a fourth fundamental unit representing temperature. The second is based on the thermal units; quantity of heat then is mass X temperature ; in this system the specific heat of water is unity by definition and is therefore suppressed in the formula. In the third system volume is sub- stituted for mass in the second. In the fourth system the author has . eliminated by defining temperature as energy per unit of mass, that is, a specific mass energy. Most of the data in columns 3, 4, 5, and 6 have been taken from the Smithsonian Physical Tables prepared by Prof. Thomas Gray; the present author has extended them and supplied some omissionsg; the new matter has been approved by Prof. Gray. t By giving 6 the dimensional formula L 2 T~ 2 , those based en the dynam- ical units and on the thermal units become identical; they are given in the last column. Or by making it L~ 1 MT~ 2 , those based 011 the dynam- ical and the thermometric units become identical. L 2 T~ 2 is the dimen- sional formula of energy per unit of mass, and L~ 1 MT~ 2 is that of pres- sure. Ruecker suggests giving the formula L 2 MT~~ 2 , which is energy. No great importance should be attached to attempts to give temperature a formula, as they are all merely speculative. TABLES OP CONVERSION FACTORS. GENERAL REMARKS. In the following set of tables every unit which is used to measure quan- tities is given with its value in terms of the other units of its kind. Such numbers are variously termed values, reduction or conversion factors, equivalents, relations, ratios, constants, etc.; they give the relations be- tween the different units and enable one to reduce each one (or a quantity measured by it) to the other (or a quantity measured by it) by means of a single multiplication. The table includes all the values and relations usually given in books under the title of "Weights and Measures, " although thase form only a very small part, as by far the greater number have never before been published together as a complete set of values. The term "Weights and Measures" has not been used in connection with these tables, partly because it is not apparent why weights are not also measures, and partly because the present tables consist largely of various measures rarely if ever given in books under the title of weights and meas- ures. Owing to the very large number of units belonging in such a table, their classification and arrangement becomes important in order to enable one to find any particular unit readily. All units for measuring the same quantity, that is, all units of length for instance, or all units of volume, etc., have here been brought together in one group, and in each group they are arranged in the order of their size ; all the different values of each unit are also arranged in order of size. To give the value of each unit in terms of each of the others would have made the tables many times as long, and they would have become unnec- essarily cumbersome, as the majority of the values are never used; the number of values or reduction factors have therefore been limited to those likely to occur in practice ; if the others are ever needed they can readily be found from these. To avoid unnecessary repetition of values, all units capable of being reduced from one to the other have here been put in the same group. Thus the group of units of volumes includes both cubical units such as cubic feet, etc., as well as capacity measures, such as gallons, liters, etc. All energy units are similarly grouped together, whether they be mechanical, thermal, or electrical. In a few cases, however, such units have been separated into different groups; forces, for instance, have not been given together with weights, although they are interconvertible, and electrical resistances have been given separately and not with velocities, although in the C. G. S. electromagnetic system of units they are velocities. Sometimes some of the units in one group are for measuring entirely dif- ferent kinds of quantities, and even have entirely different dimensions, yet their equivalents are the' same, and they have therefore been brought together to save repetition; for instance, pounds per square foot may de- note either a pressure or the weight of sheet metaj; in either case the equiv- alents in other units, such as kilograms per square meter, will be the same. Similarly, grams per centimeter may denote either the weights of a wire or 27 28 TABLES OF CONVERSION FACTORS. a surface tension, yet its equivalents in other units are the same. Power and momentum, or energy and torque, are other illustrations. In the case of lengths, surfaces, volumes, and weights (masses) there are so many unusual, special trade, obsolete, or foreign units the values or reduction factors of which are not often needed, that they have been grouped separately, so as to make the main table less cumbersome to use. The reciprocal values have been given in all cases in which it was thought they were likely to occur in practice, thus reducing all calculation to a mere multiplication, as distinguished from a long division. They are given in their proper places, and one should therefore first look for the units that one wants. Thus to convert meters into feet, see value under Meters, and to convert feet into meters, see value under Feet. Special attention is here called to the approximate values which have been given in nearly every case. These have been carefully chosen with a view to reduce the calculation to the smallest possible, generally to a multi- plication by one digit and a division by another single digit, followed by pointing off the decimal. They are believed to be the simplest values that exist. The accuracy of all these approximate values is within 2% and often within 1%; they are therefore sufficient for most calculations. The symbols or abbreviations which are given are either those in common use or those which have been recommended by societies, journals, or indi- viduals, and in a few unimportant cases where none existed they have been supplied by the author in conformity with the others. The advantages and desirability of a uniform and universal system of abbreviations or sym- bols are too evident to need further comment here. This whole system of tables, which is not a compilation but a complete recalculation, has been based on the very best fundamental values that are obtainable. Their numerical values are given in their respective places or in the introductory notes; they are printed in bold-faced type. When- ever legally adopted values existed, as for instance for the relations between the metric and the older units, they have been used. In other cases the fundamental values have been obtained from the best existing sources, and wherever possible those values have been chosen which have been deter- mined upon by the best authorities and are used by them. The chief of these authorities was the National Bureau of Standards, from which the author has obtained the legal values and all the other standard values used or recommended by it, besides much valuable assistance; among the others were the International Congresses, the Director of the Nautical Almanac, the Coast Survey Department, etc. The greatest care was taken in obtain- ing all these fundamental values, and it is believed that they are the best which exist at the present time. They are limited to only those which are absolutely necessary, as was explained above under the inter-relations of units, and none of them is therefore inconsistent with any other, nor can any derived value then have two values depending upon which way it has been calculated from the fundamental values; all the values together form a single, uniform, stable system. The fundamental values have here been given to as many places of figures as in the original source. The derived values have been given throughout to six significant figures. In some cases the fundamental value itself may not have six figures, or its possible error may not warrant so many places of figures in the derived units; but in such a completely recalculated set of values it was thought best to retain six places of significant figures throughout in all the derived values, as the correction due to any subse- quently adopted more accurate fundamental value can then be made by mere proportion instead of by a complete recalculation. The derived values are exactly correct for the particular fundamental values used here, except- ing of course the usual slight inaccuracy (1) of the last right-hand digit in the numbers and in the logarithms. The calculations have all been made with the greatest care; each was checked by calculating it in another way and in many cases the values were thus checked twice; it is therefore believed that there are no errors. There is, of course, the unavoidable uncertainty of one unit in the sixth place of figures or in the seventh place of the logarithms, although in most of these cases the next place was also calculated in order to insure the accuracy of those that were retained. Many of the values in the first few groups have been carefully checked by L. A. Fischer, Assistant Physicist of the National Bureau of Standards, and may therefore he Assumed to be those used by LENGTHS. 29 that Bureau. The author takes this opportunity to acknowledge his appre- ciation of Mr. Fischer's very valuable revision of those values. The obsolete and foreign units were compiled from various sources. They therefore involve any inaccuracies or inconsistencies that may exist in the original sources; but when the inconsistency was very great, care was taken to find which value was the correct one. In the cases in which units are sometimes used incorrectly or ambigu- ously, they have here also been placed where they might be looked for, and are there accompanied by a reference to the place where they properly be- long and where their values are given. A comparison of the values in these tables with those in other books will show that few of the latter have been based on the legal standards of this country. Moreover, the older published values will often be found to be inconsistent with each other, as they seem to have been compiled instead of being recalculated throughout from the same fundamental values, as was done here. Special attention is called to the compound uuits, most of which are seldom, if ever, found in other books; as most of these save from two to four separate calculations, they will often be found useful. LENGTHS. The fundamental standard of length of the United States is the Inter- national Meter, a bar made of an alloy of 90 percent platinum and 10 per- cent iridium and preserved at the International Bureau of Weights and Measures, near Paris. Copies of this bar are possessed by each of the twenty countries contributory to the support of the International Bureau of Weights and Measures, and these copies are known as National Prototypes. The United States owns two of these bars, whose values in terms of the Inter- national Meter are known with the greatest accuracy. One of these bars, No. 21, is used as a working standard, and the other, No. 27, is kept under seal and only used to check the value of No. 21. One of the objects of the maintenance of the International Bureau is to provide for the recomparison at regular intervals of the various National Prototypes with the International Meter, thus insuring the use of the same standard throughout the world. According to the Act of Congress of July 28, 1866, which was the first general legislation in the United States upon the subject of fixing the stand- ards of weights and measures, the relation 1 meter ==39. 370 inches was legalized, and it is the only legal relation which exists in the United States concerning the meter. This value is the one used by the National Bureau of Standards in Washington and is in general use in this country. Since 1893 the Office of Standard Weights and Measures has been authorized to derive the yard from the meter in accordance with this relation; the legal yard, foot, inch, etc., are therefore derived from the meter, and consequently are fixed and definite units. The relation between the U. S. yard and the meter is therefore no longer to be determined by measurement, as is often supposed, but is fixed definitely by precise definition. There is no legal authority in this country for the old Kater relation of 1818, namely, 39.37079, which is still in use, notably by one well-known maker of gauges. In Great Britain the relation, legalized in 1896, between this same Inter- national Meter and the British Imperial yard of 36 inches is 1 meter = 39.370 113 inches. The Kater value, 39.370 79, was used there for trade purposes until 1896. In 1867 the Clark value, 39.370 432, was recom- mended by the Warden of Standards, London, to supersede the Kater value, and was generally used in scientific work in Great Britain until 1896. There exists therefore a very slight difference between the present U. S. and British legal yard, foot, inch, etc. But these differences are only about 3 parts in one million, the U. S. yard being the longer; it is therefore abso- lutely negligible except in the most refined physical measurements. Unless otherwise stated, the values in the following tables are based on the U. S. legal relation. The value of the nautical mile in meters (1 853.25) is the one adopted many years ago by the U. S. Coast and Geodetic Survey, and has not been changed since its first adoption. It is the, length of one minute of arc of a great circle of a true sphere whose area is equal to that of the earth. The "Committee Meter" used by the U. S. Coast and Geodetic Survey prior to 1889 is equal to the international meter of 39.370 inches (U. S.). 30 LENGTHS LENGTHS, Usual. ** Accepted by the National Bureau of Standards. * Checked by L. A. Fischer, Asst. Phys. National Bureau of Standards. Aprx. means within 2%. Logarithm 1 mil = 0.025 400 05* millimeter. Aprx. *4o 2-404 8348 " = 0.001 inch 3-000 0000 I millimeter [mm] = 39.370 0* mils. Aprx. 40 1-595 1654 = 0.039 370 0* inch. Aprx. V 25 2-5951654 0.001 meter 3-000 0000 1 centimeter [cm] = 0.393 700* inch. Aprx. Vio 1-595 1654 = 0.032 808 3* foot. Aprx. Vao 2-515 9842 = 0.01 meter 2-000 0000 1 inch [in] = 1 000. mils 3-000 0000 = 25.400 05* millimeters. Aprx. \i X 100 1.404 8346 = 2.540 005* centimeters. Aprx. 1% 0-404 8346 = 0.083 333 3* foot or Vi 2 2-920 8188 = 0.027 777 8* yard or y 36 . Aprx. 11 A + 100 2-443 6975 = 0.025 400 05* meter. Aprx. *4 2-404 8346 1 decimeter [dm] = 10. centimeters 1-000 0000 3.937 00* inches. Aprx. 4. = 0.328 083* foot. Aprx. V 8 . - - = 0.109 361* yard. Aprx. V 9 . . 0.1 meter... 1 foot [ft] (Brit.)= 0.999 997 1* foot (U. S.). Aprx. 1.. 0.304 800* meter. Aprx. 8 A . . . . -595 1654 -515 9842 -038 8629 -000 0000 999 9988 -484 014C 1 foot [ft] (U. S.) = 304.801* millimeters. Aprx. 300 2-484 0158 30.480 1* centimeters. Aprx. 30 1-484 0158 12. inches 1-079 1812 3.048 01* decimeters. Aprx. 3 0-484 0158 = 1.000 002 9* feet (Brit.). Aprx. 1 ' 0.333 333 yard or V 3 0.304 801* meter. Aprx. %o = 0.000 304 801* kilometer. Aprx. 3 -*- 10 000 . = 0.000 189 394* mile. Aprx. 19-4-100 000. . . . 1 yard [yd] (Brit.) = 0.999 997 1* yard (U. S.). Aprx. 1 = 0.914 399 2* meter. Aprx. /io or i/n 1 yard [yd] (U. S.) = 91.440 2* centimeters. Aprx. 90 = 36. inches 1-556 3025 = 3. feet 0-477 1213 = 1.0000029* yards(Brit.). Aprx. 1. ... 0-000 0012 = 0.914 402** meter. Aprx. %o or i%i . - 1-961 1371 = 0.000914402* kilomt'r. Aprx. Hi * 100. 4-961 1371 = 0.000568182* mile. Aprx. # -r- 1 000 4-7544873 1 meter [m]= 1 000. millimeters 3-000 0000 = 100. centimeters 2-000 0000 = 39.370113* inches (Brit.). Aprx. 40 1-5951668 = 39.370000** inches (U.S.). Aprx. 40 1. 595 1654 = 10. decimeters 1-000 0000 = 3.280 83** feet. Aprx. 1% 0-515 9842 = 1.093 61** yards. Aprx. 1^0 Q-038 8629 = 0.546 806* fathom. Aprx. ^20 1-737 8329 = 0.001 kilometer 3-000 0000 = 0.000 621 370* mile. Aprx. %-s- 1 000 4-793 3503 1 kilometer [km] = 3 280.83* feet. Aprx. % X 10 000 3-5159842 = 1 093.61* yards. Aprx. 1100 3-0388629 = 1 000. meters 3-000 0000 = 0.621 370* mile. Aprx. % 1-793 3503 = 0.539 611 knot (Brit.). Aprx. %i 1-732 0806 44 =0.539 593 nautical mile (U. S.) Aprx. 6 /ii. . T.-732 0660 LENGTHS. 31 1 mile [ml] == same as statute mile or land mile. = 5 280.* feet. Aprx. 5 300 3.722 6339 = 1 760.* yards. Aprx. % X 1 000 3-2455127 = 1 609.35* meters. Aprx. 1 600 3-206 6497 = 1.60935* kilometers. Aprx.add%o 0-2066497 = 0.868 421 knot (Brit.). Aprx. subtract Vs 1-938 7303 = 0.868 392 nautical mile (U. S.). Aprx. subt. Vs. . 1-938 7157 1 knot or nautical mile (Brit.): = 6 O8O. feet. Aprx. 6 000 3-783 9036 = 2 026.67 yards. Aprx. 2 000 3-306 7823 = 1.853 19 kilometers. Aprx. H6 0-267 9194 = 1.151 52 miles. Aprx. add ty 0-061 2697 = 0.999 966 nautical mile or knot (U.S.). Aprx. 1 1-999 9854 1 nautical mile or knot (U.S.) same as geographical or sea mile: = 6 080.20 feet. Aprx. 6 000 3-783 9182 = 2 026.73 yards. Aprx. 2 000 3-306 7969 = 1 853.35 meters. Aprx. H'e X 1 000 3-267 9340 = 1.853 25 kilometers. Aprx. 1% 0-267 9340 = 1.151 55 miles. Aprx. add Vr 0-061 2843 = 1.000034] knots (Brit.). Aprx. 1 0-0000146 = 1 minute of earth's circumference 0-000 0000 LENGTHS (continued). Unusual, Special Trade, or Obsolete. 1 Angstroem unit (spectroscopy) = 0.1 milli-micron, or micro-milli- meter = 0.000 1 micron = 0.000 003 937 00 mil = 0.000 000 1 millimeter. 1 mil = 254 000.5 Angstroem units. 1 milli-micron [^^] (spectroscopy)or micro-millimeter (microscopy) = 10 Angstroem units = 0.001 micron = 0.000 039 370 mil = 0.000 001 milli- meter. 1 mil = 25 400.1 milli-microns. The term micro-millimeter is also used, though incorrectly, in biology for 0.001 millimeter, which length is wiore properly called a micron or micro-meter. 1 wave length of blue light is of the order of about 5000 Angstroem Units or 500 milli-microns or 0.5 micron or 0.02 mil. For accurate values see below under meter. 1 micron or microne or micro-meter [/*] (spectroscopy and micros- copy) =10 000 Angstroem units = 1000 milli-microns or micro-milli- meters =0.039 370 mil (aprx. % 5 ) = 0.001 millimeter. 1 mil = 25.400 1 microns. 1 terze (Brit.) = Vi2 second = Vi44 Hne = : H728 inch. 1 second (Brit. ) = 12. terzes = Vi2 1^16 = ^44 inch. 1 point = 0.008 inch. 1 point (typography) = % line = Vr2 inch. 1 line (U. S.) = Vi2 inch. 1 line (Brit. ) = 144. terzes= 12. seconds=Ha inch. Also given as Ho inch. 1 hairsbreadth = M line = Ms inch. 1 barleycorn = J^ inch. 1 nail [na] (cloth) = 2M inches = M span. 1 palm = 3. inches. 1 hand = 4. inches. 1 link [li] (surveyor's) = 7. 920 inches = 0.201 17 meter. link [li] (engineer's) = 12. inches = l. foot = 0.304 80 meter, span = 9. inches = 4. nails =1. quarter = ^4 yard. quarter [qr] (cloth) = 9. inches = 4. nails = l. span = ^4 yard. cubit =18. inches=iy 2 feet. cubit (in Bible) = 21.8 inches. vara (California; legal) = 33. 372 inches. pace = 3. feet. military pace = 3. feet. common pace = 2^ feet. meter used by Pratt & Whitney Co. = 39.370 79 inches -1.000 020 meters (int.). 32 LENGTHS. 1 meter = 1 553 163.5 wave lengths of red light. t = 1966249.7 " " green " f = 2083 372.1 ' blue " t 1 Committee Meter (of U. S. Coast Survey) = 39. 370 inches =1 meter (int.;. 1 ell [E, e] (cloth; Brit.) = 45. inches -1,143 meters. 1 fathom (U. S.) = 6. feet = 1.828 8 (aprx. %) meters. 1 fathom (Brit.) = 6.080 feet = 1.853 2 (aprx. 1%) meters = Hooo nauti- cal mile (Brit.). 1 rod [r], pole [p], or perch [p] (surveyor's) = 5^ yards = 5.029 2 meters. 1 decameter or dekameter [dkm]=10. meters. 1 chain [ch] (Gunter's or surveyor's) = 100. links (surveyor's) = 66. feet = 20.117 meters = 4. rods, poles, or perches = Vio furlong^/ko mile. 1 chain [ch] (engineer's) = 100. links (engineer's) = 100. feet = 30.480 meters. 1 chain [ch] (Philadelphia standard) = 100*4 feet = 30.556 meters. bolt (cloth) = 40. yards. Ibo 1 hectometer = 100. meters. 1 furlong [fur] = 660. feet = 20 1.1 7 meters = 40. rods, poles, or perches = 10. chains (Gunter's or surveyor's) = V 8 mile. 1 cable or cable's length (Brit, navy) = 608. feet (sometimes stated as 608.6 feet) = Vio nautical mile (Brit.). 1 cable's length (U.S. Navy) = 720. feet = 219.457 meters = 120. fathoms (U. S.). 1 car-mile, see under units of Energy. 1 knot (telegraph; Brit.) = 2 029. yards=l 855.32 meters. 1 geographical mile, sometimes used for nautical mile; see also under international geographical mile. 1 international nautical or sea mile = 6 076.10 feet = 1 852. meters = %o of 1 of meridian. 1 of latitude at equator = 60. (aprx.) nautical miles. = 68.70 miles (statute), lat. 20 = 68.78 40 = 69.00 60 = 69.23 80 = 69.39 90 = 69.41 1 of longitude at equator = 60. (aprx.) nautical miles. = 69.16 m les (statute). " lat. 20 = 65.02 " " 40 = 53.05 " 60 = 34.67 80 = 12.05 1 legua (California; legal) = 2.633 5 miles = 5 000. varas. 1 league (U. S.) = 4.828 05 kilometers = 3. miles (statute); also given ad 3. nautical miles. 1 international geographical mile = 24 350. 3 feet = 7 422. meters = 4.611 80 miles (statute) = 4. (aprx.) nautical miles = ii5 of 1 at equator. The term geographical mile is sometimes used also for nautical mile. 1 myriameter or miriameter=10. kilometers. 1 mean diameter of the earth J (astronomy ) = 12 742.0 kilometers = 7 917.5 miles. 1 mean diameter of earth's orhit (astronomy) = 149 340 870 9G 101 kilometers = 92 796 950 59 715 miles. t Michelson; cadmium light waves for air at 15 C. and a pressure of ^0 mm. mercury. t Based on the accepted value of a nautical mile as defined above. Harkness. LENGTHS. 33 LENGTHS (concluded). Foreign. These are mostly obsolete, as the metric system is now used in most foreign countries. The British measures are included among the U. S. measures, being very nearly, and sometimes quite the same. The trans- lated terms are merely synonymous, and not the exact equivalents. Germany. Prussia. Legal May 16, 1816. 1 Fuss ['] (foot) = 12 Zoll ["] (inches) of 12 Linicn ['"] (lines). 1 Fuss (also called " rheinlaendischer Fuss," that is, Rhineland foot) = 0.313 853 5 meter. Road measure: 1 Meile (mile) -2 000 Ruthen (rods) of 12 Fuss (feet); 1 Meile = 7.5325 kilometers; 1 Ruthe = 3. 766 25 meters. From Jan. 1, 1872, to Jan. 1, 1874: 1 deutsche Meile (German mile) = 7. 500 kilometers. Trade measure: 1 Elle(yard) = 2H Fuss (feet) = 25^ Zoll (inch) = 0.666 939 meter. 1 Lach- ter = 80 Zoll (inches) = 2.092 36 meters. Bavaria. 1 Fuss (foot) = 12 Zoll (inches) of 12 Linien (lines). More rarely 1 Fuss= 10 Zoll of 10 Linien. 1 Fuss = 0.291 859 meter. Saxony. 1 Fuss = 0.283 19 meter; subdivisions like in Bavaria. WiLertemberg. 1 Fuss = 0.286 49 meter; subdivisions like in Baden. Baden. 1 Fuss (foot) = 10 Zoll (inches) of 10 Linien (lines). 1 Fuss = 0.3 meter. Hanover. 1 Fuss = 0.292 1 meter. The following values of various German feet in inches are given in Nys- trom's Mechanics: Bavaria, 11.42; Berlin. 12.19; Bremen, 11.38; Dres- den, 11.14; Hamburg, 11.29; Hanover, 11.45; Leipsic, 11.11; Prussia, 12.36; Rhineland, 12.35; Strasburg, 11.39. Also the following road meas- ures: Germany, mile, long, 10126 yards; Hamburg mile, 8244 yards; Hanover mile, 11 559 yards; Prussia mile, 8 468 yards. France. "Old measures" (systeme ancien) used prior to 1812: 1 toise (fathom) = 6 pieds (du roi) (feet) of 12 pouces (inches) of 12 lignes (lines) of 12 points. In geodesy: 1 pied=10 pouces of 10 lignes of 10 points. 1 toise = 1.949 037 meters. Road measure: 1 lieue (league) = 2 283 toises = 4449.65 meters; 1 lieue marine = 2854 toises = 5 562.55 meters; 1 lieue moyenne = 2 534 toises = 4 938.86 meters. Field measure: 1 perche (perch) = 18 or 22 pieds. For depths of the sea: 1 brasse = 5 pieds. Trade: 1 aune (yard) de Paris= 1.188 45 meters. According to Nystrom's Me- chanics: 1 pied du roi= 12.79 inches; 1 league (lieue) marine = 6 075 yards; 1 league, common = 4 861 yards; 1 league, post, = 4 264 yards. "Usual" measures (systSme usuel) used from 1812 to 1840: 1 toise (fathom) = 2 meters; 1 pied (foot) = M meter; 1 aune (yard) = 1.2 meters. For subdivisions into other units see above under "old measures." 1 noeud = 1 knot or nautical mile. Austria. 1 Ruthe (rod) = 10 Fuss (feet) of 12 Zoll (inches) of 12 Linien (lines). 1 Fuss = 0.316 10 meter. 1 Meile (mile) = 7.586 kilometers = 4 000 Klafter of 6 Fuss. 1 Elle (yard) = 2.46 Fuss. According to Nys- trom's Mechanics: 1 Vienna foot = 12. 45 inches. Sweden. 1 meile (mile) = 6 000 famn of 3 alnar of 2 fot (foot) of 10 turn (inches) (or 12 vorktum) of 10 linier (lines). 1 meile = 10.688 4 kilometers. 1 fot = 0.296 901 meter. 1 ruthe (rod) = 16 fot. 1 corde=10 stangen of 10 fot. According to Nystrom's Mechanics: 1 Swedish foot = 11.69 inches; 1 Swedish mile =11 700. yards. Russia. 1 werst = 500 saschehn of 3 arschin of 4 tschetwert of 4 wer- schock. 1 werst = 1.066 8 kilometers = 3 500 feet = 0.662 88 mile. 1 sa- schehn =2. 133 6 meters = 7.0 feet (U.S.). 1 arschin = 7 1.1 2 centimeters = 28.0 inches. 1 tschetwert = 17.780 centimeters = 7.0 inches. 1 werschock = 4.445 centimeters =1.75 inches. The British foot is also used; 1 foot = 12 inches of 12 lines. According to Nystrom's Mechanics: 1 Russian foot=13.75 inches; 1 Moscow foot = 13.17 inches; 1 Riga foot = 10.79 inches; 1 Warsaw foot = 14.03 inches; 1 verst = l 167. yards. Switzerland. 1 Fuss (foot) = 10 Zoll (inches) of 10 Linien (lines). 1 Fuss = 0.3 meter. According to Nystrom's Mechanics: 1 Geneva foot = 19.20 inches; 1 Zurich foot = 11.81 inches; 1 Swiss mile = 9 153 yards. Holland. 1 Ruthe (rod) = 12 Fuss (feet). 1 Ruthe = 3.767 36 meters; 1 Fuss = 0.313 947 meter. According to Nystrom's Mechanics: 1 Amster- dam foot = 11. 14 inches; 1 Utrecht foot = 10.74 inches; 1 Flanders mile = 34 LENGTHS. 6869 yards; 1 Holland mile = 6 395 yards; 1 Netherlands mile = 1093 yards. Spain. 1 vara (yard) = 32.874 8 inches = 0.835 022 meter. According to Nystrom's Mechanics: 1 Spanish foot = 11. 03 inches; 1 toesas = 66.72 inches; 1 palmo = 8.64 inches; 1 Spanish common league = 7 416 yards. Italy. According to Nystrom's Mechanics: 1 Florence braccio = 21.69 inches; 1 Genoa palmo = 9. 72 inches; 1 Malta foot = 11. 17 inches; 1 Naples palmo = 10.38 inches; 1 Rome foot = 11. 60 inches; 1 Sardinia palmo = 9.78 inches; 1 Sicily palmo = 9.53 inches; 1 Turin foot = 12. 72 inches; 1 Venice foot = 13.40 inches; 1 Rome mile = 2 025. yards. Japan. Long measure: 1 ri = 36 cho of 60 ken of 6 shaku of 10 sun of 10 bu. 1 jo = 10 shaku. 1 ri = 3.93 kilometers = 2. 44 miles. 1 kilometer = 0.255 ri. 1 mile = 0.410 ri. For cloth measure: 1 jo = 10 shaku of 10 sun of 10 bu; the unit is the jo; in this measure the units are ^longer than those of the same name in the long measure. Miscellaneous (from Nystrom's Mechanics): Antwerp foot =11. 24 inches; Brussels foot = 11.45 inches. Denmark mile = 8 244 yards; Copen- hagen foot =12. 35 inches. Portugal league = 6 760 yards; Lisbon foot = 12.96 inches; Lisbon palmo = 8.64 inches. Ireland mile = 3038 yards; Scotland mile =1984 yards. Hungary mile = 9 113 yards. Bohemia mile = 10 137 yards. Poland mile, long, =8101 yards. Turkey berri = 1826 yards. Persia parasang = 6 086 yards; Persia arish = 38.27 inches. Arabia mile = 2 148 yards. China li = 629 yards; mathematic foot=13.12 inches; builder's foot = 12. 71 inches; tradesman's foot = 13. 32 inches; surveyor's foot = 12. 58 inches. Ancient. Biblical: 1 digit = 0.912 inch; 1 palm = 4 digits = 3. 648 inches; 1 span = 3 palms =10.94 inches; 1 cubit = 2 spans = 21. 888 inches; 1 fathom = 3.46 cubits = 7.296 feet. Egyptian: 1 finger = 0.737 4 inch; 1 nahud cubit = 1.476 feet; 1 royal cubit = 1.722 feet. Grecian: 1 digit = 0.754 inch; 1 pous=16 digits = 1.007 3 feet; 1 cubit = 1.133 feet; 1 stadium = 604. 375 feet; 1 mile = 8 stadiums = 4 835. feet. Hebrew: 1 cubit = 1.822 feet; 1 Sabbath day's journey = 3 648. feet; 1 mile = 4 000 cubits = 7296 feet; 1 day's journey = 33. 164 miles; 1 sacred cubit = 2.02 feet. Roman: 1 digit=0.725 7 inch; 1 uncia (inch) = 0.967 inch; 1 pes (foot) = 12 uncias=11.60 inches; 1 cubit = 24 digits = 1.45 feet; 1 passus = 3. 33 cubits = 4.835 feet; 1 millarium (mile) = 4 842 feet. Arabian: 1 foot = 1.095 feet. Babylonian: 1 foot = 1.14 feet. Lengths in which Overhead Telegraph Lines Are or Were Ex- pressed in Different Countries. (From Munro and Jamieson's Pocket Book.) The lengths here given are in English or statute miles of 5 280 feet. Arabia, mile 1.2204. Austria, mile 5.7534. Bohemia, mile 5.7596. Brabant, league 3.452 2. Burgundy, league 3.516 6. China, li 0.359 1. Denmark, mile 4.684 1. Flanders, league 3.90. Hamburg, mile 4.684 1. Hanover, mile 6.567 6. Hesse, mile 5.992 6. Holland, mile 4.602 8. Hun- gary, mile 5.1778. Italy, mile 1.1505. Lithuania, mile 5.5573. -Nor- way, mile 7.018 3. Oldenburg, mile 6.147 7. Poland, long mile 4.602 8: short mile 3.4517. Portugal, league 3.8409. Prussia, mile 4.8068; Rome, mile 0.925 0. Russia, verst 0.663 0. Saxony, mile 5.627 8. Silesia, mile 4.0244. Spain, common legua of 8000. varas, 4.2136; legal legua of 5 000 veras, 2.753 4. Swabia, mile 5.633 5. Sweden, mile 6.647 7. Switzerland, mile 5.200 5. Turkey, berri 1.037 5. Tuscany, mile 1.027 2. Westphalia, mile 6.903 9. LENGTHS. 35 Inches in Fractions, Decimals, Millimeters, and Feet. For every 64th of an inch up to 1 inch ; for every 32d of an inch up to 6 inches; for every 16th of an inch up to 12 inches; for every inch ap to 10 feet. The equivalents of other intermediate values, or of values beyond the table, may be found by adding together two or more values from the table; thus for 11%2 inches add that for 11 inches to that for %2- Milli- meters. Inches. Feet. Milli- meters. Inches. Feet. 0.396 876 H4 0.015 625 0.001 302 19.446 9 4 %4 0.765 625 0.063 80 0.793 752 J-152 0.031 250 0.002 604 19.843 8 25 /82 0.781 250 0.065 10 1.190 63 %4 0.046 875 0.003 906 20.240 7 % 0.796 875 0.066 41 1.587 50 %6 0.062 500 0.005 208 20.637 5 13 /16 0.812 500 0.067 71 1.984 38 %4 0.078 125 0.006 510 21.034 4 5 %4 0.828 125 0.069 01 2.381 25 %2 0.093 750 0.007 813 21.431 3 27 /32 0.843 750 0.070 31 2.778 13 %* 0.109 375 0.009 115 21.828 2 5 %4 0.859 375 0.071 61 3.17501 y% 0.125 000 0.010 42 22.225 H 0.875 000 0.072 92 3.571 88 %4 0.140 625 0.011 72 22.621 9 57 /64 0.890 625 0.074 22 3.968 76 %2 0.156 250 0.01302 23.0188 2 %2 0.906 250 0.075 52 4.365 63 1 M* 0.171 875 0.014 32 23.415 7 5 %4 0.921 875 0.076 82 4.762 51 fc 0.187 500 0.015 63 23.812 5 15 /16 0.937 500 0.078 13 5.159 39 18 /64 0.203 125 0.016 93 24,209 4 % 0.953 125 0.079 43 5.556 26 %J 0.218 750 0.018 23 24.606 3 % 0.968 750 0.080 73 5.953 14 15 /64 0.234 375 0.019 53 25.003 2 63 /04 0.984 375 0.082 03 6.350 01 M 0.250 000 0.020 83 25.400 1 1.000000 0.083 33 6.746 89 17 /64 0.265 625 0.022 14 26.1938 IMa 1.031 250 0.085 94 7.143 76 %2 0.281 250 0.023 44 26.987 6 IVio 1.062 500 0.088 54 7.540 64 A%4 0.296 875 0.024 74 27.781 3 1%2 1.093750 0.091 15 7 93752 5 Ae 0.312 500 0.026 04 28.575 1 1H 1.125000 0.093 75 8 334 39 2l /64 0.328 125 0.027 34 29.368 8 1%2 1.156250 0.096 35 8 731 27 % 0.343 750 0.028 65 30.162 6 1%0 1.187500 0.098 96 9 128 14 23 /64 0.359 375 0.029 95 30.956 3 ! 7 /32 1.218750 0.101 6 9.52502 y s 0.375 000 0.031 25 31.750 1 1'4 1.250000 0.1042 9.921 89 2 %4 0.390 625 0.032 55 32.543 8 1%2 1.281 250 0.106 8 10.3188 13 /32 0.406 250 0.033 85 33.337 6 ! 5 /4e 1.312 500 0.109 4 10.715 6 2T /64 0.421 875 0.035 16 34.131 3 1% 1.343 750 0.1120 11.1125 7 /4e 0.437 500 0.036 46 34.925 1 w 1.375000 0.1146 11.5094 29 /64 0.453 125 0.037 76 35.7188 l 18 /32 1.406250 0.1172 11.906 3 1B /32 0.468 750 0.039 06 36.512 6 IVie 1.437 500 0.1198 12.303 1 % 0.484 375 0.040 36 37.306 3 ! 15 /32 1.468750 0.1224 12.700 H 0.500 000 0.041 67 38.100 1 VA 1.500000 0.1250 13.096 9 33 /64 0.515625 0.042 97 38.893 8 ! 17 /32 1.531 250 0.1276 13.493 8 17 /32 0.531 250 0.044 27 39.687 6 1%6 1.562 500 0.1302 13.890 7 8 %4 0.546 875 0.045 57 40.481 3 ! 19 /32 1.593750 0.1328 14.287 5 ttS 0.562 500 0.046 88 41.275 1 VA 1.625000 0.1354 14.684 4 37 /64 0.578 125 0.048 18 42.068 8 1% 1.656250 0.1380 15.081 3 !%2 0.593 750 0.049 48 42.862 6 1^6 1.687500 0.140 6 15.478 2 3 %4 0.609 375 0.050 78 43.656 3 1 2 %2 1.718750 0.1432 15.875 X 0.625 000 0.052 08 44.450 1 IX 1.750 000 0.1458 16.271 9 4 y&4 0.640 625 0.053 39 45.243 8 1 25 /S2 1.781 250 0.1484 16.668 8 % 0.656 250 0.054 69 46.037 6 1 13 /16 1.812 500 0.151 17.065 7 4 %4 0.671 875 0.055 99 46.831 3 ! 27 /32 1.843 750 0.1536 17.462 5 iVlG 0.687 500 0.057 29 47.625 1 I-H 1.875000 0.156 3 17.859 4 45 /64 0.703 125 0.058 59 48.418 8 1 2 %2 1.906250 0.1589 18.256 3 23 /32 0.718 750 0.059 90 49.212 6 I 15 /ie 1.937 500 0.161 5 18.653 2 47 /64 0.734 375 0.061 20 50.006 3 1% 1.968750 0.164 1 19.050 fe 0.750 000 0.062 50 50.800 1 2 2.000 000 0.1667 36 LENGTHS. Milli- meters. Inches. Feet. Milli- meters. Inches. Feet. 51.593 9 2^2 2.031 250 0.169 3 96.043 9 325/32 3.781 250 0.315 1 52.387 6 21/16 2.062 500 0.171 9 96.837 7 313/16 3.812 500 0.317 7 53.181 4 23/32 2.093 750 0.1745 97.631 4 327/32 3.843 750 0.320 3 53.975 1 2X 8 2.125 000 0.177 1 98.425 2 3j/s 3.875 000 0.322 9 54.768 9 2%2 2.156 250 0.1797 99.218 9 329/32 3.906 250 0.325 5 55.562 6 2/16 2.187 500 0.182 3 100.013 315/ 10 3.937 500 0.328 1 56.356 4 27/32 2.218 750 0.1849 100.806 3% 3.968 750 0.330 7 57.150 1 2M 2.250 000 0.187 5 101.600 4 4.000 000 0.333 3 57.943 9 2% 2 2.281 250 0.190 1 102.394 4H 2 4.031 250 0.335 9 58.737 6 2%6 2.312 500 0.192 7 103.188 4^6 4.062 500 0.338 5 59.531 4 2% 2.343 750 0.195 3 103.981 4%2 4.093 750 0.341 1 60.325 1 2H 2.375 000 0.197 9 104.775 4^ 4.125000 0.343 8 61.118 9 213/32 2.406 250 0.200 5 105.569 4% 2 4.156 250 0.346 4 61.9126 27/16 2.437 500 0.203 1 106.363 43/ lfl 4.187 500 0.349 62.706 4 215/32 2.468 750 0.205 7 107.156 47/32 4.218 750 0.351 6 63.500 1 2H 2.500 000 0.208 3 107.950 4M 4.250 000 0.354 2 64.293 9 2' % 2 2.531 250 0.2109 108.744 4%2 4.281 250 0.356 8 65.087 6 2*ia 2.562 500 0.213 5 1 09-. 538 4%6 4.312 500 0.359 4 65.881 4 219/32 2.593 750 0.216 1 110.331 4% 4.343 750 0.362 66.675 1 2^ 2.625 000 0.2188 111.125 4% 4.375 000 0.364 6 67.468 9 2% 2.656 250 0.221 4 111.919 4i% 2 4.406 250 0.367 2 68.262 6 2Hle 2.687 500 0.224 112.713 47/16 4.437 500 0.369 8 69.056 4 22% 2 2.718 750 0.226 6 113.506 415/32 4.468 750 0.372 4 69.850 1 2 3 4 2.750 000 0.229 2 114.300 4M 4.500 000 0.375 70.643 9 225/ 32 2.781 250 0.231 8 115.094 417/32 4.531 250 0.377 6 71.4376 213,16 2.812 500 0.234 4 115.888 4 ft / 16 4.562 500 0.380 2 72.231 4 22% 2 2.843 750 0.237 116.681 419/ 32 4.593 750 0.382 8 73.025 1 2^ 2.875 000 0.239 6 117.475 4H 4.625 000 0.385 4 73.818 9 22% 2 2.906 250 0.242 2 118.269 4% 4.656 250 0.388 74.612 7 215/16 2.937 500 0.244 8 119.063 4H/16 4.687 500 0.390 6 75.406 4 2% 2.968 750 0.247 4 119.856 423/32 4.718 750 0.393 2 76.200 2 3 3.000 000 0.250 120.650 4M 4.750 000 0.395 8 76.993 9 3M 2 3.031 250 0.252 6 121.444 425/32 4.781 250 0.398 4 77.787 7 31/16 3.062 500 0.255 2 122.238 413/16 4.812 500 0.401 78.581 4 3% 2 3.093 750 0.257 8 123.031 427/32 4.843 750 0.403 6 79.375 2 3H 3.125 000 0.260 4 123.825 4Ji 4.875 000 0.406 3 80.168 9 35/32 3.156 250 0.2630 124.619 429/32 4.906 250 0.408 9 80.962 7 33/ 16 3.187 500 0.265 6 125.413 415/i 6 4.937 500 0.411 5 81.756 4 3V32 3.218 750 0.268 2 126.207 4% 4.968 750 0.414 1 82.550 2 3^ 3.250 000 0.270 8 127.000 5 5.000 000 0.416 7 83.343 9 3% 2 3.281 250 0.2734 127.794 5^2 5.031 250 0.419 3 84.137 7 3&/16 3.312 500 0.276 128.588 5^6 5.062 500 0.421 9 84.931 4 3% 3.343 750 0.278 6 129.382 5% 2 5.093 750 0.424 5 85.725 2 3^ 3.375 000 0.281 3 130.175 5H 5.125000 0.427 1 86.518 9 313/32 3.406 250 0.283 9 130.969 55/32 5.156 250 0.429 7 87.312 7 37/16 3.437 500 0.286 5 131.763 53/ 16 5.187 500 0.432 3 88.106 4 315/32 3.468 750 0.289 1 132.557 57/32 5.218 750 0.434 9 88.900 2 &A 3.500 000 0.291 7 133.350 5M 5.250 000 0.437 5 89.693 9 317/S2 3.531 250 0.294 3 134.144 5%2 5.281 250 0.440 1 90.487 7 3/16 3.562 500 0.296 9 134.938 5&/16 5.312 500 0.442 7 91.281 4 319/32 3.593 750 0.299 5 135.732 5% 5.343 750 0.445 3 92.075 2 3^8 3.625 000 0.302 1 136.525 5% 5.375 000 0.447 9 92.868 9 3% 3.656 250 0.304 7 137.319 513/32 5.406 250 0.450 5 93.662 7 3H16 3.687 500 0.307 3 138.113 57/16 5.437 500 0.453 1 94.456 4 323/32 3.718 750 0.309 9 138.907 515/32 5.468 750 0.455 7 95.250 2 3M 3.750 000 0.3125 139.700 5H 5.500 000 0.458 3 LENGTHS. 37 Milli- meters. Inches. Feet. Milli- meters. Inches. Feet. 140.494 5i% 2 5.531 250 0.460 9 217.488 89/16 8.562 500 0.7135 141.288 59/ie 5.562 500 0.463 5 219.075 8^ 8.625 000 0.7188 142.082 5i% 2 5.593 750 0.466 1 220.663 8ii/i 8.687 500 0.724 142.875 5% 5.625 000 0.468 8 222.250 8M 8.750 000 0.729 2 143.669 5% 5.656 250 0.471 4 223.838 813/lfl 8.812 500 0.734 4 144.463 5ii/i6 5.687 500 0.474 225.425 8 7 /8 8.875 000 0.739 6 .145.257 5*3/32 5.718 750 0.476 6 227.013 8i5/ lfl 8.937 500 0.744 8 146.050 5H 5.750 000 0.479 2 228.600 9 9.000 000 0.750 146.844 525/ 32 5.781 250 0.481 8 230.188 9Vio 9.062 500 0.755 2 147.638 5*4i 5.812 500 0.484 4 231.775 9^ 9.125 000 0.760 4 148.432 62% a 5.843 750 0.487 233.363 93/4 6 9.187 500 0.765 6 149.225 5y 8 5.875 000 0.489 6 234.950 9M 9.250 000 0.770 8 150.019 52% 2 5.906 250 0.492 2 236.538 95/16 9.312 500 0.776 150.813 515/10 5.937 500 0.494 8 238.125 m 9.375 000 0.781 3 151.607 5% 5.968 750 0.497 4 239.713 9Vl6 9.437 500 0.786 5 152.400 6 6.000 000 0.500 241.300 9H 9.500 000 0.791 7 153.988 6Vi 6 6.062 500 0.505 2 242.888 99/16 9.562 500 0.796 9 155.575 6'/8 6.125 000 0.510 4 244.475 9 5 / 9.625 000 0.802 1 157.163 63/16 6.187 500 0.5156 246.063 9H46 9.687 500 0.807 3 158.750 Q 1 A 6.250 00( 0.520 8 247.650 9M 9.750 000 0.812 5 160.338 65/16 6.312 500 0.526 249.238 913/ 16 9.812 500 0.817 7 161.925 Wi 6.375 000 0.531 3 250.825 V/s 9.875 000 0.822 9 163.513 6V16 6.437 500 0.536 5 252.413 915^6 9.937 500 0.828 1 165.100 VA 6.500 000 0.541 7 254.001 10 10.000 000 0.833 3 166.688 69/16 6.562 500 0.546 9 255.588 101/16 10.062 500 0.838 5 168.275 6^ 6.625 000 0.552 1 257.176 10H 10.125 000 0.843 8 169.863 611/16 6.687 500 0.557 3 258.763 103/16 10.187 500 0.849 171.450 6M 6.750 000 0.562 5 260.351 10& 10.250 000 0.854 2 173.038 6i%o 6.812 500 0.567 7 261.938 105/16 10.312 500 0.859 4 174.625 6^ 6.875 000 0.572 9 263.526 IO-H 10.375 000 0.864 6 176.213 6i5/ 16 6.937 500 0.578 1 265.113 lOJie 10.437 500 0.869 8 177.800 7 7.000 000 0.583 3 266.701 l<& 10.500 000 0.875 179.388 7Vio 7.062 500 0.588 5 268.288 109/ 16 10.562 500 0.880 2 180.975 7H 7.125000 0.593 8 269.876 10^ 10.625 000 0.885 4 182.563 73/10 7.187 500 0.599 271.463 ioiyi6 10.687 500 0.890 6 184.150 7M 7.250 000 0.604 2 273.051 10 3 4 10.750 000 0.895 8 185.738 75/ 10 7.312 500 0.609 4 274.638 1013/16 10.812 500 0.901 187.325 7-Hi 7.375 000 0.614 6 276.226 10^ 10.875 000 0.906 3 188.913 7Vl6 7.437 500 0.619 8 277.813 1015/ 16 10.937 500 0.9115 190.500 7^ 7.500 000 0.625 279.401 11 11.000000 0.9167 192.088 79/ 16 7.562 500 0.630 2 280.988 HVl6 11.062 500 0.921 9 193.675 7% 7.625 000 0.635 4 282.576 iiH 11.125000 0.927 1 105.203 7H10 7.687 500 0.640 6 284.163 113/46 11.187 500 0.932 3 106.850 7''4 7.750 000 0.645 8 285.751 11M 11.250000 0.937 5 108.438 7'3/ 10 7.812 500 0.651 287.338 H 5 /16 11.312500 0.942 7 200.025 TX 7.875 000 0.656 3 288.926 11% 11.375000 0.947 9 201.613 715/io 7.937 500 0.661 5 290.513 llVie 11.437500 0.953 1 203.200 8 8.000 000 0.666 7 292.101 UH 11.500000 0.958 3 204.788 81/16 8.062 500 0.671 9 293.688 ii 9 /46 11.562 500 0.963 5 206.375 8H 8.125 000 0.677 1 295.276 UH 11.625000 0.968 8 207.963 Ss/ie 8.187 500 0.682 3 296.863 imie 11.687500 0.974 209.550 8^ 8.250 000 0.687 5 298.451 liH 11.750000 0.979 2 211.138 &% 8.312 500 0.692 7 300.038 1113/16 1.812500 0.984 4 212.725 8-H 8.375 000 0.697 9 301.626 IV4 1.875000 0.989 6 214.313 8% 8.437 500 0.703 1 303.213 lli 5 /4 6 1.937500 0.994 8 215.900 8^ 8.500 000 0.708 3 304.801 12 2.000 000 1.000 38 LENGTHS. Millimeters Feet and Inches. Millimeters. Feet and Inches. 304.801 1 foot inches 1 676.40 5 feet 6 inches 330.201 1 1 1 701.80 5 7 355.601 1 2 1 727.20 5 8 381.001 1 3 1 752.60 5 9 406.401 1 4 1 778.00 5 10 431.801 1 5 1 803.40 5 11 457.201 1 6 1 828.80 6 482.601 1 7 1 854.20 6 1 508.001 1 8 1 879.60 6 2 533.401 1 9 1 905.00 6 3 558.801 1 10 1 930.40 6 4 584.201 1 11 1 955.80 6 5 609.601 2 feet 1 981.20 6 6 635.001 2 1 2 006.60 6 7 660.401 2 2 2 032.00 6 8 685.801 2 3 2 057.40 6 9 711.201 2 4 2 082.80 6 10 736.601 2 5 2 108.20 6 11 762.002 2 6 2 133.60 7 787.402 2 7 2 159.00 7 1 812.802 2 8 2 184.40 7 2 838.202 2 9 2 209.80 7 3 863.602 2 10 2 235.20 7 4 889.002 2 11 2 260.60 7 5 914.402 3 2 286.00 7 6 939.802 3 1 2 311.40 7 7 965.202 3 2 2 336.80 7 8 990.602 3 3 2 362.20 7 9 1 016.00 3 4 2 387.60 7 10 1 041.40 3 5 2413.00 7 11 1 066.80 3 6 2 438.10 8 1 092.20 3 7 2 463.80 8 1 117.60 3 8 2 ^SO./'O 8 2 143.00 3 9 2 51 Uil 8 3 168.40 3 10 2 /il-P.Ol 8 4 193.80 3 11 2505.11 8 5 219.20 4 2 500.81 8 6 244.60 4 1 2 6 JO. 21 8 7 270.00 4 2 2 641.61 8 8 295.40 4 3 2 C.67.01 8 9 320.80 4 4 2 692.41 8 10 346.20 4 5 2 717.81 8 11 371.60 4 6 2 743.21 9 397.00 4 7 2 768.61 9 1 422.40 4 8 2 794.01 9 2 447.80 4 9 2819.41 9 3 473.20 4 10 2 844.81 9 4 498.60 4 11 2 870.21 9 5 1 524.00 5 2 895.61 9 6 549.40 5 1 2921.01 9 7 574.80 5 2 2 946.41 9 8 600.20 5 3 2971.81 9 9 625.60 5 4 2997.21 9 10 651.00 5 5 3 022.61 9 11 3 048.01 10 feet LENGTHS. Conversion Tables for Lengths. Ins. = milim't Mnis = inches. Feet = meters. Mtrs = feet.. . . yards. Yds.= meters. Miles= klmtrs. Kims- miles. 1 25.400 1 0.039 370 0.304 801 3.280 83 0.914402 1.09361 1.60935 .621 370 2 50.800 1 0.078 740 0.609 601 6.561 67 1.82880 2.18722 3.218 69 1.24274 3 76.200 2 0.118110 0.914402 9.84250 2.74321 3.28083 4.828 04 1.86411 4 101.600 0.157480 1.219 20 13.1233 3.657 61 4.374 44 3.437 39 2.485 48 5 127.000 0.196850 1.52400 16.4042 4.57201 5.468 06 i. 046 73 3.10685 6 152.400 0.236 220 1.82880 19.6850 5.48641 6.561 67 3.65608 3.728 22 7 177.800 0.275 590 2.133 60 22.965 8 6.40081 7.655 28 11.2654 4.34959 8 203.200 0.314 960 2.43840 26.2467 7.31521 8.748 89 12.8748 4.97096 9 228.600 0.354 330 2.74321 29.527 5 8.229 62 9.84250 14.484 1 5.59233 10 254.001 0.393 700 3.04801 32.808 3 9.14402 10.936 1 16.093 5 6.21370 11 279.401 0.433070 3.35281 36.089 2 10.058 4 12.029 7 17.7028 6.835 07 12 304.801 0.472 440 3.65761 39.3700 10.9728 13.1233 19.3122 7.45644 13 330.201 0.511810 3.96241 42.650 8 11.8872 14.2169 20.921 5 3.077 81 14 355.601 0.551 180 4.26721 45.931 7 12.801 6 . 