otj, PHYSICAL UNITS BY MAGNUS MACLEAN, M.A., D.Sc, F.R.S.E. LECTURER ON PHYSIC'S, AMD ASSISTANT TO THE PROFESSOR OF NATURAL PHILOSOPHY, IN THE UNIVERSITY OF GLASGOW. UN: ITY '"*\ , >* 1 i ' ^"^ litfjht of Translation and Reproduction is Reserved. LONDON : BIGGS AND CO. 139-140. SALISBURY COURT, FLEET STREET, E.C M 3 This book is mainly a reprint of a series of articles contributed, at the request of the Editor, to the Electrical Engineer during the summer and autumn of 1894 Reference to the books and papers in which the different parts of the subject are treated in detail is given in the text Various Tables of Physical Constants, collated from the most recent sources, are added. I have to thank various friends for reading the proof sheets, and especially Walter Stewart, M.A., B.Sc,, who has also drawn up the Index. MAGNUS MACLEAN The University, Glasgow. 1896. CONTENTS. PAGE. Fundamental Units 9 Geometrical and Kinematica I Units ... 23 Dynamical Units ... ... 25 Electric Units ... ... 49 Electrostatic System of Units ... ... ... ... 55 Magnetic Units ... ... 60 Electromagnetic System of Units ... ... ... ... 63 Practical Electric Units 70 Measurement of Electric Units by their Electromagnetic Effects ... 77 Measurement of Electric Units by their Electrostatic Effects ... ... 90 Practical Magnetic Units ... ... ... ... ... 92 Heat Units... ... 93 Useful Physical Constants. . . ... ... 98 Order in Council on Electrical Units and Standards ... 103 Table I. Surfaces and Volumes of Solids 113 Table II, The Acceleration due to Gravity at Different - Places ... ... 114 Table III. Moments of Inertia of Symmetrical Bodies 1 1 5 Table IV Densities of Solids and Fluids ... 116 Table V Moduluses of Elasticity . . , 118 Table VI. Velocity of Elasticity Waves. . . ... 1 1 9 Table VII. Surface Tensions ... ... 1 20 -Table VIII Viscosity of Liquids ... 121 -Table IX. Viscosity of Gases ... ... 1 22 Table X. Molecular Data for Gases ... 123 Table XL Specific Inductive Capacity ... ... ... 124 Table XII. Conductivity and Resistance of Pure Copper at Temperatures from C. to 100 C. ... 125 UONTENTS. I'AGfc Table XIII. Resistance, Conductivity, and Weight of Hard-Drawn Pure Copper Wires at 15 C. ... .... 126 Table XIV. Specific Resistance of Various Pure Metals at Different Temperatures 12 ( J Table XV. Specific Resistance of Various Alloys at Different Temperatures ... ... ... ... 1 30 Table XVI. Specific Resistance of Carbon at Various Temperatures ... ... ... ... .- 131 Table XVII. Specific Resistance of Liquids and Insulators ... ... ... ... ... 132 Table XVIII. Magnetic Elements in Different Parts of the British Isles for January 1, 1886 ... ... ... 133 Table XIX. Electro-chemical Equivalents ... ... 134 Table XX. Linear Expansions of Solids... ... ... 135 Table XXI. Volume and Density of Water and Mercury 136 Table XXII. Specific Heats of Solids ... ... 137 Table XXIII. Specific Heats of Liquids ... ... 138 Table XXIV. Specific Heats of Gases 139 Table XXV. Thermal Conductivities of Solids, Liquids, and Gases ... ... ... ... ... ... 140 Table XXVI. Melting Points and Latent Heats of Fusion ... ... ... .. ... ... ... 141 Table XXVII. Boiling Points and Latent Heats of Evaporation ... ... ... ... ... ... 142 Table XXVIII. Critical Temperatures and Pressures ... 143 Table XXIX. Vapoui Pressure of Water and Mercury... 144 Table XXX. Emissivity ... ... ... ... ... 115 Table XXXI. Indices of Refraction ... ... ... 1 46 Table XXX11. Wave-Lengths and Wave-Frequencies ... 147 PHYSICAL UNITS. Fundamental Units, 1. In physical science it is essential to have certain definite units in terms of which results of observation, and physical quantities dealt with generally, can be expressed. The object of the writer of this series of articles is to discuss these physical units, and the relations between them, in an elementary, clear, and concise manner. All physical units may be divided into funda- mental units and derived units. That is, a few units are chosen and fixed arbitrarily, so that all other units in physics can be expressed in terms of them; then the former are said to be funda- mental units, and the latter to be derived units. Theo- retically, any three independent units* are necessary and sufficient, so that the selection of the three * See a paper by A. W. Kucker, M.A , F.K S., " On the Sup- pressed Dimensions of Physical Quantities," Phil Mag , vol. 27, year 1889, page 104; a paper in the same volume, page 178, by T. H. Blakesley, M.A., M.I C.E., "On Some Facts Connected with the Systems of Scientific Units of Measurement"; also a paper by W. Williams, "On the Relation of the Dimensions of Physical Quantities to Direction in Space," Phil. Mag. t vol. 34, year 1892, page 234. B 10 PHYSICAL UNITS. fundamental units is one of convenience rather than of necessity. The three fundamental units so chosen are: the unit of length, the unit of mass, and the unit of time. 2. Unfortunately for students in physics, there are in general use two systems of units the British system and the metric or French system. The latter is so much simpler that experimental results will generally be expressed in metric units (centi- metre-gramme-second units, or C.G.S. units). But there are many results which are so well known and so conveniently expressed in British units (foot-pound-second units), that we shall follow the usual custom, and give them in British units. 3. The British unit of length is the Imperial yard. It was enacted by Parliament (18 and 19 Viet,, c. 72, July 30, 1855) "that the straight line between the centres of the transverse lines in the two gold plugs in the bronze bar deposited in the office of the Exchequer shall be the genuine standard yard at 62 F., and if lost it shall be replaced by means of its copies," which are preserved at the Eoyal Mint, the Boyal Society of London, the Boyal Observatory at Greenwich, and the new palace at Westminster. The unit generally used is a foot, which is a third of a yard. 4. The French unit of length is a metre, and is defined as the distance between the ends of a rod of platinum made by Borda, the rod being at the temperature of melting ice. This length was originally intended, according to the measurements of Delambre and Mechain, to be the ten-millionth part PHYSICAL UNITS. 11 of the earth's meridian passing through Paris. But the measurements of Delambre and Mechain were subsequently found not to be quite accurate, so that the actual definition of the metre is that above stated. The unit generally used is a centimetre, - which is the one-hundredth part of a metre. 5. The British unit of mass is the Imperial pound. It is the quantity of matter in a piece of platinum marked "P.S., 1844, lib." deposited in the office of the Exchequer, and the Act quoted above states that it " shall be the legal and genuine standard measure of weight, and shall be and be denominated the Imperial standard pound avoirdupois, and shall be deemed to be the only standard measure of weight from which all other weights and other measures having reference to weight shall be derived, computed, and ascertained, and one equal seven- thousandth part of such pound avoirdupois shall be a grain, and five thousand seven hundred and sixty such grains shall be and be deemed to be a pound troy. If at any time hereafter the said Imperial standard pound avoirdupois be lost or in any manner destroyed, defaced, or otherwise injured, the Com- missioners of Her Majesty's Treasury may cause the same to be restored by reference to or adoption of any of the copies aforesaid, or such of them as may remain available for that purpose." 6. The French unit of mass is the kilogramme des Archives, and though originally intended to be the mass of a cubic decimetre of distilled water at its maximum density (approximately at the tempera- ture of 4 C.), it is really defined as the mass B2 12 PHYSICAL UNITS. of a piece of platinum constructed by Borda in accordance with a decree of the French Kepublic in 1795. The metric system is decimal, Greek prefixes denoting multiples, and Latin prefixes denoting sub- multiples. 1 kilometre = 1,000 metres 1 hektometre = 100 1 dekametre = 10 1 metre 1 metre 1 decimetre = ^V 1 centimetre = 1 millimetre = . (> u 1 kilogramme = 1 hektogramme = 1 dekagramine = 1 gramme = 1 decigramme = 1 centigramme = 1 milligramme = 1,000 grai 100 10 1 grai 117 IffTT nme& nme i 9 I The relations between the British and metric units of length and mass are : 1 yard = 0*91439179 metre. 1 metre = 1-09362311 yards = 39'370432 inches, 1 pound = 0-453593 kilogramme. 1 kilogramme = 2-2046212 pounds. The following are, however, near enough approxi- mations : 1 foot = 30-48cm. 1 pound = 453 "6 grammes. 7. It will be noticed that in the British Act of Parliament the word " weight " is used to signify the quantity of matter within a body, whereas when it is used in its proper scientific sense, it signifies the downward force of gravity on the body. The distinction between weight and mass has been excellently put by Clerk Maxwell in his "Theory of PHYSICAL UNITS. 13 Heat," tenth edition,* page 79, from which I quote: "From these legal definitions it will be seen that what is generally called a standard of weight is a certain piece of platinum that is, a particular body, the quantity of matter in which is taken and defined by the State to be a pound or a kilogramme. " The weight strictly so called that is, the tendency of this body to move downwards is not invariable, for it depends on the part of the world where it is placed, its weight being greater at the poles than at the equator, and greater at the t level of the sea than at the top of a mountain. t "What is really invariable is the quantity of matter in the body, or what is called in scientific language the mass of the body, and even in commercial transactions what is generally aimed at in weighing goods is to estimate the quantity * See also Nature, May, 1894, correspondence columns, signed "K," '' The Reviewer," " Minchin." f One pound weight = 32-17 (1-0-00257 cos 2 X) H-- | -\ poundals. Where X is the latitude of the place of observation, h is the height above the sea-level, and r is the radius of the earth approximately 21 x 10" feet. One gramme weight = 980-5 (1-0-00257 cos 2 X) ( l ~ 7.) dynes. average value in the British Isles may be taken as one pound weight = 32 -2 poundals. one gramme weight = 981 dynes. 14 PHYSICAL UNITS, of matter, and not to determine the force witb which they tend downwards. "In fact, the only occasions in common life in which it is required to estimate weight considered as a force is when we have to determine the strength required to lift or carry things, or when we have to make a structure strong enough to support their weight. In all other cases the word 1 weight ' must be understood to mean the quantity of the thing as determined by the process of weigh- ing against 'standard iveights.' "As a great deal of confusion prevails on this subject in ordinary language, and still greater con- fusion has been introduced into books on mechanics by the notion that a pound is a certain force, instead of being, as we have seen, a certain piece of platinum, or a piece of any other kind of matter equal in mass to the piece of platinum, I have thought it worth while to spend some time in denning accurately what is meant by a pound and a kilogramme." 8. On Saturday, April 2, 1892, the standard yard measure and the standard pound weight, which for twenty years previously had lain in the wall of the great staircase leading to the Committee corridor of the House of Commons, were taken out for the purpose of comparison with the Imperial yard and pound in common use. The standards were first deposited in the wall in .1853, and this was the third test to which they were subjected. The standard yard, which is a bar of bronze, was enclosed and sealed in an oak box, and the standard PHYSICAL UNITS. 15 pound, which is of platinum, was deposited first in a silver-gilt case, next in one of solid bronze, then in a mahogany box, fourthly, in a leaden case, and, lastly, in a mahogany box. The weight is first wrapped in a peculiar kind of Swedish paper, quite free from silica, and therefore not liable to cause friction. The observer at the end of the test said: "I have finished the comparison of the immured standards with the Imperial standards, and I find that they are in the same condition as when they were deposited in 1872. The comparison has been made within the following limits: the pound has been compared within the ten-thousandth part of a grain, and the yard measure within the one -hundred -thousandth part of an inch." S 9. The unit of time universally adopted is a mean solar second. The reader should notice that this unit of time differs from a solar second and from a sidereal second. A sidereal day is the time that elapses between two successive transits of any fixed star at any chosen meridian, or it is the time of the earth's rotation on its own axis. A sidereal second is, there- fore, the time that the earth takes to rotate through an angle of one two hundred and fortieth of a degree.* * "In our first edition it was stated in this section that Laplace had calculated from ancient observations of eclipses that the period of the earth's rotation about its axis had not altered by 1/10 7 of itself since 720 B.C. In 830 it was pointed out that this conclusion is overthrown by farther in-formation from ' Physical Astronomy ' acquired in the interval between the printing of the two sections, in virtue of a correction which Adams had made as early as 1863 upon Laplace's dynamical investigation of an acceleration of the 16 PHYSICAL UNITS. 10. A solar day is defined as the time that elapses between two successive transits of the sun at any given place on the earth's surface. For two reasons this time is of variable length. Firstly, the path of the earth round the sun is an ellipse, and hence its motion is sometimes quicker and sometimes slower. Secondly, the motion of the earth is in the plane of the ecliptic and not in the plane of the equator. 11. To understand the first cause, Newton's uni- versal law of attraction will help us : " Every particle in the universe attracts every other particle with a moon's mean motion, produced by the sun's attraction, 'showing that only about half of the observed acceleration of the moon's mean motion relatively to the angular velocity of the earth's rotation was accounted for by this cause." (Quoting from the first edition, 830.) ''In 1859 Adams communicated to Delaunay his final result that at the end of a century the moon is 5"-7 before the position she would have, relatively to a meridian of the earth, according to the angular velocities of the two motions at' the beginning of the century, and the acceleration of the moon's motion truly calculated from the various disturbing causes then recognised. Delaunay soon after verified this result, and about the beginning of 1866 suggested that the true explanation may be a retardation of the earth's rotation by tidal friction. Using this hypothesis, and allowing for the consequent retardation of the moon's mean motion by tidal reaction ( 276), Adams, in an esti- mate which he has communicated to us, founded on the rough assumption that the parts of the earth's retardation due to solar and lunar tides are as the squares of the respective tide-generating forces, finds 22s. as the error by which the earth would in a century get behind a perfect clock rated at the beginning of the century. If the retardation of rate giving this integral effect were uniform ( 35, 6), the earth as a timekeeper would be going slower by '22 of a second per year in the middle, or -44 of a second per year at the end than at the beginning of a century." Thomson and Tait, "Treatise on Natural Philosophy"; footnote to 405, page 460. PHYSICAL UNITS. 17 force which is proportional to the product of the masses of the two particles directly, and to the square of the distance between them inversely." Take two uniform spheres of masses, m and m 1 , and let the distance between their centres be d, then the force of attraction between them is F = k -, where Jc is CL the mutual force of gravitational attraction between unit masses at unit distance, and its value is calculated below for both the metric and British units. Now, the mass of the earth is approximately 6*14 x 10 27 grammes, and the radius of it is approxi- mately 637 x 10 6 centimetres. Hence, if we take the force of gravity on one gramme to be 980 dynes, we get * The force with which two homogeneous spheres of matter of one gramme would attract one another, when their centres are 1cm. apart, is approximately a fifteen-millionth part of a dyne. To find out what would be the mass of two spheres* that would attract with the force of one dyne, when their centres are 1cm. apart, we have F = 1, m = m 1 , and d = 1, ./. m = V 1/k = 3,928 grammes. * A uniform sphere attracts external matter as if its whole mass were concentrated at its centre. The problem in the text is un- realisable in practice, but it can be asked what would be the mass of each sphere to have a mutual force of one-hundredth of a dyne when their centres are 10cm. apart. Thus a sphere of gold weighing 3,928 grammes would be 3 -65cm. in radius. 18' PHYSICAL UNITS. If we take the British units Jc = 32 x (21 x 10 6 ) 2 " 13-5 x 10 24 965-7 x 10 l poundal ; and a calculation similar to the above gives the masses that would have, one foot apart (see footnote as to a practical problem), a mutual force of attraction of one poundal, as 31,0751b. FIG. 1. 12. Let Fig. 1 represent an ellipse, S the sun in one focus, and E the earth. The earth when at A 1 is about three million miles nearer the sun than when at A. Hence the velocity of the earth at A 1 * is greater than the velocity at A. In passing from A 1 to A, in the direction shown by the arrow, the velocity of the earth is continually decreasing, and in passing * Centrifugal force is also greater at A 1 , but a complete dis- cussion of the problem is not attempted here. The velocity at any point is inversely proportional to the length of the perpen- dicular from S to the tangent at that point. PHYSICAL UNITS. 19* from A to A 1 it is continually increasing. (The average velocity of the earth in its path is approxi- mately 18-J miles per second.) At A 1 when the earth is in the winter solstice of the northern hemisphere, the solar day is longer than when it is at A in the summer solstice. That is, a solar day in mid-winter is longer than a solar day in mid-summer. In the further understanding of this fact, one of Kepler's three laws of planetary motion is of great help : " The radius vector of any planet sweeps over equal areas in equal times." In the figure, if the earth takes the same time to pass from A to B as from A 1 to B 1 , then area S A B is equal to the area S A 1 B 1 . The other two laws of Kepler, though not necessary to the present discussion, may be stated here : " The planets revolve in ellipses around the sun, and the centre of the sun occupies one of the foci." " The squares of the periodic times of any two planets have the same ratio as the cubes of their maximum or mean distances from the sun." 13. These three laws are purely kinematical. Their dynamical interpretation was discovered by Newton. From the law that the areas swept over by the vector drawn from the sun to a planet are proportional to the times of describing them, Newton inferred that the direction of the force acting must always pass through the sun. From the law that the orbit of a planet with respect to the sun is an ellipse the sun being in one of the foci and in conjunction with the previous law, Newton inferred that the magnitude of the 20 PHYSICAL UNITS. force acting is inversely as the square of the distance from the sun. From the law that the squares of the periodic times of different planets are proportional to the cubes of their mean distances from the sun, Newton inferred that at the same distance gravity acts equally on equal masses of substances of all kinds. 