/ and volume to/ (point B). Work is done equal to pjtoj log (to/AO = the area ABV/ \ r This work is done at the ex- pense of heat-energy supplied to the working substance from an external source. 2. Starting from the con- dition j/, to/, TJ (point B) we allow the working substance to expand adiabatically, until it assumes the temperature T 2 and ^ ^ Vg the corresponding condition p 2 ', BC is a part of the adiabatic line passing through B and to/, T 2 (point C). cutting the r 2 isothermal in C. Work equal to the area BCV 2 'V/ is done by the expanding substance, but at the expense of its own heat-energy, for no heat is supplied to it. 3. The substance is now compressed until it assumes the condition p 2 , to 2 , T 2 that is, until it runs from C so far up the isothermal line r 2 as to meet at D an adiabatic line, which passes through the original point A. Work is done equal to the area CDV 2 V 2 ' = p 2 ( Q 2 log (to 2 '/to 2 ) : but it is done upon the working substance, for that substance is compressed : and heat to a corresponding amount is lost by the working substance, for it passes to sur- rounding objects, and may be wasted by conduction and radiation into all the universe. 4. The body, from which no more heat is allowed to escape, is now sup- posed to be still further compressed until it has regained its original condi- tion p v to r r/ 3 . Work is done on the working substance thus compressed, but appears as heat in the substance, not as external work either positive or negative, and the temperature rises, for no heat is allowed to escape. During the adiabatic expansion in Step 2, the change of temperature is from r l to T 2 ; and (see footnote, p. 373) in that case T 1 /r 2 = (PI /p 2 r y k ~ c ^ k = (b 2 '/to/) ( *~ c) /*. Similarly, in Step 4, the temperatures are again r 2 and TJ ; and r l /T 2 = (to 2 Ai) ( *~ c)/ *- Therefore too'/to/ = to 2 /to L ; or to 2 '/to 2 = D/Ar The whole energy supplied to the working substance from the source is p^ log (to/Xtoj) ; that wasted is /> 2 to 2 log (to 2 '/to 2 ) =jt> 2 b 2 log (to//toj) ; that utilised is [(Iog(to//b0-(^ 1 -p 2 to 2 ))^(rtlog(to//to^]-^:^ of the whole. But j 1 to 1 = m T&r l ; jt? 2 to 2 = m - 2&r 2 , where r l and T 2 are the respective temperatures. Hence the proportion of energy utilised is {(wi&Tj ml&To) -4- m&Tj} or (r l T 2 )/r l of the whole. The working substance operating in such a cycle acts as a distributor of energy ; it divides 5H, the heat-energy supplied to it from the Source of heat, into two parts : one part, 5'H, passing to the Condenser, is lost by conduction and radiation ; the remainder, W, is usefully converted into external Work. The heat SH is supplied at the higher temperature T I ; the quantity of xiii.] CARNOT'S CYCLE. 397 heat S'H is lost to surrounding objects at the lower temperature T O ; the Efficiency of such an ideal arrangement is the ratio Heat utilised _ 8H - 8'H _ W r 1 -r 2 Heat supplied 5H 8H r l Thus, so far as Carnot's cycle is concerned, even though we could find a working substance and construct a machine which could carry the cycle out in practice, yet there would be a great waste of heat-energy, unavoidable unless we had a condenser at a temperature of absolute zero. If the tem- perature of the boiler of an ideal engine competent to work out Carnot's cycle were 120 C. (393 Abs.), and that of the condenser C. (273 Abs.), the work done by such an engine could not exceed t ~ " ' , or about 30-6 per cent of the whole energy supplied as heat. The cycle above considered is reversible ; each step in it can be retraced or could be retraced if we could construct an engine capable of working without waste of energy in noise, friction, excessive conduction and radi- ation of heat, and the like work being done not by but upon the engine as it is driven backwards. The effect of reversing such a cycle would be that work W being done upon the engine, the quantity 8'H of heat would be taken from the con- denser, and the quantity 8H of heat would be communicated to the source. Any engine which operates through periodic cycles must be a reciprocat- ing engine : and in every reciprocating engine there is an absolutely neces- sary waste of energy arising from the necessity of restoring the engine to its primitive position in order that its piston may repeat its effective thrusts. Carnot's ideal " perfect " engine is one which, with a working substance capable of returning to its primitive condition, will work out the reversible cycle above described, and thus attain the efficiency above indicated : an engine which wastes no energy otherwise than by restoring the primitive condition of its working substance. The perfection of a perfect engine depends not on the nature of the working substance, but on the reversibility of the cycle which it operates, and the efficiency of such a reversible engine depends only on the temper- atures between which it works. Carnot's Principle, as enounced by himself, is the motive power of heat is independent of the material agents employed to realise it ; its quantity is determined solely by the temperatures between which the " transport of Caloric " * is effected. The efficiency ^ = < (r, T f ), where f = T I r 2 . * " oH =/(T,T)-/'(T,T)=O-,KT)T. The efficiency depends upon f, the difference of temperatures between the source and condenser, and upon \j/ (T), a function of T which is called Carnot's function, C. We have also seen that efficiency = difference of temperatures = f temperature of source T Hence C = l/r; and Carnot's function is the reciprocal of the Absolute Temperature of the Source. f * An expression implying, as in his day, the material theory of heat. 398 HEAT. [CHAP. The Efficiency of a Carnot's Reversible Reciprocating Engine is greater than that of any other reciprocating engine. If it were possible to devise a more efficient reciprocating engine it might be employed with the expenditure of a certain amount of heat to drive a reversible reciprocating engine backwards ; the source and the condenser of the Carnot's engine might be the same as those of the more efficient en- gine : the Carnot's engine would be occupied in restoring to the source the heat taken from it by the better engine ; on the whole, a surplus of work would during each cycle be done by the conjoined mechanism a surplus not accounted for by heat lost by any body a creation of energy. If the better engine were employed in driving a larger Carnot's engine backwards, there might be no surplus, no external work done ; but a greater amount of heat would be conveyed to the source by the reversed Carnot than would be taken from it by the more efficient but smaller engine, and the whole heat of the universe might be, step by step, induced to travel through the condenser into the source of the conjoined mechanism a conclusion evidently absurd. That this conclusion is absurd, or at any rate contrary to experience, so long as we cannot deal like Clerk Maxwell's Demon (p. 52) with single molecules, it is the aim of the Second Law of Thermodynamics to state : Heat cannot of itself pass from a colder body to a hotter one, nor can it be made so to pass by any inanimate material mechanism : and no mechanism can be driven by a simple cooling of any material object below the temperature of surrounding objects. The word simple, or some equivalent word, is necessary in the above statement of the second law for the following reason : A quantity of com- pressed gas can do external work, and in so doing cool itself below the temperature of surrounding objects ; but its cooling is not a simple loss of heat-energy ; there is a concurrent change of condition of the gas, a change which cannot be reversed without the expenditure of heat exceeding in amount the heat converted into work by the expanding gas. This being admitted, we may reason backwards arid arrive at the ratio of efficiency - - in a reversible engine as a direct corollary of the prop- osition ; and the statement of that ratio of efficiency in a reversible recipro- cating engine is also known as the Second Law of Thermodynamics. This Protean law assumes another form, apparently different from but essentially identical with both the preceding. Temperature being assumed proportional to the total heat-energy, the amount of heat-energy, SH ergs, supplied at the higher temperature r v is proportional to T T ; 8H = dmr 1 ; = . Similarly S'H, the heat lost to the condenser at the lower tem- perature T 2 , is 8'H = ^wr 2 ; 8'H/mr 2 =:c.* Hence 8H/mr 1 = 8'H/mr 2 ; and from this we may not only derive the former equation - ~ = -i-^l^, but also Oil T the equation SH/mrj S'H/??ir 2 = ; an equation which, in the most general case, takes a form applicable to the most complex reversible cycle, namely, 2(SH/mT) = 0, or, when the mass of gas referred- to is a unit-mass, JWH/T = (Lord Kelvin) ; an expression very convenient for mathematical purposes, but difficult to translate into words. In a perfect, a reversible * The value of $ is the same in both these cases, because in both cases the change of entropy is the difference between the entropies of the isentropic or adia- batic lines AD and BC, Fig. 138. xiii.] SECOND LAW OF THERMODYNAMICS. 399 cycle, the Entropy,* the sum of the equivalences of all the transformations, is zero (Clausius). In a non-reversible process the sum of the transforma- tions is positive, and since all processes are non-reversible, the sum of the entropies in the universe tends to a maximum. According to Rankine's mode of expression, substantially identical with the preceding, the second law is : If the absolute temperature of a uniformly-hot substance be divided into any number of equal parts, the effect of each of those parts in causing work to be performed is equal. This implies that the absolute temperature is proportional to the total heat-energy, and so merges into the preceding form of the second law. Lastly, Carnot's Principle itself is often called the Second Law of Thermodynamics. We have already studied the direct transformation of heat into work in the radiometer. In the steam-engine the heat of the steam may be in part converted into work ; the piston is bombarded by the particles of the steam, and if the resistance to its onward movement be not excessive, it is thrust forward by the joint impact of the particles which impinge on it, their several components of motion parallel to the piston-rod being effective in this respect. Even under the most favourable circumstances which can be conceived, heat cannot be wholly converted into work by any form of continuously-acting mechanism. The efficiency of the ideal perfect engine small though that efficiency be is never approached in practice ; and the efficiency of the human body considered as a machine one-fifth of the total energy supplied to it being capable of utilisation is remarkable when we con- sider the narrow limits within which it operates. Work can thus be wholly converted into heat, but heat can never be wholly converted into work ; whence a universal ten- dency to the Degradation of Energy into Heat, the lowest of its forms. MEASUREMENT OF HEAT. Temperature we have now seen to be, when measured from an absolute zero a zero of absolute cold (1) proportional to the absolute amount of molecular kinetic energy, and (2) the reciprocal of Carnot's function. What is meant by equal degrees of heat ? Why is the dif- ference between C. and 1 C. supposed to be equal to that * Clausius, dealing always with unit-masses, has applied the term Entropy to the expression S(5H/r)= () ; and it will not be difficult to see that where the sum is positive, more heat is given to the engine by the source than is giyen when that sum = 0, the work done, W, remaining unchanged; but this excess is wasted bypassing through the condenser to the external universe. 400 HEAT. [CHAP. between 100 C. and 101 C.? In a perfect gas equal differ- ences of temperature correspond to equal increments of energy. In a diagram containing a system of adiabatic and isothermal lines, the isothermal lines must be so drawn as to cut off equal areas between the adiabatic lines. Absolute zero would correspond to total absence of molecu- lar kinetic energy. If we had a perfect gas at command we might measure temperature by its means in either of two ways : (1) We might observe its pressure at constant volume : equal increments of pressure correspond to equal increments of temperature. (2) We might observe its varying volume at constant pressure : the volume is proportional to the absolute tempera- ture, and equal small increments of volume approximately cor- respond to equal small increments of temperature. The former is the more accurate method. We have no perfect gases to experiment upon : air, etc., are not perfect gases. Yet we may perform either of the above operations on a quantity of air confined in a flask, and thus construct an air thermometer. The former method that of observation of pressure is here doubly preferable to the lat- ter that of observation of expansion because in the former there is no waste of energy in doing either internal or external work, and the increase of pressure is appreciably the same as that of a perfect gas. The indications of an air thermometer used in this way may hence be assumed as an- approximate standard of comparison. For the corrections necessary, see the table in Tait's Heat, p. 340. By the air thermometer we find that for a fall of 1 C. (from 1 to C.) on the mercurial thermometer, the pressure sinks in the ratio of 274 to 273 ; hence the temperature sinks in the same ratio, absolute zero is 273 C., and Carnot's function has the numerical value of ^75 for -j &G/8x) C., in time S/. The rate of increase of temperature per unit of time will therefore be (,9//xr)(8#/&i;) C. per sec.; and this = p-8G/8x, = p-SG when 8x = 1 cm. ; which is the proposition stated above. If a bar be heated at one extremity, the amount of heat which will arrive at a sectional area a given distance along the bar will depend upon the thickness of the bar and its propor- tional surface. A thin iron wire may be melted at one end but not have its temperature raised by 1 C. at a distance of 6 feet; so much heat is lost on the way, being spent in warming the surrounding air arid in keeping up radiation from the surface. For the same reason, the most volatile oil may be burned in a lamp with a sufficiently long wick-tube. In such a bar, maintained at a uniform distribution of temperature, the heat flowing across a given cross-section can be measured by a process of summation or integration. The temperatures at different points beyond the cross-section are observed ; the rates of cooling of a similar bar at differ- ent known temperatures are also observed ; from these data the loss of heat by radiation and convection can be ascertained ; and this is kept up by, and is equal to, the flow of heat across the sectional area. The flow H is thus known; so is $r/d, the temperature-gradient at the cross-section; so is A the area : whence the value of ?9 can be calculated. In bars of different thicknesses, the distances from the heated extremity at which the same temperature can be kept up by heating the extremity of the bars to the same temperature are to one another as the square roots of the thicknesses ; and in bars of the same thicknesses but of different lengths the flow of heat into the bar varies as the square root of the cube of the length. A hot point in space conceived to be maintained permanently hot will be the centre of a flow of heat symmetrical in all directions. The points in the surrounding space which are at the same temperature may be connected and found to lie on concentric spheres, or spherical Isothermal Surfaces. xiii.] CONDUCTION OF HEAT. 409 The heat travels by the shortest path from one surface to another, by Lines of Propagation, or Lines of Flow, at right angles to both ; and there is on the whole no lateral propagation over an isothermal surface. The whole system of surfaces and lines closely resembles a system of equipo- tential surfaces and lines of force. The difference of temperature per unit of distance along the lines of spherical propagation decreases with the dis- tance, being proportional to (1 /radius 2 ). The greater the curvature of a hot body, the greater will be its loss of heat by conduction. Hence an ellipsoidal body maintained at a uniform temperature loses most heat where the curvature is greatest a proposition closely resembling one in the theory of electricity. We must distinguish a Flow of Heat from a Flow of Temperature. The latter depends, inversely, on the specific heat pa- per unit of volume ; and if we compare the passage of heat through two substances similarly heated, we find that even though the one substance have a greater conductivity than the other, yet, if its specific heat per unit of volume be greater in a still greater proportion, a given temperature may take a longer time, travelling in the better conductor, to reach a point at a given distance from the source of heat, than it does in the worse conductor. The rate of propagation of a given temperature depends upon the ther- mometric conductivity p = ?!//pcr. Thus in copper, a given Temperature travels faster than it does in iron : and so does it in still air, though the actual quantity of Heat carried by conduction in still air is extremely small. When a body is exposed to a superficial periodic variation of temperature, the variations are propagated as waves of temperature according to the same law as if they were dis- placements in a vibrating but more or less viscous solid. The waves diminish in amplitude that is, in thermometric range as they penetrate, and that in geometrical progression ; and the depth at which the amplitude is reduced in a given ratio varies asVT, and also as Vp or V#/po-. Yearly variations of temperature are thus felt at depths beneath the earth's surface 19-11 times as great as the daily variations are ; for V365 = 19-11. Where a substance is not ph}^sically similar in all directions, as in the case of crystals, the conductivity may be unequal in three directions. Thus, a plate cut out of any crystal belonging to the binaxial system, and covered with a film of wax, will, if heated by a hot wire passed through its centre, so conduct the heat that the wax melts not in a uniform circle as in glass or a crystal of the regular system it will do but in an ellipse. Some solids are extremely bad conductors of heat. Down is perhaps the worst of all conductors; hare's fur, sand, asbestos, 410 HEAT. [CHAP. are examples of substances within which warm objects may be placed and remain without losing their heat to any material extent for some time. Flannel, cork, etc., appear warm when they are touched by the bare skin, because they carry away by conduction less heat than the air had been removing before these materials had been touched. Wood is, in the radial direc- tion, a bad conductor : this has a certain effect in preserving the tree in life. The actual amount of the loss of heat suffered by a cooling body depends directly on the effective cooling surface : whence the natural tendency in warm weather to lie at full length, in winter to roll the body up into small compass. The conductivity of the skin as a whole is greatly diminished by a layer of fatty tissue. The muscles are exceedingly bad conductors. When a hot body is surrounded by one or more concentric jackets with layers of air between them, the loss of heat is remarkably diminished. A single layer of linen diminishes the loss of heat from the human body by about two-thirds ; a double layer effects a much greater economy of heat, and so forth. The practice in cold countries of using double windows pro- ceeds on this principle, and hence also the hygienic advice to multiply the number of light garments in cold weather rather than their weight. The conductivity of liquids is as a rule greater than that of gases, which in the form of true conduction of heat-energy, as distinguished from convection, is very small. It is impossible to keep the hands in water at 52 C., while it is quite possible, as observed by Banks, to remain for five minutes in air near the boiling point of water. When a hot body is placed in air it sets up a number of Convection currents. Air becomes heated and rises, carrying away the heat of the hot body : colder air takes its place. Newton's law of cooling in a current of air is, that at each instant the amount of heat lost varies as the difference of temperature between the solid and the air. This law seems to be adhered to within narrow limits. In an undisturbed atmosphere the law of cooling by convection is, that the velocity of cooling is proportional to j a r L233 , where a is a constant (45 for air), p the pressure, and r the excess of temperature (Dulong and Petit). In hydrogen the process of cooling is very rapid. The carbonic acid, etc., of the atmosphere are mixed thor- oughly and equably, not by diffusion, which would take several hundred thousand years to accomplish the task, but by con- vection currents. Convection currents, as they pass colder or warmer strata of air, exchange molecules with them by diffusion ; the temper- ature of the whole mass thus rapidly becomes uniform. xiii.] CONVECTION OF HEAT. Convection currents may be demonstrated by throwing some coloured powder into cold water and proceeding to heat the liquid over a lamp ; by looking at distant objects through the heated gases which arise from a heated boiler or wall : the rise of smoke itself is an example of solid par- ticles borne upwards by convection currents particles which, when the ascending air has become cool, again fall, and may aid in producing fogs by the condensation of water around them. Though two bodies be not in contact with one another, they may yet exchange heat across the intervening space, and the hotter body, giving out more heat than it receives, is said to radiate heat to the colder body. This transfer of heat is effected by means of the Ether of space, and we shall, in the meantime, defer the consideration of the transfer of heat by radiation until we can take a general view of waves in the Ether. Dulong and Petit found that between C. and 200 C., the aggregate amount of radiation is proportional to (1-0077 1 " 1), where r is the excess of temperature above the surrounding enclosure. With diminishing pressures of air or gas, the rate of loss of heat falls, at first more rapidly, then less so; it then remains sensibly constant (Kundt), but with still further exhaustions the rate again falls (Crookes). Transport of Heat from place to place may be effected by storing up work-energy in springs which, on being released, set a mechanism at work which evolves heat by friction ; or by stor- ing up heat as " latent heat," or by raising the temperature of a substance whose specific heat is high. The former method is not effective, because so large a number of units of work corre- spond to so small an amount of heat ; the latter are exemplified in heating by hot water or by steam. A hot-water bottle con- tains several calories of heat, according to its size and its tem- perature; these can be liberated by conduction at any desired situation. If filled with crystallised acetate of soda, melted by heat, the cooling is protracted, for the melted salt, as it slowly solidifies, gives out its latent heat. Steam at 100 C., when condensed, liberates at the point of condensation 546 calories of heat for every gramme of water condensed, and can still, in the form of hot water, surrender more heat to surrounding objects. CHAPTER XIV. ON SOUND. THE word Sound is used in four different senses : 1. The physiological sensation perceived by means of the ear. 2. The complex harmonic motion of sounding bodies the Fourier-motion, the periodic or vibratory motion of elastic masses whose vibration is the physical cause of sound. 3. The disturbances of the air which affect the ear. " Sounds," says Newton (frincip. ii. Prop. L, Prob. xii. Schol.), "since they arise in tremulous bodies, are no other than waves (pulsus) propagated in the air." 4. The energy of a sounding body. " Heat converted into Sound," etc. It is better in this sense to say explicitly, " the Energy of Sound." A sounding body is a vibrating body. Cause a tuning-fork to sound in the usual way by striking it on the knee or drawing a violin-bow across it, or by forcing a steel rod between its prongs and drawing it through the point of the fork. Apply the point of the vibrating tuning-fork to the lips, to the surface of water, to a piece of glass. Bring a vibrating tuning-fork under a light splinter of wood lying upon two points of support ; on contact, the light body will be hurled upwards. Cautiously bring a vibrating tuning-fork or bell into contact with a pith-ball suspended by a thread. Pluck one of the strings of a violin : look at it as it vibrates : touch it. Look at a harmonium or concertina reed while it is in action. Observe the distinct tremor caused by a large organ pipe while sound- ing, or even by a large drum. Relatively deep, grave sounds are produced by slower vibrations; higher, shriller sounds by more rapid vibra- tions. Take a long strip of iron say a strip 4 feet long ; fix it in a vice ; pull it aside and let it go ; it will oscillate transversely at a rate such that the oscillations can be counted ; remove it, and refix it so that only 2 feet of it are now free to move ; it will now oscillate four times as frequently : 1 foot 412 CHAP, xiv.] SOUND-WAVES. 413 free sixteen times as frequently as at first ; 6 inches free sixty-four times as frequently, and so on. The oscillations now become so rapid, the number of them in a second (z.e., their frequency) becomes so great, that they can no longer be counted directly; now we hear a sound; the shorter the vibrating part, the more rapid become the vibrations, the shriller the sound. The transmission of sound from a vibrating body to the ear involves, as a rule, the formation of sound-waves in the air. This may be rendered impossible, e.g., where the sounding body a bell suspended or placed upon wadding within the bell of an air-pump from which the air is exhausted has no contact with air, and therefore no means of transferring its own vibra- tion to air; in such a case the ear perceives no sound, even though the bell be struck, for there are no air- waves set up. But it may be impossible for another reason. Air will not oscillate in waves such as can be propagated to a distance, unless there be some well-marked compression or rarefaction produced at the centre of disturbance. Take as extreme instances of sound produced by well-marked compressions or rarefactions the effect of the discharge of a cannon, which abruptly adds a mass of gas to the already-present atmosphere, and thereby produces great and sudden compression ; or the rarefaction produced by the sudden collapse of a weak boiler when the steam contained in it has cooled down. Thus a vibrating body, before it can act as a sounding body, must produce alternate compressions and rare- factions in the air, and these must be well marked. If, however, the vibrating body be so small that at each oscillation the sur- rounding air has time to flow round it, there is at every oscil- lation a local rearrangement a local flow and reflow of the air, but the air at a little distance is almost wholly unaffected by this. The same result follows if the medium surrounding 1 the o vibrating body be rare {e.g., hydrogen) or rarefied (e.g., rarefied air ) ; then, on account of the small inertia of the medium, it is easily induced to flow round the vibrating body ; in such cases there is but little wave-motion caused at any distance, and thus there is but little sound produced. A string stretched between two points of a rigid and massive framework produces surprisingly little sound when caused to vibrate : it does not act upon the air otherwise than by setting up local flow and reflow. If the same string be stretched over bridges upon a sounding-board, the string gives, at each oscillation, an impulse to the sounding-board which causes it to yield slightly; and thus the string causes the sounding-board >to vibrate. But though the amplitude of its vibration is small, the sounding-board is broad, 414 ON SOUND. [CHAP. and the air cannot, by flowing round its edge, evade compression and rare- faction ; the air is, accordingly, alternately compressed and rarefied, and thus a system of waves is effectively set up in it. Thus the loudness of the sound produced by a string may, by the use of a sounding-board, be multiplied many thousandfold. A similar experiment may be performed with a vibrat- ing tuning-fork suspended in the air by a string, and the same fork vibrating while its shank is pressed against the panel of a door. In these cases the energy of vibration of the string or tuning-fork is very much more rapidly dissipated, while the large-surfaced sounding-board is enabled to produce an i n t e n s e r or louder sound than is produced when the string or the fork vibrates alone ; and the vibration sooner comes to an end. The speaking-trumpet is in part an application of the same principle. Instead of a comparatively small surface, the oral aperture, being the source of sound, the much broader aperture of the trumpet is practically converted into the source, and the broad sound-waves thence issuing are only slightly weakened at their origin by lateral flow. As a general rule it is therefore advisable, when sound is to be heard at a distance, to make the sources of sound of the largest size convenient. Smallness of size may, however, be compensated by quickness of vibration. Thus the chirp of certain insects is produced by such extremely rapid movements as many as 12,000 to-and-fro vibrations per second that the air is alternately compressed and rarefied on each side of the wings or in the neighbourhood of the stridulating organs, without having time to flow round them. Characteristics of Sounds. The Fourier-motions which may produce sounds differ amongst themselves in their (a) Frequency the number per second of the slowest component-oscillations. An oscillation is a complete oscillation, once to-and-fro. The fre- quency of a seconds pendulum is ; in one second it performs half a com- plete oscillation. In French works we find that a " vibration simple " is half a complete oscillation, a swing over from one side to the other ; and a seconds pendulum is held to effect such " vibrations simples " at the rate of one per second. The reason for the apparently more artificial mode of defining an oscillation here used will be seen on considering the meaning of period in S.H.M. (p. 82); a complete oscillation restores the oscillating body to its starting point. (5) They differ as to their Energy. Proportional to the energy are the Intensity and the Square of the Amplitude. (c) They also differ as to the Relative Amplitudes of their Components. Of these three particulars, the first, the frequency, depends on the vibrating body itself, its form, its material, etc., and upon its tension, but is very slightly affected by its viscosity ; the second depends entirely on external causes ; the third depends xiv.] CHARACTERISTICS OF SOUNDS. 415 partly on the form, the tension, the rigidity, etc., of the vibrat- ing body, partly on the manner in which it is set in motion. By variations in these particulars an infinite variety of Fourier-motions may be produced in vibrating or sounding bodies ; and as a natural consequence we might expect to find, as we do find, an infinite variety of musical sounds actually occurring in nature. Musical sounds may differ from one another in three cor- responding respects, viz. Pitch, Loudness, and Quality or Character. Pitch. The pitch of a clear musical sound depends on the Frequency of the Fundamental Vibration of the sounding body. Suppose a string to vibrate harmonically, and its component vibrations to occur 261, 522, 783, 1044, etc., times per second: then that string would have a fundamental vibration whose frequency is 261 per second ; and a sound of this fundamental frequency is recognised by our musical sense as the note 3j=pi The loudness of a sound increases with the amplitude of oscillation of the vibrating body ; if two strings, otherwise simi- lar and similarly circumstanced, oscillate through ranges of |- and | inch respectively, the latter has twice the amplitude and tends to produce four times as much sound as the former : the loud- ness or intensity of sound being, among sounds of the same pitch, proportional to the energy of vibration, and therefore to the square of the amplitude. Mark, however, that the relative loudness of different sounds as perceived by the ear is not to be measured by their physical intensity or the square of the amplitude of the vibrations at their source, for the ear is not necessarily, and is not in fact, equally sensitive to sound of every pitch. Vis co sity of a sounding body, while it scarcely affects the pitch, aids in causing the amplitude of the vibration, and there- fore the loudness of the sound produced, gradually to dwindle away. As to their Quality or Character, we find among sounds an infinite variety. We can distinguish a sound produced by a violin from one of the same pitch and loudness produced by a clarionet, a flute, or a pianoforte ; we can distinguish the sound of a viola from that of a violin ; one violin from another; one player from another on the same violin ; one person's voice from that of another ; the voice of the same person in different 416 ON SOUND. [CHAP. moods or states of health. The basis of all this variety lies in the endless differences that may exist between Fourier-motions which, though they agree as to the frequency of their funda- mental or slowest component and as to the total energy involved in their movement, do not necessarily coincide in the relative amplitudes of their component harmonic motions. But if, as this theory indicates, an extended series of com- ponent vibrations go to make up the aggregate vibration of a sounding body, ought we not, in the sound produced by a sound- ing body, to hear a series of tones corresponding to the series of vibrational components ? If a string produce the note gpEqE, corresponding to a fundamental vibration whose frequency is 261 per second, ought we not, at the same time, to hear other sounds corresponding to 522, to 783, to 1044, etc., vibrations per second? The reply is that we do actually hear such tones; but we do not attend to them, and for practical purposes we are therefore deaf to them. We are accustomed to interpret a sound produced by a single sounding-body the voice of a person, for example as a single sound; from earliest infancy we uncon- sciously train ourselves to listen only to the fundamental tone of any single note: and the presence of the other tones of the really- compound sound produced by a single vibrating-body has the apparent result of determining the Character of that tone to which alone we consciously listen. In many cases, when we listen for the higher component sounds, knowing what to listen for, we can hear them, even with the unaided ear : after practice the ear acquires the power of recognising the presence of these harmonics with great readiness a power which may easily become oppressive to its possessor. The special training which confers this power differs only in degree from that which enables one to discriminate the different notes which make up a chord, sounded in harmony ; for to the untrained ear even a chord, if it be well in tune, seems to be a single mass of sound. Noise. If all the keys of a piano within the compass of one or two octaves be simultaneously struck, the result is a con- fused jangle, a Noise. Here we have the Superposition of Fourier-motions resulting in an apparently -irregular disturb- ance of the air. This may go still farther ; the Fourier-motions, which are superposed on one another, may have no relation of frequency and little or no individual persistence. The more markedly this is the case, the less musical will be the sound xiv.] NOISE. 417 produced, and the more markedly will it bear the character of noise. The general hum of a town is made up of sounds and cries, each of which, taken singly, may perhaps not be unmusi- cal; but because they are not related to one another by any simple numerical ratio of frequency, they together produce the disagreeable effect of a noise. Noises, then, such as the sound of steam escaping from a boiler, wind rushing through trees, the clatter of falling objects, and so forth, may be considered to be produced by the superposition of a number of distinct musical sounds. Some of these may predominate in intensity and in persistence ; and thus a noise may have a distinguishable pitch. We may recognise differences in pitch between the noises pro- duced by drawing the thumb-nail at various speeds over the cover of a book bound in cloth, by blowing across the mouth of keys or tubes or flasks of various sizes, by letting boards of vari- ous sizes fall on a wooden floor, by blowing through glass tubes on which bulbs of various sizes have been blown, and so forth. Even where the original disturbance is in the highest degree irregular, as where bricks are pitched out of a cart, the elasticity of the bricks, small though it be, affects the pitch of the noise produced, for the thuds produced by soft porous bricks are graver than the clinks produced by hard glazed-bricks of the same size. If we listen to a continuous noise with the aid of a resonator (p. 430) tuned to some particular tone, we can often recognise the presence of that tone as a component of the noise ; the resonator will, if that tone be present as a component, sound it forth continuously if it be continuously present; intermit- tently if it occur at. intervals only. Even a single vibrating-body may, when struck, produce a noise. A bell is not, with ease, so cast as to be perfectly uni- form ; when struck it tends, if not quite uniform, to divide into unequal sectors, each of which pulsates at its own rate ; the physical result is a number of simultaneous vibrations bearing no simple relation to one another, and the physiological result is a mixed sensation, a jangle, a kind of noise. Thus sounds originate in Fourier-motions ; a musical note in a single Fourier-motion ; a noise in a number of simulta- neous Fourier-motions whose fundamental frequencies bear to one another no simple numerical relation ; and, as we shall afterwards see, the sensation of Harmony in a number of simul- taneous Fourier-motions whose fundamental frequencies have simple numerical relations to one another. 2E 418 ON SOUND. [CHAP. The simplest possible sound would be one produced by a vibration in which the Fourier-motion was represented by one component ; such a sound would be a pure Tone. The pitch of the sound or note produced by a vibrating body is the pitch of the gravest component, the fundamental Tone; and it may be specified in two ways : (1.) Physically, by stating the number of vibrations per second which correspond to that fundamental tone ; (2.) Musically, by referring the tone to its place in an arbitrary scale of pitch in conventional use among musicians. To find the frequency of vibration corresponding to any given note : As the note in question let us take, for the sake of example, that produced by an ordinary " A " tuning-fork. A card or strip of metal is placed so as to touch at one end the cogs of a little cog-wheel, while the other end is firmly fixed ; the wheel is rotated slowly each cog makes one click ; more rapidly the clicks blend into a hum ; still more rapidly the hum rises in pitch, and the faster the rotation the shriller becomes the sound; at a certain rate of rotation the sound is neither graver nor shriller than that produced by the tuning-fork ; this rate of rotation is such that the card is struck 435 times per second ; 435 impulses per second given to the card, and by the card to the air, produce the sound jE^aiE, "a' = 435." Higher sounds are due to more rapid, lower sounds to slower, vibrations than this. This arrangement is known as Savart's Wheel. Another contrivance, devised to the same end, is the Syren. A rotating disc is pierced by holes arranged equidistantly in a circle, whose centre is in the axis of rotation of the disc. A tube brings a current of air to a spot near the disc, so situated that in some positions of the disc the air can blow clear through one or other of the holes, while in others the current of air is almost cut off by the disc itself. Rotate the disc ; the current of air is alternately cut off by the disc and allowed to blow through it. If there be 87 holes in the circle of holes, and if the disc rotate five times per second, there are then produced 435 puffs of air per second, and the note " a' " is heard : its quality is, however, decidedly inferior, for the principal sound heard is the noise made by the current of air when it strikes the disc. If the current be divided by 87 pipes, so as to blow through the 87 holes simul- taneously, and to be simultaneously cut oft' from them all, the sound is very much clearer and louder than when there is only a xiv.] PITCH. - 419 single stream of air blowing through one hole at a time. Instead of 87 pipes issuing from a wind-chest, we may employ a wind- chest capped by a fixed disc containing 87 holes, arranged in a circle like that of the rotating disc : the rotating disc rotates in the immediate vicinity of the fixed one : simultaneously the air rushes through all the apertures of the rotating disc, simul- taneously it is cut off from them all. The number 87 is in practice never used; some such number as 24 or 48 is chosen. Connected with the rotating disc is some form of mechanism for recording the number of rotations effected by it in a given time. The rotating disc is caused to rotate at such a speed as causes the desired sound to be produced: the number of apertures in the disc, multiplied by the number of rotations per second, gives the num- ber of impulses per second imparted to the air, and thus deter- mines the frequency of the tone in question. The syren works under water as well as it does in air. The experiment already described on page 412 also gives roughly the means of finding the frequency of any given tone. The thin strip of metal is, by successive trial, carefully with- drawn into the vice, until its free part gives, when set in vibra- tion, a sound of precisely the same pitch as the tone whose frequency is to be determined. Say that this length is 1 inch ; and also that if 30 inches of the strip be free, it executes 29 complete oscillations per minute. The number of oscillations varies inversely as the square of the length ; whence (1 inch) 2 : (30 inches) 2 : : 29 : #, or x = 26,100 vibrations per minute, 435 per second. Still another method of determination of the frequency of vibration of sound, of a given pitch, is graphically to record the actual vibrations of the sounding body. A tuning-fork has a little feather-barb attached by cement to one of its prongs : the extremity of the barb is brought into contact with slightly- smoked paper spread over the surface of a cylinder. The cyl- inder is caused to rotate ; the point of the barb draws a straight line on the smoked .paper. The fork is caused to vibrate : the barb now describes, on the rotating cylinder, a sinuous line which records the oscillations of the tuning-fork. An indepen- dent mechanism can be made to mark the cylinder once every second, and thus the absolute number of oscillations made by the tuning-fork during each second can be counted on the permanent record. The same principle may be applied to many forms of vibrating body, such as strips of metal, membranes, etc. 420 ON SOUND. [CHAP. Musical Pitch. The arbitrary scale of pitch in common use, and typified by the white keys of a pianoforte, is the follow- ing: Thirty-two foot Octave Subcontra Octave. g* II y A,, B,, British ? D,, E,, F,, G,, German C,, D M E,, F M G N A,, H,, French ut_, re_, mi_, fa_, sol_. la_, Sl_, No. of Vi- ) 1fi .qi 9 c brations f 163125 18-3515625 20'390625 21'75 24'46875 27-1875 30-5859375. Ratius 16 : 18 ; 20 : 21-3 : 24 : 26-6 : 30. Sixteen-foot Octave Contra Octave. A, B, British C, ^ E, F, G, German C, D, E, F, G, A, H, French ut re mi fa so! Ia So No. ofVi-) .,., hrations f 32 625 36703125 40-78125 43'5 48'9375 54-375 61-171875. Ratios 32 : 36 : 40 : 42-6 : 48 : 53-3 : 60. Eight-foot Octave Great Octave. British ~C~ D E F G A B German C D E F G A H French ut, re, mi, fa, sol, la, si, No. of Vi- ),.