15.3106 22.5309 8.699 18 15 381.001 0.590 550 4.572 01 49.2125 13.7160 16.4042 24.1402 9.320 55 16 406.401 0.629 920 4.87681 52.493 3 14.6304 17.497 8 25.749 6 9.941 92 17 431.801 0.669 290 5.18161 55.7742 15.5448 18.591 4 27.3589 10.5633 18 457.201 0.708 660 5.48641 59.0550 16.4592 19.6850 28.968 2 11.1847 19 482.601 0.748 030 5.79121 62.3358 17.3736 20.778 6 30.577 6 11.8060 20 508.001 0.787 400 6.09601 65.6167 18.2880 21.8722 32.1869 12.4274 21 533.401 0.826770 6.40081 68.897 5 19.2024 22.965 8 33.796 3 13.0488 22 558.801 0.866 140 6.70561 72.1783 20.1168 24.059 4 35.405 6 13.670 1 23 584.201 0.905510 7.01041 75.459 2 21.0312 25.153 1 37.015 14.2915 24 303.601 0.944880 7.31521 78.7400 21.9456 26.2467 38.624 3 14.9129 25 635.001 0.984 250 7.62002 82.020 8 22.860 27.3403 40.233 7 15.534 2 26 660.401 1.02362 7.92482 85.3017 23.774 4 28.4339 41.8430 16.1556 27 685.801 .062 99 8.229 62 88.582 5 24.688 9 29.527 5 13.452 4 16.7770 28 711.201 .10236 8.534 42 91.8633 25.603 3 30.621 1 15.061 7 17.398 4 29 736.601 .14173 8.83922 95.1442 26.517 7 31.7147 16.671 1 18.0197 30 762.002 .181 10 9.14402' 98.425 27.432 1 32.808 3 48.280 4 18.641 1 31 787.402 ' .220 47 9.44882 101.706 28.3465 33.901 9 49.889 8 19.2625 32 812.802 .259 84 9.753 62 104.987 29.2609 34.995 6 51.499 1 19.8838 33 838.202 .299 21 10.058 4 108.268 30.1753 36.089 2 53.1085 20.505 2 34 863.602 1.33858 10.3632 111.548 31.0897 37.182 8 54.7178 21.1266 35 889.002 1.37795 10.6680 114.829 32.004 1 38.276 4 56.327 2 21.7479 36 914.402 1.41732 10.9728 118.110 32.9185 39.3700 57.9365 22.369 3 37 139.802 1.45669 11.2776 121.391 33.8329 40.463 6 59.545 8 22.990 7 38 365.202 1.43606 11.5824 124.672 34.747 3 41.5572 51.1552 23.612 1 39 990.602 1.53543 11.8872 127.953 35.661 7 42.6508 62.7645 24.233 4 40 1 016.00 1.57480 12.1920 131.233 36.576 1 43.744 4 64.3739 24.8548 41 1 041.40 1.61417 12.4968 134.514 37.4905 44.838 1 65.983 2 25.4762 42 1 066.80 1.65354 12.8016 137.795 38.4049 45.931 7 67.592 6 26.097 5 43 1 092.20 1.69291 13.1064 141.076 39.3193 47.025 3 69.2019 26.7189 44 1 117.60 1.73228 13.4112 144.357 40.233 7 48.1189 70.8113 27.3403 45 1 143.00 1.77165 13.7160 147.638 41.1481 49.2125 72.420 6 27.961 6 46 1 168.40 1.81102 14.0208 150.918 42.062 5 50.306 1 74.0300 28.583 47 1 193.80 1.85039 14.3256 154.199 42.9769 51.3997 75.639 3 29.204 4 48 1219.20 1.88976 14.6304 157.480 43.891 3 52.4933 77.248 7 29.8258 49 1 244.60 1.92913 14.9352 160.761 44.805 7 53.5869 78.858 30.447 1 50 1 270.00 1.96850 15.2400 164.042 45.720 1 54.680 6 80.467 4 31.0685 40 LENGTHS. Conversion Tables for Lengths (CONCLUDED). Ins. = milimM Mms = inches. Feet = meters. Mtrs = feet . . yards. Yds.= meters. Miles= klmtrs. Klms= miles. 51 1 295.40 2.007 87 15.5448 167.323 46.634 5 55.774 2 82.0767 31.6899 52 1 320.80 2.047 24 15.849 6 170.603 47.5489 56.807 8 83.686 1 32.3112 53 1 346.20 2.08661 16.1544 173.884 48.463 3 57.9014 85.295 4 32.932 6 54 1371.60 2.12598 16.4592 177.165 49.3777 59.055 80.904 7 33.5540 55 1 397.00 2.16535 16.7640 180.446 50.292 1 00.148 6 88.514 1 34.1753 56 1 422.40 2.20472 17.0688 183.727 51.2065 61.2422 90.1234 34.796 7 57 1 447.80 2.24409 17.3736 187.008 52.1209 62.3358 91.7328 35.418 1 58 1 473.20 2.283 46 17.6784 190.288 53.035 3 63.429 4 93.342 1 36.039 5 59 1 498.60 2.32283 17.9832 193.569 53.949 7 64.523 1 94.951 5 36.660 8 60 1 524.00 2.362 20 18.2880 196.850 54.864 1 65.6167 90.5008 37.282 2 61 1 549.40 2.40157 18.5928 200.131 55.7785 66.7103 98.1702 37.903 6 62 1 574.80 2.44094 18.897 6 203.412 56.692 9 07.8039 99.779 5 38.524 9 63 1 600.20 2.48031 19.2024 206.693 57.607 3 68.897 5 101.389 39.1463 64 1 625.60 2.51968 19.5072 209.973 58.5217 09.991 1 102.998 39.767 7 65 1 651.00 2.559 05 19.8120 213.254 59.436 1 71.0847 104.608 40.389 66 1 676.40 2.598 42 20.1168 216.535 60.350 5 72.1783 106.217 41.0104 67 1701.80 2.63779 20.421 6 219.816 61.2649 73.2719 107.826 41.6318 68 1 727.20 2.677 16 20.7264 223.097 62.1793 74.3050 109.436 42.253 2 69 1 752.60 2.71653 21.0312 226.378 63.093 7 75.459 2 111.045 42.8745 70 1 778.00 2.75590 21.3360 229.658 64.008 1 76.552 8 112.654 43.495 9 71 1 803.40 2.795 27 21.6408 232.939 64.922 5 77.6464 114.264 44.1173 72 1 828.80 2.834 64 21.9456 236.220 65.8369 78.7400 115.873 44.738 6 73 1 854.20 2.87401 22.2504 239.501 66.751 3 79.833 6 117.482 45.3600 74 1 879.60 2.91338 22.555 2 242.782 67.665 7 80.927 2 119.092 45.981 4 75 1 905.00 2.952 75 22.8600 246.063 68.580 1 82.020 8 120.701 46.602 7 76 1 930.40 2.992 12 23.1648 249.343 69.4945 83.1144 122.310 47.224 1 77 1 955.80 3.031 49 23.469 6 252.624 70.408 9 84.208 1 123.920 47.8455 78 1981.20 3.07086 23.7744 255.905 71.3233 85.301 7 125.529 48.466 9 79 2 006.60 3.11023 24.079 2 259.186 72.2377 80.395 3 127.138 49.0882 80 2 032.00 3.14960 24.3840 262.467 73.1521 87.488 9 128.748 49.709 6 81 2 057.40 3.18897 24.688 8 265.748 74.0065 88.5825 130.357 50.331 82 2 082.80 3.228 34 24.9936 269.028 74.981 89.070 1 131.966 50.952 3 83 2 108.20 3.26771 25.298 4 272.309 75.8954 90.709 7 133.576 51.5737 84 2 133.60 3.30708 25.603 3 275.590 76.809 8 91.8033 135.185 52.1951 85 2 159.00 3.34645 25.908 1 278.871 77.7242 92.9509 136.795 52.8164 86 2 184.40 3.38582 26.2129 282.152 78.638 6 94.050 138.404 53.4378 87 2 209.80 3.425 19 26.5177 285.433 79.553 95.1442 140.013 54.0592 88 2 235.20 3.46456 26.8225 288.713 30.467 4 90.2378 141.623 54.680 6 89 2260.60 3.50393 27.1273 291.994 81.3818 97.331 4 143.232 55.301 9 90 2 286.00 3.54330 27.432 1 295.275 82.2962 98.425 144.841 55.923 3 91 2311.40 3.58267 27.7369 298.556 83.2106 99.5180 140.451 56.544 7 92 2 336.80 3.62204 28.041 7 301.837 84.1250 100.012 148.000 57.1660 93 2 362.20 3.66141 28.3465 305.118 85.039 4 101.700 149.009 57.787 4 94 *> 387.60 3.700 78 28.651 3 308.398 85.953 8 102.799 151.279 58.408 8 95 2413.00 3.740 15 >8.956 1 311.679 86.868 2 103.893 152.888 59.030 1 96 2438.40 3.77952 '9.2609 314.960 87.782 6 104.987 154.497 59.651 5 97 2 463.80 3.81889 9.565 7 318.241 88.6970 100.080 150.107 60.272 9 98 489.20 3.858 26 9.8705 321.522 89.6114 107.174 157.716 60.894 3 99 514.60 3.897 63 0.1753 324.803 90.525 8 108.208 159.325 61.5150 100 2 540.01 3.937 00 0.480 1 328.083 91.4402 109.301 160.935 62.1370 SURFACES. 41 SURFACES. There are no fundamental standards of surfaces or areas, as these meas- ures are all based on the linear measures, which see for their fundamental values. The legal U. S. yard, foot, inch, mile, etc., being about 3 parts in one million larger than the legal values in Great Britain, the U. S. square yard, square foot, etc., will be about 6 parts in one million larger than the British, a difference which is absolutely negligible in all but the most refined physical measurements. The values given in the following tables are based on the U. S. legal relation. To reduce the values in these tables to British square miles, square yards, square feet,,and similar units, when very great accuracy is required subtract 6 parts for every million from all those values of a sq. mile, sq. yard, etc., which are in terms of other units than miles, yards, etc.; for instance, add this correction in the value of one sq. mile in sq. meters, but not in sq. feet, as the latter is the same for both ; or add 6 parts for every million to all the values of other units which are given in terms of sq. miles, sq. yards, etc. This will never affect the 5th place of figures by more than one unit and generally less. A circular unit or measure (such as a circular mil) is the area of a circle whose diameter is one unit or measure (as one mil). It is used for cross-sections of round wires, pipes, rods, etc., and avoids the necessity of using the value of ir ( = 3.141 592 65). If the diameter of a circle is d, its area in circular units is simply d 2 . Some of the specific relations given in the table may also be expressed as follows: If d is the diameter of a circle expressed in one kind of a unit, then the area in the other unit will be: if d s in mils the area in sq. millimeters =d 2 X 0.000 506 709* if d itd if d if d itd itd itd s in millimeters the area in sq. mils =d 2 X 1 217.36* s in centimeters the area in sq. inches = d 2 X 0.121 736* s in centimeters the area in sq. feet = d 2 X 0.000 845 391* s in inches the area in sq. centimeters = d 2 X 5.067 09* s in inches the area in sq. feet =d 2 X 0.005 454 15* s in feet the area in sq. centimeters = d 2 X729.662* s in feet the area in sq. inches = d 2 X 113.097* SURFACES; Circular or Cross -section Measures. Usual. Aprx. means within 2% ; "sq." means square and is often used as a suffix instead of the exponent ( 2 ), being simpler for printing and type-writing; thus sq. cm for era 2 ; sq. ft for ft 2 . * Checked by L. A. Fischer, Asst. Phys. National Bureau of Standards. 1 circular mil [CM]= 0.785 398* sq. mil. Aprx. % . - . -0.000 645 163* Cmm. Aprx. ifa -*- 1 000 .. = 0.000 506 709* sq. mm. Aprx. V-f- 1 000 . = 0.000 001 circular inch 1 sq. mil [mil 2 ] = 1.273 24* circular mils. Aprx. 1% = 0.000 821 447* circ. mm. Aprx. Vi 2 * 100 = 0.000645 163* sq. millimeter. Aprx. %i-*-l 000 J_. = 0.000 001 sq. inch 6-000 0000 1 circular millimeter [Cmm]: = 1 550.00* circular mils. Aprx. i# X 1 000 3.190 3308 = 1 217.36* sq. mils. Aprx. % X 1 000 3.085 4207 = 0.785 398* sq. millimeter. Aprx. % 1-895 0899 = 0.01 circular centimeter 2-000 0000 1 sq. millimeter [mm 2 ] = 1 973.52* cir. mils. Aprx. 2 000. . 3.295 2409 = 1 550.00* sq. mils. Ap. 1% X 1 000 3-190 3308 = 1.273 24* circ. mm. Aprx. 1%. . . 0-104 9101 = 0.012 732 4* circ-. cm. Aprx. Ho 2-104 9101 = 0.01 sq. centimeter 2-000 0000 =0.001 550 00* sq. in. Aprx. % 8 * 100 . 5.190 3308 42 SURFACES. 1 circular centimeter [Ccm]: = 155 000.* circular mils. Aprx. Ufa X 100 000 5-190 3308 121 736.* sq. mils. Aprx. 12 X 10 000 : 5-085 4207 = 100. circular millimeters 2-000 0000 78.539 8 * sq. millimeters. Aprx. 80 1-895 0899 = 0.785 398 * sq. centimeter. Aprx. 8 /lo 1-895 0899 = 0.155 000 * circular inch. Aprx. % 3 1-190 3308 = 0.121 736 * sq. inch. Aprx. 12-^100 1-085 4207 = 0.001 076 39 * circular foot. Aprx. 108-^-100 000 3-031 9683 = 0.000 845 391 * sq. foot. Aprx. 1/12-^ 100 4-927 0582 1 sq. centimeter [cm 2 ]: = 197 352.* circular mils. Aprx. 2 X 100 000 5-295 2409 155 000.* sq. mils. Aprx. *Vr X 100 000 5-190 3308 127.324 * circular millimeters. Aprx. l /$X 1000 2-1049101 100. sq. millimeters 2. 000 0000 1.273 21 * circular centimeters. Aprx. 1% 0-104 9101 = 0.197 352 * circular inch. Aprx. ^ - . - 1-295 2409 = 0.155 000 * sq. inch. Aprx. % 8 1-190 3308 = 0.01 * sq. decimeter 2-000 0000 = 0.001 370 50 * circular foot. Aprx. 1! *-s- 1 000 3-136 8784 = 0.001 076 387 * sq. foot. Aprx. 108 -=-100 000 3-031 9683 t circular inch [Gin]: = 1 000 000. circular mils 6-000 0000 = 785 398.* sq. mils. Aprx. % X 1 000 000 5-895 0899 = 645.103 * circular millimeters. Aprx. 1% X 100 2-809 6692 = 506.709 * sq. millimeters. Aprx. ^X 1 000 2-704 7591 = 6.451 63 * circular centimeters. Aprx. *% 0-809 6692 = 5.067 09 * sq. centimeters. Aprx. 5 0-704 7591 = 0.785 398 * sq. inch. Aprx. % 1-895 0899 = 0.006 944 44 * circular foot. Aprx. 7-*- 1 000 3-841 6375 = 0.005 454 15 * sq.foot. Aprx. /n-v- 100 3-736 7274 I sq. inch [in 2 ]= 1 273 240.* cir. mils. Aprx. HX 10000 000 6-104 9101 = 1 000 000. sq. mils 6-000 0000 = 821.447 * cir. mm. Aprx. %X 1 000 2-914 5793 = 645.163 * sq. mm. Aprx. 1% X 100 2-809 6692 = 8.214 47 * cir. centimeters. Aprx. % X 10. . 0-914 5793 = 6.451 63 * sq. cm. Aprx. 1% or 6M 0-809 6692 = 1.273 24 * circular inches. Aprx. 1% 0-104 9101 = 0.064 516 3 * sq. decimeter. Aprx. %i-i- 10.... 2-809669? = 0.008841 94 * circular foot. Aprx. 9-=- 1 000.. . 3-9465476 = 0.006 944 44 * sq. foot. Aprx. ftooo 3-841 6375 I sq. decimeter [dm 2 ] = 100. sq. centimeters 2-000 0000 = 15.5000* sq. inches. Aprx. iVrX 10. 1.1903308 = 0.107 638 7* sq.foot. Aprx. 11 -=- 100.. 1-031 968? = 0.01 sq. meter 2-000 000( = 0.011 9599* sq. yard. Aprx. 12-4-1000 2-0777253 1 circular foot [Cft] = 929.034* cir. cm. Aprx. 1^2 X 1 000.. . 2-968 0317 = 729.662* sq. cm. Aprx. s/u X 1 000 2-863 1216 144. circular ins. Aprx. M- X 1000. 2-1583625 = 113.097* sq. inches. Aprx. % X 1 000. . 2-0534524 = 0.785 398* sq. foot. Aprx. % 1-895 0893 1 sq. foot [ft 2 ] (Brit.) = 0.999 994 2 sq. foot (U. S.). Aprx. 1 L999 9975 = 0.092902 9 sq. meter. Aprx. Hia-*- 10. . 2-9680292 1 sq.foot [ft 2 ] (U.S.) = 1 182.88* cir. cm. Aprx. %X 1 000. . 3-0729410 = 929.034* sq. cm. Aprx. n/12 X 1 000. 2-968 0317 = 183.346* cir. in. Aprx. ^X 100 2-2632723 = 144. sq. ins. Aprx. V 7 X 1 000. .. 2-158 3625 = 9.290 34* sq. dm. Aprx. H1 2 X 10. .. 0-968 0317 = 1.273 24* circular feet. Aprx. 1%. . .. 0-104 9101 = 1.000005 7 sq. feet (Brit.). Aprx. 1.. . 0-000 0025 = 0.111 111* sq. yard, or V 9 1-0457575 = 0.092 903 4* sq. meter. Aprx. iMa-s-10. 2-968 0317 Isq. yard [yd 2 ] (Brit. )= 0.999 994 3 sq. yard (U.S.). Aprx. 1... 1-999 9975 = 0.836 126 sq. meter. Aprx.% 1.922 2717 SURFACES. 43 1 sq. yard[yd 2 ](U. S.) = 8 361.31 * sq. cm. Ap.%X 10000 3-922 2742 1 296.* sq. ins. Aprx. 13 X 100 3. 112 6050 = 83.613 1 * sq. dm. Aprx. %X100 1-922 2742 9. sq.feet 0-9542425 = 1.0000057 sq. yds. (Brit.). Ap. 1. 0-000 0025 = 0.836 131 * sq. meter. Aprx. % . . f.922 2742 = 0.008 361 31 * are. Aprx. %-s-100 3.922 2742 = 0.000 206 612 * acre. Ap. 21 -*- 100 000. 4.315 1546 1 sq. meter [m 2 ] = 10 000. sq. centimeters 4-000 0000 = 1 550.00 * sq. ins. Aprx. i# X 1 000. ... 3-1903308 = 100. sq. decimeters 2-000 0000 = 10.76393 sq.ft. (Brit.). Aprx. 12 Ai X 10 1-031 9708 = 10.763 87 * sq. ft. (U. S.). Aprx. i%i X 10 1.Q31 9683 = 1.19599 sq. yds. (Brit.). Aprx. add % 0-0777283 = 1.195 99 * sq. yds. (U.S.). Aprx. add % 0-0777258 = 0.01 are 2-000 0000 = 0.000 247 104 * acre. Aprx. \i + \ 000 4-392 8804 1 are or ar [a] = 1 076.387* sq. feet. Aprx. i%i X 1 000 3-031 9683 = 119.599* sq. yards. Aprx. 120 2-077 7258 = 100. sq. meters 2-000 0000 = 10. meters square 1-000 0000 " = 1 . sq. decameter 0-000 0000 = 0.024 710 4* acre. Aprx.M^-10 2-3928804 = 0.01 hectare 2-000 0000 1 acre [A] = 43 560.* sq. feet. Aprx. HsX 1 000 000 4-639 0879 = 4 840.* sq. yards. Aprx. 4 800 3-684 8454 = 4 046.87 * sq. meters. Aprx.4XlOOO 3-6071196 = 208.710 * feet square. Aprx/210 2-319 5440 = 40.468 7 * ares. Aprx. 40 1-6071196 = 0.404 687 * hectare. Aprx. y 10 1-607 1196 = 0.00404687 * sq. kilometer. Aprx. 4 -f- 1000 3-6071196 ==0.001 562 50* sq. mile. Aprx. i#-s- 1 000 3-1938200 1 hectare [ha]= 107 638.7* sq. feet. Aprx. i2/ lt x 100 000. . .. 5-031 9683 = 11 959.9* sq. yards. Aprx. 12 X 1 000 4-077 7258 = 10 000. sq. meters 4-000 0000 = 100. ares 2-000 0000 = 2.471 04* acres. Aprx. 1% 0-392 8804 = 0.01 sq. kilometer 2-000 0000 = 0.003861 01* sq. mile. Aprx. ^ 6 -^10 3-586 7004 lsq.kilometer[km2]= 10763 867.* sq.ft. Ap. i% X 10 000000 7-031 9683 = 1 195 985.* sq. yds. Aprx. 12 X 100 000 6-077 7258 = 1 000 000. sq. meters 6-000 0000 = 10 000. ares 4-000 0000 = 247.104 * acres. Aprx. MX 1 000.. .. 2-392 8804 = 100. hectares 2-000 OOOC = 0.386 101 * sq. mile. Aprx. Ke X 10. . . 1-586 7004 1 sq. mile [ml 2 ] = 27 878 400. = 3097600. = 2589999. = 25 900.0 = 640. = 259.000 = 2.590 00 sq. feet. Aprx. ^ X 10 000 000 . 7-445 2678 sq. yards. Aprx. 31 X 100 000. . . 6-491 0253 sq. meters. Aprx. 26 X 100 000 . . 6-413 2996 ares. Aprx. 26 X 1 000 4-413 2996 acres 2-806 1800 hectares. Aprx. 260 2-413 2996 sq. kilometers. Aprx. 26 -^ 10 0-413 2996 SURFACES (continued). Unusual, Special Trade, or Obsolete. 1 milliare = 0.1 sq. meter. 1 ceii tare, centar, or centaire = 10.763 87 sq. feet = l. sq. meter. 1 square (building) = 100. sq. feet. 1 declare = 10. sq. meters = 0.1 are. 1 sq. rod, or sq. pole, or sq. perch = 625. sq. links (survey or 's) = 272M sq. feet = 30M sq- yards = 25. 293 sq. meters = Vi60 acre. 1 sq, chain (Gunter's or surveyor's) = 4 356. sq. feet = 484. sq. yards = 404.687 sq. meters = 16. sq. rods, poles, or perches = 4.046 87 ares = Mo acre = 0.0404687 hectare = 0.000 404 687 sq. kilometer = 0.000 156 250 sq. mile. 44 SURFACES. 1 sq. meter = 0.002 471 04 sq. chain. 1 are = 0.247 104 sq. chain. 1 acre = 10. sq. chains. 1 hectare = 24. 7 10 4 sq. chains. 1 decare (not used) = l 000. sq. meters =10. ares. 1 rood ("R]=10 890. sq. feet = 1 210. sq. yards = 1 011.72 sq. meters = 40. sq. rods, poles, or perches = 3^ acre. l.fardingdeal (Brit.)-l. rood. 1 circular acre = 235.504 feet diameter. 1 section (of land) = 1. mile square. 1 township = 36. sq. miles. 1 sq. myriameter or miriameter = 100. sq. kilometers = 38.610 1 sq. miles. SURFACES (concluded). Foreign. These are mostly obsolete as the metric system is now used in most foreign countries. The British measures are included among the U. S. measures, being very nearly, and sometimes quite, the same. The trans- lated terms are merely synonymous, and not the exact equivalents. Germany. Prussia 1 Morgen=180 sq. Ruthen (rods). 1 Morgen = 25.53225 ares = 2 553.225 sq. meters = 0.630 912 acres. According to Nystrom's Mechanics, 1 Berlin Morgen, great, = 6 786 sq. yards; ditto, small, = 3 054 sq. yards. 1 Hamburg Morgen =11 545 sq. yards. 1 Han- over Morgen = 3 100 sq. yards. 1 Prussian Morgen = 3 053 sq. vards. France. 1 arpent (acre) = 100 sq. perches (French, nearly same as U. S.) = 34.19 are, or 51.07 are = 0.844 849 acres, or 1.261 96 acres. Austria. 1 Joch=1600 sq. Klafter = 5755 sq. meters = 1.422 08 acres. 1 Vienna Joch = 6 889 sq. yards. Sweden. 1 Tunnland = 56 000 sq. fot = 49.364 1 ares=1.21981 acres. According to Nystrom it is equal to 5 900 sq. yards. Russia. 1 Dessaetine = 2 400 sq. saschehn = 10 925 sq. meters= 2.699 61 acres. 13 066.2 sq. yards (Nystrom). Switzerland. 1 Geneva arpent = 6 179 sq. yards; 1 faux = 7 855 sq. yards; 1 Zurich acre = 3 875.0 sq. yards. Spain. 1 fanegada (since 1801) = !. 5871 acres = 69 134.08 sq. feet; according to Nystrom it is equal to 5 500 sq. yards. Japan. 1 cho (apparently not 1 cho length squared) = 10 tan of 10 se of 30 tsubo. 1 tsubo = l ken square = 3. 31 square meters = 35. 6 square feet. 1 square meter = 0.302 tsubo. 1 square foot = 0.028 1 tsubo. 1 cho = 2. 45 acres. Miscellaneous. (From Nystrom's Mechanics.) 1 fanegada, Canary Isles, = 2422 sq. yards. 1 acre, Ireland, = 7 840 sq. yards. 1 acre, Scot- land, =6 150 sq. yards. 1 moggia, Naples, = 3998 sq. yards. 1 Pezza, Rome, = 3 158 sq. yards. 1 Geira, Portugal, = 6 970 sq. yards. Conversion Tables for Surfaces. Sq. in.= Sq.cm.= Sq. ft.= Sq. m.= Sq.yd.= Acres =* Hect'r.- sq. cm. sq. ins. sq. met. sq ft sq. yds. hect'res. acres. sq.met. 1 2 3 4 5 6 7 8 9 10 6.451 63 12.903 3 19.3549 25.806 5 32.258 1 38.709 8 45.1614 51.6130 58.064 6 64.5163 0.155000 0.309 999 0.464999 0.619999 0.774 998 0.929 998 1.08500 1.23999 1.39500 1.55000 0.092 903 0.185807 0.278 710 0.371 614 0.464517 0.557 420 0.650 324 0.743227 0.836 131 0.929 034 10.763 9 21.5277 32.291 6 43.0555 53.8193 64.583 2 75.347 1 86.1110 96.874 8 107.639 0.836 13 1 67226 2.508 39 3.344 52 4.18065 5.01678 5.85291 6.689 05 7.525 18 8.36131 1.19599 2.391 97 3.58796 4.783 94 5.979 93 7.17591 8.371 90 9.567 88 10.7639 11.9599 0.404 687 0.809 375 1.21406 1.61875 2.023 44 2.428 12 2.83281 3.237 50 3.642 19 4.04687 2.471 04 4.942 09 7.41313 9.884 18 12.3552 14.8263 17.297 3 19.768 4 22.239 4 24.7104 VOLUMES. 45 VOLUMES. Cubic and Capacity Measures. The fundamental standard measures of volumes in the United States are; (a) the cubes of linear dimensions in terms of units based on the international meter; (b) the liter which is the volume of the mass of one international kilogram of pure water at its maximum density, and at 760 mm. barometric pressure; (c) the gallon, which is equal to exactly 231 cubic inches; (d) the bushel (the old Winchester bushel of England) which is equal to exactly 2 150.42 cubic inches. The inch here referred to is the one derived from this meter. The liter, being based on this kilo- gram, is the same in all the countries participating in the international con- vention. These four measures of volumes and capacities are those now used by the National Bureau of Standards in Washington and are in general use in this country. They are definite and accurate, except that the pre- cise volume of the liter expressed in cubic centimeter is still slightly un- certain as explained below. The liter is, according to the decision some years ago of the Interna- tional Committee of Weights and Measures, defined as "the volume of the mass of one kilogram of pure water at its maximum density, and under normal atmospheric pressure." The accepted temperature, at present, at which the density of water is a maximum, is 4 C., but this may be altered slightly by subsequent determinations. The normal atmospheric pressure referred to is that exerted by a vertical column of mercury 760 mm. high, at latitude 45 and at sea level, the mercury having a den- sity of 13.595 93. This definition of the liter has been adopted by the National Bureau of Standards. It was originally intended that the kilo- gram should be the weight of a cubic decimeter of water, thus making the liter exactly a cubic decimeter. The kilogram, however, being now fixed definitely as the weight of a certain piece of metal, and as this estab- lishes the liter, it still remains to determine by measurement the precise volume of the liter in cubic decimeters. The International Bureau began this determination some five years ago, and the results thus far obtained indicate that the liter is about 25 parts in 1 000000. greater than the cubic decimeter. f As this relation will always be subject to slight corrections, jvnd as the difference is of no consequence in practice, the National Bureau of Standards has for the present assumed that the liter and the cubic decimeter are equivalent; this identity is assumed also in all the tables in this book. The difference above mentioned would at most affect only the fifth place of figures slightly* and generally only the sixth. The capac- ities of vessels are determined by weighing the water necessary to fill them, and not by measuring their dimensions (see the table of the weights and volumes of water). The Customs Service and the Internal Revenue Bureau uss the gallon of 231 cubic inches above referred to; it is the old British wine gallon. In these tables this gallon and the measures based on it are indicated as "liquid; U. S." to distinguish them from the "dry; U. S." measures and the Imperial or "Brit.'* measures. The gallon and liter are the units for measuring liquids, while the bushel and liter are the units for dry measures, as for grain, fruits, vegetables, etc. The "dry" measures are about one sixth greater than the corre- sponding "liquid" measures; accurately: dry measures XI. 163 65 (aprx. add ^) = liquid meas. (log 0-065 8213) liquid measures X 0.859 367 (aprx. subtr. Vr) = dry meas. (log 1.934 1787) Th3 fluid ounce is in use chiefly by apothecaries. The barrel is no fixed unit, and no value of a barrel has ever been adopted by Congress; in the Customs Service and Internal Revenue Bureau every barrel is gauged. Concerning the bushel tho Revised Statutes of the United States, under the ascertainment of duties on grain, chapter 6, sec. 2919, says: "For the purpose of estimating the duties on importations of grain, the num- t Guillaume in n, recent book on the International Bureau of Weights and Measures states that this Bureau has found the mass of a cubic deci- meter of water at 4 C. to be 0.999 955 kg. This makes a liter about 45 parts in 1000000. greater than a cubic decimeter. But it seems that no formal adoption of any value has yet been made. 46 VOLUMES. ber of bushels shall be ascertained by weight, instead of by measuring; and sixty pounds of wheat, fifty-six pounds of corn, fifty-six pounds of rye, forty-eight pounds of barley, thirty-two pounds of oats, sixty pounds of pease and forty-two pounds of buckwheat, avoirdupois weight, shall respectively be estimated as a bushel." In Great Britain the liter is exactly the same as in the United States. The gallon, which in this country is often called the Imperial or British gallon, was originally defined as the volume of "ten imperial pounds of distilled water, weighed in air against brass weights, with the water and the air at the temperature of 62 Fahrenheit, the barometer being at 30 inches." According to the latest computations of the Standards Office in London, the British or Imperial gallon is equal to 4.545963 1 liters; this value was made legal in 1896 t m Great Britain, and is the one used as a basis throughout these tables, even for computing such values as the gallon in cubic inches ; the figures are therefore quite consistent through- out, except that the U. S. yard, foot, inch are used, which are about 3 parts in one million larger than the British, a difference too small to be considered in any but the most refined physical measurements. Owing to this difference, the U. S. cubic yard, cubic foot, etc., are about 9 parts in one million greater than the British. Hence to reduce these values in those tables to British yards, feet, and inches, when very great accuracy is required, subtract 9 parts for every million from all the values of a cubic yard, cubic foot, and cubic inch given in terms of other units than yards, feet, and inches; for instance, add this correction .in the value of a cb. foot in cb. meters, but not in cb. inches, as the latter is the same for both; or add 9 parts for every million to all the values of other units given in terms of cubic yards, cubic feet, and cubic inches. This will never affect the fifth place of figures by more than one unit, and generally not that much. In Great Britain the measures of capacities (gallons, quarts, etc.) are the same for liquid as for dry materials. The British measures of capacity differ but little from the corresponding dry measures in the United States; accurately: Brit. meas. X 1.032 02 (aprx. add Mo) = dry U. S. meas. (log 0-013 6888) dry U.S. meas. X 0.968 972 (aprx. sub. Mo) = Brit. meas. (log 1-9863112) The British measures of capacity are about 20% greater than the corre- sponding liquid measures in the United States; accurately: Brit. meas. XI. 200 91 (aprx. add %} = liquid U.S. meas. (log 0-079 5101) liquid U.S. meas. X 0.832 602 4 (aprx. i% 2 ) = Brit. meas. (log 1.920 4899) The U. S. and the British apothecary fluid measures, smaller than the pint, differ very little from each other, almost exactly 4%, the former being the greater; for conversion add 4% to the quantity expressed in the U. S. measures, or subtract 4% when expressed in British measures. VOLUMES. Cubic and Capacity Measures. Usual. ** Accepted by the National Bureau of Standards. * Checked by L. A. Fischer. Asst. Phys. National Bureau of Standards. Aprx. means within 2%. ap. means apothecary measures: cb. means cubic, and is often used as a suffix instead of the exponent ( 3 ), being simpler for printing and type writing; thus cb. cm. for cm. 3 , or cb. ft. for ft. 3 1 cb. centimeter [cm 3 ] or 1 milliliter [ml]: Logarithm = 1 000. cb. millimeters 3-000 0000 = 0.061 023 4* cb. inch. Aprx.6-M00. 2-7854962 = 0.01 deciliter 2-000 0000 = 0.00211336 pint* (liquid; U.S.). Aprx. 21 -s- 10 000. . 3-3249742 = 0.001 816 16* pint (dry; U.S.). Aprx. %i -MOO 3-259 1529 = 0.001 759 80 pint (Brit.). Aprx. %-M 000 3-245 4641 - 0.001 056 68* quart (liquid; U. S.). Aprx. 2l/ 2 -^10 000. 3-023 9442 0.001 liter or cb. decimeter 3-000 0000 -0.000 908 078* quart (dry; U. S.). Aprx. 9-5- 10 000 4-958 1229 = 0.000 879 902 quart (Brit.). Aprx. K-M 000 4-944 4341 t Formerly the legal value is said to have been 4.543 46 liters. VOLUMES. 47 1 cb. inch [in 3 ] = 16 387.16* cb. mm. Aprx. MX 100 000. .. 4-214 5038 = 16.387 16* cb. cm. or ml. Aprx. 2^X100.. 1-2145038 = 0.163 871 6* deciliters. Aprx. Y & = 0.0346320* pt. (liq.; U.S.). Aprx. %-*- 100. = 0.029 761 6* pt. (dry; U. S.). Aprx. 3-^100. = 0.0288382 pt. (Brit.). Aprx. % + 10 ' = 0.017 316 0* qt. (liq.; U.S.). Aprx. %-s- 100. " = 0.01638716 liter or cb. dm. Aprx. Y & -r- 10.. " = 0.014880 8* qt. (dry; U.S.). Aprx. %*- 100 " = 0.0144191 qt. (Brit.). Aprx. ^-J- 10 2-1589379 = 0.004 329 00* gal. (liq. ; U. S.). Aprx. % -4-100 3-636 3880 " = 0.00360477 gal. (Brit.). Aprx. fn -*- 100. .. 3-5568779 " =0.000 578 704* cb. foot. Aprx. # -r- 1 000 4-762 4563 1 deciliter [dl] = 100. cb. centimeters 2-000 0000 6.102 34* cb. inches. Aprx. 6Ho 0-785 4962 0.211 336* pt. (liq. ; U. S.). Aprx. 21 H- 100. 0.181 616* pt. (dry; U. S.). Aprx. %i 0.175 980 pint (Brit.). Aprx. % + 10 0.105 668* qt. (liq.; U. S.). Aprx. 2^-5- 100. 0.1 liter or cb. decimeter, 324 9742 -259 1529 -245 4641 -023 9442 -000 0000 = 0.0908078* quart (dry; U.S.). Aprx. Ml-. -- 2-958 1229 = 0.0879902 quart (Brit.). Aprx. ]/$ + 10. . . . 2-9444341 = 0.026 417 0* gal. (liq. U. S.). Aprx. %-*- 100. . 2-421 8842 = 0.0219975 gal. (Brit.). Aprx. 22 -^ 1 000. . . 2-3423741 " =0.003 531 45* cb. foot. Aprx.%-^-1 000 3.547 9525 1 pint [pt] (liquid; U. S.): = . 473.179* cb. centimeters. Aprx. Hi X 10 000 2-675 0258 = 28.875* cb. inches. Aprx. ft X 100 1-460 5220 = 16. fluid ounces (ap. U. S.) 1-204 1200 = 4.731 79* deciliters. Aprx. 4% 0-675 0258 = 4. gills (liquid; U. S.). . . '. 0-602 0600 = 0.859 367* pint (dry; U. S.). Aprx. % 1-934 1787 = 0.832 702 4 pint (Brit.). Aprx. % 1.920 4899 = 0.5 quart (liquid; U. S.); or \i 1-698 9700 = 0.473 179* liter or cb. decimeter. Aprx. Ml X 10 1-675 0258 = 0.125 gallon (liquid; U.S.): or H 1-0969100 = 0.016710 1* cb. foot. Aprx. M- 10. . 2-2229783 = 0.004 731 79* hectoliter. Aprx. Hi -* 10 3-675 0258 = O-.OOO 618 891* cb. yard. Aprx. ^g -7-1 000 4-7916145 = 0.000473 179* cb. meter or stere. Aprx.Ki^-100 4-6750258 1 pint [pt] (dry; U. S.): = 550.614* cb. centimeters. Aprx. 550 2-740 8471 = 33.600 312* cb. inches. Aprx.^XlOO 1-5263433 = 5.506 14* deciliters. Aprx. 5^ or ^ 0-740 8471 = 1.16365* pints (liquid; U.S.). Aprx.l^ory 6 0-0658213 = 0,968 972 pint (Brit.). Aprx. subtract ^ 1-986 3112 =. 0.550 614* liter or cb. decimeter- Aprx. "^ 10 . 1-740 8471 = 0.5 quart (dry; U.S.); or ^ 1-6989700 - 0.121 122 gallon (Brit.). Aprx. % * 10 1-083 2215 = 0.062500* peck (U.S.); or ^10 2-7958800 0.019 444 6* cb. foot. Aprx. 2y n -=- 100 2-2887996 = 0.015 625* bushel (U. S.). Aprx.H fluid ounce (ap. Brit.)=*46o pint (Brit.). 1 cb. centimeter = 0.281 57 fluid dram (ap. Brit.). 1 fluid dram (ap. U. S.) = 60. minims (ap. U. S.) = 0.225 59 (aprx. %) cb. inch = ^8 fluid ounce (ap. U. S.)=Vi 2 8 pint (liquid; U. S.) = 3.69671 milliliter or cb. centimeter. 1 cb. centimeter = 0.270 51 fluid dram (ap. U. S.). . .. 1 centiliter [cl] = 10. cb. centimeters = 0.610 234 cb. inch = 0.01 liter. 1 fluid ounce [fl ozl (ap. Brit.) = 480. minims (ap. Brit.) = 28.412 27 cb. centimeters or milliliters = 24. fluid scruples (ap. Brit. ) = 8. fluid drams . . . . . (ap. Brit.) = 1.733 81 cb. mches = 0.960 73 fluid ounce (ap. U. S.) = ^ pint (Brit.) = 0.028 412 27 liter. 1 liter = 35. 196 fluid ounces (ap. Brit.). 1 fluid ounce [fl oz] (ap. U. S.) = 480. minims (ap. U. S.) = 29.573 7 cb. centimeters = 8. fluid drams (ap. U. S.) = 1.804 69 (aprx. %) cb. inches = 1.04088 fluid ounces (ap. Brit.) = yi 6 pint (liquid; U. S.) = 0.029 573 7 liter. 1 liter = 33.813 8 fluid ounces (ap. U. S.). 1 gill [gi] (liquid; U. S.) = 118.295 cb. centimeters = 7.218 750 cb. inches = 1.18295 deciliters = 0.832 7024 gill (Brit.) = M pint (liquid; U. S.) = Y % quart (liquid; U S.) = 0.118 295 liter. 1 cb. inch = 0.138 528 gill (liquid; U. S.). 1 liter = 8.453 44 gills (liquid; U. S.). 1 gill [gi] (Brit.) = 142.061 35 cb. centimeters = 8.669 05 cb. inches = 1.4206135 deciliters =1.200 91 gills (liquid; U. S.) = M pint (Brit.) = J/g quart (Brit.) = 0.142 061 35 liter. 1 cb. inch = 0.1 15 352 8 gill (Brit.). 1 liter = 7.039 20 gills (Brit.). 1 pint[pt] (ap. U. S.) = 128. fluid drams (ap. U. S.) = 16. fluid ounces (ap. U. S.) = 0.473 179 liter = same as liquid U. S. pint. 1 pintfpt] (ap. Brit.) = 480. fluid scruples (ap. Brit.) = 160. fluid drams (ap. Brit.) = 20. fluid ounces (ap. Brit.) = 0.568 245 4 liter = same as ordi- nary Brit. pint. 1 inillistere = 1 cb. decimeter =1 liter. 1 pottle (Brit.) = H gallon Brit. 1 board foot =144. cb. mches = M2 cb. foot. 1 gallon [gal] (ap. U. S.), same as ordinary liquid U. S. gallon of 231. cb. inches. 1 gallon [gal.] (wine; old Brit.), same as ordinary liquid, U. S. gallon of 231. cb. inches. 1 gallon (dry; U. S.) (obsolete; term "^ peck" used instead) = 268.8025 cb. inches = 8. pints (dry; U. S.) = 4.40491 liters = 4. quarts (dry; U. S.) = H peck (U. S.). 1 gallon [gal] (ap. Brit.), same as ordinary Brit, or Imperial gallon. 1 lieer gallon (obsolete) = 282. cb. inches = 8. beer pints = 4. beev quarts. 1 decaliter or dekaliter [dkl] = 10. liters =1 centistere. 1 centistere = 10. cb. decimeters or liters =1 decaliter = ^ioo stere or cb. meter. VOLUMES. 53 1 gram-molecule of any gas at C. and 760 mm. pressure has a vol- ume of 22 380. cb. centimeters. 1 foot (solid, timber) = 1 cb ft. 1 solid foot=l cb. ft. 1 hektoliter [hi], same as hectoliter. 1 decistere [ds]=100. cb. decimeters or liters =1 hectoliter = Mo stere or cb. meter. 1 firkin = 9. gallons (liquid; U. S.) = 7.4943 gallons (Brit.) = 34. 068 9 liters = % barrel. Of butter = 56. pounds (av.). 1 Winchester bushel [bu], same as ordinary bushel in U. S. = 2 150.42 rb. inches. 1 struck bushel, same as ordinary bushel in U. S. = 2 150.42 cb. inches. 1 heaped bushel = 1*4 struck bushels or 1M ordinary bushels (U. S.). 1 bushel = 60 pounds of wheat = 56 pounds of corn or rye = 48 pounds of barley = 32 pounds of oats = 60 pounds of peas = 42 pounds of buck- wheat. (U. S. Customs; legal.) 1 coomb (Brit. ) = 16. pecks (Brit.) = 4. bushels (Brit. )= 1.454 708 hecto- liters = H quarter (Brit.). 1 barrel [bbl], has no legal or fixed value ; it varies between 30 and 43 gallons (liquid; U. S.). Barrels should therefore always be gauged. The following are the most usual values, about in the probable order of im- portance: 1 barrel (liquid; U. S.) = 31^ gallons (liquid; U. S.) = 4.211 cb. feet = 1.192 4 hectoliters = ^ hogshead. 1 barrel (wine and brandy; Brit.) = 31J/ gallons (Brit. ) = 1.431 98 hectoliters. Also given as 36 gallons (Brit.). 1 barrel (liquid; U. S.) = 31. gallons (liquid; U. S.) = 1.173 5 hectoliters. 1 barrel (refined oil) = 42. gallons (liquid; U. S.; used by Standard Oil Co.). 1 barrel (flour; U. S.) = 3. bushels (U. S.); also given as 3.75 cb. feet; " legal" (?) weight given as 196 pounds (av.). 1 barrel (dry; U. S.) = 21^ bushels (U. S.) = 26.756 cb. feet = 7.576 45 hectoliters. 1 barrel (beer; U. S.) = 36. beer gallons of 282 cb. inches = 1.663 6 hectoliters. 1 barrel (liquid; Penna.) = 32. gallons (liquid; U. S.). 1 sack (coal; Brit.) = 3 bushels (Brit.) =1/12 chaldron. 1 decistere = 1 hectoliter = 0.1 stere or cb. meter. 1 tierce (liquid; U. 8.) -42. gallons (liquid; U. S.) = 1.5899 hecto- liters = % hogshead (U. S.). 1 tierce (Brit.) = 42. gallons (Brit.) = 1.909 30 hectoliters = ^ hogs- head (Brit.). 1 hogshead [hhd] (beer; U. S. obsolete) = 54 beer gallons of 282 cb. inches. 1 hogshead [hhd] (liquid; U. S.) = 63. gallons (liquid; U. S.) = 2. bar- rels of 31H gallons (liquid; U. S.) = 2.384 8 hectoliters =1% tierces (liquid; U.S.). 1 hogshead [hhd] (Brit.) = 63. gallons (Brit.) = 2.863 96 hectoliters. 1 quarter [qr] (dry; U. S.) = 8. bushels (U. S.) = 2.819 1 hectoliters = M (aprx.) tun (Brit.). 1 quarter [qr] (Brit.) = 8. bushels (Brit.) = 2.909 416 hectoliters = 2. coombs (Brit.) = K (aprx.) tun (Brit.) = 1/10 last (Brit.). 1 puncheon (liquid; U. S.) = 84. gallons (liquid; U. S.) = 3.17976 hec- toliters =2. tierces (U. S.). 1 puncheon (Brit.) = 84. gallons (Brit.) = 3.818 61 hectoliters = 2. tierces (Brit.). 1 pipe or butt (liquid; U. S.) = 126. gallons (liquid; U. S.) = 4.7696 hectoliters = 2. hogsheads (U. S.). 1 pipe or butt (Brit.) = 126. gallons (Brit.) = 5.727 91 hectoliters = 2. hogsheads (Brit.). 1 cord foot (wood) = 4X4X1 foot = 16. cb. feet = 0.453 07 cb. meter or stere = J/g cord. 1 tun (liquid; U. S.) = 2f>2. gallons (liquid; U. S.) = 9.539 3 hectoliters = 6. tierces (liquid; U. S.)=4. hogsheads (U. S.) = 3. puncheons (U. S.) = 2. pipes or butts. 1 tun (Brit.) = 252. gallons (Brit.) = 11.455 8 hectoliters = 6. tierces (Brit.) =4. hogsheads (Brit.) = 3. puncheons (Brit.) = 2. pipes or butts (Brit.). 54 VOLUMES. 1 perch (masonry) = 16^X1^X1 foot = 24% cb. feet (generally 25 cb. feet, sometimes 22 cb. feet) = 0.916 67 cb. yard =0.700 85 stere or cb. meter. 1 solid yard = l cb. yard. 1 stere [s]= 1 cb. meter, which see above in other table. 1 kiloliter [kl] = 1 000. liters =10. hectoliters = 1 cb. meter or stere. 1 gross ton (2240 pounds av.) displacement of water = 35. 881 3 cb. feet = 1.328 93 cb. yards = 1.016 05 cb. meters. 1 shipping ton (for cargo; U. S.) = 40. cb. feet = 32.1426 bushels (U. S.) = 31.145 2 bushels (Brit.) = 11.326 8 hectoliters = 1.132 68 cb. meters. 1 shipping ton (for cargo; Brit.)=42. cb. feet = 33.7497 bushels (U. S.) = 32.702 5 bushels (Brit.) = 11. 893 1 hectoliters = 1.189 31 cb. meters. 1 chaldron (dry; U. S.) = 44.8006 cb. feet = 36. bushels (U. S.) = 12.686 1 hectoliters. 1 chaldron (coal; Brit.) = 12. sacks-=36. bushels (Brit.) = 58.658 cb. feet; weight 3 136. pounds (av.). 1 chaldron (Canada) = 58. 64 cb. feet or about 45 bushels (Brit.); stated also as 25.64 cb. feet or about 20 bushels (Brit.). 1 chaldron (Newcastle) weight 5 936. pounds (presumably coal). 1 wey (Brit.) = 51. 372 4 cb. feet = 40. bushels (Brit.) = 5. quarters (Brit.)=^ last (Brit.). 1 register ton (shipping; for whole vessels) = 100. cb. feet = 2. 831 7 cb. meters. 1 last (Brit.) = 102.745 cb. feet =80. bushels (Brit.) = 10. quarters (Brit.) = 2. weys (Brit.). 1 cord [c] (wood) = 4X4X8 feet = 128. cb. feet = 8. cord feet = 3.624 58 cb. meters or stere. 1 stere or cb. meter = 0.275 894 cord. 1 toise (Canada) = 261^ cb. ft. = 9.685 18 cb. yds. = 7.404 90 cb. meters. 1 rod (brickwork; Brit.) = 16> feet square X 14. inches = 272*4 sq. feet of 14. inch wall; conventionally 272. sq. feet of 14. inch wall = 317M cb. feet = 8.985 9 cb. meters. 1 rod (engineering works; Brit.) = 306. cb. ft. = HM cb. yards = 8.665 cb. meters. 1 myrialiter or myrioliter= 10 000. liters =100. hectoliters = 10. cb. meters or steres=l decastere. 1 decastere or dekastere [dks]=100. hectoliters = 10. cb. meters or steres = 1 myrialiter. 1 hectostere or hektostere [hks]=100. steres or cb. meters. 1 acre-foot (irrigation) = 325 851. gallons = 43 560. cb. feet = 1 613.33 cb. yards = 1 233.49 cb. meters. VOLUMES. Cubic and Capacity Measures (concluded). Foreign. These are mostly obsolete, as the metric system is now used in most foreign countries. The British measures are included among the U. S. measures, being very nearly, and sometimes quite, the same. The trans- lated terms are merely synonymous, and not the exact equivalents. Germany. Prussian. 1 Fuder = 4 Oxhoft of 1J^ Ohm of 2 Eimer (bucket) of 2 Anker of 30 Quart (Prussian). 1 Wispel = 6 Tonne (tun) of 4 Scheffel of 16 Metzen of 3 Quart (Prussian). 1 Quart (Prussian) = 64 cub. Zoll = 3^ 7 cub. Fuss = 1.14503 liters; 1 Wispel= 13.191 hectoliters; 1 Tonne = 2. 198 46 hectoliters; 1 Scheffel = 0.549 61 hectoliter. 1 Schacht- ruthe=144. cub. Fuss = 4.451 9 cub. meters. 1 Klafter=108 cub. Fuss = 3.338 9 cub. meters. The following values are given by Nystrom; they are in cubic inches. For liquid measures: 1 Stubgen in Bremen = 194.5, in Hamburg 221, in Hanover 231. For dry measures: 1 Scheffel in Berlin 3 180, in Bremen 4 339, in Hamburg 6 426. 1 Hanover Matter = 6 868. France. "Old Measures" (systeme ancien) used prior to 1812. 1 muid (hogshead) = 2 feuillettes of 2 quartants of 9 setiers or veltes of 4 pots of 2 pintes (pint, though more nearly equal to a quart) of 2 chopines 9f 2 demi-setiers of 2 possons of 2 demi-possons of 2 roquilles (gill). 1 pinte ound was definitely adopted by Congress in 1828 for coinage purposes and was supposed to be an exact copy of the British troy pound; formerly the avoirdupois pound was derived from this according to the relation 1 avoirdupois pound 700 %7ao troy pound; this relation is still correct, but in this country both the troy and the avoirdupois pounds as now standardized by the Government are derived from the kilogram as stated above, and hence both values are fixed definitely. The old troy pound, "although totally unfit for such purpose, "is the legal standard for coinage purposes in this country; according to the National Bureau of Standards, the troy pound of the Mint, and the troy pound of that Bureau (based on the kilogram) are the same. The kilogram was originally intended to be the mass of a cubic deci- meter or liter of pure water at the temperature of its maximum density. The present International Prototype Kilogram is an exact copy of the original kilogram of the archives, which when made was supposed to be equal to the weight of one cubic decimeter of water. It is a certain mass of platinum-iridium. The determination of the precise relation of this adopted kilogram to the mass of a cubic decimeter of water is now in progress, and it may be several years before the final results are announced. The results thus far indicate that the kilogram is heavier than it would be according to the original definition by about 25 milligrams, or about 25 parts in 1 000 000. The present kilogram, however, is definite, and the result of such a discrepancy would make the liter, which is the volume of a kilogram of water, very slightly larger than the cubic decimeter by about 25 parts in 1 000 000. The question of correcting the kilogram to agree with its original theoretical definition may be considered when the deter- mination now being made at the International Bureau has been com- pleted. For all but the most refined measurements, however, this slight discrepancy is absolutely negligible. The National Bureau of Standards has for the present assumed that the liter and the cubic decimeter are equivalent; this identity is assumed also in all the tables in this book. The relation, legalized in Great Britain in 1898, between the avoirdupois pound and the kilogram, is precisely the same as that in this country, hence the pounds are exactly the same in both countries. All the avoirdupois, troy and apothecary weights are therefore also the same in the United States and in Great Britain. WEIGHTS OR MASSES. 