14. The second cause of the variable length of the solar day is that the apparent motion of the sun, or the real motion of the earth, is in the plane of the ecliptic, and not in the plane of the equator. The angle between the plane of the equator and the plane of the ecliptic is nearly 23 J. The points in which the sun in his apparent motion cuts the plane of the celestial equator are called nodes. If we suppose an imaginary sun to start from one of these nodes at the same time as the actual sun, and to move in the equator with such a uniform velocity as to arrive at the node again at the same instant as the real sun, then time indicated by this imaginary sun is mean solar time. A mean solar second is ^ ^(ro- of a mean solar day. A mean solar day is three minutes fifty-six seconds, or nearly four mean solar minutes, longer than a sidereal day. Further information on this important subject ought to be obtained by referring to any modern text-book on astronomy. 15. It is evident from 8 that the choice of a definite length and of a definite mass, as two of the fundamental units, fulfils the most important con- siderations necessary in their selection. It should be possible to compare them easily, directly, and accu- PHYSICAL UNITS. 21 rately with other quantities of the same kind, and this at all times and at all places. 16. The unit of time is subject to a slow varia- tion (see footnote 9), and hence other fundamental units of time have been suggested. In Thomson and Tait's " Natural Philosophy," vol. i., 406, page 461, the period of vibration of a metallic spring kept at a constant temperature in an exhausted and hermeti- cally sealed glass vessel is suggested, and the state- ment is made that "it would almost certainly be more constant from age to age than the time of rotation of the earth (cooling and shrinking, as it certainly is, to an extent that must be very con- siderable in fifty million years)." In 223, page 226, it is said that " the period of vibration of a piece of quartz crystal of specified shape and size, and at a stated temperature (a tuning-fork, or bar, as one of the bars of glass used in the ' musical glasses ') gives us a unit of time which is constant through all space and all time, and independent of the earth." Other two units of time are suggested in the same article viz., (1) the time of revolution of an infinitesimal satellite close to the surface of a globe of water, at standard density ; and (2) the time of the gravest simple harmonic infinitesimal vibration of the same globe. Lastly, it has been suggested that the time of vibration of some widely diffused substance, such as sodium or hydrogen, might be taken as the unit of time, and that its wave-length in vacuum might be taken as the unit of length. 22 PHYSICAL UNITS. " The recent discoveries due to jbhe kinetic theory of gases and to spectrum analysis (especially when it is applied to the light of the heavenly bodies) indicate to us natural standard pieces of matter, such as atoms of hydrogen, or sodium, ready made in infinite numbers, all absolutely alike in every physical property. The time of vibration of a sodium particle corresponding to any one of its modes of vibration is known to be absolutely inde- pendent of its position in the universe, and it will probably remain the same so long as the particle itself exists. The wave-length for that particular ray i.e., the space through which light is propa- gated in vacuo during the time of one complete vibration of this period gives a perfectly invariable unit of length ; and it is possible that at some not very distant day the mass of such a sodium par- ticle may be employed as a natural standard for the remaining fundamental unit."* 17. The symbols to be used for the three funda- mental units are : for length [L] ; for mass [M] ; and for time [T] . The symbols L, M, T, without brackets, will be used to denote the number of such units in the quantity to be expressed. We have now to deter- mine the relations of the other physical units to these chosen fundamental units, and to examine the laws according to which the derived units vary when the * Thomson and Tait, "Natural Philosophy," 223, page 227; see also Maxwell, " Electricity and Magnetism," second edition, vol. i., 3; and "Popular Lectures and Addresses," by Lord Kelvin, vol. i., a lecture at the Institution of Civil Engineers on " Electrical Units of Measurement." UNIVF PHYSICAL UNITS. 23 fundamental units are changed from one system to the other system. Every physical quantity essentially consists of two factors : (1) the fundamental unit employed, and (2) a numeric* which expresses the number of times the fundamental unit is to be taken to make up the physical quantity considered. Thus the numeric is the ratio of the physical quantity con- sidered to the unit chosen as fundamental unit. In symbols, if [Q] is the fundamental unit, and Q the physical quantity considered, we have Q = n [Q] , where n is a numeric as above defined, and may be integral or fractional. Thus in the expression 56 metres, 56 is the numeric n, and metres is the denominational part represented by [Q] . If n is unity, the quantity expressed is the unit. Geometrical and Kinematical Units. 18. Geometrical quantities are independent of the units of mass and time, and kinematical quantities are independent of the unit of mass. These may be first considered. Kinematics may be defined as the geometry of motion. It differs from kinetics in not considering (1) the masses of the bodies moved, (2) the forces causing the motion, and (3) any mutual forces called into action between the bodies moved, on account of the motion. * Fourier, "Theoriedela Chaleur," ICO, page 157; English transla- tion, by Alex. Freeman, M.A., page 126. Thomson's "Arithmetic," ed. Ixxii., page 4. In this connection a reading of a paper by A. Lodge, in Nature, July 19, 1888, page 281, on "The Multiplication and Division of Concrete Quantities," is recommended. 24 PHYSICAL UNITS. Unit area or surface, is that of a square which has- unit length for edge. Hence its dimensional formula is[S] = [L2]. 1 sq. in. = 6'4515 sq. cm. 1 sq.ft. = 929-01 Unit volume is that of a cube which has unit length for edge. Hence its dimensional formula is [V] = [L 8 ]. 1 cub. in. = 16-387 cub. cm. 1 cub. ft. = 28316-0 1 gallon = 4541-0 i=4J litres. Unit plane angle, called a radian, is the angle sub- tended at the centre of a circle by an arc equal in length to the radius. Its dimensional formula is therefore length by length = [L]. One radian = 57'2958 = 206265" Unit solid angle is the angle subtended at the centre of a sphere of unit radius, by unit spherical surface of that sphere. Generally the measure of a solid or conical angle is obtained by dividing the area cat off by a cone from a sphere whose centre is at the vertex of the cone, by the square of the radius of the sphere. Its dimensional formula is therefore [L 2 ]/[LT = [L]. Curvature is the angle turned through by the tangent, per unit of length of the curve. Its dimensional formula is therefore I/ [L] = [Lr 1 ] . Tortuosity is the angle turned through by the osculating plane, per unit of length of the curve. Its dimensional formula is therefore [Lr 1 ] . [For an analogy between lines and surfaces as PHYSICAL UNITS. 25 regards curvature, for definitions of curvatura Integra, average curvature, specific curvature, see Thomson and Tait, " Natural Philosophy," second edition, 136 and 138, page 105 and page 109.] $ 19. Velocity is rate of change of position in space, and a body is said to have unit velocity when, if the motion be uniform, it passes over unit space in unit time. Its dimensional formula is therefore |>] = [L] / [T] = [L T" 1 ] . Thirty miles an hour = 44ft. per second. 1 mile per hour = 44*704 cm. per sec. 1 knot (nautical mile per hour) = 51 '453 ,, The dimensional formula for angular velocity is [co I = [T-i] . 1 radian per sec. = 0*15915 turn per sec. 1 turn = 6'2832 radians Acceleration is rate of change of velocity, and its dimensional formula is [a] or [v] = [v] / [T] = [L T~ 2 ]. Similarly, angular acceleration [] = [T~ 2 ] . Dynamical Units. 20. Momentum of a body is the product of its mass into its velocity, and its dimensional formula is therefore I M v\ = [L M T- 1 ]. Force, by Newton's second law of motion, is pro- portional to the time rate of change of momentum. Fa M a = constant x M a. It we define unit force as that force which will produce unit acceleration in unit mass, the constant c 26 PHYSICAL UNITS. in the equation will be unity. This is what is done, so that unit force is that force which, acting on unit mass for unit time, will produce in it unit velocity. In the British system unit force is called a poundal. A poundal is that force which, acting on one pound for one second, generates in it a velocity of one foot per second. In the metric system, unit force is called a dyne. A dyne is that force which, acting on one gramme for one second, generates in it a velocity of one centimetre per second. If gravity be allowed to act on a mass during unit time, the velocity acquired is g units. Hence the force of gravity is g times the unit of force as above denned. 1 pound weight = g poundals. 1 gramme weight = g dynes. 21 . The force of gravity varies from place to place on the earth's surface. There are two main reasons why the force of gravity at sea-level should be different at different parts of the earth's surface (1) the shape of the earth, and (2) the rotation of the earth. The earth is not an exact sphere, but more like an oblate spheroid. Its polar diameter is approximately twenty- seven miles less than its equatorial diameter. Hence by Newton's universal law of attraction, stated in 11, the force of gravity at the equator is less than at either pole. Centrifugal force, however, gives the predominant cause of the variation of gravity. At the equator gravity and centrifugal force act in opposite directions, and at other places on the earth's surface the angle between the direction of gravity and the direction of centrifugal force is equal to PHYSICAL UNITS. 27 the latitude of the place of observation, and their resultant is to be found by the parallelogram of forces. 22. Let Fig. 2 represent a diametrical section of the earth, supposed here to be spherical. E Q is an equatorial diameter, and N S the polar diameter. 2 r.cos 2 X, c2 28 PHYSICAL UNITS. and o> 2 r.cos X.sin X. At the equator X is zero, so that the centrifugal force at the equator per unit mass is w 2 r. Using the two systems of funda- mental units, the centrifugal force at the equator per gramme = (27T/86,164) 2 x 637 x 10* dynes, = 3-38 dynes ; and per pound = (2 7r/86,164) 2 x 21 x 10 6 poundal, = -111 poundal. This is approximately 1/289 of the force of gravity at the equator, so that gravity at the equator is diminished by 1/289 of itself by centrifugal force. The two causes together diminish it by T ^, or about \ per cent. At sea-level* on the earth's surface the force of gravity is very approximately given by the following formula : g = 32-088 (1 + '00513 sin 2 A) poundals, = 978 (1 + -00513 sin 2 A) dynes, where X is the latitude of the place of observation. The dimensional expression for force is therefore [F] = [M o] = [M . v/T] = [LMT- 2 ]. 23. " Wet must next consider observed terres- trial gravity, or, as it is sometimes called, apparent gravity. The observed gravity of a body is not precisely the gravitational attraction of the earth upon it. If the earth were not in motion round * See footnote 7. f "Theoretical Mechanics," by J. T. Bottomley, page 6ft. PHYSICAL UNITS. 29 Its own axis, then the gravity of a body would be simply the gravitation attraction of the earth on it. But bodies that we commonly speak of as at rest at the surface of the earth are not really so. Each of them is moving in a circular orbit of daily revolu- tion round the earth's axis. Now it follows from Newton's laws of motion that a body moving in a circular path with uniform velocity must be acted on constantly by a force directed to the centre of the circle in which it moves. This force has its effect in overcoming the inertia of the body, and causing the body to deviate from the rectilinear path which, in virtue of inertia, it would at any instant tend to pursue. Thus, in order that a body at the earth's surface (a plummet hanging by its 'Cord, for example) may move in its daily circle, along with other neighbouring objects, it must receive from without a certain definite force directed always to the centre of the circle, and depending for amount on the mass of the body, the velocity of the motion, and the radius of the circle. Now the body is acted on by gravitational attraction, which is directed very approximately towards the centre of the earth, and this attraction resolves itself so as to supply a component which is the centreward force necessary to keep the body moving in its daily circle. The residue of the attraction, .after this component has been abstracted from it, is the force which the body transmits to whatever holds or suspends it, and which, in the case of the plummet, applies itself in pulling on the string. It is this residue which we perceive by our senses as 30 PHYSICAL UNITS. a downward force exerted by the plummet, and 1 which we call the gravity of the plummet. Let P be the plummet acted on by attraction in the line P C towards the centre of the earth.* It is revolving in a daily circle whose centre is A, a point on the earth's axis, N S, and the radius, P A, of the circle obviously depends on the latitude of the place at FIG. 3. which the plummet is situated. Now, according to what has been said, it is requisite, in order to keep * Gravitational attraction is directed very approximately towards the centre of the earth ; so nearly so, that we may, with sufficient exactitude for our present purpose, speak of it as if it were directed towards the centre exactly, and may represent it so in the diagram. Strictly, however, the attraction is not directed precisely towards the centre of the earth, and the deviation is different at different places. This is due principally to the fact that the earth is not an exact sphere, but an oblate spheroid, being flattened at the poles and protuberant at the equator, and also that it has local irregularities, such as mountains and valleys, as well as- irregularities in geological structure. PHYSICAL UNITS. 31 P in its circular orbit, that a definite force, though small in amount in comparison with the whole attraction, should constantly act on P in the direc- tion P A. Let us represent the gravitational attrac- tion in magnitude and direction by the line P B ; and let us resolve it into two components, of which one, being the centreward force necessary for the daily orbital motion of the plummet, is determinate in amount and in direction, and is represented in the figure by the line P D. The other component found by completing the parallelogram of forces, will be seen to be represented by P E. This is the force which manifests itself to us by acting on the plummet cord P O, and which is called the gravity of the plummet. The line of the plummet cord is the line P E F ; and it will be seen from the diagram that gravity is not, either in direction or in amount, exactly the same as the gravitational attraction on the plummet."* 24. The measure of work done is the product of the force acting into the distance through which it acts. This assumes that the displacement takes place in the direction in which the force acts. If it * The figure shows that this is a remarkable arrangement of forces. Though the plummet hanging at rest by its cord is gene- rally thought of as being under a system of forces in equilibrium, yet it will be seen on proper consideration that the forces acting on the body are not in equilibrium ; and that really the gravita- tional attraction on the plummet, represented by P B, and the pull of the cord 011 it, P O, give jointly a resultant P D, which is unbalanced, and which is the force required to keep it moving in its daily circle. This resultant goes constantly to overcome the inertia of the plummet, and to deviate it into its circular path. 32 PHYSICAL UNITS. does not, the work done is measured by the product of the displacement into the effective component of the force in the direction of the displacement. The dimensional expression for work is therefore The units for force and work are for clearness, put into the following tabular form : Gravitation Unit. Kinetic Unit. Force / British system e \Metric system -TTT , f British system Pound weight Gramme weight Foot-pound Poundal Dyne Foot-pou ndal Work \Metric system Centimetre - gramme Centimetre-dyne or erg In practical electricity a unit of work, called a joule, is employed, and is equivalent to 10 7 ergs. 25. Energy of a body is the capacity it has for doing work. The gain or loss in the energy of a body is equivalent to the work done upon the body, or to the work done by the body. This is true whether the gain or loss of energy be due to change of configuration or position, or to change of motion. Or, in the usual nomenclature, the work done by the body, or done on the body, is a measure of its loss or gain of energy, whether that energy be what is generally called potential energy or kinetic energy. Hence the dimensional formula for energy is the same as that for work : [E] = [L 2 MT- 2 ]. The relations between the British and metric units of force and work are approximately : PHYSICAL UNITS. 33 One pound weight = 453 '59 grammes weight; = 445,052 dynes. One poundal = 14'1 grammes weight ; = 13,825 dynes. One foot-pound = 13,825 contimetre-grammes ; = 13-56 x 10 6 ergs; % thirteen and a half million ergs. One foot-poundal = 429'5 centimetre-grammes; = 421,390 ergs. 26. Moment of a couple is the product of either force into the arm. Its dimensional expression is therefore [G] = [F . L] = [L2 M T-2]. Moment of inertia of a set of bodies round any axis is the sum of the products of each mass into the square of its distance from the axis. If the masses be m v m. 2 , m 3 , . . . , and the distances be **i, r 2 , r 3 , . . . , then 2 2 m = M & 2 , where k is called the radius of gyration, and may be defined as the radius of an infinitely thin rim in which all the masses may be supposed to be con- centrated, so as to leave the moment of inertia of the system unaltered. Its dimensional formula is therefore [I] = [L 2 M]. Moment of momentum of a moving body round any point is the product of its momentum into the perpendicular distance from the point on its line of motion. Its dimensional formula is there- 34 PHYSICAL UNITS. fore = [M.-y.L] = [Lr M T" 1 ]. If we make use of the analogues, that moment of a couple in rotational motion is analogous to force in translational motion, and moment of inertia to mass, we get the following convenient and useful relations : Translational Motion. Rotational Motion. F ft ^ _ moment of couple _G T M M v = F t s = i a t 2 , F M 2 v 2 moment of inertia P 0) = 0) t = . t e = i w * 2 _ ; G i 2 o> 2 f M F 2 .-.Fs = J M v 2 * I " G 2 .'. G^ = JIO.2 If the linear velocity be V, and if the angular velocity be 2, at the zero of reckon mg, then the resulting equations would be : M (v - V) = F t ; J M (v 2 - V 2 ) = F s ; I (w - H) = G ; J- 1 (w 2 - f2 2 ) = G 0, (See 25.) Take the kinematical equations : v a t. \ I* 2 = as.) -, Multiply the translational equations by M, and the rotational equations by I, then M v = M a t = F t.\ bnd Ia = Ici=G*. 1 M v 2 = M a s = F s.J 1 1 w 2 = I w = G 0.J PHYSICAL UNITS. 