- brations I" 6525 73-40625 81-5625 87 97'875 108-75 122-34375. Ratios 64 : 72 : 80 : 85"3 : 96 : 106-6 : 120. Four-foot Octave Little Octave. ~ ^~1 C d e f g a B b ut 2 re 2 rm* 2 fa 2 sol la, S1 2 No.ofVi.) iqo .: brations | 1305 146-8125 163-125 174 195'75 217-5 244-6875. Ratios 128 144 : 160 : 170-6 : 192 : 213-3 : 240. Two-foot Octave One-stroked Octave. _ M / .^y. 7 d 1 e 1 f g 1 a 1 b' ut, No.ofVi-) 9fi ; brations j Mi re, mi, fa, so! 3 293-625 326-25 348 391'5 la, 435 S1 3 489-375. Ratios 256 : 288 : 320 : 341'3 : 384 : 426-6 : 480. One-foot Octave Two-stroked Octave ^ . Cj G &- -(=2- =H c" d" e" f 11 g" a" -^B b" ut 4 . re 4 mi 4 fa 4 so! 4 Ia 4 si 4 No. of Vi- ) , 22 brations f 522 587-25 652-5 696 783 870 978-75. Ratios 512 : 576 : 640 : 682-6 : 768 : 853-3 : 960 ] MUSICAL PITCH. 421 Six-inch or Three-stroked Octave. Three-inch or Four-stroked Octave. c 1 " d 1 " e 1 " f " g" 1 a 1 " b" 1 c"" d"" e"" f"" g"" a"" b"" c"" ut 5 re g mi g fa 5 so! 5 Ia 5 si, ut 6 re 6 mi 6 fa 6 so! 6 Ia 6 si 6 ut, No. of 1 No. of) Vibra- V tions ) 1044 1174-5 1305 1392 1566 1740 1957'5 2088 2349 2610 2784 3132 3480 3915 4176. Ratios 1024:1152 : 1280: 1365 "3: 1536 : 1706'6 : 1920 : 2048 : 2304: 2560:2730 6 : 3072: 341 3 '3: 3840:4096. The starting-point of this notation is the a' tuning-fork, made to vibrate 435 times per second, or the second string of the violin, made to vibrate in unison with such a fork. Under this system the c" tuning-fork makes 522 complete oscillations per second. This is entirely a matter of convention. The num- ber 435 was chosen by the Academic des Sciences of Paris ; 433 by the Philharmonic Society under Sir George Smart in 1826; 440 by the German Society of Nature-researchers at Stuttgart in 1834; 452 is used in the British Army; while a pitch a' = 426-6 has been highly recommended, on the ground that under such a system the tones C u , C t , C, c, c', etc., are produced by 16, 32, 64, 128, 256, etc., vibrations per second an arrangement which has the advantage of giving very simple numbers to deal with, but which has, on the other hand, the practical dis- advantage of giving a pitch which is too low to please instru- mentalists, and the didactic disadvantage of tending to conceal the real arbitrariness of the convention which assigns to the a f or the c" fork the particular number of vibrations chosen in practice. In practice there is, indeed, a great lack of agree- ment; instrument-makers are constantly raising the pitch for the sake of increasing the brilliancy of orchestral music, while vocalists are made to suffer. Modern concert .pitch has thus risen as high as a' = 460 vibrations per second, about 1J semi- tone above what it was in England in the time of Handel (V = 424), while the organ-pitch in England was, in the mid- dle of the eighteenth century, as low as a' = 388. If the stand- ard number of vibrations chosen for a' be any other than 435, the whole series of numbers given in the table must suffer a proportionate increase or reduction. The accuracy of such a scale depends not upon precision of absolute numbers of vibra- tions so much as upon correctness of the ratios of the several numbers to one another. The successive tones of the scale of C are related to one 422 ON SOUND. [CHAP. another, with respect to their frequency, in the following manner : f -_, r -Si n C=> ' f ts> S '.. 3 1 t d 1 256 : 288 : 320 : 341-3 : 384 : 426-6 : 480 : 512. 1 : 8 : 4 : 1 : I : \ : 15 8 2. Here C (V = 256) is a keynote, and upon it we have raised a diatonic major scale, d r n f S 1 t d 1 . Such a scale is found by experience to be satisfying to the ears of the Western nations ; and whatever tone be chosen as the keynote, there can always be sung or played on instruments of the violin or of the trombone class a scale of this kind, in which the intervals are felt to be pleasing and in tune, in which the intonation is felt to be just, and in which each tone, when it is carefully listened to while the keynote is borne in rnind, is felt to have its own peculiar mental effect, this depending on its relative place in the scale, and not on its absolute vibrational frequency. Singers who have sung much together, string play- ers who have practised together without pianoforte accompani- ment, naturally use the tones of such a scale without knowing or even caring what the numerical ratio of the frequencies of the various tones of the scale may be. Intervals. We may now identify the various intervals occurring within the diatonic scale Minor second, " semitone " n : f or t : d 1 . . . 15 16 Grave major-second . r : m or s : 1 . . 9 10 Major second . . . d : r, f : s, 1 : t . .89 Grave (or Pythagorean) minor-third . . r : f . .. . . . 27 32 Minor third . . . n : S or 1 : d' . .56 Major third . . . d : n, f : 1, s : t . .45 Perfect fourth d : f, r : S, m : 1, S : d', t : rc' 3 4 Acute fourth . . . 1: r 1 . . . " . . 20 27 Augmented fourth . . f : t . ... . . 32 45 Grave diminished fifth . t, : f . V . . . 45 64 Grave fifth . . . r:l . ... . . 27 40 Perfect fifth . d: S, m : t, f : d 1 , S : r 1 , 1 : m' 2 3 Minor sixth . . . t, : s, n : d 1 , 1 : f ' . . 5 8 xiv.] MUSICAL INTERVALS. 423 Major sixth . I . . d : 1, r : t, S : n' . 8 Acute major-sixth . . f : r 1 . . . " .16 Grave minor-seventh . r : d', S : f , t : 1' . Minor seventh . . n : r', 1 : S* , . .5 Seventh. . . . . d : t, 'fTn* . ., . 8 Octave . . . . d : d 1 , r : r 1 , etc. . . 1 5 27 16 9 15 2 Musical intervals are equal to one another when the con- stituent tones in each have the same relative frequency. Thus d : s : : 1 : |, and n : t : : f : -^ ; the ratio of 1 to | is equal to that of | to -^ that is, it is 2 : 3 ; whence the musical inter- val between d and s is equal to that between n and t. Transition. Any tone may be chosen as a keynote. Let us choose g' = 384 as our keynote, and then compare the tones of the scale of the key of G with those of the scale of C. Retaining the same ratios, the scalre of G is d : r : m : f : s : 1 : t : d 1 . 1 : f : 4 : f : f : f : -^ : 2. 384 : 432 : 480 : 512 : 576 : 640 : 720 : 768. Comparing the two scales we find : Scale of C (-Key C"). c' d' e' f g' a' b f c" d" e" f" g", etc. . . . . 384 : 426-6 : 480 : 512 : 576 : 640 : 682-6 : 768. Scale of G ("Key G"). 384 : 432 : 480 : 512 : 576 : 640 : 720 : 768. The tones agree with the exception of the as and the/'s. The a' of the scale -of C and the a 9 of the scale of G differ from one another in the ratio of 426-6 : 432, or 80 : 81. The two tones are perfectly distinct, and an ear that has become accus- tomed to the pure scale of C is pained, especially in harmony, by the substitution, for the proper a 1 in that scale, of the slightly sharper a' which belongs to Key G. The difference between the two tones is called a Comma; and they may be respec- tively written a' and 'a'. The/" of Key C and the correspond- ing tone in the scale of G differ more widely from one another ; their frequency-ratio is 682-6 : 720, and the interval between them, y|-|, is sometimes called a semitone. In order to play in correct tune music written in Key G as well as music written in Key C, we would require^ not only the tones of the Key of C, but also two additional tones in each 424 ON SOUND. [CHAP. octave. Every transition from one Key to another "more remote from " the Key of C multiplies the demand for neAV tones ; and that to an extent twice as great as the current notation, which neglects differences of a comma, would seem to indicate. In the table, pages 426 and 427, are given the tones of the scale of C, together with a number of tones derived from related keys. The relative, not the absolute, number of vibra- tions has been shown in each case. If a singer were called upon to produce a note of 324 vibrations per second, the feat would be impossible. This number is, however, 1-265625 X 256 ; and hence if c' have 256 vibrations per second, the note required is the re of Key D. A tuning-fork c' = 256 is set in vibration ; call the note of the fork do ; sing do, re ; fix the attention on re (d f ) ; call it do without changing its pitch ; dwell on it a moment ; then sing some such phrase as do, mi, sol, do, mi, re, do; and the desired note, 'e', a note differing by a comma from e', the mi of Key C, has been produced, the sense of tonality and key-relationship having carried the singer into the correct sound. The column headed " logarithmic increments " contains fig- ures which measure the intervals between the successive tones ; for it gives the logarithms of the frequency-ratios between each tone and its predecessor ; and the most convenient method of comparing ratios is to compare their logarithms. When the logs. are equal the ratios are equal; when the ratios are equal the inter- vals are equal. Thus the intervals between C and Cft, Db and T), T> and v Dft, D and Dft, Eb and E, E and Eft, F and Fft, 'F and 'Fft, ^Gb and r G, G\> and G, G and Gft, Ab and A, A and Aft, 'A and 'Aft, 'Bb and 'B, Bb and B, B and Bft, are all equal, being measured in that column by the logarithm -0177288, which is the log. of the ratio ||. Again, we find a number of lesser intervals whose log. is -005395, and whose ratio is -f^: these are C and 'C, Cft and 'Cft, v Db and Db, ^D and D, r Dft and Dft, T> and Eb, E and 'E, F and 'F, Fft and 'Fft, r Gb and Gb, r G and G, G and 'G, v Gft and Gft, v Ab and Ab, A and 'A, Aft and 'Aft, ^Bb and Bb, V B and B, B and 'B, V and c. Between these the interval is a Comma. The scale may be seen to be roughly divisible into 53 steps or divisions ; but these are not equal to one another ; if they were equal, the logarithms would at each step -acquire an equal increment; for the ratio between each tone and its predecessor would be equal throughout the scale. Roughly and for dia- grammatic purposes it is, however, convenient to represent the interval between C and D by 9 steps, while that between D and xiv.] MUSICAL INTERVALS. 425 E is represented by 8 : and the table is so arranged. A thor- oughly accurate table of this kind would be one engraved on metal, the intervals between any two tones in column 3 being made directly proportional to the log. increment between them. The intervals marked Pythagorean in the table are thus derived : Start from c and go upwards by successive fifths, ) The columns in the table, pp. 426-427, headed "Equally- tempered Scale," show the nature of the system of Equal Temperament, which is, as nearly as practicable, applied to the pianoforte and organ. The intervals are equal; the ratio between a tone and its predecessor and successor is in every case the same ; between each pair of tones the logarithmic increment is equal: it is = '- = -0250583. The result differs widely from pure intonation ; but we are accustomed to it. On a pianoforte equally tempered the fifths are not appre- ciably out of tune, though they are a little flat: but the thirds, three of which are forced to make an octave instead of extend- ing only from C to B$, are too sharp ; and though this be not offensive on the pianoforte, to which indeed their sharpness lends somewhat of brilliancy, yet in slow sustained harmony these sharp thirds are really discordant, as may be well heard on a loud harmonium tuned in the usual manner, and on which thirds alone are played. Loudness. The physical Intensity of a sound depends initially on the square of the amplitude of the vibration of the sounding body ; but the corresponding sensation of loudness depends not only upon peculiarities of sensitiveness of the ear, but also on the amount of physical disturbance of its drum, and 426 ON SOUND. [CHA Q fcM O i^ {CN '-rHO5O 00 lO-^l OO lOCOlO^flft OO OO OS O CO O CO OS O CO O rC ^ OSO CM O5COOSOOS (N 11 *^ CO CO OJ CO CO CO II rH 9 OO rH Or-lOOO rH 99 9 9999 9 O rH rH O rH O p p poop J^ ft o o OOOCJ>.t"- 1OO <7^COGMC-(N CO T 1 CO T-H O O O OO t^ !> l5 O *O O CO CO CO J>- J3 3 O5 (NCNCQCN lO^O OOCOOOCO rH O rH CO CO CO CO CO J>.rHO-* t^. i 1 Ttl OO !>. i 1 OS CO (N CO 05 CO CO fjl O O ^T-HCNCNCO I Tfl l5 'COCOJ>.1>. ' OOOO OO OOOO 02 g 5 o o 0000 00 OOOO OOOO O O O H , s CO O 'CO O rH O 'O 1O 'ifll O 8 o 'So 'c? o 'co P || O *o o 9 CO OO CO O rH if$ Tt< CO *rH O trHCOCOO I'-'OS J-t^r-ikriO ; ^-<*"COOO T Ij .lOt^COO OlOOp rH . .-.cOrHCvI <*-^ CO O O CO O GO : in ^o o c-j : co o : oo . fo CO CO O CO *O -00 fN CM CO -CO CO H rH ,!( rH r " 1 ,1| ^H M 05^ l 1 r- 2n ^ OO CC H g-S r-l CO CMrHrHtN OSQO rHCOCMUS ^ 1O CO O OO Tl< CO rH *-* 'iOvOCOJ^. 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' ,, rH O O 10 : o 10 3010300 1 o o o o 000 o o o o o o o 71 o o co O 1O rH GO CO 1 i i i rH 1O *t>- O rH CN X^ O m> o S3 g c | o : * S to oo o ! i g : CO CO *^* O io >> . jp oo * 00 OJ : co 10 9 rH " rH CO rH l-H rH O OO OO 01 ia o tv. tO rH tO rH o CN CN _ ^^ H rH co ON CN C? rH OO rH OO O CO rH t- rH < 10 > 03 :-?* T pq *^ ^^ O3 b ^^ S 1? *P c, , 2 j-3 ; M : r ~ . w _s ' w ^ . $4 2 : M : : : S W O 4H ^ M i^ r o ^3 o O fl o fl o M CQ < P=H fe S VH n3 f- 1 S -1^ ^ TJ <(H B r-H r p o 1 PO ?^= ^^^ 1 1 < * \ T?? 1 MM l^' % * * "d " ,q * " N ^ 'd**^ S 45 .B g 43 It-H 5 , ff ^ " o CO ^ B ^ * ? - 7> 43 "S 'e? ^ CD bO^r; > g 01 J! B S ^M CQ s S^g^ 5 g CO * '5 T3 O) 5 *3 be 1 X (_, 1 bb^ 1 1 S 5 CS 3 *s 1 1 * | 1 S -s S ^ o 'S o i * s g | S o o c ^ B be B 0) O) S a O S g 15 O Pn g g . O cc bo 1 428 ON SOUND. [CHAP. if the sound be conducted to the ear by the air, it depends on the intensity of vibration of the air near the ear; and this varies riot only (1) as the square of the amplitude of the origi- nal vibration, but also, in the open air, (2) inversely as the square of the distance of the sounding object. To compare the relative loudnesses of two sounds of nearly the same pitch, place the sounding bodies at such distances that they become just inaudible, and no more : say that the one becomes inaudible at 10, the other at 50, yards : then the loudness of the one at 50 yards' distance is at the ear equal to that of the other at 10 yards : their initial intensities must be as 10 2 : 50 2 , or 1 : 25. If the sound be not propagated in free air, but be confined in a tube, the loudness of sound may diminish at a much less rate, for ultimately the waves become plane-fronted, and move down the tube without any loss of intensity other than what is due to such loss of energy as is brought about by friction against the sides of the tube or by the viscosity of the air itself. Hence sounds can be carried along sewers, speaking- tubes, etc., to great distances without great diminution of loudness. Similarly, if sound be propagated by parallel or convergent waves in the air, as when it issues from a wide aperture, or after reflexion from a curved surface, it may lose little of its inten- sity, or may even concentrate its intensity on some particular place. The loudness of a sound also depends, if it be conveyed by a gaseous medium, on the density of that medium at the place where the vibration is imparted to it. The denser the medium the greater its inertia, and the more readily it is compressed against itself: the greater the compression, the greater the amount of energy imparted to the medium, and the louder the sound produced. A body vibrating in vacuo produces no sound : in rarefied air or hydrogen, or any other rare or rarefied gas, it produces a comparatively feeble sound ; in carbonic acid it pro- duces a louder sound than in air. A cannon fired on a moun- tain-top produces little sound ; one fired beneath is heard dis- tinctty and loudly from a balloon, even at a great height. Concentration of sound-waves renders sounds louder, as in ear-trumpets and in those stethoscopes the auditory extremity of which fits into the ear. Quality of Sound. If a body vibrate so as to produce a xiv.] QUALITY OF SOUND. 429 sound of the fundamental pitch C = 64, and if all the harmonics be present, the series is the following : 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15, etc. 64 128 192 256 320 384 448 512 576 640 704 768 832 896 960 C c g c' e' g' b\>'- c" d" e" f"+ g" a" + b\>"- b", etc. * * * # These are all tones of the scale of C, with the exception of the 7th and its octave the 14th, the llth, and the 13th. The 7th and the 14th correspond to a very flat B of 112 vibrations, lying between Aft and 'Aft ; the llth to a sharp F of 88 vibra- tions, tying between 'F and Fft ; the 13th to a flat A, lying between Ab and A. Analysis of a sound into its components may be effected by several methods, of which we shall first consider one due to Prof. Mayer. As our example we take the sound produced by a vibrating organ-reed-pipe, a sound which we recognise as peculiar and characteristic. We are provided with a set of tuning-forks, one of which vibrates in exact unison with the fundamental tone of the organ-pipe, and the rest of which respectively vibrate 2, 3, 4, 5, etc., times as rapidly. As to the organ-pipe, a part of its wall has been replaced by a piece of inelastic thin morocco leather, or some similar sub- stance, which vibrates exactly as the air within the pipe does. To one point of this is attached a bundle of silkworm-cocoon- threads, 40 inches or so in length : each of these is attached to one of the tuning-forks and tightened somewhat. The organ- pipe is caused to sound ; the leather vibrates ; the silk fibres are all set in motion, and each alternately tugs and releases its own tuning fork. If the vibration appropriate to any one of the tuning-forks be present in the original compound vibration, the corresponding fork is set in motion : if it be not present, that fork remains silent : if the vibration be ample, the fork sounds out loudly : if it be not, the sound is feeble. This arrangement analyses the sound into its components, for it can be seen which of the tuning-forks are set in vibration ; and if the organ-pipe cease sounding, the forks go on sounding for some time, and by their joint action produce a compound sound closely resembling the sound of the reed-pipe which had been the means of setting them in vibration. This action is very exact : the slightest difference between the natural rate of any tuning-fork and that of the corresponding organ-pipe vibration JTI7BRSITT, 430 ON SOUND. [CHAP. causes the fork to sound with comparative feebleness, or not to sound at all. Resonators are extensively used as a means of analysis of sound. A resonator consists in its most usual form of a bulb, Fig us g enera lly of glass or of brass, with a large aperture, a, at one side and a small one, 6, at the other. The air within such a bulb has a natural period of vibration which depends upon the cubic contents of the resonator and upon the size of the orifices. This period can be found by the pitch of the sound produced, on tapping the resonator with a soft substance, or by blowing brief blasts of air across its mouth. If the air convey a system of waves which agree in period, either absolutely or approximately, with the natural free vibration of the air in the resonator, the air in the resonator will absorb the energy of those waves, will be set in motion, and will act as a sounding body. If we be provided with a set of such resonators, the air in one of which freely vibrates in unison with an a' tuning-fork, and in the others respectively 2, 3, 4, 5, 6, 7, etc., times as rapidly, then, on listening to an a' organ -reed-pipe, one ear being closed and the other adapted to each resonator in succession (this being done by fitting the nipple b of the resonator (Fig. 142) into the ear), we shall, if the proper sound of any of the resonators be contained in the complex sound to which we listen, hear that resonator loudly sing out its proper tone ; while, if it be not present, we shall simply hear the ordinary sound of the pipe through the reso- nator, without any reinforcement. And further, if we fill our ears with the sound of the tone thus sung out by the resonator, and remember its pitch, we shall, when the note is again sounded out by the organ-pipe, have no difficulty, even without a reso- nator, in hearing the harmonic tone : and by dint of practice we may hear at will, or even independently of will, many if not all of those component harmonic tones which, by accompanying that fundamental tone to which alone in ordinary circumstances we are accustomed to listen, help to make up the note of the organ-pipe. A very convenient form of resonator may be made of a common tall lamp chimney or a similar piece of tubing. If it be held vertical, as its lower end is immersed in water to various depths its natural pitch varies : and a tube thus gradually lowered into water is capable of resounding in succession to the different harmonics of a fundamental note, so that the ear, XIV.] RESONATORS. 431 placed near the tube, can recognise their several presences. In Wintrich's resonator an aperture at the side may be closed by the finger. By aid of the same resonator an observer is thus enabled to listen alternately to the grave-pitched muscle-sound of the heart and to the sharper valve-stretching sound. See Gscheidlen, Physiol. Methodik. Resonators may be otherwise employed. If the small aper- ture b be stopped with wax, and if the resonator be brought near a sounding body, it will absorb the energy of any vibration corresponding to its own proper tone, and may then be removed and listened to : thus each one of a set of resonators may be made to select one tone out of the group of tones present in an ordinary musical sound, and to bring it to the ear to be listened to. A resonator and a sounding body to which it is in response seem to be mutually repelled, in consequence of the stresses set up in the intervening air (Dvorak). Resonators may further be used to transfer the energy, which they thus take up, to relatively-massive bodies such as tuning- forks. A resonator may be made in the form of a thin wooden box with open ends : a tuning-fork precisely in tune with it is fitted on its upper surface ; a sound causes the air in the box to vibrate, the air acts upon the box, and the box upon the tuning-fork ; if all be in exact unison, the energy accumulates in the tuning-fork, which comes to vibrate energetically and to produce a loud sound. Again, the oscillation of the air in a resonator may be rendered visible by the following device: A cavity in a block of wood '(Fig. 143) is divided into two parts by a mem- brane, such as thin gold- beater's skin. The one moiety of the cavity is connected with the cavity of a resonator : the other is connected with a sup- ply of coal-gas which enters at B and passes out Gas at C on its way to be burned at the jet D. This contrivance is called Koenig's manometric capsule. When a sound is pro- duced outside F, containing as one of its component tones the proper tone of the resonator, the air in the resonator oscillates in sympathy with that component, the diaphragm vibrates with it, and the flame at D is rendered alternately higher and lower 432 ON SOUND. [CHAP. by the action of the vibrating diaphragm on the stream of gas. The flame obviously alters its character : and the change under- gone by it can be studied by looking at it while the head is turned rapidly from side to side, the eyes being kept fixed rela- tively to the head ; or by looking at the flame through an opera- glass, which is rapidly moved across the field of view ; or, best of all, by looking not directly at the flame but at its image in a rapidly-rotating mirror : in all which cases the flame or its image appears to spread out, not into a uniform band of light, but into a band with serrated edges, or even into a chain of bead- like separate images. A sufficiently-extensive set of resonators would thus enable us to effect the analysis of sounds of any degree of complexity : but resonators do not furnish us with as delicate a means of investigation as the means first described, unless indeed they be each allied with a tuning-fork ; they respond in general with excessive readiness to any tone in proximity to their own natural tone. Synthesis of Sound. Von Helmholtz showed that any quality of sound may be built up by the superposed effect, upon the ear, of simultaneously sounding tuning-forks of the proper number, pitch, and relative loudness. Complex Sound-Waves. The pitch, the loudness, and the quality of a sound may be studied together by causing sound- waves to impinge directly upon some sensitive body without any intermediate process of selection or filtration. Thus, if instead of a resonator, as in Fig. 143, a cone be adapted to a mano- metric capsule, and if sound be produced at the mouth of the cone, sound-waves will impinge directly upon the membrane in A. The membrane will go through a complex motion some- what resembling the original compound-vibration of the sounding body, and the flame will demonstrate this by its variations of height. The image of this oscillating flame will appear in a mirror, if the mirror be made to revolve, as a band of light, serrated by large teeth, whose outline is broken by subsidiary serrations ; the number and size of the greater serrations indi- cate the frequency and amplitude of the fundamental vibration : those of the subsidiary serrations vary with the number and variety of the subsidiary vibrations. This experiment may be roughly carried out, if there be no revolving mirror at hand, by whirling the gas-flame itself (a rat's-tail jet at the end of a flexible tube) before the eye. xiv.], COMPLEX SOUND-WAVES. 433 It is interesting to carry, out this experiment by singing into the open end of the cone ; even among notes of the same pitch sung to the same vowel, the association of different forms of the flame-image with different qualities of tone and different subjective sensations is very striking ; and it is possible for a singer to attain to the production of very pure tone such pure tone having, however, a somewhat hollow quality by finding out for himself how to control the larynx so as to keep the serrations visibly open and simple. If such a membrane have a small mirror attached to it, the mirror will share in the vibrations of the membrane, and, if it be jointed on a hinge, will reflect a beam of light in such a fashion as to produce a curve upon photographic paper uniformly rolled past the vibrating membrane ; this curve will indicate the fre- quency, the amplitude, the complexity, of the vibrations of the membrane. Sound-waves, however complex, may again be caused per- manently to record the succession and variation of their own impulses. Ldon Scott's Phon autograph is a conical vessel, closed at its narrower end by a membrane ; to the membrane is attached a writing-point ; the extremity of the writing-point is brought into contact with a smoked revolving-cylinder. So long as there is no sound, the writing-point describes a uniform line on the rotating cylinder: when sound-waves enter the cone the membrane is set in vibration, and the writing-point now describes an undulating line, which varies in its form according to the frequency, the amplitude, and the complexity of the origi- nal vibration. It must be observed that membranes thus made use of do not exactly reproduce the original motion at any point of their surface : those compo- nents are exaggerated which approximately or exactly coincide in frequency with some normal mode of free vibration of the membrane itself. Edison's Phonograph is a phonautograph whose writing- point is somewhat blunt; and it records the vibrations of its membrane by being driven through variable distances into a sheet of soft tin fixed on a rotating cylinder, or into the sub- stance of a rotating. cylinder or disc of wax: it leaves a perma- nently-deforming mark, a groove of varying depth. If the membrane, after having made such a mark, be raised from the rotating cylinder, and the cylinder turned back to its initial posi- tion ; if the membrane be now readjusted in its former position, or, better, a little nearer the cylinder ; and if the cylinder be again rotated in the former direction, with the same velocity as at first, the depressions in the groove previously produced, 434 ON SOUND. [CHAP. being of variable depth, cause the blunt writing-point, under which they pass, to move alternately towards and away from the cylinder ; this compels the membrane to execute vibrations, and in so doing to set up vibrations and sound-waves in the air, which, being received by the ear, produce a sound similar to the original. Not exactly, however: the process is not perfectly reversible. Some consonants are not well reproduced, especially the explosives (b, p, t, d, k, g) and the sibilants (s, z, th); and further, it is generally found that there has been some exaggera- tion of some of the higher components in the course of trans- mission through the membrane, the effect of which is to render the sound reproduced one whose quality is somewhat metallic, nasal, or even squeaky and Punchinello-like. LAWS OF VIBRATION OF SOUNDING BODIES. These laws form properly a part of Kinetics ; but the means of research into the phenomena of Vibration which lies most readily at our disposal is the observation of the pitch of the sound produced by vibrating bodies ; for which reason some part of the consideration of these laws has been deferred to this place. In general, any vibration of a vibrating or sounding body is a periodic motion, a Fourier-motion ; though in particular cases we may find that the vibration is not a single Fourier-motion either simple or complex, but may be resolved into a number of such motions, simultaneous and superposed. Transverse Vibrations of Strings. If a string be stretched and drawn aside from its mean position, it tends to return to that position. In Fig. 144 let the string AB, subjected to a Fig. 144. tension of T dynes (= the Weight of T/g grammes), be drawn into the position ACB, the particle C being supposed for sim- plicity's sake to have been initially at O, the centre of AB : the tension tending to bring back the particle C to O is the com- ponent of the total tension resolved in the direction CO : this varies directly as CO that is, the restitution-force varies directly as the displacement the criterion of Harmonic xiv.] TRANSVERSE VIBRATIONS OF STRINGS. 435 Motion : and it can be shown as a consequence of the fact that the string is fixed at A and B, that the string will oscillate in some such manner that its aggregate motion can be analysed into a number of simple oscillations whose periods are commen- surable : in other words, that the motion of the string is a Fou- rier-motion. According to the mode of disturbance striking, plucking, bowing of the string, or the duration of these opera- tions, there may be an infinite variety in the relative amplitudes of these component simple oscillations. Some of these com- ponents may even be altogether absent : where, for example, a string is plucked at its centre, it is not possible that any of those components which have a point of rest at the centre of the string should be present, and the vibration of a string so plucked is one in which all the even components are absent. In gen- eral, a vibrating string does not present any component-oscil- lation any one of whose nodes would be at the point of disturbance. Frequency of Oscillations. The velocity of propagation of a transverse wave along a uniform stretched string, perfectly flexible, is v ~\/t/p (p. 268) ; here t = T/?rr 2 , where r is the cross-sectional radius of the string. Hence v = VT/TT/O H- r. But the length of each wave is fixed by the condition of the string ; the string is bound at each end, and if we con- fine our attention to the slowest component, the fundamental tone, we see that A, the wave-length, is equal to 2AB or 2/, twice the length of the string. Then, since n, the number of complete oscillations per second, is equal to i// A, we have n = v/\ = v/'2l = VT/irp -r- 2rl ; or, if m =T/g be the number of grammes whose Weight stretches the string, n = Vmg/Trp -H 2rl. Problem. A wire of steel (p = 7-8), 1 m. long and 1-2 'mm. thick, is stretched by the Weight of 40 kilogrammes and set in transverse vibra- tion : what will be the frequency of its fundamental vibration? What its pitch? Ans. n = Vmg/wp -f- 2rl = {V(40000 x 98_l)_-i- (3-1416 x 7-8) -f- (2 x 0-06 x 100)} = 10545 vibrations per sec. ; H==EE when c' = 253-1. So for a perfectly-flexible string: the effect of rigidity of wire or string is to diminish the number of vibrations, and to cause the motion to assume the character of a number of superposed harmonic motions of incommensurable period. The vibrations of a violin string differ much from those of a pianoforte string. In the violin the oscillating string sometimes travels in the same direction as the bow, sometimes away from it. When the bow and the string travel in the same direction, the bow drags the string with it, distorts it, pulls it out to an extent greater than that which it would have travelled if allowed freely to vibrate. When the string returns the bow fails to retain it, loses it, and as it is returning bites and catches 436 ON SOUND. [CHAP. it again by means of some rough resinous particle at some other part of the bow. The friction and, consequently, the adhesion between the string and the bow are relatively somewhat greater when both move in the same direction, for at low relative-speeds friction tends to increase. The string is thus distorted and assumes successively a number of forms, of which no one is curved : and the form of the vibrating string at any instant presents an angle between two straight lines, a form differing considerably from the curve of sines. But it is periodic, and it is true Fourier-motion. The mathematical problem is What superposition of commensurate S.H.M.'s (compare Fig. 48) will produce a vibration-curve such that, for a certain distance, the flexures so balance one another as to produce a straight line, and then so aid one another as to produce an abrupt angle, again fol- lowed by a straight line ? This can be solved, and the result is that tke vibration of a bowed string must be composed of a fundamental vibration, of weak components 2d to 6th, and of ample higher components. This agrees with the result of resonator-analysis of the sound of a violin. In the sound of a violin the upper harmonics are loud and piercing ; the nearer harmonics are feeble, and the fundamental tone stands apparently alone, but rendered penetrating in qual- ity by the high mass of harmonics. Purity of violin tone depends upon perfect periodicity of the peculiar motion of the string ; this is difficult to attain, for a good elastic violin, uniform strings, a uniform bow, uniformly resined and evenly handled, are necessary: and any stumbling of the bow over the string, or any irregular movement of the string under the bow, is revealed by scratchiness of tone. The sharper the angle made at the point of disturbance, the richer the tone in high harmonics. A string plucked with a quill, as in the old harpsichord, has thus a metallic tinkling quality, and its fundamental tone is relatively very feeble. A string struck suddenly at one point has a form differing greatly from that of the curve of sines. Part of the string remains unaffected, while the part struck is distorted. This dis- tortion travels along the string, and results in a periodic motion abounding in high components ; the tone produced is tinkling. If the same cord or wire be struck gently by a soft elastic hammer, the blow being deliberate, and, as it were, gradually insisting upon the displacement of the string at the point struck, the disturbance is more evenly spread over the whole string, the fundamental component-vibration is more prominent, the higher components are relatively more feeble, and the tone is purer. xiv.] TRANSVERSE VIBRATIONS OF STRINGS. 437 In a pianoforte string struck by an elastic soft-hammer the harmonics up to the sixth are present; the seventh is obliterated, or nearly so, by the hammer being made to strike the string at a spot one-seventh of its length from the end of the string that is, at a spot which would have been a node of the seventh component if that component had been allowed to exist in the compound vibration: and the components beyond the seventh are feebly represented. The Monochord is a box of thin light wood, containing air which communicates with the exterior air by lateral aper- tures. Upon this box rest two bridges ("banjo-bridges"), one near each end. Over the bridges is stretched a wire; of this, one end is firmly fixed to one end of the box, while the other is either passed over a pulley and made to support a weight, or else is connected with a tuning-peg, which may be turned by a tuning-key. The tension on the stretched wire may thus be varied. Experiments with the Monochord. For experiments it is better to use a form of monochord in which there are two wires, of which one is tightened by a peg, the other by the weight of a suspended mass ; in the latter of the two wires the total tension on the wire can be directly measured, in the former it must be inferred. 1. Suppose a wire 1*2 mm. thick, whose free vibrating part is 1*2 metres long, to be stretched by the weight of 48 kilogr., and the pitch of the sound produced to be G = 96 vibrations. What weight ought to be added in order to raise the pitch to d? The pitch is raised G : d i.e. a fifth : the vibrations are rendered more numerous in the ratio 2:3; the tension must be increased in the ratio 2 2 : 3 2 , or 4 : 9 ; the stretching-weight must be increased from that of 48 to that of 108 kilogr. ; the mass which would have to be added is 60 kilogrammes. 2. Two wires of equal thickness are stretched one by the tuning-peg and the other by the weight of a heavy mass so as to vibrate in unison. The weighted wire is removed and replaced by one of the same length, but of a different thickness, stretched by the same weight. A thinner wire gives a higher note, a thicker one a lower. If in Ex. 1 a wire 1 mm. thick be employed, what will be the pitch of the sound produced? The frequency varies inversely as the radius : it therefore exceeds that of a wire 1*2 mm. thick in the ratio of 6 : 5 ; the note produced will be Bb. 438 ON SOUND. [CHAP. 3. A brass wire and a steel wire of equal gauge are equally stretched : they are free to vibrate in equal lengths. Brass has a density p = 8-38, steel = 7*8. The brass wire gives a sound lower in pitch than that given by the steel wire : the respective frequencies are in the ratio of V7-8 to V8-38 or 1 : 1-03651. A catgut string, whose density is small, gives a higher note than a steel wire. 4. In order to vary the free vibrating-length of wire, a movable bridge is arranged under the string. If this be so placed that 60 cm. of the wire are free to vibrate instead of 120 as before, the sound produced will be the Octave; if 40 (=i|^), the Twelfth; if 30 C^ 1 ^), the Fifteenth; if 24 ( = 1 f a ), the Seventeenth and so forth above the fundamental note emitted by the freely-vibrating string of 120 cm. length. The number of vibrations of a string varies inversely as its length. Hence if we wish, with a string which sounds C, to pro- duce the note D, whose frequency is f x that of C, we must allow the string to vibrate not as a whole, but only in -| of its length. To produce the scale on one string, the parts of the string which are allowed to vibrate are as follows : d r m f s 1 t d 1 i t i t t A * The application of this principle is familiar in violin playing. 5. Nodes and Loops can also be shown on the monochord. If the wire, 120 cm. in its vibrating length, be lightly touched Fig.145. at 20 cm. from the end, and if the twenty-centimetre-part of the wire be set in vibration by a bow, the whole wire is found to be in vibration from end to end ; but not as a whole. It divides itself into segments or vibrating loops, separated by nodes or' points of rest. Each segment is 20 cm. long: and the sound given out is that which might be emitted by half-a-dozen separate wires each 20 cm. long that is, it bears to the note emitted by the whole string the same proportion as g 1 does to C XIV.] MONOCHORD. 