57 The metric weights are now in use everywhere for all accurate scientific measurements. In this country they are coming into more general use; chemists use them entirely. The avoirdupois weights (abbreviation av.) are used for most purposes, including merchandise in general. The troy weights are used for weighing gold, silver, etc.; they are used by the U. S. Mint; quantities, even much larger than an ounce, are usually stated in ounces and not in pounds. In the apothecary weights (abbreviation ap.) only the grain, scruple, and dram are in general use in this country; the ounce is used only when called for in prescriptions; apothecaries almost always use the avoirdupois pound and ounce. The apothecary ounce and pound are the same as the troy. The grain is the same in all three systems. No fixed rules can be given concerning the distinction between the use of the short or net ton of 2 000 Ibs. and the long or gross ton of 2 240 Ibs., but the following general rules may serve as a guide. The long ton seems to be the only official one; section 2951 of the Revised Statutes of the U. S., 2d Ed., 1878, Collection of Duties upon Imports, Chapter 6. says that by the word "ton" in that chapter is meant 2 240 pounds. With freight on railroads, a ton of 2 240 pounds seems to be generally used. In the iron and steel trades, pig iron, steel rails, and iron ore are bought and sold by the ton of 2 240 pounds. Coal seems to be weighed in long tons also; coke, however, is weighed in short tons of 2 000 pounds. The short ton of 2 000 pounds seems to be in general use for weighing chemical prod- ucts, or in general for the more expensive products; but in shipping them on railroads the ton of 2 240 pounds is often used. The short ton seems to be in general use also in traction engineering calculations. What are popularly termed weights should more correctly be called masses, but for all practical purposes the two terms are the same. The mass of a body is the same anywhere in the universe, but its weight depends on the attraction of gravitation. The mass of a body is most conveniently measured by its weight, and if the attraction of gravitation is the same (and it varies but slightly at different parts of earth) the weight will be a correct measure of the mass. With the usual beam balance, the com- parison of two masses by means of their weights is absolutely exact and quite independent of the value of gravity, which acts equally on both; but with spring scales the same mass may have slightly different weights, depending on the force of gravity. When weights ate considered as such, and not as masses, they are forces and have the dimensions of forces. The reduction factors in the following table are correct whether the units are considered as masses or as forces. The reduction factors between them and the true units of force (such as dyne and poundal) are given in a separate table of forces so as to avoid confusion. See also note under Forces and under Acceleration. WEIGHTS or MASSES. (See also FORCES.) Usual. ** Accepted by the National Bureau of Standards. * Checked by L. A. Fischer, Asst. Phys. National Bureau of Standards. av. means avoirdupois. Aprx. means within 2%. Logarithm 1 milligram [mg] = 0.1 centigram 1-000 0000 = 0.0154324* grain. Aprx. % 3 -*- 10 2-1884322 = 0.001 gram jj.QOO 0000 1 centigram [cg]= 10. milligrams 1-000 0000 = 0.154 324* grain. Aprx. 2/ 18 1.188 4322 = 0.01 gram 2-000 0000 1 grain [gr] = same in avoirdupois, troy, or apothecary weights. = 64.798 9* milligrams. Aprx. 65 1-811 5678 = 6.479 89* centigrams. Aprx. 6> 0-811 5678 = 0.064 798 9* gram. Aprx. 1% -r- 100 2-811 5678 = 0.002 285 71* ounce (av.). Aprx. /4^- 1 000 3.359 0219 1 decigram [dg]= 1.543 24* grains. Aprx. Ufa 0-188 4322 = 0.1 gram LOOO 0000 58 WEIGHTS OR MASSES. 1 gram [g]= 1 000. milligrams 3-000 0000 = 100. centigrams 2-000 0000 = 15.432 356 39** grains. Aprx. 15 1 A 1-188 4322 = 0.0352740* ounce (av.). Aprx. %-s- 100 2-5474541 = 0.0321507* ounce (troy). Aprx. 32 -- 1 000 2-5071910 = 0.002 204 62* pound (av.). Aprx. 22-=- 10 000. . . 3-343 3342 = 0.001 kilogram 3-0000000 1 ounce [oz] (av.) = 2 834.95* centigrams. Aprx. 2 A X 10 000 . 3 452 5459 = 437.500* grains. Aprx. 1% X 100 2-640 9781 = 28.349 5* grams. Aprx. ft X 100 1-452 5459 = 16. drams (av.) 1-204 1200 = 0.911 458* ounce (troy). Aprx. i/n 1-959 7369 = 0.062 500* pound (av.) or y 80 2-7958800 -0.028 349 5* kilogram. Aprx. 2 A -^ 10 2-452 5459 1 pound [lb](av.)= ? 000.** grains 3-845 0980 = 453.592 427 7 ** grams. Aprx. % X 100. . . . 2-656 6658 = 256. drams (av.) 2-408 2400 = 16. ounces (av.) 1-204 1200 = 14,5833* oz. (troy). Aprx. Vr X 100. 1-1638569 = 0.4535924* kilogram. Aprx. %-* 10 .. 1-656 6658 1 kilogram or kilo [kg]: = 15432.35639** grains. Aprx. 3 H XI 000 4-1884322 = 1 000. grams 3-000 0000 = 35.274 0* ounces (av.). Aprx. % X 10 1-547 4541 = 32.150 7* ounces (troy). Aprx. 32 L507 1910 = 2.204 62* pounds (av.). Aprx. 2% 0-343 3342 = 0.0220462* hundredweight (sh.). Aprx. 22 -s- 1000. 2-343 3342 = 0.0196841* hundredweight (long). Aprx. 2 * 100. . 2-2941162 = 0.001 102 31* short or net ton. Aprx. % * 100 3-042 3042 = 0.001 metric ton 3-000 0000 = 0.000 984 206 long or gross ton. Aprx. 1 -* 1 000 4-993 0862 1 hundredweight [cwt] (short): 100. pounds (av.) 2-000 0000 = 45.359 24* kilograms. Aprx. % X 10 ... 1-656 6658 = 0.892 857* hundredweight (long). Aprx. % 1-950 7820 0.05 short or net ton 2-698 9700 = 0.045 359 24* metric ton. Aprx. ^2 2-6566658 = 0.044 642 9* long or gross ton. Aprx. % -*- 100 2-649 7520 1 hundredweight [cwt] (long): 112. pounds (av.). Aprx. 1/9 X 1 000 2-049 2180 = 50.802 4* kilograms. Aprx. 50 1-705 8838 = 1.120 00* hundredweights (short). Aprx. 1% 0-049 2180 = 0.056 000* short or net ton. Aprx. % -5- 10 2-748 1880 = 0.050 802 4* metric ton. Aprx. Ho - 2-7058838 0.05 long or gross ton or Ho 2-6989700 I short or net ton [tn]: = 2 000. pounds (av.). . . . 3-301 0300 = 907.185* kilograms. Aprx. 900 2-957 6958 = 20. hundredweights (short) 1-301 0300 = 17.857 1* hundredweights (long). Aprx. % X 10 1-251 8120 = 0.907 185* metric ton. Aprx. subtract Vio 1-957 6958 = 0.892 857* long or gross ton. Aprx. subtract % 1-950 7820 1 metric ton, tonne, tonneau, millier, or bar [t]: = 2 204.62* pounds (av.). Aprx. 22 X 100. . 3-343 3342 = 1 000. kilograms 3-000 0000 = 22.046 2* hundredweights (short). Aprx. 22 1-343 3342 = 19.684 1* hundredweights (long). Aprx. 20 1-294 1162 = 1.102 31* short or net tons. Aprx. add Mo 0-0423042 = 0.984 206* long or gross ton. Aprx-. 1 1-993 0862 1 long or gross ton [tn]: = 2240 pounds(av.). Aprx.22XlOO. 3-3502480 = 1 016.05* kilograms. Aprx. 1 000 3-006 9138 = 22.400 0* hundredweights (short). Aprx. 22 1-350 2480 = 20. hundredweights (long) 1-301 0300 = 1.12 short or net tons. Aprx. 1% 0-049 2180 = 1.016 05* metric tons. Aprx. 1 0-006 9138 WEIGHTS OR MASSES. 59 Conversion Tables for Weights. Note. By pounds and ounces are meant avoirdupois pounds and ounces. Grains = Mil'grs = Ounces = Grams = Pounds = Klgms = Sh.ton = Lg. ton = Mt.ton = 1 2 3 4 5 6 7 8 9 10 mlgrs grains gram ounces klgrms pound met. tons met. tons long tons 0.984 1.908 2.953 3.937 4.921 '5.905 0.889 7.874 8.857 9.842 shrt tons 04.799 129.00 194.40 259.20 323.99 388.79 453.59 518.39 583 19 047.99 0.015432 0.030 805 0.040 29 / 0.001 730 0.077 102 0.092 594 0.10803 0.12340 0.13889 0.15432 28.350 )(>.(;:)',) S5.049 113.40 141.75 170.10 198.45 220.80 255.15 283.50 0.035 274 0.070.518 0.10582 0.141 10 0.17037 0.21104 0.24092 0.282 19 0.31747 0.352 74 0.453 59 0.907 18 1.3008 1.8144 2.2080 2.721 3.1751 3.0287 4.082 3 4.5359 2.204 4.4092 0.0139 8.8185 11.023 13.228 15.432 17.037 19.842 22.040 0.907 1.814 2.722 3.029 4.530 5.443 0.350 7.257 8.105 9.072 1.102 2.205 3.307 4.409 5.512 0.014 7.710 8.818 9.921 11.02 1.010 2.032 3.048 4.004 5.080 0.090 7.112 8.128 9.144 10.10 WEIGHTS or MASSES (continued). Unusual, Special Trade, or Obsolete. av. means avoirdupois weight; ap. means apothecary weight; aprx. means approximately. 0.001 milligram [ r ] (has no name, symbol used instead) =0.000 015 432 4 grains. 1 jeweller's grain = *4 carat (diamond) of various weights. 1 carat (diamond) = 4. jeweller's grains = (according to Streeter) in U. S. 205.500 milligrams or 3.171 4 grains, and in England 205.409 milli- grams or 3.1700 grains; other authorities give 3.168, 3.18, and 3.2 grains in U. S., and 3.17 in England. For values in other countries see below under Foreign Weights. 1 scruple O] (ap.) = 20. grains -1.295 978 (aprx. %) grams = M dram (ap.) = ^4 ounce (troy or ap.). 1 gram = 0.771 618 (aprx. %) scruple. 1 pennyweight [dwt] (troy) = 24. grains= 1.555 17 (aprx. i^fr) grams = %o ounce (troy or ap.). 1 gram =0.643 015 (aprx. ^4i) pennyweight. 1 drachm, same as dram. 1 dram (av.) = 27% or 27.34375 (aprx. 27^) grains =1.771 85 (aprx. %) grams = 0.455 729 (aprx. ^li) drams (ap.)== 1 /i6 ounce (av.). 1 gram = 0.564 383 'aprx. #) dram (av.). 1 dram [ 3 ] (ap.) = 60. grains = 3.887 934 grams = 3. scruples = 2. 194 29 (aprx. !%) drams (av.) = Vs ounce (ap.). 1 gram =0.257 206 drams (ap.). 1 decagram or dekagram [dkg]= 154.323 6 grains = 10. grams = 0.352 74 ounce (av.). 1 ounce (troy, silk) = 360. grains = 23. 327 6 grams. 1 ounce [oz] (troy) (used chiefly for gold and silver) = 480. grains = 31.1035 grams = 20. pennyweights = 1.097 14 (aprx. H4o) ounces (av.) = l ounce (ap.)=}4 2 or 0.0833333 pound (troy or ap.) = 0.068 571 4 pound (av.). 1 gram = 0.032 150 7 ounce (troy). 1 ounce (av.) =0.911 458 (aprx. 10 /ii) ounce (troy). 1 pound (av.) = 14.583 3 (aprx. i%) ounces (troy). 1 kilogram = 32. 150 7 ounces (troy). 1 ounce [ ] (ap.) (used only in prescriptions) =480. grains = 31. 103 5 grams = 24. scruples = 8. drams (ap.) = 1.097 14 (aprx. 1: Ho) ounces (av.) = 1 ounce (troy)=Vi 2 or 0.083 333 3 pound (ap. or troy) =0.068 571 4 pound (av.). 1 gram = 0.032 1507 ounce (ap.). 1 ounce (av.) = 0.911 458 (aprx. 1( Hi) ^unce (ap.). 1 pound (av.) = 14.583 3 ounces (ap.). 1 kilogram = 32.1507 ounces (ap.). 60 WEIGHTS OR MASSES. 1 hectogram [hg] = 100. grams = 3.527 40 ounces (av.). 1 pound (troy) (seldom used ; troy ounces used instead), = 5 760. grains = 240. penny weights = 12. ounces (troy or ap.) = l pound (ap.)= 576 %ooo or 0.822 857 (aprx. %) pound (av.) =0.373 242 (aprx. %) kilogram. 1 pound (av.) = oo/ 5760 or 1.215 28 (aprx.%) pounds (troy). 1 kilogram = 2. 679 23 pounds (troy). 1 mint pound (U. S.), same as troy pound. 1 pound (troy, silk) = 16. ounces (troy, silk) = 5 760. grains = 1 pound (troy). 1 pound (ap.) (obsolete) = 5 760. grains = 288. scruples = 96. drams (ap.) = 12. ounces (ap. or troy) = l pound (troy )= 570 %ooo orO. 822 857 (aprx. %) pound (av.) = 0.373 242 (aprx. %) kilogram. 1 pound (av.)= %7eo or 1.215 28 (aprx. %) pounds (ap.). 1 kilogram =2.679 23 pounds (ap.). 1 stone (Brit.) = 14. pounds (av.) = 6.350 29 kilograms. 1 myriagram = 10 000. grams = 22.046 2 pounds (av.) = 10. kilograms. 1 quarter [qr] (short) = 25. pounds (av.) = 11.3398 kilograms = ^ hundredweight (short). 1 quarter [qr] (long)=28. pounds (av.) = 12.700 6 kilograms = M hun- dredweight (long). 1 firkin (butter) = 56. pounds (av.) (really a capacity measure). 1 bushel (salt) = 70. pounds (av.) (really a capacity measure). 1 quintal (av.) = 100. pounds (av.) = 45.359 24 kilograms =1 hundred- weight (short). 1 barrel of flour (' 'legal"?) = 196. pounds (av.). 1 barrel of beef or pork = 200. pounds (av.). 1 quintal (metric) = 100. kilograms = 220.462 pounds (av.). 1 pig (metal) = 301. pounds (av.) = 21H stones. 1 fother (iron, lead, etc.) = 2 408. pounds (av.) = 172. stones = 8. pigs. 1 bloom ton = lVio long tons = 2 464. Ibs. Relative Weights (used in chemistry). 1 millimol =0.001 mol or gram molecule. 1 mol or mole = 1 gram molecule, which see below. 1 gram molecule = as many grams of a substance as is represented numerically by its molecular weight. A gram molecule of any gas at C. and 760 mm pressure occupies a volume of 22 380. cubic centimeters. 1 kilogram molecule = 1 000. gram molecules. 1 gramatom = as many grams of an elemental substance as is repre- sented numerically by its atomic weight. WEIGHTS or MASSES (concluded). Foreign. These are mostly obsolete, as the metric system is now used in most foreign countries. The British measures are included among the U. S. measures, being very nearly, and sometimes quite, the same. The trans- lated terms are merely synonymous, and not the exact equivalents. Germany. Prussia. To 1839 inclusive: 1 Centner (hundred weight ) = 110 Pfund (pound) of 32 Loth of 4 Quentchen. 1 Pfund = 0.467 711 kilo- gram. From 1840 for customs and from 1858 for trade: 1 Centner or Z.C. = 100 Pfund (pound) of 30 Loth of 10 Quentchen of 10 Cent or zent of 10 Kern or Korn (grain); 1 Pfund = 0.5 kilogram. Apothecaries weight: 1 Pfund = 12 Unzen (ounces) of 8 Drachmen of 3 Skrupel of 20 Gran (grain) (signs the same as in U. S. apoth. measure; values slightly smaller); 1 Pfund = 0.350 783 kilogram (see also under Baden); 1 Schiffslast (shipping weight) = 40 Centner = 2000 kilograms. 1 carat (diamond) in Berlin 205.440, in Frankfort o. M. 205.770, and in Leipsic 205.000 milligrams \Streeter). Bavaria. 1 Pfund (pound) = 32 Loth of 4 Quentchen; 1 Pfund = 0.560 kilogram. Saxony. 1 Pfund (pound) = 4 Pfenniggewicht (pennyweight) of 2 Hellergewicht ; 1 Pfund = 0.467 6 kilogram. Wurtemberg. To 1850: 1 Pfund (pound) = 32 Loth of 4 Quentchen of 4 Richtpfennig; 1 Pfund = 0.467 7 kilogram. Since 1850 like in Baden. Baden. 1 Pfund (pound) = 2 Mark of 2 Vierlingen of 4 Unzen (ounces); Dr 1 Pfund = 10 Zehnlingen of 10 Centas of 10 Dekas of 10 As; or 1 Pfund = WEIGHTS OR MASSES. 61 32 Loth of 4 Quentchen; 1 Pfund = 0.5 kilogram. (Another authority states that these are Prussian.) Hanover. 1 Pfund = 0.4696 kilogram. The following additional values of various German, pounds in pounds avoirdupois are given by Nystrom: Berlin 1.033; Bremen 1.100; Bruns- wick 1.029; Hamburg 1.068; Hanover 1.073; Leipsic 1.029. France. Old weights (systeme ancien or poid de marc) used prior to 1812. 1 millier (ton) = 10 quintaux of 100 livres (pounds); 1 livre (pound) = 2 marcs of 8 onces (ounces) of 8 gros (drams) of 3 deniers (scruples) of 24 grains; 1 livre = 0.489 506 kilogram. "Usual" weights (systeme usuel) used from 1812 to 1840: 1 livre (pound) = 0.5 kilogram. Apothecaries' weights: 1 livre (romain) (pound) = 12 onces (ounces) of 8 dragmes (drams) of 3 scrupules (scruples) of 20 grains; 1 livre romain = 0.75 livre de marc = 0.367 129 kilogram. Nystrom gives 1 Lyons pound (silk) = 1.012 2 pounds av. Streeter gives 1 carat (diamond) = 205. 500 milligrams. Austria. 1 Centner (hundred weight ) = 100 Pfund (pound) of 32 Loth of 4 Quentchen of 4 Pfennig; 1 Pfund = 0.560 01 or 0.56006 kilogram. 1 Schiffstonne (shipping ton) = 20 Centner=l 120.12 kilograms. 1 Meter- centner (metric hundredweight ) = 100 kilograms. Nystrom gives 1 Vienna pound = 1.235 pounds av. Streeter gives 1 carat (diamond) in Vienna = 206.130 milligrams. Sweden. 1 centner = 100 schalpfund or skalpund or mark (pound) of 32 lod of 4 kvintin of 69^s ass; or 1 skalpund = 100 korn of 100 art; 1 skal- pund =100 korn of 100 art; 1 skalpund = 0.425 1 or 0.425 339 5 kilogram. 1 schiffspund (shipping pound) = 20 liespund = 400 skalpund. Nystrom gives 1 Swedish pound = 0.937 5 pound av. ; 1 miner's pound = 0.828 6 pound av. Russia. 1 pfund (pound) = 32 loth of 3 solotnick of 96 doli; lpfund = 0.409 512 or 0.409 531 kilogram. Shipping weights: 1 berkowitz = 10 pud or pood of 40 pfund (pounds); 1 berkowitz = 163.81 kilograms. Nystrom gives 1 Russian pound = 0.902 pounds av. ; 1 Warsaw pound = 0.891 pound av. Switzerland. Like Baden except that 1 Pfund = 32 Loth of 16 Ungen. Nystrom gives 1 Geneva pound, heavy, = 1.214 pounds av. Holland. 1 Amsterdam or Rotterdam pound = 1.089 pounds av. 1 carat (diamond) in Amsterdam = 205. 700 milligrams. Spain. 1 marco or mark = 50 castellanos of 8 tomines of 12 Spanish gold grains. 1 marco = 0.507 6 pounds av. in Spain = about 0.506 5 pounds av. in South America; other values up to 0.54; 1 castellano = 71.07 to 71.04 grains (U. S.). 1 tonelada (ton) = 20 quintal (hundred- weight) of 4 arroba (quarters) of 25 libra (pounds); 1 tonelada of Castile = 2032.2 pounds avoirdupois; 1 libra = 0.461 kilogram = 1.016 1 pounds av. ; 1 arroba of Castile or Madrid = 25. 402 5 pounds av. ; it has various values in different parts of Spain. Nystrom gives 1 Barcelona pound = 0.888 1 pound avoirdupois. Streeter gives 1 carat (diamond) = 205. 393 milligram. Italy. Nystrom gives the following values in pounds avoirdupois: Bologna pound 0.798; Corsica pound 0.759; Florence pound 0.749; Genoa pound 1.077; Leghorn pound 0.749; Naples Rottoli 1.964; Rome pound 0.748 ; Sicily pound 0.700 ; Venice pound, heavy, 1 .055, light, 0.667. Streeter gives 1 carat (diamond) in Florence = 195. 200, in Leghorn 215.990 milli- grams. Japan. 1 kwan = 1000 momme of 10 fun. 1 kwan = 6^ (aprx.) kin of 160 momme. 1 kwan = 3.76 kilograms.= 8.29 pounds (av.). 1 kilo- gram =0.266 kwan. 1 pound = 0.121 kwan. ' Miscellaneous. Nystrom gives the following values which have here been reduced to pounds avoirdupois: Antwerp pound 1.034; Copenhagen pound 1.101; Madeira pound 0.698; Tangiers pound 1.061; Cairo rottoli 0.952; Alexandria rottoli 0.935; Algiers rottoli 1.190; Damascus rottoli 3.96; Tunis rottoli 1.110; Tripoli rottoli 1.120; Cyprus rottoli 5.24; Can- dia rottoli 1.164; Aleppo rottoli 4.89; Aleppo oke 2.79; Constantinople oke 2.81; Smyrna oke 2.74; Mocha maund 3.00; Morea pound 1.101; Ben- al seer 1.867; Batavia catty 1.302; China catty 1.326; Japan catty 1.300; umatra catty 2.81. Streeter gives 1 carat (diamond) in Lisbon 205.750, in Borneo 105.000, and in Madras 207.353 3 milligrams 62 WEIGHTS AND LENGTHS. WEIGHTS or MASSES and LENGTHS; WEIGHTS of WIRES, RAILS, BARS; FORCES and LENGTHS; FILM or SURFACE TENSION; CAPILLARITY. (Mass -=- length ; force ~- length.) In this group of units the masses, forces, and the weights considered as ^oth masses and forces have all been combined to avoid repetition of their Delations to each other. When the units involve masses or weights con- >idered as masses, as in pounds per foot, they are used to measure such quantities as the weights of rails, bars, wires, etc.; while when the units involve forces or weights considered as forces, they are used to measure >uch quantities as surface tension, capillarity, etc. The dimensions of the units in the two cases are different. Weights considered as forces involve the value of gravity, but masses and forces do not. Aprx. means within 2%. Logarithm I dyne per centimeter [dyne/cm]: = 0.039 973 9 grain per inch. Aprx. 4-^100 2-6017764 = 0.001 019 79 gram per centimeter. Aprx. 1 -5- 1 000 3-008 5098 = 0.000 183 719 poundal per inch. Aprx. i%-5- 10 000 4-264 1530 I iTomid (av.) per mile [lb/ml]: = 0.281 849 kilogram per kilometer. Aprx. % 1-450 0161 = 0.110 480 grain per inch. Aprx. % 1-043 2829 = 0.002 818 49 gram per centimeter. Aprx. % -5- 100 3.450 0161 = 0.000 568 182 pound per yard. Aprx. 44-4-1 000 4-754 4873 = 0.000 281 849 kilogram per meter. Aprx. 2 A -*- 1 000 4-450 0161 = 0.000 189 394 pound per foot. Aprx. 19 -f- 100 000 4-277 3661 1 kilogram per kilometer [kg/km] or gram per meter [g/m] or milligram per millimeter [mg/mm]: = 9.805 97 dynes per centimeter. Aprx. 10. 0-991 4904 = 3.548 00 pounds per mile. Aprx. % 0-549 9839 = 0.391 983 grain per inch. Aprx. ^lo 1-593 2668 0.01 gram per centimeter 2-000 0000 = 0.002 015 91 pound per yard. Aprx. 2-^1 000 3-304 4713 = 0.001 801 54 poundal per inch. Aprx. %-*-! 000 3-255 6434 0.001 kilogram per meter 3-000 0000 = 0.000 671 970 pound per foot. Aprx. %*-! 000 4-827 3500 * .rain per inch [gr/in]: = 25.016 3 dynes per centimeter. Aprx. MX 100 1-398 2236 = 9.051 4 pounds per mile. Aprx. 9 0-956 7171 = 2.551 14 kilograms per kilometer. Aprx. MX 10. . . . 0-406 7332 = 0.025 511 4 gram per centimeter. Aprx. J-5- 10 2-4067332 = 0.005 142 86 pound per yard. Aprx. 51 -5- 10 000 3-7112045 = 0.004 595 96 poundal per inch. Aprx. 6 /is -s- 100 3-662 3766 = 0.002 551 14 kilogram per meter. Aprx. }<-*- 100 3-406 7332 = 0.001 714 3 pound per foot. Aprx. 17-s-lO 000 3-2340832 '=0.000 142 857 pound per inch. Aprx. 3 /<7 * 1 000 4-154 9020 I gram per centimeter [g/cm]: = 980.597 dynes per centimeter. Aprx. 1 000 2-991 4904 = 354.800 pounds per mile. Aprx. %X100 2-549 9839 100. kg per km or g per m or mg per mm 2-000 0000 = 39.198 3 grains per inch. Aprx. 39 1-593 2668 = 0.201 591 pound per yard. Aprx. % 1-304 4713 = 0.180154 poundal per inch. Aprx. % -4- 10 1-2556434 = 0.1 kilogram per meter 1-000 0000 = 0.067 197 pound per foot. Aprx. %-*-10 2-827 3500 = 0.005 599 75 pound per inch. Aprx. % H- 100 3.748 1688 1 pound per yard [lb/yd]: 1 760. pounds per mile. Aprx. % X 1 000 3-245 5127 = 496.054 kilograms per kilometer. Aprx. MX 1 000.. . 2-6955287 = 194.444 grains per inch. Aprx. %i X 10 000 2-288 7955 = 4.960 54 grams per centimeter. Aprx. 5 0-695 5287 = 0.496 054 kilogram per meter. Aprx. % 1-695 5287 = 0.333 333 pound per foot or M 1-522 8787 = 0.027 777 8 pound per inch. Aprx. i& -4-100 2-443 6975 PRESSURES. 63 1 poundalper inch = 5 443.11 dynes per cm. Apr*. ! KX1 000 3-735 8470 = 217.582 grains per inch. Aprx. HfcXIOO 2-337 6234 = 5.550 81 grams per centimeter. Aprx. ^ 0-744 356$ 1 kilogram per meter [kg/m]: = 3 548.00 pounds per mile. Aprx. % X 1 000 3.549 983$ = 1 000. kilograms per kilometer 3-000 OOOC = 391.983 grains per inch. Aprx. 400 2-593 266$ = 10. grams per centimeter 1-000 0000 = 2.015 91 pounds per yard. Aprx. 2 0-304 4713 = 0.671 970 pound per foot. Aprx. ^ 1.827 3500 =0.055 997 5 pound per inch. Aprx. ^-s-lOO 2-748 168ft 1 pound per foot [lb/ft]: = 5 280. pounds per mile. Aprx. Vi 9 X 100 000 3.722 6340 = 1488.16 kilograms per kilometer. Aprx. %X 1 000. .. 3-1726500 = 583.333 grains per inch. Aprx. HT X 10 000 2-7659168 = 14.881 6 grams per centimeter. Aprx. % X 10 1.172 6500 = 3. pounds per yard 0-477 1213 = 1.48S 16 kilograms per meter. Aprx. % 0-172 6500 = 0.083 333 3 pound per inch. Aprx. % -*- 10 5-920 8188 1 pound per inch [lb/in]= 7 000. grains per inch 3-845 0980 = 178.579 gram/cm. Aprx. %X 100. 2-2518312 = 36. pounds per yard 1-556 3025 = 17.8579 kgpermetr. Aprx. %X 10 1.251 8312 12. pounds per foot 1-079 1812 Ton per mile. A term popularly though incorrectly used for "ton-mile, " which see under units of Energy. It is never used io the sense that a mile of something weighs a ton, except possibly in referring to submarine cables, rails, etc. PRESSURES j PRESSURES of WATER, MERCURY, and ATMOSPHERE; STRESS or FORCE per UNIT AREA. WEIGHTS or FORCES and SURFACES; WEIGHTS of SHEETS, DEPOSITS, COATINGS, etc. (Force -v- surface; mass ^-surface.) The pressures of water columns have all (including those involving only non-metric units) been calculated on the uniform basis that a cubic deci- meter of water weighs one kilogram (see notes under Volumes and Weights). The pressures of mercury columns have all been calculated on the basis that the specific gravity of mercury is 13,595 93, which is the value accepted by the International Bureau of Weights and Measures, and by the (U. S.) National Bureau of Standards. This is the value used in the legal defini- tion of the liter in reference to the atmospheric pressure. The pressure of the atmosphere here used as a standard is that equal to 760 millimeters of mercury of the specific gravity given above. To convert barometric pressures from millimeters .to inches or the reverse, use the reduction factors for one millimeter of mercury in inches of mer- cury or the reverse. In this group of units, the forces, masses, and the weights, considered as both forces and masses, have all been combined to avoid repetition of their relations to each other. When the units involve forces, or weights con- sidered as forces, they are used to measure pressures or stresses per unit area; while when the units involve masses, or weights considered as masses, they are used to measure the weights of sheets, as those of metals. For instance, pounds per square foot may represent a pressure or the weight of a sheet of metal; in the former case the pound is a force and in the latter a mass. The dimensions of the units in the two cases are different. Weights considered as forces involve the value of gravity, but masses and true units of force do not. Aprx. means within 2%. Hg means mercury. 64 PRESSURES. Logarithm Q.OOO 0000 2-8273501 2-008 5098 2-000 0000 3-319 8754 5-875 1006 4-668 9876 1 dyne per square centimeter [dyne/cm 2 ]: = 1. barief i = 0.067 197 poundal per square foot. Aprx. ^ -=- 10 . . . = 0.0101979 kilogram per square meter. Aprx. ^oo . . = 0.01 megadyne per square meter = 0.002 088 70 pound per square foot. Aprx. 21 -4- 10 000. = 0.000 750 068 millimeter of mercury. Aprx. % -r- 1 000. . . = 0.000 466 646 poundal per square inch. Aprx. iy 3 * 10 000 1 barie f (Fr. barye) == 1. dyne per square centimeter. 1 gram per square decimeter [g/dm 2 ]: 0.1 kilogram per sq. m, which see for other values. = 0.020 481 7 pound per square foot. Aprx. 205 * 10 000 . . . 1 poundal per square foot: = 14.881 6 dynes per square centimeter. Aprx. % X 10. . = 0.151 761 kilogram per square meter. Aprx. %-r- 10.. . . =0.031 083 2 pound per square foot. Aprx. ^2 = 0.011 162 2 millimeter of mercury. Aprx. % + 10 1 kilogram per square meter [kg/m 2 ]: = 98.059 66 dynes per square centimeter. Aprx. 100. = 10. grams per square decimeter = 6.589 32 poundals per square foot. Aprx. % X 10. 0.204 817 pound per sq. foot. Aprx. 205 -*- 1 000. . . = 0.1 gram per square centimeter = 0.073 551 4 millimeter of mercury. Aprx. %-r- 10 . . . = 0.045 759 2 poundal per square inch. Aprx. % -J- 100. = 0.003 280 83 foot of water. Aprx. Y z -4- 100 = 0.002 895 72 inch of mercury. Aprx. % * 100 = 0.001 422 34 pound per square inch. Aprx. ^ * 100 . . = 0.001 meter of water = 0.000 102 408 ton (short )/sq. ft. Aprx. 102 -4- 1 000 000 = 0.000 1 kilogram per square centimeter =0.000 096 778 2 atmosphere. Aprx. 29/ 30 -=_ 1 00 000 = 0.000 091 436 1 ton (long) per sq. ft. Aprx. ^4i -=- 1 000 . . 1 megadyne per square meter: = 100. dynes per sq. cm, which see for other values 1 pound per square foot [lb/ft 2 ]: 478.767 dynes per square centimeter. Aprx. 480. . . = 48.824 1 grams per square decimeter. Aprx. 49 .... = 32.171 7 poundals per square foot. Aprx. ^ X 1 000. = 4.882 41 kilograms per square meter. Aprx. 4 % . . . = 0.359 108 millimeter of mercury. Aprx. ^ii = 0.016 018 4 foot of water. Aprx. % -*- 100 = 0.014 138 1 inch of mercury. Aprx. % -r- 100 = 0.006 944 44 pound per square inch. Aprx. 7-4-1 000 . . . = 0.004 882 41 meter of water. Aprx. 49 * 10 000 0.000 5 ton (short) per square fopt or ^ -5- 1 000 .... = 0.000 488 241 kilogram per sq. cm. Aprx. 49 -4- 100 000. . . = 0.000 472 511 atmosphere. Aprx. 47-4-100 000 = 0.000 446 429 ton (long) per sq. foot. Aprx. % -4- 1 000 . . . 1 gram per square centimeter [g/cm 2 ]: = 10. kilograms per sq. m, which see for other values 1-000 0000 2-000 0000 t Recommended by a Committee of the International Physical Congress of 1900, in Paris, for the absolute unit of pressure, that is, for one dyne per sq. centimeter. It seems it was not officially adopted by that Congress. The recommendation includes that the megabarie (or megabarye) is repre- sented with sufficient accuracy for practical purposes by the pressure of 75 cm of mercury at C. ; this latter is nearly what is usually accepted as the pressure of one atmosphere, namely 76 cm of mercury. It was origi- nally proposed to the Congress to adopt this name barie for the atmos- pheric pressure, making it equal to a megadyne per sq. centimeter, but this was changed by the Committee of that Congress; it is, however, some- times used in this sense. PRESSURES. 65 1 millimeter of mercury column [mm Hg]: = 1 333.21 dynes per sq. cm. Aprx. % X 1 000 ....... 3-124 8994 = 89.587 9 poundals per square foot. Aprx. 90 ....... 1-952 2495 = 13.595 93f kilograms per sq. meter. Aprx. % X 10 ---- 1.133 4090 2.784 68 pounds per square foot. Aprx. l i ........ 0-444 7749 = 0.622 138 poundal per square inch. Aprx. % ........ 1-793 8870 = 0.044 606 foot of water. Aprx. %-* 10 ............. 2-649 3932 = 0.039 370 inch of mercury. Aprx. 4-^100 .......... 2-595 1654 = 0.019 3380 pound per square inch. Aprx. 19^-1 000. . 2-286 4124 = 0.013 595 93 meter of water. Aprx. %-:-100 ........... 2-133 4090 = 0.001 392 34 ton (short) per square foot. Aprx. % -* 1 000. 3-143 7449 = 0.001 359 59 kilogram per sq. cm. Aprx. % + 1 000 ..... 3-133 4090 = 0.001 315 79 atmosphere. Aprx.Va^-l 000 ........... 3-119 1864 = 0.00124316 ton (long) per sq. foot. Aprx. ^-s- 100 ---- 3-0945269 1 poundal per square inch: - 2 142.95 dynes per sq. cm. Aprx. % 4 X 10 000 ........ = 21.853 6 kilograms per square meter. Aprx. 1%'XlO. - = 1.607 36 millimeters of mercury. Aprx. % ........... = 0.031 083 2 pound per square inch. Aprx. 3/32 .......... 1 foot of water column = 304.801 kgpersq. m. Aprx. 300.. - 62.428 3 Ibs/sq. ft. Aprx. 5/^xlOO 22.4185mm Hg. Aprx.%X 10... . . . 0.882 617 inch Hg. Ap. subtr. 1/9. 0.433 530 Ib per sq. inch. Aprx. 3-331 0124 1-339 5220 0-206 1130 2-492 5253 2-484 0158 1.795 3817 1.350 6068 1.945 7722 . . . . - 1-637 0192 0.304 801 meter of water. Aprx. 3 /io 1-484 0158 = 0.031 214 2 sh. ton/ft 2 . Aprx. M 2 . - - 2-494 3517 " =0.030 480 1 kg/cm 2 . Aprx. 3-^100. . 2-484 0158 = 0.0294980 atm. Aprx. 3 -MOO.. . . 2-4697932 = 0.027 869 8 1. ton/ft 2 . Ap. l li-t-WQ.. 2-445 1337 1 inch of mercury column [in Hg]: = 345.337 kilograms per sq. meter. Aprx. % X 100 ..... 2-538 2436 = 70.731 pounds per square foot. Aprx. 70 .......... L849 6095 = 25.400 05 millimeters of mercury. Aprx. MX 100 ...... L404 8346 = 1.132 99 feet of water. Aprx. 1% ................... 0-054 2278 = 0.491 187 pound per square inch. Aprx. }/>, ........... 1-691 2470 = 0.345 337 meter of water. Aprx. % ................. 1-538 2436 -0.035 365 5 ton (short) per sq. foot. Aprx. %-t- 100 ...... 2-548 5795 = 0.034 533 7 kiiogram per sq. centimeter. Aprx. % * 100. . 2-538 2436 = 0.033 421 1 atmosphere. Aprx. Mo .................... 2-524 0210 = 0.031 576 3 ton (long) per sq. foot. Aprx. V^ ........... 2-499 3615 1 pound per square inch [lb/in 2 ]: = 703.067 kilograms per sq. meter. Aprx. 700 ........ 2-846 9966 = 144. pounds per square foot. Aprx. ^ X 1 000 ---- 2-158 3625 = 51.711 6 millimeters of mercury. Aprx. 3 Ve X 10 ..... 1-713 5876 = 32.171 7 poundals per sq. inch. Aprx. Mi X 1 000 ---- 1.507 4746 = 2.306 65 feet of water. Aprx. % ................... 0-362 9808 = 2.035 88 inches of mercury. Aprx. 2 ............... 0-308 7530 = 0.703 067 meter of water. Aprx. 7 /lo ................ 1-846 9966 0.072 ton (short) per square foot. Aprx. ^4 ...... 2-857 3325 = 0.0703067 kilogram per sq. centimeter. Aprx. 7 -5- 100. . 2-8469966 = 0.068 041 5 atmosphere. Aprx. Mo ................... 2-832 7740 : =0.0642857 ton (long) per sq. foot. Aprx.%1^-10 ....... 2-8081145 1 meter of water column or ) metric ton per square meter [t/m 2 ]: = 1 OOO. kilograms per square meter ............ .... 3-000 0000 = 204.817 pounds per square foot. Aprx. 205 ......... 2-3113659 = 73.551 4 millimeters of mercury. Aprx. MXlOO ..... 1.866 5910 = 3.280 83 feet of water. Aprx. MX 10 ............... 0-515 9842 = 2.895 72 inches of mercury. Aprx. 2 % ............. 0-461 7564 = 1.422 34 pounds per sq. inch. Aprx. # X 10 ......... 0-1530034 = 0.102 408 ton (short) per sq. foot. Aprx. 4 H+ 100 ..... 1-010 3359 0.1 kilogram per square centimeter ............ 1-000 0000 -0.096 778 2 atmosphere. Aprx. 97-^1 000 ............. 2-985 7774 = 0.091 436 1 ton (long) per square foot. Aprx. Hi ...... 2-061 1179 t This is the specific gravity of mercury used throughout in these tables. 66 PRESSURES. 1 ton (short) per square foot [tn/ft 2 ]: = 9 764.82 kilograms per square meter. Aprx. 9 800. . 3-989 6641 = 2 000. pounds per square foot 3-301 0300 = 718.216 millimeters of mercury. Aprx. % X I'OOO . . 2-856 2551 = 32.036 7 feet of water. Aprx. 32 1.505 6483 = 28.276 2 inches of mercury. Aprx. % X 100 1-451 4205 = 13.888 9 pounds per square inch. Aprx. % X 10. ... = 9.764 82 meters of water. Aprx. 98 -^ 10 = 0.976 482 kilogram per sq. cm. Aprx. subtract Mo . = 0.945 021 atmosphere. Aprx. subtract ^o = 0.892 857 ton (long) per sq. ft. Aprx. subtract % = 0.006 944 44 ton (short) per sq. inch. Aprx. 7 + 1 000... . = 0.006 200 40 ton (long) per sq. inch. Aprx. % -5- 100 1 kilogram per square centimeter [kg/cm 2 ]: = 10 000. kilograms per square meter 4-000 0000 = 2 048.17 pounds per square foot. Aprx. 2 050 3.311 3659 = 735.514 millimeters of mercury. Aprx. MX 1 000 2-866 5910 = 32.808 3 feet of water. Aprx. MX 100 1.515 9842 = 28.957 2 inches of mercury. Aprx. ty X 100 1-461 7564 = 14.223 4 pounds per sq. inch. Aprx. Vr X 100 1-153 0034 = 10. meters of water 1-000 0000 = 1.02408 tons (short) per sq. foot. Aprx. add ^o 0-0103359 = 0.967 782 atmosphere. Aprx. subtract Mo 1-985 7774 = 0.914 361 ton (long) per square foot. Aprx. i%i 1-961 1179 = 0.001 metric ton per square centimeter 3-000 0000 1 barie f = 75 centimeters of Hg (aprx.). Accurately 75.0068. " =1 megadyne per square centimeter. 1 me^aburiet = 1 megadyne per square centimeter 0-000 0000 1 megadyne per sq. cm. =750.068 mm. of Hg. Aprx. ^XlOOO 2-8751006 0.986 931 atmosphere (stand.) Ap. 1 1-9942870 1 atmosphere [atm] (standard): = 10 332.9 kilograms per square meter. Aprx. 10 300 4-014 2226 = 2 116.35 pounds per square foot. Aprx. 2 100 3-325 5885 76O. millimeters of mercury. Aprx. MX 1 000 2-880 8136 = 33.900 6 feet of water. Aprx. } JX 100 1-530 2068 = 29.921 2 inches of mercury. Aprx. bJ 1-475 9790 = 14.696 9 pounds per square inch. Aprx. *% 1-167 2260 = 10.332 9 meters of water. Aprx. 10M 1-014 2226 = 1.058 18 t9ns (short) per sq. foot. Aprx. add Mo 0-024 5585 = 1.033 29 kilograms per sq. cm. Aprx. add Mo 0-014 2226 = 1.013 24 megadynes per sq. centimeter. Aprx.l 0-005 7130 = 1.013 24 megabaries.J Aprx. 1 0-005 7130 = 0.944801 ton (long) per sq. foot. Aprx. subtract Mo- ... 1-9753405 1 ton (long) per square foot [tn/ft 2 ]: = 10 936.6 kilograms per sq. meter. Aprx. 1 1 000 4.038 8821 2 240. pounds per square foot. Aprx. % X 1 000 . . 3-350 2480 = 804.402 millimeters of mercury. Aprx. 800 2-905 4731 = 35.8811 feet of water. Aprx. %i X 100 1-5548663 = 31.669 3 inches of mercury. Aprx. 32 1-500 6385 = 15.555 6 pounds per square inch. Aprx. 3 M 1-191 8855 = 10.936 6 meters of water. Aprx. 11 1.038 8821 = 1.12 tons (short) per sq. foot. Aprx. add 1 A - - . 0-0492180 = 1.093 66 kilograms per square cm. Aprx. add Vio . . . 0-038 8821 = 1.058 42 atmospheres. Aprx. add Mo 0-024 6595 = 0.007 777 78 ton (short) per sq. inch. Aprx. % - 100 -890 8555 = 0.006 944 44 ton (long) per sq. inch. Aprx. 7-M 000 . . . 3-841 6375 1 kilogram per square millimeter [kg/mm 2 ]: = 100. kilograms per sq. cm, which see for other values. . . . 2-000 0000 1 ton (short) per square inch [tn/in 2 ]: 144. tons (short) per sq. foot. Aprx. ty X 1 000 . . = 140.613 kilograms per sq. cm. Aprx. % X 100 = 136.083 atmospheres. Aprx. % X 100 = 0.892 857 ton (long) per sq. inch. Aprx. subtr. Mo ... = 0.140 613 metric ton per sq. centimeter. Aprx. ty t Not authoritative; see foot-note on page 64. t Authoritative; see foot-note on page 64. WEIGHTS AND VOLUMES. 67 1 ton (long) per square incli [tn/in 2 ): = 157.487 kilograms per sq. centimeter. Aprx. i# X 100 . 2-197 2446 = 152.413 atmospheres. Aprx. % X 100 2-183 0220 144. tons (long) per sq. foot. Aprx. V 7 X 1 000 2-158 3625 1.12 tons (short) per square inch. Aprx. add Y% . . . 0-049 2180 = 0.157 487 metric ton per sq. cm. Aprx. 1$*10 1-197 2446 1 metric ton per square centimeter [t/cm 2 ]: = 1 000. kilograms per sq. cm, which see for other values.. 3-000 0000 -967.782 atmospheres. Aprx. 970 2-985 7774 = 7.111 70 tons (short) per sq. inch. Aprx. 5 /rX10 0-8519734 = 6.349 73 tons (long) per sq. inch. Aprx. % X 10 0-802 7554 Conversion Tables for Pressures. Pounds per sq. inch = kilogram atmos- Kilograms persq.cm. Ibs. per pheres. per sq. cm = sq. in. atmos- Atmosph's = Ibs. per kg. per pheres. sq. in. sq. cm. 1 0.070 307 14.223 14.697 0.068042 1.0333 0.967 78 2 0.14061 28.447 29.394 0.13608 2.066 6 1.9356 3 0.21092 42.670 44.091 0.204 12 3.099 9 2.903 3 4 0.281 23 56.894 58.788 0.272 17 4.1332 3.871 1 5 0.351 53 71.117 73.485 0.34021 5.1665 4.8389 6 0.421 84 85.340 88.181 0.408 25 6.1997 5.8067 7 0.492 15 99.564 102.88 0.47629 7.2330 6.774 5 8 0.562 45 113.79 117.58 0.544 33 8.266 3 7.742 3 9 0.632 76 128.01 132.27 0.61237 9.2996 8.7100 10 0.70307 142.23 146.97 0.680 42 10.333 9.6778 WEIGHTS or MASSES and VOLUMES ; DENSITIES; WEIGHTS of MATERIALS; MASSES per unit of VOLUME. (Weight ~ volume.) Only the more usual units are given here, as the table would otherwise have become very long and cumbersome. The relations between such compound units as these are the same as those between their individual units whenever one of the latter is the same in both; for instance, the rela- tion between pounds per cubic yard and kilograms per cubic yard is the same as between pounds and kilograms, and as these are given in the tables of weights they are not repeated here. In such a reduction multiply the pounds per cubic yard by the value of 1 pound in kilograms. Similarly, the relation between pounds per cubic yard and pounds per cubic meter is the same as that between a cubic meter and a cubic yard, but in this case care must be taken in the reduction on account of the word " per," not to multiply the former by the value of one cubic yard in cubic meters, but to divide instead, as a cubic yard is smaller than a cubic meter, hence *he weight per cubic meter is larger. To avoid such a long division use instead the reciprocal relation, namely, the value of one cubic meter in cubic yards and then multiply. The general rule for all compound units is that if the individual unit to be changed is preceded by the word " per," then divide by the value of the old unit in terms of the new one (or multiply by its re- ciprocal); in all other cases multiply, even when the unit follows a hyphen, as, for instance, in the case of pounds in foot-pounds. In this group of units the weights are always masses and never forces; no unit exists having the dimensions of force divided by volume. The value of gravity is therefore not involved in these values. "Weights of Materials. In the metric system the number represent- ing the density or specific gravity also represents the actual weight in grams of a cubic centimeter of the material. Hence the actual weight of any other 68 WEIGHTS AND VOLUMES. unit of volume of that material in terms of any other unit of weight is deter- mined by merely multiplying the specific gravity or density (when based on water, as is usual for all materials except gases) by the value of 1 gram per cubic centimeter in terms of those units as given in the table below. Thus the weight of any material in pounds per cubic foot is 62.43 multiplied by its specific gravity, this figure 62.43 being the value of 1 gram per cubic centimeter in terms of pounds per cubic foot. Similarly, the specific gravity or density is easily calculated by means of the figures in this table, when the weight of a unit of volume is given in terms of any of the usual non-metric units. Thus if the weight of any material is, say, 100 pounds per cubic foot, its specific gravity is 0.016 02 (which is the value of 1 pound per cubic foot in terms of grams per cubic centimeter in the table) multi- plied by 100, that is 1.602. Aprx. means within 2%. Logarithm 1 pound per cubic yard [lb/yd 3 ]: = 0.593 273 kilogram per cubic meter. Aprx. %o ...... 1-773 2545 = 0.037 037 or HT Pound per cb. ft. Aprx. %-* 10 2-5686362 = 0.000 593 273 gram per cb. cm or kg per lit. Ap, .0006. . . 4-773 2545 = 0.000 593 273 ton (met.) per cb. meter. Aprx. 6-*-. 10 000. 4.773 2545 0.000 5 ton (short) per cubic yard or ^-^1 000. . . . 4-698 9700 = 0.000 446 429 ton (long) per cubic yard. Aprx. % + l 000. 4-649 7520 1 kilogram per cubic meter [kg/m 3 ]: 1.685 56 pounds per cubic yard. Aprx. *% 0-226 7455 = 0.062 428 3 pound per cubic foot. Aprx. ^-f-10. ..... 2-795 3817 0.001 gram per cb. cm or kilogram per liter 3-000 0000 =-= 0-001 ton (met.) per cubic meter 3-000 0000 = 0.000 842 782 ton (short) per cb. yd. Aprx. %-*- 1 000. . . 4-925 7155 = 0.000 752 484 ton (long) per cb. yd. Aprx. %*- 1 000 4-876 4975 1 grain per cubic inch [gr/in 3 ]: = 0.246 857 pound per cubic foot. Aprx. }i 1-392 4457 = 0.003 954 25 gram per cb. cm or kg per lit. Aprx. "H ooo... 3-597 0640 1 pound per bushel [Ib/bu] (U. S.): = 12.871 8 kilograms per cubic meter. Aprx. 9 AX10 1-1096387 = 1.287 18 kilograms per hectoliter. Aprx. % 0-1096387 = 1.032 02 pounds per bushel (Brit.). Aprx. add Ko- 0-013 6888 = 0.803 564 pound per cubic foot. Aprx. % 1-905 0204 1 pound per bushel [Ib/bu] (Brit.): = 12.472 4 kilograms per cubic meter. Aprx. YsX 100. . . . 1-095 9499 = 1.247 24 kilograms per hectoliter. Aprx. 1% 0-095 9499 = 0.968 972 pound per bushel (U.S.). Aprx. subtr. Mo L986 3112 = 0.778 630 pound per cubic foot. Aprx. % 1-891 3318 1 kilogram per hectoliter [kg/hi]: = 10. kilograms per cubic meter, which see for other values. 1-000 0000 1 pound per cubic foot [lb/ft 3 ]: = 27. pounds per cubic yard. Aprx. % X 10 1-431 3638 = 16.018 4 kilograms per cubic meter. Aprx. %X 10. .. 1-204 6183 = 4.050 93 grains per cubic inch. Aprx. 4 0-607 5543 = 1.601 84 kilograms per hectoliter. Aprx. % 0-2046183 = 1.284 31 pounds per bushel (Brit.). Aprx. % 0-108 6684 = 1.244 46 pounds per bushel (U. S.). Aprx. 1M 0-094 9796 = 0.160 538 pound per gallon (Brit.). Aprx. %-hlO 1-205 5784 = 0.133681 pound per gallon (liquid; U.S.). Aprx.%-5- 10 Ll26 0683 = 0.0160184 gram per cb. cm or kg per lit. Aprx. % -MOO.. 2-2046183 = 0.0160184 ton (met.) per cubic meter. Aprx. %-^ 100. . 