35 27. Activity is a word introduced by Lord Kelvin to signify " time-rate of doing work." Its dimensional formula is therefore [A] = [W T' 1 ] = [L 2 M T~ 3 ]. The unit rate of working used by engineers in this country is one horse-power that is, 550 foot-pounds per second. In the metric system this is equivalent to 76 metre- kilogrammes per second. Unit rate of working, or unit activity, on this system is one erg per second. In practical electricity a watt is generally employed as the unit of activity, and is equivalent to 10 7 ergs per second. One watt is a joule per second. One horse- power = 746 watts. 28. Density of a body is the quantity of matter in unit volume of it. Specific gravity of a body is the ratio of the weight of any volume of it to the weight of an equal volume of water. In the metric system unit volume of water is unit mass, but this is not so on the British system. Hence in the metric system the same number expresses the density and the specific gravity of the body, whereas on the British system the density of a body is equal to its specific gravity multiplied by 62*5, since one cubic foot of water weighs approxi- mately 62'51b. Specific gravity being a ratio, its value must be the same whatever system of units we use, and its dimensional formula must necessarily be unity. The dimensional formula for density is lib. per cub. ft. =0 1 01602 gm. per cub. cm. 1 grain per cub. inch = 0*00395 ,, 29. In the theory of elastic bodies any modulus is stress divided by strain, and the different moduluses 36 PHYSICAL UNITS. are obtained by defining the stress and the strain in particular ways. A strain is any change in the shape or in the bulk of a body, and stress is the name given to the forces which produce this strain. Thus bodies can have two distinct kinds of elasticity namely, elasticity of shape, or rigidity modulus, and -elasticity of bulk, or bulk modulus. Fluids have a bulk modulus in perfection, but have no rigidity modulus. Solids have both a bulk modulus and a rigidity modulus, but neither of them in perfection. Strain is a mere ratio of length to length, or volume to volume, and hence its dimension is unity. Stress is invariably given in units of force per unit of area, and its dimensional formula is therefore [L-^MT- 2 ]. Rigidity modulus, n, is the tangential forces (shear- ing forces) per unit area, divided by the shear pro- duced. Here shear is defined as an angle. If a cube be taken, then the deformation from a right angle -expressed as a fraction of a radian is the shear. Bulk modulus, k, is the hydrostatic forces per unit area divided by the change in volume per unit volume. Young's modulus, M, is the stretching force per unit area divided by the elongation produced per unit length. 1 9n* M = _L + A 3 k + n (See Thomson and Tait, " Natural Philosophy," 83, page 221.) PHYSICAL UNITS. 37 Hence the dimensional formula for rigidity modulus, n, bulk modulus, k, and Young's modulus, M, is, in each case, [L~ J M T- 2 ]. 30. Torsional rigidity and flexural rigidity depend not only on the material, but also on the shape and on the axis round which torsion or flexure is taken, Torsional rigidity of a bar may be defined as the couple that produces unit twist in it per unit length r and its flexural rigidity in any given plane as the couple that would produce in it unit curvature in this plane. Let N denote torsional rigidity and N l flexural rigidity, then [N] or [N 1 ] = [L 3 M T~ 2 ]. For a symmetrical rod the torsional rigidity is equal to the rigidity modulus of the material of which it is made multiplied by the moment of inertia of the cross-section round the proper diameter. Hence its torsional rigidity is proportional to the product of the rigidity modulus and the square of its cross- section, and therefore the dimensional formula is [N] = [L- 1 M T- 2 ] x [L 4 ] = [L :{ M T~ 2 ]. The flexural rigidity is proportional to the product ot the Young's modulus and the square of the cross- section. Hence its dimensional formula is [N 1 ] = [L-iMT- 2 ] x [I/] = [L'MT- 2 ], the same as otherwise found above. lib. per square inch = 70'31 gms. per sq. cm. 1 ton =1-575 x 10 5 gms. 38 PHYSICAL UNITS. 1 atmosphere =76 cms. of mercury at C. = 1033*3 gmfe. per sq. cm. = 29'92 inches of mercury at C. = 14*71bs. per sq. inch. 31. Surface tension of liquids is given in units of force per unit of length. Its dimensional formula is therefore [r] = |L M T-2] ^ [L] = [M T~ 2 ]. This is equivalent to energy per unit area. 1 grain per inch = 25 dynes per cm. 32. Viscosity is a property possessed not only by liquids and gases, but by the most elastic of solids. To define the viscosity of a fluid, suppose you have a layer of unit thickness of it between two planes of indefinite extent to each of which the fluid adheres. Let one plane be fixed : then the tangential forces per unit area necessary to move the other plane with unit velocity is the measure of the viscosity of the fluid.* It is, therefore, tangential forces per unit area divided by shear per unit time. The dimensions of viscosity will therefore differ from the dimensions of stress simply by the time unit, just as the dimensions of velocity differ from those of acceleration. Hence the dimensional formula for viscosity is * "The viscosity of a liquid (or the numerical reckoning or measure of the viscous quality) is denned as the tangential force per unit area, in either of the mutually perpendicular planes of zero elongation, of a simple distortion, required to produce change of shape at unit speed." Kelvin, "Mathematical and Physical Papers," vol. iii., page 437. PHYSICAL UNITS. 39 The viscosity of water diminishes with rise of temperature. Poiseuille, who investigated the flow of water through capillary tubes, gives Q = 1836-724 (1 +0-0336793 t + 0-0002209936* 2 ) H P V L where Q is the quantity of water in cubic milli- metres that at temperature t C. flows in one second through a capillary tube of length L mm., and diameter D mm., under a pressure of a mercury column of height H mm. Reducing to C.G.S. units, and neglecting figures after the fifth decimal place, this gives Q = 1350 (1 x 0-03368 t + 0-00022 * 2 ) -5^, L where Q = quantity in cubic centimetres that flows through in one second, p -- -- the pressure at the mouth of the capillary tube in grammes weight per square centimetre (approximately equal to the head of the water), D == diameter of the capillary tube in centi- metres, and L = length of the capillary tube in centimetres. But it can be shown that for laminar motion 128/* L where /* is the viscosity of the water. 1-818 x 10- 5 wt ' per sq ' cm " * Annalen tier Physik, Poggendorff, band Iviii., page 436; or Repertorium der Physik, Dove, band vii., page 131. 40 PHYSICAL UNITS. Thus HQ = 1*818 x 10~ 5 grammes weight per sq. cm. Ahoo = <277 x 10 ~ 5 The viscosity of water at 100 C. is less than a sixth of the viscosity of water at C. 33. "The diffusivity of one substance in another is the number of units of the substance which pass in unit of time through unit of surface, across which the gradient of concentration is unit of substance per unit of volume per unit of length."* "Diffusivity is essentially to be reckoned in units of area per unit of time,"* and its dimensional formula is therefore [ K ] = [IrT- 1 ]. In art. xcviii. of Kelvin's " Mathematical and Physical Papers," vol. iii., page 428, it is shown that the same law governs the following five cases of diffusion : I. Motion of a viscous fluid. II. Closed electric currents within a homogeneous conductor. III. Heat. IV. Substances in solution. V. Electric potential in the conductor of a sub- marine cable when electromagnetic inertia can be neglected. See also, in this connection, Tables A, B, and C, pages 207 and 208 of the same volume. It may be remarked that the diffusivity of fluid for viscous motion is its viscosity divided by its density, and therefore M = [L' 1 M T->] - [L~ 3 M] = [L 2 !* 1 ]. * Tait, " Properties of Matter," page 257. * Kelvin, "Mathematical and Physical Papers," vol. iii., page 205. Name. Symbol. Dimensions in LMT. Plane angle e Solid angle 12 Strain Specific gravity Curvature Tortuosity I IT \lr l L- 1 L- 1 Area S L 2 Volume V L^ - Linear velocity v, L L T" 1 Angular velocity w, T _i Linear acceleration en, v L L T~ 2 Angular acceleration . . . Moment of inertia Density w, I D T -2 / L 2 M L~ 3 M Momentum M v\ Impulse F t \ L M T- 1 Force * V j F L M T~- Moment of momentum... Torque &9 G L 2 M T- 1 L 2 M T~ 2 Work ) *{5:'* 1 L 2 M T~- Energy / Energy per unit area . . . Energy per unit volume Activity . . x {|KJ A M T- 2 L-i M T- 2 L 2 M T~ 3 Stress ^1 Pressure P L -i M y-2 ' Surface tension . . 7- M T~ a Rigidity modulus n } Bulk modulus ... k \ L-I M T~ 2 Young's modulus . r M Torsional rigidity ITi J N 1 Flexural rigidity Viscosity V Ni) n, L 3 M T- 2 L-I M T" 1 Diffusivitv . r K L 2 T-i' J) 42 PHYSICAL UNITS, 34. The units hitherto discussed, the symbols used for them, and the dimensional expressions obtained for them, may be put into a tabular form y as shown on the preceding page. 35. These dimensional formulae may be used ; (1) to find the change that takes place in any derived unit when the fundamental unit is altered ; (2) to check any result we may have arrived at in an in- vestigation by counting the dimensions ; and, to a limited extent, (3) to anticipate results of a dynami- cal investigation by reasoning based on dimensions. Examples are appended to illustrate these statements. W. Williams, in a paper entitled " On the Relation of the Dimensions of Physical Quantities to Direction in Space," printed in the Proceedings of the Physical Society, vol. xi., 1890-92, uses three rectangular directions, X, Y, Z, and hence shows that " the dimensional formulae may be taken as representing the physical identities of the various quantities, as indicating, in fact, how our conceptions of their physical nature (in terms, of course, of other and more fundamental conceptions) are formed just as the formula of a chemical compound indicates its composition and chemical identity." 36. 1. How many ergs are there in a foot-pound ? The dimensional formula for work is [L 2 MT~ 2 ], Let x be the number of ergs corresponding to 32*2 foot-poundals, then (ft.) 2 x Ib. = 32 . 9 (30-48)3 x 453-6 . (sec.) 2 I 2 .\JB- 13-56 x 10 6 ergs; U PHYSICAL UNITS. 43 or, one foot-pound = 30*48 cm.-lbs. ; = 30-48 x 453-6 cm.-gms. ; = 30-48 x 453-6 x 981 cm.-dynes; = 13-56 x 10 6 ergs. 2. Determine the unit of time in order that, the foot being the unit of length, g may be expressed by unity. The dimensional formula for acceleration is [L T~ 2 ]. ft. ft. Therefore, 32 x (sec.) t = _ _ second ; 4 x/2 = i V^ = '177 second. 3. Supposing the unit of force to be the weight of 61b., and the unit of acceleration referred to foot and second as units to be 24, find the unit of mass. 24 (sec.) 2 ft ' (sec.) 2 P ' 6 x 32 -^4- ; - 81b. 4. A body moving uniformly passes over a mile in 15 minutes. If 64 be the measure of velocity, and lift, the unit of length, find the unit of time. D2 44 PHYSICAL UNITS. QQ A mile in 15 minutes = ft. per second. 15 ?? JL = 64 L 1 15 sec. t ' .'. t = G4x * lxl5 = 120 sees. = 2 minutes. 5. If the force of gravity be taken as the unit of force, and a rate of 10 miles an hour as the unit of velocity, what must be the units of time and length ? The dimensions for force and velocity are [L M T~ 2 ] and [L T 1 - 1 ] and 10 miles an hour = Yft. per second. 32X = ..... ! and I 4 x JL = (2) 3 (sec.) t where I and t are the required units of length and time. From (2),Z=Y t, and substituting in (1),*=V*& = # second. Therefore /=Wft. This is equivalent to saying that a falling body acquires a velocity of Wft. per J| second at the end of IT second of its fall, or 32ft. per second per second = Wft- P er U second per U second. 6. A shot weighing a Ibs. is fired from a gun with a velocity of v ft. per second. If its kinetic energy, the impulse it has received, and the acceleration due to gravity be denoted by k, i, and / respectively, find the units of length, mass, and time. PHYSICAL UNITS. 45 (*.) =/t l2- ' ' ' (1) . / m i (sec.) t \ "/ -/* / Q \ # -7- Co" == / -To ' ' W (sec.) z ^ a v 2 ft avt From (l)and(2), m = z A; ^ 1 1 7 .. ' . . . (4) 2kP " i/ ; . I vi t 2k and from (3) = *L t f vi f nni From (4), J . x I - i a v ^ 2 a h hence m = _ x - = _ . i I $ For example, if a = 2oz., v = 800ft. per second, and k = i =f = unity, then t = - = 12-5 sees., 1= = 5,000ft.,andm = 2a = ilb. ^^ 9 37. 1. A person deduces from some investigations the following equations. Determine whether they are possible or not. (a) 6 M 2 V + 2 g* F L 2 D T 8 - 3 F 2 L T 4 = 0; (b) v* T - 4 M a L + 3 F = 0. 46 PHYSICAL UNITS. Substituting the dimensional expressions for each of the terms in equation (a) we get , T9T o /L\ 3 LM T9 M > /L M\ 2 ' (TV x ~r^ x I* x ' VT^V x x ' which reduces to L 3 M 2 , L 3 M 2 , I/M 2 . The three terms are of the same dimensions, there- fore the equation is possible. Substituting dimensions in equation (b) we get which reduces to L 3 M L 2 LM _ , M _ , _ . rp 2 ' "J" 2 T 2 The three terms are not of the same dimensions, and therefore the equation is physically impossible. 38. 1. The centrifugal force acting on a body depends on the mass of the body, the velocity of the body, and the radius of the circle the body is describing. Let the index of each of these be x, y, z, then F = m x . vy . r 2 . Substituting dimensions L M T- 2 = M* x LT- 127 x if Equating coefficients of L, M, T, we get 1 = y + z - } 1 = X ; - 2 = - y; .'.x= 1; y = 2; 2= - 1; PHYSICAL UNITS. 47 2. The velocity of a distortional wave through a material depends on the rigidity modulus of the material and on its density. Let us suppose that it varies as the #th power of n and as the ytla. power of D. Then v = n* . D V ; L T- 1 = (L- 1 M T- 2 )* x (L~ 3 Mf . Equating the indices of L, M, T : 1 = - x - 3y; = x + y } - I = - 2x. .'. x = J, and y = - J. In / V D If the modulus be expressed in gravitation units this could be written I 9 n - V D~ " where L is the length modulus of rigidity. Generally the corresponding length modulus is the modulus in gravitation measure divided by the density. Hence we can say that the velocity of any wave-motion through any material is the velocity a body would acquire in falling through a height equal to half the proper length modulus.* The proper length modu- luses for the following wave-motions are : 1. Longitudinal vibrations along a bar or cord, L = M/D, where M is the Young's modulus of the * See Lord Kelvin's "Mathematical and Physical Papers," vol. iii., xciu, page 40, and civ. , page 519, 48 PHYSICAL UNITS, material and D is its density. If the Young's modulus is given in grammes weight per square centimetre and the density in grammes per cubic centimetre, the length modulus will be given in centimetres. Multiplying this by the value of g in centimetres per second per second, and taking the square root of the product, the velocity of the wave-motion is found in centimetres per second. Similar remarks apply to the following several cases of wave-motion. 2. Waves of simple distortion, Ii = n/~D. 3. Waves along a bar or cord which is prevented from lateral shrinking, or waves of simple longitudinal extension and contraction in a homogeneous isotropic infinite solid that is, waves analogous to sound,. 4. Sound-waves in a fluid, L = &/D. [For fluids, 0.] 5. Sound-waves in air, L = &/D, or (by Boyle' & law) P/D = height of the homogeneous atmosphere,. where P is the atmospheric pressure. 6. Long waves in water of uniform depth, L = depth. 7. Deep-water waves, L = wave-length /2 TT. 8. Transverse pulse in stretched cord, L = length of cord amounting in weight to the stretching weight = W/m, where W is the stretching weight, and m the mass of the cord per unit length, PHYSICAL UNITS. Electric Units. 39. All electrical quantities can be expressed in> terms of the mechanical, magnetic, chemical, or thermal effects produced by electricity. Magnetic , chemical, and thermal effects are, however, measured in terms of some mechanical unit, and hence it is only necessary to state the relations between the electrical quantities and the mechanical units. We may start (a) from the experimental results of Coulomb, by defining the force between two quan- tities of static electricity, or the force between two quantities of magnetism ; (b) from Ampere's formula, by denning the force between two elements of 'currents of electricity. We thus arrive at what is commonly called the electrostatic units, the electro- magnetic units, and the electrodynamic units. It will be shown subsequently that by giving a certain value to the constant in Ampere's formula, the electrodynamic unit current is equal to the electro- magnetic unit current. One of the clearest statements of the relations between the electric quantities is given in a report by Clerk Maxwell to the British Association meeting at Newcastle- on -Tyne, 1863, and reprinted with various other reports by Spon, London, 1873 : " . . . The electrical phenomena susceptible of measurement are four in number current, electro- motive force, resistance, and quantity. . . . Their 50 PHYSICAL UNITS. relations one to another are extremely simple, and may be expressed by two equations. " First, by Ohm's law, experimentally determined, we have the equation C = E/R (1) where C = current, E = electromotive force, and B = resistance. From this formula it follows that the unit electromotive force must produce the unit current in a circuit of unit resistance ; for if units were chosen bearing any other relation to each other, C would be equal to x E/K, where x would be a useless and absurd factor, complicating all -calculation, and confusing the very simple con- ception of the relation established by Ohm's law. " Secondly, it has been experimentally proved by Dr. Faraday that the statical quantity of electricity conveyed by any given current is simply propor- tional to the strength of the current, whether electro- magnetically or electro-chemically measured, and to the time during which it flows ; hence, in mathe- matical language, we have the equation Q - C t (2) where t = time, and Q = quantity. From this equa- tion it follows that the unit of quantity must be the quantity conveyed by the unit current in the unit of time; otherwise we should have Q = y C t, where y would be a second useless and absurd coefficient. From equations (1) and (2) it follows that only two of the electrical units could be arbitrarily chosen, ]; = [L 2 M T- 2 x L~^ M~i T K~^] ; = [L2 M^ T- 1 K~] ; Electric potential at a point due to a quantity, q, of electricity is the quotient obtained by dividing the quantity of electricity by the distance and by the inductive capacity. Thus [v] = [q L- 1 K- 1 ] = [L* M* T' 1 K"^]. 45. The capacity of a conductor, c, is the quotient obtained by dividing the quantity of electricity with which it is charged by the potential which this charge produces in it. Hence its dimensional formula is [c] = [q v-i] = [L* M^ T-i K* x L~^M~^ T K*] = [L K]. Hence, if, as in the ordinary electrostatic system, we suppose [K] = l, then the dimensions of capacity will be that of length, and is correctly expressed as so many centimetres. PHYSICAL UNITS. 59 46. The resistance of a conductor, r, is by Ohm's law measured by the difference of potential at its extremities divided by the current produced in it thereby. Therefore its dimensional formula is = [L- 1 T K- 1 ]. The resistance of a conductor is also measured by the time a unit of electricity requires to pass through it when unit difference of potential is maintained between its ends. Hence [r] = [v . T . 0-i] = [L- 1 T K- 1 ]. 47. Conductance is the reciprocal of resistance, and its dimensional expression is therefore [L T" 1 K]. 48. These electrostatic quantities may now be put into a tabulated form for convenience of reference. Name. I Relation to other Quantities. Dimensions in L M T K. 3 , j x Quantity Surf ace density Electric dis- placement... Current Intensity of electric field Potential . . f 7 V \J force x (distance) 2 x K quantity -=- area quantity -f- time force -r quantity. / work -r- quantity | L l ^2 T~ 2 K 2 ^ L -i M *T-iK-i Capacity Resistance . . . Conductance... c r l/r \quantity -f- distance x KJ quantity -r potential potential -r current LK L- 1 T K- 1 E 2 60 PHYSICAL UNITS. Magnetic Units. 