439 two octaves and a fifth. Similarly for other fractional divi- sions of the wire or string. Lightly stopping the string has the effect of destroying or of checking the formation of all those modes of vibration which have not a node at the point touched ; hence the 6th, 12th, 18th, etc., components of the vibration of the whole string are unchecked, while the other components are rendered impossible. The nodes and loops of a string vibrating in this way are rendered manifest by paper riders placed, some at the nodes, some on the loops ; when the string enters into vibration those riders which had been placed on the nodes will retain their place, while those on the loops will be jerked off. Nodes and loops on vibrating strings may be illustrated on a large scale as follows: Take an indiarubber tube 10 feet long, filled with sand, or a long spiral of iron or brass wire, and fix one end of this to a wall ; hold the other end in the hand. On moving the hand gently, the natural period of oscillation of the cord can be easily found. Give with the hand a series of transverse impulses, so timed as to aid the natural oscillations : the tube or spiral will enter as a whole into wide oscillations. Now give such impulses twice as often : the cord will divide into two segments, pivoting in a striking manner round the central point. Do so three times as often as at first: the cord will divide into three segments or loops, pivoting on two nodes. By increasing the frequency of the movements of the hand, the cord can be made to oscillate in 4, 5, 6, 7, or even 8 or 9 segments, according to the dexterity of the experimenter. The experi- ment is a striking one ; and it may be varied by causing the hand to move in a circle or an ellipse instead of a straight line. Melde's Experiments. A tuning-fork is provided with a little hook on one of its prongs ; to this hook is attached a fine white silk thread. This thread is passed over a wheel and at- tached to a suspended mass partly immersed in water ; the quantity of this mass can be coarsely adjusted by the addi- tion or removal of sand ; its effective weight can be finely adjusted by varying the quan- tity of water in W (Fig. 146). The fork is set in vibration; waves appear to travel up and Fig.HS. 440 ON SOUND. [CHAP. down the thread ; if the string be illuminated by a beam of light in a dark room, the effect is singularly beautiful. As the ten- sion is increased, the segments vary : at length the thread vibrates as a whole, and seems to form an opalescent spindle. Its fre- quency of vibration is half that of the fork ; the thread, when at its limit, is pulled back by the retreating fork into its mean position, but is relaxed and allowed to swing over upon the return of the fork ; whence two oscillations of the fork corre- spond to one of the thread. If the tension be reduced to ^, the vibrating part of the string must be shortened to J in order to keep time with the fork, or else, if the string be not shortened, the string will divide into two equal segments or loops separated by a node : if the tension be reduced to -J, the string divides into three loops with two nodes ; and so forth. If the tuning-fork be turned round through 90, so as not now to tighten and relax the thread, but to give it a series of transverse impulses, a similar series of phenomena will be observed; but the fundamental vibration is now simply synchro- nous with that of the tuning-fork. When the thread is suspended between two tuning-forks whose frequencies bear an aliquot ratio, the tuning-forks being placed at such distances from one another as to tighten the thread to the required amount, the motion of the thread becomes periodic, and presents a complex of beautiful loops and nodes which are obtained with comparative ease. Transverse vibrations of cords may be studied with respect to the motion of each particle by casting a beam of light along a vibrating cord, and looking at a particular bright or bright- ened spot on the cord. The bright spot appears, when the cord is looked at end-on, to give in quick succession a large variety of such forms as we have already seen to be produced by the composition of S.H.M.'s. A bright spot on the cord may also be looked at through a microscope whose object-glass is borne upon a vibrating tuning-fork ; the apparent motion of the spot produced by the motion of the object-glass (this being parallel to the length of the cord) is compounded in the eye with its real motion ; the apparent up-and-down motion of the spot, as looked at transversely, is spread out into an open curve, and thus becomes more intelligible, for the eye can more readily comprehend open curves than simple up-and-down movements. Longitudinal vibrations of a string may be excited by xiv.] LONGITUDINAL VIBRATIONS. 441 drawing one point of a violin bow along the string: a very shrill tone is produced. The velocity of propagation is v = v / g7p ; the wave-length is twice the length of the string, or A. = 21 ; the number of fundamental vibrations per second, n v/\ = Vg/p -=- 2/. Problem. A steel wire (elasticity g = 2, 520,000000 #, and density 7-8), of one metre in length, is clamped at the two ends and set in longitudinal vibration. What will be the pitch of the sound produced? Ans. The frequency is n = ^%fp + 21 = (V2520,000000 x 981 H- 7-8 -=- 200} =. 2815 vibrations per second = /"" + . As a rule the longitudinal vibrations of a string or wire are much more frequent than the transverse ones, and thus produce a much shriller sound, and further, they are not so much affected by tension applied to the string, for a variation of tension which would materially modify the frequency of transverse vibration would have little effect upon either the elasticity g, or the density p, upon which the longitudinal vibrations depend. Before the transverse vibrations could be as frequent as the longitudinal ones, it would be necessary that the tension should be t = g, the amount ideally necessary to double the length of the wire. A violin e"-string gives, when 'rubbed longitudinally by one point of the bow, a sound in the neighbourhood of ('/$)"" ; while, when let down so as to sound only e', it gives out a longitudinal vibration-sound not so low as (/$)"" ; the longitudinal vibration hardly falls a comma, while the trans- verse falls an octave. All the catgut strings of a violin may be observed to give out nearly the same longitudinal note, for this does not depend on their thickness. From this we see how important it is to use the bow in such a way as to bring out transverse vibrations only, and by no means to wield it so that any component of its motion over the string can excite longitudinal vibrations, resulting, as these do, in shrill discordant tones. By means of the monochord we may learn that a string, while vibrating longitudinally, divides into loops separated by nodes, just as it does while executing transverse vibrations. Longitudinal vibrations of rods resemble those of strings or wires. A glass rod grasped by its centre and rubbed longitu- dinally by a resined cloth will enter into longitudinal vibration and will produce a shrill sound. A glass tube treated in the same manner may be made to vibrate so vehemently that it shivers into segments. Transverse vibrations of rods obey the rule that if be the thickness of the rod, I its length, g its Young's Modulus of elasticity, and p its density, then n , c, and the secondary differential tones c, bb, d', g'b,Vb, c" ; and the primary chord is thus embedded in a mass of combination-tones comprising, among others, the discordant series xiv.] HARMONY AND DISSONANCE, 475 The consonance of c 1 , e'b, g' is therefore necessarily much harsher than that of c\ e\ and g'. For further developments of this subject the reader must be referred to von Helmholtz's Sensations of Tone, translated by Mr. Ellis. When just intonation is possible, as among glee-singers or quartette-players, each listens to his fellow-performers as well as to his own voice or instrument, and gives out that note which he feels to belong to the key in which the party is performing, and learns to do so in such a way as to avoid beats : thus, as is said, the performers rub off one another's asperities. In this way mathematically-exact ratios of considerable complexity are accurately attained without necessary knowledge of them on the part of the performers. VOICE VOWELS, The voice is produced by vibrations of the larynx, especially of the vocal chords, in whole or in part. Above these is placed the mouth-cavity, which may assume various forms under the action of the various muscles which regulate the position of the tongue, the soft palate, the floor, and the sides of the mouth. This mouth-cavity acts as a resonator and reflector. According to the number of upper harmonics which are reinforced, and the extent to which they are severally reinforced, will vary the quality of the sound emitted. Upon this quality, and upon nothing else, depends that Character which we recognise as some particular Vowel ; for every vowel is a particular Quality of Sound. An elementary example of this is furnished by a common pocket tuning- fork ; when set in vibration and the broad face of one of the prongs pre- sented to the ear, the fork seems to emit the vowel u or oo ; when its shank is pressed against a table the fork seems to say ; now the octave becomes prominent. The reason is that the fork swings in circular arcs, and not in transverse straight lines ; it consequently presses against the table at the end of each half -oscillation, and causes it to emit the octave as well as the fundamental tone: A tone almost pure gives the hollow sound of the vowel u ; one accompanied by its octave gives the brighter sound of. the vowel o. Each vowel gives a particular form of indentation in a phonograph. Vowel sounds can be analysed by means of resonators ; and when a particular vowel is sung in presence of an open piano (loud pedal down) that vowel is repeated by the strings : each component of the complex vibration is taken up by that string which is in unison with it. On the other hand, von Helmholtz 476 ON SOUND. [CHAP. showed that by causing a number of resonators of a series whose frequencies were as 1:2:3:4:5:6:7, etc., to vibrate with independent intensities, he could at will produce by synthesis not only a great number of qualities of tone widely differing from one another, such as clarionet- tone, etc., but could also build up the different vowels themselves. To each vowel corresponds a different form of the resonating mouth-cavity ; to each such form corresponds a different natural pitch of vibration. When the larynx emits a complex sound containing as one of its components a tone of this natural pitch, this tone is strongly reinforced, and the quality of tone some- what affected. TRANSFORMATIONS OF THE ENERGY OF SOUND. Sound being in its physical aspect a kind of motion, in the course of which work is done against elasticity and inertia, it is superfluous to speak of the conversion of the energy of sound into that of mechanical work. The transmission of sound is a transmission of energy, and the sound produced by a sounding- body is mechanically equivalent to a definite amount of work. When a heavy tuning-fork is attached to the piston of a little pump, as in Edison's harmonic engine, it can be made to do work; but then it produces somewhat less sound than when vibrating freely. The mechanical equivalent of sound may be estimated, as it has been by Mayer, by comparing the sound pro- duced by a free tuning-fork with the sound produced by the same fork on equal excitation when its prongs are connected by a thin strip of indiarubber, and by finding the amount of heat developed in the rubber in the latter case. Work may be, on the other hand, converted into sound. In general there are two methods of accomplishing this transforma- tion, firstly, by storing potential energy in an elastic body, which is then liberated ; secondly, by transforming uniform into intermittent motion through the agency of friction. We have already studied the mode of excitation of a violin string. A pointed slate-pencil pushed across a slate at a certain angle pro- duces a well-known shrill scream, and the mark produced by 'it will be found on close examination to consist of a train of sepa- rate dots ; the action is not unlike that of the violin string. The scream of unoiled bearings in a machine may be accounted for in the same way, and in such a case much of the energy of rotation of the machine is wasted in the form of sound. xiv.] TRANSFORMATIONS OF SOUNP-F^NERGY. 477 Heat may be, in some cases, transformed into the energy of Sound. Trevelyan's rocker and singing-flames we have already studied ; the singing of a kettle is due to the rhythmical agita- tion produced by the formation and collapse of bubbles; the roar of steam issuing from a boiler is produced by the disturb- ance of the surrounding air by steam which, after thrusting aside the surrounding air, collapses into water-drops ; the roar of a chimney is due to the oscillation to-and-fro, within the chimney, of heated columns of air or smoke which set the air within the chimney in vibration, of which the deep roar heard by us is generally a high harmonic : in all such cases the energy of the sound produced is obtained at the expense of the Heat supplied. But, like other forms of energy, that of Sound is ultimately dissipated. When sound is produced in a room, every particle of the walls and contents of the room is set in vibration ; there is, indeed, no way of protecting bodies surrounding a source of sound from this influence, except perhaps by placing them upon several alternate layers of caoutchouc and soft putty within a vacuum. At last the sound degenerates, after repeated reflexion within each object, into irregular molecular motion, and its energy is converted into Heat. So when a tuning-fork is set in motion and sounded in the open air, part of the energy which was initially communicated to the tuning-fork when it was first set in vibration is lost, in consequence of the viscosity of the fork, which becomes slightly warmer ; while part of that energy is expended upon the external air, which, by reason of its own viscosity, gradually extinguishes the sound, beginning with the highest components, and the whole at length dies away, the energy of sound-motion becoming converted into the degener- ated form of Heat, which ultimately becomes diffused through- out the entire Universe. CHAPTER XV. OF ETHER-WAVES. IN this chapter a variety of phenomena fall to be considered which can be explained as phenomena of undulation in the all-pervading Ether, and may thus be said to be due to Ether- Waves. It is necessary, however, to make a reservation of opinion, and to point out that all we are really entitled to affirm is that the phenomena in question are transferences of energy through the Ether, accompanied by variable disturbances of that medium disturbances whose variations follow the same laws as those of wave-motion, but which may in themselves be due to changes not necessarily of position within the Ether, but possibly of its stress, of its electric condition, or of some other property of the interstellar medium as yet unknown to us. Their theory has been chiefly developed by those who considered the phenomena of Light, Radiant Heat, etc., as phenomena of Wave-Motion in the Ether; and, with this preliminary explanation, we shall in the sequel speak unreservedly of these phenomena as due to Ether-Waves. NATURE OF RADIATION. The all-pervading Ether can be set in vibration by the vibra- tion of the molecules of ordinary matter. This local disturbance sets up waves; and by these waves energy may be transferred from one place to another. This process of transference of energy by Ether-waves is the process of Radiation. The Radiation of Energy by the Sun amounts, according to the results of Prof. Langley's experiments on Mount Whitney, Southern California, to about 16500 horse-power per square foot of the sun's surface. Of this about one 2125,000000th part meets the earth ; this, at the earth, amounts to about 150 ft.-lbs. per sq. ft. per sec., or 3 ca per sq. cm. per minute. Of this about one-third is always spent in heating the atmosphere ; the rest may be more or less cut off from access to the earth's surface by clouds, dust, water- 478 CHAP, xv.] NATURE OF RADIATION. 479 vapour, etc. The energy of the waves comprised within a cubic mile of the Ether near the earth's surface, or, to use Lord Kelvin's phrase, the " mechani- cal value of a cubic mile of sunlight," is accordingly about 23000 ft.-lbs. Ether- waves can also be produced by electric methods, for which see p. 741. These waves are, as yet, longer and of less frequency than those produced by the vibration of molecules. Heat-waves and light-waves in Ether are not waves of com- pression and rarefaction, like those of sound in air. The propa- gation of an ether-wave is effected after a different fashion, some- what difficult to realise. The analogy of a transverse vibration running along a cord, or of a wave of up-and-down oscillation running over the surface of water or over a thin membrane, must be extended to the Ether, with its three dimensions in space. At any point where the movement of the Ether is examined, it is found to be an oscillation at right angles to the direction in which the wave is being propagated, and therefore parallel to the wave-front. The vibration of the Ether, when due to molecular vibration, is initially of the nature of a forced vibration; it is probably excited by the oscillation of a part of the Ether, which is in some way entangled within, or which envelopes, the vibrating molecule. The molecular vibration which excites the ether-waves is a true vibra- tion of the molecule, not a translational oscillation from place to place. The molecules of ordinary matter must be supposed, in virtue of their small size, to vibrate very rapidly. We have already stated that the average diameter of molecules is perhaps the 2>50 1 0000 th part of a millimetre, and that they may perhaps consist of ether rolling within ether in vibratile vortices. A steel tuning-fork 2 inches (50 mm.) long may, if it be of the proper form, vibrate 240 times a second ; if it were 2(50 1 0000 mm. long, and of the same shape, it would vibrate 30,000,000000 times per second ; if made not of steel, but of ether, its frequency would be greater in the ratio of the velocity of propagation in ether to that in steel, and would therefore amount to about 2600,000000,000000 oscillations per second. The vibration of a molecule may be more like that of a disc than that of a tuning-fork ; but the rough analogy just mentioned may serve to show that it is, even a priori, possible that some such number might denote the average frequency of molecular oscillations, an average modified in the direction of retardation by the formation of heavier molecules through the coalescence of smaller molecules, or perhaps by the reaction of the Ether which is set in forced vibration, or modified, on the other hand, in the direction of acceleration by the forma- tion of higher-pitched vibrations, which may be, to use the musical analogy, dissonant with one another when the structural arrangement of the molecules is unsym metrical. The molecule of sodium-vapour acts somewhat like a disc which is slightly unsymmetrical : such a disc would give out^two tones very near one another in pitch : and a vibrating sodium-molecule gives rise to two sets of ether-waves which differ only slightly in frequency. 480 OF ETHER-WAVES. [CHAP. Frequency. The ether-waves which are produced by the mechanical vibrations of molecules have frequencies which range between about 20,000000,000000 (Langley) and about 4000,000000,000000 oscillations per second a range, to use a musical analogy, of about eleven octaves : but of these our eyes are sensitive to scarcely one octave to those, namely, which range between about 392,000000,000000 per second (extreme red of the spectrum), and about 757,000000,000000 per second (extreme violet). Those ether-waves which have been produced by electric methods have frequencies ranging between about 500 per second and 7500,000000 per second. Velocity and Wave-Length. These waves all travel through the Ether of space at the same rate, namely, about 30057,400000 cm. (186,680 miles) per second. Ether-waves while traversing the Ether present no essential differences, except in. respect of their wave-lengths; the wave-length X is equal to v/n, where v is the velocity of propagation and n the frequency. Those ether-waves which are produced by the oscillations of molecules vary accordingly in length, in a vacuum, from about -fa cm. to about lt go^ooo cm., and those waves to which our unassisted eyes are sensitive, the waves of Light, have wave-lengths ranging between TJIJTS cm> an( l syinf cm. These wave-lengths are usually specified in terms of "tenth- metres " ; a tenth-metre being 1 metre -=- 10 10 , or 0-000000,01 cm. Extreme red and extreme violet have thus, in a vacuum, the respective wave-lengths of 7667 and 3970 tenth-metres. Those ether-waves which have been pro- duced by electric methods have wave-lengths ranging between 60,000000 cm. (373 miles) and 4 cm. Ether-waves do not traverse all substances with equal speed: hence their wave-lengths in different substances vary; if any particular kind of radiation have to be spoken of, it may be denned by specifying its wave-length in some specified medium, but it is better to state its numerical frequency. To do the lat- ter implies, however, that we assume and we are apparently justified in assuming that all kinds of radiation pass through a vacuum that is, through the ether of space with equal speed. Kinds of Radiation. When a succession of ether-waves impinges on a mass of ordinary matter, the effect varies accord- ing to the nature and the condition of the body which receives their shock ; if it be an ordinary opaque mass, that mass may be warmed, the energy of wave-motion being transformed into heat, and the waves which have impinged upon the opaque mass are xv.] KINDS OF RADIATION. 481 ex post facto called a beam of Radiant Heat ; if they fall upon the eye, they may produce a sensation of light, and the wave- system is then called a beam of Light : falling upon a sensitised photographic plate, or a living green leaf, they may operate chemical decomposition, and the wave-system is then called a beam of Actinic rays. The word "rays" in the last phrase may be understood to mean, not imaginary lines at right angles to the wave-front, but kinds of radiation; and hence we speak of Heat rays, of Light rays, of Chemical or Actinic rays ; these names being given to one and the same train of waves according to the effects which it is found competent to pro- duce. But while ether-waves are in course of traversing the ether, there is neither Heat, Light, nor Chemical Decomposition ; merely wave-motion, and transference of energy by wave-motion. Hence none of these names can in strictness be applied to a train of waves while these waves are actually travelling through the Ether. Ether-waves which differ in their frequency differ to some extent in their degree of power of producing the motion of heat or the sensation of light, or of doing the work of chemical decom- position. All ether-waves can produce heat, for their energy is converted into heat when they fall upon and are absorbed by such a substance as a thick layer of lampblack, which for the most part arrests and extinguishes them. The long and slow ether-waves which have been produced by electric methods are not so readily arrested and extin- guished as those produced by molecular vibration; they can, therefore, to a large extent traverse such an obstacle as a brick partition or a deal door ; but the difference in this respect between them and ordinary radiation-waves is a question of degree, and by a sufficient obstacle their energy can be reduced to Heat. Those ether-waves which take their origin in vibration of ordinary matter, and whose frequency is less than 392,000000,- 000000 per second, are too slow either to affect the eye with the sensation of light, or, in the ordinary case, to impart to mole- cules an agitation brisk enough to shake them to pieces, and thus to operate chemical decomposition. Such slow waves, whose presence can only be recognised after their impact, by the conversion of their energy into Heat, are called Dark-Heat- Waves. If they fall upon an ordinary photographic plate they do not operate chemical decomposition; but if the molecules 2i 482 OF ETHER-WAVES. [CHAP. upon which they impinge be specially heavy and complex, even these slow heat-waves may be found to toss and shake them with briskness sufficient to break them up. Radiant Heat of sufficient intensity is thus found to operate chemical decompositions of a different order from those brought about by contact- heating; e.g., the formation of olefines from paraffin-vapours (W. Young), instead of that of aromatic hydrocarbons with deposition of carbon. A dark nebula in the Pleiades was first photographed and then with difficulty seen. Major Abney has been able to photograph with heat-rays down to a frequency of about 160,000000,000000. The waves may, on the other hand, be so rapid above 757,000000,000000 per second as to produce no visual effect on the eye ; the eye is normally, physiologically, blind to them, and is unable to feel their impact ; but they may effect chemical decomposition; their successive impulses may aid the natural free vibrations of the molecule, which thus become increasingly ample : and just as a resonant tumbler into which its own note is steadily sung vibrates, shivers, and breaks into fragments, so a molecule, quivering under the steady, regular, and continu- ously well-timed blows of the rapid ether-waves, may yield and break up into its constituent atoms, or into groups of atoms, which constitute simpler molecules. Such rapid waves are called Invisible or Ultra-violet Chemical Rays. According to Lubbock, ants and Daphnia seem to be able to see ultra- violet rays. The power of operating chemical decomposition possessed by the more rapid waves depends more upon their frequency than upon their intensity. The rays which are most active in decomposing carbonic acid by chloro- phyll are visible, being the yellow and the red. For waves of given length but different intensities, the quantity of chemical decomposition is, within wide ranges of intensity, the same when the product (Time of Exposure x Intensity of Light) is the same. For very feeble intensities this product must be increased ; in some cases, as in star- photography, very considerably. This law is, however, the basis of calcu- lations of " exposure " in photography. The slower waves may thus produce heat, or perhaps chemical decomposition of heavy complex-molecules ; waves of medium rapidity may produce heat, the sensation of light, or chemical effect; the more rapid ones may produce heat or chemical effect according to the substance upon which they fall. The invisible chemical rays, though they can operate chem- ical decomposition, are yet of very feeble physical intensity; xv.] KINDS OF KADIATION. 483 their aggregate kinetic energy is, in the radiations from the sun, as we receive them filtered through our atmosphere, mil- lions of times less than that of the slower red or dark heat rays: even those rays which are visible are effective not so much in virtue of their intensity, which is but small, as in virtue of the extraordinary sensitiveness of the eye to light that is, to the impact of ether-waves of a certain range of frequency. Colour. Within the limits of visibility 392 billions to 757 billions there is an indefinite variety of integral and frac- tional numbers, each of which represents the frequency of a particular kind of radiation, a particular kind of light. Physi- cally there are as many kinds of light as there are possible frequencies between the limits mentioned. These kinds of light, each physically characterised by the number of waves which strike the eye during a second, are recognised by the normal eye as being distinct, not as the result of any conscious process of counting the number of impulses suffered by the eye during a second, which would be absolutely impossible, but in consequence of the distinct and peculiar Sensation attending the reception, in the eye, of wave-motion of each particular fre- quency a sensation known in each case as that of a particular Colour. Thus, when we look at a Bunsen burner, the flame of which is caused to emit a dingy-yellow light by contact with common salt, we recognise the sensation as one of yellow light. Colour is a sensation : it is not a material existence ; but the physical basis and cause of the special sensation of yellow light is, in this case, the joint simultaneous impact on the eye of two kinds of ether-waves, which have the respective frequencies of 508,905810,000000 and 510,604000,000000 per second, or the re- spective wave-lengths in air of 5895 and 5889-04 tenth-metres. Either of these trains of waves impinging singly on the eye would produce (see, however, p. 575) a sensation of yellow, the slower one giving a yellow very slightly more orange in its tint than the other does, The term " yellow light," which means primarily a certain sensation, means, secondarily, the physical cause of this sensation that is, a train of ether-waves of a par- ticular frequency. Any particular colour is best specified by a statement of the frequency of the single wave-motion, which can produce that colour when it enters the eye ; and the analogy between light of any given Colour and a sound of any given Pitch is obvious. 484 OF ETHER-WAVES. [CHAP. When there fall successively, upon the normal eye, trains of light-waves which differ only slightly in their frequency, the respective colour-sensations produced by them may resemble one another generically, though not precisely. When, in grad- ual succession, luminous waves of all possible frequencies are caused to strike the eye, we obtain in successive gradation the sensations of all the colours of the spectrum. The slowest waves which can affect the eye produce a sensation of red, those somewhat more rapid a sensation of scarlet; then in succession we find, as the frequencies increase, that the sensa- tions produced are those of orange-red, reddish-orange, orange, yellow-orange, orange-yellow, yellow, greenish-yellow, yellowish- green, green, bluish-green, greenish-blue, blue, blue-violet, vio- let. Waves of still greater rapidity than those which produce the sensation of violet are practically invisible ; but it must be admitted that they are not perfectly so. Even beyond the ordinary range of visibility some eyes are affected by ultra-violet ether-waves ; a sensation of lavender-gray colour results : a spectrum is often seen, especially if the dispersion be small, to contain three bright bands of lavender-gray in the ultra-violet region. This light is, in intensity, about l-1200th part of that which shines in the same region of the spectrum when it is rendered visible by fluorescence. Beyond the red there is, similarly, a crimson. The table, page 485, modified from Ogden Rood's Modern Chromatics and Lord Kelvin's Royal Instit. Lecture, Feb. 2, 1883, gives the frequencies and the wave-lengths in air of the several undulations which correspond to the several leading colours of the spectrum, and to some of the so-called Fraun- hofer Lines. When a source of light is receding from the eye, fewer waves per second strike the eye ; the light approximates towards red. Conversely, the light of an approaching luminous object is, as it were, sharpened in pitch. The characteristic lines in the spectrum are thus somewhat displaced ; and by this application of Db'ppler's principle, the speed of relative approach or recession of the earth and many fixed stars has been estimated. That which we call white light is, in the state in which we receive it from such a body as a white-hot bar of iron or, per- haps in its purest form, from the crater of the positive pole of the -electric arc, a mixture of long and short waves; waves of all periods within the range of visibility are either continuously present, or, if absent for a time, are absent in such feeble propor- tions or for such short intervals that they are not appreciably missed by the eye. White light of this kind is comparable to an XV.] COLOUR. 485 utterly-discordant chaos of sound of every audible pitch ; such a noise would produce no distinct impression of pitch of any kind; and so white light is un coloured. Frequencies. Wave-lengths in centimetres. (Angstrom.) Line A Centre of red ..... 395,000000,000000 429,400000,000000 00007604 00007000 Line B 437,300000,000000 00006867 457,700000,000000 00006562 Centre of orange-red .... Centre of orange Line D 1 484,000000,000000 503,300000,000000 508,905810,000000 00006208 00005972 00005895 Line D 2 Centre of orange-yellow 510,604000,000000 511,200000,000000 517,500000,000000 00005889 00005879 00005808 Centre of green ..... Line E 570,200000,000000 570,500000,000000 580,000000,000000 00005271 00005269 00005183 Centre of blue-green .... Centre of cyan-blue .... Line F 591,400000,000000 606,000000,000000 617,900000,000000 635,200000,000000 00005082 00004960 00004861 00004732 Centre of violet-blue .... Line G ...... Centre of puce-violet . Line H 1 Line H 2 ...... 685,800000,000000 697,300000,000000 740,500000,000000 756,900000,000000 763,600000,000000 00004383 00004307 00004059 00003968 00003933 If a parallel beam of light of one kind, one wave-length, one colour, homogeneous or monochromatic light, be caused to pass through a slit in an opaque screen, it may be received upon a white screen, and it will cast upon that screen a coloured image of the slit. If the light, on issuing from the slit, instead of being received directly upon a screen, be made to pass through a glass prism, the narrow edge of which is held parallel to the slit, it will be refracted by that prism, and the image of the slit will now be found in a new position on the screen. If a beam of white light be so dealt with, a number of coloured images of the slit will be formed, each in its proper place on the screen, each image overlapping its neighbour if the slit be of appreciable width; there will thus be formed a many- coloured band of light, in which the colours are marshalled in the order of the frequency of their waves, the slpwest waves, the red, being least refracted by the glass prism ; the quickest 486 OF ETHER-WAVES. [CHAP. waves, the violet, being most refracted. This is the spectrum : every component of the original white-light is displayed in the spectrum, each in its distinct place ; and thus the prism fur- nishes us with a means of analysing light that is, of finding what its components are. But the spectrum extends be}^ond the visible part of it ; the more rapid invisible rays, being more refrangible than the violet, form an invisible part an ultra-violet region which we detect by the phenomena of fluorescence (p. 504), or by casting the whole spectrum upon a sensitive photographic-plate, upon which we afterwards find a record of a region of the spec- trum invisible to the eye ; and the slower dark-heat rays form an invisible part of the spectrum beyond the red, the heat spectrum or ultra-red region, not visible, but demonstrable by means of any apparatus, such as a thermometer or a thermo- pile (Fig. 212), which is sensitive to heat. If the prism used be made of quartz, or if the spectrum be produced by reflexion from a diffraction-grating (p. 549), it will be found that the ultra-violet region is, if the light analysed be that of the elec- tric arc, from six to eight times as long as the whole of the visible part of the spectrum; while if the prism used be of glass, it absorbs to a remarkable degree these rapid ultra-violet waves. If the light analysed be that of the sun, the ultra-violet part of the spectrum is comparatively very short, on account of absorption by the atmosphere. This effect of the atmosphere is of extreme importance. Sunlight is originally bright blue, and is extremely rich in the more refrangible rays, but filtration through two absorbent atmospheres that of the sun and that of the earth renders it a yellowish-white (Langley). The ultra-violet part of the spectrum is enormously brighter at high altitudes. Compound Coloured-Light. Let us now cast a beam of sunshine or of electric light, shining through a slit in an opaque screen, upon a piece of greenish-blue glass, and receive upon a white screen the light which passes through this coloured glass: by the aid of a lens we may obtain a greenish-blue image of the slit upon the screen. So far as we have yet learned, such coloured light, whatever be the mechanism of its production, is a single kind of light perhaps due to waves of only a single frequency: whether this be so in the particular case may be tested by interposing a prism in the path of the coloured beam of light: if the greenish-blue light be homogeneous, we shall again have on the screen an image of the slit, altered in position, xv.] COLOUR. 487 but not in colour. This is not what we find : a short and imper- fect spectrum is produced ; the transmitted greenish-blue light is analysed by the prism into green light, blue light, yellow light, with perhaps some other colours, more or less faintly represented. This phenomenon is very singular. It shows that two widely-differing physical causes are capable of producing exactly the same colour-sensation : the one being, as we have already seen, the impact of ether-waves of a single definite frequency, the other being the joint impact on the retina of a number of wave-systems, each of which is capable, if it were to act inde- pendently, of producing a distinct sensation ; and the colour-sensation which is produced by the joint action of these wave-systems may differ from that which characterises any one of them. It is as if a listener to concerted music were to hear the strains of an orchestra compounded into some sort of loud melody of average pitch, he being wholly unable, by his unaided ear, to recognise the really compound nature of the sound heard by him. Then, whether the instruments all played in unison or diverged into pre- calculated harmony, the effect on his ear might remain the same. Further, many such mixtures may produce the same apparently simple sensation ; and, accordingly, such a phrase as " green light " or " orange light " is perfectly vague, unless it be accompanied by a specification of its physical cause. Complementary Colours. The greenish-blue glass, in the instance just alluded to, has in whole or in part prevented the transmission of violet light, of red, of orange, and of other kinds of light which are present in white sunlight ; the complex of undulations thus denied transmission would, if collectively allowed to impinge on the eye, have produced a single sensation of red light. If this compound red-light had not been obstructed by the coloured glass, the transmitted beam would have been white ; this compound red-light thus obstructed by the greenish glass, and the compound greenish-light transmitted by it, will pass together through a piece of clear glass, and will together produce the sensation of white light. To the eye it is a matter of indifference whether the red or the greenish light be monochromatic or compound ; monochromatic red-light and monochromatic greenish-blue light, allowed to fall upon the same spot in the eye', will mingle, and, if they be of the proper hue, will produce the compound sensation of white light. These colours, red and greenish-blue, each of the proper hue, are thus complementary to one another ; together they make up white light. The following pairs of colours are, among others, thus complementary to one another : Red and a very greenish blue, orange and cyan-blue (a rather greenish blue), yellow and ultramarine blue, greenish-yellow and 488 OF ETHER-WAVES. [CHAP. violet, green and " purple," the latter being a colour not in the spectrum, but formed by the superposition of blue and red. The expression " white light," standing alone, is thus also wholly vague ; physiologically it means light which produces the sensation of white ; physically it may mean (1) a mixture of all possible light-waves, long and short, in certain proportions ; or (2) a mixture of two complementary simple colours ; (3) a mix- ture of two complementary compound colours ; or (4) a simple colour blended with a complementary compound one of any degree of complexity. The white light of sunlight at sea-level is made up (Vierordt and Rood) by a mixture ( = 1000) of the following coloured lights : Red, 54 ; Orange- red, 140; Orange, 80; Orange-yellow, 114; Yellow, 54; Greenish-yellow, 206 ; Yellowish-green, 121 ; Green and blue-green, 134 ; Cyan-blue, 32 ; Blue, 40 ; Ultramarine and blue-violet, 20 ; Violet, 5. RADIATIONS OF A HOT BODY. The hotter a body, the greater the intensity of the aggre- gate disturbance which it sets up in the Ether ; and further, the greater the frequency of the most rapid components of that dis- turbance. A white-hot iron ball is visible in a dark room ; it emits dark heat-rays, light-rays, and also the rapid ultra-violet rays : it can be seen and photographed, and its warmth can be felt at a distance. If it be intensely hot it may emit so great a proportion of violet and blue light that it appears bluish ; it is " blue-hot." As it cools down, the more rapid vibrations die away ; the ultra-violet waves cease to be formed ; the mass becomes some- what less easy to photograph by its own light. Gradually the violet rays cease to be emitted; the light radiated is now apparently tinged with yellow : the apparent colour becomes orange, then red ; a body at a red-heat is difficult to photograph, though it continues perfectly visible in the dark. When its temperature sinks to a point below 525 C., it ceases to radiate light and becomes invisible in the dark ; it continues, however, to radiate heat, as may be felt for some time by the cooler hand placed near it. H. F. Weber points out that platinum at 390 C., gold at 417 C., and iron (not quite free from rust) at 377 C., become faintly visible, first fog- gray, then ash-gray, then yellowish-gray, then faintly red, then red-hot, and so on. xv.] RADIATIONS OF A HOT BODY. 489 The luminous radiations of an Argand oil-lamp are \ % of the whole : of a gas-flame, 0-3 to 1; a Welsbach incandescent gas-lamp, 1; an electric glow-lamp, 5-6 ; a small electric arc, 5-10 ; a 5000-candle arc, at 3000 C., 25%. Of the solar radiation, 25 % is luminous (Sir C. W. Siemens). It never ceases to radiate heat; it could not cease to do so unless it were cooled down to absolute zero. Since the molecules of all bodies are in repeated collision with their fel- low-molecules, as they rebound at each collision, they shiver and they vibrate. They must therefore continuously originate ether-waves waves which, when the temperature of the body is below 525 C., are too slow to affect the eye. Exchange of Radiations. Two bodies placed opposite to one another, with intervening Ether, of which we cannot get rid, and with or without intervening air, may present the two fol- lowing cases : 1. Both may be of the same temperature, in which case the one loses by imparting to the other exactly as much energy as it takes up from those ether- waves which strike it, having been origi- nated by the other hot body; whence two bodies equally hot exchange their energies by radiation, but do this to an equal ex- tent, and there is thus no change in their relative temperatures. 