2-2046183 = 0.013 5 ton (short) per cubic yard. Aprx. 4 / 3 -H 100 .. 2-1303338 = 0.012 053 6 ton (long) per cubic yard. Aprx. %-r-lOO . . 2-081 1158 1 pound per gallon [Ib/gal] (liquid; U. S.): = 7.480 52 pounds per cubic foot. Aprx. MX10.. 0-8739317 = 1.200 91 pounds per gallon (Brit.). Aprx. add % 0-0795101 = 0.119 826 gram per cb. cm or kg per lit. Aprx. 12 -i- 100... 1.078 5500 1 pound per gallon [Ib/gal] (Brit.): = 6.229 05 pounds per cubic foot. Aprx. 634 0-794 4216 = 0.832 702 4 pound per gallon (liquid; U. S.). Aprx. %.. . 1.920 4899 = 0.099 779 2 gram per cb. cm or kg per liter. Aprx. Ho... . 2-9990399 1 pound per quart [lb/qt]=4. pounds per gallon 0-602 0600 WEIGHTS AND VOLUMES OF WATER. 69 1 gram per cubic centimeter [g/cm 3 ] or 1 kilogram per liter [kg/1] or 1 ton (met.) per cubic meter [t/m 3 ]: = 1685.57 pounds per cubic yard. Aprx. K X 10 000. . . 3-2267455 1 000. kilograms per cubic meter , . . . . 3-000 0000 = 252.893 grains per cubic inch. Aprx. MX 1 000 2-402 9360 100. kilograms per hectoliter 2-000 0000 = 80.177 1 pounds per bushel (Brit.). Aprx. 80 1-904 0501 = 77.6893 pounds per bushel (U. S.). Aprx. %X100.. 1-8903613 = 62.428 3 pounds per cubic foot. Aprx. ^>( 100 1-795 3817 = 10.022 1 pounds per gallon (Brit.). Aprx. 10 1-000 9601 = 8.34545 pounds per gal (liquid; U.S.). Aprx. Vi 2 X 100 0-921 4500 = 0.842 783 ton (short) per cubic yard. Aprx. subtr. K 1-925 7155 = 0.752 484 ton (long) per cubic yard. Aprx. 1-876 4975 = 0.0361275 pound per cubic inch. Aprx. "Hi -f- 10 2-5578380 0.001 kilogram per cubic centimeter 3-000 0000 1 ton (short) per cubic yard [tn/yd 3 ]: = 1 186.55 kilograms per cubic meter. Aprx. % X 1 000. . . 3-074 2845 = 118.655 kilograms per hectoliter. Aprx. %X 100 2-074 2845 = 95.133 7 pounds per bushel (Brit.). Aprx. 95 1-978 3346 = 92.181 9 pounds per bushel ( U. S.). Aprx. Vn X 1 000. . . 1-964 6458 = 74.074 1 pounds per cubic foot. Aprx. MX 100 1-869 6662 = 1.186 55 tons (met.) per cb. m or kg per lit. Ap. add %... 0-074 2845 = 0.892 857 ton (long) per cubic yard. Aprx. 9 /i 1-950 7820 1 ton (long) per cubic yard [tn/yd 3 ]: = 1 328.93 kilograms per cubic meter. Aprx. % X 1 000 3-123 5025 = 132.893 kilograms per hectoliter. Aprx. %X 100 2-123 5025 = 106.550 pounds per bushel (Brit.). Aprx. 107 2-027 5526 = 103.244 pounds per bushel (U. S.). Aprx. 103 2-013 8638 = 82.963 pounds per cubic foot. Aprx. % X 100. , 1-918 8842 = 1.328 93 tons (met.) per cb. m or kg per lit. Ap. add K-- 0-123 5025 = 1.12 tons (short) per cubic yard. Aprx. add % 0-049 2180 1 pound per cubic inch [lb/in 3 ]: = 27.679 7 grams per cb. centimeter. Aprx. ^X 10. ... 1-442 1620 ==0.027 679 7 kilogram per cubic centimeter. Aprx. ^-^ 100 2-442 1620 1 kilogram per cubic centimeter [kg/cm 3 ]: = 1 000. grams per cb. cm or tons (met.) per cb. m 3-000 0000 = 36.127 5 pounds per cubic inch. Aprx. 36 1-557 8380 WEIGHTS and VOLUMES of WATER. Factors for calculating weights or volumes of materials from their specific gravity. The following two groups of numbers give the weights (W) of all the different units of volume of water occurring in practice; also the volumes (V) of all the different units of weight of water occurring in practice; the latter are, of course, the reciprocals of the former and are given so as to avoid the long divisions by the former. Besides their direct application to hydraulics and to the calibration of vessels and for measuring, or for the indirect determinations of irregular volumes by means of weights, they are also of use for determining th-3 weights of materials, as the weight of a unit of volume of any material, whether solid or liquid, is its specific gravity or density multiplied by one of these factors, W; or the volume of a unit of weight of any material, whether solid or liquid, is one of these factors, V, divided by its specific gravity or density. They are applicable also to gases provided the value of the specific gravity or density which is used is based on water and not on air or hydrogen. For the weights of columns of water, mercury, or the air, see under Pressures. All these values, even those given entirely in English units, have been calculated from the uniform bases that one liter of water weighs one kilo- gram, and that a liter is equal to a cubic decimeter. (See notes on the liter in the introductory remarks on units of Volume and Weight.) 70 WEIGHTS AND VOLUMES OF WATER. WEIGHTS of WATER, W. Aprx. means within 2%. Logarithm 1 cubic centimeter =* 15.423 4 grains. Aprx. 3 3^or 151^. . 1.188 4322 = 1. gram 0-000 0000 = 0.0352740 oz (av.). Aprx. %-?- 100 .. 2-547454] = 0.00220462 Ib (av.,). Aprx. % + 100. .. 3.348 3342 I cubic inch = 252.893 grains. Aprx. MX1 000 2-402 9360 = 16.387 2 grams. Aprx. % X 100 or % 1.214 5038 = .578 040 ounce (av.). Aprx. ^ 1.761 9579 -=0.036 127 5 pound (av.). Aprx. #10 2-557 8380 1 pint (liquid; U. S.) = 1.043 18 pounds (av.). Aprx. add Ho. . 0-018 3600 =0.473 179 kilogram. Aprx. Ki X 10 1-675 0258 1 pint (dry; U. S.) = 1.213 90 pounds (av.). Aprx. % 0-0841813 = 0.550 614 kilogram. Aprx. ^^-10 1-7408471 1 pint (Brit.) = 1.252 77 pounds (av.). Aprx. % 0-097 8701 = 0.568 245 39 kilogram. Aprx. ty? 1-754 5359 1 quart (liquid; U. S.) = 2.086 36 Ib (av.). Aprx. 2V W or2y 10 . . Q-319 3900 = 0.946 359 kilogram. Aprx. subt. Ho- . . 1-9760558 1 liter = 2.204 62 pounds (av.). Aprx. s^ 0-343 3342 1. kilogram 0-000 0000 1 quart (dry; U.S-) = 2.427 79 pounds (av.). Aprx. 2^0 or 1%.. 0-3852113 = 1.101 23 kilograms. Aprx. add Ho 0-041 8771 1 quart (Brit.)= 2.505 53 pounds (av.). Aprx. 1% 0-398 9001 = 1.136 490 8 kilograms. Aprx. add Vr 0-055 5659 1 gallon (liquid; U. S.) =8.345 45 pounds (av.). Aprx. 5% Q-921 4500 = 3.785 43 kilograms. Aprx.^XlO 0-5781158 1 gallon (Brit.) = 10.022 1 pounds (av.). Aprx. 10 1-0009601 = 4.545 963 1 kilograms. Aprx. % or A 1 A 0-657 6259 1 peck (U. S.) = 19.422 3 pounds (av.). Aprx. %i XI 000 1-2883013 = 8.809 82 kilograms. Aprx.J^XlO 0-9449671 1 peck (Brit.)= 20.044 3 pounds (av.). Aprx. 20 1-301 9901 = 9.091 926 2 kilograms. Aprx. 9 or .i, which is found directly from a table of cosines. Sometimes the power factor is stated in percent, in which case it is equal to the above figure multiplied by 100. 80 POWEK. Load factor is a term commonly applied to electric, steam, or hydraulic ower stations to show how much of the total possible amount of power as actually been generated or used during a limited time, It is the ratio of the mean power used during a limited time (generally 1 day) divided by p h the total power that the station could have generated during that time; as it is usually stated in percent, this ratio must be multiplied by 100. If the average power generated during a day is % of that which the station is capable of generating, the load factor is 25%. A 100% load factor means that the station is running at its full output all the time. In water- power installations or in stations having storage batteries, this quantity is of use in determining the amount of storage capacity desired. POWER; RATE of ENERGY; RATE of DOING WORK ; MOMENTUM. Aprx. means within 2%. Logarithm 1 erg per second or 1 dyne-centimeter per second; = 0.000 000 1 watt .................................... 7-000 0000 1 grain-centimeter per second [g-cm/s]: = 0.000 098 059 7 watt. Aprx. ^o ooo .................... 5-991 4904 1 foot-grain per second [ft-gr/s]: = 0.000 193 675 watt. Aprx. %i -4- 100 ................... 4-287 0740 1 foot-pound per minute [ft-lb/min]: = 0.022 595 4 watt. Aprx. 9 /i -4- 100 .................. 2-3540208 = 0.011 363 6 mile-pound per hour. Aprx. 8 A -4- 100 ..... 5-055 5174 = 0.000 030 723 4 metric horse-power. Aprx. Vis * 10 000. . 5-487 4691 = 0.000 030 303 horse-power. Aprx. 3 -4- 100 000 ......... 5-481 4860 = 0.000 022 595 4 kilowatt. Aprx. % -s- 100 000 ........... 5-354 0208 1 calorie (small) per minute [cal/min]: = 0.069'769 5 watt. Aprx. Vioo ............. '. ........... 2-843 6653 1 kilogram-meter per minute [kg-m/min]: = 0.163 433 watt. Aprx. Y & ......................... 1.213 3391 = 0.082 193 2 mile-pound per hour. Aprx. % -4- 10 ....... 2-914 8357 = 0.000 222 222 metric horse-power. Aprx. % -4- 1 000 ...... 4-346 7875 = 0.000 219 182 horse-power. Aprx. % -4- 1 000 ............ 4-340 8044 -0.000 163 433 kilowatt. Aprx.K-J-1 000 ............... 4-213 3391 1 watt [w] or 1 joule per second: = 10 000 000- ergs per second ......................... 7-000 0000 = 10197.9 gram-centimeters per second. Aprx. 10000. 4-0085096 = 5 163.28 foot-grains per second. Aprx. 5 200 ....... 3-7129260 = 44.256 7 foot-pounds per minute. Aprx. % X 100 ____ 1.645 9793 = 14.332 9 small calories per minute. Aprx. Vr X 100. . . 1-156 3347 = 6.118 72 kilogram-meters per minute. Aprx. 6 ...... 0-786 6609 = 0.737 612 foot-pound per second. Aprx. % .......... 1-887 8279 = 0.502 917 mile-pound per hour. Aprx. H ............ 1-7014965 = 0.238 882 small calorie per second. Aprx. 24-4-100. . . 1-378 1834 = 0.101979 kilogram-meter per second. Aprx.^o ...... 1-0085096 = 0.056 877 6 thermal unit per minute. Aprx. #-4-10 ____ 2-754 9414 = 0.0315987 Ib-Centgr. heat unit per min. Aprx. 2%^- 100 2-4996689 = 0.014 332 9 large calorie per minute. Aprx. ^70 ........ 2-156 3347 = 0.001 359 72 metric horse-power. Aprx. % * 1 000 ...... 3-133 4483 = 0.001 341 11 horse-power. Aprx. % -4-1 000 ............ 3-127 4653 0.001 kilowatt ............................... 3-000 0000 watts = volt-amperes X cos angle of lag. volt-amperes = volts X amperes. = watts -4- cos angle of lag. 1 foot-pound per second [ft-lb/s]: 60. foot-pounds per minute .................. 1-778 1513 = 8.29532 kilogram-meters per minute. Aprx. %X 10. 0-9188330 = 1.355 73 watts. Aprx. % ........................ 0-132 1721 = 0.001 843 40 metric horse-power. Aprx. *% -4- 1 000 ..... 3-265 6204 0.001 818 18 horse-power. Aprx. 34i-^lOO ............. 3-259 6373 355 73 kilowatt. Aprx. %-s- 1 000 ............... 3-182 1721 , POWER. 81 1 mile-pound per hour 88. foot-pounds per minute. Aprx. % X 100 1 .944 4827 = 12.1665 kilogram-meters per minute. Aprx. 12 1-0851644 = 1.988 40 watts. Aprx. 2 0-298 5035 = 0.002 666 67 horse-power. Aprx. 8 / 3 ^- 1 000 3-425 9687 1 calorie (small) per second [eal/s] = 4.186 17 watts. Ap. e% 2 . 0-621 8166 1 kilogram-meter per second [kg-m/s]: 433.980 foot-pounds per minute. Aprx. % X 1 000 . 2-6374696 = 60. kilogram-meters per minute 1-778 1513 = 9.805 97 watts. Aprx. 10 0-991 4904 = 0.0133333 metric horse-power. Aprx. %-* 100 2-1249387 = 0.013 150 9 horse-power. Aprx. 4 / 3 -*- 100 2-118 9557 = 0.009 805 97 kilowatt. Aprx. 1-4-100 3-991 4904 1 thermal unit per minute [lb-F/minl: = 17.581 6 watts. Aprx. % X 10. /. 1-245 0586 = 0.023 906 metric horse-power. Aprx. 1%-f- 100 1-378 5069 = 0.023 578 9 horse-power. Aprx. %-^-100 2-372 5239 = 0.017 581 6 kilowatt. Aprx. % + WQ 2-245 0586 1 pound-Centigrade heat unit per minute [lb-C/min]: = 31.646 9 watts. Aprx. 32 1.500 3311 = 0.043 030 8 metric horse-power. Aprx. % -*- 10 2-633 7794 = 0.042 442 1 horse-power. Aprx. % -* 10 2-627 7964 ' =0.031 646 9 kilowatt. Aprx. 32-r-l 000 2-500 3311 1 watt-hour per minute =60. watts 1-778 1513 1 calorie (large) per minute [Cal/min]: = 69.769 5 watts. Aprx. 70 1-843 6653 = 0.094 866 7 metric horse-power. Aprx. %i 2-977 1136 = 0.093 568 7 horse-power. Aprx. % 2 2-971 1306 = 0.069 769 5 kilowatt. Aprx. Vioo 2-843 6653 1 mile-pound per minute [ml-lb/min]: = 119.304 watts. Aprx. 120 2-0766548 = 0.162 220 metric horse-power. Aprx. %n-10 1.210 1031 = 0.160 000 horse-power. Aprx. %-f- 10 1-204 1200 -0.119 304 kilowatt. Aprx. %-r-lO. . , 1-076 6541 1 kilogram -kilometer per minute [kg-km/min]: = 163.433 watts. Aprx. Y & X 1 00 2-213 339] = 0.222 222 metric horse-power. Aprx. % 1-346 7875 = 0.219 182 horse-power. Aprx. % 1-340 8044 = 0.163 433 kilowatt. Aprx. % 1-213 3391 1 metric horse-power [hp] or French horse-power or chevalvapeur or force de cheval or Pferde-kraft : = 7.354 48 X10 9 ergs per second. Aprx. 2 %X10 9 9-8665517 = 32 548.5 foot-pounds per minute. Aprx. 33 000 4-512 5309 = 4 500. kilogram-meters per minute. Aprx. / 2 X 1000 3-653 2125 735.448 watts. Aprx. 2 ^X 100 2-866 5517 = 542.475 foot-pounds per second. Aprx. /ii X 1 000 .. 2-7343797 = 75. kilogram-meters per second or 9-4 X 1 00 1-875 0613 = 41.830 5 thermal units per minute. Aprx. 42 1-621 4931 = 23.239 2 Ib-Ctg. heat units per minute. Aprx. % X 10. 1-366 2206 = 10.541 1 large calories per minute. Aprx. 2 K 1-022 8864 = 0.986 318 horse-power. Aprx. 1 1-994 0170 = 0-750 000 poncelet 1-875 0613 = 0.735 448 kilowatt. Aprx. 2 %-^-10.. 1-866 5517 1 horse-power [hp]: = 7.456 SOX 10 9 ergs per second. Aprx. MX 10* 9-872 5348 = 33 OOO. foot-pounds per min. Aprx. MX 100 000. . . 4-518 5139 = 4562.42 kg-meters per minute. Aprx. % XJ. 000 3-6591956 = 745.650 watts. Aprx. MX 1 000 2-872 5348 = 550. foot-pounds per second. Aprx. 1^X100. . . 2-740 3627 = 375.000 mile-pounds per hour. Aprx. %Xl 000... . 2-574 0313 = 76.040 4 kg-meters per second. Aprx. %X 100 1-881 0444 = 42.4108 thermal units per min. Aprx. 3 A X 100 1-6274762 = 23.5615 Ib-Ctg. heat units per min. Aprx. % X 10. . 1-3722037 = 10.687 3 large calories per min. Aprx. 3 %j 1-028 8695 = 1.013 87 metric horse-powers. Aprx. 1 0-005 9830 = 0.760 404 poncelet. Aprx. % 1-881 0444 = 0.745 650 kilowatt. Aprx. M 1-872 5348 82 POWER. I poncelet = 100. kilogram -meters per second 2-000 0000 41 = 1.33333 metric horse-powers. Aprx. % 0-1249387 = 1.315 09 horse-powers. Aprx. % : 0-118 9557 = 0.980 597 kilowatt. Aprx. 1 L991 4904 1 kilowatt [kw] = IX 10 10 ergs per second 10-000 0000 = 44256.7 ft-lbs per min. Aprx. %X 100 000.. 4-6459793 = 6118.72 kg-met. per min. Aprx. 6 X 1 000 . . 3-7866609 = 1 000. watts 3-000 0000 = 737.612 ft-lbs. per sec. Aprx. MXl 000... . 2-867 8279 = 101.979 kilogram-metr. per sec. Aprx. 100. 2-0085096 = 56.8776 thermal u. per min. Aprx. # X 100. 1-7549414 = 31.598 7 Ib-Ctg. heat u. per min. Aprx. <*%.. 1.499 6689 = 14.3329 large cal. per min. Aprx. 14 X 100.. 1.1563347 = 1.35972 metric horse-powers. Aprx. add}^. 0-1334483 = 1.341 11 horse-powers. Aprx. add 1 A 0-127 4652 = 1.019 79 poncelets. Aprx. 1 0-008 5096 1 watt-lrour per second = 3 600. watts. Aprx. ^Xl 000. . . 3-556 3025 = 3.600 kilowatts. Aprx. 1 H 0-556 3025 1 metric horse-power-hour per minute: = 60. metric horse-powers 1-778 1513 1 hors-power-hour per minute =60. horse-powers 1-778 1513 1 kilowatt-hour per minute = 60. kilowatts 1-778 1513 1 metric horse-power-hour per second: = 3 600. metric horse-powers. Aprx. ^X 1 000 3-556 3025 1 horse-power-hour per second: = 3 600. horse-powers. Aprx. ^ XI 000 3-556 3025 1 kilowatt hour per second: = 3600. kilowatts. Aprx. ^X 1000 3-5563025 Conversion Tables for Power. Ho r se-powers = kilowatts . metrhp';-* Meti'ic hp's kilowatts horse- Kilowatts = horse- powers metrhp's powers 1 2 3 . 4 5 6 7 8 9 10 0.74565 1.4913 2.2370 2.9826 3.7283 4.4739 5.2196 5.965 2 6.7109 7.456 5 1.341 1 2.682 2 4.023 3 5.3644 6.705 6 8.046 7 9.387 8 10.729 12.070 13.411 0.735 45 1.4709 2.206 3 2.941 8 3.6772 4.4127 5.1481 5.8836 6.6190 7.3545 1.3597 2.7194 4.079 2 5.438 9 6.7986 8.1583 9.5180 10.878 12.237 13.597 1.0139 2.027 7 3.041 6 4.055 5 5.069 4 6.0832 7.097 1 8.1110 9.1248 10.139 0.986 32 1.9726 2.9590 3.945 3 4.9316 5.9179 6.9042 7.890 5 8.876 9 9.8632 FORCES. 83 FORCES 5 WEIGHTS Considered as Forces. (See also Weights.)" Only true units of force (dynes and poundals) are given here, together with their values in terms of weights, and the reciprocals of these values. The relations between two weights when considered as forces are, of course, the same as when they are considered as masses; these have been given under Weights and are therefore not repeated here. A true unit of force is independent of the value of gravity and is the same throughout the universe. But a weight considered as a force in- cludes the value of gravity and is therefore different for different values of gravity. The value of gravity used in these tables is 980.596 6 (see note on this value under the units of Acceleration). A dyne is that force which, acting on a mass of one gram for one second, produces a velocity of one centimeter per second; this definition refers to a space which is free from the attraction of other bodies. A poundal is similarly that force w^iich, acting on a mass of one pound for one second, produces a velocity of one foot per second. The attraction of gravity of the earth is really a force, but it cannot be used as a unit of force, as the amount of this force which acts on anybody depends on the mass on which it acts. When reduced to the force per gram mass, it becomes the same thing as the force represented by one gram considered as a weight, and this together with all similar values is given in the table. The attraction of gravitation becomes a constant quantity when it is stated as an acceleration, as in this form it is independent of the mass on which it acts; it can then be used as a unit and is included as such in the table of accelerations, which see for its reduction factors. For tractive forces or tractive efforts, see under this title at the end of units of Energy. Aprx. means within 2%. Logarithm 1 microdyne = 0.000 001 dyne 6-000 0000 1 milligram = 0.980 596 6 dyne. Aprx. 98-=- 100 1.991 4904 = 0.000 070926 5 poundals. Aprx. 7 -=-100 000. .. 5-850 8088 1 dyne = 1 .019 79 milligrams. Aprx. 1 0-008 5096 = 0.015 737 7 grain. Aprx. *# -f- 100 2-196 9418 = 0.001 019 79 gram. Aprx. 1 -=- 1 000 3.008 5096 = 0.000 035 971 9 ounce (av.). Aprx. 4 -=-110 000 5-5559637 = 0.000 072 330 poundal. Aprx. ^-=-10 000 5.859 3184 -0 000 002 248 25 pound (av.). Aprx. % -=- 1 000 000. . . . 6-351 8437 1 grain = 63.541 6 dynes. Aprx. Vii X 100 1-8030582 = 0.001 595 96 poundal. Aprx. 46^-10 000 3-662 3766 1 gram = 980.596 6 dynes. Aprx. 1 000 2-991 4904 = 0.070926 5 poundal. Aprx.7-^100 2-8508088 1 kilodyne = 1 000. dynes 3-000 0000 1 myriadyne = 10 000. dynes 4-000 0000 lpoundal= 14 099.1 milligrams. Aprx. ^r X 100 000 4-1491912 - 13 825.5 dynes. Aprx. *HX 10 000 ...4-1406816 = 217.582 grains. Aprx. 220 2-337 6234 = 14.099 1 grams. Aprx. Vi X 100 1-1491912 = 0.497 331 ounce (av.). Aprx. ^ 1-696 6453 = 0.031 083 2 pound. Aprx. 31-4-1 000 2-492 5253 = 0.014 099 1 kilogram. Aprx. ^o 2-1491912 1 ounce (av.) = 27 799.5 dynes. Aprx. ^ X 10 000 4.444 0363 = 2.010 73 poundals. Aprx. 2 '. 0-303 3547 1 pound (av.)= 444 791. dynes. Aprx.%Xl 000000 5-648 1563 = 32.171 7 poundals. Aprx. 32 1-507 4746 I kilogram =980 596.6 dynes. Aprx. 1 000 000 5.991 4904 == 70.926 5 poundals. Aprx. 70 1-850 8088 = 0.980 597 megadyne. Aprx. 1 1.991 4904 1 megadyne = 1 000 000. dynes 6-000 0000 = 72.330 poundals. Aprx. 5 / 7 2-571 5016 86 ANGULAR VELOCITIES; ROTARY SPEEDS. 1 kilometer per minute [km/min]: = 54.680 6 feet per second. Aprx. 55 1.737 8329 = 37.282 2 miles per hour. Aprx. 37 , 1-571 5016 = 16.666 7 meters per second. Aprx. HX 100 1-221 8487 = 0.621 370 mile per minute. Aprx. y% 1.793 3503 1 mile per minute [ml/min]: 88. feet per second , 1-944 4827 60. miles per hour 1-778 1513 = 26.822 4 meters per second. Aprx. 27 1-428 4984 = 1.609 35 kilometers per minute. Aprx. % 0-206 6497 Miscellaneous concrete units: Average velocity of molecules about 500. meters per second (Woodward). Velocity of light about 300 000. kilometers per second. ANGULAR VELOCITIES; ROTARY SPEEDS. (Angle -r- time.) For simple reductions or relations between angles, see values under Angles. Aprx. means within 2%. Angular velocity = angle moved through divided by time. in degrees per second = angle in degrees -r- time in seconds. = revolutions per second X 360. " in revolutions per minute = revolutions -5- time in minutes. ' ' in radians per second = 2 n X revolutions per second. Logarithm 1 revolution per hour [rev/h or rph]: ==0.1 degree per second 1 .000 0000 1 radian per minute: = 9.549 30 revolutions per hour. Aprx. 19 / 2 0-979 9714 = 0.954 930 degree per second. Aprx. subtract >^o 1-979 9714 = 0.002 652 58 revolution per second. Aprx. %'-*- 1 000 3-423 6689 1 degree per second: 10. revolutions per hour 1 000 0000 1.047 20 radians per minute. Aprx. add J^o 0-020 0287 = 0.166 667 revolution per minute or Y & 1-221 8487 = Moo or 0.002 777 78 rev per second. Aprx. %i -* 100 3-4436975 1 revolution per minute [rev/min or rpm]: 6. degrees per second 0-778 1513 = 0.104 720 radian per second. Aprx. 2 j^ -7-100 1. 020 0287 = 0.016 666 7 rev per second or Ko 2-221 8487 1 radian per second [aj]: = 57 295 8 degrees per second. Aprx. 57 . . 1-758 1226 = 0.159 155 revolutions per second Aprx. 1( Hoo 1-201 8201 I revolution per second [rev/s or rps]: 60. revolutions per minute. 1-778 1513 = 6.283 185 radians per second. Aprx.^sXlO 0-7981799 FREQUENCY; PERIODICITY; PERIOD; ALTERNA- TIONS. (l--time; time.) Frequency or periodicity is the number of recurrences of some periodic or wave motion during a given time; this time is always understood to be a second unless otherwise stated; the frequency always refers to the number of complete waves. The "number of alternations" however, refers to the number of changes of the direction of the motion or to the reversals, and therefore refers to half waves, and is always equal to double the frequency, if the time is the same. The period is the time of one com- plete wave or oscillation and is therefore the reciprocal of the frequency. The term frequency is the one most generally used, and always refers to a second; the term number of alternations is unfortunately preferred by some and when it refers to electric currents the time is usually a minute; the term period is used comparatively rarely as a measure, its use being generally limited to scientific discussions. FREQUENCY. LINEAR ACCELERATIONS. 87 In mathematical discussions of electric alternating-current problems the frequency is often replaced by an angular velocity, generally represented by (a and measured in radians per second (see under Angular Velocities above). Then to = 2irn t in which o> is in radians per second and n is the true frequency in cycles per second, a cycle being here considered the same thing as a complete revolution. The frequency is also equal to the velocity of propagation divided by the wave length. The wave lengths are therefore measured in units of length, but when the velocity for a class of waves is a constant (as those of light or the electromagnetic waves), the wave lengths may also be in- dicated in units of time, in which case a wave length becomes equal to the period of the wave. Wave length should not be confounded with the amplitude, which latter measures the intensity of the wave and has noth- ing to do with the frequency, period, or wave length. If n is the frequency per second [], then: the period in seconds = l/n; the number of alternations per minute = 120n. If n is the number of alternations per minute, then: the frequency per second = n/120; the period in seconds = 120/n. If n is the period in seconds, then: the frequency per second = l/n; the number of alternations per minute = 120n. If n is the frequency in cycles per second, a> the angular velocity in radians per second, and if a cycle is represented by one revolution, then: o) = 2xn; or n = 0.159 155 2-255 2725 = 3.141 59 radians. Aprx. 2Jft 0-497 1499 2. quadrants 0-301 0300 = 0.5 circumference, or Yi 1-698 9700 1 circumference or revolution [rev] = 21 600. minutes 4-334 4538 = 360. degrees 2-556 3025 = 6.283 185 radians. Aprx. %X10 0-798 1799 = 4. quadrants 0-602 0600 = 2. TT'S (considered as an angle of 180) 0-301 0300 1 electrical degree =the 360th part of a cycle; see p. 121. SOLID ANGLES. (Surfaces radius.) A solid angle is an angle, like that at the point of a cone, which is sub- tended by a spherical surface. The unit solid angle is that angle which, at the center of a sphere of unit radius, subtends a unit area on the surface of the sphere; this unit is sometimes called a steradian. A spherical right angle is assumed to be an angle, like at the corner of a rectangular block, which is bounded by three planes perpendicular to each other. Logarithm 1 unit= 0.636 620 spherical right angle, or 2A 1-803 8801 " = 0.159 155 hemisphere, or l-^27r 1-2018201 " =0.079577 5 sphere, or 1+4* 2-900 7901 1 steradian = 1 unit solid angle ; ( see above) 000 0000 1 spherical right angle =1.570 80 units, or x/2 0-196 1199 = 0.25 hemisphere, or M 1-397 9400 = 0.125 sphere, or H 1-096 9100 1 hemisphere = 6-283 19 units, or 2?r 0-798 1799 = 4. spherical right angles 0-602 0600 = 0.5 sphere, or M 1-698 9700 1 sphere = 12.566 4 units, or 4* 1-099 2099 8. spherical right angles 0-903 0900 44 = 2. hemispheres 0-301 0300 90 GRADES; SLOPES; INCLINES. GRADES; SLOPES; INCLINES. (Angle; length -4- length.) Grades are indicated in terms of so many different kinds of units, some of which involve trigonometric relations, that the relations between some of them become complicated. Those given in the following table are mathe- matically correct and are strictly proportional; they may therefore be used like those in the other tables; for instance, a 1% grade from the table is 52.8 feet rise per mile, hence a 5% grade will be 5 times this; or from the table, 1 foot p t er mile is a 0.018 9% grade, hence 100 feet per mile will be a 1.89% grade. Much unnecessary labor and confusion would be avoided if grades were uniformly represented in percent, that is, in the rise per hundred. There seems to be a tendency to adopt this unit generally. In the following table all the relations are given in terms of the percent unit as a basis ; the relation between any two others is readily found by reducing both to the same percentage value. In using percentage values it should be remembered that they mean the rise per hundred, and it may therefore be necessary sometimes to multiply or divide by 100 when the actual or total distances or rises are involved. Thus a rise of 12 feet in 600 feet is a 2% grade, as the 600 feet must first be divided by 1OO to reduce it to hundreds, before dividing it into 12. Or if the total rise on a 2% grade is given as 12 feet, it means 12-^2 = 6 feet per hundred, which must therefore be multiplied by 100 ! total dis irises from the incorrect way in which perc _ are not infrequently written. Thus fifty percent should be written 50% to get the total distance. Much confusion arises from the incorrect way in which percentage values and not .50%, which latter means half of one percent, or fifty huridredths of one percent. Much confusion also arises from the fact that sometimes the sloping or inclined distance is meant instead of the horizontal distance, and gener- ally it is not stated which one is understood. In referring to profiles or to distances on a map, the horizontal distance is always understood, thus involving the tangent of the angle; while in formulas for the traction on grades, or when the distances are the actual lengths of track or road, the sloping or inclined distance is generally irmplied, as the traction formulas then become simpler; they then involve the sine of the angle. The pres- ent table gives the correct reduction factors for both. Some of these are the same in both systems; but it should be understood that a grade of a given percent based on horizontal distances is slightly different from a grade of the same percent based on sloping distances; the latter is always the larger angle or steeper grade. The difference is however generally negligibly small for all but exceptionally steep grades; up to a 14% grade, which is about the limit for traction on rails and for the usual roads, the difference is less than 1%. The following reduction factors are the same whether the units are based on the horizontal or on the sloping distances. Logarithm 1 inch per mile [in/ml] =0.001 578 28% 3-198 1849 1 foot per mile [ft/ml] = 0.018 939 4% 2-277 3661 1 %o 0.1% 1.000 0000 1 per mil [% ~\ 0.1% 1. 000 0000 1 millimeter per meter [mm/m] = 0.1% 1-000 0000 1 foot per thousand feet [ft/M] = 0.1% 1-000 0000 1 foot per 1OO feet [ft /C] 1.% 0-000 0000 1 foot rise per foot [ft /ft] 100.% 2-000 0000 1% =633.6 inches per mile 2-801 8152 " = 52.8 feet per mile 1-722 6339 " = 10. % 9 , or 10 per mil.; 1-0000000 " = 10. millimeters per meter ' 1-000 0000 " = 10. feet per thousand feet 1-000 0000 " = 1. foot per hundred feet.. 0-000 0000 " = 0.01 foot rise per foot 2-000 0000 GRADES; SLOPES; INCLINES. 91 n miles per foot rise = 1/n feet rise per mile. = (0.0189394 -s-n)%. n % =(0.018 939 4 -*-n) miles per foot rise. n feet (rise) per mile = 1/n miles per foot rise. n feet per foot rise = 1/n feet (rise) per foot. = (iop/n)%. n% = (I00/n) feet per foot rise. n loot (rise) per foot = 1/n foot per foot rise. The following reduction factors are only for units based on the hori- zontal distances. n% = 7: % based on sloping distances. n% = 100Xsin. n degrees rise = (100 tan n)%. n% = number of degrees whose tangent is 0.01 n. If n is the tangent of angle, then the rise in % = 100n. If n is the rise in %, then the tan = O.Oln. If n is the sine of angle, then the rise in % = Or find from a vi -n 2 table the tangent corresponding to this sine, then the rise in % = 100 X tan. If n is the rise in %,then the sine of the angle = , . Or Vl00 2 + n 2 the sine may be found from a table as that corresponding to tan = 0.01n. The following reduction factors are only for units based on the sloping distances. n% = % based on horizontal distances. V100 2 -n 2 n% = 100Xtan. n degrees rise = (100 sin n)%. n% = number of degrees whose sine is O.Oln. If n is the sine of angle, then the rise in % = 100 n. If n is the rise in %, then the sine = 0.01 n. ' If n is the tangent of angle, then the rise in %= 100n _. Qr find from a table the sine corresponding to this tangent, then the rise in % = 100 X sin. If n is the rise in %, then the tangent of the angle = 7^ l Or Vl00 2 -n 2 the tangent may be found from a table as that corresponding to sine = O.Oln. Approximate: for small angles and for most engineering calculations concerning grades, the sine and the tangent are very nearly equal, hence the simpler of the formulas given above can in most cases be used for both, and no tables are then necessary. Up to a 14% grade (about 8 degrees) the error made thereby is less than 1%. The actual values given below in the fifteen-column table avoid the calculations with the above reduction factors. Intermediate values suffi- ciently accurate for most purposes may be found from this table by ordinary interpolation. 92 GRADES; SLOPES; INCLINES. For Sloping Distances Only. Sine Functions. Equivalent Percent Based on Horizontal Distances. OOi-HCOCO r-H>COt>"3 i"~ Ci i-< "*< l> OOOO i-H*-H-MOO^lO COr^XC5r-l i-t i-H r-t r-l r-t 1 1 SO-HCO^) >-lt>.COl> OOOO T-Hrt(NCClO l^CTlrH^t> OOOOO OOOOO OOr-Hr-lrH i-i(MCO^tiO Ot^OOO5O i-H(NCOrfiO OOOOO OOOOO OOOOO a o3 .-KMCOTFiO O1>XO;O r-KNCO-^iO OOOOO OOOOO OOOOO !si -tOiCCt^r>.QOoo For Horizontal Distances Only. Tangent Functions. Equivalent Percent Based on Sloping Distances. o>O5OiC5 co oo i> co 10 O5CJOOCOX O O5 O5 Oi C5 O O5 Ol O5 O r- C<| CO ^ i QQ OO5t^-^ OiCOiO^tiO OOiOSOS XQOI^CC'O CO'-iOJCOCO OOOiO5O5 O5O5O5O5C5 O5 C5 CO 00 00 i-iCt^ rH O5COCOOOCO CO COUO r-i OCOWMCO CO COQO I-H O5COO3-^CO CO COT-iGOOO iOO'tCOO l^.OJOOCOO OJ O5 CO Tf t> CO >O t^. -^ CO i X O 0 id CO OiT^COl>t>. 1-1 1>. CO 1-1 CO t^iO^fCO^ r^oooo 66 coo oocoo OOOOO OOOOO OOOOO OOOOO OOOOO OOOOO i s ig COO^fX i-HiOCOOOO OCOCOOJN lOOiOi-^O 1-1 CO (N 1^. t>. 13*1 i-i(NCO^O COt^XOSO ^(NCOTt^iO OOOOO OOOOO OOOOO Per Mil (%o) or Millimeters per Meter, or Feet per 1000 Feet. 2gg^SSgSi82^^^g Percent (%) or Feet per 100 Feet. rH(NCO^>O CO|>XO5O rH(NCOTflO TIME. 93 TIME. There are in use two different systems of units of time, the mean solar time and the sidereal time. The mean solar time is based on the appar- ent motion of the sun with respect to the earth, that is, on the motion of the earth with respect to the sun. The sidereal time is based on the appar- ent motion of the stars with respect to the earth, that is, on the motion of the earth with respect to the stars. The mean solar time is that indicated by the clocks in common use; it is that, which is furnished by the U. S. Naval Observatory and is used in ail physical research. In all derived units in use, such as velocities, forces, power, electrical quantities, etc., which involve the element of time, this mean solar time is understood to be meant. According to the National Bureau of Standards, Lord Kelvin (formerly Sir William Thomson), Prof. Walter S. Harshrnan the Director of the Nautical Almanac, Prof. R. S. Woodward, and other authorities, it is the second of mean solar time which is the unit of time in tho ccntimctcr-gram-second system of units. Mean solar time is always understood to be meant in all designations of time un- less otherwise specifically stated. In all the units in this book which involve time, the mean solar time is understood. Sidereal time is used only for astronomical purposes. It is considered as possessing more nearly the essential qualification of a standard unit, namely invariability, but the mean solar time has been adopted instead. However, the relation between the two is known to such a degree of pre- cision, that mean solar time is also perfectly uniform. Tables for inter- changing sidereal and mean solar time are given in the American Ephemeris. Besides these two sets of units of time, there are probably hundreds of other terms used to designate various periods of time, chiefly different kinds of years, months, cycles, etc. These are generally used only in astronomy and history, and many of them are obsolete; they have there- fore not been included in tha following table. All the important values in the table have been checked through the kindness of Prof. Walter S. Harshman, Director of the Nautical Almanac; many of these have been accepted as the best obtainable values, and aa such are used in the American Ephemeris and Nautical Almanac. The International Bureau of Weights and Measures has established an important distinction in the notation of time. When it refers to the epoch, that is, the date or time of day, the reference letters are used as indices; and when it refers to the duration of a phenomenon, they are on the same line with the numbers. For instance, an experiment began at 2 h 15 m 46" lasted 2h 15m 46s, and ended at 4 h 31 m 32 s . Standard Hail way Time in the United States and Canada. On November 18, 1883, a new system of railway time called "Standard Time " went into effect on most of the railroads of the United States and Canada, and has since been adopted by most of the principal cities. According to this system the country is divided into five strips or zones running north and south, each 15 in width. Throughout each strip the time of the clock is the same, and it differs from that in the two neighboring strips by pre- cisely one hour; for instance, when it- is 4 o'clock in one strip it is 5 o'clock in the next one east, and 3 o'clock in the next one west. The following table gives the longitude of the middle line of each strip; the time in that strip is the correct time for that longitude. The actual lines midway between these, where th? time changes by one hour, do not always correspond exactly with the theoretical ones, for obvious reasons. The table also gives the name by which that time is designated in each strip and the conventional color by which it is indicated on maps. Eastern time is exactly 5 hours later than Greenwich time. Me f a romG?len < w 3 icY. eSt Name of Standard Time ' Conventional Color. 60 Intercolonial time Brown 75 Eastern time Red 90 Central time Blue 105 Mountain time Green 120 Pacific time Yellow 94 TIME. TIME. Mean solar time is the time in universal use except in astronomy. Un- less otherwise stated, mean solar time is understood in this table. * Accepted by Prof. Walter S. Harshman. Director of the Nautical Almanac. 1 sidereal second -0.997 269 57* second (mean solar) I- 99^8 126* 1 second [s] (mean solar) = 1.002 737 91* sidereal seconds 0-001 1874* 1 sidereal minute = 60.* sidereal seconds. 1 minute [min or m] (mean solar) = 60.* seconds (mean solar). 1 sidereal hour = 60.* sidereal minutes, or 3 600.* sidereal seconds. 1 hour [h] (mean solar): = 60.* minutes (mean solar), or 3 600.* seconds (mean solar). 1 sidereal day : = 86 164.1* seconds (mean solar). = 86 400.* sidereal seconds. = 1 440.* sidereal minutes. = 24.* sidereal hours. 23. h, 56. m, 4.091* s (mean solar). = 0.997 269 57* day (mean solar) I.ggg 8126* = 1 mean solar day less 3 m and 55;909* s (mean solar). 1 day (mean solar) : 86 400.* seconds (mean solar). = 86 636.555* sidereal seconds. 1 440.* minutes (mean solar). = 24.* hours (mean solar). = 24. sidereal hours, 3m, 56.555* s sidereal time. = 1.002 737 91* sidereal days 0-001 1874* = 1 sidereal day plus 3. minutes 56.555* seconds sidereal. 365.242 20* mean solar days = 366. 242 20* sidereal days. 1 civil or calendar day: = 1 day (mean solar) ; is reckoned from mean midnight to mean midnight. An apparent solar or a natural day is variable. An astronomical or nautical day is reckoned from mean noon to mean noon. I week = 7. days (mean solar). 1 anomalistic month ; = 27. days, 13. hours, 18. minutes, 37.4 seconds (mean solar ?). 1 civil or calendar month [mo]: = 28, 29, 30, and 31 days (mean solar). = aprx. Ma year (mean solar). 1 average lunar or synodic month: = 29. days, 12. hours, 44. minutes, 2.8* seconds (mean solar). = 29.530 59* days (mean solar). 1 average sidereal month = 27. days, 7. hours, 43. minutes, 11.5* seconds. 1 lunar year = 354. days, 8. hours, 48. minutes, 34. seconds (mean solar), = 12. lunar or synodic months. 1 common lunar year = 354. days. 1 year [yr] (mean solar): = 366.242 20* sidereal days. = 365.242 20* days (mean solar). 365. days, 5. hours, 48. minutes, 46.* seconds (mean solar). = 1. sidereal year less 20. minutes, 26.9* seconds (sidereal). 1 sidereal year: = 366.256 399 2* sidereal days. = 365.256 360 4* days (mean solar). = 365. days, 6. hours, 9. minutes, 9.5* seconds (mean solar), 1. mean solar year plus 20. rnin, 23.6* sec (mean solar). 1 civil or calendar year, ordinary = 365.* days (mean solar), leapt =366.* days (mean solar). 1 ccmmon year = 1. ordinary civil or calendar year. 1 Julian year = 365.25* days (mean solar). t A leap year is one whose number is divisible by 4, except when the number ends in two ciphers, then it must be divisible by 400. DISCHARGES; IRRIGATION. 95 1 Gregorian year = 365. days, 5. hours, 49. minutes, 12. sec (mean solar). tropical or natural year = l. year (mean solar). anomalistic year = 365. d, 6. h, 13. m, 53.* s (mean solar). A legal year is obsolete. solar cycle = 28.* Julian years. century t= 100.* civil or calendar years. lunisolar cycle = 532. years. 1 milleuium = 1 000. calendar years. DISCHARGES; FLOW of WATER; IRRIGATION UNITS j VOLUME and TIME. (Volume -* time.) (See also Volumes.) Discharges (as of water) are generally measured in terms of some volume per second as cubic feet per second, gallons per second, cubic meters per second, etc. The relations between them are therefore the same as between those respective volumes, which see under the units of Volumes. The same is true if they are given per minute. When one is per second and the other per minute, reduce either to the same time as the other and then use the table of volumes. The only unit differing from these is the miner's inch which is sometimes used in the western United States for measuring the flow of water in streams, particularly for mining. It is a somewhat vague unit, generally insuffi- ciently denned, and has so-called "legal" values in different States, which values differ. Its value varies from about 1.20 to about 1.76 cubic feet per minute; the mean is generally taken as about 1.5; for this value the reduc- tion factors are given in the following table. Aprx. means within 2%. Logarithm 1 miner's inch = 1.5 cubic feet per minute 0-176 0913 = 0.187 013 gallon per second. Aprx. 3/i 6 .. . 1-271 8717 = 0.025 cubic foot per second. or*4o .... 