49. The unit magnetic pole, m, is defined in exactly the same manner as the unit quantity of electrostatic electricity was defined in 40. The analogue to electrostatic inductive capacity is mag- netic inductive capacity, generally called magnetic permeability, and denoted by /UL. Hence the dimensional formula for strength of magnetic pole is 50. Magnetic surface density, cr 1 , intensity of magnetic field, H, and magnetic potential, V, are defined similar to the corresponding quantities in electrostatics in 41, 43, and 44 respectively, and hence their dimensional formulae are : [o- 1 ] [H] = [L"*M^ T- 1 /T^]; [V] = [L^ M^ T- 1 /*""]. 51. The moment of a magnet, M, is the strength of either pole into the distance between the poles. Hence its dimensional formula is [M] = [m x L]; If a bar magnet is placed in a uniform field of force, it experiences a couple tending to set it with the lines joining its poles in the direction of the force of the field. When the magnet is so placed PHYSICAL UNITS. 61 that the line joining its poles is at right angles to the direction of the lines of force of the field, the couple is equal to the product of the magnetic moment into the strength of the field. Hence [M] = [G.H-i]; =. [L 2 M T~ 2 x L* M~i T pk] ; = [L* M^ T- 1 t$]. ' 52. Intensity of magnetisation, I, of a uniformly magnetised magnet is measured by its magnetic moment divided by its volume. Hence its dimen- sional formula is [I] = [L~*~ M* T- 1 $ x L- 3 ] ; = [L~i M* T- 1 /*i]. This is equal to the surface distribution of the magnetism over the ends of the bar, and its dimen- sional formula is therefore the same as that of magnetic surface density. 53. Magnetic susceptibility, K, is the ratio of the intensity of magnetisation to the magnetising force. Its dimensional formula, therefore, is K = ^^ - 1 *~2 54. Magnetic induction, B, is proved to be equal to the resultant of H and 4 ?r I. If the substance be isotropic and has no magnetism except what exists on account of the magnetising force, H, then B = H + 47rl; PHYSICAL UNITS. The dimensional formula for magnetic induction is therefore [B] = [L~ * M^ T- 1 (jT? x p] ; the same as that of intensity of magnetisation. 55. These magnetic quantities may now be put into a tabulated form for convenience of reference. Name. 1 I Relation to other Quantities. Quantity of } magnetism . . . Magnetic pole... Magnetic sur- face density . . . Intensity of magnetic field Magnetic force Magnetic poten- tial }- o- 1 }" V *J 'force x (distance)' 2 x /x magnetic pole 4- area force -f strength of pole work -7- strength of pole Magnetic moment Intensity of magnetisation . Magnetic induc- tion M I R strength of pole x dist- ance moment 4- volume magnetic force x per- Magne t i c permeability... Magne t i c susceptibility .. P K meability intensity of magneti- 1 sation 4- magnetising force Dimensions in LMT/i. 31 1 ** ~ 2 PHYSICAL UNITS. 63 Electromagnetic System of Units. 56. In the electromagnetic system all the units are derived from the definition of magnetic pole given in 49. From the experiments of Oersted and of Ampere we are led to regard a linear conductor carrying a current as producing a magnetic field. If the linear conductor be bent into a circle of unit radius, then unit magnetic pole placed at the centre is acted on by unit force per unit length of the conductor when unit current is flowing in it. The total force acting on the unit pole is therefore 2 TT units. If r be the symbol used to represent current, then = [L M T- 2 x L x IT* If we have, as in a practical experiment we would have, a magnet of length I and strength of pole m suspended at the centre of a coil of mean radius r, large in comparison with I, and if n be the number of turns in the coil, and the angle between the plane of the coil and the axis of the magnet, then neglecting the depth and breadth of the coil, the moment G tending to turn the magnet when a current T is in the coil, is Gr = rxmx27r?irx x I cos [F] = [Gm- 1 ] which is the same as above. 64 PHYSICAL UNITS. Let us suppose the coil to be fixed so that its- plane is vertical and in the magnetic meridian, that the intensity of the magnetic field at the magnet is H ; then when the deflection of the magnet due to the current T in the coil is 9, the moment G tending to bring the magnet back to its original position is G = m H Zsin 0. Equating these two expressions for G we get T =_*! H tan B. 2irn . ' . [T] = [L H] = [L x L~i M* T- 1 /T*] = [L* M* T- 1 /*"*]. 57. The quantity of electricity, Q, is equal to the product of the current into the time during which it lasts. 58. The work done in urging a quantity, Q, of electricity through a difference of potential, V, is, as in 44 = Q V. .-.[V] = [W Q-!] = [L 2 M T- 2 x L~2 M~2>2] = [L* M2 T- 2 p i]. 59. The capacity of a conductor is, as defined in 45, the quotient obtained by dividing the quantity of electricity with which it is charged by the potential which this charge produces in it. Hence THK PHYSICAL UNITS. 65 60. The dimensions of a resistance are [R] = [vr-i] ; = I J-j JVl'" _L " fji" x J_j ~ JM X yu.*" j = [L T-i /,]. 61. The coefficient of self-induction of a circuit, in a medium of constant permeability, is the ratio of the counter electromotive force of the circuit to- the rate of variation of the current producing it, In symbols where L x is the coefficient of self-induction, /. [LJ = [L^M^ T- 2 i$ -f- L^ M* T- 1 ^ x T" 1 ] The simplest way of denning the coefficient of self- induction is to say that it is the ratio of the total induction in the circuit to the current producing it. If N be the total induction or total flux, /. [LJ = [L~M2 T~V x L 2 x L~M~ T If Mj be the coefficient of mutual induction between two circuits, then the counter electromotive force 66 PHYSICAL UNITS. 4 l , in one circuit, due to a current T in the other f T j circuit, is e 1 = Mj - , a t 62. These electromagnetic quantities may now be put into a tabulated form for convenience of reference : Name. "o ^ Relation to other Quantities. Dimensions LMT/x. in Current r (Length x magnetic^ i i -j Quantity o \ force J current x time L* M* ^ i Potential v f Electromotive force F, work -r- quantity Lv -M"? T 2 - M^ 1 L j u* "Capacity n \. quantity -r potential L -l T 2 -! Resistance . . . Self-induction R L! C Electromotive force ^ \ -f- current J f Electromotive force ^ Mutual induc- tion M, ^ rate of variation of V current L/, i 63. The dimensions of these electromagnetic quantities have been obtained by starting with the dimensional formula of unit quantity of mag- netism, and from it deriving that of unit current. Similarly we might start with the dimensional PHYSICAL UNITS. 67 formula of unit current in electrostatics, and from it derive the dimensional formula of unit quantity of magnetism, and hence of all other magnetic and electromagnetic quantities. For, as in 56, M-pr.Ly- 1 ]; = [L M T-' 2 x L x L~^ M * T 2 K~ 2 ] ; = [L* M* K~^J. The rest of the quantities need not be worked out in detail. They are all given in the tables below. The reader is also referred to the following papers for further information on this subject : " On the Different Systems of Measures for Electric and Magnetic Quan- tities " by Prof. K. Clausius (Phil. Mag., June, 1882, page 381) ; " On the Dimensions of a Magnetic Pole in the Electrostatic System of Units," by Prof. J. D. Everett (Phil. Mag., May, 1882, page 376, and June, 1882, page 431) ; also articles on the same subject by J. J. Thomson and Prof. J. Larmor in Phil. Mag., June, 1882, pages 427 and 429 ; " Kemarks on Abso- lute Systems of Physical Units," by A. F. Sundell (Phil. Mag., August, 1882, page 81) ; " On the Dimen- sions of a Unit of Magnetism in the Electrostatic System of Measures," by B. Clausius (Phil. Mag., August, page 124), also J. J. Thomson (Phil. Mag., September, 1882, page 225), and Oliver J. Lodge, D.Sc. (Phil. Mag., November, 1882, page 357) ; " On Systems of Absolute Measures for Electric and Magnetic Quan- tities," by Prof. H. Helmholtz (Phil. Mag., December, 1882, page 430). | 64. Table of dimensions of electric and magnetic quantities : 68 PHYSICAL UNITS. Name. I LMTK. LMT/x. Quantity of elec- tricity Q 31 1 L*M 2 T- 1 K 2 L 2 M 2 fT% Current 2 *% y r Magnetic potential... Surface density 7 5 ? Electric displace- ment I L -i M i T -iK 2 L^M* fk Magnetic surface density cr 1 _3 1 _1 L~ 2 M 2 ^ T- 1 fj* Electric potential . . . Total electromotive force e E 1 L i M i T - 1K -i L^ M 2 T~ 2 i& Quantity of mag- netism I J Y in L 2 M 2 K~ 2 L! M^ T- 1 /x^ Magnetic pole Intensity of mag- netic field L^M 2 ^ T~ 2 K 2 L~2 ]^i T- 1 u~i Magnetic force i Magnetic moment . . . Intensity of magneti- sation M I 1 L l M J R - J 31 1 L^ M i T -i ^ Magnetic induction .. Intensity of electric field B) f L 2 M 2 K 2 L 2 M 2 T x /x 2 L 2 M 2 T~- fi^ Specific inductive capacity K K L~ 2 T 2 /x" 1 Capacity cC L K L- 1 T 2 /A- 1 Resistance r R LT- 1 , Magnetic permea- bility n 1 Magnetic suscepti- bility - L-* T^ K-i / Self-induction Mutual induction . . . L 1 TV IT 1 J. w JA. L/. PHYSICAL UNITS. 69 65. It will be noticed from these expressions <1) that mass appears always under a square root, and does not appear in any of the expressions involving K and ^ to integral powers ; (2) that the ratio of the dimensions in terms of K to the dimensions in terms of /u, is a power of L T" 1 K^ JUL*. Hence, if we suppose that every electric and magnetic quantity has definite dimen- sions in terms of length, mass, and time, K^ ^ must be of the dimensions of a slowness or the inverse of a velocity, and "the two systems become identical as regards dimensions, and differ only by a numerical coefficient, just as centimetres and kilometres do." [See "On the Suppressed Dimensions of Physical Quantities," by A. W. Kiicker, M.A., F.K.S., Proceedings of the Physical Society of London, vol. 10, page 37 ; and " On the Dimensions of Electromagnetic Units," by Prof. Gr. F. Fitzgerald, F.R.S., same volume, page 95.] 66. By supposing the dimensions of K and /x to be zero, the ordinary electrostatic and electro- magnetic systems of units are obtained. The quantities most generally used in the two systems are given in the following table: 70 PHYSICAL UNITS. Dimensions Dimensions Name. o H in Electrostatic 1 in Electro- magnetic Ratio. & System. System. 3 1 j Quantity . . . Q L 2 M 2 T" 1 L2~ ]y[2" 2 _ L T _! Q Current y i r L 2 M 2 T~i .Z = L T" 1 = r Potential V I> M* T-' v L! ^i T~ 2 v L- 1 T l V v Capacity c L o L-I T 2 c L 2 T~ 2 = v 2 C Resistance . . . r L- 1 T R L T- 1 I = Lr 2 T 2 = - i One electromagnetic unit of quantity = v electrostatic units ; current = w potential = _ v i) capacity = v 2 if resistance = - ^vbere v (determined by experiments to be referred to subsequently) is a velocity of 3 x 10 10 centimetres per second, or 300,000 kilometres per second. Practical Electric Units. 67. Let a conducting circuit be placed in a magnetic field, and let tbe current in tbis circuit be r under the action of an electromotive force E. Let tbe circuit be so moved tbat in time t tbe number PHYSICAL UNITS. 71 of lines of force passing through it is increased by N, then the work done against the electromagnetic forces is measured by the product T N. But the work done on the current is E F t. .-. ET t + TN = 0; t Hence, if the number of lines of force enclosed by a circuit be increased, an electromotive force in the B FIG. 4. negative direction is set up, and the measure of this- electromotive force is the number of lines of force added per unit of time. 68. In Fig. 4, let the circuit consist of two long parallel metallic rails A E, D F, connected per- manently by a wire, A D. Let B C be a slider free to move along the rails either towards AD, or E F. Suppose the plane of A B C D to be vertical, the rails to run from west to east, and that 72 PHYSICAL UNITS. you are looking northwards at the figure, then the direction of the lines of force due to the horizontal component of the earth's magnetism is from the front of the page to the back, and these lines are practically parallel to one another. If a current in the circuit is in the direction A D C B, the slider will move eastwards towards E F, and cause an electromotive force of the magnitude H Iv to act round the circuit in the direction A B C D, where H is the number of lines of force per unit area perpendicular to A B C D, I is the length of the slider B C, and v is the velocity it has in the eastward direction. This illustrates the direction of the resulting electromotive force. Now if there be no current in the circuit, if the magnetic field be unity (one line of force per square centimetre), and if the distance between the rails be one centimetre, unit electromotive force is induced in the circuit when the velocity of the rail is one centi- metre per second. Thus unit electromotive force is produced in any body which moves so that it cuts one line of magnetic force per second. This unit is found to be too small for practical purposes, and a unit one hundred million times greater, called the volt, is taken as the practical unit. Thus, if we still suppose the rails to be one centi- metre apart, and to be placed at right angles to a magnetic field of force of unit intensity, then when the slider moved with a velocity of one hundred million centimetres per second, the electromotive force induced in the circuit would be one volt. An electromotive force of one volt = 10 8 C.G.S. units. PHYSICAL UNITS. 73 69. It is seen from 60 that the dimension of resistance is a velocity, and Fig. 4 will, as first suggested by Lord Kelvin, show how a velocity may express a resistance. Instead of the linear conductor A D joining the ends of the rails, let us have a coil of vertical wire of radius r, with its plane in the magnetic meridian and a short magnet suspended at its centre. Let this magnet be deflected to an angle 0, when the uniform velocity of the slider towards E F is v units of length per unit time. Then if R be the resistance of the slider and circle of wire (the resistance of the rails being so small as to be negligible in comparison, so that the resistance of the whole circuit before the slider begins to move is E, and is still R after it has moved for some time) the current, by 56, is T = Z_ . H tan 6. 2 7T But by the preceding section F = ; R ,. R = iJLi". r tan 6 If the radius of the coil be equal in length to the circumference of a circle whose radius is the distance between the rails, and if is 45, then ~R = v. Hence the resistance of the circuit in electromagnetic units is the velocity with which the slider of length I must move so as to give a deflection of 45 to the magnetic needle suspended at the centre of the coil. Note that this velocity is independent of the moment of the suspended magnet, and of the intensity of the F 74 PHYSICAL UNITS. field. It is also independent, as was pointed out by Weber, of the fundamental unit of length and of time chosen. In the centimetre-gramme-second system of units, unit velocity is a centimetre per second, hence unit resistance on this system is a centimetre per second. But for practical purposes this unit is too small, and a unit one thousand million times greater is taken as the unit of resist- ance, and is called an ohm. A resistance of one ohm = 10 9 O.G.S. units. 70. The practical unit of current derived by Ohm's law from these practical units of electro- motive force and resistance is called an ampere. A current of one ampere = 10 8 -=- 10 9 C.G.S. units ; = Id" 1 C.G.S unit. 71. The quantity of electricity conveyed by an ampere in one second is called a coulomb. An ampere is a coulomb per second. A quantity of one coulomb = 10" 1 C.G.S. unit. 72. The practical unit of capacity is called the "farad," and by 45 a condenser has unit capacity when a charge of one coulomb raises it to an electromotive force of one volt. A capacity of one farad = 10 -1 -r 10 8 C.G.S. units ; = 10- 9 C.G.S. unit. 73. The work done in raising a quantity of elec- tricity Q, through a difference of potential E, is equal to the product of these two quantities. Hence PHYSICAL UNITS. 75 If these quantities are expressed in the absolute electromagnetic centimetre-gramme-second system of units, the work done in any time t is expressed in centimetre-dynes or ergs. If they are expressed in terms of the practical units (for example, E in volts and Q in coulombs) W = E x 10 s x Q x 10- 1 = EQ x 10 7 ergs. The work done by raising a coulomb of electricity through a difference of potential of one volt, is, on the suggestion of Sir William Siemens, called a joule. One joule of work = 10 7 ergs. Thus, if the difference of potential be expressed in volts, the current in amperes, and the resistance in ohms, the work done in any time t in an electric circuit is W = E T t joules ; 74. If a joule of work be done in the circuit during each successive second of time, and if the work be done uniformly, then the rate of doing work is called a watt. One watt = one joule per second ; = 10 7 ergs per second. Hence, expressing E in volts, T in amperes, and B in ohms, the activity in an electric circuit is A = E T watts ; To reduce this activity to horse-power, we have F2 76 PHYSICAL UNITS. to divide by 746, for 1 h.p. = 550 foot - pounds per second = 746 x 10 7 ergs per second ; = 746 watts (see 27) One kilowatt = 1,000 watts = 1 J h.p. The Board of Trade supply unit is one kilowatt for one hour, or a kilowatt-hour. It is, therefore,, equal to 3,600 x 1,000 joules. = 3-6 x 10 6 joules. This is a departure from the decimal system, and many people specially wedded to the decimal notation state that the Board of Trade might have chosen a million joules as the unit, and that watt- meters and joule-meters could have been graduated to such a unit quite as readily. 75. The following table gives the ratio of these practical units to the units both in the electrostatic and electromagnetic systems : Name. Symbol. Electrostatic Electromag- Units. netic Units. Quantity of electricity.. Coulomb ?Q 3 x 10 lo- 1 Current . . . Ampere yr 3xl0 9 lo- 1 Potential difference.. Volt eE 1/300 10 s Resistance .. Ohm rR 1/9 x 10 11 10 9 Capacity. < Farad Microfarad }oC{ 9 x 10 11 9x 10 5 10~ 9 10 -15 PHYSICAL UNITS. 77 76. As these ratios are of great importance in the reduction of experimental results, they are given directly in terms of one another as follows : One ampere = 3 x 10 electrostatic units of current ; = electromagnetic unit of current. One volt = electrostatic unit of potential ; 300 = 10 s electromagnetic units of potential ; One ohm = electrostatic unit of resistance ; 9 x 10 11 = 10 electromagnetic units of resistance. One microfarad = 9 x 10 5 electrostatic units of capacity ; = electromagnetic unit of capacity. Measurement of Electric Units by their Electro- magnetic Effects. 77. Quantity. Quantity can be measured prac- tically by a ballistic galvanometer (1) when we know the steady current that gives unit deflection on it ; or (2) when we know the dimensions of the ballistic galvanometer, and the horizontal component of the earth's magnetism at the centre of its coil ( 78). Quantity can also be measured by electro-chemical effects ( 80). 78 PHYSICAL UNITS. Let I = moment of inertia of needle in ballistic galvanometer * f M = magnetic moment of needle in ballistic galvanometer ; G = galvanometer constant, 9 = , if n = number of turns, and r = mean radius ; r Q = quantity of electricity that flows in time r. Then strength of field at centre of coil = G y r where y = the current. Moment of the force = M G y. Impulse = M G y r ( 26) = M G Q = I w, if a) is the angular velocity with which the needle begins to move. /A/T (~^ f^\\ 2 . * . J I o> 2 (the kinetic energy) = J I f J M 2 G 2 Q 2 2 I c FIG. 5. Let A A 1 be the initial position of the needle and B B 1 its position at the end of the maximum swing. Then the work done on the needle, if B C be the perpendicular from B on A A 1 , = 2 H m . A C, where m is the strength of the pole ; PHYSICAL UNITS. 