2. The one may, on the other hand, be hotter than the other. The hotter body sets up a more vehement system of ether-waves than the colder one can ; in doing this it expends its energy to a greater extent than the colder one does ; the hotter loses more energy than it gains ; the colder gains more than it loses ; in course of time their energies, and therefore their temperatures, become equal : when the temperatures have become equal, though the two bodies still go on imparting energy to each other, neither profits by the exchange, and their temperatures remain relatively equal. The absolute amount of radiation of energy from a body does not depend on the condition or even on the presence of surrounding objects, but solely on the condition of the body itself. It is easy to see that the absolute physical brightness of the sun or of a candle is at any moment independent of the presence of illuminated objects ; it is not, however, at first sight so clear that not only does a fire warm the walls of a room, but these walls also warm the fire ; that the sun warms the earth while the earth to a lesser extent, it is true warms the sun ; and that the warming of a colder body by a hotter one depends upon the difference of two similar but unequally-opposed actions. 490 OF ETHER-WAVES. [CHAP. When a lump of ice is placed near an object at the ordinary temperature, that object is cooled ; it loses to the ice more heat than it gets from the ice : the ice apparently radiates cold. When one body is surrounded by another, the body enclosed and the walls of the enclosure come to have the same temper- ature, if they be relatively at rest. A thermometer whose bulb is immersed in a cavity will come to indicate the temperature of the walls of the cavity, whether it be in contact with them or not. This equalisation of temperature by radiation is quite inde- pendent of the form of the walls of the cavity; a cavity of any form acts in the same way as a spherical cavity would do. In Fig. 154 the irregular hollow body ABC sur- rounds a body E ; both E and ABC assume after some time a common temperature, and remain at an equal temperature. The irregular hollow body ABC might be replaced by the hollow spherical- body FGH, or by the hollow sphere KLM, or any other hollow sphere concentric with these. From this ensue the following propositions. 1. The amount of energy received by a receiving surface per unit of its area the amount of heat received, the brightness of light there varies inversely as the square of the distance from the source of radiation. The advantage of extensive surface possessed by the larger sphere KLM is ex- actly neutralised by its disadvantage of distance ; its surface is greater, the radiation received by it per unit of area is less, both in the ratio of the squares of the radii, and the total radiation received by it is the same, whatever be its radius. A candle at a distance of 1 foot can illuminate a printed page as brightly as a 25-candle gas-burner at a distance of 5 feet. A bright wall is equally bright at all distances when looked at through a narrow conical tube. Close at hand it appears brighter, area for area, but less of it can be seen; at a distance it appears dimmer, but more of its surface can be seen ; in all cases the amount of light falling on a given area of the retina is the same. 2. When a plane wave whose area is AB strikes squarely and simul- taneously all parts of a surface whose area is also AB the normals to the wave being also normals to the receiving surface the receiving surface receives a certain number of units of energy per second. In Fig. 155 AB is a hot or bright body radiating ether-waves towards CD ; CD receives e units of energy per second per unit of its area. If the receiving surface be tilted, say into the position DE, the wave- front, striking obliquely, is now able to cover the larger surface DE ; no more of the wave can now reach DE than would previously have reached XV.] EXCHANGE OF RADIATIONS. 491 CD. The same quantity of energy is thus distributed over a larger surface : the quantity of energy communicated to it per unit of its area is dimin- ished in the ratio or is equal to (e x cos CDE) units. In accordance with this, the intensity of sunlight at noon is greater than during the earlier and later portions of the day, when the surface of the earth is presented obliquely to the sun's radiation. Fig.155. c 3. Let AB receive energy from CD or DE ; then, whether the surface be the smaller CD vertically facing it, or the larger DE arranged obliquely, is a matter of indifference ; in either case there will be radiated towards AB the same amount of energy. DE therefore radiates towards AB, in the direction DB, less energy per unit of its surface than CD does in that direc- tion when equally heated, and that in the ratio of cos CDE : 1. Were this not so, and did a hot surface radiate equally in all directions, then a body placed within a hot enclosure might become hotter than the walls of that enclosure. This principle explains the apparent uniformity of brightness of the sun's disc. Towards the margin of the sun's apparent disc, areas which seem equal to similar areas near the centre are in reality much larger ; but we see them obliquely ; their larger superficial area exactly compensates the effect of their oblique aspect. 4. Radiation reflected from a mirror to a focus can never make an object placed at the focus radiate more energy per sq. cm. than the Source does ; the temperature of the object cannot exceed that of the Source ; but the object may, if sufficiently small, come to the same temperature as the source, after which there is between it and the source an equilibrium of radiation. Whence a thin wire in the focus of a very large mirror in sun- light ought (atmospheric absorption, etc., apart) to come up to the Sun's Temperature (3000 C., Siemens, but this is apparently too low a figure), but not to exceed it. Ericsson's Sun-motor is practically a huge parabolic mirror, in whose focus a high temperature is attained, which is utilised by an engine. The law just stated that bodies are always radiating and receiving energy that the amount of radiation depends on the temperature of the radiating body that at constant tempera- tures bodies radiate as much energy as they receive is known as Pre vest's Law of Exchanges. From this it follows that good radiators are good absorbents; and conversely, good absorbents are good radiators. If a hot- water vessel be intended to retain its heat for a comparatively long period in the open air, it must be polished externally ; a polished sur- face, being a good reflector, is a bad absorbent, and is therefore a bad radiator; while a blackened surface, being a good absorbent, is a good 492 OF ETHER-WAVES. [CHAP. radiator, and heat is with comparative rapidity lost through a coating of lampblack, provided that it be not so thick as to impede conduction of heat to the surface. Prevost's Law is not only true of the aggregate energy gained or lost by a body through radiation ; it is also true, as Balfour Stewart pointed out, with regard to each particular form or kind of radiation by means of which energy may be conveyed between neighbouring objects. If a piece of yellow glass be placed within a hot shell of iron, the glass and the iron may both shine by their own light, and the glass may be looked at through a minute aperture in the wall of the hollow shell. Yellow glass absorbs ultramarine light, and a white-hot object, looked at through it, appears yellow, provided that the glass be colder than the source of the white light ; but when the yellow glass is itself as hot as the source of white light, as it must be in this instance, in which we look through the white-hot glass at the white-hot wall of the iron shell, the glass seems per- fectly transparent to the whole white light, a phenomenon which may be interpreted as showing that while the glass only transmits yellow, it itself radiates blue light ; the aggregate radiations, the transmitted yellow and the radiated blue, produce in the eye an aggregate effect of pure white. If the yellow glass be hotter than the source of light behind it, it seems rela- tively blue. The conclusion is, that as yellow glass absorbs blue light, so when itself heated it radiates blue light. Stokes' s Law. A body which absorbs any particular kind of radiation will in general, when heated, become a source of radiation of the same kind ; just as a resonator will, when it vibrates, impart to the air the same kind of sound of which it may rob the air when it, the resonator, is relatively at rest. If a screen of strings tuned, say to the note of a, be arranged between a sounding a organ-pipe and a listener, the latter will hear comparatively little of the sound produced by the pipe ; by resonance the strings have taken up the energy, and have con- verted part of it into Heat. If a mixed sound were produced on the farther side of such a screen, the sound of a would not be transmitted to the listener ; the rest of the mixed sound would be heard by him. When mixed ether-waves strike a system of molecules of which some are tuned to particular frequencies, those molecules will take up the energy of vibrations of those frequencies : the body will appear to be opaque to the corresponding waves. From the reciprocity of absorption and radiation it follows that if a given substance be divided into portions, of which the one, A, is hot, while the other, B, is comparatively cool, radia- tions from A will be absorbed by B ; the cooler portion, B, is xv.] STOKES'S LAW. 493 opaque to radiations from the hotter portion, A. Thus, if car- bonic oxide be burned, its flame contains hot carbonic-acid; the radiations from such a flame cannot pass through pure, com- paratively cool carbonic-acid, and are checked in very large pro- portion by air containing even a very small percentage of that gas or, curiously, of CS 2 -vapour. A hydrogen flame contains hot aqueous-vapour ; the heat radiated from this very slow dark heat-waves cannot pass through comparatively cool aqueous-vapour : the result is, as Prof. Tyndall showed, that while the sun's light and heat can reach the earth's surface through the humid atmosphere, their effect is to warm the earth and cause it to produce slow waves of dark heat; these resemble in frequency the waves produced by hot aqueous- vapour in a hydrogen flame, and they cannot pass away through the aque- ous vapour of the atmosphere. The atmosphere thus acts as a kind of heat-trap, and the surface of the earth is preserved from extremes of cold produced by excessive radiation. Prof. S. P. Langley has shown that, besides this, the atmosphere is itself, independently of aqueous vapour, remarkably opaque to certain heat-waves of great length, which are radi- ated outwards from the soil, but which, being absorbed by the atmosphere and spent in warming it, are trapped by it ; that these same heat-rays, on their way from the sun, are absorbed by our atmosphere and never actu- ally reach the earth ; and that were it not for the atmosphere the earth's temperature would be below 200 C., even under the vertical rays of a tropical sun. Thick glass has also a remarkable effect of this kind. Burning sodium-vapour emits a particular yellow light ; if looked at through a mass of sodium-vapour, it can hardly be seen ; sodium-vapour absorbs the light given out by hotter sodium-vapour. Even though light, of that particular kind, do not happen, in any particular instance, to have been emitted by burning sodium, if the attempt be made to transmit it through sodium-vapour the sodium-vapour will be found opaque to that kind of light. If an electric lamp produce a beam of light which contains amongst others this particular kind of light, and if a spirit lamp have salt (NaCl) or, better, NaBr, placed in its wick so that it gives out this particular yellow light (this denoting that the spirit-lamp flame contains incandescent sodium-vapour) ; and if the electric arc be looked at through the spirit-lamp flame, then the colour of its light would appear, if the eye were suffi- ciently sensitive, to be altered ; it is bluer ; the sodium-yellow light of the electric arc is absorbed as it passes through the com- paratively cool spirit-lamp flame, which, by its own compara- tively-feeble radiation, does not repair the damage done by it, and the light which has passed through the spirit-flame is com- paratively (not absolutely) wanting in that particular kind of yellow. The beam may be made, after passing through the 494 OF ETHER-WAVES. [CHAP. sodium-vapour, to traverse a slit and a prism, and thus to form a spectrum on a screen. It will be found, if this be done, that the spectrum is discontinuous ; at the place where the particular yellow light ought to have been found, and would have been found had no spirit-lamp flame intervened, we find a dark line a dark image of the slit, which, if the slit be fine and the focussing accurate, is found to be a double line ; a line not abso- lutely lightless, but shining with the comparatively -feeble rays of the spirit-lamp, and therefore dark in comparison with its envi- ronment. If the temperature of the spirit-flame be increased, the dark line brightens up; if the temperature of the absorber be equal to that of the source, there is no dark line ; if the temper- ature of the absorber be higher than that of the source, more of the particular light is emitted than is absorbed by it, and the line is relatively bright. The prism, which resolves any com- pound light into differently-coloured linear images of a slit, images which stand side by side so closely as to blend into one another, but any defect or redundancy of brightness in any one or in any group of which can be at once detected, offers a more delicate means of investigation than the eye can afford. In Spectrum Analysis a prism or a diffraction-grating is used, to disperse into a spectrum the light which passes through a nar- row slit from a luminous body; by inspection of the spectrum we can at once see what kinds of light are emitted, and what kinds are not emitted, by a luminous body. But the kinds of light emitted by incandescent substances are generally (since they depend on the vibrational frequencies of the molecules of the substances) distinctively characteristic of each chemical element, and, to a certain extent, of each physical state of each degree of temperature and even of the chemical consti- tution of the incandescent substance. The spectrum of the limelight is continuous ; that of the sun is not. It presents dark lines; among others, the double sodium-line: the' presence of this indicates a bright central source of light, a hot region of the sun's atmosphere, containing incandescent sodium-vapour, the light from which is absorbed by the cooler sodium-vapour in the upper and cooler regions of the same atmosphere. These lines, discovered, by Fraun- hof er, and named after him, are distinguished by letters; and the best-marked of the numerous Fraunhofer-lines are known as A, B, and C in the red, D (a double line) on the orange side of yellow, E in the green, F in the blue, G at the beginning and xv.] SPECTRUM ANALYSIS. 495 Hj and H 2 near the end of the violet. The position of any col- our is often roughly specified by stating its proximity to one or other of these Fraunhofer-lines. The lines C, F, and G pertain to hydrogen, the double line D to sodium, and E to iron. The vapour of Helium, an element found in solar eruptions, and whose spectrum is a single line D 3 , a little to the violet side of the sodium-lines Dj and D 2 , seems to have no absorptive power ; this is perfectly exceptional. Particular wave-lengths do not occur among the radiations from ice; whence ice presents a dark-heat spectrum with invisible dark lines. When the body radiating energy consists of a gas, each mole- cule, as it proceeds in its free path, executes free vibrations, like a vibrating tuning-fork thrown through the air ; and the mass thus vibrating may impress upon the Ether only one kind of vibration, or perhaps, if the structure of the vibrating molecule be complex, a large though not an indefinite number of simul- taneous oscillations whose frequencies may or may not be com- mensurable. Thus a white-hot vapour may emit only a few distinct kinds of light, and may produce a line-spectrum a spectrum consisting of a few isolated, linear, diversely-coloured images of the slit. A very rare gas may emit very little heat or light, even at such temper- atures as 1500 C. : gases are bad radiators. The outer shell of a flame is non-luminous, and may sometimes cause the formation of dark axes in the bright lines of the spectrum. Vapours are nearer their points of liquefac- tion than gases are, and are better radiators. When the particles are so close together as to have no free path, or but a small one, they very frequently collide and rebound, and thus vibrate in an irregular manner ; no rate of vibration is long enough absent for the eye to detect its absence. From the radiations of an incandescent solid or liquid, no kind of radiation appears to be absent, up to the most rapid which is given out by the incandescent body ; and the spectrum of such a body is continuous, so far as it extends. It is not, however, necessarily equally bright throughout ; didymium and erbium oxides give well-marked bright bands in the spectrum of the light which they emit while incandescent. If a heated gas or vapour be compressed, the shocks between its molecules become proportionately more numerous : if its tem- perature be increased, the energy of each shock becomes greater ; in either of these cases the vibrations of the molecules tend towards irregularity and complexity; and there may, in addi- tion to the main free-vibration of the molecules which is well- 496 OF ETHER-WAVES. [CHAP. marked if there be any appreciable free path be a number of additional vibrations of all or of many frequencies : a condition which is indicated by the broadening of the lines in a linear spectrum of a gas into the bands of a band-spectrum. As pressure is relieved, the spectrum merges into that of an ordi- nary incandescent gas, or, on the other hand, as the pressure is increased, into the continuous spectrum of an incandescent liquid or solid. Even the flame of hydrogen, at high pressure, is luminous, with a continuous spectrum (Lockyer). Continuity between the gaseous or vapourous and the liquid states is thus indicated on an independent ground. Light from incandescent solids or liquids travels from some distance within the surface ; for it is polarised at right angles to the plane of inci- dence ; this shows that it has been refracted on its outward passage through the surface of the incandescent body and into the rarer surrounding medium. Light from incandescent gases is not polarised ; sunlight is not polarised ; hence sunlight is due to incandescent gas or vapour. Variations in the light emitted by one and the same sub- stance under different conditions, and therefore in the spectrum of that light, serve to indicate molecular changes in the substance which radiates light. Salts, if undissociated, have a different mode of vibration, and therefore a different spectrum, from their component elements ; heat, or, if ordinary heat fail, the extremely high temperature produced by a discharge of high-tension elec- tricity will break them up into their elements. Even the ele- ments are reduced to comparatively-simple forms of aggregation by high temperatures ; their continuous spectrum breaks up into one of bands ; a still higher temperature, such as that of a high-tension electric spark if other means fail, converts the spectrum into a line spectrum, the line spectra being perhaps due to atoms, the band spectra to molecules. The spectra of the same substance at different temperatures are often remark- ably dissimilar. At temperatures beyond our reach, such as those of some of the fixed stars, or the lower levels of the sun's atmosphere the high temperature of which may be inferred from the great amount of the highly-refrangible rays emitted by them the elements themselves appear to be broken up and reduced to simpler forms of matter. This lends probability to the belief that the various elements are modifications of one kind of mat- ter a belief somewhat strengthened by numerous coincidences between the lines of the spectra of different elements. xv.] TRANSMISSION, REFLEXION, AND ABSORPTION. 497 TRANSMISSION, REFLEXION, AND ABSORPTION. When ether-waves fall upon a transparent body, they pass through it: they are propagated through the ether which lies between the molecules. When a body is thus pervious to light it is specially said to be transparent; when pervious to dark heat, as rock-salt is, it is said to be diathermanous, ho special term being used to denote transparency to actinic radia- tion. A body impervious to light is opaque; one impervious to dark heat is adiathermanous. A perfectly-transparent body is invisible. Colourless thin glass, with a dustless, polished, clean surface, approaches this character : objects are seen beyond it, and, as we say, through it : they appear, if the glass be thin, inappreciably distorted. Light may be reflected from the polished surface of glass, and the pres- ence of the glass may thus be rendered manifest to one standing in a particular position ; the sun shining on the windows of a distant house makes the window-glass visible. Glass is almost invisible in a mixture of 6 vols. essence of cloves and 1 vol. essence of turpentine. If glass be roughened at its surface, it presents numerous facets which reflect light so as to make the glass visible in all directions ; and light passing through it is irregularly turned out of its path in all directions ; objects beyond cannot be seen dis- tinctly, though light can pass through the whole mass, and rough- ened glass, though not perfectly transparent, is translucent. When glass is powdered, the powder presents so many facets and reflects so often the light which falls upon it that the whole is practically opaque: it is a powder which reflects in every direction the light incident upon it in white light a white powder, in red light a red powder. When ether-waves of any kind impinge upon a body impervi- ous to them their progress is arrested ; in part they are reflected or scattered; in part they are absorbed by the impervious body ; the Ether loses energy, ordinary matter gains it, and the impervious body is heated to an extent corresponding with the amount of energy absorbed this heat being first communi- cated to the superficial layer of the body. If ether-waves impinge upon a body which is transparent and diathermanous, that body is not heated, for the ether- waves pass through it and are not absorbed. Thus, clear moun- 2K 498 OF ETHER-WAVES. [CHAP. tain air is not heated by the sunshine which streams through it ; in the shade it may be very cold. Sunshine may stream through clear ice, or even through hoar-frost, without melting it. If there be any particles of dust in the air or in the ice, these, being opaque, will become heated, and the air is then, by con- duction, rendered warm, or the ice is melted. Some bodies are impervious to all kinds of radiation ; others, having a power of Selective Absorption, are impervious to some kinds only. Thus radiant heat can pass, while the more rapid light-waves cannot pass, through a thin piece of black vulcanite, or through a strong solution of iodine in bisulphide of carbon : while a crystal of alum is, on the other hand, transparent to light, but is almost adiathermanous, impervious to heat-rays. Lampblack, again, is very transparent to the slowest heat-waves, and air very opaque to some of them. A soap-bubble film is remarkably adiathermanous, cutting off about half the heat of an incident beam. Glass is transparent and diathermanous, but is somewhat opaque to the ultra-violet rapid ether-waves ; a quartz prism or lens allows a great amount of ultra-violet radiation to pass through it which a glass prism or lens would extinguish. On the other hand, very long ether-waves go readily through a stone wall; and silver-leaf, just thick enough to be opaque, transmits ultra-violet rays. The absorptive power of a substance may not be so extensive as to enable it to absorb and extinguish light-rays or heat-rays of all kinds ; it may arrest some only. A piece of green glass can only allow a certain number of kinds of light to pass through it ; by their joint impact on the retina these produce the sensation of green. Sunlight contains other waves than these ; they have been absorbed; the green glass is opaque to them. These waves would together have produced a sensation of purple-coloured light. If this purple light had alone fallen upon the green glass, it would riot have been transmitted ; the glass would have appeared to be opaque. When sunlight is directed first through purple glass and then through green, the eye perceives black- ness : the two pieces of glass are together opaque, though each of them is transparent to its own kind of light. Very dark-red glass and green glass together produce a similar effect of blackness : pale-red glass allows some green light to traverse it, and so, when it is combined with green glass, the result is dark-green light. Nickel nitrate absorbs red and violet, and is therefore green when in solution. Cobalt solution is red. A mixture of strong solutions of the two metals is black : diluted it becomes, however, almost colourless. Copper, when it receives the impact of white light, emits orange light, together with superficially-reflected white light. Electrically-deposited cop- xv.] ABSORPTION. 499 per, while immersed in a solution of sulphate of copper, which does not allow the transmission of orange light, looks as white as plaster-of-Paris does in the same liquid. The colour of a coloured object, as seen by transmitted light, is produced by subtraction of the light absorbed from the light incident upon the object. The kind of light transmitted may vary with the thick- ness of the absorbing medium. In such a case the medium is said to be dichromatic, or dichroic. A solution of chloride of chromium, in a thin layer, absorbs much yellow, orange, and yellowish-green light ; in a thicker layer it absorbs all but the red and some green and blue ; in a still thicker layer the only colour transmitted is red. Thus a wedge-shaped layer of this solution appears to vary in colour, according to the thickness, from a greenish-blue, through purple, to red. Chlorophyll appears green in thin layers, red in thick. Iodine vapour trans- mits a blue group and a red group, as also ultra-violet rays ; together these produce an impression of purple : in thicker layers the blue rays alone are transmitted, and the vapour appears blue. Each wave-length has its own Coefficient of Transmission through each transparent substance ; if half the intensity be lost by transmission through a layer 1 cm. in thickness, the proportion actually transmitted is 50 per cent., and the coefficient of transmission is 0-50. The next cm. of thick- ness will transmit 0-50 x 0-50 = 0-25 ; the tenth cm. will only transmit 0-50 10 = 0-00098 times the original intensity. Small differences in the coeffi- cients for the several wave-lengths make considerable differences in the composition of the aggregate light transmitted through a thick layer of a selectively-absorbent substance. When a strong solution of blood is interposed in the path of a beam of light, no light but red is transmitted; dilute the solution gradually, and successively the solution appears more and more yellowish, and of increas- ingly paler hue. The special absorptions, of absorbent bodies are most thoroughly studied, not by means of their visible colours, but by the prismatic analysis of the light which passes through them. It is then found that some substances absorb several distinct kinds of light, belonging to different regions of the spectrum. Transparent coloured-objects, through which light is filtered, give dark bands across the spectrum the so-called " Absorp- tion-bands, " which indicate what kind of light has been stopped and extinguished by the absorbent object. These bands vary in breadth with the degree of concentration of the absorbent solu- tion employed, and they vary in position with its nature. 500 OF ETHER-WAVES. [CHAP. When a strong solution of blood is interposed in the path of a beam of light which is on its way to form a spectrum on a screen, all the spectrum, with the exception of the red part of it, disappears. As the liquid is diluted the spectrum lengthens out : orange, yellows, greens, blues, are successively added ; but there always remain two relatively-dark absorption-bands in the spectrum, in the yellow and in the green, between the Fraunhofer-lines known as D and E. If the blood be treated with sulphide of ammonium, it will be reduced ; its oxyhsemoglobin will become reduced haemoglobin ; the chemical constitu- tion changes, and with it the absorbent power ; the absorption-band is now a single band placed between the two preceding. If absorption-bands be numerous and pretty uniformly distributed throughout the spectrum, or if they be in complementary regions, the absorbent substance may present no distinctive colour, e.g., benzene. On the evidence of absorption-bands Major Abney has brought to light the existence of traces of benzene vapour between the earth and the sun, and Prof. Langley has shown that there are very peculiar gaps in the heat-spectrum, which are probably due to absorption by the upper regions of the solar atmosphere. The kind of light absorbed by a body may also vary with its molecular constitution. It is supposed (von Helmholtz) that each absorption depends on the pres- ence of a particular kind of molecule, differing from the simple chemical molecule. Chlorine has many absorption-bands in its spectrum, and it must either arrange itself in many kinds of molecules, or else its ordinary mole- cules, considered as vibrating bodies, must be extremely complex, and have many free periods of vibration. By changes in the absorption-bands we may learn that sub- stances change their molecular constitution when heated. Iodine vapour gives an extensive absorption ; when highly heated, the absorption-spectrum becomes reduced to a few bands ; when the vapour is still more highly heated, some of the absorption-bands disappear, and one of them is replaced by a group of fine lines. Sulphur-vapour changes its absorption-spectrum when its density changes at 1000 C. ; N 2 O 4 , when it becomes NO 2 , changes its spectrum, though it does not do so when it becomes a liquid ; iodine, on the other hand, when dissolved in carbon disulphide, has the same absorption-spectrum as when it is in the state of vapour. In a red solution of cobalt the chloride, for example when heat is applied to it, the salt enters into a different state of hydration ; its molecular structure is changed ; the solution becomes blue. If there be no molecular difference between a substance incandescent and the same substance absorptive of light from a hotter object, a condition probably realised in the case of didy- xv.J ABSORPTION. 501 mium, erbium, and terbium compounds, the incandescence- and the absorption-spectra will be mutually complementary ; the one presenting bright lines where the other presents dark. The Colour of a coloured object seen by reflected light is also generally due to absorption. An object seen by reflected sunlight does not appear to be coloured in any degree unless there have been absorption of some of the components of the incident white-light, and the colour of a coloured object is complementary to the colour which would have been pro- duced by these absorbed components had they jointly impinged on the eye. Some of the light incident on a piece of coloured glass is reflected at its surface ; there is no absorption ; if the incident light be white, the light reflected is also white. If a piece of green glass be laid upon black paper, and if it be looked at in such a direction that daylight is not directly reflected from it into the eye, it will be nearly invisible, and will be devoid of colour ; it will appear black. If coloured glass be ground to powder, the powder is white ; white light is reflected at every facet, while the light reflected from the lower surfaces of the fragments, and again issuing into the air, has nowhere traversed a layer of sufficient thickness to cause the extinction of all the absorbable components of the incident sunlight. The finer the powder, the whiter it is ; the coarser it is, or the more energetic its absorption, the more marked is its colour. For the same reason, froth is white. If the upper surface of a sheet of green glass be ground, it will appear almost white ; if the ground sur- face be looked at through the glass, it will appear green, for the light issuing from the glass is white light, which has undergone a certain amount of absorption. If the green powder be immersed in water or oil, there is less superficial reflexion at the several facets ; there is deeper penetration of the light into the mass, and consequently more absorption ; the colour appears to deepen. Hence the value of oil as a medium in painting. A solution of chloride of copper placed in a deep black- walled vessel will not appear to have any colour; it will seem black ; it reflects no light except from its surface. If powdered chalk be mixed with it, light is now reflected from the white particles of chalk, and passes out in every direction, through every part of the surface ; so much of the reflected light is absorbed that it appears green when it reaches the eye, the milky mass appears green. In a similar \vay a piece of malachite 502 OF ETHER-WAVES. [CHAP. is penetrated by light to a very small depth ; internal reflexion occurs ; absorption of all the outpassing light takes place, with the exception of certain kinds, which jointly appear green ; the malachite is green. A piece of polished gold reflects white light at its surface ; it also reflects interiorly, and from within the sub- stance of the gold at a very small depth there is reflected in all directions a quantity of light which, by absorption before reach- ing the surface, has become of an orange colour. If the layer of gold be very thin, that part of the light which would be absorbed by a thicker layer may, in part, pass through and issue into transparent media before its energy is wholly converted into heat. A thin piece of gold-leaf thus appears transparent and allows a greenish-blue kind of light to pass through it, which, if the leaf be rendered very thin by the action upon it of a solution of cyanide of potassium, may become violet, for both green and violet light then find their way through. The object-glass of an astronomical telescope may be covered with a thin layer of silver, which will reflect the heat and some of the light, allowing a pleasant greenish light, and also some actinic rays, to pass. When a beam of light enters the eye after undergoing repeated reflexion from gold to gold, it is of a deep-orange colour ; this is the true colour of gold. As we ordinarily see gold, the orange light coming from its deeper particles is mixed with much white light irregularly reflected from its sur- face. The true colour of copper is scarlet, of silver a yellowish-bronze colour, of brass a rich golden-red. By reason of such repeated reflexion, a deep metal-vase, equally polished within and without, appears to be of a much richer colour internally than it is externally, and silk-velvets appear of a richer colour than silks, for light undergoes repeated reflexions between the vertical fibres which constitute the outer aspect of the former. When there is little opportunity for reflexion from the inner particles of a body, as where light falls exceedingly obliquely upon a gold mirror from a white object and is reflected into the eye, the image of the white object in the polished gold-mirror appears not gold-coloured, but white. Some metals can be rendered transparent, not by being reduced to thin films, but by being reduced to the liquid state : potassium and sodium can be dissolved in anhydrous liquid-ammonia; the solution is blue, and the true colour of these metals is therefore a copper colour. If the incident light be already coloured, it may be that the whole of it is absorbed. An object, blue or red in daylight, if illuminated by a sodium-flame, may absorb all the light that falls upon it ; if it do so, it appears black ; a bunch of flowers, looked at in such a light, where it is riot yellow appears black ; it must either reflect some or none of the light which falls upon it. A piece of red cloth illuminated by the red regions of the spec- xv.] ABSORPTION. 503 trum glows with a bright red ; when moved into other regions it becomes black, for it absorbs the incident light. The blue colour of opalescent bodies, which in general present a multitude of reflecting particles embedded in a uniform matrix, and of which we may take as a type the sky-blue liquid obtained by adding to water a very small proportion of milk, is not primarily due to absorption. The principle is an established one (p. 523, near top), that where there is most refraction of light there is the greatest proportion of reflected light. A beam of mixed light falls upon a colourless transparent-body : all the rays are both refracted and reflected ; the blue and violet are the more sharply refracted, and a greater proportion of them is reflected than of the less- refrangible rays. Even after one reflexion the image of an object in a mirror is bluer than the object itself. After multiple reflexion, light may become distinctly blue. Multiplicity of reflexion is favoured by smallness of the individual particles. The light which is not reflected is wholly, or in part, absorbed ; the sun, looked at through a thin layer of dilute milk, appears yellow ; through a thicker layer, orange or red ; through a still thicker layer it cannot be seen. Similar phenomena are presented by water into which a little very dilute alcoholic-solution of resin or mastic has been dropped with stirring, by salt water into which a few drops of a very dilute solution of nitrate of silver have been stirred, by a thin haze, by smoke ; all these appear blue by reflected, yellow or red by transmitted light. Even the Sky itself is a haze of this kind, the air being rendered visible, against the dark background of black space, by sunlight reflected from its fine suspended dust- or water-particles ; while the light transmitted is always more or less yellowish, and, in the afternoon and evening, when sunlight comes to us through a greater thickness of the more dusty layers, verges towards orange or even red. Such a dust-haze is more opaque than adiathermanous. When the particles of a haze increase in size they jointly offer a greater resistance to the entry of light into the fog : light is reflected more promptly, and the reflected light presents a large proportion of white light. This phe- nomenon is familiar to the smoker ; the thick clouds of smoke produced by vigorous smoking are obviously different from the thin fine blue columns which ascend from a cigar laid aside for a moment. The colours of metals may be partly accounted for in a similar way. Steel and zinc have a normal refraction ; the violet is most refrangible ; they appear blue. Bell-metal, brass, Au, Cu, Ag, have abnormal dispersion; the red end is most refrangible and most reflected ; they appear red or reddish. Speculum-metal refracts red more than green, but also violet more than green; on the whole it is reddish (Jamin). Those rays which are absorbed in the greatest pro- portion by any substance are reflected by it in the least; when a beam of sunshine falls on a green leaf, the actinic rays are absorbed and spent in doing chemical work ; the light reflected from such a leaf is feeble in actinic ra}~s, and foli- age is consequently not easy to photograph. Light which is absorbed is generally converted into Heat ; this may presently be radiated away ; shorter, quicker light-waves strike the body ; longer, slower waves of dark heat leave it. 504 OF ETHER-WAVES. [CHAP. FLUORESCENCE, PHOSPHORESCENCE, AND CALORESCENCE. Fluorescence and Phosphorescence. The molecular dis- turbances of the interior particles of a body impinged upon by light may, however, give rise to other waves which are not so slow as to be invisible ; the ether-waves absorbed may thus give rise to Light. In this case the body may not only reflect light, but it may also seem to emit light from within ; it is fluo- rescent. The particles down to a very small depth, being set in agitation, originate a new set of ether-waves, which are propa- gated from each particle in every direction. The phenomena of Fluorescence maybe shown by a solution containing sesculin and fraxin, which may be very simply prepared by stirring some horse-chestnut twigs in water ; a beam of light is caused to pass through this solution, and then for some distance within the solution the liquid seems self-luminous and shines in a dark room with an opalescent shimmer along the track of the beam of light. This effect is partly due to the impact of the light rays, but is principally due to the rapid invisible ultra-violet waves. If a piece of paper be wetted with this solution, and if this paper be then used as a screen on which the image of a slit is thrown through a quartz prism, the ultra-violet part of the spectrum is rendered visible ; a compound blue light radiates from the paper over an area six or eight times as long as the ordinary visible coloured spectrum ; the light refracted by a prism may, with the same effect, fall on the walls of a glass vessel containing the fluo- rescent solution. Quinine chloride or disulphate, on paper or in solution, gives a blue light that blue which is seen about the edge of the upper surface of a solution of quinine in a phial ; petroleum or shale oil a green ; turmeric solution in alcohol, or much better in castor oil, a green; uranium compounds, especially uranium glass, a green light ; chlorophyll in solution, or lying undissolved in the cells of leaves, a red ; an alcoholic solution of soot or one of datura stramonium, a greenish blue. Among fluorescent sub- stances we find also such compounds as eosin (tetrabromofluorescein), fluo- rescein (resorcin-phthalein), anthracene, fluor-spar (especially chlorophane, which, when heated by conduction or by radiant heat, shines with an emer- ald-green light), many sulphides, especially those of barium and calcium, and, to a slight degree, the cornea and the crystalline lens, and the rods and cones of the retina. Very frequently a body goes on vibrating for some time after ether-waves have ceased to strike it ; this is familiar when the waves given out by it are Heat-waves. Sometimes, how- ever, the body thus vibrating produces Light, and such a body Balmain's luminous paint, for example which goes on visi- bly shining or fluorescing for some time after ether-waves have ceased to impinge upon it, is said to be phosphorescent. Among such bodies we find barium and calcium sulphides, diamonds, chlorophane, dry paper, silk, sugar, teeth, the alkalies and alkaline earths and their salts in general, and compounds of uranium. xv.] PHOSPHORESCENCE. 