2-397 9400 = 0.000707 925 cb. meters per sec. Ap. ^ooo. 4-849 9875 The acre-foot is sometimes used as a unit for measuring irrigation ; it means a body of water 1 acre in area and 1 foot in depth. It is therefore really a true unit of volume. Its chief equivalents are; 1 acre-foot = 325 851. gallons. = 43 560. cubic feet. = 1 613.33 cubic yards. " = 1 233.49 cubic meters. t The twentieth century is generally assumed to have begun with Janu- ary 1,1901. ELECTRIC AND MAGNETIC UNITS. ELECTRIC and MAGNETIC UNITS. General Remarks. In the following tables C.G.S. refers to the centi- meter-gram-second system of units; elmg means electromagnetic; elst means electrostatic; v means the velocity of light in air, which is here taken as equal to about 3X10 10 centimeters per second, which value has been included in the logarithms of those relations which involve this v; the word " about," used with such derivatives of this velocity, means that they include whatever inaccuracy there is in this velocity. Aprx. means that the simple fractions given are correct within 2%. The values of the derived figures in these tables are generally given to six significant figures and seven-place logarithms, even though the original fundamental data may sometimes not warrant such accuracy; the object is to enable the correc- tions due to any subsequently adopted more accurate fundamental values to be made by mere proportion instead of by complete recalculations. The fundamental values of the electrical units used in these tables are those adopted by the International Electrical Congress at Chicago. Besides the exact values in terms of the C.G.S. units, that Congress also defined certain concrete units as the closest approximations which existed at that time; these concrete units are the ones in use in practice at pres- ent. In calculating the relations between the electrical and the mechan- ical and thermal units like foot-pounds, horse-powers, heat-units, etc., for these tables, it had to be assumed that these concrete units are exactly equal to those defined in terms of the C.G.S. units which they represent, as it would otherwise be impossible to calculate those relations until the dif- ferences which may exist between the concrete and the exact values are known; such relations therefore must always involve these differences if they exist ; whatever they may be they are absolutely negligible in all but the most refined physical research. The absolute values of the electric units are given for both the usual electromagnetic system and the less usual electrostatic system, but the magnetic units have been confined to those in the electromagnetic system, as the others are rarely if ever used; the dimensional formulas and interrelations of the latter are given in the table of Physical Quantities in the Introduction. In a paper read before the American Institute of Electrical Engineers in July, 1903, Dr. A. E. Kennelly suggests the prefixes ab- or abs- to the names volt, ohm, etc., to designate the corresponding absolute electromagnetic units; thus abvolt, absohm, etc., mean the absolute or C.G.S. electromagnetic units. Similarly the prefix abstat- designates the corresponding absolute electrostatic units. The suggestion seems a good one, as it is often very convenient to have easily remembered specific names for the absolute electrical units. The American Institute of Electrical Engineers has adopted the rule that vector quantities when used should be denoted by capital italics. This applies chiefly to electromotive force, current, and impedance. Owing to the numerous important and very useful interrelations between the various electric and magnetic quantities, many of whbh are simple unit relations like Ohm's law or Joule's law, there have been added to the tables of these units numerous formulas giving the rela- tions between quantities measured in terms of various different electric and magnetic unite, many of which will frequently be found useful; they are correct numerically also, and may be used like any other formulas. As there is a very large number of such relations, only those likely to be used are given; the others can be readily derived from them. Such relations are usually given in the form of algebraic formulas, but the method adopted here is preferred because it shows directly for what particular units the relations are numerically correct. Treatises on alternating currents should be consulted for the limiting conditions under which these relations apply to alternating or other periodically varying quantities. The author is indebted to Prof. W. S. Franklin for important suggestions concerning the quantitative interrelations of electric and magnetic units when their intensities are varying or alternating. Also to Dr. Frank A. Wolff, Jr., Assistant Physicist of the National Bureau of Standards, for his kindness in endorsing the correctness of a number of the more important derived values of the electrical units. RESISTANCE; IMPEDANCE; REACTANCE. 97 Mean, effective and maximum values in periodically varying; functions. With alternating currents the instantaneous values of both the electromotive force and the current vary continually. When the arithmetical mean or average of all these momentary values of the electro- motive force or current is taken, it is called the mean value; this value is seldom if ever used; for a true alternating current the algebraic mean is always zero for one whole period, hence the arithmetic mean of the whole period or the algebraic or arithmetic mean of half a period is used instead. When the square root of the mean square is used it is called the effective value; this is the value almost always used and is the one meant when not otherwise specih'ed; it is this value which corresponds to the electromotive force or current of a direct current circuit in calculations of the energy. Similarly, there is a mean and an effective magnetomotive force, mag- netizing force, flux, flux density, etc., but as the arithmetic mean values of these are of less importance the effective values are the ones gen- erally understood unless otherwise specified. By the maximum value of any of these is meant the greatest value reached in one period ; this value is sometimes the important one, notably in the strain on the insulation, which depends on the maximum electromotive force, or in the calculation of the hysteresis loss, which depends on the maximum flux density. These terms, mean, effective, and maximum, are not used in practice in connec- tion with the power in watts; the mean watts are always understood and are equal to the product of the effective volts, the effective amperes, and the cosine of the angle of phase difference. The following table gives the relations between the maximum, effective, and mean values when the variations follow the sine law. By mean value is here meant that for a half period, as the algebraic mean for a whole period is always zero. Aprx. means within 2%. Mean value (half period). Logarithm -effective value X 0.900 316 (or(2-^ ir)\/2). Aprx. subt. 10%.. 1.954 3951 = maximum va'ueX 0.636 620 (or 2/7:). Aprx.f's ............ 1-803 8801 Effective value: mean value X 1.11072 (or (w-J-4)\/2). Aprx. *% .......... 0-045 6049 maximum value X 0.707 107 (or W2). Aprx. %o ......... 1-849 4850 Maximum value: = mean value XI. 570 80 (or' 7T/2). Aprx. 1^7 ............... 0-1961199 = effective value X 1.414 21 (or \/2). Aprx. l % ............ 0-150 5150 RESISTANCE [R, r]; IMPEDANCE [Z, z]j REACT- ANCE [X, x]. (Electromotive force ~ current ; lengthX resistivity ~ cross-section.) These units are used to measure the opposition offered to the passage of a current through a circuit or part of a circuit. The greater the resistan e the greater this opposition. It is similar to the mechanical friction of a moving body, like that of water in a pipe, although the analogy does not extend to the numerical laws, nor are there any specific units for mechan- ical frictional resistance, as there are for electrical resistance. According to Ohm's law the resistance in ohms is equal to the electromotive force in volts divided by the current in amperes; or according to Joule's law it is equal in ohms to the power in wiitts divided by the square of the current in amperes; or to the square of the voltage divided by the watts. These relations apply to direct currents, and refer to the true or "ohmic" resist- ance of the conductor itself, which is dependent only on the material, size, and temperature of the conductor. They also apply to alternating cur- rents when there is no reactance in circuit (caused by inductance or capacity) in which case the effective values of the electromotive force and current, and the true watts, are meant. Resistance refers to a given circuit or part of a circuit, while resistivity (which see) refers to the specific resistance of the material irrespective of the size or shape of the circuit. 98 RESISTANCE; IMPEDANCE; REACTANCE. Reactance, although not a resistance, and impedance, which is often called apparent resistance, and is a resistance combined with a react- ance, are both correctly measured and expressed in the same units as resist- ance, namely, ohms, although the reactance depends on the inductance and capacity of the circuit and on the frequency of the alternating current, and is therefore not true resistance. Both of these terms are limited chiefly to alternating current circuits. The impedance in ohms is equal to the effec- tive electromotive force in volts, divided by the effective current in amperes, regardless of what the phase difference may be (that being embraced by the vector character of the impedance). It is also equal to the square root of the sum of the squares of the resistance and the reactance, all being ex- pressed in ohms. For further explanations concerning the calculations of alternating current circuits, reference should be made to treatises on this subject. In direct current circuits, resistance (true or "ohmic") in ohms is the reciprocal of conductance in mhos; similarly, in alternating current cir- cuits impedance in ohms is the reciprocal of admittance in mhos. When various parts of a circuit are connected in series their total resistance or impedance in ohms is simply the sum of all the individual resistances or impedances in ohms. If they are in parallel or multiple, however, it is best to add their conductances or admittances in mhos and then take the reciprocal of this sum, which will then be the total joint resistance or im- pedance in ohms. The unit now universally used is the ohm, by which is here meant the international ohm of the International Congress of 1893 at Chicago, based on the value 10 9 C.G.S. units, and represented by the resistance of a column of mercury at C., 106.3 centimeters long, weighing 14.452 1 grams, and having a uniform crpss-section. It was made legal in this country by Con- gress in 1894 and is adopted by the National Bureau of Standards. The Reichsanstalt has also adopted this value, and there is therefore uniformity in the resistance standards used in those two institutions. The National Bureau of Standards at present uses 1-ohm manganin resistance standards, verified from time to time at the Reichsanstalt, so that the results of the Bureau are at present expressed in terms of the particular mercurial resist- ance standards of the Reichsanstalt. The construction of primary mercurial standards is about to be undertaken by the Bureau. The British National Physical Laboratory is also undertaking the construction of such standards, and the present definition of the unit of resistance in that country in terms of the Board of Trade ohm and the B. A. units, may be replaced by one in terms of the primary mercurial standard. In some of the relations with other units, such as the absolute or those of energy, the value of the international ohm as above defined is in the following tables assumed to be equal to the theoretical value, namely, 10 9 electromagnetic C.G.S. units, which value is sometimes called the true ohm. In the relations between the international ohm and the other mer- cury units given in the following table, it is assumed that the uniform cross- section referred to in the definition of the international ohm is one square millimeter. The legal ohm was a mercury standard in use for a number of years prior to the adoption of the international ohm ; it differs from the latter only m that the length is 106 centimeters and that its cross-section is defined to be 1 sq. millimeter, while with the international ohm it is the weight which is defined. The Siemens unit formerly used is the resistance of a column of mer- cury, at C., having a cross-section of one square millimeter and a length of one meter. It was never legalized, but has often been used as a well- defined standard of reference. The British Association unit, or B. A. unit, was formerly the standard in Great Britain ; the legal standard now used there is a Board of Trade coil or ohm equal to the international ohm, on the assumed relation that 1 inter- national ohm = 1.013 58 B. A. units. The relation accepted by the National Bureau of Standards is the mean of the relations determined by Glazebrook and by Lindeck in 1892, namely, 1 international (Reichsanstalt) ohm = 1.013 48 B. A. units. The primary standards of the Reichsanstalt, in terms of which this value is given, are themselves subject to various sources of error involved in the construction of such standards. RESISTANCE; IMPEDANCE; REACTANCE. 99 The electromagnetic C.G.S. unit (or absolute unit) is the resistance through which 1 C.G.S. unit of electromotive force will cause 1 C.G.S. unit of current to flow. The electrostatic C.G.S. unit (or absolute unit) is similarly denned with respect to the electrostatic units of e.m.f. and current. RESISTANCE; IMPEDANCE; REACTANCE. ** Accepted by the National Bureau of Standards. * Checked by Dr. Frank A. Wolff, Jr., Asst. Phys. National Bureau of Standards. Aprx. means within ,2%. By "ohm" is here meant the international ohm, unless otherwise stated, v is the velocity of light. Logarithm 1 CGS unit [elmg]= 1 absohm 0-000 0000 = 0.001 microhm 3-000 0000 = 10~ 9 ohm 9-000 0000 = l/v 2 CGS unit (elst). About % X lO" 20 .. . 21-0457575 1 ahsohm = 1 CGS unit (elmg) 0-000 0000 1 microhm = 1 000. CGS units (elmg) 3-000 0000 = 0.000 001 ohm 6-000 0000 1 Siemens unit (S.U.) = 0.940 734* ohm. Aprx. subtract 6%. 1-973 4667 1 Hi it ish Association unit [B.A.U.]: = 0.986699** ohm. f Aprx. subtract 1% 1-9941848 = 0.986 602* ohm.t Aprx. subtract 1% 1.994 1420 I eg-al ohm = 0.997 178* ohm. Aprx. 1 ' 1-998 7726 1 ohm= 10 1J CGS units (elmg).. . 9-000 0000 10 microhms 6. 000 0000 " =1.06300* Siemens units. Aprx. add 6% 0-0265333 " =1.01358* British Association units. t Aprx. add 1%. 0-0058580 " =1.01348** British Ass'n units. t Aprx. add 1% 0-0058152 1 = 1.002 83* legal ohms. Aprx. 1 0-001 2274 ' ' = 10~ megohm 6-000 0000 = I0 9 /v 2 CGS unit (elst). About Vo X 10" 11 12-045 7575 1 international ohm = 1 ohm, which see above. 1 true ohm = 10 9 CGS units (elmg) 9-000 0000 1 ohm of Reichsanstalt = 1 ohm 0-000 0000 1 Hoard of Trade (Brit.) ohiurT = 1.013 58 Brit. Ass'n unit. Aprx. add 1% 0-0058580 1 ohm 0-000 0000 1 megohm = 10 ohms 6-000 0000 . 10 15 /v 2 CGS units (elst). About % X 10~ 5 6-045 7575 1 CGS unit (elst) = v 2 CGS units (elmg) About 9X 10 20 . 20-954 2425 = 7> 2 X10- 9 ohms. About 9X10 11 11-9542425 1 abstatohm 0-000 0000 1 ahstatohm = 1 CGS unit (elst) 0-000 0000 1 absolute unit = 1 CGS unit either elst or elmg 0-000 0000 The relations to other measures are as follows: Ohms = volts -r- amperes. " = volts X seconds -r- coulombs. = volts 2 -T- watts. = watts -r- amperes 2 . = watts X seconds 2 -r- coulombs 2 . = volts 2 X seconds -=- joules. = joules -r- (amperes 2 X seconds). = joules X seconds -J- coulombs 2 . t Mean of Glazebrook's and Lindeck's values of B. A. units in terms of Reichsanstalt primary mercurial standards, accepted by the National Bureau of Standards. J Legal relation in Great Britain. Treatises on alternating currents should be consulted for the limiting conditions under which these relations apply to alternating or other period- ically varying quantities. 1 Latest value, 1903: 1 Reichsanstalt ohm = 1.000 165 Board of Trade ohms. 100 RESISTANCE; IMPEDANCE; REACTANCE. Ohms resistance: = 1 -4- mhos conductance. For direct currents only. = henrys -Mime constant in seconds. = induced volts -r- (rate of change of amperes per second X time con- stant in seconds). = henrys X final amperes X applied volts -^( joules of kinetic energy of the current X 2). = applied volts X\/[henrys-r-( joules of kinetic energy of the current X2)]. = applied volts 2 X time constant in seconds -=-( joules of kinetic energy of the current X 2). = joules of kinetic energy of the current X 2 -* (time constant in seconds X final amperes 2 ). = (maxwells X number of turns) -r- (final amperes X time constant in seconds X 10 ). When the flux is due only to the current, as in self-induction. Microhms resistance: -=l-hmegamhos conductance. For direct currents only. For alternating current circuits :f Ohms resistance : = volts energy component of e.m.f. -=- total amperes. = watts -r- amperes 2 . = V(ohms impedance 2 ohms reactance 2 ). = \/[(l -r-mhos admittance 2 ) ohms reactance 2 ]. = mhos conductance X ohms impedance 2 . = mhos conductance -j- mhos admittance 2 . =*\/( a Pphed volts 2 induced volts 2 )-;- amperes. = applied volts -s- (amperes X VlXtime constant in seconds X frequency X 6.283 19t) 2 + l ) = V[(applied volts *- amperes) 2 - (henrys XfrequencyX 6. 283 19t) 2 ]. = vTohms impedance 2 (henrys X frequency X 6.283 19$ ) 2 ]. ^induced volts -*- (time constant in seconds X amperes X frequency X 6.283 191:). Ohms reactance : = volts wattless component of e.m.f. -r- total amperes. = \/(ohms impedance 2 ohms resistance 2 ). = VT(1 -s-mhos admittance 2 ) ohms resistance 2 ]. = ohms impedance 2 X mhos susceptance. = mhos susceptance^- mhos admittance 2 . = ohms magnetic reactance ohms capacity reactance. = ( henrys XfrequencyX 6. 283 19J)- [0.159 155 -^-(farads X frequency)]. Ohms magnetic reactance = henrys XfrequencyX 6. 283 194 Ohms capacity reactance = 0.159 155 -K farads X frequency). = 159 155. -r- (microfarads X frequency). Ohms impedance = total effective volts -*- total effective amperes, ^^(ohms resistance 2 -}- ohms reactance 2 ). = 1 -r-mhos admittance. = l-*-\/(inhos conductance 2 + mhos susceptance 2 ). = \/(ohms resistance -T- mhos conductance). = v / (ohms reactance -T- mhos susceptance). = \/[ohms resistance 2 -f (henrys X freq. X 6.283 19]; ) 2 ]. t Treatises on alternating currents should be consulted for the limiting conditions under which these relations apply to alternating or other period- ically varying quantities. t Or 2*. Aprx. % x 10. Log 0-798 1799- Or 1-^2*. Aprx. %-*-10. Log 1.201 8201- RESISTANCE AND LENGTH. 101 RESISTANCE and LENGTH, for the, SAME CROSS- SECTION. (Redstance-f-lcng'th.) , For wires of the same cross-section and material the following relations exist. Aprx. means within 2%. Logarithm 1 ohm per mile: = 0.621 370 ohm per kilometer. Aprx. ^ 1-7933503 = 0.000 621 370 ohm per meter. Aprx. V s * 1 000.. . . 4.793 3503 = 0.000 189 394 ohm per foot. Aprx. 19-^-100 000 . . 4.277 3661 1 ohm per kilometer : 1.609 35 ohms per mile. Aprx. add 6 /io 0-206 6497 0.001 ohm per meter 3-000 0000 = 0.000 304 801 ohm per foot. Aprx. 3 -5-10 000 4-484 0158 1 ohm per meter = 1609.35 ohms per mile. Aprx. 1 600 .... 3-2066497 = 1 000. ohms per kilometer 3-000 0000 = 0.304 801 ohm per foot. Aprx. /io 1-484 0158 1 ohm per foot = 5 280. ohms per mile. Aprx. 5 300 3-722 6339 -=3280.83 ohms per klm. Aprx. M X 10 000 . . 3-5159842 = 3.280 83 ohms per meter. Aprx. 10 /a 0-515 9842 RESISTANCE and CROSS-SECTION, for the SAME LENGTH. (ResistanceX cross-section.) When the lengths of wires of the same material are the same and the cross-sections are different, the relations between the respective compound units representing the product of the resistance and the cross-section are the same as the relations between the different cross-section units, which see under the units of Surface. Thus if the compound unit ohm-centi- meters 2 is the product of the resistance in ohms and the cross-section in square centimeters of any wire, and if similarly ohm-inches 2 i s the product of the oliJis and the square inches cross-section of another wire of the same length, then 1 ohm-centimeter 2 = 0.1 55 000 ohm-inch 2 , in which 0.155000 is the value of 1 sq. centimeter in sq. inches. These units are used for converting the values of resistivities from one unit to another (see also under units of Resistivity, below); thus if n is the resistivity of a material in ohm, circular mil, foot, units,, and it is required to change this into ohm, sq. mil, foot, units, multiply 'n by the value of 1 circular mil in square mils as given in the table of units of Surface, namely, 0.785 398. RESISTIVITY [p]\ SPECIFIC RESISTANCE. (Resistance X cross-section ~- length.) These units are used to measure the inherent quality of a material to resist the passage of an electric current. Resistivity differs from resistance in that the latter refers to the number of ohms of any given circuit or part of a circuit, and depends on the length, cross-section, and quality of the material; while resistivity refers only to the m-fi^ of the material itself and is always the same for the same material; it is the resistance of a unit amount of the material, like a cube of one centimeter, or a mil-foot, or a meter of one sq. millimeter section. It bears somewhat the same relation to resistance as the density of a material does to the weight of any given 102 RESISTIVITY; SPECIFIC RESISTANCE. amoiAit of it; the density is always the same for that material, being the weight of a unit of volume .ifrhiie the total weight of an actual piece depends upon the size of that piece. * As the resistivity or specific resistance is a quality of a material, i^s, valuesare usually given in tables of physical con- st put s^ the refistivitv may also be calculated from the resistance of any gi /on pi^.-e by multiplying this resistance in ohms by the cross-section and dividing by the length; the result will of course be different, depending upon the units used. The resistivity, being a property of a material, is the same for direct as for alternating currents. There are several units in use. The most rational one is the resistance in ohms between two parallel sides of a cube of one centimeter of the ma- terial. This unit is sometimes called the ohm-centimeter unit, or 1 ohm per cubic centimeter, or more correctly, 1 ohm-square-centimeter per centimeter; it will here be called the ohm, cubic centimeter, unit. For most conducting materials in use, except electrolytes, the resistivity when stated in these units is a very small number, hence it is often stated in microhms (millionths of an ohm) instead of ohms. The electromagnetic and the electrostatic C.G.S. units are the same as the above except that the resistance is stated in the respective C.G.S. units instead of in ohms. Another unit in common use is the meter-millimeter unit, that is, the resistance in ohms of a wire one meter long and one square millimeter in cross-section; it also has the name 1 ohm per meter per square milli- meter, or more correctly 1 ohm-square-millimeter per meter; it will here be called the ohm, square millimeter, meter, unit. This unit has the advantage that it generally involves less calculation than the cubic- centimeter unit, as the lengths and cross-sections are in practice usually stated in meters and square millimeters when the metric system is used. A more rational unit for circular wires, although apparently not in general use, is similar to this one except that the cross-section is a circle of one millimeter diameter, and is therefore equal to a "circular millimeter" as distinguished from a "square millimeter" (see explanation of circular units under units of Surface). This has the advantage of eliminating from the calculation for the usual round wires the troublesome factor n (or 3.141 59), because when the cross-sections are stated in circular units they are directly equal to the squares of the diameters. The unit usually used when the lengths are in feet and the diameters in mils (that is, thousandths of an inch), is the mil-foot or circular mil-foot, which is the resistance in ohms of a round wire one foot long and one mil in diameter. The cross-section then is one circular mil, and the cross- sections of other round wires are then equal to the squares of their diam- eters in mils (see explanation in preceding paragraph). This unit also has the name of 1 ohm per foot per circular mil pr per mil diameter, or more correctly, 1 ohm-circular-mil per foot; it will here be called the ohm, circular mil, foot, unit. A similar unit, though less frequently used, is the square mil-foot; it differs from the other only in that the cross-section is a square mil instead of a circular mil, and it is more convenient to use with bars of rectangular cross-section. It also has the name of 1 ohm per foot per square mil, or more correctly, 1 ohm-square-mil per foot; it will here be called the ohm, square mil, foot, unit. Sometimes resistivities are denoted in terms of that of mercury or pure copper as a basis. They are then mere relative resistivities or ratios of two resistivities, that of the material divided by that of mercury or copper, and are therefore not in terms of any real units, although they might be called mercury or copper units. In the following table the resistivities of mercury and of copper have been added to facilitate making calculations with such relative resistivities. The resistivity of mercury here used is that deduced from the definition of the concrete international ohm, namely, that a column of uniform cross-section, 106.3 centimeters long, weighing 14.4521 grams, has a resistance of one ohm at C. The cross-section is for this purpose assumed to be one square millimeter. For the resistivity of pure copper the Matthiessen value is still in use, namely 1.687 microhms for one centimeter length and one square centimeter cross-section, at 15 C., according to Prof. Lindeck of the Reichsanstalt. Pure copper as now made has a lower resistivity than this; according to Prof, Lindeck the value used in Germany (presumably under the authority of the Reichsan- stalt) is 1.667 in the same units. RESISTIVITY; SPECIFIC RESISTANCE. 103 Aprx. means within 2%. v is the velocity of light. Logarithm 1 CGS unit (elmg) = 10~~ 9 ohm, cubic centimeter, unit 9-000 0000 = 1/v 2 CGS unit (elst). About % X lO" 20 . . . 21.045 7575 1 ohm, circular mil, foot, unit: = 0.785 398 ohm, sq. mil, foot, unit. Aprx. 8 /io 1-895 0899 = 0.166 243 microhm, cb. cm, unit. Aprx. Y& 1-220 7433 = 0.002 116 67 ohm, circ. mm, meter, unit. Aprx. 2 Vio ooo- 3-3256534 = 0.001 662 43 ohm, sq. mm, meter, unit. Aprx. K * 100.. 3-2207433 1 mil-foot unit. See 1 ohm, circular mil, foot, unit. 1 ohm-circular-mil per foot. See 1 ohm, circular mil, foot, unit. 1 ohm per foot per circular.mil. See 1 ohm, circular mil, foot, unit. 1 ohm per foot per mil diameter. See 1 ohm, circular mil, foot, unit. 1 ohm, sq. mil, foot, unit: = 1.273 24 ohm, circular mil, foot, units. Aprx. J % 0-1049101 = 0.211 667 microhm, cb. cm, unit. Aprx. 21-^100.. . L325 6534 = 0.00269503 ohm, circ. mm, met. unit. Aprx. % -*- 1 000. 3-4305635 = 0.002 11667 ohm, sq. mm, met. unit. Aprx. 21 -4-10 000. 3-3256534 1 ohm-sq. mil per foot. See 1 ohm, sq. mil, foot, unit. ] ohm per foot per sq. mil. See 1 ohm, sq. mil, foot, unit. 1 microhm, cb. centimeter, unit: 1 000. CGS units (elmg). . . 6.01529 ohm, circular mil, foot, units. Aprx. 6.^. . 0-7792567 1/01 1 000. CGS units (elmg) 3-000 0000 .01529 ohm, circular mil, foot, units. Aprx. 6.... 0-7792567 = 4.72440 ohm, sq. mil, foot, units. Aprx. Hi X 100. . 0-6743466 = 0.012732 4 ohm, circ. mm, met. unit. Aprx. >s -5-10... . 2-1049101 0.01 ohm, sq. mm, meter, unit 2-0000000 10~ 6 ohm, cb. cm, unit 6-000 0000 1 microhm-sq. centimeter per centimeter. See 1 microhm, cb. cm, unit. 1 microhm per cubic centimeter. See 1 microhm, cb. cm, unit. 1 ohm, circular mm, meter, unit: = 472.440 ohm, circ. mil, ft, units. Aprx. 1 oo/ 21 . 2 -674 3466 = 371.054 ohm, sq. mil, ft, units. Aprx. ^ X 1 000 2-569 4365 78.539 8 microhm, cb. cm, units. Aprx. 80 1-8950899 = 0.785 398 ohm, sq. mm, meter, unit. Aprx. 8 /io : - 1-895 0899 = 0.000 078 539 8 ohm, cb. cm, unit. Aprx. 80-4- 1 000 000. 5-895 0899 1 ohm-circular-mm per meter. See 1'ohm, circular mm, meter, unit. 1 ohm per meter per circular mm. See 1 ohm, circular mm, meter, unit. 1 ohm, sq. mm, meter, unit: = 601.529 ohm, circular mil, foot, units. Aprx. 600. . . . 2-779 2567 = 472.440 ohm, sq. mil, foot, units. Aprx. j^i X 10 000. . 2-674 3466 = 100. microhm, cb. cm, units . 2-000 0000 = 1.273 24 ohm, circular mm, meter, units. Aprx. 1% . .. 0-104 9101 = 0.000 1 ohm, cb. centimeter, unit ; 4-000 0000 1 ohm-sq. mm per meter. See 1 ohm, sq. mm, meter, unit. 1 ohm per meter per sq. mm. See 1 ohm, sq. mm, meter, unit. 1 ohm. cb. centimeter, unit: 10 CGS units (elmg) 9-000 0000 = 1 000 000. microhms, cb. cm, units 6-000 0000 = 12732.4 ohm, circ. mm, met. units. Aprx. i/s X 100 000. 4-1049101 = 10 000. ohm, sq. mm, meter, units 4-000 0000 = 10 9 A 2 CGS unit (elst). About VQ X 10~ n 12-0457575 1 ohm-sq. cm per cm. See 1 ohm, cb, centimeter, unit. 1 ohm per cubic centimeter. See 1 ohm, cb. cm, unit. 1 megohm, cb. centimeter, unit: = 1 000 000. ohm, cb. cm, units 6-000 0000 1 CGS unit (elst) = v 2 CGS units (elmg). About 9 X 10 20 20-954 2425 = v2 X 10- 9 ohm, cb. cm, units. Abt. 9X10 11 11-9542425 " =v 2 X 10~ 15 megohm, cb. cm, units. About 9X 10 5 5-954 2425 104 RESISTIVITY; CONDUCTANCE. Logarithm Resistivity of copper: f 10.027 5 ohm, circular mil, foot, units 1-001 1923 7.875 57 ohm, sq. mil, foot, units 0-896 2822 1.667f microhm, cb. cm, units 0-221 9356 = 0.021 224 9 ohm, circular mm, meter, unit 2-326 8457 = 0.017 720 2 times that of mercury 2-248 4689 = 0.016 67 ohm, sq. mm, meter, unit 2-221 9356 = 0.000001667 ohm, cb. cm, unit 6-2219358 Resistivity of copper (Matthiessen): J 10.147 8 ohm, circular mil, foot, units 1-006 3718 7.970 06 ohm, sq. mil, foot, units 0-901 4617 1.687J microhm, cb. cm, units 0-227 1151 = 0.021 479 5 ohm, circular mm, meter, unit 2-332 0252 = 0.017 932 8 times that of mercury 2-253 6484 = 0.016 87 ohm, sq. mm, meter, unit 2-227 1151 = 0.000 001 687 ohm, cb. cm, unit 6-227 1151 Resistivity of mercury : 565.879 ohm, circular mil, foot, units 2-752 7234 444.40 ohm, sq. mil, foot, units 2-647 8133 94.073 4 microhm, cb. cm, units 1-973 4667 56.432 7 times that of copper 1.751 5311 55.7637 times that of copper (Matthiessen). . . . 1-746 3516 1.197 78 ohm, circular mm, meter, units 0-078 3768 = O.94O 734 ohm, sq. mm, meter, unit 1-973 4667 = 0.000 094 073 4 ohm, cb. cm, unit 5-973 4667 The relations of resistivity to other measures are as follows: Resistivity (in ohm.cb. cm, units) = Is- conductivity (in mho, cb. cm units). = ohmsXsq. cm section * cm length. CONDUCTANCE [G, g]; ADMITTANCE [Y, y]j SUS- CEPTANCE [B, b]. (Current -=- electromotive force ; 1 -T- resistance ; cross-section X conductivity -*- length.) These units are used to measure the quality of tne resistances are generally used by preference, wnen, However, tnere are several circuits in parallel or multiple arc, the calculation of their joint action is simpler if made with conductances, as their joint conductance is then merely the sum of the individual conductances, while when resist- ances are used the joint resistance is equal to the reciprocal of the sum of the reciprocals of the individual resistances. For direct current circuits, or when there is no reactance in alternating current circuits, the conductance in mhos is equal to the reciprocal of the resistance in ohms. It follows from Ohm's law that for direct current circuits, and for alternating current circuits without reactance, the con- ductance in mhos is equal to the current in amperes divided by the elec- tromotive force in volts. The conductance in mhos is also equal to the resistance in ohms divided by the sum of the squares of the resistance in ohms and the reactance in ohms. The relations to joules and watts are rarely if ever used. t Pure copper at 15 C.; according to Prof. Lindeck. j Matthiessen's value for pure copper at 15 C. ; according to Prof. Lindeck. Pure mercury at C., based on definition of international ohm. CONDUCTANCE; ADMITTANCE; SUSCEPTANCE. 105 Admittance, which is the reciprocal of impedance, and susceptance, which together with conductance make admittance, are both correctly expressed and measured in the same units as conductances, namely, mhos, although they depend on the inductance and capacity of the circuit and on the frequency of the alternating current and are therefore not true con- ductances. Both of these terms are limited chiefly to alternating current circuits. The admittance in mhos is equal to the effective current in amperes divided by the effective electromotive force in volts, regardless of what the phase difference may be; its value in mhos is equal to the reciprocal of the impedance in ohms. It is also equal to the square root of the sum of the squares of the conductance and the susceptance, all values being in mhos. The susceptance in mhos is equal to the wattless current in amperes divided by the electromotive force in volts. It is also equal in mhos to the reactance in ohms divided by the sum of the squares of the resistance in ohms and the reactance in ohms. For further explanations concerning the calculation of alternating current circuits reference should be made to treatises on that subject. The only unit used in practice is the mho, which is the reciprocal of the ohm ; it is the word ohm written reversed to indicate the reciprocal. There is no official sanction for its use, but as there is no other practical unit, it has come into use. The electromagnetic and electrostatic C.G.S. units are the re- ciprocals of the corresponding units of resistance, v is the velocity of light. Logarithm 1 CGS unit (elst) = 10 9 > 2 mho. About % X 10" 11 12-045 7575 = 1A- 2 CGS unit (elmg). About V tt X lO" 20 . 21-045 7575 1 mho=v2x 10~-' CGS units (elst). About 9X 10 11 11-954 2125 = 10~ 9 CGS unit (elmg) 9-000 0000 1 megamho = 1 000 000. mhos 6-000 0000 = 0.001 CGS unit (elmg) 3. 000 0000 1 CGS unit (elmg) = v 2 CGS units (elst). About 9X10 20 .. 20-954 2425 = 10 9 mhos 9.000 0000 = 1 000. megamhos 3-000 0000 The relations to other measures are as follows, f (See also the reciprocals of those under resistance.) Mhos = 1-^ohms. ' = amperes-?- volts. ' = watts -r- volts 2 . '* = amperes 2 -:- watts. Mega mho 6 = 1 -^microhms. Mhos conductance: = 1 -j-ohms resistance, For direct currents only. = amperes energy component of c ur re nt-J- volts total e.m.f. = watts * volts 2 . = x/(nihos admittance 2 mhos susceptance 2 ). = VT(l-*-ohms impedance 2 ) mhos susceptance 2 ]. = ohms resistance-;- ohms impedance 2 . = ohms resistance -v- (ohms resistance 2 -f- ohms reactance 2 ). = ohms resistance X mhos admittance 2 . Mhos susceptance: = amperes wattless component of current -4- volts total e.m.f. = X/(mhos admittance 2 mhos conductance 2 ). = ^[(1 -r-ohms impedance 2 ) mhos conductance 2 ]. = ohms reactance-:- ohms impedance 2 . = ohms reactance -r- (ohms resistance 2 -f ohms reactance 2 ). = ohms reactance X mhos admittance 2 . Mhos admittance = total effective amperes -r- total effective volts. = \/( m hos conductance 2 -!- mhos susceptance 2 ). = l-7-ohms impedance. = l-r-v / (hms resistance 2 + ohms reactance 2 ), ^v^mhos conductance -T- ohms resistance). = v / (mhos susceptance -T- ohms reactance). t Treatises on alternating currents should be consulted for the limiting conditions under which these relations apply to alternating or other peri- odically varying quantities. 106 CONDUCTIVITY; SPECIFIC CONDUCTANCE. CONDUCTIVITY [ r ] ; SPECIFIC CONDUCTANCE. (Conductance X length -f- cross-section j 1 -v- resistivity .) These units are used to measure the inherent quality of a material to conduct an electric current. Conductivity, which is the reciprocal of resis- tivity, differs from conductance in that the latter applies to a given circuit or part of a circuit, and its amount depends on the length, cross section, and quality of the material; while -conductivity refers only to the nature of the material and is always the same for the same material; it is the conductance of a unit amount of the material like a cube of one centimeter. It bears somewhat the same relation to conductance as the density of a material does to the weight of a given amount of it; the density is always the same for that material, being the weight of a unit of volume, while the total weight of an actual piece depends on the size of that piece. As the conductivity is a quality of .a material, its values are usually given in tables of physical constants. The conductivity may also be calculated from the conductance of any piece or part of a circuit by multiplying the con- ductance in mhos (or 1 -f- resistance in ohms) by the length d,nd dividing by the cross-section; the result will of course be different depending upon the units used. The conductivity, being a property of a material, is the same for direct and for alternating current. The values of the conductivities of materials are not in general use, the reciprocal quantity, namely resistivity, being generally preferred, as it is simpler to use in most calculations. The use Of conductivities in practice is generally limited to electrolytes and for making comparisons of other conducting materials with copper, or for comparing different qualities of copper with each other. The most rational unit and the one which seems to be coming into more general use, is the conductivity of a material of which a cube of one centi- ' meter has a conductance of one mho between two parallel sides. It will here be called the mho, cubic centimeter, unit. This means that the resistance of a column of that material one centimeter long and one square centimeter cross-section, is one ohm, that is, it corresponds to the ohm, cubic centimeter, unit of resistivity; conductivities stated in the mho, cubic centimeter, unit are the numerical reciprocals of the resistivities stated in the ohm, cubic centimeter, unit. .This unit of conductivity is therefore the best one to use when conductivities are to be converted into resistivities or the reverse. This unit is used chiefly for electrolytes; the best conducting aqueous solutions of acids at 40 C. have a conductivity of about one, in terms of this unit. The electromagnetic C.G.S. unit is the same, except that the con- ductance is stated in C.G.S. units instead of mhos. Similarly with the electrostatic C.G.S. unit. The conductances in mhos of a wire one meter long and one square milli- meter or one circular millimeter cross-section, or one foot long and one square mil or one circular mil cross-section, may also be used as units of conductivity, each being the reciprocal of the corresponding unit of resis- tivity, which see. The most usual way of stating the conductivity of a solid material is to give the ratio of its conductivity to that of pure copper as a standard; this avoids the use of the little-known unit of conductance (mho). This ratio is usually given as a percent, but may also be stated as so-and-so many "copper units" of conductivity. The Matthiessen value for the resistivity of pure copper at 15 C. is, according to Prof. Lindeck of the Reichsanstalt, 1 687 in microhm, cubic centimeter, units. The conductivity correspond- ties greater than 100% when based on the Matthiessen standard. A better value for pure copper is the one used as standard in Germany, presumably by authority of the Reichsanstalt, which according to Prof Lindeck is a resistivity of 1 667 in microhm, cubic centimeter, units, at 15 C. The CONDUCTIVITY; SPECIFIC CONDUCTANCE. 107 conductivity corresponding to this is 599 880. in mho, cubic centimeter, units. In the following table this is called the copper unit. Mercury was formerly often used as a standard of comparison, particularly in stat- ing the conductivities of electrolytes. The resistivity determined from the definition of the international ohm is 94.073 4 in microhm, cubic centimeter, units. The conductivity corresponding to this is 10 630. in mho, cubic centimeter, units. In the following table this is called the mercury unit. Logarithm 1CGS unit (elst): 10V*' 2 mho, cb. cm, unit. About % X 10~ n 15.045 7575 = 94 073.4/7' 2 mercury unit. About 1.045 26 X 10" 10 16-019 2242 l/7> 2 COS unit (elmg). About Vo X lO" 23 21-045 7575 1 mho, cb. centimeter, unit: v 2 X 10~ 9 CGS units (elst). About 9X 10 11 11-954 2425 = 0.000 094 073 4 mercury unit 5.973 4667 = 0.000 001 687 copper unit (Matthiessen) 6-227 1151 = 0.000 001 667 copper unit 6-221 9356 1C- 9 CGS unit (elmg) 9. 000 0000 1 mercury unit: = v 2 X 1.0630X10- 5 CGS units (elst). Ab. 9.567 000 X 10 15 15-9807758 10 630. mho, cb. cm, units 4-026 5333 0.017 932 8 copper unit (Matthiessen) 2-253 6484 0.017 720 2 copper unit 2-248 4689 = 0.000 010 630 CGS unit (elrng) 5-026 5333 1 copper unit (Matthiessen): 592 768. mho, cb. cm, units 5-772 8849 = 55.763 7 mercury units. 1-746 3519 -0.000 592768 CGS unit (elmg) 4-772 8849 1 copper unit = 599 880. mho, cb. cm, units 5-778 0644 = 56.432 7 mercury units 1-751 5311 = 0.000 599 880 CGS unit (elmg) 4-778 0644 1 CGS unit (elmg) = v 2 CG,:* units (elst). About 9X 10 2 ". 20-9542425 10 9 mho, cb. cm, units 9-000 0000 = 94 073.4 mercury units 4-973 4667 = 1 687. copper units (Matthiessen) 3-227 1151 = 1 667. copper units 3-221 9356 Conductivity of mercury : f = 10 630. mho, cb. centimeter, units 4-026 5333 = 0.017 932 8 times that of copper (Matthiessen) 2-253 6484 = 0.017 720 2 times that of copper 2-248 4689 Conductivity of copper (Matthiessen): t = 592 768. mho, cb. centimeter, units - 5-772 8849 = 55.7637 times that of mercury 1-746 3516 Conductivity of copper: = 599 880. mho, cb. centimeter, units 5-778 0644 = 56.432 7 times that of mercury 1-751 5311 The relations of conductivity to other measures are as follows: Conductivity (in mho, cb. cm, units). = !* resistivity (in ohm, cb. cm, units). = mhos X cm length -r-sq. cm section. = cm length -s- (ohms X sq. cm section). t Pure mercury at C. ; based on the definition of the international ohm. t Matthiessen's value for pure copper at 15 C. ; according to Prof. Lin- deck. Pure copper at 15 C. ; according to Prof. Lindeck. 