79 2 H m I (1 -cos a), if / is half the length of the needle ; M H 2 sin 2 _ . 2 21 = M H 2 ~ 2 . a /I H Hence Q = sin A/ If the deflection is small and the motion simple harmonic, the period of the needle vibrating in the magnetic field, H, is T = 2 Q = H T a G ' -n- ' n 2 " (In any simple harmonic motion the period is equal to 2 TT x/ - Mass Now, in rota- Force per unit displacement tional motion, moment of inertia and torque are analogues to mass and force in translational motion. Hence, by analogy, T = 2 TT Moment of inertia Torque per radian For a complete discussion of the ballistic galvano- meter, see Gray's " Absolute Measurements in Elec- tricity and Magnetism," Vol. II., Part II., pp. 390 to 397.) If the galvanometer be slightly damped, and if X be the logarithm of the ratio of each half amplitude to the next, then 80 PHYSICAL UNITS. TT can be found by sending a steady current y through the galvanometer, giving a deflection 0. Then 7 = 1 tan 6. G a sm ' Q = -x (1 + ) coulombs, if 7 is in amperes. TT tan v \ 2/ 78. Current. In the electromagnetic system currents can be measured (1) by their action on a magnetic needle, as described in the last paragraph of 56. This involves a knowledge of the horizontal component of the earth's magnetic force. An exact determination of this can be made by the Gaussian method of finding the deflection produced on a mag- netometer needle by a magnet at a given distance, and then finding the period of the magnet vibrating under the earth's horizontal magnetic force when suspended in the position occupied by the magneto- meter in the deflection experiment.* 79. Currents can be measured (2) by their mutual action on one another. Ampere has investigated the laws of mechanical action between conductors carry- ing currents ; and he has shown that the action of a small closed circuit at a distance is the same as that of a small magnet placed at the centre of the closed circuit, provided that the axis of the magnet is at right angles to the plane of the closed circuit and * Gray's "Absolute Measurements in Electricity and Magnetism," Vol. II., Part L, Chapter II., pp. 70 to 79. PHYSICAL UNITS. 81 that its magDetic moment is equal to the product of the area of the closed circuit into the current. Let there be two conductors carrying currents y and y 1 . Take an elementary length of each, d s and d s l . Let be the angle which d s makes with the line joining it to d s 1 , and let O 1 be the angle which d s 1 makes with the same line. Let e be the angle between the directions of d s and d s 1 , then the force between these two elements can be shown to be 1 ( 3 ^ df= B y y 1 d s d s 1 - cos e + _ cos 6 cos 1 j- , where r is the distance between the two elements, d s and d s 1 . Ampere assumed B equal to unity, but, to make the formula correspond to the electro- magnetic unit current, B must be 2. If the two long conductors are parallel, the force of one con- ductor on unit length of the other is 2 y y l /r. Hence we might define unit current electro- dynamically as follows : Let there be two straight parallel conductors, unit distance apart, and carrying equal currents. Let one of them be of finite length, and the other very long. Then unit current flows in each of them when there is unit force between them per unit length of the finite conductor. This unit of electrodynamic current is I/ J2 of the unit of electromagnetic current, or 7'07 amperes. It is not used in practice. Hence, to make this unit of current coincide with the electromagnetic unit of current, we must define it as that current which, flowing in each of our two conductors, produces two 82 PHYSICAL UNITS. units of force per unit length of the finite conductor, when they are unit distance apart. Electro-dynamometers and electro-dynamic balances- measure currents by the mutual action of their coils- on one another. The Board of Trade has accepted instruments of this type for measuring standard currents in practice, the recommendation of the Board of Trade Committee on Electrical Stan- dards being " that instruments constructed on the principle of the balance, in which, by the proper disposition of the conductors, forces of attraction and repulsion are produced, which depend upon the amount of current passing, and are balanced by known weights, should be adopted as the Board of Trade standards for the measurement of current whether unvarying or alternating." 80. Electric currents can be measured (3) by their electro-chemical action. Faraday has shown that when a current passes through an electrolyte the quantity of each substance decomposed is pro- portional to the quantity of electricity which passes. The ampere is thus defined as the unvarying current which, when passed through a solution of nitrate of silver in water (definitely specified in the report previously referred to), deposits silver at the rate of O'OOlllS of a gramme per second. The balances referred to in 79 (2) are standardised by the elec- trolysis of a solution of silver nitrate or of copper sulphate. 81. Resistance. Various methods have been used for the determination of resistances in absolute PHYSICAL UNITS. 83> measure, and the reader is referred to the following papers and books for detailed descriptions and dis- cussions of them : " On the Methods Employed for Determining the Ohm," by G. Wiedemann, Phil. Mag., October, 1882, p. 258 ; "Comparison of Methods for the Determination of Resistances in Absolute Measure," by Lord Eayleigh, Phil. Mag., November, 1882, p. 329; "Electricity and Magnetism," by Clerk Maxwell, Vol. II., Chapter XVIII. ; "Absolute Measurements in Electricity and Magnetism," by Prof. A. Gray, Vol. II., Part II., Chapter X., pp. 538 to 602. 82. Resistance may be determined in absolute measure (1) by a calorimetric method. For the heat generated in time t, by a current y, in a resistance R, is y 2 R t, and this is equal to J H. If J were very accurately known, and y determined by any of the methods mentioned, R could thus be determined. But this method can more appropriately be used to determine J, from a knowledge of R by other methods. 83. Resistance can be measured (2) by induced transient currents. Weber used an earth-coil of effective area S; that is,, the sum of the areas of its N turns, movable round a horizontal east and west axis. If H 1 be the total number of lines of force per unit area due to the earth at the position of the coil, the quantity of electricity set in motion by O Q TTl one half turn of the coil is Q = , where R R is the resistance of the whole circuit. If the coil '84 PHYSICAL UNITS. be in circuit with a ballistic galvanometer, we have, by ( 77), Q = "G ' ? Sm 2 ' * H ' T" sin a/2 ' Kirchhoff used a coil so situated relative to the coil in circuit with the ballistic galvanometer that the .coefficient of mutual induction between the two coils could be accurately calculated. A sudden reversal of the current in the coil not in circuit with the galvanometer induces in the coil in circuit with the galvanometer a quantity, , where M is the coefficient of mutual induction between the two coils, y is the current in one coil, and B is the total resistance in the circuit of the other coil. .-. R = 2M y . G - . - . _-L_ H T sin a/2 84. Besistance can be measured (3) by the method of the rotating coil, first suggested by Weber, and first practically carried out by Clerk Maxwell, Balfour Stewart, and Fleeming Jenkin, in 1863, by an appa- ratus designed by Prof. Wm. Thomson (now Lord Kelvin). The method is sometimes known as the B. A. Committee method : " The coil of wire is made to revolve about a vertical diameter with constant velocity. The motion of the coil among the lines of force due to the earth's magnetism pro- duces induced currents in the coil which are alter- nately in opposite directions with respect to the coil PHYSICAL UNITS. 85 itself, the direction changing as the plane of the coil passes through the east and west direction. If we- consider the direction of the current with respect ta a fixed line in the east and west direction, we shall find that the changes in the current are accom- panied with changes in the face of the coil pre- seoted to the east, so that the absolute direction of the current, as seen from the east, remains always- the same. If a magnet be suspended in the centre of the coil, it will be deflected from the north and south line by the action of these currents, and will be turned in the same direction as the coil revolves. The force producing this deflection is continually varying in magnitude and direction; but, as the periodic time is small, the oscillations of the magnet may be rendered insensible by increasing the mass of the apparatus along with which it is suspended. The resistance of the coil may be found when we know the dimensions of the coil, the velocity of rotation, and the deflection of the magnet. The intensity of terrestrial magnetism enters into the measurement of the electromotive force, and also into the measurement of the current ; but the measure of the resistance, which is the ratio of these two quantities, is quite independent of the value of the magnetic intensity."* 85. Eesistance can be measured in absolute measure (4) by Lorenz's method. The method consists in rotating a circular metal disc round its- * Report of the Committee on Electrical Standards appointed by the British Association, 1863 (Spon's reprint, p. 97). The mathe- matical theory of the experiment is given by Clerk Maxwell on. pp. 101 to 109 of the same reprint. 86 PHYSICAL UNITS. axis in a magnetic field due to a coil which is coaxial with it. In practice, the electromotive force between the centre and the circumference] of the disc, Induced currents 106-29 ' 106-31 7 Wuilleumeier 1890 106-31 8 Duncan & Wilkes 1890 Lorenz 106-34 9 Jones 1891 Lorenz 106-31 10 Strecker 1885 106-32 ^ 11 Hutchinson 1888 106-30 1 mean 12 Salvioni 1890 106-33 ( 106-31 12* Salvioni 106-30 J 13 H. F. Weber 1884 Induced current 105-37 } 14 H. F. Weber Rotating coil 106-16 15 Poibi 1884 /Mean effect of in-\ \ duced current / 105-89 mean 16 Himstedt 1885 105-98 ' 105-94 17 Dorn 1889 Damping of a magnet 106-24 18 Wild 1883 Damping of a magnet 106-03 19 Lorenz 1885 Lorenz method 105-93 PHYSICAL UNITS. 89 88. Electromotive Force. Electromotive forces can be measured indirectly when resistances and currents are measured by any of tbe methods mentioned ; for, by Ohm's law, the difference of potential between any two points of a circuit is equal to the product of the current passing and the resistance between the points. Electromotive forces can be measured directly as the ratio of the work done by the transfer of a given quantity of electricity to the quantity transferred. From data supplied by Dr. Joule, Lord Kelvin has determined by this method the electro- motive force of a single cell of Daniell's battery. (See "Applications of the Principle of Mechanical Effect to the Measurement of Electromotive Forces, and of Galvanic Eesistances," by Lord Kelvin, Phil. Mag., December, 1851. Reprinted in " Mathematical and Physical Papers," Vol. I., p. 490.) As a practical standard of electromotive force, the Board of Trade has taken the electrical pressure, at a temperature of 15 C., between the poles or elec- trodes of the voltaic cell known as Clark's cell (specially prepared) as 1*434 volts. Corrections, similar to those mentioned regarding the ohm, must be applied to B.A. volts, legal volts, etc., to reduce them to Board of Trade volts or international volts. 89. Capacity. The capacity of a conductor is the ratio of the quantity of electricity in the charge to the electromotive force producing it, and hence an indirect measurement of capacity is easily per- formed. It can be measured by a ballistic galvano- meter when the resistance of the circuit in electro- magnetic measure is known. G PHYSICAL UNITS. Measurement of Electric Units by their Electrostatic Effects. 90. Quantity. The practical method of measur- ing a quantity of electricity is to measure the capacity of the conductor containing the charge ( 93), and to multiply this by the electromotive force producing the charge ( 92). 91. Current. The practical method of measur- ing current is to find the difference of potential ( 92) and the resistance ( 94) between any two points of the conductor, and to divide the one by the other. 92. Electromotive Force. The practical method of measuring electromotive force is to measure the force between two electrified bodies, for the differ- ence of potential between them is proportional to the square root of the force exerted between them. Absolute electrometers are constructed on this principle. 93. Capacity. The capacity of a conductor is measured by comparing its capacity with that of a spherical conductor insulated at a distance from other conductors. The capacity of the latter is equal to its radius. 94. Resistance. Electrostatic resistance can be measured by allowing a body of known capacity ( 93), and charged to a potential, measured as in ( 92), to discharge itself through the conductor whose electrostatic resistance we want to measure. This method is applicable only to conductors of large resistance. PHYSICAL UNITS. 91 95. Now that we know how to measure the -electric units electrostatically and electromagnetic- ally, we can find the ratio of the one to the other that is, we can experimentally find what was denoted by v i] 66. This velocity, v, has been determined experimentally by measuring in electrostatic units and in electromagnetic units : Ratio. I. Quantity, 90 and 77 v II. Current, 91 and 78, 79, 80 v III. Resistance, 94 and 82-85 I/?; 2 IV. Electromotive force, 92 and 88 1/v t: J > 1 Experimenter. Reference. Method employed, Compari- son of v in centimetres per sec. 1856 { Kohlrausch I Pofjg ' Ann " 1856 Quantity 3-107 xlO 10 ' g 1868 Maxwell. Phil. Trans. , 1868 Potential 2-842 x 10 10 o no 1869 { W a nd T KS? n }jB- A- "port, 1869 Potential 2-808 x!0 in i 1872 McKichan Phil. Trans., 1872 Potential 2-896 xlO 10 a 1879 Ayrton and Perry ^^^^ } Capacity 2-960 xlO 10 o c C 1880 Shida f^ 10 4 Potential 2-955 xlO 1 " x V^ isou j ~r- 1881 Stoletow /Soc. Franc de\ \ Phys., 1881 / Capacity 2-990 xlO 10 Ml* o 1882 F. Exner Wien. Ber., 1882; Potential 2-920 xlO 10 ~~ 1881 Klemencic Wien. Ber., 1884! Capacity 3-019 xlO 10 x 1883 J. J. Thomson Phil. Trans. 1 1883 Capacity 2-963 xlO 10 o 1888 Himstedt Wied. Ann., 1888 Capacity 3-009 xlO 10 ~ 1889 Rowland Phil. Mag., 1889J Quantity 2-9815 x 10 10 71 1889 E. B. Rosa Phil. Mag., 1889! Capacity 3 0004 x 10 10 II 1889 W. Thomson Inst.E.E. Potential 3-004 xlO 10 i 1890 fj. J. Thomson \ 1 and Searle J Phil. Trans., 1890 Capacity 2-9955xl0 10 o 92 PHYSICAL UNITS. 96. There is still considerable difference of opinion as to the names to be given to the practical magnetic units. The Committee on Notation of trie- Chamber of Delegates of the International Electrical Congress of 1893 in Chicago recommended that unit field intensity, unit magnetic induction, and unit magnetising force be each called a Gauss ' r unit flux of magnetic force a Weber ; unit magneto- motive force a Gilbert; and unit reluctance or mag- netic resistance an Oersted. 97. At the meeting of the British Association at Ipswich (1895) a discussion took place on magnetic- units, and it was suggested : (1) That, as a unit for magnetic field, 10 8 C.G.S. lines be called a weber. NOTE. A weber added per second at a steady rate to the field girdled by a wire circuit induces one volt in every turn of that circuit. Hence the webers " cut " by a closed wire circuit of n turns are equal to the quantity of electricity in coulombs thereby impelled round that circuit multiplied by - th its resistance in ohms. n (2) That the C.G.S. unit of magnetic potential or of magnetomotive force be called a gauss. NOTE. An ampere-turn corresponds to (= 1*2566) gauss. Hence the number of gausses round any closed curve linked on an electric circuit is equal to 4 Tr/10 times the number of ampere-turns in this- circuit. (3) That the termination -ance be used in general for words expressing the properties of a definite body or piece of matter; e.g., resistance, conductance,, inductance, permeance, reluctance, etc., and that the termination -ivity or -ility or the like be used for words- PHYSICAL UNITS. 93 expressing the specific properties of a material ; ^.g. conductivity, resistivity, inductivity, refractivity, permeability, etc. Heat Units. 98. Heat being a form of energy, a unit of heat may be taken as a unit of energy. In the C.G.S. system the erg is the unit of heat, and its dimensional expression is [Lr M T~ 2 ]. 99. Heat is more commonly expressed in thermal units, and the unit quantity of heat so defined is the amount of heat required to raise the temperature of unit mass of water at a given temperature through unit difference of temperature. Hence some unit of temperature is necessary, and if we call the dimen- sion of a unit or of a degree of temperature the dimensional formula for unit of heat will be [M &]. 100. The ratio of the dynamical unit defined in 98 to the thermal unit defined in 99 is called the " dynamical equivalent of heat," or " Joule's equivalent," and is generally denoted by the letter J. Thus [J] = [L 2 M T~ 2 ] -=- [M 0] = [L 2 T~ 2 6~ 1 ]. 101. No definite temperature has been chosen for defining the unit quantity of heat, but experiment shows that it takes 42 x 10 6 ergs to raise the tempera- ture of one gramme of water from 10 C. to 11 C. Reducing this to gravitation units by dividing by 981 (which is taken as the average value of g for this country) we find it takes : 94 PHYSICAL UNITS. 428 metre-kilogrammes to raise 1 kilogramme of water 1 CL 1,404 foot-pounds ,, 1 pound 1 C. 780 1F. since 1 metre=3'2S feet, and 1 C. = l-8 F. 102. Specific heat of a substance is the ratio of the quantity of heat necessary to raise any mass of it through unit difference of temperature, to the quantity of heat necessary to raise an equal mass of water through unit difference of temperature. As- the quantity of heat necessary to raise unit mass of water through unit difference of temperature is taken as unit quantity of heat, specific heat of a substance is a mere number. Thermal capacity of a body is the quantity of heat necessary to raise its temperature one degree. Thermal capacity is thus equal to the product of the mass of the body and the specific heat of its sub- stance. Otherwise, specific heat of a substance is the thermal capacity of unit mass. The dimensional expression for thermal capacity in thermal units is M]. 103. Sometimes the thermal capacity per unit volume is used instead of the thermal capacity per unit mass, and the units are then called thermo- metric units. As the volume of unit mass of water is approximately the unit of volume in the centi- metre-gramme-second system of units, thermal capa- city per unit volume is also a mere number, as it is the ratio of the quantity of heat necessary to raise the substance through unit difference of tempera- ture, to the quantity of heat necessary to raise an PHYSICAL UNITS. 95 equal volume of water through unit difference of temperature. 104. Latent heat of fusion of a substance is the quantity of heat necessary to change the state of one gramme of it from solid to liquid without any change of temperature. Latent heat of vaporisation of a sub- stance is the quantity of heat necessary to change the state of one gramme of it from liquid to vapour with- out change of temperature. The latent heat of water is approximately 80, and the latent heat of steam at 100 C. is 537. This means that it takes 80 heat units to convert 1 gramme of ice at C. to 1 gramme of water at C., the heat unit in this case being the quantity of heat necessary to raise the temperature of 1 gramme of water through 1 C., say from 4 C. to 5 C. The specific heat of water is not the same at all temperatures. Kegnault, in 1840, expressed the specific heat of water at any temperature $ C. from up to 230 C. by the formula S = 1 + 4 x 10~ 5 6 + 9 x 10- 7 0*. Similarly, it takes 537 times as much heat to convert a gramme of water at 100 C. into a gramme of steam at the same temperature as would raise the temperature of 1 gramme of water from 4 C. to 5 C. Taking the specific heat of water to be unity, the formula for the latent heat of steam at any tempera- ture 0, as given by Regnault, is L = 606-5 - 0-695 6. 