505 These substances may be placed in a Geissler tube in a dark room ; an electric current passes ; the solids commence to fluoresce in the light pro- duced by the discharge, but the observer's eyes are kept shut ; the current is stopped, and the eyes are at once opened to look at the tubes ; the solids are seen shining in the dark room. For substances the duration of whose phosphorescence is very small Becquerel's Phosphoroscope may be employed. In rapid succession a phos- phorescent body is exposed to bright light and brought against a dark back- ground before the eye of an observer situated in darkness. Most objects are found by this means to be to some extent phosphorescent ; and apparently all are markedly so when extremely cold. The compound nature of the light produced by fluorescence or by phos- phorescence can be ascertained by means of a slit and a prism. It is a very singular fact that the red rays of the spectrum and the invisible heat-rays have the effect of accelerating the exhaustion of a phos- phorescing body. If a body, phosphorescing after exposure to white light, or better, to violet and ultra-violet rays, have a spectrum instantaneously thrown upon it, the body thereafter phosphoresces more brightly in the area occupied by the ultra-red part of the spectrum ; if the exposure to the spectral image be relatively prolonged, the phosphorescence becomes exhausted in those regions on which heat-rays had fallen, and now the Fraunhofer dark lines in the invisible part of the spectrum are rendered manifest by the sur- vival of local phosphorescences in those parts of the screen which have not been affected by the impact of heat-waves (Becquerel). A similar action of these rays has been long known : they often reverse the chemical action of the actinic rays. As a rule a fluorescent or phosphorescent body emits for a longer or shorter time, on exposure to light, or, specially, on exposure to actinic rays, the same kind of light which, when light falls upon it, it absorbs ; and thus, in some instances, the light emitted by fluorescent and phosphorescent bodies presents bright bands where the absorption-spectrum of the same sub- stance presents dark bands ; but the whole series of phenomena of fluorescence is one full of anomalies ; we do not fully know the laws of the molecular groupings of different substances, simple and compound, their necessary modes of vibration, or their relations to the Ether. A mixed beam of sunlight which has passed through a fluo- rescent solution cannot affect another solution of the same kind ; fluorescent solutions rapidly absorb those rays which are the effective cause of their luminosity. We sometimes find transformation of slower waves into more rapid ones. When a solution of naphthaline-red has been shone upon by a beam of deep-red light, it emits by fluo- rescence an orange-yellow light. Chlorophyll presents an analo- gous phenomenon ; it fluoresces with a red light, even though it TOT TJITIVBRSITY 506 OF ETHER-WAVES. [CHAP. be shone upon by a slower red-light. In the case of chloro- phane, the impact of slow radiant-heat waves is competent to set up an emerald-green light. Calorescence. When a beam of light is filtered through a solution of iodine in bisulphide of carbon, so that dark heat- rays can alone pass through, these heat-rays may be brought to a focus by a lens, and absorbed by a piece of platinum placed at the focus ; this will become luminous and give rise to ether- waves of all kinds ; if its light be examined by a prism it will be found to give a continuous spectrum. This phenomenon was called by Tyndall the Calorescence of heat-rays. SOURCES OF ETHER-WAVES. Vibrations of Molecules. Light, Heat, and Chemical Radiation being primarily due to the vibration of particles of ordinary Matter in the midst of Ether, the energy of ether- waves is derived from the kinetic energy of vibrating particles ; and whatever increases the Kinetic Energy of these vibrat- ing particles increases their vibratory movement, and gives rise to increased radiation. When by any action a given amount of energy is liberated in or communicated to a system of material particles, the rapidity of their resultant vibration, and therefore that of the ether- waves set up by them, depends on the rapidity with which that action occurs. When energy is slowly imparted to or liberated among them, the vibrations of the particles may remain relatively slow, and radiant heat may alone be the result ; while if the particles be suddenly set in violent commotion, their vibration will be complex and irregular, the particles will become incandescent, and they will at once originate not only heat, but also light or even actinic waves. When a flash of lightning or an electric spark rushes through the air it jars the particles of air, and renders the air incandescent and luminous ; and it even originates actinic waves, for an electric spark can be photographed as well as seen. When the electric discharge through the air is continuous or rapidly intermittent, its light is, to the eye, apparently continuous, and we have the Electric Light. When a flint and steel are struck together, the con- cussion agitates the molecules of those particles of steel which are knocked off, and a luminous spark is produced ; so also when a b'ullet strikes a target there is a flash of light. Within a gas-flame, molecules of a hydro-carbon are robbed of part of their hydrogen by a process of destructive distillation ; the residues are heavy, almost purely-carbonaceous molecules, and these, in virtue of the energy supplied by the combustion of the hydrogen, become strongly agitated and incandescent, oscillating within the gas-flame, and therein act- xv.] VIBRATIONS OF MOLECULES. 507 ing as sources of light until the current takes them into the zone of perfect combustion in the outer region of the flame ; there they become completely oxidised into gaseous carbonic acid, and thereupon lose in great part their radiative power. The brightness of a gas-flame is favoured by external pressure, or by a relatively small internal pressure and velocity of outflow, by the long continuance of carbon particles or other solid particles (which in a candle-flame can reflect light and cast a shadow in sunlight) within the flame in which they are incandescent, and by heating the gas before it reaches the flame. When a crystal is cleft it often emits a flash of light; work is done in splitting the crystal : the energy of part of this work appears as that of ether-waves. When salts suddenly crystallise out of a liquid men- struum it not unfrequently happens that the formation of crys- tals is attended with a flash of light ; the salt leaves the water and coheres with particles of its own substance ; the agitation attending this process causes ether-waves to be set up. Even the application of 'moderate heat, falling far short of such a temperature as might produce incandescence, may cause a body to become luminous, as in the case of the fluo- rescence of fluor-spar (chlorophane) and the diamond, which shine when heat is imparted to them by conduction. We have already seen that light may result from the impact of ether- waves upon a body. Chemical union is often attended with both heat and light: as when we drop copper filings or powdered antimony into chlorine gas, or in the ordinary phenomena of combustion. Even slow combustion, such as that of eremacausis or decay, may cause light, as in the luminosity of decaying wood ; or the green luminosity visible on the surface of some fish when in a state of incipient decay; or the slow oxidation of apiece of phosphorus in the air at ordinary temperatures, or of sulphur or the metal arsenic at higher temperatures. Even during the life of organisms they may become luminous either abnor- mally, as when the skin of the human body evolves phosphu- retted hydrogen ; or. normally, as in the glow-worm, in the nocti- luca, in medusoids, and in many other invertebrate animals ; and the production of light may even be under the control of the animal, as in the fish photichthys, which can temporarily illumi- nate its prey. Light is in these cases produced at the expense of the animal heat which might otherwise have been evolved. Vibrations communicated to the Ether. Jn the cases discussed, the origin of the light plainly is in the agitation of 508 OF ETHER-WAVES. [CHAP. ordinary matter, but there is a certain deficiency of knowledge in respect of the next step in the transference of energy. How is any ether-wave set up in the Ether by the motion of any particle of ordinary matter within it ? A full answer to this question would involve a full knowledge of the constitution of the Ether, and of the relation of the Ether to the particles of ordinary matter which are embedded in it a question still under discussion. Some hold that the Ether is entirely independent of ordi- nary matter, being unaffected in density by its presence ; others hold that it is of various densities in various substances, these densities being in different transparent substances inversely proportional to the squares of the velocities of light within them. Some hold that it is so independent of ordinary matter that a moving solid body moves freely through ether like an ideal net through ideally-frictionless water ; in which case it would be difficult to understand how a vibrating molecule could set up vibrations in it. If this were so, the most rapidly-moving solid transparent object would allow the transmission of light through the ether which permeates it, as if it were itself at rest. The contrary view seems probable ; a ray of light is said to be retarded a little by being made to pass up a running stream of water ; the effect, quite perceptible in the case of water circu- lating at the comparatively-slow rate of two metres per second, is, however, imperceptible in a current of air. A beam of light was found by Fizeau to be retarded when made to pass through a cylinder of glass, rotating in such a direction that the rota- tion of the glass tended to carry back the light while in the act of passing through it. The consequence of such an adhesion between the Ether and the matter embedded in it is, that the earth must to some extent drag the Ether with it as it rolls through space ; yet Aberration (p. 511) tells somewhat against this. The whole subject is as yet one of the most recondite in Physics. PROPAGATION OF WAVES THROUGH THE ETHER. At present it is usual, in discussing the propagation of ether- waves, to assume the wave to have been effectually set up ; the wave-motion is studied as it diverges from a small wave-front formed in the immediate neighbourhood of the vibrating mole- cule ; and in discussing the transmission of ether-waves of different wave-lengths through different transparent bodies, we xv.] PROPAGATION THROUGH THE ETHER, 5Q9 shall have to take for granted that the interaction of the Ether and the ordinary Matter an action which cannot be very great, for, if it were, Transparence would be impossible is such as, in different media, unequally to retard ether- waves of different wave-lengths. This retarding effect depends somehow upon the nature of the transparent body ; and this holds good not only with regard to light in general as where a diamond is found to transmit light much more slowly than water does but also with reference to each particular kind of light. Each trans- parent substance has its own rate of transmission for ether- waves of each particular frequency; and this is found for each case only by experiment. A denser substance may some- times transmit ether-waves more rapidly than a rarer one does : light passes more rapidly through water, for example, than through alcohol or oil of turpentine. A substance through which light travels more slowly is said, however, to be optically denser. On the assumption that the density of the Ether is different in different substances, it would follow that all wave-lengths must be diminished or increased in equal proportions, that all kinds of waves must be equally retarded or accelerated, and all kinds of light, heat, or chemical rays there- fore equally refracted, on passing from one medium into another a con- clusion contradicted by the simplest experiment with a prism. Cauchy, on the arbitrary assumption that the Ether consisted of separate particles of an average size extremely minute as compared with the average distance between them, found that the amount of retardation was affected by the fre- quency of undulation (much in the same way as the speed of propagation of a wave along a cord is modified by stiffness of the cord), and that thus prismatic dispersion became explicable ; an assumption which more modern writers unwilling to admit that the Ether, which is not found to be capable of having waves of compression and rarefaction set up in it, and whose parts yet preserve or tend to preserve their mean positions, can be a fluid com- posed of separate molecules have converted by interpretation Into the following, namely, that there is some kind of discontinuity in the relations between the Ether and the ordinary matter which it permeates ; a discon- tinuity which is held to show that while the Ether may be considered to be a homogeneous jelly-like solid, which can yield to powerful stresses after the manner of a fluid, the matter, apparently homogeneous, which is embedded in it, is not truly homogeneous throughout. The index of refraction, /8, varies with the wave-length, A, being con- nected with it by the law ft = k\? + A + B/A 2 + C/A. 4 (Briot), where k, A, B, and C are constants to be determined by experiment. The Ether is analogous to a very weak solution of gelatine : to relatively- great momenta it acts as a fluid, and it closes up behind moving particles; to small stresses it acts as a solid, and it suffers tangential strain, without change of volume, under the influence of a tangential stress. Ether-vibration transverse. When any part/>f the Ether is displaced by a vibrating molecule, the displaced portion always 510 OF ETHER-WAVES. [CHAP. tends to return to its normal position ; in doing so it sets up waves. These are waves of transverse vibration, like those of an elastic string or rod plucked laterally. Some have held that the Ether is absolutely incompressible, and that it is impossible to form waves of compression in it ; according to others, waves of compression are at first formed, but very rapidly die out. If it be assumed that the Ether is analogous to an elastic solid ; that the resistance to compression, ft, is very great in comparison with the rigid- ity, n, to transverse distortion; then (Green) it can be shown that the compressional waves will travel with extreme velocity, but will die out after a few wave-lengths. In explaining Double Refraction on this basis, Green found it necessary, however, to assume that the vibrations occur parallel to the plane of polarisation, an assumption which is now considered inadmissible (p. 522) : and there are also other difficulties in working out this theory. If, on the other hand, it be assumed (Lord Rayleigh, Lord Kelvin, and Mr. Glazebrook) that the resistances of the Ether to compres- sion and distortion are the same in all media, but that in crystals the matter present acts so as to make the Ether behave as if it were itself dif- ferent in density in different crystallographic directions; that the Ether has a certain negative* resistance to compression, which means that it would dilate on pressure, arid is infinite in extent or bounded by a rigid boundary; and that the vibrations are at right angles to the plane of polarisation : then there can be no compressional or dilatational waves, and the results of mathematical calculation agree in the main with the facts, so far as these are ascertainable by experiment external to crystals, and also agree with the results deduced from Clerk Maxwell's theory. According to Clerk Maxwell's view the Ether is a homogeneous body, a non-conductor of electricity : periodic electric stresses applied to this produce waves which travel at the rate of about 300,000000 metres per second ; these waves are waves of transverse vibration, and there is no vibration longitu- dinal or normal to the wave-front. These waves, due to electric displacement, are quite competent to explain the ordinary phenomena of light, and this theory explains on mathematical grounds that absence of the normal or compressional vibration which has been a source of great perplexity in most of the mechanical theories of light. According to this view, each particle of a body through which light is shining is in rapid succession exposed to alternately-opposite electric stresses: at each half-vibration it becomes oppositely electrified ; but the ordinary effects of electricity are not generally observed when light shines through or on a body, for the electrification produced by any one half -vibration simply reverses the effect of that pro- duced by the previous half-vibration. The Velocity of propagation of ether-waves through the Ether of space is found by two astronomical methods. * Any compression-and-rarefaction waves formed will promptly die out if their velocity, \/ik ~\~ ttt) /P, be very great, or if it be very small, in comparison with the velocity, v'n/P, of transverse-distortional waves. Green assumed their velocity to be comparatively very great, whence ft would be very great in comparison with n ; the other physicists named have assumed their velocity to be comparatively very small, whence (Jt+stt) very nearly =0, and ft would be negative, and nearly equal to PL xv.] VELOCITY OF LIGHT. 511 I.Jupiter's Satellites. These pass out of sight behind the mass of Jupiter and again reappear : when the earth is nearest to Jupiter the eclipses and reappearances appear to take place 8| minutes earlier, when the earth has wheeled round to the opposite side of its orbit and is at its farthest from Jupiter 8 minutes later, than they would have appeared if the earth had been at the centre of its orbit. The suddenly commencing or ceasing light takes 16 - minutes to cross the earth's orbit, a distance of 299,270,000000 metres : it therefore travels 302,300000 metres per second. According to the latest determinations, the velocity is 299,336000 metres per second. 2. Aberration. No star is seen in its true place: every star seems to describe a little ellipse in the heavens, and seems to travel round the ellipse once a year. The reason is, that as the earth wheels onward in its orbit, bearing the observing telescope with it, rays of light, coming from distant stars, on their way down the telescope tend, short though the tele- scope tube be, to verge towards the hinder side of that tube : for which reason, in order to see the star in the centre of the field, the eyepiece must be tilted appreciably backwards in a direction opposed to that of the earth's orbital motion : the telescope, when the star is seen in the centre of its field, is therefore directed not towards the true position of the star, but towards a point in advance of it. In the course of a year, therefore, as the earth bowls round its elliptical orbit, the successive points to which it is necessary to direct the telescope are found to have been situated on the circumference of an ellipse. The size of this ellipse indicates the amount of tilting of the telescope : from this can be inferred the proportion between the length of the telescope and the distance traversed by the ocular during the time spent by the ether-waves in passing down the telescope tube ; the speed of the waves of light can be calculated from these data, and is found to be 299,300,000 metres per second. Such is the simple theory of Aberration : but the amount of aberration is the same whatever be the transparent medium e.g., water with which the telescope is filled. Hence it would appear that a diminution of the relative motion of the earth and the ether exists, and may be explained by assuming that the water carries the contained ether, wholly or partly, along with it. This is confirmed by Fizeau's experiments on the bodily transfer- ence of ether-waves in a stream of water, like that of sound-waves by wind ; but Michelson and Morley find that the earth as a whole drags the surround- ing ether with it in a way which is difficult to reconcile numerically with the ordinary theory of Aberration. Do waves of different frequencies travel through the Ether of space at the same or at different rates ? If their rates were different, then a suddenly-appearing satellite of Jupiter, or a suddenly-brightening variable star, would be first rendered visi- ble by that light which first arrives at and enters the eye, and it might consequently appear violet or blue ; and when it disap- peared it would continue for the longest time visible by that component of light which is slowest in travelling, and therefore might appear red before vanishing ; or again, aberration of light would necessarily have the effect of giving us an image of 512 OF ETHER-WAVES. [CHAP. each star drawn out into a spectrum. Nothing of the kind is observed ; all kinds of ether-waves must therefore travel through the ether of space at the same rate. Terrestrial experiments for ascertaining the velocity of light are based upon one of two principles. 1. Fizeau's principle. A ray of light is rendered intermittent by flashing between the teeth of a rotating cogwheel. It travels to a distant mirror ; each flash is there reflected along its former path. Before a flash can again reach the cogwheel, the cogwheel may have rotated so far that one of its cogs now obstructs the returning ray ; if a sufficiently-increased speed be imparted to the cogwheel, the light is allowed again to pass between the teeth of the wheel through a neighbouring notch, which has now come to occupy the position at first occupied by that notch through which the light had flashed on its outward journey. Given, then, that the light has travelled to a certain distance and back, and that in the meantime the cog- wheel has been rotated through a certain angle, it is, in principle, easy to find the speed of propagation of the light. Fizeau found this to be 314 million metres : Cornu, by similar experiments, obtained the value 300,400,000 metres in vacuo. 2. Foucault's principle. A beam of light starts from a source S; it strikes a mirror M, and is reflected to a distant mirror R, on which it is focussed by a lens between S and M : it is there reflected and retraces its journey : it is again reflected from M and returns to S. If, however, the mirror M have, in the meantime, been rotated perceptibly before the beam of light has had time to return from the distant R, the light can no longer be reflected from M towards the original point S; it illuminates some other point T. The distance between S and T can be measured ; the amount of rotation of the mirror M in the time taken by the light to go from M to R and back can be inferred from this ; the speed of rotation of the mirror M can be read off on a speed-indicator attached to the rotating apparatus : the distance traversed by light in one second can be ascertained by calculation from these data. There is no need to use instantaneous flashes of light from S ; the steady beam from S reflected from the rotating mirror M only encounters the small fixed mirror R for an instant once in the course of each revolution, and is thus rendered practically instantaneous. Michelson put the lens between M and R, and thus obtained greater brightness, which enabled M and R to be much farther apart, and a greater deflection to be produced. By this means, with a mirror rotating 1000 times in a second, Foucault demonstrated that light takes a measurable time to pass through a distance of 7 or 8 yards. Lord Rayleigh has shown that these different methods cannot be expected to give the same results, for it is not precisely the same thing which is observed in all these cases. In some (aberration method) the speed of single waves is observed ; in others (Fizeau, Jupiter's satellites) the speed of propagation of groups of waves, which is not the same as that of a single wave, unless the velocity of the wave be independent of the wave-length; in others (Foucault) these are blended. From the concord- ance of the results obtained by the different methods it would appear that the wave-velocity is, at any rate for wave-lengths between blue and red, not dependent on the wave-length. xv.] VELOCITY OF LIGHT. 513 As a mean result it may be stated that the velocity of ether-waves in a vacuum that is, in the Ether of space is 300,574000 metres, or 30,057,400000 cm. per second = 186772 miles per second. From this it follows that tt, the rigidity of the Ether, and />, its density in vacuo, are definite in amount, and bear to one another the relation n = p x (30057,400000) 2 ; for v VnTp centimetres per second. The mean velocity in air is less than that in vacuo in the ratio of 1 to 1-000294. Tt is generally believed that light of all colours travels with equal velocities through air, though some doubt has been cast on this result by the experiments of Forbes and Young (Phil. Trans. 1882), who conclude that blue light travels more rapidly in air than red light does, in the ratio of 1018 to 1000. If this were so, however, Foucault's experiment would give drawn-out coloured images ; which it does not do. By a modification of Foucault's method, above described, the relative speeds of light in two different transparent media, or in the same medium at different temperatures or under different pressures, may be compared. The light between M and R has to traverse a space in which a certain thick- ness of the medium, whose retarding power is to be examined, may be laid in the path of the beam : the beam may be exposed, by having to pass through this medium, to a retardation, which is rendered manifest and measurable by an alteration of the position of the image T. The Physical Intensity of light at a place is measured by the energy transmitted through that place, Der unit of cross- sectional area, in a second of time ; for light of constant colour, this intensity is also proportional to its brightness as perceived by the eye. Hence there are two methods of measuring the intensity of a beam of light: 1. Calorimetrical: allow the beam to fall upon a thermopile, and estimate the intensity of the light by the amount of heat into which it is converted upon absorption ; the beam in this case having undergone a preliminary sifting through some adiathermanous medium. 2. Photo- metrical: two sources of light are placed at such distances from an illuminated body that they appear to produce the same effect, such as equal shadows, or equal illumination of the two sides of a disc ; but this method is only accurate when the two lights to be compared are of exactly the same colour. The intensity of .actinic radiation may be estimated by observing the depth of tint produced in a piece of photographic paper exposed for a given time. The total radiation may be measured calorimetrically. As a unit of photometric intensity the Paris Electrical Standards Committee recommended (May 1884) the light emitted by 1 sq. cm. of melted platinum at its solidification-temperature. The twentieth part of this is the normal " decimal candle " (International Electrical Congress, 1889). For many purposes of mathematical calculation it is more Advantageous to measure the intensity of radiation by the average energy in ergs per cub. cm. See p. 142. 2L 514 OF ETHER-WAVES. [CHAP. MODE OF PROPAGATION POLARISATION. Waves of light have the peculiarities of propagation char- acterising waves whose wave-length is generally small in com- parison with the breadth of their wave-front. They do not usually diverge laterally from the directions mapped out by the normals to their wave-fronts ; or, as it is commonly expressed, Light travels in straight lines; they can only so diverge when they are made to pass through apertures or round obsta- cles not very much greater in breadth than their own wave- length. The light from a single luminous point is propagated in spherical waves; that from such an extended object as a candle-flame, in waves which, at some distance from the source, are approximately spherical. If light from a wide source be made to pass through a narrow tube, or successively to traverse equal apertures in two opaque screens, at such a distance from the source that the wave passing through the second screen has a plane front (see Fig. 57 a), then on the farther side of the sec- ond screen there may be an unwidening or parallel beam of light. Such a parallel beam of light, as it traverses space, may be compared to a bundle of vibrating strings of ether, iso- lated in the ether, vibrating independently, and practically unaffected by the ether situated laterally with respect to them. Each such imaginary individual cord may enter into transverse vibrations of different kinds, analogous to the vibrations of strings. 1. It may transversely vibrate simply up-and-down, or from side-to-side, or in any other single direction, its vibrations are restricted to one plane ; the whole beam is then called a beam of Plane-Polarised Light. 2. Its vibration may be resoluble into equal simultaneous transverse-vibrations in two planes at right angles to one another. (#.) These may be of equal period, and the vibration in one plane may be J period behind or in front of that in the other ; looked at endwise, any part of the ether in such a beam would necessarily be seen if it could be rendered visible like a bright point on a vibrating string to execute small circular vibrations. Such a beam of light is said to be Circularly- Polarised. Looked at from one side the vibration would apparently progress like a screw. XV.] POLAEISATION. 515 Fig.156. A common corkscrew is a right-handed spiral. A simple experiment with a string, one end of which is fixed to a wall, while the other is held in the hand, will show that, in order to impress upon the string the right-handed spiral form, we must rotate the free end in a direction opposed to that of the hands of a watch. Such is the movement in a so- called right-handed circularly-polarised beam of light. When the rotation is in the opposite sense, the circularly-polarised ray is left-handed. Fig. 156 shows the direction of propagation and of rotation, and the forms assumed by the vibrating ether in a right (R) and in a left-handed (L) ray respectively. (6.) The circle may, by a difference of phase other than J period, be converted into an ellipse. A beam, the ether in which rotates in ellipses, is one of Elliptically-Polarised light ; this again may be right or left-handed. ( 0, < oo, d' is less than/, and also less than d. When d = +/, d f = + \f; Virtual Image at half focal distance. If rays converge upon a point behind the lens, so that d is negative, then, so long as d is not numerically greater than /, the lens will make them con- verge upon a point at a greater negative distance. If d = /, the converg- ent rays are rendered parallel. If d be negative and numerically greater than /, the convergent rays are made to diverge as if from some positive distance d e , < oo, >/. When d = -2f,d' = + 2/. The focus of a thick-edged lens is most conveniently found by coupling it with a thin-edged one, found by trial among a sufficiently extensive series, so that together they shall produce no change in the apparent size of an object seen through them. As to the inversion or erectness of the image produced by a thin-edged lens, an object at O (Fig. 170), at a distance exceeding twice the focal length, produces a smaller inverted Fig.170. image, a real image, at I, and an eye placed beyond I that is, at a sufficient distance from the lens will perceive the real image of a distant object, inverted, smaller than the object, and apparently situated in space between him and the lens; the contrary being the general impression. The eye must be so far beyond I that the real image in space can be looked at in the same way as any ordinary object of vision. In all cases, an object and its real image are interchangeable, so that an object at I will produce a real image at O. If the distance be greater than the focal length, but less than twice that length, the image is still real and inverted, but is larger than the object. XV.] LENSES. 537 When, however, the object is brought so near the lens as to lie at a distance from the lens less than the focal length, then, to an eye situated at any distance on the other side of the lens, a virtual image will be apparent, erect, magnified, and more distant than the object. Hence these lenses are commonly used as magnifying-glasses (Fig. 171). The relative linear sizes of object and image are in all cases proportional to their respective distances from the " optical centre " of the lens. Fig.171. Fig.172. In divergent or thick-edged lenses, Fig. 172 shows that the image of a real object is erect, diminished, virtual, and nearer to the lens than the object itself ; and, since there is no subsequent crossing of the rays beyond the lens, there is no inversion of the image. The virtual image of such an object is always at a distance from the lens less than the focal length. Lenses of this kind may be used as diminishing-glasses. For some purposes flexible lenses may be used in which the curvature may be slightly varied. Cusco's ophthalmoscopic lens consists of two pieces of thin microscopic cover-glass fixed in a frame : water fills the cavity between them ; by forcing more or less water into the cavity the curvature may be varied. Even when the light transmitted through convergent lenses is monochromatic, the focussing can never be exact if their sur- faces be spherical ; each point of an extended object forms a slightly-blurred image. This effect can be reduced somewhat by the use of Diaphragms, which allow only the central part of a beam to pass through the centre, and the marginal rays to pass through the marginal part of a lens ; they thus diminish the Spherical Aberration of the lens (as in the pupil of the eye), but this can only be brought to a minimum by modifying the curvatures of the lenses used. This could be done, for parallel incident rays, or for rays coming from a pre- determined distance, by making the anterior face of a single 538 OF ETHER- WAVES. [CHAP. lens ellipsoidal or hyperboloidal and the hinder face spherical ; but such surfaces cannot well be produced. If lenses could be produced diminishing in density towards the centre, the same effect might be attained. Both these refinements are present in the human eye. What the optician does is to combine lenses, which have spherical and plane surfaces, so as approximately to bring about this desired result, and then to correct the curvatures by a process of systematic trial and error. Some combinations of lenses are so devised as to bring all the points of an extended image into the same plane, and thus to produce a flat field; others to bring points differing in distance to foci which differ very little from one another, and thus to secure penetration. The calculation of the various curvatures necessary for these ends often involves considerable mathematical skill. In general a lens with spherical surfaces is equivalent to a series of prisms whose angles vary with the varying distances from the axis. In a thin-edged lens the marginal rays are gen- erally more refracted than the axial ; thus a square object yields a Distorted Image, the corners of which appear squeezed in, and the boundaries of which are convex. Similarly, a thick- edged lens draws out the corners and produces concave bound- ing lines. This tendency is obviated by using lenses in pairs, symmetrically disposed, so that the distortions produced by the one lens may be reversed by the other. Again, if the screen on which the image is received be not parallel to the object, the image is apparently so distorted that lines parallel in the object appear to diverge or converge in the image ; whence the use of the Swing-back, maintained vertical, in photographic cameras. The general principle underlying calculations relating to systems of lenses is that the image formed or tending to be formed by the first lens is taken as the object (real or virtual) of the second, and so on. The upshot is, that for every arrange- ment of any lens-system, there is always an image formed some- where, corresponding in size (but not in its position) with that which might have been produced by an Equivalent single Lens. The adjustment of a system of lenses (e.g., the focus- sing of a telescope or microscope) is for the purpose of causing the image to be formed in a place where it will be convenient to inspect or to use it. The action of a system of lenses is, approximately, equivalent to the for- mation of an image by a simple lens plus a determinate shifting of the image formed. This shifting being allowed for, it is often convenient to xv.] LENSES. 539 represent a system of lenses by an equivalent lens. For instance, the Eye may be ideally reduced for many purposes to a single lens composed of aqueous or vitreous humour, having its back coincident with the retina, and its anterior aspect a spherical surface of 5-1248 mm. radius, situated at its most anterior point 2-3448 mm. behind the actual anterior surface of the cornea. Such a lens would refract incident light and bring images of dis- tant points to a focus upon the retina in the same way as the actual Eye does. Gauss's Method. Gauss, followed by Listing, starting from this con- sideration as to the equivalence of any system of lenses to a single lens plus a determinate shift, found that every possible system of lenses could, if well centred, be reduced to a Region of Space to be traversed by the incident light, and presenting six characteristic or Cardinal Points, ranged along the axis of the system. These are the Incidental Focus F, the Incidental Principal Point P, the Incidental Nodal Point N, the Refractional Principal Point P', the Refractional Nodal Point N', and the Refractional Focal Point F'. The incidental points become refractional, and vice versa, when the direction of the rays is reversed. The rays are assumed to travel all near the axis. All rays proceeding from F become after refraction parallel to each other and to the axis ; all parallel incident rays, parallel to the axis of the system, pass through F'. An object at P, or in the same plane (at right angles to the axis of the system) with it, forms an equal and erect image at P' or in the same plane with it ; there are only two such points. Any ray apparently making for N before refraction is, after refraction, parallel to its former course, but appears to be coming from N' ; there are only two such points. The distance PF is the Incidental, while P'F' is the Refractional Principal Focal Distance. These six points, all in one line, are closely related. The distance FN is equal to the distance FT' ; and N'F' = FP.' Therefore PN = P'N' = FP - F'P' ; and PP' = NN'. Further, if ft be the ratio between the index of refraction of the medium nearer the source and that of the medium beyond the lens, FP = F'P' (1//3) ; in the case of a lens in air, these two principal focal distances are equal, and further, P coincides with N and P' with N'. Planes passing through F and F' at right angles to the axis of the sys- tem are called its Focal Planes. Rays diverging from any point in one of these focal planes (of which rays one might be towards the corresponding nodal point) emerge parallel to one another ; and since the ray from the divergence-point to the corre- sponding nodal point would have emerged parallel to its original direction, all the rays must necessarily emerge parallel to the original direction of that ray. Thus they retain parallelism with that ray, the direction of which is, on its emergence, determinate. Rays parallel in the firs't medium converge on some point in the second focal plane. That ray which travels towards N emerges as if it had come from N' parallel to its former course. Hence a line drawn from N' parallel to the originally parallel rays will cut the second focal plane in a certain point ; towards that point in the second focal plane all the rays originally parallel must converge. The artifice of Gauss's method (for which see his Collected Works, or, for an elementary exposition, von Helmholtz's Physiol. Optik, andf Clerk Max- well, Qu. J. Malhem., 1858, p. 233; or Pendlebury, Lenses and Systems of 540 OF ETHER-WAVES. [CHAP. Lenses) is, so to speak, the identification of an incident set of rays as they cross the first Focal Plane; the rays are then traced until they arrive at the second plane ; from the data thus obtained their subsequent course can be ascertained. Mathematical difficulties are thus minimised, for the problem becomes mainly one of finding these cardinal points for a lens-system of any given form and of any degree of complexity. In the case of a single lens surrounded by a single medium, such as air, let the radius of the right-hand surface be r ( -f if the centre be towards the right, if towards the left), and let that of the left-hand surface be r' (simi- larly + or ) ; and if J3 stand for /?,//?, the relative refractive index of the lens as compared with that of the medium ; and if A and B be the left- and right-hand axial points of the lens, so that AB = t, the axial thickness of the lens : then, all measurements being reckoned on the footing that + AX is a distance from A towards the right and AX the same distance towards the left, we have AF = (- firr' -((3 - 1) tr') + (j3 - 1) [0 (r - r') + ( _ 1) f] ; BF' = (firr 1 - (/? - 1) fr)-*- the same divisor ; AP = - tr 1 + [/?(r-r') + ) Fig. 172 a. (FO AXIS lenses are only truly reversible when the principal focal planes are made to exchange places; which, since in the case of thin-edged convexo-concave lenses the principal planes mostly lie outside the convex face and in that of thick-edged convexo-concave lenses outside the concave face, often involves considerable shifting of the lens in its setting. If the object O be at any distance d = PO from the incidental principal xv.] LENSES. 