108 ELECTROMOTIVE FORCE; POTENTIAL. ELECTROMOTIVE FORCE [e.m.f., E, e]; POTENTIAL} DIFFERENCE OR FALL OF POTENTIAL [p. d., U, u]; STRESS; ELECTRICAL PRESSURE ; VOLT- AGE. (Magnetic flux -f- time; current X resistance.) These units are used to measure the electrical pressure, stress, or motive force which produces or tends to produce a current, just as pounds measure the pressure of air either in the form of compressed air or wind, or as the differences of level of water measure the force which causes the water to now and which might similarly be called the hydraulic motive force. Ac- cording to Ohm's law the electromotive force in volts is equal to the current in amperes multiplied by the resistance in ohms; or according to Joule's law it is equal to the power in watts divided by the amperes; or to the square root of ths product of the watts and tha ohms; these apply to the electromotive forces or differences of potential of direct currents; they apply to Alternating current electromotive forces also when there is no reactance in tho circuit and therefore no phase shifting (caused by induct- ance or capacity), in which case they refer to the effective electromotive force; when th?re is re >ctunce in such circuits the electromotive force in volts equals the current in amperes multiplied by the impedance in ohms. The terms electromotive force and potential are used synonymously in practice, although the use of the latter term is not to be commended owing to its more general meaning in physics. Difference, of potential means in general the difference betweon two absolute potentials, e.m.f.'s, or potentials in general, the actual values of which need not be known ; this term is often distinguished from electromotive force, in that the latter applies to tho total whi -h is generated in a battery or dynamo while the difference of potential applies only to a portion of it, liko that available at the terminals, or that between any two pointe on a circuit. Voltage means any e.m.f. or difference of potential when expressed in volts. Ab- solute potential is sometimes used to denote the potential above or below some assumed zero, which is usually taken as that of the earth. The unit universally used is the volt, by which is here meant the inter- national voit of the International Congress of 1893 in Chicago, denned as that electromotive force which will maintain one international ampere through one international ohm, represented for practical purposes by 1-^-1 .434 of that of a Clark cell at 15 C.f It was made legal in this country by Congress in 1894, and is adopted by the National Bureau of Standards. The "saturated" Weston or cadmium standard cell (with excess of crystals) may eventually be substituted for the Clark cell as the official standard because it has a much smaller temperature coefficient (see table below); the relation between the Clark and this Weston cell being known quite definitely (see table below), either may be used as the standard; in this table this ratio and the value of the Clark cell are used as the funda- mental values. There is another type of cadmium cell called the ' unsatu- rated " Weston cell, in which there is no excess of crystals at ordinary temperatures, as the solution is saturated at 4 C. ; this has the advantage of having a still lower temperature coefficient, which can be neglected en- tirely at ordinary temperatures; the National Bureau of Standards does not regard it safe to assign a definite value to this unsaturated Weston cell owing to ths possibility of the seal being imperfect and the consequent change in the concentration of the solution, and also the impossibility of ascertaining the exact temperature at which the solution was saturated. The e.m.f. of this cell, with a solution saturated at 4 C., is, however, 1.019 8 international volts, this value being the same as that of the saturated cell at the 'same temperature. A number of such cells belonging to the Bureau have been intercompared; and were found to differ by a number of units t This is the value of the volt used in calibrating the Weston voltmeters. ELECTROMOTIVE FORCE; POTENTIAL. 109 in the last decimal place; hence the cell should not be employed as a stand- ard of reference, although as a working standard it can hardly be improved upon. The value adopted at the Reichsanstalt for the electromotive force of thb Clark cell is based upon a determination of its e.m.f. in terms of the elec- trochemical equivalent of silver and the unit of resistance, and also upon i* similar determination of the e.m.f. of the Weston or cadmium cell, togethev with a determination of the ratio of the values of these two cells. As the values thus obtained for the Clark and Weston or cadmium cells by the silver voltameter did not agree with the directly determined ratio, each of the silver voltameter determinations was given equal weight and the two separate values adjusted so as to give the ratio directly determined. The Reichsanstalt's value thus obtained for the Clark cell is 1.432 85 instead of 1.434 as defined by the International Congress and legal in this country, f This Reichsanstalt value may be more accurate, but is not legalized here. As the cells are the same, this makes a very slight difference between the volt used by the Reichsanstalt and that legal and used in this country (the international volt). The National Bureau of Standards uses as the funda- mental units those of resistance and electromotive force, obtaining the ampere from them, thus bringing all three into agreemen' with each other. According to Weston the international concrete volt, an^ore, and ohm, as defined by the Chicago Congress, agree with each other. In some of the relations with other units, such as the absolute, the mag- netic, the energy units, etc., the value of the international volt as above defined in terms of the Clark cell is in the following tables assumed to be equal to the theoretical value, namely 10 8 electromagnetic C.G.S. units, which value is sometimes called the true volt. According to the theoret- ical definition of a volt, it is the difference of potential generated in a con- ductor which cuts 10 s C.G.S. units of magnetic flux (or 10 s maxwells or lines of force) per second; or it is the difference of potential generated per centimeter length of a conductor moving transversely through a magnetic field of a density of 1 C.G.S. unit of density (or 1 gaiiss) at a velocity of 10 s centimeters (or 1 000 kilometers) per second. A difference of potential thus generated is moreover directly proportional to the amount of magnetic flux traversed per second. In the older literature the e.m.f. of a Daniell cell (about 1.1 volt) was often used as a unit. The electromagnetic C.G.S. unit (or absolute unit) is the difference of potential generated at the ends of a conductor 1 centimeter long moving through a magnetic field of unit density (one gauss) at a speed of 1 centi- meter per second perpendicularly to the direction of the field; or more briefly, it is the difference of potential induced in a conductor which cuts one unit of magnetic flux (one maxwell or one line of force) per second. The electrostatic C.G.S. unit (or absolute unit) is that difference of potential through which one electrostatic unit of quantity falls when the work done by it is one erg ELECTROMOTIVE FORCE. ** Accepted by the National Bureau of Standards. * Checked by Dr. Frank A. Wolff, Jr., Asst. Phys. National Bureau of Standards. Aprx. means within 2%. By "volt" is meant international volt unless otherwise stated, v is the velocity of light. Logarithm 1 t GS unit (elmg) = 1 abvolt 0-000 0000 = 0.01 microvolt 2-000 0000 = 10- vo i t g-000 0000 = 1/v CGSunit(elst). About ^ XIO" 10 . .. II. 522 8787 1 abvolt = 1 CGS unit (elmg) 0-000 0000 1 microvolt = 100. CGS units (elmg) 2-000 0000 = 0.000 001 volt 6-000 0000 1 millivolt = 0.001 volt 3-000 0000 1 legal volt =0.997 178* volt. Aprx. 1 1-998 7726 1 ampere X 1 legal ohm. t The difference corresponds almost exactly to that due to one Centigrade degree difference of temperature; it is about 8 hundredths of one percent 110 ELECTROMOTIVE FORCE; POTENTIAL. 1 volt [V, v] : Logarithm 10 s CGS units (elmg) 8-0000000 = 1 000 000. microvolts . . . 6-000 0000 = 1 000. millivolts 3-000 0000 = 1.002 83* legal volts. Aprx.'l 001 2274 = 0.999 198** volt of Reichsanstalt. t Aprx. subtr. 8 /ioo%.. 1-999 6518 = 0.980962** Weston (satur.) cell at 20 C. Ap.sub.2%... 1-9916523 = 0.980567 Weston (unsat.) cell at any temp.t Aprx. subtr. 2% 1.991 4774 = 0.697 350* Clark cell at 15 C. Aprx. i/ w 1.843 4508 108 /v CGS unit (elst). About Men 3-522 8787 0.001 kilovolt f-000 0000 1 international volt = 1 volt, which see above , 000 0000 1 true volt = 10 s CGS units (elmg) 8-000 0000 1 volt of Reich8aiistalt = 1.000 803** volts.f Ap. add 8 /loo%. 0-000 3484 1 Weston (cadmium; saturated) cell at 2O C. with exce.ss of crystals: = 1.019 4** volts. Aprx. add 2% 008 3477 = 1.018 6** volts of Reichsanstalt. Aprx. add 2% 007 9993 = 0.71088* Clark cell at 15 C. Aprx. % 1-8517985 1 Weston (cadmium; unsaturatedf)cell at any ordinary temperature: = 1 .019 8* volts. Aprx. add 2% 008 5226 = 1.019 0* volts of Reichsanstalt. Aprx. add 2% 008 1742 1 Daiiiell cell = 1.1 volts approximately (unreliable) 0-041 3927 1 Clark cell at 15 C.: = 1.434** volts. Aprx. 1% 0-1565492 = 1.432*85** volts of Reichsanstalt. Aprx. *% 0-156 2008 = 1.4O6 T** Weston cells at 20 C Aprx. % 0-1482015 1 CGS unit (elst)= v CGS units (elmg). Ab. 3X10 10 . 10-477 1213 = v X 10-< s volts. About 300 2-477 1213 1 abstavolt 000 0000 = vX 10~ n kilovolt. About s/lo I 477 1213 1 abstavolt = 1 CGS unit (elst) 000 0000 1 kilovolt = 1000. volts 3-000 0000 = W n /v CGS units, (elst). About io/ 3 0-522 8787 1 meg-avolt = 1 000 000. volts 6 000 0000 1 absolute unit - 1 CGS unit, either elmg or elst 0-000 0000 The relations of volts to other measures are as follows: J Volts amperes X ohms. . = ohms X coulombs -=- seconds. = watts -r- amperes. = kilowatts X 1 000. -T- amperes. =- >/( watts X ohms). = watts X seconds -5- coulombs. = joules -r- coulombs. = joules -r- (amperes X seconds). = \/( joules X ohms -r- seconds). = coulombs -j- farads. = coulombs X 1 000 000. ^-microfarads. = \/( joules of stored energy X 2 -4- farads). = 1 000. X \/( joules of stored energy X 2 -5- microfarads). = maxwells X number of turns -*- (seconds X 1 8 ). = gausses X sq. centimeters -J- (seconds X 10 8 ). Induced volts: = henrys X rate of change of amperes per second. = time constant in seconds X ohms X rate of change of amperes per sec. Applied volts: = henrys X final amperes -Mime constant in seconds. = joules of kinetic energy of the current X ohms X 2 -T- (henrys X final amperes). = ohm sX\/[ joules of kinetic energy of the current X 2 -r-henrys J. = joules of kinetic energy of the current X 2 -*-( time constant in sec- onds X final amperes). = >/[( joules of kinetic energy of the current X ohms X 2) * time con- stant in seconds . 1 See explanatory note above. } Consult treatises on alternating currents for the limiting conditions. ELECTROMOTIVE FORCE; POTENTIAL. Ill For alternating current circuits: f Volts = watts -r- (am peresX cos ^J) = \/[( watts Xohrns)-r- cos is the phase difference in degrees. Or 2*. Aprx. ^XIO. Log. 0.798 1799. 112 ELECTRICAL CURRENT. ELECTRICAL CURRENT [I, i]; CURRENT STRENGTH or INTENSITY. (Electromotive force ~ resistance; quantity -v- time. ) These units are used to measure the rate of flow or passage of units of electricity per second, just as the rate of flow of water or air is measured in -- , ! power in watts divided by t volts; or to the square root of the quotient of the watts divided by the ohms. These anply to direct currents without counter-electromotive forces ; they apply to alternating currents also when there is no reactance in the circuit (caused by inductance or capacity), in which case they refer to the effective current. In any alternating current circuit with reactance the current in amperes is equal to the electromotive force in volts divided by the impedance in ohms. An ampere is also equal to a passage of one coulomb per second. The unit universally used is the ampere, by which is here meant the international ampere of the International Congress of 1893 at Chicago, defined as equal to Vio of the C.G.S. electromagnetic unit of current; it was made legal in this country by Congress in 1894. For practical purposes it is defined by that congress as the current which (under specified conditions) deposits 0.00 1 118 gram of silver per second. f The ampere is, however, preferably determined from the ohm and the voltage of a standard cell. Owing to the slight discrepancy in the units of current, electromotive force, resistance, and Ohm's law, the National Bureau of Standards has selected two of these as the fundamental units, namely those of resistance and elec- tromotive force established by the International Congress and legalized in this country, and from these the ampere is derived, thus bringing all three into agreement with Ohm's law. Individual Clark standard cells agree with each other to within at least 2 parts in 10 000., and by the use of care- fully purified materials this agreement can be still closer, while different determinations of the electro-chemical equivalent of silver differ by con- siderably larger amounts, unless repeated under perfectly definite condi- tions and made with great care. The Reichsaristalt measured the electro- motive force of the Clark cell in terms of the ohm and the ampere based on the silver voltameter, but obtained a slightly different value for this cell from that defined by the International Congress; hence the ampere of the Reichsanstalt, which is in agreement with the volt and ohm there used, is slightly different from the ampere used in this country, which is based on the international volt and ohm, although both these amperes were originally intended to be the same. But this discrepancy, which is only about 8 him- dredths of one percent, is quite negligible in ordinary practice. According to Weston the concrete volt, ampere, and ohm as defined by the Chicago Congress are in agreement with each other to within one pan in 1 000. In some of the relations with other units, such as the absolute, the mag- netic, the energy units, etc., the value of the international ampere as above defined is in the following tables assumed to be equal to the theoretical value, namely Vio of the electromagnetic C.G.S. unit, which value is some- times called the true ampere. The electromagnetic C.G.S. unit (or absolute unit) is that current which, flowing in the circumference of a circle of one centimeter radius, will, for every centimeter length of circumference, exert in air a force of one dyne on a unit magnetic pole placed at the center; one whole circumfer- ence therefore exerts a force of 2n dynes on that pole. Or under the same conditions, every centimeter of the circumference will in air produce at the center a magnetic field of one unit density (one gauss), that is, one unit of magnetic flux (one maxwell) per square centimeter. The electrostatic C.G.S. unit (or absolute unit) is the current which flows when one electrostatic C.G.S. unit of quantity passes per second. f This is the value used in calibrating the Weston amperemeters. ELECTRICAL CURRENT. 113 ELECTRICAL CURRENT. ** Accepted by the National Bureau of Standards. * Checked by Dr. Frank A. Wolff, Jr., Asst. Phys. National Bureau of Standards. Aprx. means within 2%. By "ampere" is meant the international am- pere, assumed to be equal to the international volt divided by the interna- tional ohm, unless otherwise stated, v is the velocity of light. Logarithm 1 CGS unit (elst) = 1 abstatampere = 10 7 /v microampere. About % -r- 1 000 = 10/v ampere. About \i X 10~ 9 = l/v CGS unit (elmg). About Y z X lO" 10 .. . 1 abstatampere = 1 CGS unit (elst) 1 microampere = v/10 7 CGS units (elst). About 3 000. . = 0.000 001 ampere 1 milliampere =v/10 000 CGS units (elst). About 3 000 000. = 0.001 ampere = 0.0001 CGS unit (elmg) '. 1 ampere [A, a]= v/10 CGS units (elst). About 3 X 10 a . = 1000. milliamperes = 0.999 198** ampere of Reichsanstaltf 0.1 CGS unit (elmg) 1 international ampere = 1 ampere, which see above 1 true ampere =0.1 CGS unit (elmg) 1 ampere of Reichsai^talt - 1.000 803** amperesf 1 ampere of Nat. 1$ urea. a of Standards = 1 volt-r- 1 ohm. 1 CGS unit (elmg) = v CGS units (elst). About 3 X 10 10-477 1213 = 10 amperes 1 .000 0000 = 1 absampere 000 0000 1 absampere = 1 CGS unit (el Tig) 0-000 0000 1 kiloampere = 1 000. amperes 3 000 0000 = 100. CGS units (elmg) 2-000 0000 1 absolute unit = l CGS unit, either elst or elmg 0-000 0000 The relations to other measures are as follows: J Amperes = volts -f- ohms = coulombs -f- seconds. = watts -r- volts. = 1 000 X kilowatts -4- volts. = \/( watts -r- ohms). = joules -H( volts X second ). " =\/[joules-:- (ohms X seconds)]. 'flute of change of amperes per second: = induced volts -r- henrys. = induced volts -5- (ohms X time constant in seconds). l'i;il amperes: = applied volts X time constant in seconds -4- henrys. --= joules of kinetic energy of the current X ohms X 2 ^-(henrys X applied volts). = \/( joules of kinetic energy of the current X 2 -s- henrys). = joules of kinetic energy of the current X 2 -s- (applied volts X time con- stant in seconds). = vT Joules of kinetic energy of the current X 2 -5- (ohms X time constant in seconds) . f See explanatory notes above. j Treatises on alternating currents should be consulted for the limiting conditions under which these relations apply to alternating or other period- ically varying quantities 114 ELECTRICAL CURRENT; CURRENT DENSITY. When the flux is due only to the current, as in self-induction, and when there is no magnetic leakage: Final amperes: == (max well sX number of turns) -:- (henrys X 10 s ). = (maxwells X number of turns) *- (time constant in seconds X ohms X 10 s ). = ergs of kinetic energy 6f the current X 20 -- (maxwells X no. of turns). = joules of kinetic energy of the current X 2 X 10 s -?- (maxwells X number of turns). When there is magnetic leakage, substitute in the above for the quantity "max wells X number of turns," the mean flux turns, that is the "mean max wells X number of turns." Thus: Final amperes: = (mean maxwells X number of turns) -r- (henrys of self-induction X 10 s ). When the flux is from an external source, and independent of the current as in mutual induction, and when there is no magnetic leakage: Final amperes: = ergs of kinetic energy of the current X 10 -5- (maxwells X no. of turns). = joules of kinetic energy of the current X 10 s -s- (maxwells X no. of turns) When there is magnetic leakage, make the same substitution as described above. Final amperes in primary: = mean maxwells through secondary X secondary turns-:- (henrys of mutual induction X 10 s ). For alternating-current circuits f: Amperes: = watts -4- (volts X cos $ J ). = >/[ watts -s- (ohms X cos 0J)]. = \/[applied volts 2 induced volts 2 ] -r-ohms resistance. farads X volts X frequency X 6. 283 19 . = microfarads X volts X frequency X 6.283 19 * 1 000 000. = induced volts -s- (henrys X frequency X6.283 19 ). = induced volts + (time constant in seconds X ohms resist anceX fre- quency X 6.283 19). ^applied volts -i-VIX henrys X frequency X 6. 283 19 ) 2 +(ohms res.) 2 J. CURRENT DENSITY. (Current -surface.) These units are used to measure the current flowing through a unit cross- section of a wire or other conductor; or the amount of current flowing into or out of a unit surface of an electrode in an electrolyte. Aprx. means within 2%. 1 ampere per sq. meter: Logarithm = 0.092 903 4 ampere per sq. foot. Aprx. H12 -*-10 29680317 == 0.01 ampere per sq. decimeter 2-000 0000 = 0.000 645 163 ampere per sq. inch. Aprx. Vn -*- 1 000 4-809 6692 1 ampere per sq. foot: = 10.763 87 amperes per sq. meter. Aprx 12 /n X 10 1 031 9683 = 0.107 6387 ampere per sq. decimeter. Aprx.i%i -5-100... 1-0319683 = 0.006 944 44 ampere per sq. inch. Aprx. 7 /iooo 3-841 6375 1 ampere per sq. decimeter: 100 amperes per sq. meter 2-000 0000 = 9.290 34 amperes per sq. foot. Aprx. 11/12 X 10 968 0317 = 0.064 5163 ampere per sq. inch. Aprx.Vii-^10 2-8096692 t Treatises on alternating currents should be consulted for the limiting conditions under which these relations apply to alternating or other peri- odically varying quantities. | is the phase difference in degrees. Or 2*. Aprx. ^iX 10. Log 0-798 1799- CURRENT DENSITY; ELECTRICAL QUANTITY. 115 Logarithm 1 ampere per sq. inch: = 1 550.00 amperes per sq. meter. Aprx. ity X 1 000 3. 190 3308 = 144. amperes per sq. foot. Aprx. Vr X 1 000 2-158 3625 = 15.500 amperes per sq. decimeter. Aprx. ity X 10. . . 1-1903308 = 0.785 398 ampere per circular inch. Aprx. s /io 1-895 0899 = 0.155 000 ampere per sq. centimeter. Aprx. 2 /ia 1-190 3308 = 0.121 736 ampere per circular cm. Aprx. 12-5-100 1-085 4207 1 ampere per circular inch: = 1.273 24 amperes per sq. inch. Aprx. *% 0-104 9101 = 0.197 352 ampere per sq. centimeter. Aprx. 2 Ao 1-295 2409 = 0.155 000 ampere per circular cm. Aprx. %s 1-190 3308 1 ampere per sq. centimeter: = 929.034 amperes per sq. foot Aprx. HiaXl 000 2-968 0317 = 100. amperes per sq. decimeter 2 000 0000 = 6.451 63 amperes per sq. inch. Aprx. 6^2 809 6692 = 5.067 09 amperes per circular inch. Aprx. 5 0-704 7591 = 0.785398 ampere per circular centimeter. Aprx.%o. ... 1-8950899 1 ampere per circular centimeter: = 8.214 47 amperes per sq inch. Aprx. % X 10 0-914 5793 = 6.451 63 amperes per circular inch. Aprx. *% = 1.273 24 amperes per sq. centimeter. Aprx. *% 1 ampere per sq. millimeter: = 0.785 398 ampere per circular mm. Aprx. 8 /io = 0.000 645 163 ampere per sq. mil. Aprx. V\i -*- 1 000. . . = 0.000 506 709 ampere per circular mil. Aprx. >-* 1 000 . 1 ampere per circular millimeter: 1.273 24 amperes per sq. millimeter. Aprx. '%.... = 0.000 821 447 ampere per sq. mil. Aprx. M 3 -5- 100 = 0.000 645 163 ampere per circular mil. Aprx. 7 /n * 1 000 1 ampere per sq. mil: = 1 550.00 amperes per sq. millimeter. Aprx. *# X 1 000.. = 1 217.36 amperes per circular mm. Aprx. % X 1 000. . . = 0.785 398 ampere per circular mil. Aprx. 9io 1 ampere per circular mil: = 1 973.52 amperes per sq. millimeter. Aprx. 2 000 3-295 2409 = 1 550.00 amperes per circular mm. Aprx. Hfr X 1 000 . . . 3-190 3308 = 1.273 24 amperes per sq. mil. Aprx. 1% 0-104 9101 ELECTRICAL QUANTITY [Q, q]j CHARGE. (Current X time ; capacity X electromotive force. ) These units are used to measure the amount of electricity as such, just as a quantity of matter might be measured in the number of units called molecules, which it contains. The number of units of electricity in a given quantity of electricity is the same Avhether or not it is flowing in the form of a current, or whether or not it is subjected to an electrical pressure, just as the number of molecules in a given weight of air is the same whether or not it is in motion, as in a wind, or whether or not it is under pressure, as in compressed air. The quantity of electricity in coulombs is equal to the current in amperes multiplied by the time in seconds; or to the energy in joules divided by the voltage; or to voltage multiplied by the capacity of a condenser in farads. The unit generally used is the coulomb, by which is here meant the international coulomb of the International Congress of 1893 at Chicago, denned as "the quantity of electricity transferred by a current of one international ampere in one second." It was made legal in this country by Congress in 1894, and is accepted by the National Bureau of Standards. For practical purposes it is that quantity which will deposit 0.001 118 gram of silver in a silver voltameter. In some of the relations with other units, such as the absolute, the energy units, etc., the above value is in the following tables assumed to be equal to the theoretical value, namely ^4o the electromagnetic C.G.S. unit. Another unit frequently used, especially with batteries and in electrochemistry, is the ampere- 116 ELECTRICAL QUANTITY; CHARGE. hour; it is the quantity of electricity transferred by a current of one ampere in one hour. in one second. ine B*Bjruww \;.VJT. .-. unii< ^ur ausomie uiui; is that quantity of electricity which in air exerts a force of one dyne on an equal quantity one centimeter distant. ELECTKICAL QUANTITY; CHARGE. Aprx. means within 2%. By "coulomb" is meant the international coulomb, v is the .velocity of light. Logarithm 1 CGS unit [elst]: 1 abstatcoulomb 000 0000 = 10 7 /y microcoulomb. About J^-s- 1 000 4 522 8787 = 10/v coulomb. About MX 10~ 15-522 8787 = l/v CGS unit (elmg). About ^X 10~ 10 11-5228787 = 1-4- (360 v) ampere-hour. About 9.259 X 10" 14 14-9665762 1 abstatcoulomb = 1 CGS unit (elst) 0-000 0000 1 microcoulomb: = vX 0.000 000 1 CGS units (elst). About 3 000.. 3-4771213 = 0.000 001 coulomb 6-000 0000 = 0.000 000 1 CGS unit (elmg) 7-000 0000 = 2.777 78 X 10- 10 ampere-hour. Aprx. ! MX 10" 10 10-443 6975 1 coulomb [C, c] : v/10 CGS units (elst). About 3 000 000 000... . Q.477 1213 = 1 000 000. microcoulombs 6 000 0000 = 0.1 CGS unit (elmg) 1-000 0000 = 0.000 277 778 ampere-hour. Aprx. "4+ 10 000 4 443 6975 1 international coulomb = 1 coulomb, which see above. 1 true coulomb -0.1 CGS unit (elmg) 1-000 000^ 1 ampere-second == 1 coulomb, which see above. 1 CGS unit (elmg): = 1 abscoulomb 000 0000 = v CGS units (elst). About 3 X 10 10 10-447 1213 = 10 coulombs 1-000 0000 = 0.002777 78 ampere-hour. Aprx ^-s-1 000 3-443 6975 1 abscoulomb = 1 CGS unit (elmg) 000 0000 1 ampere-hour [ah]: = vX 360. CGS units (elst). About 1.08 X 10 13 13-033 4238 = 3600. coulombs 3-556 3025 = 360. CGS units (elmg) 2-556 3025 1 absolute unit = 1 CGS unit, either elst. or elmg 0-000 0000 The relations to other measures are as folio ws:f Coulombs = amperes X seconds. = amperes X hours X 3 600. = volts X seconds -4- ohms. = watts X seconds -4- volts. = \/( watts X seconds 2 -*- ohms). " = joules -T- volts. = \/( joules X seconds -4- ohms). 41 = farads X volts. = microfarads X volts -4- 1 000 000. Ampere-hours = coulombs -f- 3 600. = amperes X hours. = volts X hours -r- ohms. = watts X hours -4- volts. \/( watts X hours 2 -4- ohms). = joules -*- (volts X 3 600). = Vfjoules X hours -4- (ohms X 3 600)]. Microcoulombs = microfarads X volts. t Treatises on alternating currents should be consulted for the limiting conditions under which these relations apply to alternating or other peri- odically varying quantities. ELECTRICAL CAPACITY. 117 ELECTRICAL CAPACITY [C, c]. (Quantity ^ electro- motive force.) These units arte used to measure the ability of a body (like a condenser) to hold charges of electricity (measured in coulombs) under electrical stress, pressure, or potential (measured in volts). The capacity of a con- denser is greater the greater the charge in coulombs that it will hold for the same pressure in volts, or the less the pressure in volts required for the same charge. The capacity in farads is equal to the charge in coulombs divided by the electromotive force in volts. In some respects an electrical capacity is analogous to the capacity of a closed vessel to hold air under pressure, the greater the pressure the greater the quantity of air, yet the capacity of the vessel remains the same and could be measured in terms of the air and its pressure. The unit universally used is the microfarad or millionth of a farad; by farad is here meant the international farad of the International Congress of 1893 at Chicago, defined as equal to one international coulomb divided by one international volt. It was made legal in this country by Congress in 1894, and is accepted by the National Bureau of Standards. As the farad is an inconveniently large unit, never occurring in practice, the microfarad is generally used. The electromagnetic C.G.S. unit (or absolute unit) is the capacity of a condenser which when charged at one C.G.S. unit of potential will hold one C.G.S. unit of quantity. The elec- trostatic C.G.S. unit (or absolute unit) is similarly defined with respect to the electrostatic units of potential and quantity. By "farad" is meant the international farad, v is the velocity of light. Logarithm 1 COS unit [elst]= 1 abstafarad 0-000 0000 = 10 15 /v 2 microfarad. About V X 10~ 5 6 045 7575 = 10 9 /?; 2 farad. About V 9 X lO" 11 12-045 7575 = l/v 2 CGS unit (elmg). About V 9 X 10" 20 . 21-045 7575 1 abstafarad => 1 CGS unit (elst) 0-000 0000 I microfarad =v 2 X 1Q- 15 CGS units (elst). About9X10 5 5.954 2425 1 microcoulomb -- 1 volt 0-000 0000 = 0.000001 farad 6 000 0000 = 10~ 15 CGS unit (elmg) 1 5 000 0000 1 farad [F]= v 2 X 10~ 9 CGS units (elst). About 9X 10 11 11-954 2425 = 1 000 000. microfarads 6 000 0000 1C- 9 CGS unit (elmg) 9.QOO 0000 1 international farad = 1 farad, which see above 0-000 0000 1 CGS unit (elmg)= u 2 CGS units (elst). About 9X10 20 .. . 20-954 2425 = 10 15 microfarads 15 000 0000 44 = 10 9 farads 90000000 = 1 abfarad 0-000 0000 1 abfarad = 1 CGS unit (elmg). 0-000 0000 1 absolute unit = 1 CGS unit, either elmg or elst 0-000 0000 The relations to other measures are as follows: t Farads = coulombs -T- volts. " = joules of stored energy X 2 rf- volts 2 . Microfarads = coulombs X 1 000 000. -* volts. = microcoulombs -J- volts. = joules of stored energy X 2 000 000. -f- volts 2 . For alternating -current circuits -f Farads = amperes -5- (volts X frequency X 6.283 19 t ). " = 1 -v- (ohms reactance X frequency X 6.283 19 J ) Microfarads = amperes X 1 000 000 -*- (volts X frequency X 6. 283 19 t). = 1 000 000. 4- (ohms reactance X frequency X 6.283 19 j ). t Treatises on alternating currents should be consulted for the limiting conditions under which these relations apply to alternating or other period- ically varying quantities. t Or 2n. Aprx. ^X 10, Log 0-798 1799. 118 INDUCTANCE. INDUCTANCE [L, 1]; COEFFICIENT of SELF- or MUTUAL INDUCTION. (E,m.f. -f- (current -f time) ; re- sistance X time; number of turnsXflux-7-currentj kinetic energy -f- square of current.) These units are used to measure the intensity of that property by virtue of which an electromotive force is produced by changes of current in a neighboring circuit (as in a transformer) or by changes of current in the circuit itself; in the former case the phenomenon is called mutual induc- tion, and in the latter self-induction. An electric current has a property analogous in some respects to the inertia of a heavy moving body; a cur- rent resists momentarily any change in its strength, just as a heavy moving body resists any change in its velocity. When a current is started, it en- counters for a short time (see time-constant below) a counter-electromotive force in its own circuit due either to itself (self-induction) or to a current in the opposite direction which it is inducing in a neighboring circuit (mutual induction). Similarly, if stopped it tends to prolong itself either in its own circuit or (by induction) in a neighboring circuit; the quicker the change the greater this tendency. This property is due to the inductance. The term induction in electrodynamics applies broadly to the general phenomenon of the generation of an electromotive force (which may or may not produce a current) by magnetic flux, whether it be that of a magnet or that surrounding a current. The terms self and mutual induction apply to the special cases of this phenomenon when the induction is produced by a current in its own circuit or in a neighboring circuit respectively, there being no mechanical motion. The terms coefficient of self or coefficient of mutual induction apply to the numerical value of the intensity of this phenomenon, by which it is measured; the terms self-inductance and mutual inductance are now more generally used instead. The term inductance applies to both of these coefficients, and is therefore the measure of self or mutual induction. The inductance factor is the ratio of the wattless volt amperes to the total volt amperes; or the ratio of the wattless component of the current or e.m.f . to the total current or e.m.f. The inductance depends for its value on the geometric conditions of the circuit, and varies with the size and shape of the circuit or of a coil, with the square of the number of turns of a coil, with the distance between the wires, etc.; it also varies according to an irregular law with the presence of iron or other magnetic material. An inductance, in henrys, is measured by and is equal to the electromotive force induced, in volts, divided by the rate of change of current in amperes per second/which causes it; it is also equal in henrys to twice the kinetic (magnetic) energy of the circuit in joules divided by the square of the final current in amperes; a coefficient of self-induction in henrys is also equal to the resistance of the circuit in ohms multiplied by its time-constant (see below) in seconds. The induct- ance of any particular circuit is also the constant relation of the product of the magnetic flux and th3 turns, to the current producing the flux. In the C.G.S. system of units, on which the practical unit is based, this measure happens to be of the same kind as a length; this is a consequence of what is called a "suppressed factor" in the dimensional formula, and as it is misleading and answers no useful purpose, the coincidence should not be given any importance. While it is not incorrect to express inductances in centimeters, it misleads and is not good practice. The electromotive force produced by inductance is generated precisely as in dynamos by the cutting of magnetic flux or lines of force and at the rate of 10 s such lines (maxwells) per second for each volt, the magnetic flux in inductance being that produced by the current, and is proportional in amount to the change in the current strength. In the case of dynamos the wire moves and the flux is at rest, while in inductance the wire is at rest and the flux moves. The unit now most generally used is the henry. By henry is here meant that of the International Congress of 1893 at Chicago, defined as the induc- a circuit when the electromotive force induced iu this circuit ia one INDUCTANCE. 119 international volt, while the inducing current varies at the rate of .one inter- national ampere per second. It was made legal in this country by Con- gress in 1894, and is accepted by the National Bureau of Standards. In some of the relations in these tables, this value is assumed to be equal to 10 9 C.G.S. electromagnetic units of inductance. This unit was formerly, and for some years officially, called a "quadrant." If the self-inductance of a given circuit (usually a coil) is one henry, it means that the magnetic lines of force corresponding in that circuit to one ampere will form 10 8 linkages of unit magnetic lines of force (maxwells) with that circuit. The unit called sec-ohm was at one time used, as in self-induction the inductance is equal to the product of the resistance in ohms, and the time-constant of the circuit in seconds. The electromagnetic C.G.S, unit (or abs9lute unit) is that induction which will induce one C.G.S. unit of electromotive force by a change of cur- rent at the rate of one C.G.S. unit of current per second. It happens to be numerically equal to 1 centimeter of length and is sometimes so represented. The electrostatic C.G.S. unit (or absolute unit) is similarly denned with respect to the electrostatic units of electromotive force and current. INDUCTANCE. v is the velocity of light. Logarithm 1 CGS unit (elmg) = 1 centimeter 000 0000 = 0.001 microhenry 3-000 0000 = 10- fJ henry g.QOO 0000 = l/v 2 CGS unit (elst).. About % X 10" w . . 21-045 7575 1 centimeter inductance = 1 CGS unit (elmg) 000 0000 1 microhenry = 1 000. CGS units (elmg) 3-000 0000 = 0.000 001 henry 6 000 0000 1 millihenry = 0.001 henry 3-000 0000 1 henry [H]= 10'' CGS units (elmg) 9-000 0000 = 10 000. kilometers, or 1 earth's quadrant 4 000 0000 = 10 9 A 2 CGS unit (elst). About % X 10-" 12-045 7575 1 quadrant = 1 henry 0-000 0000 1 quad = 1 quadrant or henry 0-000 0000 1 sec-ohm = 1 henry 0-000 0000 1 CGS unit (elst) = v 2 CGS units (elmg). AboutQXlO 2 " 20-9542425 = r 2 centimeters. About 9 X 10 20 . . . . 20-9542425 = v 2 X 10~ 9 henrys. About9X10 n 11-9542425 The relations to other measures are as folio ws:f Henrys = induced volts-;- rate of change of amperes per second. = time constant in seconds X ohms. = time constant in seconds X applied volts -f- final amperes. = joules of kinetic energy of the current X 2 -r- final amperes 2 . = joules of kinetic energy of the current X ohms 2 X 2 * applied volts 2 . When the flux is due only to the current, as in self-induction, and when there is no magnetic leakage: Henrys of self-induction: = (maxwells X number of turns 2 ) -r- (ampere-turns X 10 8 ). = (maxwells X number of turns) -r-( final amperes X 10 8 ). = (number of turns X 0.000 1) 2 X 1.256 64 } -s- oersteds. When there is magnetic leakage, substitute in the above for the quantity " maxwells X number of turns," the mean flux turns, that is, the " mean maxwells X number of turns." Thus: Henrys of self-induction: = (mean max wells X number of turns) -r-( final amperes X 10 s ). When the flux is from an external source and independent of the current, as in mutual induction, and when there is magnetic leakage: Henrys of mutual induction: = (mean maxwells through the secondary X secondary turns) -s- (final amperes in primary X 10 s ). t Treatises on alternating currents should be consulted for the limiting conditions under which these relations apply to alternating or other period- ically varying quantities. t Or 4 rr+10. Aprx. 1%. Log 099 2099- 120 INDUCTANCE; TIME-CONSTANT. For alternating current circuits :f Henrys = induced volts -s- (amperes X frequency X 6. 283 19 J). = ohms reactance -r- (frequency X 6. 283 19J). = v / (ohms impedance 2 ohms resistance 2 ) -s- frequency X 6. 283 19 J. 44 =vT( applied volts -r- amperes) 2 ohms resistance 2 ] -5- frequency X 6.283 19 1 Inductance factor: = wattless component of current or e.m.f. -r- total current or e.m.f. = V(1 power factor 2 ). TIME-CONSTANT (of inductive circuits). (Inductances resistance ; time.) When a current is started in a circuit containing inductance of the kind called self-induction, as is the case for instance in coils, particularly if they have many turns and iron cores, the current will not reach its full value Instantly, but owing to the self-induction it will at first be opposed by a jounter-electromotive force due to the inductance; this opposition will grow less and less until the current has reached its full strength. It is similar to what takes place when a heavy weight like a street car is started to move; its inertia will at first oppose the moving force, but this opposition will grow less and less as the speed increases until the full, constant speed is attained, and the inertia will then offer no further opposition. It is sometimes of importance to know how long it takes before the cur- rent has reached its ultimate value, but theoretically it takes an infinite time, and therefore it is usual to state the time that it takes the current to rise to a certain definite fractional part of its full value, namely nearly % (the exact figure is given below), and this time is called the "time-constant" of that circuit. This time in seconds (often a very small fraction of a second) is equal to the self-induction in henrys divided by the resistance in ohms; or instead of the ohms one may of course use the applied volts divided by the final steady current in amperes. This time-constant is therefore greater the greater the self-induction and the less the resistance. It gives more information about a circuit than the mere inductance does, as it includes the resistance; the self-inductance of a coil, for instance, is the same whether the wire is made of copper or of a high resisting metal, but the time-constant is less in the latter case. The exact fractional part of the full value of the current, above referred to, is (e l)-=-e, in which e is the base of the Naperian logarithms. Numer- ically this is equal to 63.212%, or nearly %. The unit in which time-constants are always given is the second, hence there are no reduction factors. The most important relations to other measures are as follows: Time-constant in seconds = henrys -f- ohms resistance. = henrys X final amperes -r- applied volts. For further relations see those for henrys under inductance, and divide them by the ohms resistance. t Treatises on alternating currents should be consulted for the limiting conditions under which these relations apply to alternating or other period- ically varying quantities. I Or 2*. Aprx. % X 10. Log 0-798 1799- FREQUENCY. 121 FREQUENCY; PERIODICITY; PERIOD; ALTERNA- TIONS. (1-i-time; time.) Frequency or periodicity is the number of recurrences or cycles of some periodic or wave phenomenon or oscillation during a given time which is always understood to be a second unless otherwise stated; the frequency always refers to the number of complete cycles. The number of alterna- tions, however, refers to the number of changes of the direction or to the reversals, and therefore refers to half- waves, and is always equal to double the frequency, if the time is the same. The period is the time of one com- plete wave or oscillation and is therefore the reciprocal of the frequency. The term " frequency" is the one most generally used, and always refers to a second; the term " number of alterations" is preferred by some, and when it refers to electric currents the time is usually a minute; the term "period "is used comparatively rarely as a measure, its use being gen- erally limited to scientific discussions; the unit is generally the second. In mathematical discussions of electric alternating-current problems the frequency is often replaced by an angular velocity, generally represented by a) and measured in terms of radians per second (see under Angular Veloc- ities above). Then w = 2,7in, in which a> is in radians per second and n is the true frequency in cycles per eecond, a cycle being here considered the same thing as a complete revolution. The frequency is also equal to the velocity of propagation divided by the wave-length. The wave-lengths are therefore measured in units of length, but when the velocity for a class of waves is a constant (as those of light or the electromagnetic waves), the wave-lengths may also be in- dicated in units of time, in which case a wave-length becomes equal to the period of the wave. Wave-length should not be confounded with the amplitude, which measures the intensity of the wave and has nothing to do with the frequency, period, or wave-length. If n is the frequency per second [CO], then: the period in seconds = 1/n; the number of alternations per minute = 120n. If n is the number of alternations per minute, then: the frequency per second = n/120; the period in seconds = 120/n. If n is the period in seconds, then: the frequency per second = 1/n; the number of alternations per minute = 120n. If n is the frequency in cycles per second, and (o the angular velocity in radians per second, and if a cycle is represented by one complete revolu- tion, then: a) = 2i:n\ or n = 0.159 155w; and . 