105. The linear, superficial, or cubical expansion of a body is the ratio of the increase of length, surface, 96 PHYSICAL UNITS. or volume produced by a rise of temperature of one degree to the original length, surface, or volume. If 1 be the length of a rod at C., and 10 its length at C., then the coefficient of linear expansion, , * -# /. I ff = l (l+ a 0). Similarly, if y be the coefficient of cubical expansion, V0=V (1 + yB). y and a are, for the same substance, approximately in the ratio of 3 to 1. 106. Thermal conductivity is the quantity of heat that passes through unit area of a slab of unit thickness in unit time when the difference of tem- perature between the two sides is unity. It is obvious that the quantity of heat that passes through a slab of material when the steady state has been reached viz., when the distribution of tempera- tures in the slab does not vary with the time, is directly proportional to (1) the conductivity of the material, (2) the area considered, (3) the difference of tempera- ture between the two sides, and (4) the time, and is inversely proportional to the thickness of the slab. Let H be the quantity of heat, in thermal units, that passes through an area A, of thickness d, in time t, when the difference of the temperatures of the two sides is 6. Then if k denote the conductivity of the material, H = &. A. L . t, d T, Hd * " OF THH UNIVERSIT PHYSICAL UNITS. Now the dimensions of H in thermal units by 99 = [M0]. =[L-IMT-I.] (i) If H be expressed in dynamical units, [F] = [L 2 M T~ 2 x L x L- 2 x 0^ x T" 1 ] = [LMT-30~i] (2) Again, if H be expressed in thermometric units, ['] = [L 3 6 x L x L- 2 x 0-i x T- 1 ], = [L 2 T-i] (3) Maxwell called the conductivity defined in (3) thermometric conductivity, to distinguish it from thermcd conductivity as defined in (1). Lord Kelvin called it diffusivity . Diffusivity of a substance for heat is thermal conductivity divided by thermal capacity per unit bulk. In symbols, k' = k/C, where C is the thermal capacity of the substance per unit volume. It is properly specified in units of area per nnit of time, as we have already seen in 33. 107. Emissivity of a body is the quantity of heat radiated by it per unit area, per unit time, per unit difference of temperature between the surface and the surrounding medium. If e denote emissivity, then H & /. [e] = [M 6 x L- 2 x tf- 1 x T- 1 ] = [L~ 2 M T- 1 ]. , 108. The entropy of a body is directly proportional 98 PHYSICAL UNITS. to the quantity of heat in it and inversely proportional to its temperature. If < denote entropy, then [>] = [M6> x<9-i] = [M]. 109. Some of the more useful physical constants and relations are, for convenience of reference, given here in alphabetical order. Air weighs -0807281b. per cubic foot at C. and 76 cms. of mercury; Occupies 12-387 cubic feet per pound at C. and 76 cms. of mercury. Ampere = 1/10 C.G.S. unit of electric current; = a coulomb per second ; deposits 1*177 grammes of copper per hour; deposits 4-025 grammes of silver per hour. Atmosphere = 76 centimetres of mercury ; = 29-922in. of mercury; = 14'6961b. weight per square inch; == 1033-3 gms. weight per square centimetre; = 1013600 dynes. " Candle. The British standard candle is of spermaceti. It is ^in. in diameter, weighs Jib., and burns at the rate of 120 grains per hour. Candle-power. The illumination given by a candle at a distance of a foot is taken as a standard. In glow lamps the activity is about four watts per candle-power. Mr. Vernon Harcourt has devised a burner to give an " average candle" when a mixture of pentane vapour and air is burnt from an orifice of Jin. diameter, so as to form a steady conical flame 2jin. high. PHYSICAL UNITS. 99' Centimetre = -010936 yard; = -032809 foot; = -393708 inch ; ,, (square) = "155006 square inch; (cubic) = -061027 cubic inch. Centimetre-gramme = 72-33 x 10~ 6 foot-pound; = 2328-0 x 10~ G foot-poundal ; = 98-14 x 10~ 6 joule. Centimetre-gramme per second = 98-14 x 10~ 6 watt; = -00434 foot-pound per minute. Day, mean solar = 86400 mean solar seconds ; sidereal = 86164 Degree of longitude = 4 minutes of mean solar time ; ,, at the equator = 69 '170 British statute miles ; at London = 42*67 latitude at equator = 68'693 ,, at London = 69'16 ,, ,, Dyne = the force which, acting for one second on a mass of one gramme, generates in it a velocity of one centimetre per second. Electrical unit of energy = 1 kilowatt-hour ; = 1 J horse-power for one hour ; = 3-6 x 10 joules. It will supply sixteen 1 6-candle-power lamps for nearly one hour, and will give as much light as 100 cubic feet of gas Erg = Work done by a dyne acting through a centimetre ;. = 1/421390 foot-poundal. 100 PHYSICAL UNITS. Farad = the electrostatic capacity of a condenser which requires a coulomb of electricity to raise its potential by one volt; = 10~ 9 C.G.S. unit. Foot = 30-48 centimetres. Foot per second = 0-68182 mile per hour ; = 1*0972 kilometres per hour. Foot-pound = 0-13825 metre-kilogramme; = 1-356 x 10 7 ergs. Foot-pound per minute = 1/44-232 watt ; = 230*42 centimetre - grammes per second. Foot-pound per second = 1/550 horse-power ; ., = 1-3565 watts. Foot-poundal = 421390 ergs. Gallon = the volume of lOlb. of pure distilled water at 62 F., and pressure 30in. of mercury ; = 277*463 cubic inches ; = 4-546 litres. Grain per inch =25 dynes per centimetre. Gramme = 15*43235 grains; = 1/453-6 pound. Gramme per sq. cm. = 1/70 "31 pound per sq. in. Henry, 01 Quadrant, or Secohm = proposed unit of self-induc- tion = 10 9 C.G.S. units. Horse-power = 550 foot-pounds per second ; = 746 watts ; = 76 metre-kilogrammes per second. Horse-power hour =3/4 Board of Trade electric unit or kilowatt-hour. PHYSICAL UNITS. 101 Inch = 25-39954 millimetres. Joule = 10 7 ergs ; = 23-73 foot-p.oundals. Joule's Equivalent = 42 x 10 6 ergs. Kilogramme = 2'20461b. Kilogramme per sq. cm. = 14'22281b. per sq. in. Kilometre = 0*6214 mile. Kilometre per hour = 54'68ft. per minute. Kilowatt =1-34 horse-power ; = 737 '2 foot-pounds per second. Knot = 1 nautical mile per hour ; = 6087ft. per hour ; Admiralty = 6080ft. per hour. Litre = 1 cubic decimetre. = 6T0253 cubic inches. Megohm = one million ohms. Mercury, inch of = "491b. per sq. in. Metre = 39'37in. Microhm = one-millionth ohm ; = 1000 C.G.S. units. Micromillimetre = 10~ 7 centimetre. Mile per hour = 88ft. per minute ; = 44*7 centimetres per second. Ohm = 10 9 C.G.S. unit ; = 1-0136 B.A. unit; = 1-0028 legal ohm; = 1-0630 Siemens's unit. Pound = 7000 grains ; = 453-6 grammes. 102 PHYSICAL UNITS. Pound per sq. in. = 70'31 grammes per sq. cm. Poundal = that force which, acting on lib. for one second, generates in it a velocity of 1ft. per second ; = 13825 dynes. Radian = 57'2957S = 57 17' 45" = 206265". Ton = 1-016 metric tonnes. Ton per sq. in. = 157 '5 kilogrammes per square centimetre. Water weighs 62'4151b. per cubic foot at 4 C. ; occupies 0*016022 cubic foot per Ib. gallon of, weighs lOlb, at 62 F. ; ton of sea, occupies 35 cubic feet. Watt = 10 7 ergs per second ; = 44'23 foot-pounds per minute. Yard = *9144 metre. Year, civil = 365-24224 mean solar days ; = 365 days, 5 hours, 48 minutes, 49 '7 seconds ; = 31*557 x 10 6 mean solar seconds. sidereal = 365-25635 mean solar days. PHYSICAL UNITS. 103 APPENDIX. "Order in Council on Electrical Units and Standards, dated August 23, 1894. Whereas by the Weights and Measures Act, 1889, it is among other things enacted that the Board of Trade shall from time to time cause such new denominations of standards for the measurement of electricity as appear to them to be required for use in trade to be made and duly verified. And whereas it has been made to appear to the Board of Trade that new denominations of stan- dards are required for use in trade based upon the following units of electrical measurement viz. : 1. The ohm, which has the value 10 9 in terms of the centimetre and the second of time, and is repre- sented by the resistance offered to an unvarying electric current by a column of mercury at the temperature of melting ice 14*4521 grammes in mass of a constant cross-sectional area and of a length of 106'3 centimetres. 2. The ampere, which has the value T \^ in terms of the centimetre, the gramme, and the second of time, and which is represented by the unvarying electric current which when passed through a solu- tion of nitrate of silver in water in accordance with the specification appended hereto, and marked A, deposits silver at the rate of 0*001118 of a gramme per second. 104 PHYSICAL UNITS. 3. The volt, which has the value 10 s in terms of the centimetre, the gramme, and the second of time, being the electrical pressure that if steadily applied to a conductor whose resistance is one ohm will produce a current of one ampere, and which is represented by *6974 (TT?T) of the electrical pres- sure at a temperature of 15 C. between the poles of the voltaic cell known as Clark's cell set up in accordance with the specification appended hereto, and marked B. And whereas they have caused the said new denominations of standards to be made and duly verified. Now, therefore, her Majesty, by virtue of the power vested in her by the said Act, by and with the advice of her Privy Council, is pleased to approve the several denominations of standards set forth in the schedule hereto as new denomina- tions of standards for electrical measurement. C. L. PEEL. SCHEDULE. I. Standard of Electrical Resistance. A standard of electrical resistance denominated one ohm being the resistance between the copper terminals of the instrument marked " Board of Trade Ohm Standard Verified 1894 " to the passage of an unvarying electrical current, when the coil of insulated wire forming part of the aforesaid instrument and con- nected to the aforesaid terminals is in all parts at a temperature of 15 *4 C. PHYSICAL UNITS. 105 II. Standard of Electrical Current. A standard of electrical current denominated one ampere being the current which is passing in and through the coils of wire forming part of the instrument marked "Board of Trade Ampere Standard Verified 1894," when, on reversing the current in the fixed coils, the change in the forces acting upon the suspended coil in its sighted position is exactly balanced by the force exerted by gravity in Westminster upon the iridio-platinum weight marked A, and forming part of the said instrument. III. Standard of Electrical Pressure. A standard of electrical pressure denominated one volt being one-hundredth part of the pressure which, when applied between the terminals forming part of the instrument marked " Board of Trade Volt Standard Verified 1894," causes that rotation of the sus- pended portion of the instrument which is exactly measured by the coincidence of the sighting wire with the image of the fiducial mark A before and after application of the pressure, and with that of the fiducial mark B during the application of the pressure, these images being produced by the sus- pended mirror and observed by means of the eyepiece. In the use of the above standards the limits of accuracy attainable are as follows : For the ohm, within one-hundredth part of 1 per cent. For the ampere, within one-tenth part of 1 per cent. H 106 PHYSICAL UNITS. For the volt, within one-tenth part of 1 per cent. The coils and instruments referred to in this- schedule are deposited at the Board of Trade Standardising Laboratory, 8, Richmond-terrace, Whitehall, London. SPECIFICATIONS referred to in the foregoing Order in Council. SPECIFICATION A. In the following specification the term silver voltameter means the arrangement of apparatus by means of which an electric current is passed through a solution of nitrate of silver in water. The silver voltameter measures the total electrical quantity which has passed during the time of the experi- ment, and by noting this time the time average of the current, or if the current has been kept con- stant the current itself, can be deduced. In employing the silver voltameter to measure currents of about one ampere the following arrange- ments should be adopted : The cathode on which the silver is to be deposited should take the form of a platinum bowl not less than 10 centimetres in diameter, and from four to five centimetres in depth. The anode should be a plate of pure silver some 30 square centimetres in area and two or three millimetres in thickness. This is supported horizontally in the liquid near the top of the solution by a platinum wire passed through holes in the plate at opposite PHYSICAL UNITS. 107 corners. To prevent the disintegrated silver which is formed on the anode from falling on to the cathode, the anode should be wrapped round with pure niter paper, secured at the back with sealing- wax. The liquid should consist of a neutral solution of pure silver nitrate, containing about 15 parts by weight of the nitrate to 85 parts of water. The resistance of the voltameter changes some- what as the current passes. To prevent these changes having too great an effect on the current, some resistance besides that of the voltameter should be inserted in the circuit. The total metallic resist- ance of the circuit should not be less than 10 ohms. Method of Making a Measurement. The platinum bowl is washed with nitric acid and distilled water, dried by heat, and then left to cool in a desiccator. When thoroughly dry, it is weighed carefully. It is nearly filled with the solution, and con- nected to the rest of the circuit by being placed on a clean copper support to which a binding screw is attached. This copper support must be insulated. The anode is then immersed in the solution so as to be well covered by it, and supported in that position ; the connections to the rest of the circuit are made. Contact is made at the key, noting the time of contact. The current is allowed to pass for not H2 108 PHYSICAL UNITS. less than half-an-hour, and the time at which con- tact is broken is observed. Care must be taken that the clock used is keeping correct time during this interval. The solution is now removed from the bowl, and the deposit is washed with distilled water and left to soak for at least six hours. It is then rinsed successively with distilled water and absolute alcohol, and dried in a hot-air bath at a temperature of about 160 C. After cooling in a desiccator it is weighed again. The gain in weight gives the silver deposited. To find the current in amperes, this weight, expressed in grammes, must be divided by the number of seconds during which the current has been passed, and by 0*001118. The result will be the time average of the current, if during the interval the current has varied. In determining by this method the constant of an instrument the current should be kept as nearly constant as possible, and the readings of the instru- ment observed at frequent intervals of time. These observations give a curve from which the reading corresponding to the mean current (time average of the current) can be found. The current, as calcu- lated by the voltameter, corresponds to this reading. SPECIFICATION B. ON THE PREPARATION OF THE CLARK CELL. Definition of the Cell. The cell consists of zinc or an amalgam of zinc with mercury and of mercury in a neutral saturated PHYSICAL UNITS. 109 solution of zinc sulphate and mercurous sulphate in water, prepared with mercurous sulphate in excess. Preparation of the Materials. 1. The Mercury. To secure purity it should be first treated with acid in the usual manner, and subsequently distilled in vacuo. 2. The Zinc. Take a portion of a rod of pure redistilled zinc, solder to one end a piece of copper wire, clean the whole with glass-paper or a steel burnisher, carefully removing any loose pieces of the zinc. Just before making-up the cell dip the zinc into dilute sulphuric acid, wash with distilled water, and dry with a clean cloth or filter paper. 3. The Mercurous Sulphate. Take mercurous sulphate, purchased as pure, mix with it a small quantity of pure mercury, and wash the whole thoroughly with cold distilled water by agitation in a bottle ; drain off the water, and repeat the pro- cess at least twice. After the last washing, drain off as much of the water as possible. 4. The Zinc Sulphate Solution. Prepare a neutral saturated solution of pure ("pure recrystallised") zinc sulphate by mixing in a flask distilled water with nearly twice its weight of crystals of pure zinc sul- phate, and adding zinc oxide in the proportion of about 2 per cent, by weight of the zinc sulphate crystals to neutralise any free acid. The crystals should be dissolved with the aid of gentle heat, but the temperature to which the solution is raised should not exceed 30 C. Mercurous sulphate treated as described in 3 should be added in the 110 PHYSICAL UNITS. proportion of about 12 per cent, by weight of the zinc sulphate crystals to neutralise any free zinc oxide remaining, and the solution filtered, while still warm, into a stock bottle. Crystals should form as it cools. 