541 plane, the distance of the image I from the other principal plane, d' = P'l, is determinable numerically by the equation 1/PO + I/ P'l 1/PF, or \/d + l/d' = I//, f being taken as -f in divergent, in convergent lenses. If we put the thickness of the lens out of view, making t = in the formulae, we arrive at the usual lens-forinulse. For example, in the convexo- concave lens discussed above, AF = firr' -r- {(ft 1) @(r r')} = {(ft 1) (1/r- l/r')}- 1 = -72; BF'=+72; AP = ; BP=0; AB=0; that is, the focal distances measured from the imaginarily coinciding principal planes or, in other words, from the optical centre of the lens, are each equal to 72 cm. It will be seen how widely these values depart in this case from the true values as given by the Gauss-formulae above. The accompanying diagram (Fig. 172 a), in which the obliquity of the marginal rays is exaggerated, may serve to illustrate the method as applied to a biconvex lens in air : in this case the focal distances are equal and the principal points, which coincide with the nodal, are within the lens. Chromatic Aberration. When mixed coloured-light is passed through a thin-edged lens, violet light is most refracted, and comes to a focus sooner than the red rays do ; beyond the red focus is the heat-focus ; between the violet focus and the lens is the region of the photographic focus. If a beam of white light be passed through a single con- vergent-leris, a screen placed at the violet focus will give an image with a red border the red rays not having yet con- verged ; if it be placed a little farther off, at the red focus, the image is now surrounded -by a violet border, for the violet rays are already divergent. Consequently no clear definition can be obtained by the use of such simple lenses, and it is neces- sary to render them Achromatic. A biconvex lens of flint glass, more convergent than is necessary, is coupled with a bicon- cave lens of crown glass of proper curvature ; the latter destroys the dispersion, by bringing two colours to the same focus, with- out wholly doing away with the deviation ; the couplet acts on the whole as a single lens, producing a somewhat smaller refrac- tion than either of the lenses. This arrangement may be seen in the object-glass of any common telescope. For still further accuracy three, four, or even a greater number of lenses may be combined, by which three, four, or more colours are brought to the same focus ; as in 'the achromatic objectives of microscopes. Makers of photographic lenses have shown much skill in making the photographic and the visual focus coincide ; for special photographic work, such as Rutherford's lunar photography, lenses have had to be constructed whose curvatures are calculated with reference to the focus of the highly- refrangible actinic rays alone; and, while nothing can be distinctly seen through such lenses, photographs of extraordinary clearness have been taken by their aid. 542 OF ETHER-WAVES. [CHAP. Radiant Heat may be shown to be reflected and refracted like Light, by concentrating rays of dark heat upon a Thermo- pile by means of a lens or a mirror, or by refracting them by means of a prism into a new path, in the course of which the thermopile must somewhere be placed before it will indicate the impact of Heat-waves : by photography of the infra-red region of the spectrum ; by Langley's Bolometer (p. 717) ; by Becquerel's Phosphorescence-effect, p. 505. INTERFERENCE. Ether-waves are capable of Interference. Two systems of equal waves, arriving at the same point in opposite phases, will produce at that point no effect, either of light or of heat or of photographic action: at that point the ether will be at rest; and thus light added to light may produce darkness. In Fig. 75 the two points A and B are centres of wave-motion, and at the points &', d',f, on the screen MN, there is no disturbance, while at intervening points, a', f , e\ etc. Let us now consider a monochromatic beam of plane-polar- ised light. Such a beam may be divided into two parts by reflex- ion from a silvered or platinum mirror bent in the middle at an angle very nearly equal to 180, or else by refraction through a biprism whose angle is very nearly 180. The last case is shown in Fig. 173. S is a source of light; the light from it is transmitted through a polariser N : it is now a polarised beam. The rays are received by a convergent lens, which makes them converge upon S". In its course it is passed through a piece of glass R, coloured red with suboxide of copper : it is now to a rough approximation monochromatic. It is then passed through the biprism P, which refracts it in such a way that it seems td come from two equal and equally -distant foci at S'" and S"". xv.] INTERFERENCE. 543 The light may then be received, either on a screen, or directly in the observer's eye placed in the onward path of the beam. A series of dark and bright fringes will be seen, corresponding to the alternate fringes of rest and disturbance of Fig. 75. The two beams apparently travelling from S'" and S"" are polarised in the same plane, and any irregularity of amplitude charac- terising the one is participated in by the other. Hence they are in a position to interfere fully and regularly with one another. If, on the other hand, they had been polarised in planes at right angles to one another, they could not have extinguished one another at any point. When common light is used, it may be at once filtered through a piece of red glass and then passed through a convergent lens. Light from two different sources cannot show interference-phenomena well ; any irregularities in the one vibration ought to be participated in by the other ; and hence even light from the same source, if one of the beams have been very much delayed, may be rendered unable to show these phe- nomena, through the irregularities having had time to produce a difference between the two beams of light, originating from the same source at dif- ferent times. To procure monochromatic light it is better to project a spectrum upon a screen in which there is a slit, and then, behind the screen, to make use of that part of the spectrum whose light falls upon and traverses the slit. It is very easy to procure a bright spot which may represent a simple luminous point, by making a small hole in a metal screen, and in this insert- ing a drop of glycerine. This acts as a powerfully-convergent lens, and if sunlight be concentrated upon it there will appear on the dark side of the Fig.174. screen an intensely bright little spot of light which may be used as a source of light for many experiments; with such a source of light Fresnel dis- covered the laws of diffraction. More elaborately, the same result may be better attained by means of the electric light made to converge by an achro- matic lens of exceedingly short focus, a high-power microscopic objective. When monochromatic common light, proceeding from a luminous point, is passed through a biprism, its vibrations in each of two planes, at right angles to one another, produce the 544 OF ETHER-WAVES. [CHAP. effects of interference independently of one another, but pro- duce their respective fringes and bands in coincident positions on the screen. When mixed coloured-light or white light is treated in this way, the red fringes do not coincide with the violet fringes ; the violet fringes are more numerous than the red fringes, and are closer together. This will be understood from Fig. 75 ; if the wave-length be increased, the points a', 6', c', d', e\ must become farther distant from one another. A violet fringe is seen near the axial line of the beam ; it is over- lapped by a blue, the blue by a green, and so on : each coloured fringe produced by the interference of white light presents a complete spectrum. The number of such spectra is limited; at a little distance from the axial line of the beam the fringes overlap one another so as to produce what appears to the eye to be simply white light, but the spectrum of which shows a series of alternately dark and light bands: all the colours being equally encroached upon by dark bands, the result seems white. Michelson has been able to observe 200,000 fringes. A bent mirror used instead of a biprism produces, by reflexion of white light upon a screen, alternate fringes of white light and darkness. Measurement of Wave-length. If S'"S"" be the apparent position of the two images or apparent sources of light, which must be monochro- matic ; N the position of the central fringe, illuminated by the joint action of S'" and S"" ; the angle S"'NS"" = 23; N' the position of, say, the fourth bright fringe ; S'"N' is shorter than S""N' by four wave-lengths ; this dif- ference is very nearly equal to (NN' x S"'S"" -*- AN } = NN' X 2 tan 8. Fig.175. The angle 28 can be measured with a theodolite ; the distance NN' can be measured with a micrometer ; the value of the four wave-lengths, and therefore of one wavfi-length, can be determined from these data. Fig. 75 shows that the line of propagation of these fringes in space is hyperbolic ; the foci of these hyperbolas being the two apparent sources. The bands vanish when one-half of the biprism or mirror is covered. If the light from one of the sources be retarded by being made to pass through a layer of a substance in which light travels more slowly than in air, the whole of the fringes will be shifted somewhat towards the side on which the retardation xv.] INTERFERENCE. 545 takes place. From the amount of this shifting may be calcu- lated the amount of retardation ; and by means of this the rela- tive velocities of light in (and therefore the refractive indices of) such things as hot air, cold air, hydrogen gas, normal glass, compressed glass, compressed liquids, and so forth, may be estimated. Colours of thin films. Thin films of transparent sub- stances, such as oil upon the surface of water, iron oxide upon the surface of tempered steel, oxides deposited upon metals by the galvanic batteiy, soap bubbles, glass blown out to an extreme tenuity or exfoliating under the influence of slow decomposition, present curious colours when shone upon by a comparatively bright light. Such films may be rendered permanent ; a solution of bitumen and a little caoutchouc in a mixture of benzene and oil of naphtha, dropped upon water, forms films which solidify and may be caused to adhere to a sheet of paper. In Fig. 176 monochromatic light from S is incident upon a thin transparent-film AB of uniform thickness. A part of the light is at once reflected to R from Fig.i?6. the first surface of the film. Another part is refracted to R' after having undergone one reflexion at the second surface. If the path of the beam in the film be an even * number of half wave-lengths, the beam travelling to R' is opposed in phase to that travel- \\T' ling to R, and an eye placed at RR' (these points being supposed very close together) will receive no impression of light ; or, rather, it will receive but a feeble impression, for tlie ray to R' cannot be quite equal in intensity to that travelling to R. Again, an eye placed at XT' will per- ceive but a feeble impression of light ; not absolute darkness, for the ray to T is considerably more intense than that to T', and is not completely, neutralised by it. There is, however, complete interference for any one wave-length if multiple reflexion be taken into account. * This seems strange ; we might have expected a retardation of an odd number of half wave-lengths to produce a difference in phase of half a period ; but it will be remembered that the beam reflected at one of the surfaces of the>film that sur- face, namely, which separates an optically denser from a rarer medium suffers a loss of half a wave-length, which is independent of the thickness of the film. 546 OF ETHER-WAVES. [CHAP. Let the film be of variable thickness; a film of air between a glass plate and a biprism, or between a convex lens and a plate of glass, varies in thickness with the distance from the centre ; in the former case the thickness of the film of air varies as the dis- tance, in the latter approximately as the square of the distance from the central point. Monochromatic light reflected from such a system presents the appearance of alter- nately dark and bright bands or circles bright where the directly-reflected light and the light reflected from the second surface of the film are similar in phase dark where they are opposed. In the case of a lens pressed against a plate they are known as Newton's rings. The less the curvature of the lens the greater the distance between two consecutive rings. The distance between the consecutive rings is, approximately, inversely proportional to the radius, so that the external rings are most crowded together. If such a substance as water be used between the lens and the glass, the rings are closer together; the width of the rings varies inversely as /3, the refractive index of the substance thus employed ; for a shorter distance in an optically-denser medium is equivalent to a longer distance in air. On inclining the incidence of the light, the rings become dilated. By transmission, a second system of rings is produced, complementary but dimmer. If mixed coloured or white light be employed, the dark and bright rings of the several components cannot coincide, and the result is a series of circular spectra, in each of which the violet circle is the nar- rowest. These spectra overlap one another at a little distance from the centre, and blend into what appears to the eye to be white light. A series of dark rings or fringes may be obtained by rubbing a film of soap on black glass, drying it, and breathing gently upon one point of this through a glass tube ; this, done in the sunshine, gives rise to bright colours. It is not possible actually to obtain monochromatic light ; even that emitted by incandescent sodium-vapour, in which some five hundred rings can be seen, is not quite monochromatic. The centre of Newton's rings is dark if there be approximate contact ; perfect contact there never can be, for a dustless surface.it is impossible to obtain ; even when there is no appreciable thickness of film traversed, the fact that one ray is reflected from the upper, and the other from the lower surface, the one at the bounding surface of an optically-denser, the other at the surface of an optically-rarer medium, causes the one to lose, while the other does not lose, half a wave-length on reflexion; they thus become XV.] INTERFERENCE. 547 opposed in phase, and the centre is dark. If, however, both reflexions be made to take place from the surface of an optically-denser medium, as in Young's experiment, in which light travelling through a lens of crown glass was reflected first from the upper surface of a film of oil of sassafras, lying between that lens and a plate of flint glass, sassafras being inter- mediate in its refractive power between crown and flint glass, there is no such relative retardation, and the centre of the system of rings is bright. The Iridescence of mother of pearl and of objects with a finely-grooved or striated surface, such as butterfly's scales, is an effect of interference. Sunlight falls upon their surface; some of this is reflected from the ridges, some from the grooves, and in this way a difference of path is set up among the reflected rays, which causes differences of phase among them, and, in the case of some of them, opposition of phase and extinction. When the incidence of the reflected light is very oblique, the ridges alone may reflect, the differences of phase and of path produced will be very small ; there will be little iridescence and very considerable reflexion. The propagation of light "in straight lines" within the same isotropic medium is itself a result of interference. From it is derived the power of making a geometrical Shadow. In Fig. 178 a real focus at F acts as a source of light. It casts a sharply-defined shadow of an object O upon a screen. If the source of light be an extended one, not a mere point, the shadow consists of two regions, a central umbra and a marginal penumbra. In Fig. 179 the sun, S, shines upon the earth, E : the earth being smaller than the sun, there is formed a cone of darkness behind the earth ; if the moon travel wholly or partially into this cone of shadow, it will be wholly or partly unillumined, and we have a total or a partial eclipse of the moon. But outside this shadow there is a penumbral region, in which a body, or any point of a body, will be in " half -shadow," not fully illuminated, because able only to see a portion of the illuminating body. Fig.178. Fig.179. 548 OF ETHER-WAVES. [CHAP. Fig.180. When light radiating from an extended object passes through a small aperture, the waves arriving at the aperture from , the object traverse the aper- ture, and there cross each other; they then diverge, and a screen placed on the opposite side of the aperture receives an inverted image of the object, whose size varies with the distance of the screen, as in the well-known Camera Obscura. An aperture of no appreciable breadth would, at whatever distance the screen might be placed, give a perfect image in the natural colours, an image of which no part would be out of focus ; one of ^V^ 110 ^ diameter will give on a screen at 40 inches distance an image which, though wanting in brightness, is as perfectly defined an image as any possible lens placed at the aperture can produce : one of y^-inch will produce the same definition at a distance of 250 inches ; or, in general (Lord Rayleigh), if A be the wave-length, r the semi-diameter of the aperture, d the least distance of good definition, d (2r 2 /A). When the screen is nearer than this, each point of the object makes on the screen an image which has the same shape as the aperture, and the superposition of these makes a blurred image. When the diameter is ^-inch (No. 10 steel sewing-needle), d 8-57 inches (pin-hole photography) ; but with smaller apertures than this, Diffraction begins to confuse the result. Light thus travels in straight lines, and is incapable of passing round corners under ordinary circumstances, and as examined by our ordinary senses. A closer examination of the subject shows, however, that light does to a certain degree pass even round corners. The phe- nomena of Diffraction, in which this is observed, are explicable on the ordinary principles of interference. Let S (Fig. 77) be the source of light ; waves diverge from this as a centre. These waves impinge upon a screen AB. Fig. 77 shows that beyond the screen AB there is a series of fringes within the geometrical shadow ; that even in the part directly in view of the source of light there are bands of relative darkness ; that the central point of the shadow may be nearly as brightly illuminated as if there had been no screen AB ; that the broader the object AB, the narrower will be the fringes ; that the forms in space of the regions of approximate darkness are hyperboloids ; while if the source of light be removed to an infinite distance, the hyperbolic lines of relative rest in the illuminated region are practically xv.] DIFFRACTION. 549 reduced to straight lines, but sweep past the obstacle without touching it. When the obstacle is circular a minute circle of tinfoil pasted on a piece of clear glass the shadow cast upon a screen, or received in the eye directly or by the aid of a lens or telescope focussed on the obstacle, is seen to be surrounded by a series of dark and bright rings ; or, if the light from S be mixed-coloured or white light, by a series of spectra ; while the shadow is also modified by a series of such bands or spectra, and its centre is bright. A similar construction for a little cir- cular aperture in an opaque screen at AB will show that the bright spot produced on a screen beyond AB will have fringes blurring the sharpness of its edges, and that at certain distances of the second screen from AB the centre of the bright spot will be dark. When the obstacle or chink is linear and parallel-sided, the fringes or spectra are parallel to one another ; when it is not so they assume a curved form ; when it is angular the fringes may assume a great variety of remarkable and beautiful forms. The phenomenon of diffraction can be roughly observed by looking at a distant gas-flame, edge on, with the half-closed eyes ; the sun shining on the eye-lashes will also produce a simi- lar effect ; the morning sun, shining on twigs of trees situated between the sun and the eye, causes the shadows of some of them to become bright in the centre, and a curious silvery appearance results. The image of any point seen through a telescope or micro- scope has its clearness of definition interfered with by the dif- fraction of rays of light round the edges of the diaphragm, or round the edges of the lens. This effect is generally insignifi- cant in terrestrial telescopes ; it is very noticeable in astronom- ical telescopes, where the source of light, a distant star, ought to appear reduced to a point, but is apparently enlarged into a perceptible disc surrounded by rings ; and in the microscope it sets a limit to the powers attainable, for high powers involve small lenses and small apertures, and these bring diffraction in their train. The limit of microscopic definition is about 3 ^ QO mm. with white, and about -g-oVo mm - w ith blue light. If a very large number of parallel equidistant lines be ruled upon glass or metal, plane-fronted light issuing from a slit or from the image of a slit will, if transmitted' through or reflected from this so-called Diffraction-grating, and focussed 550 OF ETHER-WAVES. [CHAP. upon a screen or in the eye, be found to be resolved into a cen- tral bright image of the slit, on each side of which is a dark space, and then a series of successive spectra, overlapping or separated by dark spaces, according to the fineness of the grating: these spectra have their violet ends turned towards the central bright image (see Fig. 77a). By multiplying the number of lines in the Diffraction-grating, as in Prof. Rowland's gratings, which have 43,000 equidistant lines to the inch, the spectra may be rendered almost perfectly pure, so that Fraun- hofer's lines may be easily seen in them. A microscopical preparation of muscular tissue will often be found to act as a more or less efficient diffraction-grating ; the striations of the mus- cular fibres take the place of the grooves engraved on the glass. The value of diffraction-spectra is that the deviation in the successive spectra depends directly upon the wave-length ; their disadvantage the mechanical difficulties of uniform grooving of the grating, and of making clean-cut grooves. If any kind of light have, in air, the wave-length A centimetres, and if N be the average number of lines per centimetre engraved on the grating ; and if 8 be the angular deviation of any particular coloured light (or, better, of any particular Fraunhofer line), then sin 8 is equal to NX for the first spectrum, to 2xA for the second spectrum, and so forth ; and since N and 8 can be measured, A can be accurately found. At the spot where light of wave-length A. appears in the third spectrum, that of wave-length 3A./2 in the second and that of wave-length 3A. in the first spectrum coincide. The definition in the diffraction-spectrum is best in the position of minimum deviation (p. 141); and the normal spectrum is pro- duced when the angle of incidence is so regulated that the angle of diffraction is zero (see p. 141). The Twinkling of Stars is another effect of interfer- ence : light, coming to the eye from a star so distant as to be practically a single luminous point, arrives in rays which have traversed slightly unequal distances in an irregularly-refracting atmosphere and thus enter the eye in irregularly-unequal phases. Now one colour is extinguished, now another ; the eye perceives coloured light complementary to that momentarily lost. No two persons can, as a rule, see any star twinkling in precisely the same manner. The planets twinkle only at their edges: their discs present many points or sources of light, whose scin- tillations, on the whole, mask one another. If a planet and a twinkling star say Jupiter and Sirius be severally looked at through an opera-glass which is rapidly whirled across the field of view, the image of the planet will appear to be drawn out into a continuous xv. INTERFERENCE. 551 streak, while that of the star will be broken up into a chain of unequally- bright and differently-coloured spots of light. The colours of light from a bright point twinkling through a dusty haze, or shining through a piece of glass covered with lycopodium ; the Corona (red externally) which surrounds the moon as it shines through an atmos- phere charged with particles of condensed aqueous vapour; the coloured rings seen when particles float in the vitreous humour of the eye, these are all different diffractive effects of interference ; and the smaller the size of the particles which produce them, the greater the breadth of the coloured rings. Each particle acts as a partially opaque small screen. The interference of Actinic Rays may be shown by pho- tography; of Dark Heat, by passing a delicate Thermopile, a Tasimeter (p. 636), or a Bolometer (Langley's Thermic Balance, p. 717) through an invisible diffraction-fringe system of dark heat-waves, obtained by treating rays of dark heat with a bent mirror or a biprism ; under these circumstances the instru- ment employed will alternately indicate and cease to indicate the impact of heat-waves. DOUBLE REFRACTION. If a transparent medium have the same properties in all directions it is homogeneous, or, optically, isotropic. A wave of mechanical disturbance starting from a single point of disturb- ance in it will be spherical. The properties of the ether-waves within transparent substances are, in some fashion, correlated with the molecular structure ot the substance, and thus any ether-waves propagated from centres within homogeneous or isotropic substances are themselves also spherical. Substances in which the propagation of light is in spherical waves are either amorphous, or else belong to the cubical system of crystals, the system in which the three crystallographic axes of the crystal are equal. In some crystalline substances one of the crystallographic axes differs from the other two ; the crystal is then symmetrical in reference to this axis only, and is said to be uniaxial. A mechanical disturbance is propagated in such a crystal in the form of an ellipsoid. A slice cut out of such a crystal in such a way that its faces are parallel to this principal axis, is said to have been cut parallel to the Principal Section of the crystal. The propagation of an ether-wave in a uniaxial crystal is peculiar. Fig. 181 shows an equal-sided rhombohedron cut out of a crystal of Iceland spar by splitting it along its natural cleay- 552 OF ETHEE- WAVES. [CHAP. age-planes ; its axis AB joins the opposite obtuse-angles. Let a point C on this axis be a centre of optical disturbance. Then two concentric sets of waves are produced; the one spherical just as in glass, the other ellipsoidal ; one of the axes of the ellipsoid coincides with the axis of the crystal, and is equal for a given in- terval of time to the diameter of the sphere developed in an equal time ; the other two axes, which, to avoid circumlocu- tion, we shall here call the extraordinary axes, are equal to one another, and are either longer or shorter than the former, according to the nature of the crystal. The next question is, Which part of a general disturbance at C is propagated in the spherical, and which in the ellipsoidal wave ? It may roughly be stated that just as we have seen beams of polarised light differently affected by simple reflexion and refrac- tion according to the plane of their polarisation, so in double refraction the behaviour of a beam of light depends upon its state of polarisation. On referring to Fig. 59 we find that the construction there given for the course of a refracted plane-fronted wave may be .reduced to the following construction (due to Huyghens) for a single ray refracted at the surface of an ordinary isotropic medium. AB is an incident ray travelling through the medium M ; CD a circular arc, drawn from centre B, with radius proportionate to the velocity of light in the medium M. Continue the arc CD into the second medium M'; produce AB until it cuts that arc in E ; from E draw a tangent line (or plane) cutting the re- fracting surface in T. From B as centre draw a semicircular arc in the medium M', with a radius proportionate to the velocity of light in M'. From T draw a tangent to this arc ; the tangent touches the arc at B f ; join BB'. BB f is the refracted ray. XV.] DOUBLE REFRACTION. 553 Fig.183. A series of somewhat similar constructions will enable us to study a certain number of cases of double refraction. Suppose a block to be cut out of a crystal of Iceland spar in such a way that one of its cut surfaces is parallel to the axis ; and suppose an incident beam to fall upon that surface in a direction at right angles to the axis. Fig. 183 shows that if GH represent such a block, and if the incident beam be in the plane of the paper, the axis is in such a case looked at end- on ; and then we find that the incident ray is divided into two parts, which travel at different rates, the slower one, BO, in the central sphere, the more rapid one, BE, in the outer ellipsoid, which, looked at in this aspect, has a circular section ; the former, BO, the Ordinary Ray (which obeys the ordinary law sin L = /3 sin g), being more refracted than the latter, BE, the Extraordinary Ray. Both these rays are in this case in the plane of the paper, like the original incident-ray. The relative radii of the two circles may be found from the respective amounts of refraction of the two rays at this kind of incidence. For the light emitted by sodium-vapour, the ordinary index and the extraordinary index of Iceland spar are respectively 1-65850 and 1-48635; the reciprocals of these numbers represent the relative velocities of the ordinary and the extraordinary rays in Iceland spar as compared with that of light in air, this being reckoned as unity. In such crystals as those of Iceland spar the ordinary ray is more retarded than the extraordinary. Let us now turn the block of spar round so that its axis is brought into the plane of the paper that is, into the same plane with the incident light ; the incident light now travels in a principal section of the crystal. One of the extraor- dinary axes of the ellipsoid, being at right angles to the axis of the crystal, is at right angles to the refracting surface ; its semi-axis, BF in the sectional figure (Fig. 184), bears to the Fig.184. 554 OF ETHER-WAVES. [CHAP. Fig.185. radius of the circle the ratio of 1-65850 to 1-48635 if the light used be that emitted by sodium-vapour. If the block of spar be cut by a plane at right angles to the principal sections, but not parallel to the axis, we obtain the result shown in Fig. 185. A The incident light is in the plane of the paper; the axis of the crystal is also in the plane of the paper. When the surface which receives the incident beam has been cut at right angles to the axis, and the light falls upon it normally (that is, at right angles to the surface, or parallel to the axis), there is no double refraction ; the ordinary and the extraordinary rays coincide. A parallel-sided slice of Iceland spar cut in any other direction than at right angles to the axis will divide an incident ray into an ordinary and an extraordinary ray, except in the case in which one of the rays is so refracted as to become parallel with the axis, in which case the other ray coincides with it. In Fig. 186 the incidence is normal, and an observer at A will see two images of a spot at B, of which one, the ordinary, is produced as it would have been by ordinary glass : while if he turn the slice round, the extraordinary image will rotate round the ordinary one. This can be readily observed with an ordinary crystal of Iceland spar. Light striking on a plate or a common crystal of Iceland spar is thus split into two rays, and a single point or a page of , A print looked at through such a crystal gives a double image. Conversely, a pair of points, C, D, if looked at by an observer at A, will have their images blended, and by finding for various distances between C and D the angle ABE, at which these points appear to blend, the two refractive indices may be found: the rays CB, DB, and BA being caused to lie all in a principal section of the crystal. BD represents the ordinary and BC the extraordinary ray. Fig. 186. Fig.187. C D xv.] DOUBLE REFRACTION. 555 When the incident ray is oblique to the principal section, the extraor- dinary ray is no longer in the same plane with the incident and the ordinary refracted ray, but is deflected to one or the other side : the tangent plane to the ellipsoid does not touch it in the plane of incidence. The above figures are drawn for crystals such as Iceland spar, beryl, emerald, mica, ruby, sapphire, tourmaline, the ordi- nary index of refraction of which is greater than the extraor- dinary, and in which the ordinary ray travels more slowly than the extraordinary, and lies between the extraordinary ray and the axis ; such crystals are called Negative Crystals. In others, such as ice, quartz, boracite, the extraordinary ray lies between the ordinary ray and the axis ; such crystals are called Positive Crystals. In the latter, the extraordinary axes of the ellipsoid are shorter than the diameter of the sphere, which thus encloses the ellipsoid : the extraordinary index of refraction is in them greater than the ordinary index. The two rays, the ordinary and the extraordinary, are found to be polarised in planes almost exactly at right angles to one another. The ordinary ray is polarised in a plane contain- ing both the incident ray and the crystalline axis. If the incidence be that of Fig. 184, the incident ray, the reflected ray, and both refracted rays are in the same plane, the plane of the paper, and the axis is parallel to that plane ; the ordinary ray is said to be polarised in that plane ; light polarised in such a plane of incidence passes through the spar as an ordinary ray. The extraordinary ray, when the whole three rays thus travel in a principal section of the crystal, is found to be polarised in a plane exactly at right angles with the plane of polarisation of the ordinary ray. The second face of the block of crystal may be so cut that it receives the ordinary and the extraordinary rays at such an angle as to transmit the one, but totally to reflect the other. In Nicol's prism a long rhomb of Iceland spar is cut in this way, and the portions are so cemented by Canada balsam that when common light enters the Nicol it is divided into two rays, of which one, the Ordinary, is totally reflected when it meets the cemented surface, while the Extraordinary ray is trans- mitted and emerges (the faces of the prism having been, in order to permit this, cut down to the proper angle) in a direc- tion parallel to that of the incident ray. The whole arrange- ment is thus capable of acting as a polariser ; and if polarised light be sent through it in one rotational position, the Nicol will 556 OF ETHER- WAVES. [CHAP. transmit it freely ; while if the Nicol be rotated through 90 in either direction, on either side of the most favourable position, it will transmit none of it. It can thus serve as a means not only of producing polarised light, but also of detecting polarised light, and of finding in what plane it is polarised ; and when it does this duty it is called an analyser. Foucault's prism is shorter than Mcol's, and O an air-film replaces the balsam. In Rochon's prism two similar pieces of quartz make up a parallel-faced block (Fig. 187 a): the ordinary ray emerges without deviation : the extraordi- Axis nary is sent away to one side. In tourmaline there is double refraction ; but one of the rays, the ordinary, is absorbed, and the extraordinary alone passes through. Thus a thin plate of tourmaline acts as a polariser of common light incident upon it, and another plate rotating in front of it may act as an analyser, a convenient arrangement, were it not that tourmaline is always dark in colour, and absorbs much of the light incident upon it. For this reason Nicol's prisms are commonly used as sources of polarised light. Crystals of sulphate of iodo-quinine act like tourmaline, but are useless because they are dark, small, and brittle. We may here recall the different modes of obtaining a beam of plane- polarised light. 1. Reflexion of ordinary light from glass at the angle of complete polarisation. 2. Transmission through a pile of glass plates with parallel sides ; the angle of incidence being the angle of complete polarisation, or an angle approximating to it. 3. Separation of the ordinary from the extraordinary ray produced by double refraction ; this being done (a) by tourmaline, which extinguishes the ordinary ray ; (6) by a Mcol or a Foucault prism, which turns aside the ordinary ray; (c) by a Rochon prism, which turns aside the extraordinary ray. Some crystals, such as topaz and arragonite, have two axes, and are called Binaxial Crystals : in these the wave-surface is very complex, and they have three indices of refraction. In general, in these crystals, the wave-front is oblique to the rays, and there is no ray which obeys the ordinary law of refraction that sin i = /3 sin g; but that ray which does so most nearly in general, and which does so per- fectly when the incidence is in one of the principal sections, is called the ordinary ray ; while the other of the two rays, into which a ray of incident XV.] DOUBLE REFRACTION. 557 light is divided on non-axial incidence, is called the extraordinary ray. In such crystals the positions of the optic axes, which have no invariable rela- tion to the crystallographic axes, are variable ; they vary with the tempera- ture of the crystal, and with the kind of light employed ; and in some cases a crystal is found to be binaxial for one, uniaxial for another kind of light ; Fig. 1876. Fig. 187 c. 187 d. Glauberite (native sulphate of soda and lime), for example, being binaxial to red, uniaxial to violet light. If we take a crystal of arragonite, which happens to have its three indices of refraction in directions at right angles to one another, we find that in the three principal sections the circles and ellipses, which represent the simulta- neous propagations of the different parts of the wave from a central point O, are differently related to one another. Let v t be the greatest velocity (the velocity in air being taken as equal to unity), v ul the least, and v n the intermediate, these three velocities determining the three indices of refrac- tion for any particular colour ; then the propagation of a disturbance from a central point O results in the formation of a complex wave-surface which may be understood by looking at Figs. 187 b, c, and d, which represent the three principal sections : or better, by studying the models of this kind of surface which are now procurable. One of these principal sections (Fig. 187 d), that of greatest and least elasticity, presents the peculiarity that the ellipse is partly inside, partly outside the circle. If a line NM be drawn, touching both ellipse and circle at N and M, it will be seen that the disturb- ance from O reaches M and N" at the same time ; and after successive intervals of time, as ellipse and circle expand, the successive tangent-lines M'N', M"N", etc., remain parallel to MN. The portions of the wave-front at M and N Fig. 187 e. therefore move forward, with respect to the direction OM, with equal velocity; and this direction is one of the Optic Axes of the crystal, for the particular colour employed; there being another, OM', in the same principal section. If a single ray of natural monochromatic light, SO, Fig. 187 e, fall upon a plate of arragonite at such an angle that the ordinary ray travels along the optic axis OM, the common tangent-plane NM advances parallel to itself ; 558 OF ETHER- WAVES. [CHAP. and a separate ordinary and extraordinary ray, O and E, might be expected to emerge, producing two images of S. Internal Conical Refraction. So far as the diagram 187 d can show, we would not expect more than these two images : but if we refer to a model of the wave-surface, and if, instead of applying a mere tangent-line MN, we apply a tangent-plane such as a piece of flat glass, we shall find that the tangent-plane is in contact with the wave-surface along the whole periphery of a circle of contact. Each and every point of this circle satisfies the same conditions as the points M and N" ; and if non-polarised monochromatic light reach the point O (Fig. 187 e), there to be refracted so that the ordinary ray tends to go along the optic axis OM, and the extraordinary along ON, the result is that the two images of S, visible at other angles of incidence of SO, open out into a complete circle of light, which has been produced by the splitting of the incident ray of natural light, SO, into a hollow cone of rays, ONM, within the crystal ; the rays composing this cone all pass through the above-mentioned circle of contact. The light at every point of this circular image is polarised, and at opposite points it is polarised in planes at right angles to one another. External Conical Refraction. In Fig. 187 d we see a point P there being four such points at which rays from O in both waves arrive at the same time. The direction of vibration in the two simultaneous rays is not, however, the same. If the light from O emerged at P into air, the rays OP would be refracted, so far as the diagram can show us, in two directions. But on referring to a model of the wave-surface we would find that at P there was a conical dimple, into the bottom of which a hollow tangent-cone might be fitted, with its apex at P. Every radial line in this tangent-cone would determine a different refraction of a pencil of light travelling along OP and emerging at P into air. Light travelling in the direction OP there- fore opens out, when it emerges into the air, into a hollow cone of light ; and conversely, a solid cone of rays concentrated by a lens may in part be collected and made to run in the common direction PO. When light has passed through a crystal of Iceland spar and been divided into an ordinary and an extraordinary ray, if it be caused to fall upon a second crystal whose faces are par- allel to those of the first, the two rays pass through, suffering no further division; the ordinary ray emerging from the first crystal is still the ordinary ray in the second crystal, which acts like a mere prolongation of the first. If the second crystal be turned 90 round a longitudinal axis parallel to the line AB in Fig. 188, there is still no Fig.iss. division of the rays ; but ^u^^m^a A J Kffih ffiffii the ordinary ray on emer- gence from the first crystal is an extraordinary ray relative to the second crystal, and is refracted as such in that crystal ; and the converse applies to the extraordinary ray emerging from the first crystal. If the second crystal occupy any rotational position intermediate between XV.] DOUBLE REFRACTION. 