74); the present table is limited to a few specific values and to some relations between the electrical unit of energy and other electrical units. The energy in joules delivered to a circuit is equal to the electromotive force in volts multiplied either by the quantity of electricity in coulombs, or by the product of the current in amperes and the time in seconds. These apply to direct currents; in alternating-current calculations involving energy, the power and not the work done is generally the important con- sideration; the energy of alternating currents of any wave form delivered per cycle, in joules, is equal to the power in mean watts divided by the frequency. For the energy stored in a current see the preceding section on Kinetic Energy. The unit universally used is the joule, by which is here meant the joule of the International Congress of 1893 in Chicago, defined as equal to 10 7 C.G.S. units of work (ergs) and represented sufficiently well for practical use by the energy expended in one second by an international ampere in an international ohm ; in the relations in these tables this defined value is used. It was made legal in this country by Congress in 1894 and is accepted by the National Bureau of Standards. Sometimes the ampere-hour is used as the unit of electrical quantity, in which case the corresponding unit of energy becomes the volt-ampere-hour, usually called the watt-hour; the kilowatt-hour ( = 1000. watt-hours) is also common. The electro- magnetic C.G.S. unit (or absolute unit) is the erg, defined as the work of one dyne acting through one centimeter. The electrostatic C.G.S. unit (or absolute unit) is this same erg. ELECTRICAL ENERGY. 123 ELECTRICAL ENERGY. Aprx. means within 2%. Logarithm 1 CGS unit (elmg) = 1 erg 0-000 0000 1 CGS unit (elst) = 1 erg 0-000 0000 1 absolute unit = 1 erg 000 0000 1 erg = 1 CGS unit (elmg) 0-000 0000 4 = 1 CGS unit (elst) 0-000 0000 ' ' = lO- 7 joule 7.000 0000 1 microioule = 10. ergs 1-000 0000 1 joule [J] = 10 000 000. ergs y.QOO 0000 = 0.000 277 778 watt-hour. Aprx. 3 /u -*- 1 000 4-443 6975 1 kilo.joule = 1 000. joules 3. 000 0000 1 watt-hour = 3 600. joules 3-556 3025 = 3.6 kilojoules 0-556 3025 1 kilowatt-hour = 3 600 000. joules 6-556 3025 = 1 000. watt-hours 3-000 0000 For further conversion factors, see table of units of Energy, page 74. The relations to other measures are as follows: f Joules = volts X coulombs. = volts X amperes X seconds. = volts 2 X seconds -4- ohms. = amperes 2 X ohms X seconds. = ohms X coulombs 2 -f- seconds. " = watts X seconds. Joules of stored energy = farads X volts 2 -f- 2. -microfarads X volts 2 -=-2 000 000. Ergs of stored energy = microfarads X volts 2 X 5. Joules of kinetic energy of the current: = henrys X final amperes 2 -4- 2. = henrys X applied volts 2 -4- (ohms 2 X 2). = time-constant in seconds X ohms X final amperes 2 -4- 2. = time-constant in seconds X final amperes X applied volts -f- 2. = time-constant in seconds X applied volts 2 -4- (ohms X 2). Ergs of kinetic energy of the current = henrys X final amperes 2 X5X 10 6 . When the flux is due only to the current, as in self-induction, and when there is no magnetic leakage: Joules of stored energy = max wells X ampere-turns -4- (2 X 10 8 ). Ergg of stored energy = maxwells X ampere-turns -4-20. When there is magnetic leakage: Joules of stored energy = mean maxwells X ampere-turns -4-2 X 10 s . Erg.s of stored energy = mean max wells X ampere-turns -4- 20. When the flux is from an external source, and independent of the cur- . rent, as in mutual induction, and when there is no magnetic leakage: Joules of stored energy = maxwells X ampere-turns -4- 10 s . Ergs of stored energy = maxwells X ampere-turns -4- 10. When there is magnetic leakage substitute "mean maxwells" for " maxwells." For alternating current circuits: t Joules per cycle = watts -4- frequency. = effective amperes X effective volts X cos $-4- frequency. t Treatises on alternating currents should be consulted for the limiting conditions under which these relations apply to alternating or other peri- odically varying quantities. 124 ELECTRICAL POWER. ELECTRICAL POWER [P]. (Current X electromotive force ; energy -~ time j energy X frequency.) The units of electrical power are all convertible directly into units cf other kinds of power, and the electrical and absolute units have there- fore been included with the mechanical, thermal, and other units in the general table of all the units of Power (p. 80). The present table is lim- ited to a few specific values and to 391116 relations between the electrical unit of power and other electrical units. The power in watts is equal to the energy in joules divided by the time in seconds; this applies to both direct and alternating currents whether there is phase shifting in the latter case or not. According to Joule's law, the power in watts is also equal to the square of the current in am- peres multiplied by the resistance in ohms, or to the current in amperes multiplied by the electromotive force in volts, or to the square of the electromotive force in volts divided by the resistance in ohms. These apply to direct currents; they apply to alternating currents also, but only under certain conditions, the chief one of which is that there is no reactance in the circuit and therefore no phase shifting (caused by in- ductance or capacity), in which case the relations refer to the effective values of the current and electromotive force. For further information treatises on alternating currents should be consulted. In any case, these relations give the true power when the result is multiplied by the power factor (see below). The unit universally used is the watt, by which is here meant the watt of the International Congress of 1893, in Chicago, defined as equal to 10 7 C.G.S. units of power (erg per second) and represented sufficiently well for practical use by the work done at the rate of one joule per second; in the relations in these tables this defined value is used. It was made legal in this country by Congress in 1894, and is accepted by the National Bureau of Standards. The kilowatt ( = 1000 watts) is also common, The electromagnetic C.G.S. unit (or absolute unit) is an erg per second. The electrostatic C.G.S. unit (or absolute unit) is the same erjj per second. Power Factor is a term used to show the amount of true power con- tained in a given amount of apparent power. It is the ratio of the true power to the apparent power. Its use is limited chiefly to electric power generated by alternating currents. With direct electric currents, the power is equal to the product of the volts and the amperes, and is called watts; with alternating currents, however, this is true only when the volts and amperes are exactly in phase with each other, which often is not the case. When there is such a difference in phase, that is when the current lags behind or precedes the voltage, their product is only apparent power and is usually measured in terms of the product of the volts and the amperes and called volt-amperes. If the true power in such a casa is measured in watts, then the power factor will be the number of watts divided by the number of volt-amperes, and it will always be less than unity, in practice usually between 07 and 0.95. For true sine waves, the real power in watts is equal to the voltage X current X cos 0, in which is the angular phase difference; hence it follows that in such cases the power factor is numerically equal to cos , whose numerical value is found directly from a table of cosines. Sometimes the power factor is stated in percent, in which case it is equal to the above figure multiplied by 100. The inductance factor is the ratio of the wattless volt-amperes to the total volt-amperes; or the ratio of the wattless component of the current or the e.m.f. to the total current or e.m.f. The sum of the squares i)f the inductance factor and the power factor is equal to unity . ELECTROCHEMICAL EQUIVALENTS. 125 .ELECTRICAL, POWER. Logarithm 1 CGS unit (elmg) = 1 erg per second 0-000 0000 1 CGS unit (elst) = 1 erg per second 0-000 0000 1 absolute unit = 1 erg per second 000 0000 1 erg per second == 1 CGS unit (elmg) 000 0000 = 1 CGS unit (elst) 0-000 0000 = 10- 7 watt 7.000 0000 1 microwatt = 10. ergs per second 1 000 0000 1 watt [W, w] = 10 000 000. ergs per second 7-000 0000 1 joule per second 000 0000 1 kilowatt = 1 000 watts 3-000 0000 For further conversion factors, see table of units of Power, page 80. The relations to other measures are as follows: f "Watts = volts X amperes. = amperes 2 X ohms. = volts 2 -T- ohms. = coulombs X volts -5- seconds. = coulombs 2 X ohms -v- seconds. 2 = joules -s- seconds. For alternating-current circuits: f Watts = volts X amperes X cos . = amperes 2 X ohms X cos 0. = volts 2 X cos $ -H ohms. =effec. volts Xeffec. amp. X ohms resistance -5- ohms impedance. = joules per cycle X frequency. Mean watts = effective volts X effective amperes X cos <. Power factor = true power in watts H- apparent power in volt-amperes. = energy component of current or e.m.f. -f- total current or e.m.f. = \/( 1 inductance factor 2 ) . Inductance factor: = wattless component of current or e.m.f. -s- total current or e.m.f. = >/(! power factor 2 ). ELECTROCHEMICAL EQUIVALENTS and DERIVA- TIVES. (Weight -7- quantity of electricity; quantity of electricity -:- weight ; weight -r- energy j energy -r- weight.) The electrochemical equivalent of any chemical element or ion is the amount by weight which changes its chemical combination per coulomb of electricity during electrolysis. For these and their derivatives various compound units are used such as milligrams per coulomb, grams per ampere-hour, pounds per ampere-hour, etc. The relations between most of these are simply the relations between the respective units of weight, which see under Weights. The reciprocals of these are also used frequently. The electrochemical equivalents of any elements or ions are, according to Faraday's law, proportional to their atomic weights and inversely proportional to their changes of valency. For determining the actual values, that of some one element must be determined experimentally, after which all the others can be calculated. In the following relations the value taken as a basis is the electrochemical equivalent of silver adopted by the International Electrical Congress of Chicago in 1893, in the defi- nition of the ampere, and legal in this country, namely 0.001 118 gram per coulomb. The atomic weight of silver used in these relations is 107.93, t Treatises on alternating currents should be consulted for the limiting conditions under which these relations apply to alternating or other peri- odically varying quantities. 126 ELECTROCHEMICAL EQUIVALENTS. DEPOSITS. which is the usually accepted value on the basis of = 16. These funda- mental values correspond with the usually accepted value of the ionic charge, 96 540. coulombs per monovalent gram ion, within the limits of accuracy of the data. For a complete table of the equivalents and their derivatives, of all the various elements and for various changes of valency, accompanied by descriptions of how to use them, see the author's Table of Electrochemical Equivalents and their Derivatives, in Electrochemical In- dustry, Jan., 1903, p. 169. The following relations apply to all elements or compounds. (The atomic weights are all based on oxygen = 16, but the electrochemical equivalents are independent of whether the atomic weights are based on = 16 or on H = l.) Milligrams per coulomb = 0.010359X] atomic weight Grams per ampere-hour = 0.037 291 X \ , . weignt . Pounds per ampere-hour = 0.000 082 21 X J cnan S e of valency Grams per watt-hour =0.037 291 X 1 Kilograms per kilowatt hour =0.037 291 X atomic weight Kilograms per horse-power hour = 0.027 806 X !* weignt Pounds per kilowatt-hour = 0.082 21 X I cnan g e of val. X volts. Pounds per horse-power hour = 0.061 30 X J Coulombs per milligram = 96.54 X 1 c hanee of valencv Ampere hours per gram =26.816 X \ -^ F^. Ampere-hours per pound = 12 164. X J atomic weight. Watt-hours per gram =26.816X1 Kilowatt-hours per kilogram =26.816X ohane-p n f val Kilowatt-hours per pound = 12.164 X [ - ^ X volts. Horse-power hours per kilogram =35.964 X atomic weight Horse-power hours per pound =16.313X J Ionic charge for a monovalent gram ion = 96 539. coulombs. For the amount of gas in cubic centimeters at C. and 760 mm mer- cury pressure, developed at one electrode, on the basis that one gram mole- cule of a gas has a volume of 22.38 liters, the following relations exist, in which n is the number of atoms per molecule : Cb. centimeters per ampere-hour =834. 6-5- (nX change of valency). Ampere-hours per cb. centimeter = 0.001 198 XnX change of valency. ELECTROLYTIC DEPOSITS. (Mass -f- time; mass -5- surface.) Two kinds of units are used for measuring deposits, such as those in elec- trolysis. One is for measuring the weight of the deposit on a limited sur- face, and includes such units as grams per square decimeter, ounces per square foot, etc., these are all given above in the same table as that for Pressures (page 63) and are therefore not repeated here The other kind of unit is for measuring the rate of deposition, and in- cludes such units as milligrams per second, pounds per day, tons per year, etc.; the reduction factors for these are given in the following table. The year is taken as equal to 365M days and the ton as equal to the short ton of 2000. pounds. Aprx. means within 2%. Logarithm 1 pound (av) per year [Ib/yrJ: = 0.002 737 85 pound per day. Aprx. i^-f- 1000 3-4374098 = 0.000 862 409 gram per minute. Aprx. % *- 1000 4-9357131 I kilogram per year [kg/yr]: = 0.006 035 93 pound per day. Aprx. 6 -f- 1 000 3-780 7440 = 0.002 737 85 kilogram per day. Aprx. ^A^\ 000 3-437 4098 = 0.001 901 29 gram per minute. Aprx. 19^-10 000 3-279 0473 0.666 667 ounce per hour, or? 0.453 592 kilogram per day. Aprx. % *- 10 0.314 995 gram per minute. Aprx. 31 -* 100 0.182 625 ton (short) per year. Aprx. H'e * 10 0.165 675 metric ton per year. Aprx. ELECTROLYTIC DEPOSITS. 127 Logarithm 1 grain per hour [g/h]: = 0.016 666 7 gram per minute, or Veo, which see .......... 2-221 8487 1 milligram per second [mg/s]: = 0.06 gram per minute, which see for other values ......... 2-778 1513 1 pound (av) per day [Ib/day] = 365.25 P9unds per year. Aprx. 11 AX 100 ....... , . . . 2-562 5902 165.675 kilograms per year. Aprx. Ve X 1 000 ........ 2-219 2560 "-823 9087 -656 6658 498 3033 -261 5602 -.--.- . ._ .-219 2560 = 0.041 6667 P9und per hour, orV 2 4 ..................... 2-619 7888 = 0.018 899 7 kilogram per hour. Aprx. 19 ^ 1 000 ....... 2-276 4548 I ounce (av) per hour [oz/h]: 1.5 pounds per day, or% ....................... 0-176 0913 = 0.680 389 kilogram per day. Aprx. 68-*- 100 ............ 1-8327571 = 0.47 2 492 gram per minute. Aprx. "/a * 10 ............. 1-6743948 I kilogram per day [kg/day]: = 805.238 pounds per year. Aprx. 800 ............... 2-905 9244 = 365.250 kilograms per year. Aprx. l \i X 100 ......... 2-562 5902 = 0.694 444 gram per minute. Aprx. Vio ............... 1-841 6375 = 0.402619 ton (short) per year. Aprx. Mo ............. I 604 8944 = 365 250 metric ton per year. Aprx. iff-flO ......... 1-562 5902 = 0.091 859 3 pound per hour. Aprx. Vn ................ 2 963 1230 = 0.041 6667 kilogram per hour, or 1/24 .................. 2-619 7888 I giam per minute [g/min]: = 1159.54 pounds per year. Aprx. %Xl 000 ........... 30642869 = 525.96 kilograms per year. Aprx. Vi 9 X 10 000 ....... 2-720 9527 = 60 grams per hour ............................ 1-778 1513 = 16.6667 milligrams per second, or ioo/ 6 ............ ---- 1-2218487 = 3 174 66 pounds per day. Aprx. s'4'w ................ 0-501 6967 = 1.440 00 kilograms per day. Aprx. 1% ................ 0-158 3625 = 0.579772 ton (short) per year. Aprx. ty .............. 1-763 2569 = 0.525 96 metric ton per year. Aprx. 10 /i9 ............. 1-720 9527 = 0.132277 pound per hour. Aprx. % * 10 .............. 1-1214855 = 0.06 kilogram per hour. ........................ 2-778 1513 1 ton (short) per year [tn/yr]: = 5.475 70 pounds per day. Aprx. % . . . : ............. 0-738 4398 = 2.483 74 kilograms per day. Aprx. 1% ................ 0-395 1058 = 1.724 82 grams per minute. Aprx. % ................. 0-236 7431 = 0.228 154 pound per hour. Aprx. %-i-lO ........... . . 1-358 2288 = 103 489 kilogram per hour. Aprx. 3 >i-5-10 ............ 1-0148944 1 metric ton per year [t/yr], = 6.035 93 pounds per day. Aprx. 6 ............... ---- 0-780 7440 = 2.737 85 kilograms per day. Aprx. 3 /n X 10 ........... 0-437 4098 = 1.901 29 grams per minute. Aprx l %o ............... 279 0473 = 0.251 497 pound per hour. Aprx. M .................. I 400 5328 = 0.114 077 kilogram per hour. Aprx. 8 / 7 ................ 1-057 1986 1 pound (av) per hour [lb/h]; = 8 766. pounds per year. Aprx. J^X 10 000 ........... 3-9428015 = 3 976.19 kilograms per year. Aprx. 4 000 ............. 3-599 4672 = 10.886 2 kilograms per day. Aprx. 11 ................ 1-036 8770 . 7.559 87 grams per minute. Aprx. ^X 10 ............. 0-878 5145 = 4.383 tons (short) per year. Aprx. % X 10 .......... 0-641 7715 = 3.976 19 metric tons per year. Aprx. 4 ............... 0-599 4672 1 kilogram per hour [kg/h]: = 19 325.7 pounds per year. Aprx. 19 000 .............. 4-286 1356 = 8 766. kilograms per year. Aprx. H X 10 000 ......... 3-942 8015 = 52.910 9 pounds per day. Aprx. 53 ................... 1-723 5454 = 16.666 7 grams per minute, or i% .................... 1-221 8487 = 9.662 86 tons (short) per year. Aprx. 2% or %i X 100 ---- 0-985 1056 = 8.766 metric tons per year. Aprx. 7 /sX 10 ........... 0-942 8015 1 ounce (av) per minute [oz/min]: = 3^ pounds per hour, which see for other values .......... 0-574 0313 128 ELECTROCHEMICAL ENERGY. ELECTROCHEMICAL ENERGY. The term electrochemical energy is used to refer to electrical energy when it performs chemical work, or to chemical energy when it is set free as electrical energy. The most convenient unit for expressing electrochemical energy is the joule, as the calculations are then the simplest; the calorie is, however, the one generally used in 'tables which give the energy of com- bination of chemical compounds; values in calories or any other units of energy are readily reduced to joules with the aid of the reduction factors given under Energy, page 74. Care must be taken to distinguish between the large and the small calorie, a distinction which is often not made in text-books and tables. There is a very simple and direct way of calculating how many volts will be required to decompose a chemical compound electrolytically, or how many volts will be generated in a battery in which chemical compounds are formed, when the heats or energies of combination of the compounds are known. It is sometimes called Thomson's law, and can be directly deduced from Faraday's law and the principle of the conservation of energy. It is important to remember, however, that the number of volts thus calculated is limited to that required to supply the necessary energy of decomposition, or that generated in a battery by the formation of compounds; it includes nothing more. The actual voltage involves some further correction factors, which, though generally small as com- pared with the voltage of decomposition, are sometimes of importance. Among these correction factors is the voltage required to overcome the resistance of the electrolyte; this depends on the current flowing and on the resistance of the electrolyte. Another correction factor is the Gibbs or Helmhpltz temperature coefficient, namely the rate of change of the voltage with the temperature; this falls out when there is no such change, or in practice when this change is inappreciable. Another correction fac- tor is what is called the "over-voltage" ; this depends on the fact that it takes different voltages to set free the same gas at electrodes of different metals. For these correction factors treatises on electrochemistry should be consulted. In this rule for calculating the voltage due to the heat of combination, which is given below, the constant is based on the relation between the joule and the calorie given in the table of units of Energy, and on the same fundamental electrochemical and chemical constants for silver which were used to calculate the reduction factors for Electrochemical Equiva- lents above, namely 0.001 118 gram per coulomb as the electrochemical equivalent of silver, and 107.93 as the atomic weight of silver, on the basis ofO = 16. These fundamental values correspond with the usually accepted value of the ionic charge, 96 540. coulombs per monovalent gram ion, within the limits of accuracy of the data. For the heats of com- bination of compounds, see reference-books on this subject; the data are usually given in calories per gram molecule, but care must be taken to find out whether the large calorie or the small calorie is the one meant; also whether it is a gram molecule or a kilogram molecule that is meant ; care must also be taken to see that the given number of calories apply to the exact decomposition or combination under consideration, whether gases are set free and escape as such, or whether they recombine, and if the latter, whether this recombination enters into the electrochemical reaction or whether it is purely chemical, as by local action, also whether water or some other accompanying product is formed or decomposed, etc. ELECTROCHEMICAL ENERGY. RELUCTANCE. 129 Rule for calculating the voltage of decomposition or composi- tion, from the heat of combination. Logarithm For monovalent ions, or for one equivalent weight: The number of volts: = the number of kilogram calories per gram molecule X 0.043 363 Aprx. % -i- 10. ... 2. 637 n 62 = the number of kilogram calories per kilogram molecule, or gram calories per gram molecule, X 0.000 043 363. Aprx. %-MO 000 5.637 1162 = tne number of kilogram calories per gram molecule -r 23.061. Aprx.%XlO . 1-3628838 = the number of kilogram calories per kilogram molecule, or gram calories per gram molecule, -^23 061. Aprx. % X 10 000 4.362 8838 For miiltivalent ions divide the volts thus obtained by the valency. That is, for bivalent ions calculate the voltage from the above and then divide by 2; for trivalent ions divide by 3, etc. Or the number of caloriea may be reduced to that for one equivalent weight (by dividing the calories by the valency), and the above rules will then give the volts directly. MAGNETIC RELUCTANCE [61, R]; MAGNETIC RESIS- TANCE. (Magnetomotive force -v- flux; length -~ (surface X permeability) .) Reluctance measures the amount by which the material in a magnetic circuit or part of a circuit resists or opposes th^ flux; the more it resists, the greater the reluctance ; it is the opposite to permeance and numerically it is equal to the reciprocal of permeance. As the reluctance of a centi- meter cube of air (or more correctly, of a vacuum) in th3 C. G.S system is, by definition, equal to unity (that is, to one oersted), it follows that the amount of reluctance of any magnetic circuit or any part of it also represents the number of times that its reluctance is greater than that of the air of the same volume and shape. Reluctance is analogous to resistance in an electric circuit, but differs in these three important features: (1) in circuits con- taining magnetic materials it generally varies very greatly with the flux density, being constant only for air or other diamagnetic materials ; (2) there is no such a thing as a magnetic insulator, that is, an infinitely great reluct- ance; (3) maintenance of a flux through a reluctance does not necessarily require the continuous expenditure of energy. A reluctance in oersteds is equal to the magnetomotive force in gilberts divided by the flux in max- wells The reluctance of a circuit in oersteds is equal to the length of the circuit in centimeters divided by its cross-section in square centimeters and then either multiplied by the reluctivity or divided by the permeability; but in magnetic materials the reluctivity and the permeability vary with the density of the flux in the circuit, hence this method of calculation is practicable only for non-magnetic materials like air, the permeability of which is constant and is equal to unity; it may be applied to air-gaps, for instance. The reluctance is not used very often in magnetic calculations, these beino; usually based on the property called permeability, as is ex- plained bek)w under the units of magnetizing force. The only unit used is the C.G.S. unit called an oersted, which is the reluctance through which a C.G.S. unit of magnetomotive force (called a gilbert) will produce a C.G.S. unit of flux (called a maxwell). 1 CGS unit (elmg) = l oersted. \ oersted = 1 CGS unit (elmg). 130 MAGNETIC RELUCTIVITY. The relations to other measures are as follows: Oersteds: = gilberts -r- maxwells. = gilberts -5- (gausses Xsq. centimeters section). = ampere-turns X 1.256 64f -=- max wells. = ampere-turns X 1.256 64f * (gausses X sq. centimeters section). = CGS unit of current-turns X 12.566 4J -^-maxwells = CGS unit current-turns X 12. 566 4 J-r- (gausses Xsq. cm. section). = 1 -5- permeance in CGS units. = centimeters length -5- (sq. centimeters section X permeability). = centimeters length X reluctivity -r-sq. centimeters section. = inches length X 0.393 700 *- (sq inches section X permeability). = inches length X reluctivity X 0.393 700 -r-sq. inches section. = (number of turns X 0.000 1 ) 2 X 1 .256 64f -5- henrys. Oersteds in iron = oersteds in air -r- permeability, in air = oersteds in iron X permeability MAGNETIC RELUCTIVITY [v]; SPECIFIC MAGNET- IC RELUCTANCE; MAGNETIC RESISTIVITY; SPECIFIC MAGNETIC RESISTANCE. (Imperme- ability j magnetizing force -f- magnetic induction j reluc- tance X surface length ; reluctance -h reluctance.) Reluctivity measures the number of times that a material resists or opposes magnetic flux, more than air (or vacuum) does. It is a property of a material and numerically it is equal to tha reciprocal of the permea- bility, which see. It corresponds in some respects to resistivity or specific resistance in electrical circuits. It is specific reluctance based on air; but as the reluctivity of air is unity, that of any other material is numerically equal to the reluctance in oersteds of a centimeter cube of that material. Reluctivity is seldom used in magnetic calculation, its reciprocal, the permeability, being used instead. Like permeability, it is a property of the material in a magnetic circuit, and its values are different with different flux densities. There are no units, as it is a mere ratio. The relations of reluctivity to other magnetic measures are the reciprocals of those for permeability. The chief relations are: Reluctivity = 1 + permeability.' = sq. centimeters section X oersteds -r- centimeters length. = sq. inches section X oersteds X 2. 540 01 ^-inches length. MAGNETIC PERMEANCE; MAGNETIC CONDUCT- ANCE j MAGNETIC CAPACITY. (1 -reluctance j flux -f- magnetomotive force j surface -~- (length X permea- bility).) Permeance measures the amount by which a given magnetic circuit con- ducts the flux, the better it conducts, the greater the penmeance; it is the opposite to reluctance, and numerically it is equal to the reciprocal of reluct- ance. It is analogous to conductance in an electrical circuit Just as the reluctance of a given magnetic circuit refers to that particular circuit or its parts, so permeance refers to a particular magnetic circuit or its parts. In referring to the property of the material itself, independently of its t Or 4;r/10. Aprx. Y* X 10. Log 099 2099- J Or 4*. Aprx. Y% X 100. Log 1-099 2099 PERMEANCE . PERMEABILITY. 131 size and shape, the term permeability or specific permeance is used, which see below. When a given magnetic circuit contains a magnetic material like iron, the permeance does not remain constant like the conductance of an electric circuit, but it varies very greatly with the density of the flux; it is constant only for air. Permeance is seldom used in magnetic calcu- lations (see under magnetizing force). It can always be avoided by using the reciprocal of the reluctance instead ; both may be avoided by basing the calculations on the permeability. The unit is the C.G.S. unit, which is the reciprocal of the absolute or C.G.S. unit of reluctance, that is, the reciprocal of an oersted; it has no name. The permeance of a centimeter cube of air (or vacuum) between two parallel sides is unity in the C.G.S. system. Hence the permeance of a magnetic circuit or of a part of it, also represents the number of times that its permeance is greater than that of an equal volume of air of the same size and shape. It is equal, in C.G.S. units, to the permeability multiplied by the cross-section in square centimeters and divided by the length in centimeters. The relations to other measures are as follows: Permeance = maxwells * gilberts. = 1 -T- oersteds. = permeability Xsq. centimeters sections centimeters length. = permeability X sq. inches section X 2.540 01 -r- inches length. Permeance of iron = permeance of air X permeability. of air = permeance of iron -5- permeability. See also the reciprocals of the relations given under Rel uctance. MAGNETIC PERMEABILITY [u] ; SPECIFIC PERI&E, ANCE j MAGNETIC CONDUCTIVITY. (Magnetic induction -v- magnetizing force \ flux density -r- flux density ; permeance -j- permeance ; 1 -~ reluctivity.) Permeability is a very important quantity in most magnetic calculations ; it measures the number of times that a material conducts or aids magnetic flux better than air does; if, for instance, the permeability of a certain kind of iron under certain conditions is 300 , it means that it conducts magnetic flux or lines of force 300. times as well as an equal amount of air would under the same conditions, that is, it is 300. times as permeable to the flux . It may be said to be magnetic conductivity compared to air as a standard. If a given magnetomotive force produces a certain amount of flux in a given circuit of air, it will produce 300. times this flux if the air were replaced by this kind of iron. It follows from this that the magnet- izing force (usually represented by H) multiplied by the permeability (n) gives the induction (B) in the iron, that is, the flux density in the iron; or permeability = induction -s- magnetizing force; this is further explained under the units of magnetizing force. Permeability is an inherent property of materials and its values are usually given in tables or curves. It is also the same as specific permeance, which is the permeance of a material as compared with that of air; as the permeance of a centimeter cube of air is unity, the law ,above given follows. Permeability is the reciprocal of reluctivity. It is analogous to conductivity or specific conductance in electrical calculations, with a very important difference, however, namely that while the electrical conductivity of a material is constant and does not change- with the current which flows, the permeability of the chief magnetic materials varies very greatly with the density of the flux, and its values must therefore be given for each flux density, for which reason they are usually given in the form of curves called permeability curves or magneti- zation curves. For air, however, the permeability is always constant an^ is numerically equal to unity. Paramagnetic bodies are those whojH* permeability is greater than unity, and diamagnetic bodies those whose permeability is less than unity. 132 PERMEABILITY. MAGNETOMOTIVE FORCE. There are no units of permeability, as it is a mere relation, number, or ratio between two quantities of the same kind. In this respect it is like specific gravity. For a brief description of calculations involving permeabilities see under the units of Magnetizing Force. The relations to other measures are as follows: Permeability: = gausses in iron -5- gausses in air. = inch gausses in iron-:- inch gausses in air. = permeance of iron -f- permeance of air. = oersteds of air -J- oersteds of iron. = 1-7- reluctivity. = 1 + [susceptibility X 12.566 4 (or 4w)]. = gausses -v- gilberts per centimeter. = gausses X 0.7 95 77 5 (or 10-^-4^) -J- ampere-turns per centimeter. = gausses X 2.540 01 -i- gilberts per inch. = gausses X 2. 021 27 -=- ampere-turns per inch. = inch gausses X 0.393 700 -h gilberts per inch. = inch gausses X 0.3 13 297 -f- ampere-turns per inch. = gausses X 0.07 9 577 5 (or 1 -f-4)-*-CGS unit current-turns per cm. = centimeters length -j-(sq. centimeters section X oersteds). = inches length X 0.393 700-r-(sq. inches section X oersteds). = permeance X centimeters length -j-sq. centimeters. For the relations with maxwells substitute for "gausses," in any of the above, "maxwells -j-sq. centimeters section"; and for "inch gausses," sub- stitute "max wells -r-sq. inches section." MAGNETIC SUSCEPTIBILITY [>]. (Intensity of mag, netization H- magnetizing force.) This quantity is used chiefly in physical conceptions; it is somewhat similar to permeability in that it expresses the magnetizability of a sub- stance. It is equal to the intensity of magnetization (see below) divided by the magnetizing force which produces it. There are no units, as it is a mere ratio or number. Its relation to permeability is as follows: Susceptibility = (permeability -1)X 0.07 9 577 5 (or l-Mjr). MAGNETOMOTIVE FORCE [m.m.f. SF, F]j AMPERE- TURNS [a-t]; MAGNETIC POTENTIAL; DIFFER- ENCE OF MAGNETIC POTENTIAL; MAGNETIC PRESSURE. (Current X turns ; flux X reluctance ; energy -~ pole strength.) These units are used to measure the magnetic pressure or "motive force" which produces or ends to produce a magnetic flux in a magnetic circuit, just as an electromotive force tends to produce a current of electricity, or a pressure of water tends to produce a flow of water and might similarly be called the hydraulic motive force. In practice magnetomotive forces are generally produced by, and are often measured in terms of, what are called ampere-turns; this term means the product of an electric current in am- peres and the number of turns or windings of the coil through which it flows; such a current-carrying coil produces a definite magnetomotive force, which in turn produces an amount of flux dependent on the amount of reluctance in the whole magnetic circuit. According to the laws of electromagnetism, the magnetomotive force in C.G.S. units (gilberts) is always numerically equal to the ampere-turns multiplied by 4^-7-10, whether MAGNETOMOTIVE FORCE. 133 there is iron in the magnetic circuit or not. The magnetomotive force in gilberts in any closed loop encircling a long straight wire through which a current passes is 4n times the C.G.S. unit of current, or 4?r-:-10 times the number of amperes. It is always directly proportional to the current pro- ducing it. The law of the magnetic circuit is similar to Ohm's law for the electric circuit, namely that the flux (corresponding to the current) is equal to the magnetomotive force (corresponding to the electromotive force), divided by the reluctance (corresponding to the resistance); hence the magnetomotive force in C.G.S. units (gilberts) is equal to the flux in C.G.S. units (maxwells) multiplied by the reluctance in C.G.S. units (oersteds). This is true whether there is iron in the magnetic circuit or not. Most calculations occurring in practice are simplified by using the quan- tity called the magnetizing force instead of the magnetomotive force, as is explained below under the units of Magnetizing Force. There are three units of magnetomotive force in use; the more general, and often the more convenient one, is the ampere-turn, which is equal to the magnetomotive force produced by one ampere flowing once around a magnetic circuit; this is irrespective of the shape or size of the electric or magnetic circuit; the latter affect only the reluctance of the magnetic cir- cuit and thereby the resulting flux. This unit bears an incommensurate relation to the absolute magnetic units owing to the factor 4?r. The second usual unit is the electromgaiietic C.G.S. unit (or absolute unit), called a gilbert. Its definition is based on the fact that the magneto- motive force produced in any one closed loop around a long straight wire through which a C.G.S. unit of current (or 10 amperes) flows, is 4?r C.G.S. units of magnetomotive force; hence one such unit is equal to l-r-4;r of this magnetomotive force. It may also be defined as that magnetomotive force which will produce a flux of one C.G.S. unit (maxwell) through one C.G.S. unit of reluctance (oersted). The third unit is like the ampere-turn except that the current is the C.G.S. unit of current (10 amperes) instead of the ampere. This unit is therefore equal to 10 times the ampere-turn unit. It has no name other than the C.G.S. unit car rent-turn. It is never used in practice. Logarithm 1 CGS unit (elmg) = 1 gilbert, which see for other values. 1 gilbert: 1 CGS unit (elmg) of magnetomotive force. = 0.795 775 (or 10/4?r) ampere-turn. Aprx. subtr. V 5 . . . . 1-900 7901 = 0.079 577 5 (or l/4;r) CGS unit of current-turn. Aprx. 8 /ioo 2-900 7901 1 ampere-turn = 1.256 637 (or 4?r/10) CGS units. Aprx. add Y 0-099 2099 = 1.256637 (or 4 7T/10) gilberts. Aprx. add M- 0-0992099 0.1 CGS unit of current-turn 1-000 0000 1 CGS unit of current-turn: = 12.566 37 (or 4*) CGS units (elmg). Aprx. 10% 1-099 2099 = 12.566 37 (or 4;r) gilberts. Aprx. 10% 1-099 2099 10. ampere-turns 1-000 0000 The relations to other measures are as follows : Gilberts: == ampere-turns X 1.256 64. t = CGS unit current-turns X 12. 566 4.J = maxwells X oersteds. = max wells -^permeance (in CGS (elmg) units). = maxwellsXcm length -5- (sq. cm section X permeability). = maxwells X inches length X 0.393 700 -H (sq. inches section X permea- bility). = maxwells X centimeters length -=-sq. centimeters section. For air. = maxwells X ins. length X 0.393 700-n sq. ins. section. For air. = gausses X centimeters length -r- permeability. = gausses X inches length X 2 540 01 -f- permeability. = inch gausses X inches length X 0.393 700 * permeability. = gausses X centimeters length. For air. = gausses X inches length X 2.540 01 . For air. = inch gausses X inches length X 0.393 700. For air. = gausses X oersteds X sq. centimeters section. t Or 47T/10. Aprx. add M- Log 0-099 2099- \ Or 4*. Aprx. y&X 100. Log 1.099 2099- 134 MAGNETOMOTIVE FORCE. MAGNETIZING FORCE. Gilberts for iron = gilberts for air -5- permeability. Gilberts for air = gilberts for iron X permeability. Ampere-turns: = gilbertsX 0.795 775.f = COS unit current-turns X 10, *= max wells X oersteds X 0.795 775. t = maxwells X 0.795 775f-*- permeance (in CGS (elmg) units). = maxwells X cm length XO. 795 775f-^(sq. cm section X permeability). = maxwells X ins. length X 0.313 296-Ksq. ins. section X permeability), = maxwells X cm length X 0.795 775f *- sq. cm section. For air. = maxwells X ins. lengthXO.313 296-5-sq. ins. section. For air. = gausses X centimeters length X 0.795 77 5f-*- permeability. = gausses X inches length X 2. 021 27 -5- permeability. = inch gausses X inches lengthXO.313 296 -s- permeability. = gausses X centimeters length X 0.795 775. t For air. = gausses X inches length X 2. 021 27. For air. = inch gausses X inches length X 0.313 296. For air. = gausses X oersteds X sq. centimeters section X 0.795 775. f Ampere-turns for iron = ampere-turns for air -s- permeability. Ampere-turns for air = ampere-turns for iron X permeability. CGS unit of current-turns = ampere-turns X 0.1. = gilberts X 0.07 9 577 54 (For further relations divide those for ampere-turns by 10.) When the flux is due only to the current, as in self-induction, and when there is no magnetic leakage: Ampere-turns =ergs of kinetic energy of the current X 20 -5- max wells. = joules of kinetic energy of the current X 2 X lO 8 -?- max wells. = maxwells X number of turns 2 -T- (henrys X 10 8 ). When the flux is from an external source, and independent of the current as in mutual induction, and when there is no magnetic leakage: Ampere-turns =ergs of kinetic energy of the current X 10 -T- maxwells. = joules of kinetic energy of the current X 10 s -4- maxwells. MAGNETIZING FORCE [X, H]J MAGNETOMOTIVE FORCE per CENTIMETER j MAGNETIC FORCE; FIELD INTENSITY. (Turns X current -f- length j magneto- motive force -T- length j induction -f- per me ability j flux den- sity -5- permeability j forces- pole strength.) This quantity, which is one of the most important in the more usual mag- netic calculations, is used to measure the magnetomotive force produced per unit length of a coil or solenoid carrying an electric current ; or the magnetomotive force required per unit length of any part of a magnetic circuit to produce the desired flux density in that part. The usual calcu- lations of magnetic circuits then often become simpler than they would be if the whole magnetomotive force itself is used. In electric circuits it has its analogy in the electromotive force produced per centimeter length of active wire in an armature of a dynamo or in a transformer, or the difference of potential required per centimeter length of a conductor in order to pro- duce the desired current density in that conductor. There are no specifically named units of magnetizing force. The one most frequently used when the metric system is employed, is an ampere- turn per centimeter length. When inches are used the unit is an am- pere-turn per inch. The absolute or C.G.S. unit is one gilbert per centimeter. When the magnetic circuit consists of air, the magnetizing t Or 10H-4;r. Aprx. % . Log 1-900 7901- i Or l-*-4ff. Aprx. 8-s-lOO. Log 2-900 7901- MAGNETIZING FORCE. 