5. The Mercurous Sulphate and Zinc Sulphate Paste. Mix the washed mercurous sulphate with the zinc sulphate solution, adding sufficient crystals of zinc sulphate from the stock bottle to ensure saturation, and a small quantity of pure mercury. Shake these up well together to form a paste of the consistence of cream. Heat the paste, but not above a temperature of 30 C. Keep the paste for an hour at this temperature, agitating it from time to time, then allow it to cool ; continue to shake it occasionally while it is cooling. Crystals of zinc sulphate should then be distinctly visible, and should be distributed throughout the mass ; if this is not the case, add more crystals from the stock bottle, and repeat the whole process. This method ensures the formation of a saturated solution of zinc and mercurous sulphates in water. To Set up the Cell. The cell may conveniently be set up in a small test tube of about two centimetres diameter, and four or five centimetres deep. Place the mercury in the bottom of this tube, filling it to a depth of, say, "5 centimetre. Cut a cork about *5 centimetre thick to fit the tube ; at one side of the cork bore a hole through which the zinc rod can pass tightly ; at the other side bore another hole for the glass PHYSICAL UNITS. Ill tube which covers the platinum wire ; at the edge of the cork cut a nick through which the air can pass when the cork is pushed into the tube. Wash the cork thoroughly with warm water, and leave it to soak in water for some hours before use. Pass the zinc rod about one centimetre through the -cork. Contact is made with the mercury by means of a platinum wire about No. 22 gauge. This is pro- tected from contact with the other materials of the <;ell by being sealed into a glass tube. The ends of the wire project from the ends of the tube ; one end forms the terminal, the other end and a portion of the glass tube dip into the mercury. Clean the glass tube and platinum wire carefully, then heat the exposed end of the platinum red hot, and insert it in the mercury in the test tube, taking care that the whole of the exposed platinum is covered. Shake up the paste and introduce it without con- tact with the upper part of the walls of the test tube, filling the tube above the mercury to a depth of rather more than one centimetre. Then insert the cork and zinc rod, passing the glass tube through the hole prepared for it. Push the cork gently down until its lower surface is nearly in contact with the liquid. The air will thus be nearly all expelled, and the cell should be left in this condition for at least 24 hours before sealing, which should be done as follows : Melt some marine glue until it is fluid enough to pour by its own weight, and pour it into the test PHYSICAL UNITS. tube above the cork, using sufficient to cover com- pletely the zinc and soldering. The glass tube containing the platinum wire should project some way above the top of the marine glue. The cell may be sealed in a more permanent manner by coating the marine glue, when it is set, with a solution of sodium silicate, and leaving it to harden. The cell thus set up may be mounted in any desirable manner. It is convenient to arrange the mounting so that the cell may be immersed in a water-bath up to the level of, say, the upper surface of the cork. Its temperature can then be deter- mined more accurately than is possible when the cell is in air. In using the cell sudden variations of temperatures- should as far as possible be avoided. The form of the vessel containing the cell may be varied. In the H form, the zinc is replaced by an amalgam of 10 parts by weight of zinc to 90 of mercury. The other materials should be prepared as already described. Contact is made with the amalgam in one leg of the cell, and with the mercury in the other, by means of platinum wires- sealed through the glass. PHYSICAL UNITS. TABLE I. Surfaces and Volumes of Solids. Geometrical Form. Surface. Volume. Rectangular parallele- piped, a, b, c Right circular cylinder, radius r, and height h Solid ring, D outer \ 2 (a 6 + 6 c + c a) 1 f(Tr DV J (TT d\ 2 \ ab c wr-h diameter, and d V inner diameter ) 32 Regular pyramid -I Lateral surface = peri- meter of base x ^ slant j- Base x ^ altitude Right circular cone, j height h j Frustum of a cone, radii R and r, and h = slant height, and hi = vertical height height Lateral surface = cir - | cumference of base slant height Sphere 4 TT r- - TT D- * ^ r 3 _ 1 TJ. J)a 3 Spherical cap, height ) h, and radius of > 2 7T r A + 7T TV = *(f- + t*) base T\ \ J Base x 4 height = Jr*A R Theorems of Pappus. If a plane closed curve revolve through any angle about an axis in its plane, the solid content of the surface generated is equal to the product of the area of the curve into the length of the path described by its centre of gravity. If the arc of any plane curve revolve through any angle round an axis in its plane, the area of the surface generated is equal to the product of the length of the curve into the length of the path described by its centre of gravity. 114 PHYSICAL UNITS. TABLE II. Values, in centimetres per second per second, of the Accelera- tion due to Gravity at different places reduced to sea level, and of the length of the second's pendulum in centi- metres. Latitude. Value of g. Value of I. Equator 0' 978-10 99-103 'Constantinople 41 0' 980-26 99-321 Paris 45 0' 48 50' 980-61 980-94 99-356 99-390 Greenwich 51 29' 981-17 99-413 Berlin 52 30' 981-25 99-422 Dublin 53 21' 981-32 99-429 Glasgow 55 52'42"'8 981-44 99-442 )&''& Edinburgh ... 55 57' 981-54 99-451 Pole 90 0' 983-11 99-610 PHYSICAL UNITS. 115 TABLE III. Moments of Inertia of Symmetrical Bodies, Rod round perpendicular axis through middle (J = half length) ................................. I = M . P/3 Rod round perpendicular axis through end (f = half length) ................................. I = M . (2 /) 2 /3 'Circular wire about a diameter (radius r) . . . I = M . r' 2 /'2 Rectangular plate (sides 2a and 26) round axis passing through centre of gravity, and parallel to side 2a .............................. I = M . 6 2 /3 Rectangular plate (sides 2a and 26) round axis passing through centre of gravity, and parallel to side 26 .............................. I = M . a 2 /3 Rectangular plate (sides 2a and 26) round axis passing through centre of gravity, and perpendicular to its plane ..................... I = M . (a 2 + 6 2 )/3 Circular plate about a diameter (radius r) ... I = M . r' 2 /4 Circular plate about a perpendicular axis through centre ................................. I = M . r 2 /2 Elliptic plate about minor axis .................. I = M . a 2 /4 Elliptic plate about axis through centre and perpendicular to plane ........................ I = M . (a 2 + 6-)/4 Cylinder about axis (radius r) .................. I = M . r" 2 /2 Hollow cylinder (inner radius r, outer radius Cylinder round axis through middle, and perpendicular to length (2 I) .................. I = M,(/ 2 /3 + r-/4) Sphere about a diameter ........................ I = M. 2 r' 2 /5 Ellipsoid about a principal diameter ......... I = M . (a 2 + 6 2 )/5 Spherical shell about a diameter ... I = M . 2 (r 5 -r 1 5 )/5 (^-r, 3 ) 116 PHYSICAL UNITS. TABLE IV. Tables of Densities of Solids and Fluids. The number for solids and liquids is approximately the weight in grammes of a cubic centimetre of the substance. In the table for gases another column is added to give specific gravity relative to air. (a) Solids. Aluminium 2-67 Ivory 1*92 Amber . . 1-1 Lead 11 '4 Antimony 6'7 Lithium 59 Arsenic * 5-7 Magnesium 1*7 Bamboo . 0-4 Manganese 8*0 Barium . . ^ 3'75 Marble 2*7 Basalt 2-8 Mica 2*7 to 3-1 Bismuth ... 9-8 Nickel 8-6- Bone 1-8 to 2-0 Opal _ 1*9 to 2*3 Boron 2-7 Palladium 12-1 Brass 8-4 Paraffin 0*87 Brick 2-1 Pearl 2-7 Bromine 3-0 Phosphorus 1-84 Bronze coinage . 8-66 Platinoid 8*7 Cadmium 8 '67 Platinum 0'3 to 22*1 Calcium . . . _ . . . 1-58 Porcelain ... 2-15 to 2-38 Carbon (diamond) ... 3'3 to 3-5 Potassium 0-88 (srraohite) .. 2-2 to 2-3 Pumice stone 2*2 to 2*5 Chalk 1-8 to 2-8 Quartz (rock crystal) 2-65 Chromium 6-5 Rocks 2*5 to 3*0 Clay . .. . 1-92 Sand (dry) 1-42 Cobalt 8'6 Selenium 4*3 Copper 8'95 Silicon 2-6 Coral 2'69 Silver 10-57 Cork 0*24 (Mint) 10-38 Earth (average den- Slate 2*1 to 2*8 sity) .. 5-5 Sodium 0-97 Emery 4-0 S permaceti 0-94 German silver 8-5 Strontium 2-54 Glass (crown) 2'5 to 2*7 Sulphur 2-07 (flint) 3'0 to 3'5 Talc 2*5 Gold 19-3 Tellurium '. 6-4 allov (Mint) 17*49 Tin 7*3 Granite 2-7 Tourmaline 2 -9 to 3*3 Graphite 2-2 \Vax (bees') 0'9ft Gut taper ch a . 0-97 Wood ash .. . 0*75 Human body (mean) Ice 1-07 0'917 beech 0'69 to ebony 1*1 to 0*80 1-2 Iceland spar 2-7 elm 66 Indiarubber 0'92 to 0'99 lignum vitsB . 1*3 Iodine 5'0 oak 0*69 to 0-99 Iridium 22'38 pine 0*5ft Iron (cast) 6'95 to 7'3 teak ... . 0-8, . (wrought) . 7'6 to 7-8 Zinc .. 7*2 PHYSICAL UNITS. TABLE IV. (continued). 117 (6) Liquids at C. Alcohol (absolute) 0-791 ,, (proof spirit) 0'916 Bisulphide of carbon 1-293 Blood (human) T06 Chloroform ...._ 1*527 Ether _ 0'736 Glycerine T27 Hydrochloric acid 1 '27 Mercury ... 13-596 Milk Nitric acid Oil, linseed ,, olive ,, petroleum .. ,, turpentine .. Sulphuric acid .. Water ... 1-03 1-56 0-94 92 0-76 to 0-83 0-87- 1 85 0-999884 1 -026 (c) Gases at C. and 76 cms. of Pressure. Gas Mass of a Cubic Centimetre in Grammes. Density compared with Air. Air dry 0-0012932 1*0000 Ammonia 0-0007697 0-5952 Carbonic acid 0-0019774 1-5291 ,, oxide 0-0012344 0-9546 Cyanogen Chlorine 0-0023304 0-0031329 1-8019 2-4226 Hydrogen 0-0000896 0-0693 Alarsh gas 0-0007271 0-5622 Nitric oxide .. . 0-0013434 1-0388 Nitrogen 0*0012561 0*9714 Nitrous oxide 0-0019697 1-5231 defiant o^as 0-001274 0*9852 Oxvfiten . 0-0014299 1-1057 Sulphurous anhydride.. 0-0027289 2-1101 118 PHYSICAL UNITS. PHYSICAL UNITS. U\N TABLE VI. Velocity of Elasticity Waves in kilometres per second and in British statute miles per second. (1 kilometre = 0'62 14 British statute mile.) Substance. Longi nal, ir ro tudi- i free i. Distortional, Condensational rarefactional, in infinite solid or fluid. v = Den- sity, D V iJ gn/D ?M/D Jff(k + i)ID Aluminium ... Brass Kilos, per Sec. 4-96 3-41 3-72 3-92 4-53 4-32 5-11 5-26 3-70 2-83 2-66 3-47 3-47 4-96 Miles per Sec. 3-08 2-12 2-31 2-44 2-81 2-68 3-17 3-27 2-30 1-76 1-65 2-16 2-16 3-08 Kilos, per Sec. 3-04 2-08 2-24 2-35 2-85 2-71 3-17 3-25 2-31 1-71 1-61 2-23 2-55 Miles per Sec. 1-89 1-29 1-39 1-46 1-77 1-39 1 97 2-02 1-43 1-06 1-00 1-39 1-58 Kilos, per Sec. 5-99 4-24 5-01 5-56 5-02 4-81 5-92 6-18 4-20 3-77 3-47 3-69 3-49 2-00 1-49 Miles per Sec. 3-72 2-63 3-12 3-46 3-12 2-99 3-68 3-84 2-61 2-34 2-16 2-29 2-17 1-24 926 2-76 8-47 8-9 8-8 2-94 7-24 7-68 7-73 8-78 20-81 10-21 7-11 2-68 2-74 13-6 1:0 Copper . . German silver Glass, flint ... Iron, cast ... Iron, wrought Pianoforte steel wire... Platinoid Platinum Silver Zinc Clay, rock ... Slate Mercury Water 120 PHYSICAL UNITS. TABLE VII. Surface Tensions at 20 C., in dynes per lineal centimetre (Quincke, Wied. Ann., 1886, p. 219). Liquid. Density. Tension o the P Surface Liquid fr< separating >m Air. Water. Mercury. Water G'9982 81 o 418 ; jMercurv 13-5432 540 418 Bisulphide of carbon . . . Chloroform 1-2687 1-4878 32-1 30-6 41-75 29-5 372-5 399 Alcohol 0-7906 25-5 399 Olive oil 0-9136 36-9 20-56 335 Turpentine 0-8867 29-7 11-55 250-5 Petroleum 0*7977 31-7 27-8 284 Hydrochloric acid ri 70-1 377 Solution of hyposul- phite of soda 1-1248 77'5 44-7.5 1 PHYSICAL UNITS. 121 TABLE VIII. Viscosity of Liquids, in dynes per square centimetre, at the temperatures stated. Viscosity diminishes in the case of liquids, and increases in the case of gases, for elevation of temperature. I.I * *; ++ ++ Sj n- g 'o -i h 3 * "rt <\ 3 1 1 S S-H S '1 S ^ 1 1 ; o 3 M fS si S O 0181 0169 5 0154 10 0133 0162 00156 00283 00746 15 0116 00141 00271 20 0102 00127 00258 00641 00370 00567 25 0091 '-00114 00245 00357 00539 30 0081 00104 00233 00555 00344 00513 35 0073 00332 00489 40 0067 00086 00488 00467 45 0061 50 0056 00072 00433 60 0047 80 0036 90 OQ32 99 0123 154 0109 197 0102 249 00964 315 00918 340 00897 * Based on results obtained by Sprung (1876), Meyer (1877), and Slotte (1883). f Obtained by Koch (1881). t Wijkander (1879). I 122 PHYSICAL UNITS. TABLE IX. Viscosity of Gases, in dynes per square centimetre. Gas. Values at C. by Obermayer. Values at 20C. by Meyer, front Graham's Experiments. Air 000168 000190 Ammonia 000108 Carbonic acid . 000141 000160 Carbonic oxide 000163 000184 Chlorine ... 000147 Hydrogen . . ^0000822 000093 Marsh gas 000120 Nitric oxide 000164 000186 Nitrogen 000163 000184 Nitrous oxide 000141 000160 Olefiant gas .... 000109 Oxvfiren . 000187 000212 Steam 000097 Sulphuretted hydrogen 000130 Sulphurous acid ... ... 000138 Dry air being taken as the standard, the viscosity of argon is 1-21, and that of helium is 0'96 (Rayleigh, Brit. Ass. Rep. y 1895, p. 609). PHYSICAL UNITS. 123 C+H _J O 02 ^ o o ;o o O 00 sis rO. OQ 0) X X XX O O O s s SQ IO XO C5 (M ^0 | t>- t Oi C5 i-H 11"* T T T ? o o o o "S 03 ^ M ' ' ' ^* _ 3 o ^ r"} X X XX " cS -2 CO OO OO O5 Q *^-i .S O 00 I-H ^ 00 X 1 *" o o o o 1 1 II o o o o 00 00 O5 0*0 gO i-H r 1 i 1 i-H X X XX t- a * " .s OO t^ O5 GO H ^ ^ S'~ Oi 10 ^ co a filial (M lO i ( CO (M (M -HH GO CO Oi O5 i 1 11 -T3 | 13 'o fc|n ^ g > Jg 2 00 -* ^ ^ CO CO -M *Q S t? o <8 ^3 ^ .3 O 3 SH 3 ^ O CO t^ t>- O CO X) CO GO (M 00 1 -^ - J ^o c3 o ^? ^ ^ ^ y <^> Ol C^l (7 s ! C*Q i H 02* *> ^-^^^ ^ I-H > 1 .i ^_; CO t>- CO i CO 02 '35 II CO O *O Ol C^ O fe s ^ O O I-H Ci O -^H rj o .5 r^H fin r^H : : : ^ : a g I : '3 : '1 : C3 0> O . CD 02 ^5 tt ^jjQ ^ *J~* ^ S-i s n^j * p^> ^ ?2 r r^ i W ^ o o do i 2 124 PHYSICAL UNITS. TABLE XL Specific Inductive Capacity (air taken as unity). Substance. Specific Inductive Capacity. Authority. Air at '001 mm. pressure Carbon dioxide 0-9985 1-0008 Ayrton and Perry Hydro sen 0-9998 Olefiant gas 1-0007 Sulphur dioxide Benzene 1-0037 2-20 jj Silow Carbon disulphide 1-81 Gordon Castor oil 4-78 J. Hopkinson Petroleum, common 2-10 Sperm oil 3-02 Turpentine, commercial Water at 25 C 2-23 75-7 E. B Rosa Chatterton's compound. . . Ebonite . 2-55 2-28 Gordon Glass, flint (density 3 '6 6) crown (density 2*485) Guttapercha 7-38 6-96 2-46 J. Hopkinson ?) Gordon Indiarubber pure 2-34 Siemens Mica 5-0 Faraday Paraffin wax 2-55 Gibson and Barclay Resin 2-74 Boltzman Shellac 1-997 Gordon Sulphur 2-58 PHYSICAL UNITS. TABLE XII. 125 Conductivity and Resistance of Pure Copper at Temperatures from C. to 100 C., calculated from Matthiessen's formula for conductivity at t C. 1 - -0038701 1 + -000009009 P Tempera- ture. Conduc- tivity. Resistance. Tempera- ture. Conduc- tivity. Resist- ance. 1-0000 1-000 27 0-9020 1-1085 1 0-9961 1-00388 28 0-8987 1-1127 2 0-9923 1-00776 29 0-8953 1-1169 3 0-9885 1-0116 30 0-8920 1 1211 4 0-9847 1-0156 31 0-8887 1-1253 5 0-9809 1-0195 32 0-8854 1-1295 6 0-9771 1-0234 33 0-8821 1-1337 7 0-9734 1-0274 34 0-8788 1-1379' 8 0-9696 1-0313 35 0-8756 1-1421 9 0-9559 1-0353 36 0-8723 1-1464 10 0-9622 1-0393 37 0-8691 1-1506 11 0-9585 1-0433 38 0-8659 1-1548 12 0-9549 1-0473 39 0-8628 1-1591 13 0-9512 1-0513 40 0-8596 1-1633 14 0-9476 1-0553 45 0-8441 1-1847 15 0-9440 1-0593 50 0-8290 1-2063 16 0-9404 1-0634 55 0-8144 1-2279 17 0-9368 1-0675 60 0-8002 1-2497 18 0-9333 1-0715 65 0-7865 1-2715 19 0-9297 1-0756 70 0-7732 1-2933 20 0-9262 1-0797 75 0-7604 1-3151 21 0-9227 1-0838 80 0-7480 1-3369 22 0-9192 1-0879 85 0-7361 13585 23 0-9158 1-0920 90 0-7247 1 3799 24 0-9123 1-0961 95 0-7136 14013 25 0-9089 1-1003 100 0-7031 1-4222 25 0-9054 1-1044 126 PHYSICAL UNITS. . eS -*a -J 3% 3 -tJ II 61 w c ^ 9 fi: H ! 2: g'is ts ^ 3 8 &S CJ 00 o 13 c 5 OS I-H OS (M CO I OS O CO CO O 2} ^ 22 co co l O O 888 t O ^ CO CO lO >-H CO OO IT .CO U3 * Tfl CO t>-os to os r- T*H t GO ( Ol ' CO GO JO 09 O OS rJH CO (M * ~H o i QO ost^co TC CO CO CO CO QO O O t >O .(M i I O 30 CO U7> . ^3 00 0} I s OQ & '^ be ;^| CO CO CO CO CO r-H cb cb LO r^- os cb il- TH CO CO ^. t- VO (N t- f-H ^ O '"H CO GO 1C I-H O CO lO O CO CO id t ' CO CO ^H (M ^H (N OS 00 OS r-t oo co 10 T* <* TJH O O co os to fM ^H F-H t (N CO OS 2S^c^ CX5 t-CO IO QO t 00 O I i co r- os 10 oo 10 (M o I'M t CO QO 1C CO O CO CO CO r- ( -H f-H OS CO TjH CO O OS GO r-H F < f-H i-H f-H O O -* o co (N 00 O OS t CO VO T* CO O^ IH . t- CO lO 1O ' CO O-l CO CO (N CN CO 5^ O O O O O i-H CO CO * U7> COtQOOSO PHYSICAL UNITS. 127 (M CO O SS CO ^H O C^l 10 i i I-H Ol 00 O t co co co GO o ^ in co m co o 01 co "* in t- GO o 01 in GO 01 o o o o o o > 1 1 1 1 i i 01 It CO GO i os in GO Tt^ ^ os 1 10 j^ os cO os co -- *-H TT co o cp o 01 cp 01 7-1 p cp GO n gs T* -HH in t os o os co < 01 GO t GO o cor~ 4t< r^ os t cb n rh os^moom OIOSCOT^CO co 01 < t r-< I-H CO CO 01 rH r-H i-H ^H co co co os m GO t^ co o (3 8 6 t-< COOOT^CO^OSOt-OlOSCO' oo"S 99T* H 9V' tl cpOl-^oiGO cpincp^HGO ^OIGOGC GOT^Ot-in TtO ^HOSOS-*O COCO^HOSt V -r 1 ^cooio-* Tt< < 01 m i i oico Hcooi o *-H >n GO i in < H t -* 01 i 83222 8S SSSoS Soo 00 uv P-H _> &3 ^rj r^ i so co cooioioiH < < r^ 10 o init- GO co^oioos GO t co >n in -^ ^ co i f-H ^^^^9 99999 999 23 CO Ol GO ^ Ol i CO CO 01 JS^ci^ s i i IN co -^ m Ol 01 01 01 W ! 128 PHYSICAL UNITS. CM O OS O O CO CM i i iO OS CO O OO GO CO <* O t- lO CM ii rH CM GO t CM CM CO rf< Tt< pt-giCMT*H !> rH CO CO CO GO * CM >O O3 rH 4-1 rH CM CM CO * ^"^ 22 OJ CM CM CO CO Tf 1O CO OO rH CO CO OS W O OS CO !> rH p O rH rH rH rH CM CO CO b CO i5 iO GOO CMCOCOCMCM O001OO4OS. OS "^ OS lO O> 1^* CO O t^ CO lO "^ CO CO CM ""* "^ p 9 9 COCOCMCMCN rHrHrH* t^O CO COT^ lOt^OirHCO COOSCMCOCM lOi>COCOQO I>OSCO1>TJH CMrHrHCOlO OOCMGO^CO CO1>-CM1>-CM OO'^'-HOOlr COUtirfiCOCM '~ 1 7^999 ^COCOCMCM rHrH^-l" CO CO OOCOOO^COi-HrHCOOO li CO !> t^ CO T*H CM rH CO rH CM CO CO CO ~- lO 8 i rH t^ Oi Jh- CM CO O (.-^ O i i Oi Oi O CO 1^ i CD O CO CM CM - i cr QQ COlOlOT* TflCOCOCMrH (>1 CO CM O I OrHOr-i rHGOCOiO ipcOOO-HOO iCCMOQOCO rH CO CM rH TJH CO CM O Thl-^T^CO COCOCMCMCM CMCMrHrHrH rH' rH OSI>-IQCOCM rHOOS ppppp pop : 9999 CO CM O : 8| rH CM CO -^ Ifi COI>GOOSO rH CM CO -^ IO CO !> CO OS O co co co co co co co co co -^ T* rti -^ T* * <* -^ rt< * ira PHYSICAL UNITS. a -= P ^O S 3 ill! PH jo - 1 1 1 I Sin J ' 1 5 1 ^ < 82 3 PV 5m Sui SI 01 jo - fs jo - II II O CL II II O - II II o <* 130 PHYSICAL UNITS. I o I ." I 03 S * u & *. .5 s c S a 2 3 I s 8 3 O CO CJ . -a=l 2 *--H ^ rt i-H - a . C . CD CO J^COCO 2'o || || 08^^ S CO O r^ CO S 00 O i s 8 T" )=i oo O f C. P- *C. p. _ 99 3835 o 100 6168 Temperature of boiling water. 18-9 4049 18-9 3911 18-9 6303 air. 1 4090 1 3953 0-8 6360 melting ice. -80 4189 -80 4054 -80 6495 ,, melting carbonic acid. -100 4218 -100 4079 -100 6533 ,, boiling ethylene. -182 4321 -182 4180 oxygen. i 132 PHYSICAL UNITS. TABLE XVII. Specific Resistance (approximate) of Liquids and Insulators. It decreases with increase of temperature. Substance. Specific Resistance in ohms per cubic centimetre. Tempe- rature C, Bichromate of potash (saturated solution) 29-6 4-7 1-4 29-3 21-5 8-0 3-4 x 10 5 37-6 x 10 5 28-0 x 10 15 6-2 x 10 15 450-0 x 10 12 284-0 x 10 6 840 x 10 12 34-0 x 10 15 9-0 x 10 15 18 10 14 12 11 46 46 24 - 0'2 20 46 28 Common salt, saturated Nitric acid (density 1 '36) Sulphate of copper, saturated Sulphate of zinc ,, ,. Sulphuric acid (density 1'2) Water distilled Ebonite Glass (below 40 C., very large) Gruttapercha Ice Mica Paraffin Shellac PHYSICAL UNITS. 133 TABLE XVIII. Magnetic Elements in Different Parts of the British Isles for January 1, 1886, according to a recent survey by Profs. Riicker and Thorpe (Bakerian Lecture, Phil. Trans., 1890). Declination. Dip. Horizon- tal force in dyne. Vertical force in dyne. Lerwick / 20 30 / 72 47 1471 4748 Inverness 21 43 71 31 1564 4680 Edinburgh 20 47 70 38 1618 4606 "^Glasgow 21 12 70 45 1606 4600 Carlisle 20 26 69 54 1662 4543 Leeds 19 9 69 11 1708 4492 Nottingham . 18 45 68 38 1747 4464 Cambridge 18 5 68 2 1778 4410 Oxford 18 34 67 58 1789 4419 Greenwich 17 55 67 29 1816 4376 Kew 18 16 67 37 1809 4395 Exeter 19 29 67 26 1826 4395 Dublin 21 41 69 16 1709 4513 Cork 22 18 68 46 1751 4507 * The value of the horizontal component of the earth's mag- netism in the Physical Laboratory of the University of Glasgow determined by the Gaussian method, is '154 dyne. 134 PHYSICAL UNITS. TABLE XIX. Electro-chemical Equivalents, calculated on the basis of Lord Rayleigh's electro-chemical equivalent of silver. Elements. c g> 'rf Atomic Weight. Chemical Equivalent. Electro -chemi- cal Equivalent (Milligrammes per Coulomb). Coulombs per Gramme. Grammes per Ampere-hour. Aluminium 3 2 1 3 1 3 2 2 2 2 1 2 1 1 I 4 2 2 1 1 1 3 2 27-3 63-0 63-0 196-2 1-0 55-9 55-9 206-4 23-94 199-8 199-8 58-6 39-04 107-66 22-99 117-8 117-8 64-9 79-75 35-37 126-53 14-01 15-96 9-1 31-5 63-0 65-4 10 18-64 27-95 103-2 11-97 99-9 199-8 29-3 39-04 107-66 22-99 29-45 58-9 32-45 79-75 35-37 126-53 4-67 7-98 0-0925 0-3271 0-6542 0-6791 0-0104 0-1936 0-2904 1-0716 0-1243 1-0374 2-0747 0-3043 0-4054 1-1180 0-2387 0-3058 0-6116 0-3370 0-8281 0-3673 1-3139 0-0485 0-0829 10583-0 3058-6 1529-3 1473-5 96293-0 5166-4 3445-5 933-2 8045-0 964-0 482-0 3286-8 2467-5 894-4 4188-9 3270-0 1635-0 2967-1 0-3402 1-1770 23540 2-4448 0-0374 0-6968 1-0448 3-8578 0-4475 3-7345 7-4690 1-0953 1-4595 4-0250 0-8594 1-1009 2-2018 1-2133 Copper (cupric)... (cuprous) . Gold Hydrogen Iron (ferric) (ferrous) ... Lead Magnesium ..... Mercury (mer- curic) . . . Mercury (mer- curous) Nickel Potassium Silver Sodium Tin (stannic) (stannous) ... Zinc . . . Electro-negative : Bromine Chlorine Iodine Nitrogen ...... Oxveren PHYSICAL UNITS. TABLE XX. Linear Expansions of Solids. 135 Substance. Mean Expan- sion per Degree C. into 10~ 6 . Range, in Degrees. C. Authority. Aluminium, com- mercial 22-2 to 100 Calvert, Johnson, Low& Antimony 10-56 to 100 Matthiessen Arsenic, sublimed Bismuth 5-59 13-16 40 to 100 Fizeau Matthiessen Brass, cast 18'75 to 100 Smeaton Cadmium 31-59 to 100 Matthiessen Carbon, graphite... Cobalt 7-86 12-36 40 40 Fizeau Fizeau Copper Ebonite Glass, plate 18-66 770 8-91 to 100 16-7 to 25-3 to 100 Matthiessen Kohlrausch Lavoisier and Laplace* Gold, annealed ... Granite 14-70 8-68 to 100 to 100 Matthiessen Bartlett Ice 52-36 27' 5 to T25 Schumacher Indium, cast Iridium 41-7 7-0 40 40 Fizeau Fizeau Iron 11 -90 to 100 Calvert Johnson Low& Lead 27*99 to 100 Matthiessen Magnesium, cast... Marble, white Nickel 26-94 10-72 12-79 40 to 100 40 Fizeau Dunn and Sang Fizeau Palladium 11 -04 to 100 Matthiessen Platinum Platinum - iridi u m (one-ten thiridium) Quartz fibre Sandstone 8-86 8'84 8-7 11-74 to 100 40 46 -5 to 71 -3 to 100 Matthiessen Fizeau Threlfall Adie Slate 10-38 to 100 Adie Silver 19-43 to 100 Matthiessen Sodium 68-0 Speculum metal ... Steel, soft 19-33 10-3 to 100 Smeaton Calvert, Johnson Lowe* Sulphur, Sicily ... Tin 64-13 22'96 40 to 100 Fizeau Matthiessen Type metal (lead and antimony)... Zinc 20-33 29-76 16-6 to 100 to 100 Daniell Matthiessen P SF > CM' 1 ! UNIVERoIT 136 PHYSICAL UNITS. TABLE XXI. Volume and Density of Water and Mercury (water results reduced by Prof. W. H, Miller, from Kupffer's results, which gives the absolute density, 1 '000013 at 4 C.). Water. Mercury. Tempe- rature. po True Density in grammes per cubic Volume in cubic centi- metres per Density in grammes per cubic Volume of Mercury at C. taken as o . centimetre. gramme. centimetre. unity. 