559 these, each ray incident on it is decomposed into an ordinary and an extraordinary ray. There are thus, in the ordinary case, four images of a bright point seen through a pair of crystals arranged end to end, at a distance from one another, and these images blend into two when the crystals are, by rotation, placed parallel or at right angles to one another. Interposed Lamina. When a polariser and an analyser of any kind are arranged at right angles, so that a plane-fronted beam incident on the system is wholly cut off or deflected by it, an eye placed beyond the analyser can perceive no light; but if a thin film of mica, or other double-refracting substance, uniaxial or bin axial, of uniform thickness, be caused to intervene between the polariser and the analyser, the field may become filled with light, coloured or white, according to the position of the interposed film. In Fig. 189 the line AB represents a plane vertical to the paper, and cutting the paper in AB : we call this the vertical plane, or the plane AB. Then let us by any convenient means produce a beam of plane-polarised monochromatic light, polarised in the plane AB, and let us suppose this beam to be seen end-on, travelling away from the observer's eye. Interpose a thin plate of some birefringerit substance in the path of the beam : let the axis of this lie in the plane CD. The beam AB is broken up by the inter- posed plate into two : one in which the plane of polarisation is parallel to CD, one in which it is at right angles to that plane. The former is trans- mitted through the interposed plate as an ordinary ray, the latter as an extraordinary. The lines Oa, Of, Oc, indicate the relative amplitudes of vibration in the incident polarised beam, in the extraordinary, and in the ordinary transmitted beams respectively. The interposed plate may be so thin that although the incident beam is divided into two transmitted beams, these have not perceptibly separated from one another, and on emergence are not only parallel, but are also practically coincident. In a wide-fronted wave-system this coincidence may be held to be absolute except at the edges of the beam. Though the two beams coincide in direction, their undulations do no x coin- cide in phase ; in positive crys- tals the extraordinary, in nega- tive crystals the ordinary ray is more retarded than its compan- ion. Let us suppose that the more retarded ray has lost one Avave-length : then the result of superposition of the two emer- gent rays will be a plaiie-polar- ised beam similar to that which had originally fallen upon the interposed Fig.189. 560 OF ETHER-WAVES. [CHAP. Fig.190, plate, and in the same plane AB ; if half a wave-length (= |-\) be lost, the result will be an equal plane-polarised beam, polarised in the plane EE'. If, again, CD coincide with AB that is, if the principal plane of the interposed crystalline plate be -parallel to the plane of polarisation of the incident light there is no extraordinary beam O/; and the light, hav- ing been transmitted through the interposed film as an ordinary ray, emerges as it entered, plane-polarised in the original plane AB. If, again, CD be at right angles to AB, the incident beam is wholly transmitted as an extraordinary ray, and emerges polarised in the original plane. Let us now suppose that CD is inclined to AB at an angle of 45 : if one of the rays be retarded by some even multiple of |X, the result is plane- polarised light, either polarised in the original plane (when the retardation may be measured in whole wave-lengths), or in one at right angles to it (when the retardation is some odd number of half wave-lengths), for EE' is at right angles to AB when CD makes 45 with it. Again, if the retardation be some odd mul- tiple of |A, the extraordinary and E ordinary rays are compounded into a circularly - polarised ray of light; and if the retardation be of any value other than some mul- tiple of a quarter wave-length, the result is an elliptically-polarised beam, the ellipse being, according to the amount of retardation, some one of those indefinitely numerous ellipses which may be described within the rectangle EaE6'. In the general case, AB (Fig. 190) being the plane of the incident beam, CD the principal section of the interposed plate, the angle AOC having any value, and Oa, Oc, Of being respectively the relative amplitudes of the incident ray polarised in the plane of AB, and of the ordinary and extraor- dinary rays emergent from the interposed plate; compounded, their result is an Elliptically-Polarised beam, of which the limits are : (a) A plane-polarised beam, whose plane of polarisation is AB and whose amplitude is represented by Oa. (1) When CD coincides with AB. (2) When CD is at right angles to AB. (3) When the relative retardation of cd and fg is 0, or an even multiple of ^A. (&) An equal plane-polarised beam whose plane of polarisation is EE' ; the angle AOE' being equal to twice AOC : this is the result when the relative retardation is an odd number of half wave-lengths. (c) A circularly-polarised beam when the angle AOC is equal to 45, and the relative retardation is some odd multiple of ^A. (1) Right-handed (rotation contrary to hands of a watch) when the component polarised in fg loses (together with any number of whole wave-lengths) one quarter wave-length or gains three quarters relatively to that in cd. (2) Left-handed when it relatively gains one quarter or loses three. xv.] DOUBLE REFRAC' ,7 561 Fig.lDl. Elliptically-polarised light. is produced in every other relative position of CD. This is right-handed if the relative retardation of the extraordinary T&jfg transmitted through CD lie between and |A or between n\ and n\ + |A, where n is any whole number ; left-handed if its relative retar- dation lie between \\ and A, or between n\ + \\ and (n + 1)A. If the plane CD lie so that the angle AOC lies to the left of AB (the observer being, as hitherto, supposed to be stationed near the source of light), these conditions of left- and right-handedness respectively are reversed. A plate of birefringent substance of such a thickness, that when it is interposed in the path of a beam of plane-polarised light of a particular colour, with its principal section at an angle of 45 to the plane of polari- sation, it converts that plane-polarised light into circularly-polarised light, is called a quarter-undulation plate. Quarter-undulation plates are of two kinds: (a) Where the thickness is just such as to cause a relative retardation equal to A, or to (A + 5 A) ; (6) Where the plates are thicker, but are opposed in their action. In Fig. 191 two plates cut out of a doubly-refracting crystal are shown fitted together; the one is cut so that its axis is parallel to the plane of the paper ; the other has its axis at right angles to the paper. Incident light arrives already polarised ; it is divided by double refraction into two rays, an ordinary and an extraor- dinary; then, since the second plate has its axis at right angles to the axis of the first, the ordinary ray of the first plate is refracted in this as an extraordinary ray, while the extraordinary ray of the former passes through as an ordinary ray. On emergence both rays are parallel and practically coincident ; and the amount of relative retardation is equal to that produced by a thin plate equal in thickness to the difference between the thicknesses of the two plates. When light, plane-polarised, is totally reflected from glass, it is found to be elliptically-polarised, unless it had been originally polarised in the plane of incidence, or in a plane at right angles to this. Reflexion from metals presents this peculiarity at all angles of incidence. The vibratory movement actually extends beyond the surface of the glass into the rarer medium beyond, as may be proved on bringing a second piece of glass close to the totally reflecting surface, when interference- colours will be seen. As a result of this, a differ- ence of phase is set up between the two components (polarised in and at right angles to the plane of incidence) into which the incident light may be resolved. A similar result occurs in metallic reflexion, for some of the light penetrates to a slight depth below the reflecting surface. A wave cannot have its direction abruptly changed ; and during the gradual change of its direction, its phase becomes altered to a slight extent : and this effect differs in amount according to the direction of vibration of the incident waves. When the angle of incidence is such that the difference of phase set up corresponds to a relative retardation |A, two such total reflexions would convert a plane-polarised ray into a circufarly-polarised one. If a rhomb of glass be cut in such a form that a ray of light may pass 2o Fig.192. 562 OF ETHER-WAVES. [CHAP. normally through one surface, strike a second surface at the appropriate angle of incidence and be there totally reflected, strike the third surface at an equal angle, and pass out normally through a fourth surface, a ray so travelling through it will, on emergence, be found to be circularly-polarised. Such a rhomb is known as a Fresnel's Rhomb, and acts as a quarter-undu- lation plate for every kind of light, while a film of mica, real or virtual, can only act as such towards light of one kind. If plane-polarised light pass successively through two similar quarter- undulation plates, similarly placed, the emergent light is plane-polarised in a plane at right angles to the original plane of polarisation ; whereas, if the two quarter-undulation plates be opposed in their action, the light is restored by the second to its original state, plane-polarised in the original plane. A second quarter-undulation plate of known action affords us a means of distinguishing right- from left-handed elliptically- or circularly- polarised light. In metallic reflexion there is always a particular angle of incidence, at which circularly-polarised light is converted by reflexion into plane-polarised light. The two vibrations which make up the circular or elliptic vibration of the ether in a circularly or elliptically-polarised beam of light are not in a condition to interfere with one another on account of their difference of phase, because they are executed in planes at right angles to one another. If a beam circularly or elliptically polarised by an interposed lamina be received upon a birefringent analyser, it is split into two parts, one an ordi- nary ray, the other an extraordinary ray, and each of these is plane-pola- rised. In Fig 193 AB is the plane of original polarisation, CD a principal section of the interposed lamina, EE' a principal section of the analysing crystal. Then a plane-polarised ray whose amplitude is represented in magnitude by the line Oa, and whose plane of polarisation is AB, is resolved by the interposed lamina into two, Oc and Of, which are upon emergence compounded into a plane- or an elliptically- or circularly-polarised beam, according to their relative retarda- tions. When this strikes the analyser, its components Oc and Of are them- selves resolved each into a pair of com- ponents parallel and at right angles to EE' ; these are respectively Oe f and Oh from Oc, and Oe and Og from Of. In the plane EE' we have there- fore two vibrations, Oe' and Oe ; in the plane at right angles to EE' we have the vibrations Og and OA. But Oe' and Oe differ in phase ; so do Og and Oh. These are therefore in a condition for interference. The ordinary ray, passing through the analyser, is made up of the mutually- interfering components, Oe' and Oe, and the extraordinary of Og and Oh ; the effect of interference is to cause a distribution of energy such that the ordinary ray gains or loses as much energy as the extraordinary loses or gains, and thus the energies of the Fig.193. xv.] DOUBLE REFRACTION. 563 ordinary and the extraordinary rays are, taken together, equal to the energy of the incident plane-polarised ray. The amount of relative retardation caused by the interposition of the doubly-refracting lamina, when meas- ured in wave-lengths, depends upon the particular kind of light employed. Hence when the original plane-polarised light is a white light, each colour obeys its own law ; each colour, if strong in the ordinary, is weak in the extraordinary ray, and vice versa ; thus the extraordinary ray and the ordi- nary are coloured, and their colours are complementary. The following are the limiting cases : 1. There is no extraordinary ray when (a) AB, CD, and EE' (Fig. 193) coincide. (6) AB and EE' coincide, and CD is at right angles to them. 2. There is no ordinary image when (a) AB and CD coincide, and EE' is at right angles to them. (b) CD and EE ; coincide, and AB is at right angles to them. 3. The two images are equal for every colour, and are therefore white (a) When AB and CD coincide, and the angle AOE' = 45. (&) When AB and CD are at right angles, and the angle AOE' = 45. (c) When CD and EE' coincide, both being at an angle of 45 with AB. In every other position the two images are complementarity coloured. Determination of the character of a Beam of Light. A crystal of Iceland spar capable of rotation round a longitudinal axis may be used as an analyser, and will enable one, with the intervention of a doubly-refracting lamina, to determine the character of a beam of light falling upon it. Plane-polarised light : as the prism is rotated, the ordinary and the extraordinary images appear and alternately wax and wane, disappearing and reappearing. In this instance the doubly-refracting lamina is dispensed with. Elliptically-polarised light and partially-polarised common light : the two images never entirely disappear, though they become alter- nately brighter and dimmer. Circularly-polarised light, and natural light: the two images do not vary in their relative intensity with the rotation of the prism ; they continue nearly equal. Elliptically and circularly-polarised light on the one hand, and common light unpolarised or partially polarised on the other, are distinguished by the respective actions upon them of a quarter-undulation plate, interposed between the source and the analyser ; the former are converted by this plate into plane-polarised light, the latter are not ; and the former then produce only one image in some positions of the analyser, while the latter always produce two. Colours produced by interposed film. When a polariser and an analyser are so placed that the latter quenches the light which the former transmits, the interposition between them of a plate of mica or selenite, or any other doubly-refracting sub- stance, will cause light again to reach the eye, provided that the principal section of the interposed substance be neither par- 564 OF ETHER-WAVES. [CHAP. allel nor at right angles to the principal sections either of the polariser or analyser. In Fig. 193 above, let the angle AOE' be made a right angle ; Og and O^ come to coincide in direction with AB ; Oe and Oe' with GH, at right angles to AB. The polariser allows ab to pass : the analyser cuts off all components polarised in the plane AB ; whence crossed prisms produce per- fect darkness. But the intervention of the doubly-refracting substance resolves the light which cannot traverse the analyser into two rays, of each of which there is some part that can traverse that obstruction. If the doubly- refracting substance interposed be uniform in thickness, the whole field under crossed prisms becomes filled with uniform coloured light ; if the polariser, or the analyser, or the interposed film, be turned round, the light first becomes white, and then passes into the complementary colour. The colours produced by a given film depend upon the amount of relative retardation produced by it in light of each kind. This depends upon (a) the substance of the film and its refractive indices ; (6) its thickness ; (c) the inclination at which the ray traversing it strikes it; (c) the relation of its optic axis or axes to the plane of its surface. When an irregular film of mica or selenite, flaked off with a penknife from a large mass, is interposed between crossed prisms, the eye, looking through the analyser, sees the darkness of crossed prisms transformed by the interposition into a series of gorgeously brilliant colours ; and as the analyser is turned round, these fade away into white light, and reappear in comple- mentary hues. If the film be a very thin wedge, each thickness of it produces its own colour, and a kind of spectrum is thus produced. A double wedge of quartz, known under the name of Babinet's compensator, and shown in Fig. 194, acts as a virtual film of graded thickness, and gives a series of fringes pi 1M or spectra. This chromatic property of a Axis doubly-refracting film and an analyser may be made use of to detect polarised light: if the light looked at through such a system be wholly or even partially polarised, the phenomena of polarisation-colours come into view ; and while, for example, natural light in such a case gives two nearly equal white images when a crystal of Iceland spar is used as the analyser, circularly-polarised light, on the other hand, gives two complementary coloured-images of almost exactly equal intensity equal, that is, from the physical point of view, though to the eye these coloured images may not seem equally bright. xv.] DOUBLE REFRACTION. 565 When a divergent or a convergent beam of white light passes normally through an interposed film cut at right angles to its axis, the centre of the ordinary image is, when the analyser is parallel to the polariser, found to be bright and colourless, while round this there is a series of annular fringes or spectra, the local colours of which depend upon the local relative retardations ; the whole being traversed by a colourless cross, whose branches are parallel and at right angles to the plane of polarisation. At the same time, the extraordinary image presents the complementary appear- ances a black centre, a black cross, and complementary col- ours. When the analyser is turned round through 90, so that the ordinary image becomes an extraordinary one, it reverses its appearance. This cross is really a coincidence of two crosses, one parallel and at right angles to the primitive plane of polarisation, and the other parallel and at right angles to the principal section of the analyser. When a lamina is interposed whose axis is not at right angles to its surface, the coloured (or isochromatic) lines are modified into hyperbolic curves, or even into lines nearly straight. When the lamina used has been cut from a binaxial crystal, the isochro- matic lines are converted into a series of curves known as lemniscates, and the dark or colourless crosses are represented by a pair of hyperbolic curves. The doubly-refracting power of a body may be de- tected when it is placed between crossed prisms, and by this means it is found that substances which are ordinarily iso- tropic become doubly refracting when they are exposed to compression, or to dilatation, or flexure, or torsion, or vibra- tion (especially at the nodes), or to molecular stress, as where they are heated and then suddenly cooled, or to electrical stress ; and crystals ordinarily isotropic become double-refract- ing when exposed to mechanical stress, or when they crystallise irregularly or are not homogeneous. Organic tissues are by this means for the most part found to be double-refracting, and they seem, when placed between crossed prisms, to shine by their own light again.st a dark background a circumstance favourable to definition, for there is no diffraction of light round the fibres, but practically of little utility, for it is difficult to get prisms of Iceland spar sufficiently clear to be interposed in the path of the rays coining from a high-power objective. It has been proposed to make use of a dynamometer which measures forces by the compressions exerted on glass which is interposed between crossed prisms, these compressions being estimated by the colours produced : 566 OF ETHER-WAVES. [CHAP. the greater the compression, the greater the difference of phase set up between the ordinary and the extraordinary rays, and the greater the wave- length corresponding to that colour which is cut out of the emergent light. It has also been found that slices of different minerals placed between crossed prisms act in very characteristic manners, and are thus, in many cases, easily identified. Andrews proposed as a test for sodium to make sodium-platinum chlo- ride, which produces, when placed between crossed prisms, colours so vivid and characteristic that the millionth part of a grain of sodium can be detected by this means. ROTATORY POLARISATION. When natural white-light is passed through a polariser, then through a film of mica or selenite cut parallel to the axis, and lastly, through an analysing prism of Iceland spar, it gives, as we have seen, two colourless images of the source of light. If now we replace the mica or selenite by a slice of quartz cut parallel to the axis, the two images produced are com- plementarily coloured. If their light be examined with a prism, it is found that the spectrum of the light of the extraordinary image is lacking in a particular region, which presents a dark band, while that particular region is bright in the spectrum of the ordinary image. Further, as the analyser is turned round, the dark band in the spectrum of the extraordinary ray seems to travel up or down the spectrum ; and if the piece of qiiavtz used be very thin, this dark band may traverse the whole spectrum while the analyser is rotated through an angle of less than 180. That particular kind of light which is absent in the extraordinary ray leaves the quartz plate in a condition of polarisation in a plane parallel to the principal section of the analyser. Each position of the analyser cuts off a distinct kind of light in the extraordinary ray : hence light of each colour must have become polarised in a special plane, and the plane of polarisation of the light incident upon the quartz has been rotated, that of each component colour to a specific extent. Rotation is easier to detect with polarised than with common light; but common light is similarly rotated, as interference-experiments may be made to show. Biot found that a, the amount of angular rotation of the plane of polarisa- tion of each colour, was, very roughly, proportional to the square of its wave- frequency, or inversely proportional to the square of A., the wave-length. Boltz- mann showed that the true law is that a =(A -r- A 2 ) + (B -4- A 4 ) : in quartz, for example (Stefan), a = [(7-07018/10 6 ) f A 2 ] + [(0-14983/10 12 ) -*- A 4 ], where A. is the wave-length in mm., and a the rotation produced by a slice 1 mm. thick, this rotation being measured in degrees of angle. We have seen that a plane-polarised beam is equivalent to two equal and opposite circularly-polarised beams ; but quartz allows a right-handed circularly-polarised beam to travel faster througn it than a left-handed one ; at any given point the right-handed component is therefore not so advanced in its phase as its left-handed companion : this is equivalent to a relative gain of phase by the so-called left-handed component (see definition, p. 515); this causes the plane of the plane-polarised ray gradually to turn to the right, in the same direction as the hands of a watch when the ray is looked at from xv.] ROTATORY POLARISATION. 567 behind, from polariser towards analyser. Ward thinks the periods are altered, not the velocities. A piece of quartz 1 mm. thick thus turns the plane of polarisation of yellow rays about 22 ; a piece about 16-36 mm. thick will turn it through 360, for the amount of rotation is proportional to the thickness of the rotating medium. For the Fraunhofer line B the specific rotatory power of quartz (1 mm. thick) is 15-55; for line D, 21-67 ; for line II r 50-98. A substance which acts in the same sense as quartz is said to be dextro-rotatory or positive; one which, causing a relatively- slow pro- pagation of right-handed circularly-polarised light, rotates the plane of polarisation to the left, is laevo-rotatory or negative. This property is not confined to crystals. The following list comprises a few examples of bodies of each kind : Dextro-rotatory. = Some samples of quartz ; cane sugar, grape sugar, camphor ; many essential oils, such as oil of orange, oil of caraway ; cincho- nine, quinidine ; castor-oil. Lsevo-rotatory. Some samples of quartz ; oil of anise, oil of mint, oil of turpentine ; quinine ; sugar of fruits, starch ; albumin. The rotatory powers of different substances are compared by means of two constants, (a) The real specific rotatory power; the rotation, for a given colour or Fraunhofer line, produced by a layer 1 mm. thick of the substance itself. The symbol a D denotes the real rotation for the Fraunhofer line D. (6) The apparent specific rotatory power, [a], for a given line or colour ([a] D , that for the line D) ; the rotation produced by a substance in a state of dilution. It is equal to a/dp, where a is the observed rotation (measured in degrees), e the quantity of active substance per gramme of solution, I the length of the column employed, and p the density of the solution. The apparent specific rotatory power is slightly increased by rise of temperature and modified by the nature and proportion of the diluent substance. With these variations, for cane sugar [a] D is about 67; for milk-sugar a-lactose 80, /^-lactose 54-5; for crystallised grape-sugar in 7-68% solu- tion [ a ] D = 52-89, in 82-6% solution [a] D = 57-8. A column 20 cm. in length of solution of cane sugar, containing in each 100 cubic cm. 16-350 grms. of cane sugar, is equivalent in rotatory power to a plate of right-handed quartz 1 mm. thick. This fact, coupled with the fortunate circumstance that the rotatory dispersion for quartz is the same as that for cane sugar and glucose, enables the strength of solutions of sugars to be approximately determined by means of a Saccharimeter. Essential oils are found to retain their rotatory power unimpaired (due allowance being made for proportionate dilution) when in dilute solution, or even when in the state of vapour, provided that they undergo no chemical change. When substances Isevo- or dextro-rotatory are mixed with each other or with indifferent substances, and if there be no chemical change, the rotatory effect of the whole is found by multiplying the rotatory index of each substance by the proportion in which it is present, and finding the joint effect of the components of the mixture by a process of simple addition. If a rotatory substance assume the crystalline form, its rotatory action is very often masked by double refraction : whence solids, such as camphor, are 568 OF ETHEE-WAVES. [CHAP. generally best examined in solution ; exceptions to this being found in some cases, such as those of benzile and chlorate of soda, where the rotatory power depends upon the crystalline structure, and in which the crystals are gen- erally hemihedric, or, as it were, distorted towards one side. Rotatory polarisation is thus due either to crystalline arrangement of molecules or to the structure of the molecules themselves ; and it has been shown (van 't Hoff) that bodies gifted with the molecular power of rotation have, in their chemical graphic formulae, a marked want of symmetry. It is thought that wherever there is such asymmetry of the molecule there ought to be rotatory power, but that this is masked by different molecules possessing opposite asymmetries : this is illustrated by dextro-rotatory and Isevo-rotatory tartaric acid, whose crystals possess opposite asymmetries, but which, when mixed in solution, form non-rotatory racemic acid. We are now in a position to understand the pieces which make up a Soleil's saccharimeter. 1. A Nicol's prism, achromatised by a prop- erly shaped prism of glass through which the transmitted extraordinary ray passes : the achromatic Nicol, thus acting as a polariser, is so placed that the light transmitted by it is polarised in a vertical plane. 2. A double-quartz plate, or Biquartz; two semicircular plates of quartz joined by a vertical cement-line, and thus forming a circular disc of uniform thickness : the two halves have opposite rotatory power, and their thickness is so adjusted that they respectively deviate through 90 in oppo- site directions the plane of polarisation of incident plane-polarised greenish- yellow light ; they therefore both deviate greenish-yellow light, incident upon them and polarised in a vertical plane, into the same horizontal plane. 3. A Liquid-holder; a tube fitted with clear glass at each end, in which is placed a layer of the liquid to be examined, 10 centimetres in length, such being the distance between the terminal glass-plates. 4. A Compensator. This is in its effect a quartz plate of variable thickness. It consists of two pieces of quartz of a wedge shape. One of these can be made to slip over the other ; the central thickness is thus variable at will. The amount of P """"%?'%- ^^Wljjijj^ movement can be measured by means of a vernier connected with the one wedge, and a scale connected with the other. When the zero of the vernier coin- cides with the zero of the scale, the thickness of the compensator is such that it exactly neutralises the rotatory effect of one of the halves of the biquartz, while it doubles that of the other, the effect being, in both cases, to bring the light back to the original vertical plane of polarisation. 5. An Analyser: this is generally a Nicol's prism. 6. A Lens to be focussed on the biquartz. To use the instrument: Fill the liquid-holder with water, and put it in position ; focus the lens 6 so as to obtain a clear image of the biquartz ; make the vernier and the scale of the compensator to coincide ; turn the analyser round until there is observed to fill the field a particular hue, lying between the red and the blue, and called the teinte de passage ; this hue being chosen out of the many which will successively come into view, on the ground that as the instrument is constructed, the appearance of this colour denotes that the principal section of the analyser is parallel to the plane of polarisa- tion of the yellow ; if the yellow could have passed through the analyser it would have been transmitted as an ordinary ray : but a Nicol prism cuts off the ordinary ray ; the yellow is therefore cut off ; the extraordinary ray, xv.] EOTATOKY POLAEISATION. 569 which alone passes through the analyser, is thus represented in colour by white daylight, minus its bright yellow : the remainder produces in the eye the effect of a dim lavender gray, which, with great sensitiveness, merges into red on the one hand, or into blue on the other, when the analyser is slightly rotated. Both halves of the quartz plate appear of the same colour, because, from them both, the yellow light issues polarised in the same horizontal plane. If now the water in the liquid-holder be replaced by the liquid to be tested, and if that liquid have rotatory power, the two halves of the quartz will cease to appear of the same colour : the liquid aids the rotatory effect of the one, and is opposed to that of the other. The effective thickness of the compensator is now varied until the rotatory effect of the liquid is neutral- ised : the vernier shows by its displacement, how much the thickness has been increased or diminished : the graduation of the vernier is arbitrary, but a displacement of one step on the scale amounts generally to a difference of one-tenth of a millimetre in the thickness of the quartz ; and as the vernier reads to tenths, the positive or negative alteration of thickness of the quartz, found necessary to restore the uniform coloration of the field, may be meas- ured to the hundredth of a millimetre. If the thickness of the compensator have to be diminished, the liquid has rotatory power similar to that of the quartz used in making the compensator ; if it have to be increased, its action is contrary to that of the quartz. It is necessary to know of what kind this quartz is ; this being known, it can be stated that 100 mm. of the liquid are equal, positively or negatively, to so many millimetres of dextro-rotatory or Isevo-rotatory quartz, as the case may be ; and thus the rotatory power of the liquid can be specified with precision. Thus a layer of water 10 cm. thick, containing diabetic sugar in solution in the proportion of 10 grammes per litre, is equivalent to a thickness of 342 mm. of right-handed quartz : the thickness of a dextro-rotatory com- pensator of quartz would have to be diminished by an amount corresponding to 34-2 divisions of the scale on the interposition of a solution of that sub- stance of the given thickness and the given strength; while if the solution were weaker or stronger, the amount of change of thickness of the quartz, as shown by the amount of displacement of the vernier, would be approxi- mately proportional to the strength. If the liquid to be examined be coloured, a Nicol and a quartz in front of the saccharimeter will frequently enable this to be corrected, by filling the field of view with the complemen- tary colour. For other Saccharimeters in use, see Watt's Dictionary of Chemistry, Suppt. iii. p. 1198. TRANSFORMATIONS OF THE ENERGY OF ETHER-WAVES. We have already seen this energy transformed into molec- ular work in the processes of photography, and it is now merely necessary to remark that whatever increases the absorption of light by a set of molecules, increases the chemical work done by the incident ether-waves ; if, for example, a spectrum be cast upon a photographic plate prepared with collodion 'in which chlorophyll has been dissolved, the local of the chlorophyll 570 OF ETHER- WAVES. [CHAP. absorption-bands comes out most strongly in the resultant pho- tograph of the spectrum. The impact of ether-waves upon some substances gives their molecules a new arrangement : selenium is thus so acted upon by light that it becomes a better conductor of electricity than it is in the dark ; and hard rubber is superficially acted upon by light, so that when the incident beam is intermittent or har- monically variable in intensity, the rubber emits a sound which reproduces in its pitch or its complexity the peculiarities of the incident light. It has been proposed to call the last-mentioned property of hard rubber the sonorescence of that substance. As to the mechanical or molar work, the pressure exerted by the impinging ether- waves, though small, is definite. The energy in one cub. cm. of sunlight at the earth's surface is about 6 ,oooooo erg, and the pressure of direct sunlight per square cm. is about 67yiroo~oo dyne, or, roughly speaking, about the weight of 4 Ibs. on each square mile of ground. This would, if the earth were a rigid obstacle, press upon its whole sur- face with an aggregate force of about 884 x 10 n dynes. This force, acting upon the earth's mass (614 x 10- 5 grammes) would produce an outward acceleration of 1-44 x 10~ 14 cm.-per-sec. per second, or a yearly acceleration of 14-33 cm.-per-annum. The effect due to this, say 10,000 miles in 15,000 years, would be small in comparison with the uncertainties which have existed as regards the earth's true distance from the sun ; but it would pro- duce other retardations which do not appear to have occurred. The earth cannot, however, be regarded as a rigid obstacle ; it is to a great extent pervious to the ether in which it travels. The mechanical effect of ether-waves is rather to be looked for in their heating effect than in direct pressure. They may heat absorbent gases, such as ammonia, and cause them to do mechanical work, or to produce sound if the incident beam be intermittent or harmonically variable. OPTICAL INSTRUMENTS. The Eye, considered as a simple lens, brings parallel rays incident upon the cornea to a focus upon the retina. Hence, when it is at rest, as when one meditatively contemplates space, it is adapted for vision of infinitely- distant objects, or, as the phrase goes, it is accommodated for infinity. To look at nearer objects requires an effort for each -an effort of accommo- dation. This is effected by increasing the convexity of that part of the eye called the crystalline lens, which is normally flattened. The range of accommodation provided by our power of varying the form of the crystalline lens is the same as if we were provided with a set of lenses of all focal lengths between infinity and about ten centimetres. xv.] OPTICAL INSTRUMENTS. 571 The front of the eye has the form of a prolate spheroid, and parallel rays tend, after refraction thereat, to converge upon one point, without spherical aberration. The external rays are cut off by the iris, which acts as a diaphragm whose aperture can be automatically adjusted according to the brightness of the incident light, and thus again tends to counterbalance spherical aberration. The crystalline lens of the eye varies in density from surface to centre, and thus again spherical aberration is reduced. By these means the image formed on the retina is made as clear as usually need be : but spherical aberration is never completely absent. The eye presents several other faults, as we find when we expose it to severe tests. Its several parts are not truly centred. Its surfaces are never truly symmetrical round an axis. It is often too long in the bulb, so that rays are brought to a focus before arriving at the retina, and produce, instead of clear images of the several points of an object, a number of overlapping diffusion-circles, and there is consequently produced a blurred image of the whole ; this condition requires the use, in front of the eye, of thick-edged lenses, in order somewhat to diverge the incident beam. The bulb may, on the other hand, be too short, so that converging lenses are necessary. The images of equally-distant coloured objects can never appear equally distinct, for they are not in focus at the same time. The front of the cornea has frequently a somewhat cylindrical form, in consequence of which horizontal and vertical objects do not come to the same focus. The field of vision is extremely limited, and the most sensitive part of the retina is excentric. Yet for all this we are for the most part insensible to these defects ; we have the power of adjusting the eye with extreme rapidity for all the parts of an extended object, and we have been educated by experi- ence to use both our eyes, and thus, by blending the separate pictures pro- vided by the two eyes, to form judgments as to the solid form and distance of remote objects, a power which we discover to have depended greatly upon binocular vision when we try, shutting one eye, suddenly to touch any given object at arm's length, though it can be cultivated even with one eye ; just as microscopists who have long used a monocular, and cultivated a habit of keeping the fine adjustment in action, find no perspective advan- tage in the use of a binocular microscope. . With age the power of accom- modation wanes ; for near objects the image cannot be brought to a focus on the retina ; and then, in order clearly to see near objects, the aid of con- vergent lenses must be sought. The Microscope. An ordinary thin-edged lens is called a simple microscope or magnifying glass. The simplest compound microscope is formed of an objective a combination of lenses which converges the rays, divergent from the object, into an inverted real image, achromatic and aplanatic (i.e. devoid of the effects of spherical aberration), in a plane in space between itself and the eyepiece and of a convergent eyepiece, also compound, and corrected for spherical and chromatic aberration, which mag- nifies this inverted real image, and produces an inverted virtual image at an apparent distance from the eye, not less than that of the nearest distinct vision. The real image formed by the objective must be at the focus of the eyepiece ; hence, when a more highly-magnifying eyepiece is used, in order to throw the real image up to the focus of the eyepiece the objective must more closely approach the object examined. In practice the eyepiece con- sists essentially of two parts; that nearer the object is the field-glass, which catches the rays that were on their way to form a real image far up 572 OF ETHER-WAVES. [CHAP. the tube, and brings them to another focus, so that they form a real image at a point which coincides with the focus of the upper part of the eyepiece that is, at the place where the diaphragm within the eyepiece is situated. To make this real image visible, remove the upper lens of the eyepiece, and drop a disc of oiled tissue-paper down upon the diaphragm within the eye- piece. The field-glass brings the whole image of the object within the field of view of the eye-glass. The rays from the objective, instead of being received by an eyepiece, may, the eyepiece being removed, be allowed to fall upon a screen ; they will there form a real image of any size, which may be traced by hand, if the illumination be sufficient ; if the screen be a sensitised photographic plate, a photograph may be produced. In this case, the image being more remote, the object must be nearer the lens than it is in the ordinary use of the instrument. The minute virtual image of surrounding objects produced by reflexion from a globule of mercury is one of the most trying tests for an ordinary microscope. In the astronomical telescope parallel rays from a distant star are made to converge and form a small real image; this is examined by a simple achromatic eyepiece. The image is inverted like that in the micro- scope. In the terrestrial telescope rays nearly parallel are made to converge and form a small inverted real image ; this small image is magnified and reinverted by an arrangement of lenses equivalent to a compound microscope. In the opera glass a convergent lens directs incident rays towards an inverted real image, but before this is formed the rays meet a divergent lens, Fig.196. which causes them, instead of converging towards a real inverted image, to diverge as if from a virtual erect image, as is shown in Fig. 196. This combination of lenses Galileo's doublet is one of the simplest and most useful. In the ophthalmoscope, as used for the observation of an erect image of the fundus of the eye, the principle of Galileo's doublet is sometimes util- ised. In the first place, light is made to fall upon the fundus of the eye by means of a concave mirror held in the hand or fixed upon the forehead of the observer. The fundus is thus illuminated, and becomes a source of light. Rays from it pass towards the eye of the observer through a central aper- ture in the mirror, placed opposite the eye of the observer. These rays from the fundus are, if the eye observed be myopic (too long in the bulb), ren- dered convergent by the media of the observed eye itself, and a thick-edged lens, placed near the eye observed, causes them to enter the eye of the observer as if they had proceeded from an enlarged erect virtual image. The con- vergent lens of Galileo's doublet is thus represented by the observed eye itself, while the biconcave lens employed makes up the pair of lenses. If the eye observed be normal, and accommodated for an infinite distance, rays xv,J OPTICAL INSTRUMENTS. 