135 force can also be expressed and measured in terms of units of flux density, namely gausses, for, although they mean something different, they are numerically the same as gilberts per centimeter for air, as is shown below. If a current flows through a uniform coil of wire which is very long as compared with its diameter, the flux density produced in its interior will be practically uniform except near its ends; it will be nearly uniform throughout if the two ends are brought together to form a ring coil. More^ over, this flux density is independent of the diameter of the coil or the shape of its cross-section, which affect only the reluctance and the total flux. The magnetizing force produced by such a coil, whether it contains iron or not, is numerically equal in gilberts per centimeter to 4irnc, in which n is the number of turns or windings per ceritimete length of coil, and c is the current in absolute units; nc is therefore the number of current-turns per centimeter, corresponding to (but not numerically equal to) the ampere- turns per centimeter. Imagine a series of planes perpendicular to the axis and one centimeter apart; then the magnetizing force given by this formula will be the magnetomotive force in gilberts produced between each plane and the next. As an analogy, suppose the interior were replaced by a con- ductor carrying an electric current, and the coil itself were replaced by a device which induces an electromotive force in that conductor, then the volts induced per centimeter length of this device will evidently be the volts of electromotive force which exist between each of these parallel planes and the next. It is also true for iron as well as for air that the magnetizing force equals the flux density divided by the permeability, but as the permeability of air is unity by definition, it follows that for air (or more exactly for a vacuum) the magnetizing force of such a coil when expressed in gilberts per centimeter, is numerically equal to the flux density in its interior in gausses The above general formula therefore also gives, as a special case, the flux density in air in gausses. This has given rise to much confusion, as it makes it appear at first sight as tlwugh a magnetomotive force was of the same nature as a flux density, which with the electric units would be like saying that an electromotive force was of the same nature as a current density. The explanation is that the reluctance of a centimeter cube of air is unity, hence the flux density in gausses through each centimeter cube of air will be numerically the same as the magnetomotive force in gilberts acting between its two opposite faces; or in the above illustration with the imaginary parallel planes one centimeter apart, the flux density in gausses in each space between two such planes will for air be numerically equal to the magnetomotive force (in gilberts) between each plane and the next. Analogously, if a wire happens to have a resistance of one ohm per foot, the number of volts acting at the ends of each foot will be numerically the same as the number of amperes flowing. The identity is in the num- bers and not necessarily in the nature of the units; moreover, the numerical identity exists only between the units gilberts per centimeter and gauss, and not between the other units like ampere-turns. The formula 4?mc therefore always gives the magnetizing force produced in gilberts per centimeter length of coil .whether there is iron in the coil or not. It also gives the flux density in gausses in the interior of the coil, but for air only, c is the current in absolute units and must be replaced in the formula by C-^10, if C is to be in amperes. When the result given by this formula is multiplied by the entire length of the coil in centi- meters, it gives the total magnetomotive force of the whole coil, in gilberts. The same magnetizing force will produce entirely different flux densities in materials of different permeabilities, just as the same voltage will pro- duce entirely different currents in materials of different conductivities. The magnetizing force in gilberts per centimeter multiplied by the permea- bility of any material gives the flux density in gausses, or the induction in gausses, produced in that material by that magnetizing force. Or in dif- ferent terms, if a coil produces in its interior a certain flux density in gausses in air, then that flux density must be multiplied by the permeability of the material to get the flux density or induction in that material in gausses, when the air circuit is completely replaced by that material, as in trans- formers; for the case in which the circuit is partly air and partly iron, see the next paragraphs. The more usual calculations, such as those for dy- namos and transformers, start with the desired induction, and in order to avoid the calculation of the reluctance, the troublesome factor 0.4*, and 136 MAGNETIZING FORCE. the various other reduction factors when inch units are used, such calcula- tions are generally made as described in the following paragraphs. Magnetic calculations, like those for dynamos and transformers. The values of the permeabilities of the particular iron or steel which is to be used, are usually given in the form of a curve called a permeability curve or a magnetization curve, whose horizontal distances are the magnetizing forces in ampere-turns per centimeter (//), and whose vertical ones are the corresponding inductions (B) or flux densities, in gausses or lines of force per square centimeter in that quality of iron or steel. It must be decided multiply this by the axial or center-line length in centimeters, of this par- ticular part of the iron under consideration, be it the cores or the yoke- pieces, or the armature, and the result will be the total number of ampere- turns or the magnetomotive force, which is required to magnetize that part to that particular induction. Notice that the length here used is that of the path of the flux through that part of the iron, and not necessarily the length of the coil. Having done this for each of the iron parts making up the whole circuit, add them all together and the result will be the total ampere-turns required for the total iron part of the circuit. The ampere- turns required for producing the flux in the air-gap are calculated from the desired flux density in the gap, as follows: ampere-turns = flux density in gausses X length of air-path of flux (that is, twice the length of the gap) in centimeters X 0.795 775. These latter ampere-turns (which generally are by far the larger part of the whole) are then added to those required for the iron part, thus giving the total required for the complete magnetic circuit. The coils for producing these ampere-turns may have any length and may be wound around any convenient part of the circuit, for when the mag- netic circuit is chiefly of iron, the magnetizing force of the coils will act in that circuit very nearly the same way no matter how they are distributed over the iron. When the dimensions are all in inches and the flux densities in maxwells per square inch (inch-gausses), then the magnetization curve should be E lotted for those units and the calculations are then precisely the same except :>r the air-gap, for which the formula then becomes: ampere-turns = flux density in maxwells per sq. inch X length of air-path of flux in inches X 0.313 296. When the linear dimensions are in inches, and the flux densities in gausses, the curve should be plotted accordingly and the calculations are again the same except for the air-gap, for which the formula then becomes: ampere- turns = gausses X length of air-path of flux in inches X2. 021 27. It will be noticed that in this method of calculation the reluctance need not be known. The total required cross-section of the iron is determined by dividing the given total flux by the given flux density. The reverse calculation to the above is much more difficult; in that case the ampere- turns of such a composite magnetic circuit together with all the dimen- sions are given, and the flux produced by them is to be determined. In such a case perform the calculation backwards, by making trial calcula- tions as just described, using different assumed total fluxes, until one is found which will require the given number of ampere-turns. For a simple magnetic circuit like that in a transformer, in which there is no air-gap, either calculation amounts to little more than reading off the results from the magnetization curve. Logarithm 1 gilbert per inch: = 0.795 775 ampere-turn per inch. Aprx. subt. % 1-900 7901 = 0.393 700 gilbert per centimeter. Aprx. */io 1-595 1654 = 0.313 296 ampere-turn per centimeter. Aprx. %e 1-4959555 1 ampere-tuvii per inch: = 1.256 64 gilberts per inch. Aprx. add % 0-099 2099 = 0.494 738 gilbert per centimeter. Aprx. ^ 1-694 3753 = 0.393 700 ampere-turn per centimeter. Aprx- ^lo. .... 1-595 1654 1 gilbert per cm = 2.540 01 gilberts per inch. Aprx. *% 0-404 8346 = 2.02127 ampere-turns per inch. Aprx. 2. 0-3056247 1. CGS unit (elmg) 0-000 0000 = 0.795 775 amp.-turu per cm. Ap. subt. % . L900 7901 MAGNETIZING FORCE. MAGNETIC FLUX. 137 Lrgarithm 1 CGS unit (elmg) = 1. gilbert per centimeter . 0-000 0000 1 ampere-turn per cm: = 3.191 86 gilberts per inch. Aprx. 3% 0-504 0445 = 2.540 01 ampere-turns per inch. Aprx. 1( )4 0-404 8346 = 1.256 64 gilberts per cm. Aprx. add % 0-099 2099 1 CGS unit of current-turn per centimeter: = 31.918 6 gilberts per inch. Aprx. 32 1.504 0445 = 25.400 1 ampere-turns per inch. Aprx. 25 1-404 8346 = 12.566 4 gilberts per centimeter. Aprx. l / s X 100 1-099 2099 10. ampere,yturns per centimeter 1-000 0000 For air (or vacuum) only : 1 gauss = 1 gilbert per centimeter. The relations to other measures are as follows : Ampere-turns per centimeter: = gausses X 0.795 77 5t * permeability. = maxwells X 0.7 95 77 5f-Ksq. cm section X permeability). = gausses X 0.7 95 77 5f- For air. = maxwells X 0.795 77 5f *- sq. cm section. For air. Ampere-turns per inch: = gausses X 2.021 27 -* permeability. = maxwells X 0.313 297 -r-(sq. inches section X permeability). = inch gausses X 0.313 296 ^permeability. = gausses X 2. 021 27. For air. = maxwells X 0.313 297 -s- sq. inches section. For air. = inch gausses X 0.3 13 296. For air. Gilberts per centimeter = gausses -f- permeability. = max wells -r-(sq. cm section X permeability). = gausses. For air. = maxwells -r- sq. centimeters. For air. Gilberts per inch: = gausses X 2.540 01 -4- permeability. = maxwells X 0.393 700 -5- (sq. inches section X permeability). = inch gausses X 0.393 700 -* permeability. = gausses X 2.540 01 . For air. = maxwells X 0.393 700 -4- sq. inches section. For air. = inch gausses X 0.393 700. For air. CGS unit of current-turns per centimeter: = gausses X 0.079 577 5 Impermeability. = gausses X 0.079 577 54 For air. MAGNETIC FLUX [*, <]; LINES OF FORCE; FLUX OF FORCE; AMOUNT OF MAGNETIC FIELD; POLE STRENGTH [m]. (Magnetomotive force -=- reluct- ance ; magnetic induction (or flux density) X surface ; elec- tromotive force X time; length X \/f ore e.) These units are used to measure the total quantity or amount or number f magnetic lines of force or the amount of flow or flux of magnetism, just as amperes are used to measure the quantity or amount of electric current, or as cubic feet per second measure the amount of a flow of water. This magnetic flux or flow is in some respects analogous to a current of electricity, as it must always form a closed circuit upon itself, and be the same in amount in every cross-section of the circuit; it follows a law analogous to Ohm's law, as the flux is equal to the magnetomotive force divided by the t Or 10-^4*. Aprx. 8/10. Log 1. 900 7901- t Or l-fr-4*. Aprx. 8/100. Log 2-900 7901- 138 MAGNETIC FLUX. reluctance. It differs, however, in that no work is being done continuously in the circuit of a magnetic flux, and it can therefore continue to exist in- definitely as in a permanent magnet, without consuming or producing energy. Energy is stored in the flux when it is produced, and it is given out again whenever the flux ceases to exist, but no energy is necessarily required to maintain it ; it is therefore in this respect more like a mechanical pressure or stress, as that of compressed air. The kinetic energy (which see above) required to start an electric current is stored in the system as magnetic energy, and given out again as electrical energy when the current stops. From the energy standpoint , magnetic flux is more analogous to coulombs of electricity. A magnetic circuit also differs from an electric circuit in that it can never be opened, as there is no such a thing as a magnetic insu- lator; magnetic flux can cease only by contracting to a point somewhat like an extremely small rubber band which is allowed to contract after hav- ing been stretched. The space surrounding a magnet or an electric current or that between two magnetic poles, is called a magnetic field or magnetic iield of force, as it contains magnetic flux or magnetic lines of force; units of flux measure the total amount of this field (but riot its intensity or density) or the total number of such lines of force ; this flux also continues through the magnet itself. Flux is also equal to the flux density (sometimes called induction), in lines of force per square centimeter (or per square inch), multiplied by the number of square centimeters (or square inches) cross-section of its path, and is then often called the total flux to distinguish it from the flux density. An electric current is always encircled by such flux, and the two circuits, namely the electric and the magnetic, are always linked together like the two links of a chain. When the magnetic flux enclosed in a coil or loop of wire is increased or diminished, an electromotive force is produced in that wire; this is the fundamental principle of a dynamo; or stated in different terms, when a wire cuts through magnetic flux, an electromotive force is produced in the wire; or the linking and unlinking of circuits of flux and electric circuits produces an electromotive force; it is this that produces self- and mutual induction. The term flux-turns or mean flux-turns is sometimes used for denot- ing the product of the number of turns and the mean flux (in maxwells) in one turn; the magnetic leakage is thereby eliminated. The mean flux- turns in maxwell-turns are equal to the self-inductance in henrys multiplied by 10 8 times the final current in amperes. The unit universally used is the absolute or electromagnetic C.G.S. unit or single line offeree, and is defined as that amount of flux which acting on a unit magnetic pole will propel it with a force of one dyne. It can also be defined as the amount of flux passing through one square centi- meter cross-section of a field having a flux density of one C.G.S. unit. A unit magnetic pole (imaginary) is one which will exert a force of one dyne on another unit pole one centimeter distant. From each such pole there radiates a flux equal to 4;r (or 12.566 4) of these units or lines of force. A single or unit line of force in a magnetic field may be said to stand for or represent a tube of such a cross-section that it always embraces a unit of flux; a definite amount of flux may have widely different lengths of cir- cuit or cross-sections without changing its amount. This C.G.S. unit is called a maxwell, according to the International Congress of 1900. A maxwell is therefore the same thing as a single or unit line of force as above defined. Logarithm 1 COS unit (elmg) = 1 maxwell. 000 0000 1 maxwell: 1 CGS unit (elmg) 000 0000 1 line of force. 000 OGOO 1 gauss-centimeter 2 000 0000 = 0.155 000 gauss-inch 2 . Aprx. 3ia 1-190 3308 = 0.079 577 5 (or 34 TT) of the flux from a unit pole. Aprx. 8 /ioo 29007901 1 weber (obsolete) = 1 maxwell 0-000 0000 1 unit pole (flux from) = 12.566 37 (or 4 ?r) maxwells 1.099 2099 1 Kapp line (obsolete) = 6 000. maxwells 3 778 1513 MAGNETIC FLUX. 139 The relations to other measures are as follows: Maxwells: = gausses X sq. centimeters. = in -h-gaussesXsq. inches. = gilberts -r- oersteds. = gilberts X permeance (in CGS units)., = gilberts X permeability X sq. centimeters section -& centimeters length. = gilberts X permeability X sq. inches section X 2.540 01 -s- inches length. = gilberts Xsq. centimeters section -r- centimeters length. For air. = gilberts Xsq. inches section X 2. 540 01 -f- inches length. For air. = gilberts per centimeter X permeability X sq. centimeters section. = gilberts per inch X permeability X sq. inches section X 2. 540 Oi. = gilberts per centimeter X sq. centimeters section. For air. = gilberts per inch X sq. inch section X 2.540 01. For air. = ampere-turns X 1 .256 64f -*- oersteds. = ampere-turnsX 1.256 64f X permeance (in CGS units). = ampere-turns X permeability X sq. centimeters section X 1.25.664f-*- centimeters length. = ampere-turns X permeability X sq. inches section X 3.191 86 -f- inches length. ampere-turns X sq. centimeters section X 1 .256 64 f * centimeters length. For air. = ampere-turns Xsq. inches section X 3. 191 86 ^-inches length. For air. = ampere-turns per centimeter X permeability X sq. centimeters sec- tion X 1.256 64. f = ampere-turns per inch X permeability Xsq. inches section X 3. 191 86. = ampere-turns per cm X sq. centimeters section X 1.256 64. t v For air. = ampere-turns per inchXsq. inches section X 3. 191 86. For air. = CGS unit current-turns X 12.566 4J -5- oersteds. (For further relations with CGS unit current-turns, multiply those in terms of ampere-turns by 10; that is, substitute for "ampere-turns" in any of the above, the quantity "CGS unit current-turns X 10.") Maxwells = volts X seconds XIO 8 -^ number of turns. = volts X seconds X 10 8 . For a single conductor. When the flux is due only to the current, as in self-induction, and when there is no magnetic leakage: 'Maxwells = joules of stored energy X 2 X 10 s -j- ampere-turns. = joules of stored energy X2X 10 8 -s- amperes. For a single wire. = ergs of stored energy X 20 -f- ampere-turns. = henrys X ampere-turns X 10 8 -*- number of turns 2 . " = henrys X final amperes X 10 8 -5- number of turns. When there is magnetic leakage, substitute for "maxwells" in the above, "mean maxwells." The mean maxwells are the mean flux-turns divided by the total number of turns. When the flux is from an external source, and independent of the current, as in mutual induction, and when there is no magnetic leakage: Max wells = joules of stored energy X 10 8 -r- ampere-turns. = joules of stored energy X 10 s -f- amperes. For a single conductor. = ergs of stored energy X 10-4- ampere-turns. When there is. magnetic leakage, make the same substitution as above described. Mean maxwells (through the secondary): = henrys (of mutual induction) X final amperes (of primary) X 10 8 + number of turns (of secondary). t Or 47T/10. Aprx. add l /i. Log 0-099 209fJ. t Or 4n. Aprx. 1 /%X 100. Log 1.Q99 2099- 140 MAGNETIC FLUX DENSITY. MAGNETIC FLUX DENSITY [X, H]j . MAGNETIC INDUCTION [ ; B] ; LINES OF FORCE PER UNIT CROSS-SECTION; EARTH'S FIELD. (Flux^-sur- face ; magnetizing force X permeability.) This quantity, which is one of the most important in magnetic calcula- tions, measures the extent to which a body is magnetized as expressed by the amount of flux which exists per square centimeter (or square inch) of cross-section of the circuit; it gives the density of the flux in C.G.S. units (maxwells or lines of force) per square centimeter or square inch cross-sec- tion. It corresponds to current density in electrical calculations. As it specifies or determines the strength of a magnetic field , it is often called the field strength or field intensity ;t the earth's magnetic field, for instance, is expressed in these units. When it refers to air it is generally represented by 3C or simply H. When it refers to the flux density "induced" in a mag- netic material, such as iron, by an outside source, such as a current in a coil of wire, it is often called the induction, generally expressed by (B or sim- ply by B, which is one of the most important quantities in magnetic cal- culations. In the calculation and design of dynamos and transformers this induction or flux density in the iron is of prime importance. From it as a starting-point, the size of the core and the required ampere-turns are calculated, as was explained briefly under the units of magnetizing force. The saturation-point of magnetic material like iron is expressed in terms of these units of flux density. The total flux in maxwells is equal to the flux density in gausses multiplied by the total cross-section in square centi- meters. As the permeability or reluctivity of air is unity, it follows that the magnetizing force in gilberts per centimeter of a long coil is numerically the same as the flux density in gausses produced by it in the interior of the coil, when there is no magnetic material in it. For this reason the magne- tizing force is often confused with flux density or its equivalent the intensity of field, as it is then often called. This applies only to the absolute units; when ampere-turns or when inch units are used, a numerical factor must be introduced. The unit universally used when the dimensions are in the metric system is the electromagnetic C.G.S. unit, which, according to the International Congress of 1900, is called a gauss. It is defined as that field intensity which is produced at the center of a circle of one centimeter radius by 1 C.G.S. unit of current (or 10 amperes) flowing through an arc of this circle one centimeter long. This is one of the relations which connect the electric with the magnetic units. It can also be defined as the inten- sity of the field at one centimeter distance from a unit pole, that is, at the surface of a sphere of one centimeter radius, having an imaginary C.G.S. .unit pole at its center; from such a pole 4;r unit lines of force (maxwells) emanate, and as the area of the sphere is 4?r square centimeters, it follows that the flux density will be one line of force or maxwell per square centi- meter of the spherical surface. It may also be defined as that field inten- sity which will exert a pull of one dyne on an (imaginary) isolated unit magnetic pple placed in it. All three of these definitions refer to the same unit. The formulas giving the field intensity in coils, like those for mag- nets or for galvanometers, all give the result in terms of this unit, provided the current is stated in terms of the absolute unit of current which is equal to 10 amperes; great care must be taken in such formulas to use the proper unit of current; it should always be stated whether the formula has been reduced to amperes or not. The practical unit is therefore the same as the C.G.S. unit, namely the gauss, which means one maxwell (or line of force) per square centi- meter; and a flux density or induction stated in a number of gausses means that number of maxwells (or lines of force) per square centimeter. fit should be distinguished, however, from the term intensity of mag- netization (see below), which is a term sometimes used in physics and has a different meaning. MAGNETIC FLUX DENSITY. 141 When the dimensions are in inches the flux density and induction are often for convenience stated in lines of force (or maxwells) per square inch; this in .h unit has no generally accepted name; the name inch gauss is here proposed and is used in these tables. Logarithm I maxwell per sq. inch = 1. inch gauss 0-000 0000 = 0.155000 gauss. Aprx. 2,4s 1-1903308 1 inch gauss = 1. maxwell per sq. inch 000 0000 = 0.155 000 gauss. Aprx. 2 A 3 ] .190 3308 1 CGS unit (elmg) = 1 . ga.uss 000 0000 1 gauss = 6. 451 63 maxwells per sq. inch. Aprx. x % or 6^. . . . 0-809 6692 = 6.451 63 inch gausses. . Aprx. 1: % or 6^ 809 6692 = 1. CGS unit (elmg) of flux density 000 0000 1. CGS unit (elmg) of flux per sq. centimeter. . . 0-000 0000 = 1. maxwell per sq. centimeter 000 0000 1. magnetic "line of force" per sq. centimeter.. . 000 0000 1 maxwell per sq. centimeter = 1. gauss 000 0000 1 kilogauss =1 000. gausses 3-000 0000 The relations to other measures are as follows: Gausses = max wells -r-sq. centimeters. = inch gausses X 0.1 55 000. = gilberts per centimeter X permeability. = gilberts per inch X permeability X 0.393 700. = gilberts per centimeter. For air. -gilberts per inch X 0.393 700. For air. = gilberts X permeability -r- centimeters. = gilberts X permeability X 0.393 700 -5- inches. = gilberts -=- centimeters. For air. = gilberts X 0.393 700 H- inches. For air. = gilberts -r- (oersteds X sq. centimeters). = ampere-turns per centimeter X permeability X 1.256 64. f = ampere-turns per inch X permeability X 0.494 738. = ampere-turns per centimeter X 1.256 64. f For air. = ampere-turns per inchXO.494 738. For air. = ampere-turns X permeability X 1 -256 64 1 * centimeters. = ampere-turns X permeability X 0.494 738 -=- inches. = ampere-turns X 1 .256 64| -*- centimeters. For air. = ampe re-t urns X 0.494 738 -i- inches. For air. = ampere-turns X 1 .256 64t *- (oersteds X sq. centimeters). = CGS unit current-turns per centimeter X permeability X 12.566 4.t (For further relations with CGS unit current-turns, multiply those in terms of ampere-turns by 10; th -t is, substitute for "ampere-turns" in any of the above, the quantity "CGS unit current-turns X 10.") Gausses = volts X seconds X 10 s -r- (number of turns X sq. centimeters). = CGS units of intensity of magnetization X 12.566 44 Inch gausses : = max wells -i-sq. inches. = gausses X 6. 451 63. = gilberts per centimeter X permeability X 6.451 63. = gilberts per inch X permeability X 2. 540 01. gilberts per centimeter X 6.451 63. For air. = gilberts per inch X 2.540 01 . For air. = gilberts X permeability X 2.540 01 -s- inches. = gilberts X 2.540 01 ^-inches. For air. = gilberts -*- (oersteds X sq. inches). = ampere-turns per centimeter X permeability X 8. 107 35 = ampere-turns per inch X permeability X 3.191 86. = ampere-turns per centimeter X 8. 107 35. For air. = ampere-turns per inch X 3.191 86. For air. = amoere-turns X permeability X 3.191 86 -Cinches. = ampere-turns X 3. 191 86-r-inches. For air. = ampere-turns X 1.256 64f-=-( oersteds X sq. inches). = CGS unit current-turns per centimeter X permeability X 8 1.07 3 5. t Or 4;r/10. Aprx. add M- Log 099 2099, JOr4r. Aprx. i^XlOO. Log 1 099 2099- 142 MAGNETIC MOMENT. INTENSITY. (For further relations with CGS unit current-turns, multiply those in terms of ampere-turns by 10; that is, substitute for "ampere-turns" in any of the above the quantity "CGS unit current-turns X 10.") Inch gausses = volts X seconds X 10 s -s- (number of turns X sq. inches). Gausses in iron = gausses in air X permeability. ' ' in air = gausses in iron -5- permeability. Inch gausses in iron = inch gausses in air X permeability. in air = inch gausses in iron -*- permeability. MAGNETIC MOMENT [fflfe]. (Pole strength X length.) This quantity, used chiefly in magnetometry, is the product of the pole strength of a magnet multiplied by its theoretical length, that is, by the distance between the two centers at which the poles may be considered to be condensed. As a pole (see under flux) is not measured in units like force, such a moment is not directly comparable with a mechanical momer t called torque, but as the force existing between two unit poles one centi- meter apart is one dyne, a magnetic moment may be converted into a mechanical moment. As a single pole has no real existence such a cal- culation in practice always involves the action of two poles on two others. . There are no special units. The C.G.S. unit is a unit pole multiplied by a centimeter, and would therefore be called a pole-centimeter. A max- well-centimeter might also be used, as a unit pole has 4^ ( = about 12^) maxwells or lines of force issuing from it ; each line of force exerts a force of one dyne on a unit pole. Two unit poles one centimeter apart attract or repel each other with a force of one dyne; the force between any two poles is proportional to the product of the two pole strengths in terms of the above unit poles, and inversely proportional to the square of the dis- tance between them in centimeters. 1 unit- pole-centimeter unit -=1. unit pole XI. centimeter. = 1. CGS unit of magnetic moment. The relation to other measures are as 'follows: Magnetic moments (in CGS units): = CGS unit poles X centimeters. = intensity of magnetization (in CGS units) X volume in cb. cm. = gausses X 0.07 9 577 5 X volume in cb. cm. INTENSITY OF MAGNETIZATION [3, I]; MOMENT PER UNIT VOLUME; POLE STRENGTH PER UNIT CROSS-SECTION. (Magnetic moment 4- volume ; pole strength -f- surface .) This quantity, used chiefly in physical conceptions, measures the polar- '.zed state of the interior of a magnet. If a magnet were cut into small pieces (assuming that the magnetic state was not altered thereby) each piece would be a separate magnet whose magnetic moment bears the same proportion to its volume as the moment of the original magnet bears to its volume, hence the magnetic state remains the same if stated in the mag- netic moment per cubic centimeter, which quantity is called the intensity of magnetization. It is also the pole strength per square centimeter cross- section. As pole strength is convertible into flux (maxwells), it follows that the intensity of magnetization is a unit of the same nature as flux den- sity, that is, maxwells per square centimeter or gausses. They differ only in the bases on which they are defined. The C.G.S. unit is one unit moment per cubic centimeter, that is, one unit-pole-centimeter per cubic centimeter, or one unit pole per square centimeter. This unit is numerically equal to 4n gausses, as its relation to gausses is the same as the relation of a unit pole is to a unit of flux. 1 CGS unit of intensity of magnetization: 1. CGS unit of magnetic moment per cb. centimeter. = 1. unit-pole-centimeter unit of magnetic moment per cb. cm. = 12.566 4 (or 4;r) gausses. The relations to other measures are as follows: CGS units of intensity of magnetization: = CGS units of magnetic moments -^cb. centimeters. = gausses X 0.07 9 577 5. MAGNETIC ENERGY. 143 MAGNETIC WORK or ENERGY [W], (Magnetomotive force X flux 5 ampere-turns X flux.) This quantity is seldom used in calculations. When a current is started in a wire or in an electro-magnet, or when the armature of a steel magnet is pulled off, magnetic energy is stored ; it is given out again in some other form, often in the form of a spark, when the current is stopped, or as mechanical energy when a permanent magnet attracts its armature to itself. In a transformer, the energy of the primary current is all converted into magnetic energy which is reconverted into electrical energy in the secondary circuit. The magnetic energy is equal to. and in fact is the same thing as, the kinetic energy of a current (which see above). It is equal to the product of magnetomotive force and flux. It appears as heat in the hysteresis loss. It should not be confused with the power used contin- uously in exciting an electromagnet, as that power is all converted electric- ally into heat; it is only when the current is first started that any electric energy is converted into magnetic energy. Magnetic flux itself is not energy any more than coulombs of electricity or mechanical pressure; energy is required to produce a pressure, but not necessarily to maintain it, and so it is with magnetic flux (which see above). There are no specific units of magnetic energy; it is usually measured in joules or ergs, but may be measured in terms of any of the units of energy, which see. The C.G.S. unit is the erg; the practical unit is the joule. The relations to other measures are as follows: Joules of magnetic energy [J] = henrys X final amperes 2 -*- 2. = henrys X applied volts 2 -*- (ohms 2 X 2). = time constant in seconds X ohms X final amperes 2 -r- 2. = time constant in seconds X final amperes X applied volts * 2. = time constant in seconds X applied volts 2 -5- ( ohms X 2). When the flux is due only to the current, as in self-induction, and wheix there is no magnetic leakage: Joules of magnetic energy: = max wells X ampere-turns -f- (2 X 10 8 ). = gausses X sq. centimeters X ampere-turns -*- (2 X 10 8 ). = inch-gausses X sq. inches X ampere-turns -5- (2 X 10 8 ). = maxwells 2 X oersteds X 0.397 887t-^-10 8 . = maxwells X gilberts X 0. 397 887 1 ^ 10 8 . = gilberts 2 X 0.397 887 f-*- (oersteds X 10 s ). = ampere-turns 2 X permeability X sq. centimeter section X 0.628 318 (or 2 TT /I ())*-( centimeters lengthXIO 8 ). = ampere-turns 2 X permeability X sq. inches section X 1 .595 93 *- inches lengthXIO 8 . (For further relations substitute for any of the above units their equiva- lents in terms of the desired units, as given in the other tables.) When the flux is from an external source and independent of the current, as in mutual induction, and when there is no magnetic leakage, the mag- netic energy is twice as great as that given by the above relations; hence all the values above given must be multiplied by 2. When there is magnetic leakage, use the "mean maxwells" instead of the "maxwells." The mean maxwells are the mean flux turns divided by the total number of turns. Ergs of magnetic energy = henrys X final amperes 2 X5X 10. (For further relations of ergs to other units multiply those given above for joules by 10 7 .) t Or 10-*-8;r. Aprx. Sixteenth size = Twenty-fouth size = Thirty-second size = Forty-eighth size = Ninety-sixth size = = 12 inches to the foot. 6 " " " " 4 3 2 . inch PAPER MEASURE. MISCELLANEOUS MEASURES. 1 quire = 24 or 25 sheets. 1 ream =20 quires = 480 sheets. 1 ream = generally 500 sheets. 1 bundle (obs.) = 2 reams = 1 000 sheets. 1 hale (obsolete) = 5 bundles. 1 dozen =12. 1 gross =12 dozen 1 great gross = 12 gross. = 144. 1 jjreat gross = 144 dozen = 1 728. 1 score =20. FUNCTIONS OF It. 169 USEFUL FUNCTIONS OF 7t. Logarithm Aprx. means within 2%. K (called " pi" )= circumference of a circle divided by the diameter and is a constant. JT = approximately 2 %, which equals 3.142 86 or ^100% too much. TT = approximately 85 ^iia, which equals 3.141 592 9. n= 3.141 592 653 589793 238 462 643 383 279 502 88. t . . . 497 1499 2^ = 6.283 185 307 180 798 1799 3*= 9.424777960769 09742711 4?r= 12.566 370 614359. Aprx. HX 100 1-0992099 5*= 15.707 963 267 949 1-196 1199 6;:= 18.849 555 921 539 1-275 3011 7 = 21.991 148 575 129 1-3422479 8* = 25.132741 228718 1-400 2398 9^ = 28.274 333 882 308 1-451 3924 10^ = 31.415 926 535 898 1-497 1499 471/3 = 4.188 790 204 786 622 0886 jr/1 =3.141 5927. Aprx. 2% 04971499 jr/2 = 1.5707963. Aprx. !% 01961199 7r/3 = 1.047 197 6 0020 0286 ir/4 = 0.785 398 2. Aprx. 8 /io I 895 0899 */5 = 0.628 318 5 1-798 1799 */6 = 0.5235988 ' 1-718 9986 7r/7=0.4487990 1-6520518 /8 = 0.392 699 1 1-594 0599 7r/9 = 0.349 065 9 1-542 9074 7r/10 = 0.314 159 3 1-497 1499 jr/12 = 0.261 799 4 1-417 9686 jr/16 = 0.196 349 5 1-293 0298 7r/32 = 0.098 174 8 2 991 9999 ff /64 = 0.049 1)87 4 2 690 9698 K /108 = 0.0290888 2463 7261 n -/180 = 0.017 453 3 . 2 241 8774 7r/360 = 0.008 726 65 3 940 8474 1 x 7r/4 = 0.785 398 2. Aprx. 8 /io I 895 0899 2X7r/4 = 1.5707963 196 1199 3 X 7T/4 = 2.356 194 5 372 2112 4 X 7r/4 = 3.141 5927 04971499 5 x 7T/4 = 3.926 990 8 594 0599 6X;r/4 = 4.7123890 673 2411 7 X ;r/4 = 5.497 787 1 740 1879 8X;r/4 = 6.283 1853 798 1799 9 X 7r/4=7.0685835 849 3324 10X7r/4=7.853981 6 895 0899 l/7r = 0.318 309 9. Aprx. %2 I 502 8501 2/7r = 0.636 619 8 I 803 8801 3/;r = 0.9549297 1 979 9713 4/7r= 1.273 2395 104 9101 5/;r = 1.591 5494 0201 8201 6/^ = 1.9098593 0-281 0014 7/7r = 2.228 169 2 0347 9482 8/7r = 2.546 479 1 0-405 9401 9/7r = 2.864 7890 - 0-4570928 10/r = 3.183 098 9 0-502 8501 l2/;r = 3.819 718 6 0-582 0314 t Ludolph's value. For Vega's value to 140 decimal places see Ilistoire des Recherchea aur la Quadrature du Cercle, by Montucla, 1831, p. 282 170 FUNCTIONS OF it. Logarithm jrv/2 = 4.4428829 0-647.6649 ;T-H x/2 = 2.221 441 4 0-346 6349 4*-*- 10 = 1.256637061. Aprx. add M 00992099 n 2 = 9.869604401089. Aprx. 10 0-9942997 4;r 2 =39.478418. Aprx. 40 1-5963597 x2+4 = 2.46740110. Aprx.^XlO 0-3922397 7T 3 = 31.006 276 680 300 1-491 4496 ;r 4 = 97.409 091 1-988 5995 TT> =306.019 69 2 485 7494 TT C =961.389 19 2 982 8992 1 * x* = 0.101 321 2 1 005 7003 1 -J- n 3 = 0.032 251 5 2 508 55U4 X/* = 1.772453850908 ,...02485749 2V* = 3.544 907 6 549 6049 v/27r = 2.506 628 : 399 0899 tyn = 1.464 592 165 7166 1 + TT = 0.318 309 866 1-502 8501 1-^2* = 0.159154933. Aprx. %* 10 1-2018201 l-M;r = 0.07957747. Aprx.s/ioo 2-9007901 10-J-47T = 0.7957747. Aprx.s/lo 1-9007901 10-5-8* = 0.3978873. Aprx.^io 15997601 1^-N/ff = 0.5641896 1-7514251 I-J-^TT = 0.6827841 18342834 Log it = 0.497 149 872 694 133 854 35. Log,, n = 1.144 729 885 849 400 174 14. it = 180 considered as an angle. 7T/180 = 0.017 453 29 2-241 8774 ;r/360 = 0.008 726 65 3 940 8474 ISO/* = 57.2957795 1-7581226 360/7T =114.591 559 2-059 1526 USEFUL NUMBERS. Logarithm 2 = 1.414 21. Aprx. VrXlO 0-1505150 = 1.732 05. Aprx. % 0-238 5606 2= 1.259 92. Aprx. add % 0-100 3433 3 = 1.442 25. Aprx. ^X 10 0-1590404 4 = 1.587 40. Aprx.%. 0-200 6867 LOGARITHMS. PHYSICAL CONSTANTS. 171 SYSTEMS OF LOGARITHMS. The logarithm [log] of a given number is the exponent which denotes the power to which a certain fixed numerical base is raised, in order to produce this given number. Thus if the base is 10, then the log of 100 is 2, because 10 raised to the 2d power =100. To multiply two numbers, add their logs; to divide, subtract the log of the divisor from that of the dividend; then from the resulting log find the corresponding number. Log or logic means the common, usual, or Briggs' logarithm; the base of this system is 10. Logg or In means the Naperian, natural, or hyperbolic logarithm; the base of this system is about 2.718 (see below), generally denoted by e. Log of 1 =0 in any system. Log of base = l in its own system. e = base of Naperian, natural, or hyperbolic logarithms. c = 2.718281 828. logio of e- 0.434 294 481 903 252. Log<> of e = 1 . 10 = base of usual, common, or Briggs' logarithms. log* of 10 = 2.302 585 092 994 046. Logio of 10 = 1. log* 10Xlog 10 e = l. The modulus of any system is the constant by which the Naperian logarithm of a number must be multiplied to give the logarithm of the number in that system. The modulus of any system is equal to the recip- rocal of the Naperian log of the base of that system. Modulus of Naperian system = 1 -v-log* of e = 1. Modulus of common system = 1 * log* of 10 = 0.434 294 481 903 252. The logarithm of a number (n) in any system is equal to the modulus of that system multiplied by the Naperian logarithm of the number. Or: Logio n = modulus (common system) Xlog^ n. Log Russia, 165 Tschetwerik, Russia, 55 Tschetwerka, Russia, 55 Tschetwert, Russia, 33, 55 Tsubo, Japan, 44, 172 Tsun, China, 172 Turn, Sweden, 33 Turn, cubic, Sweden, 55 Tun, 53 Tunna, Sweden, 55, 173 Tunne, Germany, 54 Sweden, 55 Tunnland, Sweden, 44 U Uncia, ancient, 34 Unge, Switzerland, 61 Unit, circular, defined, 41 Unit current-turn, 133 do. per centimeter, 137 Unit pole, magnetic, 138 Unit-pole-centimeter unit, 142 Units, see under respective names absolute system, 11 absolute vs. concrete, 12 C.G.S. system, 11 changing in formulas, 6 concrete vs. absolute, 12 compound names of, 4 inter-relation of, 1 three groups of, 1 vs. quantities, 4 U.S. standards, 29, 45, 56, 98, 108 112,115,117,119,122,124,166 U.S. to British, volumes, 46 Unze, Germany, 60 Useful numbers, 170 V Valency, 125, 126, 129 Vara, California, 3-1 Argentine, Cent. America, Chile, Cuba, Curagoa, Mexico, 172 Paraguay, Peru, Venezuela, 173 Spain. 34, 173 Varying functions, 97 Vector potential, 22, 24 Vector quantities 96 Vedro, Russia, 173 Velocity i angular, 86 do-, physical, 19, 23 do., frequency, 121 do., rate of increase of , 88 concrete units, 86 light, 86 do. as a relation, 11, 14, 21, 24, 25,96 linear, 85 do., physical, 19 do., rate of increase of, 87 molecules, 86 Velte, France, 54 Venezolano, Venezuela, 166 Vergees, Isle of Jersey, 172 Verst, Russia, 33 Vierling, Germany, 60 Viertel, Antwerp, 55 Violle. 146 defined, 145 Vis-viva, 72 do., physical, 19 INDEX. 195 Vlocka, Russian Poland, 173 Volt: tables, 109 text, 108 applied, 110 electro-chemical energy, 129 induced, 110 international, 110 do., denned, 108 legal, 109 Reichsanstalt, 110 relations to other units, 110 standards, denned, 108 true, 110 do., denned, 109 to calories, 129 -ampere, 80 -coulomb (joule), 74 Voltage, 108 of decomposition, 128 do., calculation of, 129 Vorktum, Sweden, 33 Volume: table, 46 text 45 fundamental standards, 45 digit conversion tables, 51 Ehysical, 18 jreign, 54 U.S. to British, 46 water, 69, 71 and mass, 67 and time, 95 weights, 67 from specific gravities, 69 W Water, flow of, 95 foot of, pressure, 65 meter of, pressure, 65 pressures of, 63 specific heat of, 171 volume of, 71 weights, 70 Watt: table, 80, 125 defined, 124 relations to other units, 125 magnetic power 144 per candle, 149 -hour. 76, 123 defined, 122 per gram, 126 per minute 81 per second, 82 -second (joule), 74 Wave-length 31 Waves, periodicity, 86 Weber, 138 Weddras, Russia, 55 Wedro, Russia. 55 Week, 94 Weight: table, 57 text, 56 fundamental standards, 56 digit conversion tables, 59 physical, 18 bars, 62 coatings, 63. 126 forces, 83 deposits, 63, 126 foreign, 60, 172 materials, 67 rails, 62 relative, chemical, 60 sheets, 63 water, 69-71 wires, 62 and length, 62 and measures, tables, 30-173 do., text, 27 and surface, 63 and money, 167 and volume, 67 from specific gravities, 69 Werschock, Russia, 33 Werst, Russia, 33 Weston cells, defined, 108 do., voltage of, 110 do., temperature correction, 111 Wey, 54 Winchester bushel, 53 Wires, weights of, 62 Wispel, Germany, 54 Work: tables, 74 digit conversion table, 77 text, 72 electrical, 122 do., units, 74, 123 magnetic, 143 physical, 19 rate of doing, 79 Yard: table, 30 to meters, digit table, 39 cubic, 50 do., to cb. meters, digit table, 51 solid, 54 square, 42 do., to sq. met., digit table, 44 Year ( solar ^. 94 calendar, civil, common Julian, lunar sidereal, 94 anomalistic , Gregorian legal , nat- ural, tropical, 95 Yen, Japan, 165 196 INDEX. Zehnling, Germany 60 Zent, Germany, 60 Zoll, Austria, Germany, Switzer- land, 33 Zorzec, Poland, 55 n, as an angle, 89 n, useful functions of, 169 10 to the nth power, 8 % denned, 7 % grades, 90 /oo denned, 8 Voo grades, 90 - (hyphen) in names of units, 4 THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL PINE OP 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO SO CENTS ON THE FOURTH DAY AND TO $!.OO ON THE SEVENTH DAY OVERDUE. NG1NH)(4 G LIBRARY 10m-7,'44(1064s) ,YA 0307 995716 i.-- THE UNIVERSITY OF CALIFORNIA LIBRARY