999884 1-000116 13-596 1-000000 1 999941 1-000059 2 999982 1-000018 3 1-000004 999996 4 1-000013 999987 13-586 1-000716 5 1-000003 999997 13-584 1-000896 6 999983 1-000017 7 999946 1-000054 8 999899 1-000101 9 999837 1-000163 10 999760 1-000240 13-572 1-001792 15 999173 1-000828 13-559 1-002691 20 998272 1-001731 13-547 1-003590 25 997133 1-002875 30 995778 1-004240 13-523 1-005393 35 99419 40 99236 1-007682 13-449 1-007201 45 99038 50 98821 1-011928 13-474 1-009013 60 98339 1-016906 13-450 1-010831 70 97795 1-022542 13-426 1-012655 80 97195 1-028856 13-401 1-014482 90 96557 1-035662 13-377 1-016315 100 95866 1-043117 13-353 1-018153 PHYSICAL UNITS. TABLE XXII. Specific Heats of Solids. 137 Solids. Specific Heat. Temperature, Centigrade. Authority. Aluminium 2122 15 to 97 Regnault A n fcinion v 0486 13 to 106 Bede Arsenic (crystalline) Bismuth Brass (4 Cu. 1 Sn ). Bromine 0830 0298 0858 0843 21 to 68 9 to 102 15 to 98 - 78 to - 20 Bettendorf &Wiillner Bede Kegnault Regnault Cadmium 0548 to 100 Bunsen Copper . . 0933 15 to 100 Bede Glass 198 Gold 0316 to 100 Violle Graphite 1604 11 H F Weber Ice 504 - 20 to Regnault Iodine 0541 9 to 98 Regnault Iridium 0323 to 100 Violle Iron 1124 50 Bystrdm Lead 0315 19 to 48 Kopp Magnesium Manganese 245 1217 20 to 51 14 to 97 Kopp Regnault Mercury 0319 - 78 to - 40 Resjnault Nickel 1092 14 to 97 Regnault Palladium 0592 to 100 Violle Phosphorus (yellow) Platinum 1699 0323 - 78 to 10 to 100 Regnault Violle Platinum-iridium . . . Potassium 0410 1655 - 78 to Resnault Silicon 1697 22 H F Weber Silver J 0559 to 100 Bunsen Sodium 2934 - 28 to 6 Reofnault Sulphur 163 17 to 45 Kopp Tin 0559 to 100 Bunsen Zinc 0935 to 100 Bunsen 138 PHYSICAL UNITS. TABLE XXIII. Specific Heats of Liquids. Liquids. Specific Heat. Temperature, Centigrade. Acid, acetic 460 10 to 15 hydrochloric ( + 10 H 2 O) . . . ,, nitric ( + 10 H 2 O) 749 768 18 18 ,, sulphuric 332 16 to 20 ( + 5 H O) 576 16 to 20 Alcohol, amyl 564 26 to 44 ethvl 615 30 methyl . . . 601 15 to 20 Benzol 416 19 to 30 Bromine . 107 13 to 45 Chloroform 235 30 Carbon disulphide . . .. 240 30 Ether (C 9 H 5 ) 2 O 529 Mercury 0335 17 to 48 Nitro-benzol 348 10 to 15 Paraffin 683 Phosphorus (yellow) 205 49 to 98 Sulphur 235 119 to 147 Sulphur chloride (S 2 C1 2 ) 202 10 to 15 Turpentine 454 40 PHYSICAL UNITS. 139 TABLE XXIV. Specific Heats of Gases. Pressure Volume Pressure Con- stant Constant. Constant. Volume Con- stant. Air 2375 1684 1 410 Ammonia H 3 N 5356 391 1-370 Bromine Br 0555 0429 1-294 Carbon monoxide ,, dioxide... ,, disulphide Chlorine u^2 CO CO 9 cs; Clo" 2425 2169 1596 1241 1736 172 131 0928 1-397 1 261 1-218 1-337 Chloroform . . . CHC1 3 1489 140 1*064 Ether C,H 1A O 4797 453 T059 Hydrochloric acid Hydrogen 4 10 HC1 Ho 1867 3-4090 1304 2-411 1-432 1-414 sulphide Marsh gas 2 H 2 S CH 4 2451 5929 184 468 1-332 1-267 Nitrogen N 2 2438 1727 T412 Nitric oxide NO 2317 1652 T403 Nitrous oxide ... Olefiant gas N 2 O CoH, 2262 4040 181 359 1-250 T125 Oxygen . . ^24 O 2175 1551 1-402 2 H 2 O 4805 370 1-298 Sulphur dioxide... SO 2 1544 123 1-265 140 PHYSICAL UNITS. TABLE XXV. Thermal Conductivities, at Temperatures from 7 to 15 C. (a) Solids. Aluminium Antimony.. Bismuth . Cadmium Copper German silver Iron Lead Magnesium at 100 C... Phosphor bronze Silver 1 Tin Zinc .. 344 044 018 204 220 996 070 201 084 37 415 096 153 28 Calico -000139 Cork -000397 Eiderdown '000108 Glass -0021 Fir, across fibre -00026 along -00047 Ice at C -00568 Oak, across fibres '00059 Sandstone (Craigleith) '01068 Underground strata... '005 Walnut, across fibres '00029 along ... -00048 Wool '000122 Writing-paper, white '000 1 1 9 (b) Liquids (from 9 to 15 C., by H. F. Weber). Alcohol, amyl '000328 ethyl '000423 methyl '000495 Aniline '000408 Benzol -000333 Bisulphide of carbon '000343 Chloroform '000288 Ether '000303 Glycerine -000670 Oil of turpentine ... '000260 Sulphuric acid -000765 Water . . '00136 (c) Gases (Winkelmann). Air -0000516 Carbonic acid '0000273 oxide .. . -0000499 Hydrogen '00033 Marsh gas -000065 Nitrous oxide... . '0000312 PHYSICAL UNITS. TABLE XXVI. Melting Points and Latent Heats of Fusion. 141 Melting Points C. Latent Heats of Fusion. Aluminium . . . 850 Antimony ... 432 Bismuth. . 268 12-6 Bromine - 7 16 - 2 Cadmium 321 13'7 Cobalt 1500 Copper 1054 Gold 1045 Glass 1100 Ice 80-0 Iodine 115 11*7 Iridium 1950 Iron, cast (greV) . . 1200 23-0 (white)... 1100 33-0 , wrought 2000 ,, steel cast 1375 Lead . . . 326 5-4 Manganese 1500 Mercury -39 2-8 Nickel . 1450 4-6 Palladium 1500 36-3 Phosphorus (yellow) 44 5-2 Platinum 1775 27-2 Potassium 62 Silver 954 21-1 Sulphur rhombic 114 9-4 Tin ... 233 14-3 Zinc . . 433 28-1 142 PHYSICAL UNITS. TABLE XXVII. Boiling Points and Latent Heats of Evaporation. Substance. Boiling Point C. Pressure in mms. of Mercury Latent Heats of Evapora- tion. Air -191-4 Alcohol ethyl C H r O 78-4 760 208-92 methyl . C 2 H 4 66-3 263-86 Ammonia H q N - 33-7 749 297-38 Arson A, -187 Benzol 1 C^H, 80-4 760 Bromine ^6 6 Br 59-27 760 45-6 Carbon dioxide CO - 78-2 760 49*32 disulphide . . . monoxide . . . CS 2 CO 46-2 -190-0 760 86-67 Chlorine Cl - 33-6 760 Chloroform CHCL 61-2 760 Ether (C H,) O 34-97 760 90-7 Glycerine C,H a O* 290-0 760 Hydrogen ^3 8 3 -243-5 Iodine I 250-0 23-95 Mercury Hg 357-25 760 62-0 Nitric oxide NO -153-6 Nitrogen N -194-4 Olive oil 315-0 Oxvsren O - 182-0 Ozone o -106-0 Phosphorus (yellow). . . Potassium p 3 K 287-3 725-0 762 Rose oil 274-0 Sulphur S 440 362-80 Sulphuric acid H 9 SO. 325-0 760 122-1 Turpentine 159-3 760 74-0 Water H 2 O" 100-0 760 537-0 Zinc chloride . ZnCL 680-0 PHYSICAL UNITS. 143 TABLE XXVIII. Critical Temperatures and Pressures. i o " lb CO 4-> ."s *o PH g ^ M S3 g* g g S i- a -^ 02 O Air - 140-0 39-0 Olszewski Alcohol . . . + 234-6 65-0 Hannay Argon -121 50-6 -189-6 Olszewski Benzol + 280-6 49-5 Sajotschewsky Carbonic acid + 30-92 73-0 Andrews Carbon mon- oxide . . . -139-5 35-5 -207 100 Olszewski Carbon bi- sulphide + 273-0 77-9 Hannay Chloroform + 260-0 54-9 Sajotschewsky Ether + 190-0 36-9 Hydrochlo- " ric acid .. + 51-25 86-0 -116 (Olszewski) Ansdell Hydrogen .. - 234-5 20(1,5' Olszewski Nitric oxide -93-5 71-2 -167 138 }J Nitrogen ... - 146-0 35-0 -214 60 55 Nitrous oxide . . . + 36-4 73-1 Janssen Oxygen . . . -118-8 50-8 Olszewski Sulphurous acid + 155-4 78-9 Sajotschewsky Water + 365 200-5 Cailletet 144 PHYSICAL UNITS. TABLE XXIX. Vapour Pressure of Water in Centimetres of Mercury. Tempera- ture C. Pressure. Tempera- ture C. Pressure. Tempera- ture C. Pressure. -30 0-04 60 14-88 150 358-12 -25 0-06 65 18-69 155 408-86 -20 0-09 70 23-31 160 465-16 -15 0-14 75 28-85 165 527-45 -10 0-21 80 35-46 170 596-17 - 5 0-31 85 43-30 175 671-75 0-46 90 52-54 180 754-64 5 0-65 95 63-38 185 845-32 10 0-92 100 76-00 190 944-27 15 1-27 105 90-64 195 1051-96 20 1-74 110 107-54 200 1168-90 25 2-35 115 126-94 205 1295-57 30 3-15 120 149-13 210 1432-48 35 4-18 125 174-39 215 1580-13 40 5-49 130 203-03 220 1739-04 45 7-14 135 235-37 225 1909-70 50 9-20 140 271-76 230 2092-64 55 11-75 145 312-55 Vapour Pressure of Mercury in Centimetres of Mercury (first four by Hertz rest by Ramsay and Young). 000019 60 0029 140 1763 10 00005 70 0052 160 4013 20 00013 80 0092 180 8535 30 00029 90 0160 200 1-7015 40 0008 100 0270 >220 3-1957 50 0015 120 O719 i _ OF TH UNIV ., PHYSICAL UNITS. TABLE XXX. Emissivity showing loss of heat, at atmospheric pressure, per square centimetre, per second, per 1 C. of excess of tempera- ture of cooling globe above temperature of surrounding enclosure (14-5 C.) [Bottomley]. Phil Trans., Vol. 184 (1893). Tempera- ture of Globe C. Copper Globe polished bright. Copper Globe thinly covered with lampblack. Copper Globe polished bright and thinly lacquered. Copper Globe plated with silver. 21 165 x 10~ G 278x10- 246 x 10~ G 22 170 281 , 250 23 174 284 254 174x10- 24 178 287 , 257 178 25 181 290 , 260 182 26 184 293 263 185 27 187 296 , 265 187-5 28 190 299 , 268 190 29 192 301-5 271 192 30 194 304-5 273 194 31 196 307 276 196 32 198 310 , 278 198 33 199-5 313 , 280 200 34 201 316 , 282-5 201 , 35 202 320 , 285 203 , 36 203-5 323 , 287 205 , 37 205 326 , 290 206-5 , 38 206-5 329 , 292 208 , 39 207 332 , 294 209-5 , 40 208 335 297 211 , 41 338 , 299 212-5 , 42 341 , 301-5 214 , 43 304 215 , 44 ' 307 216 , 45 _ 309 218 146 PHYSICAL UNITS. TABLE XXXI. Indices of Refraction of Solids, Liquids, and Gases for the mean D line. (a) Solids. Alum 1-547 Borax 1'475 anada balsam 1*532 Diamond 2-45 to 2'97 Emerald 1-585 Felspar ... 1'764 Fluorspar 1-436 Glass (flint) 1'57 to 1-58 (crown) 1-525 to 1-534 ,, (plate) 1-514 to 1*542 Ice . 1-310 Iceland spar (ord. ) 1 '658 (ext.) 1-486 Phosphorus 2*224 Quartz (ord.) T544 (ext.) 1-553 Rock salt ..... 1-545 Ruby 1-779 Sulphur (native) 2-115 Topaz 1*610 Tourmaline 1'668 (6) Liquids. Alcohol (rectified) 1 '372 Aniline 1'57 Benzene 1'49 Carbon disulphide 1*678 Chloroform 1"44 Ether 1*358 Glycerine 1*47 Hydrochloric acid (concen- trated) 1-410 Linseed oil 1-485 Nitric acid (sp. gr. 1-48) ... 1*410 500 Nut oil Oil of almonds ,, cassia lavender Olive oil Phenol Saliva Sulphuric acid (sp. gr. 1*7) Turpentine .'... 1*46 Water .. l*33i 031 457 470 55 339 429 (c) Gases. Den- sity. Index. Den- sity. Index. Air 1-000 1-000294 Marsh gas 0*555 1 -000443 Ammonia 0-596 1*000385 Nitric oxide 1*039 1*000303 Carbon monoxide ,, dioxide... Chlorine . . 1-957 1-524 2-470 1-000340 1-000449 1*000772 Nitrogen .. Nitrous oxide Oxvsen 0-971 1-520 1-106 1-000300 1-000503 1 -000273 Cyanogen 1-806 1*000834 Sulphide of carbon 2*644 1-000150 Ether 2-580 1-000153 Sulphurous acid 2*234 1-000665 Hydros-en .. 0-069 1 -000138 PHYSICAL UNITS. 147 TABLE XXXII. Wave-lengths and Wave-frequencies, taking v = 3 x 10 10 cms. per Second (Watts). Designation and Origin. Wave-length in Air in 10~ 5 cm. Frequency per Second. Refractive Index of Air. A 7-6040 394-6 x 10 12 1-00029286 B 6-8670 436-9 1-00029350 (H) D (Na) 6-5621 5-89513 } 457-2 509-3 , 1-00029383 1-00029470 j yj.ic*y.. ^ E (Ca and Fe) 5-88912 / 5-26913 569-4 1-00029584 &i (Mg) . . 5-18310 578-8 5 /Ms) 5 17216 579-9 V 2 V 1TA & / b 3 (Ni and Fe) 5-16848 \ b. (Ms. and Fe)... 5-16688 / 580-5 F(H) O (Fe) 4-86072 4-30725 617-2 696-5 1-00029685 1-00029873 ^-" \ x / H (Ca) . . 3-9681 756-0 1 K (Ca) . 3-9330 762-8 I 1-00030028 MFe) M (Fe) .. 3-8198 3-7270 785-2 805-0 1-00030096 1-00030148 N (Fe) 3-5805 837-7 1-00030212 O (Fe double) 3-4398 872-0 1-00030336 P (Fe and Ti) 3-3592 893-2 1-00030397 Q (Fe) 3-2849 913-1 1-00030459 R (FeandCa) r (Fe double) 3-1790 3-1443 943-6 954-1 1-00030555 S x (Ni double) 3-1006 } S (Fe triple) 3-0995 / 967-7 s (Fe) 3*0464 985-4 T(Fe double) t (Fe)... 3-0197 2-9943 993-3 1002 u (Fe) 2-9478 1017 usr ID IE A. Acceleration 25 Angular 25 Due to Gravity at Various Places 114 Activity 35 Air, Collected Data for Density of 98 Refractive Index for Diffe- rent Lights 147 Specific Heat of 139 Thermal Conductivity of ... 140 Ampere ... 74, 77, 82, 98, 103, 105 Ampere's Formula 81 Angle 24 Solid 24 Area, Unit of 24 Atmosphere 38, 98 Attraction, Constant of ... 17, 18 Newton's Law of 16, 17 B. 87 78 87 142 B. A. Ohm Ballistic Galvanometer.. Board of Trade Ohm Boiling Points, Table of C. Candle, Standard 98 Candle-power 98 Capacities, Specific Inductive 124 Capacity, Electrical 58, 64 Measure of 89, 90 Specific Inductive 56 Thermal 94 Capillary Tubes, Flow through 39 Carbon, Specific Resistance of 131 Centimetre 10, 11 in British Units 99 Centimetre-gramme in British Units 99 Tentimetre-dyne 32, 99 Centrifugal Force 27 at Equator 28 O.GS. System 10 Clark's Standard Cell .. 108, 109 Coefficient of Expansion ... 95, 96 Coil, Rotating ,. .. . 84 Conductance 59 Conductivity of Copper at Different Temperatures . . 125 Conductivity of Copper Wires 126 Thermal 96 Thermometric 97 Constants, Table of Useful ... 98 Copper, Conductivity and Resistance of 125 Coulomb 74 Couple, Moment of 3& Critical Pressures 143 Temperatures 143 Current Balance 82 Current, Heat Generated by 83 Measurement of ... 80, 81, 82, 9ft Standard of 105 Unit of 57, 63 Curvature 24, 25- D. Day, Sidereal 15, 99- Solar ^.. 16, 19, 20, 99 Declination, Magnetic, at Various Places . , 13& INDEX. 11. Degrees of Longitude and Latitude ...................... 99 Densities of Solids ............... 116 Liquids and Gases ......... 117 Density ............................ 35 of Mercury ............ . ...... 136 of Water .... .................. 136 Derived Units .................... 9 Diffusion ......................... 40 Diffusivity .................... 40, 97 Dimensional Formulae ......... 23 Usesof .......................... 42 Dimensions of Electric and Magnetic Quantities ......... 68 Dimensions of Dyne 69 26, 99 E. Earth, Mass of 17 Size and Figure of 17, 26 Elasticity 36 Elasticity Waves, Table of Velocity of 119 Electric Force 57 Pressure, Standard of 105 Electric Quantities, Relations between 49 Electric Units, Measurement of 90 Electro-chemical Action 82 Equivalents, Table of 134 Electro-dynamics 81 Electro-dynamic Units 49, 81 Electro-dynamometer 82 Electro-magnetic Units, 49, 55, 63 Dimensions of 59 Table of 59 Electrostatic Units 48 Dimensions of 59 Tableof 59 Electromotive Force 50, 72 Due to Cutting Lines of Force 71 Measurement of 89, 90 Standard of 89 Emissivities, Table of 145 Emissivity 97 Energy 32 Electrical Unit of 99 Entropy 97 Equivalent, Mechanical, of Heat 101 Erg 32, 99 Examples in Theory of Units 42 to 47 F. Farad 74, 100 Field-Intensity 57. 60 Foot in Centimetres 100 Foot per Second 100 Foot-pound and Foot-poundal 32, 100 Foot-pound in Ergs 43. 100 Force, Unit of 25 32, 33 Dimensional Formula for 28 Electric 57 between Electric Current and Magnetic Pole 54 of Gravity 26,27 between Two Quantities of Electricity 52, 56 Fundamental Units 9, 10 Symbols for 22 G. #, Value of, at Various Places 114 Gallon 100 Galvanometer, Ballistic 78 Gases, Densities of 117 Molecular Data for 123 Specific Heats of 139 Specific Inductive Capacities of 124 Gauss as Unit 92 Gilbert as Unit 5)2 Grain 11 Grain perinch 100 Gramme-weight 13, 26, 100 Per Sq. Cm 100 Gravity, Terrestrial, 26, 27, 28, 29 Acceleration Due to, at Different Places 114 Gyration, Radius of ... 33 111. INDEX. H. Heat 93 Dynamical Equivalent of ... 93 Generated by Current 83 Unit of 93 Henry 100 Homogeneous Atmosphere ... 123 Horse-power 35, 100 Horse-power Hour 100 I. Ice, Density of 116 Specific Heat of 137 Inch 101 Indices of Refraction 146 Induction, Magnetic 61, 62 Inertia, Moment of 33 Moments of (Table) 115 Insulators, Resistance of 132 Intensity of Electric Field ... 57 Magnetic Field 60 Magnetisation 61 Joule as Unit 32, 75, 101 Joule's Equivalent 101 K, Kepler's Laws 19 Newton's Deductions from 19, 20 Kilogramme 11, 12 101 Per Sq. Cm 101 Kilowatt 76, 101 Kilowatt-hour 76, 99 Kinematics 23 Kinetic Energy 32 Knot 25, 101 Admiralty 101 Latent Heat 95 of Steam 95 Latent Heats of Evaporation 142 Fusion 141 Latitude Degree of 99- Length, British Unit of .... 10 French Unit of 10 Length Modulus 47 Length Moduluses 48 Light, Velocity of 147 Wave-Lengths 147 Linear Expansions of Solids. 135 Litre 101 Longitude, Degree of.. 99 M. Magnetic Elements in Diffe- rent Parts of British Isles 133 Magnetic Induction 61,62 Permeability .. 60 Pole 60 Susceptibility 61 Units, Table of 61 B. A. Discussion on 92 Dimensions of 61 Mass and Weight 12, 13 Mass, British Unit of 11 French Unit of 11 Standards of 11 Mechanical Equivalent of Heat 101 Megohm 101 Melting Points, Table of 141 Mercury, Inch of 101 Density and Volume of, at Different Temperatures.. 136 Vapour, Pressures of 144 Metre 10, 101 Microfarad 76 Microhm 101 Micromillimetre 101 Mile per Hour 101 Modulus of Elasticity 35 Bulk 36 Rigidity 36 Young's 36 Moduluses, Table of 118 Molecular Data for Gases 123 Molecules of Gases, Diameter of -. .... 125 INDEX. Molecules, Mean Path of 123 Number of Collisions per Second 123 Velocity of 123 Moment of Magnet 6C Inertia ; 33 Moments of Inertia 115 Momentum 25 Moment of 33 Mutual Induction . 65 N. Newtonian Velocity of Sound in Various Gases 123 Nodes 20 Numeric . 23 0. Oersted as Unit 92 Ohm 74, 87, 101, 103, 104 Ohm's Law .. 50 P. Pappus' Theorems 113 Pendulum, Seconds at Diffe- rent Places 114 Potential, Electric 58 Magnetic 60 Pound, in Grains and in Grammes 101 per Square Inch 102 Standard 11, 14 Troy 11 Poundal 26, 102 Pound-weight in Poundals, 13, 26 Practical Units 70 in Terms of Absolute C.G.S. Units 76, 77 Pressure, Electrical, Standard of 105 Problems on Units... .. 42 to 47 Q. Quadrant ... 100 Quantity of Electricity, Unit of 50, 53- Electromagnetic Unit 64 E lectrostatic Unit 56 Measurement of 77, 90 Work done in Raising Poten- tial of... 64 R. Radian 24, 102 Radius of Gyration 33 Ratios of Electric Units 70 Specific Heats of Gases 139 Refraction Indices 146 Resistance 59, 65 Resistance Expressed by a Velocity 73 of Copper at Different Tem- peratures 125 of Copper Wires 126 Measurement of, 83, 84, 85, 86, 90 Relation between Units of 87 Standard of 104 Rigidity, Flexural 37 Torsional 37 Rotating Coil 84 Disc 85 Rotational Motion, Analogues of , to Translational 34 S. Secohm 100 Second, Mean Solar 15, 20 Sidereal 15 Solar 16 Second's Pendulum, Length at Different Places 114 Self-induction 65 Shear 36 Siemens's Unit 87 Sound, Velocity of 123- Specific Gravities 116 Gravity 35 Heat .. 94 V. INDEX. Specific Heat of Water 95 Heats of Solids (Table) 137 Liquids , 138 Gases 139 Ratio of 139 Inductive Capacity 56 Capacities (Table) 124 Resistance of Alloys at Different Temperatures 130 of Carbon at Different Temperatures 131 of Liquids and Insulators at Different Tempera- tures 132 of Pure Metals at Diffe- rent Temperatures 129 Standard of Electric Current 105 Pressure 105 Resistance 104 Strain 36 Stress. 36 Surface Density, Electric ... 57 Magnetic 60 Surfaces of Solids 113 Surface Tension 38 Tensions, Table of 120 T. Thermal Capacity 94 Conductivity 96 Conductivities, Table of ... 140 Time, Unit of 15 Variable 15 Units Suggested 21 Ton 102 Per Square Inch.. 102 Tortuosity 24 U. Units, Geometrical and Kine- matical 23 Relation between British and Metric 12 Table of 41 Table of Dimensions of, 41, 59, 62, 66, 68, 70 V. v, Determinations of 91 Vapour Pressures of Mercury 144 Water 144 Velocity 25 Angular 25 of Light 70, 147 of Sound, Newtonian 123 of a Wave-motion 47 Viscosity 38, 39 of Gases, Table of 122 of Liquids, Table of 121 Volt , 72, 104, 105 Voltameter, Silver 106, 107 Volume of Mercury 136 of Water 136 Elasticity of 36 Unit of .. 24 W. Water, Density of 102, 186 Expansion of 136 Vapour Pressures of 144 Volume at Different Tem- peratures 136 Watt as Unit 35, 75, 102 Wave-frequencies 147 Wave-lengths 147 Weber as Unit 92 Weight and Mass 12, 13, 14 Weight not Invariable 13 Weight of Copper Wires 126 Wires, Board of Trade Gauge 126 Work 31, 32, 33 Done by Current 51, 52 in Raising Electric Potential ...... 74 Working, Rate of 35 Y. Yard 10, 102 Yard Measure, Standard ... 10, 14 Year, Civil 102 Sidereal 102 Young's Modulus 36 14 DAY USE RETURN OWED This book is due on the last date stamped below, or on the date to which renewed. Renewed books are subject to immediate recall. 7Mar'57HJ REG D LD MAR 4 1957. 1 LD 21-100m-6,'56 (B9311slO)476 General Library University of California Berkeley YB 098C2 QG <0\ THE UNIVERSITY OF CALIFORNIA LIBRARY