573 proceeding from any point of its retina emerge parallel, and a second lens is not absolutely necessary if the observing eye be normal, for the rays come to a focus on the observing retina, if the observing eye be also accommo- dated for infinity ; if the observed eye be too short in the bulb the rays are, on emergence from it, still divergent, and in this case a convex lens is necessary. The ophthalmoscope may also be used in such a way as to give an inverted image, not so much magnified as in the preceding case, but more extensive in its field, brighter, and more easy of attainment. A beam of light reflected from the mirror converges upon and passes through a focus ; it then diverges on its way towards the eye, but encounters a thin-edged lens which causes it rapidly to converge into and then to pass through a focus within the eye, and, after traversing this focus, to illuminate a wide area of the fundus of the eye. Light from the illuminated f undus is collected by the biconvex lens before mentioned, which forms a real image; the rays from this image pass on, through the aperture in the mirror, into the eye of the observer, who then perceives an inverted and magnified image of the fundus of the eye, an image which may be still further enlarged by means of a second convergent-lens placed behind the aperture of the mirror. VISUAL PERCEPTION. The retina is not a uniform surface, but is made up of elements whose average distance from one another, in the yellow spot, is about -005 mm. Distant points, whose angu- lar distance is such that their images on the retina are less than 005 mm. from one another, seem to blend into one, and thus two stars, whose angular distance is less than 70", appear to the eye as a single star. The stimulation of nerves is associated with chemical work in the nerve-ends ; and this with absorption. In this respect it is interesting to find that the retina, which is particularly sensi- tive to yellow and green light, absorbs green and yellow light, and in white light appears purple. It has been pointed out that the blindness of the eye to heat-waves and actinic waves is of advantage : for the energy of heat-radiation is relatively so great that everything would appear intensely bright, and our ordinary vision of objects would be impossible if the rays of dark heat were visible ; while if the ultra-violet rays were visi- ble, the image of every point would be shrouded in a haze due to chromatic aberration. Prof. S. P. Langley estimates that the amount of energy which is neces- sary to produce vision ranges from j^ erg for the extreme red to 100 .oooooo erg for the green of the spectrum. When two colours affect any one element of the retina at the same time, the resultant sensation is that of a single colour, 574 OF ETHER-WAVES. [CHAP. not the same as either of the components. Thus, when the stim- ulations which would separately give rise to red and yellow are superposed, the resultant impression is one of orange ; red light and yellow light together make orange light. In the same way yellow and green make yellowish-green or greenish-yellow ; and in general, colours near one another in the spectrum give rise, when compounded, to an average or intermediate sensation. But in the same way green and purple-red in different propor- tions, that is, of different intensities, will produce all hues of purplish-red, red, orange, yellow, greenish-yellow, and yellowish- green ; green and indigo-blue, all hues of greenish-blue and blue ; indigo-blue and purple-red, all hues of violet-blue, violet, and purple, up to the purple-red employed. Methods of Mixture of Colours. 1. A source of light : a prism : a screen upon which a spectrum is formed : two slits in the screen, so placed as to admit the passage of two selected colours of the spectrum : achromatic lenses behind the screen converge the two coloured beams towards a com- mon crossing point : a screen there placed indicates the mixed colour. 2. A V-shaped slit in a screen (von Helmholtz) ; a prism behind this : the two spectra produced overlap each other and produce a very extensive series of combination-colours. 3. Maxwell's discs: e.g., a disc of red- and one of green-painted card- board : each disc is slit down to the centre, and cut out at the centre so as to be fitted upon a rotating top : the one disc being slipped through the other, the relative proportions of red and green in view can be modified at will : the whole is rotated at such a rate that the successive impressions of red and green enter the eye at least from twenty-five to fifty times per second : each local impression of red in the retina is still vivid while that of green has already commenced, and vice versa ; the colours blend in the eye, and various shades of orange-red, orange, yellow, or yellowish-green are produced, according to the relative proportion of the colours blended. 4. Parallel rays are caused, by a lens, to converge upon a focal point ; the light traversing different portions of the lens is, by the interposition of transparent coloured screens, diversely coloured; all comes to the same focus ; the eye, placed axially at the focus, receives mixed rays ; the colours blend in the eye (Aitken). But a colour on one side of the green, when blended with one on the other side of it, produces always a certain amount of white light, which dilutes the resultant colour : and if we try to blend yellow with blue, we obtain nothing but white light. Yellow and blue are said, then, to be complementary to one another : and to every colour there is some complementary col- our. The complement to green is purple-red, which does not happen to be in the spectrum, but is intermediate between the spectral red and violet. xv.] VISUAL PERCEPTION. 575 That yellow light and blue light make white light is contrary to the general impression, which is that yellow and blue make green : when yellow and blue pigments are mixed, the yellow and the blue lights reflected from the mixture destroy one another, forming white light ; and the resid- ual green, never absent even from the purest pigment-reflected blue or yellow light, is perceived, somewhat wanting in brightness, and diluted by the white light produced by the complementary colours. The phenomena of Double Refraction enable us to produce an indefi- nite number of complementary colours. Any pure colour or hue may, by means of Maxwell's discs, be diluted with varying proportions of white or of black. Pure red thus treated passes through pink tints to white, or through brown shades to black. Every one of these tints or shades will be complementary to the colour (greenish-blue) which was complementary to the original pure red ; but the result of the mixture is not, in the case of the darkened shades, a pure white, but a lighter or a darker gray ; and in all cases the proportion of the pure red to the pure greenish-blue (to keep by our example) remains fixed and inde- pendent of the percentage of white or of black added to the original pure red. Any colour in nature can be matched by finding out a proper angular proportion (including the case of complete omission of one or more) for a set of five Maxwell's discs, viz., white, black, vermilion-red, emerald-green, and ultramarine-blue. From this point of view, green would appear to affect the eye as a primary colour, the others being a purple-red arid an indigo-blue. The primary colours are, according to von Helmholtz, a slightly pur- plish red, a vegetation-green, slightly yellowish (wave-length about 5600 tenth-metres), and an ultramarine-blue (about 4820). Young's original statement was red, green, and violet ; Clerk Maxwell concluded they were vermilion, emerald-green, and ultramarine-blue ; and Fick, red, green, and blue. Ilering, on the other hand, contends that there are four primary colours; Helmholtz's purplish-red, and a green, complementary to it, be- tween the Fraunhofer lines b and F ; a yellow (about 5750 tenth-metres), and a blue, complementary thereto (about 4830). The explanation of this peculiar blending power of the eye has been, that every element of the eye which is broad enough to perceive white light consists of three ultimate elements, each of which is capable of perceiving one of the physiologically primary colours ; and that relatively varying degrees of stimulation of the respective nerve-ends give rise to blended sensations of intermediate character. Then it was supposed that those who were " colour-blind " had lost their sensitiveness to one (or more) of these primary colours. But it now appears, as Pole and others have shown, that whenever sensitiveness to green is lost, that to red is lost also ; while that to yellow and that to blue generally remain, but may also be lost together, so as to leave no colour-sense at all. Hence Hering's view seems preferable to the trisensatioiial theories. A spectrum formed by light travelling from a waning source is found to modify its tints as the light fades ; the orange-red seems to become more purely red, the yellow-green more purely green, and so on ; at length the faint spectrum is approximately restricted to red, green, and violet or violet- blue bands ; of each group of nerve-ends, one is feebly stimulated by a given 576 OF ETHER- WAVES. [CHAP. xv. colour, the others are inappreciably so, though if one be stimulated the others can never remain wholly unaffected. On the other hand, if a col- oured light be rendered exceedingly bright, the other nerve-ends participate in the excitement: very bright red seems somewhat orange; violet very easily passes over into whiteness when its brilliancy is excessive. A black colour is due to the absence of stimulation of any of the nerve- ends; and between bright white and black there is a gradation of weak whites which are called grays. The purest black (Chevreul's black) is obtained on looking at a comparatively small hole in the lid of a deep black- velvet-lined box. Fatigue of the retina causes it to become insensible to a colour long looked at: when white light is then looked at, it appears of a hue comple- mentary to that colour, the sense for which has been temporarily exhausted. When some of the nerve-ends of the retina are stimulated, the stimula- tion spreads to some degree : a very narrow white-hot wire appears, especially from a little distance, to be much wider than it really is ; this phenomenon being named Irradiation. In consequence of this, the crescent moon appears larger than that part of the moon which is illuminated by light reflected from the earth ; a candle- or gas-flame appears continuous, though its incan- descent particles are by no means in contact with one another ; and the glowing filament of an electric incandescent lamp appears much thicker than it really is. Perception of Form. The two eyes receive images of different form ; these are blended by a mental operation into a compound image, which experience has taught us to associate with the distance of the several parts of the object. This is applied in the Stereoscope ; two pictures of images taken from different photographic standpoints are formed, one in each eye, and the effect is that of outstanding relief. This may be exag- gerated with singular effect where the photographs are taken from standpoints situated at a mutual distance of several feet : mountain scenery is thus brought into perspective. The same exaggerated effect may be observed when a landscape is looked at through a pair of telescopes, parallel but at several inches' distance from one another, the light traversing each being brought into the corresponding eye by an arrangement of reflecting prisms. The images in the two eyes may often differ in brightness : when this is the case, there is a struggle between the two fields of view, which causes the impression known to us as that of Lustre ; this effect being specially well marked in the case of metals. One of the most curious things in the action of the eyes is that a single mental image is not formed in binocular vision unless the images be formed on Corresponding points of the two retinae : if we displace one eye we see two images; and the relative positions of the eyes are adjusted by a system of muscles, so as to secure this correspondence. CHAPTER XVI. ELECTRICITY AND MAGNETISM. ELECTRICITY and MAGNETISM are not in themselves forms of Energy ; neither are they forms of Matter. They may perhaps be provisionally defined as properties or Conditions of Matter; but whether this Matter be the ordi- nary matter, or whether it be, on the other hand, that all-pervad- ing Ether by which ordinary matter is everywhere surrounded and permeated, is a question which has been Tinder discussion, and which is now held to be settled in favour of the latter view. At first sight it would appear that the electricity of an elec- trified body is a condition of that body itself. When a small piece of resin and a small piece of glass are rubbed together, it is found that after they are pulled asunder, the resin and the glass attract one another with a definite and measurable force ; and that this force varies (beyond a certain small dis- tance) inversely as the square of the distance between them. This attraction across an intervening space has been by some held to be due to a so-called Mutual Action at a Distance ; but when the bodies are pulled away from one another, work is done upon them, which will be restored when they are allowed to approach one another ; and it seems probable that this work has been done, not upon two isolated bodies mutually acting at a distance, but upon a system, which consists of the two bodies together with the Ether between them. This Ether has been stressed by their separation ; the tendency of the two bodies to approach one another is the elastic tendency of the Ether to recover its original condition ; and these phenomena of electric attraction and repulsion may be explained as phenomena of Ether-stress. Two masses of resin rubbed on glass are found -to repel one another ; two masses of glass which have been rubbed with resin 2p 577 578 ELECTRICITY AND MAGNETISM. [CHAP. also repel one another ; in other words, two masses in a s i m i la r electric condition generally repel one another. According to the nature, the size, the dryness, of the pieces of material exposed to mutual friction, and according to some other circumstances, it is found that after friction and separa- tion the force of mutual repulsion or attraction of two electrified bodies varies. One body may thus be more or less highly elec- trified than another ; it is said to possess or be charged with a greater or a less quantity of electricity. Two bodies are said to be equally charged or to be charged with equal Quantities of Electricity when (being of the same size and form) they can precisely replace one another in their action upon other electrified bodies. When two equally electrified bodies, at a mutual distance of one centimetre in air,* repel or attract one another with a force which balances one dyne, they are each said to be charged with a quantity equal to one C.G.S. Electrostatic Unit of Electricity. If one of these bodies, thus said to be charged with a unit of electricity, be brought to an exact centimetre's distance, in air, from a body charged with an unknown quantity of electricity, the force between the two electrified bodies may be measured directly; and if it be equal to Q dynes, the body tested is shown to bear a charge of Q units of electricity. Further, if a body bearing Q units be brought to the same distance from a body charged with Q' units, the force between them, in air, will be equal to Q x Q f = QQ' dynes. Quantity of Electricity is thus treated of as if it were analo- gous to Mass, or Quantity of Matter ; but only as a means of expression. The facts observed can, to a great extent, be stated and systematized by means of the device of attributing the phe- nomena to the existence of Electric Matter, which may be variously distributed ; but it will soon be seen that this is purely a device, and that the electric matter, with whose quantities and actions we deal, is imaginary merely. A piece of glass, after being rubbed with resin, is said to bear a charge of vitreous electricity ; the resin, on the other hand, is said to be charged with resinous electricity. If any body become electrified in any way, it must become either vitre- ously or resinously electrified. Similarly-electrified bodies repel one another; dissimi- * We shall, in the meantime, assume that the medium surrounding the electrified bodies is air in all cases. xvi.] "ELECTRIC MATTER." 579 larly-electrified bodies attract one another; these statements being, when the bodies are very near one another, subject to an exception hereafter to appear (p. 601). When a jet of water issues from a metallic nozzle connected with an electric machine, the particles of the issuing stream, being similarly electri- fied, repel one another, and the jet is broken up into spray. When the nozzle has a capillary orifice, the surface-tension at the aperture is overcome by the electric self-repulsion, and the liquid rapidly issues as if its viscosity were greatly diminished. If a body charged with resinous electricity and one equally charged with vitreous be brought into contact, the charges of both apparently disappear and the bodies resume a neutral state. Vitreous and resinous electricities are thus found to bear to one another the same relation as positive and negative quantities in algebra, and by a purely arbitrary convention charges of vitreous electricity are said to be positive, and resinous negative. The above statements are comprised within the statement that if F be the mechanical force of repulsion between two charges of electricity, these charges being Q units in one body and Q' units in another, and d the distance between them, F = k QQ'/^ 2 '> and if our units of quantity be so chosen that k = 1 when the medium between the charges is air, then F = QQ' /d' 2 . If Q and Q' be both positive or both negative, the product QQ' is positive, and the stress is expansive or repulsive ; while if one of the charges be resinous and the other vitreous, QQ,' is negative, and the stress is such that the bodies appear to attract one another. When an electrified body presents a charge of Q units uni- formly distributed over a superficial area of A sq. cm., its charge per sq. cm. is Q/A =, where Q is the charge on the sphere. (2.) The density of the surface-distribution on the sphere = charge/area =Q/4irr i s=, = Q/, but to i<|>*; hence it is, per sq. cm., $$ a- = 27TO- 2 dynes. A soap-bubble when electrified expands ; the atmospheric pressure is resisted by a self-repulsion or so-called Electric Tension over the surface, whose outward resultant, f = p Q , is equal to !. a- per sq. cm., and the Electric Force <|> = 47TO-, it follows that ;; = <|>' 2 / STT ; and this is equal to the Mechanical Force /= t, in dynes per sq. cm., across the field, in the direction of the lines of force. That is to say, the space surrounding the conductor is under tension, and pulls upon the conductor with a force or traction t =f=p ^- a = 27ro- 2 = ' 2 /S7r, all in dynes per sq. cm. of the surface of the conductor. Let it be borne in mind, however, that what is actually observed in electrostatic phenomena is the measurable repul- sion or attraction, which is found to vary as if it were due to positive or negative charges under the law of inverse squares. The actual movements or tendencies to movement are equally explainable under the assumption that they are due to Ether- stress, which implies that there shall be two localities between which the Ether may be stressed. Accordingly, if an electrified body be " insulated " by being placed on a dry glass stand within a room, the walls of the room are oppositely electrified, and bear a complementary charge, numerically equal in the aggregate to the charge of the insulated body. The-space comprised between the electrified body anxLihe oppositely-electrified walls of the room is a Field of Force, permeated by Lines of Force and Equipotential Surfaces. The lines of force traversing such a field quit the free surface of the insulated body at right angles, and strike the walls of the room, again at right angles. They * For a gravitational analogy, see problem 10, p. 189. xvi.] FIELD OF ELECTRIC FORCE. 583 are, in general, of a curved form. A certain number of lines of force may be grouped within a bundle or Tube of Force, whose cross-sectional area increases as the lines of force diverge from one another, or diminishes as they converge ; and <|>, the resultant local Electric Force, or the mechanical force on a unit of elec- tric quantity placed at any point within any such tube, must vary inversely as the local cross-sectional area of the tube. The Intensity of the Field, or the Number of Lines of Electric Force per sq. cm. of cross-sectional area of any Tube of Force, at any point, is also equal to the local value of <|>. The force upon a unit-charge brought within distance d of a charge Q would be = Q x'unity/rf 2 . The Intensity of the Field, or the local value of , is therefore Q/t/ 2 . This, taken numerically, will be the Num- ber of Lines of Force per sq. cm. cross-section of a Tube of Force cut across at distance d cm. from the charge Q. But the area of a sphere of radius d cm. is 4ird' 2 sq. cm. : hence the total number of lines of force cut by such a sphere, i.e. the Total Number of Lines of Force radiating from a charge Q, = c|> x area = 4?rQ. If the tubes of force be constant in cross-sectional area, the lines of force are parallel to one another, and the equipotential surfaces are equidistant and plane ; the field is then a Uniform Field of Force, the Ether in which is exposed to a uniform intensity of stress. Such a field we find in the central part of the space between two parallel plates insulated from one another and brought to different potentials. The imaginary electric matter or imaginary superficial film appears only at the free ends of the lines of force ; and on a conductor its imaginary local " superficial density " is always o- <)>/47r = V//27r, where/ is the actual mechanical force or traction t per sq. cm. in the field, and the actual force upon a unit electric quantity put in the field, close to the charged conductor. The lines of force tend to shorten themselves and, if they run in the same general direction, to repel one another. The Lines of Force are, it will be borne in mind, merely convenient means of representing to ourselves the actual forces or stresses within the field of electric force. Dust floating in the air between two charged surfaces finds its way along the lines of force to one surface or the other, and if sticky, it agglomerates (Lodge). " Thunder clears the air." The conception of Potential (Chapter VII.) is 'one of the highest importance in the theory of Electricity : but it must be 584 ELECTKICITY AND MAGNETISM. [CHAP. remembered that "Potential" is not in itself a physical state, nor is it an explanation of electric any more than it is of gravitational phenomena; it is a scientific concept; it is an aid to calculation, and it enables us to see and to make use of gravi- tational analogies ; it is based upon the law of the repulsion and attraction of the imaginary electric matter, which law is itself merely a mode of statement of the observed forces in the field of force. The Absolute Electrical Potential at a point is a mathe- matical expression, possessing a numerical value: it measures the tendency which the existing electric forces would have to drive an electrified particle away from or to prevent its approach to the point in question, if such a particle, bearing a charge equal to one unit in quantity, were situated at that point or were brought up to that point; and it is numerically equal to the number of ergs of work that must be done in order to bring a positive unit of electricity from a region where there is absolutely no electric force e.g., from a region at an infinite distance from all electrified bodies by any path up to the point in question; provided always that the trans- fer of the positive unit of electricity be supposed to have no effect whatsoever upon the distribution of the electricity of other bodies in the neighbourhood of that point. Difference of potential between two points. If VQ ergs of work must be done in order to remove a quantity Q of electricity from the point A to the point B against electric repulsion, then the two points A and B are at potentials which, considered absolutely, may be unknown, but which differ numerically by V: and B is said to be at a higher potential than A, by V units of potential. When there is a difference of potential between any two points in space, a body bearing a charge of positive electricity, and placed at the point at which the potential is greater, is driven towards the point of less potential, just as in the corresponding gravitation-problem, a mass tends to fall towards a lower level; and if free to move it will follow the track of the lines of force, travelling thus from each equipotential surface to the next one, indefinitely near it, by the shortest path. The path between the two points is not necessarily the shortest, for the lines of force are often curved (see Fig. 234). A positively-charged particle, placed in a region of posi- tive potential, will be repelled along the lines of force into a xvi.] ELECTROSTATIC POTENTIAL. 585 region of less or of zero or of negative potential : a negatively- charged body under the same circumstances travels in the oppo- site direction. In other words, the Lines of Force correspond at each point to the Direction in which the Potential most rapidly decreases. The Mean Electric Force <|> acting upon a Unit-Charge of electricity within an electrical field is equal to the difference between the potentials of two points within that field and situated at a mutual distance of one centi- metre, that distance being measured along the lines of force : for if V y and V y/ be the potentials of two points whose mutual distance is d, the Work done in moving a unit of electricity from the point of lower to the point of higher potential is V y V y/ ; but this Work done is also equal to <|K/, where 4> is the mean force resisting the transfer ; whence $d = V, - V /y , and 4> = (V, - V /y ) - d. When d = l cm., <|> = V, - V //; and when (V, - V,,) = l,4 = i/d. But this Electric Force on Unit Quantity, <|> = (V, - V /y ) * d, is the Potential-Slope or Potential-Gradient; and in a uniform field this is uniform, for the potential diminishes equably throughout the field. Hence in a uniform field the Electric Force, which is 47rcr near one of the charged conductors, remains equal throughout the field up to the opposed conductor ; and the superficial density a' of the opposed charge must be equal, for = 47TO- = 47TO-'. In a non-uniform field, the gradient varies as the equi- potential surfaces lie nearer or farther apart ; and as Tubes of Force widen out or become narrower, the Electric Force <|> i.e. the number of lines of force per sq. cm. of cross-area of the tubes of force is inversely propor- tional to their area ; and when they reach the opposed conductor, the tubes of force which have left an area A engage an opposing area A' ; hence 4>' = 47T0--A/A'; but this is also equal to 47ro-'; whence AV = A which determines an electric flow : to determine this the form and the dimensions of the electric conductor must be known, as well as the difference of electric potential between its extremities, just as the dimensions of a water-pipe must be known, as well as the available head of water, before we can calculate the local falls or slopes of pressure and the forces pro- ducing flow. We have already seen that = (V, V y/ ) -s- d; and this is Clerk Max- well's " Electromotive Intensity " or true Electromotive Force, the Electric Force acting upon a unit-charge of electricity at any point referred to. This is, in other words, the Potential-Slope across the point in question, in the direction of the Lines of Force. The Cause of D.P., whatever that may be, is also called E.M.F. : and this operates in a direction opposed to that in which the resulting E.M.D.P. tends to act; just as the upward pressure of a force-pump driving water into a cistern is opposed to the downward pressure and flow obtain- able from the cistern when filled. This may sometimes operate from point to point, so as to heap up the resultant D.P. : and when we sum up the effects of all the local E.M.F.'s, we may arrive at the aggregate resultant D.P. produced. Difference of Potential is often spoken of by electrical engineers as Electric Pressure : thus we hear of high-pressure and low-pressure currents. It is also known as Voltage, when measured in the particular units known as Volts. If two bodies be at different potentials, when they are con- 588 ELECTRICITY AND MAGNETISM. [CHAP. nected by a metallic wire the charge over them will be read- justed by a momentary current along the wire, and they will come to the same potential. Two bodies are said to be at the same potential when elec- tricity has no tendency to travel from one to the other, even though they be brought into communication by a metallic wire. Difference of potential is thus also analogous to difference of tempera- ture, and " electromotive force " to temperature-gradient. The earth itself is arbitrarily assumed to be at zero potential : and bodies in such a condition that when they are placed in contact or in metallic communication with the earth their electric condition is unaltered, have a potential whose value is equal to this arbitrary zero. The arbitrary or conventional potential or, briefly, The Potential of a point in an electric field of force is, numeri- cally, the number of ergs of work necessary to bring a unit of electric charge up to the point in question from a region of nominal zero-potential e.g., from the surface of the earth. Between a positively-charged body within a room and the negatively-charged walls of the room there must lie, in the inter- vening field of force, one equipotential surface which has a Zero Potential, its potential being the same as that of the earth out- side the room. Within this equipotential closed surface there is a region of Positive Potential ; exterior to it there is a region of Negative Potential. The potential of the inner region is greatest at the surface of the electrified body ; the potential in the negative region is most negative on the surface of the walls. If the walls of the room be at the same potential as the earth, then the whole field of force is at positive potentials, relatively to the arbitrary or conventional zero of potential. The stress in the field, the forces and poten- tial-difference across it, are not affected by this. The walls and the earth have still been negatively charged ; but a change has been effected in the absolute value of the nominal zero of potential. The potential cannot be a maximum or a minimum at any point within a field of force, if that point be not upon the surface of one of the con- ductors whose surfaces bound the field. Conductors and Non-conductors. In the familiar case of a lightning-conductor we see a marked distinction between the conductive copper along which a continuous current of electricity can flow, and the air or an unprotected building which can only be traversed by a disruptive discharge. A conductor is, when a charge is borne by it and retained by it in equilibrium, a sub- stance throughout the whole volume and over the whole surface xvi.] CONDUCTORS AND NON-CONDUCTORS. 589 of which the potential is uniform ; while if inequalities of poten- tial were set up within it, the conducting material of a perfect conductor would offer no resistance to the readjustment of poten- tial by means of a current. A perfect non-conductor or dielectric would, on the other hand, be a substance the different parts of which may, after an electric disturbance, remain, without any process of readjustment and for an indefinite period of time, at potentials differing to any extent. There are no bodies which are absolute non-conductors ; all conduct electricity more or less slowly. There are no bodies which are perfect conductors ; all offer more or less resistance to the flow of electricity. Bodies which conduct extremely badly are called Non-conductors or insulators : bodies which offer comparatively small resistance to the passage of electricity are in practice called Conductors. The ether in an insulator can stand exposure to a moderate stress ; that in a perfect conductor, for some reason, cannot; that in an ordinary con- ductor yields continuously, with a greater or less quasi-plasticity. When a charged body is placed upon an insulator, such as ebonite, guttapercha, indiarubber, dry glass, sealing-wax, quartz, it is said to be insulated ; its potential cannot become equal to that of the earth for a long period of time ; it is said to retain its charge for a long period. Air at a high pressure is almost an absolute insulator : cold air, damp or dry, at the ordinary pressure is one of the best insulators : but even within cold air, bodies charged with electricity gradually lose their charge ; a partial vacuum is a good conductor ; a good vacuum is again an extremely good insulator. Ice insulates, water is a bad conductor; obsidians and lavas insulate when hot, and steatite even when red-hot ; glass when diy is an insulator, but when very hot is a conductor. A body charged and sup- ported upon a dry-glass stem within a vacuum or a very dry cold atmos- phere will retain its charge for a very long period ; but if the air be damp, so that the insulating glass stem condenses upon its surface a film of mois- ture from the air, that film will slowly conduct the charge to earth. The insulating character of air seems (J. J. Thomson) to indicate that free molecules of air cannot be charged, but that there must be free atoms, or that dust or extremely minute water-drops must be present, before this can be done. A metal or a phosphorescent substance, negatively charged and exposed to ultra-violet light, readily loses its charge ; and it appears that the surface of the conductor is then to some extent broken up into dust and repelled. In a high vacuum (yrooToFo atmo.), silver negatively charged will (Crookes) evaporate in this way so vehemently as to assume a super- ficial glow, so intense is the agitation of its particles. A sufficient difference of potential will cause a spark to fly between two charged conductors across the intervening dielectric : in the case of turpen- tine, paraffin, and olive oil, the striking distance is when the discharge is continuous regularly, when the discharge is interrupted irregularly proper- 590 ELECTRICITY AND MAGNETISM. [CHAP. tional to the difference of potential ; in air the striking distance increases faster than the difference of potential, and the curve indicating the ratios of striking distances to differences of potential is a parabola. A steeper potential-slope, , is required for thin than for thick layers : but a potential- difference less than 1 to 1| electrostatic unit will not, in air, produce a spark at all. Kinds of Conductors. There are two kinds or classes of Conductors : first, those which act as conductors without any apparent displacement of their own substance, such as metals and other substances ordinarily known as good conductors ; and second, those in which the transfer of electricity is accompanied by a relative displacement of particles within the conductor. If a current of electricity be sent through a solution of common salt in water, it will be found that those particles of Na and Cl, into which the salt is likewise on other grounds believed to break up on its being dissolved in water, are displaced within the solu- tion, and that they travel, the Na towards the electrically nega- tive and the Cl towards the positive extremity of the solution, each kind of particle with its own specific velocity. Each such particle or sub-molecule or ion parts at its extremity of the solution with a certain definite quantity of electricity, positive or negative, and no more. Conductors of this kind are called Electrolytes; and they consist mainly of solutions of salts and of chemically-strong acids or bases, in which the whole or a part of the substance dissolved has been broken up into these sub-molecules or ions, but in which the water plays no part as a conductor. The phenomena of electricity present themselves within a conductor only while a current is actually passing through it; for then only are there any differences of potential within the conductor. And even this can only occur when the conductor is an imperfect one ; for within a perfect conductor there never could be any difference of potential set up. With a perfect con- ductor, or when there is no current, but a more or less perma- nent condition of Statical Equilibrium of the charge, electrical phenomena are restricted to the Field of Force' that is, to the non-conductor or Dielectric external to the conductor; for within non-conductors alone, not in conductors, can any electrical stress or difference of potential, permanently or for any length of time, be maintained. If the air had been as good a conductor as copper we would probably never have known anything about electricity, for our attention would never have been directed to any electrical phenomena. xvi.] CONDUCTION. 591 Phenomena of electricity in a state of equilibrium, associated with more or less permanent differences of potential and evinced within a dielectric, are said to be electrostatic ; those evinced during adjustment of electric potential by the passage of a cur- rent along a conductor are said to be electrokinetic. If electrostatic phenomena be due to stresses in the Ether, electrokinetic are due to movements of the same ; and a moment- ary current of electricity in a copper wire is a throb due to a readjustment of the stresses in the Ether or dielectric surround- ing that wire ; the throb is accompanied by a readjustment of the Lines of Force in the field surrounding the wire ; these lines, as it were, slip along the wire, carrying Energy with them in the Ether or other dielectric, not in the wire itself, except in so far as the imperfectly-conducting character of the wire may lead to its acting to some extent as if it itself were a dielectric or non- conductor. The lines of force in that case traverse, and slip along within, the substance of the wire itself. "Free" and ' ' Bound " Charges. A distinction is fre- quently made between afreeandabound charge of electricity. The former is understood to be a charge borne by an insulated body, and independent of surrounding objects, while the latter is such a charge as is held in position by the presence and attrac- tion of a charge of the opposite character upon a neighbouring body. In truth, however, all charges are bound charges ; the complementary distribution must be somewhere; the field of force may be great or small, but it must have its limits. It may be small, as when a little electrified body is suspended within a metal flask which is not insulated; it may be great, like the field of force between a thundercloud and the earth : in the for- mer case the complementary charge is distributed over the inner surface of the flask ; in the latter it travels about, and is at its densest upon the surface of the earth beneath the travelling thundercloud, or else upon adjacent clouds. Even when an electrified body is placed at an extremely great distance from all surrounding objects, it cannot be held to have a free charge, for its charge is bound by the complementary distribution upon the far-distant objects; and a particle isolated in otherwise vacu- ous infinite space, if such a thing were possible, could not become charged with electricity at all, for the complementary charge could, in such a case, have no locus. If the Ether be stretched or compressed, it must be stretched or compressed between at least two points, which may be near 592 ELECTKICITY AND MAGNETISM. [CHAP. or far from one another. Bearing this in mind, however, it is undoubtedly convenient in many respects to permit ourselves the use of such expressions as " a body freely charged with Q units of + electricity," and in so doing to omit, provisionally, all consideration of the complementary charge, which is sup- posed sufficiently distant. This mode of expression is also in accord with the convention that the potential of the earth is always zero. Division of Charge. When a conductor charged with electricity is brought into contact or into metallic communica- tion with another at a different potential, the electric potentials of the two conductors become equalised ; and if the two bodies be of the same form, size, temperature, and chemical nature, and if they be symmetrically arranged, they will, after separation, each bear a charge equal to one-half of the algebraical sum of the original charges of the two bodies. This change of distribution involves a readjustment of the lines of force and of the equipoteiitial surfaces throughout the surrounding dielectric, and an alteration of the distribution of the complementary charge over the oppo- site boundary of the field of force. When the bodies are unequal in size, etc., or are unsymmet- rically arranged, the division of the charge between them is not equal. Two similar but unequal spheres are found, after being brought into communication by a long thin wire, which is then removed, to bear charges proportioned to their radii. Electrostatic Capacity or Permittance. When a con- ductor has a charge of electricity imparted to it, the potential of its surface and of its whole volume is raised, positively or nega- tively (i.e., lowered), as the case may be. When a body, insu- lated in air, requires a charge of C units of electricity to be imparted to it in order to raise its potential by one unit that is, from zero to unity, or from V to V -f- 1 it is said to have a Capacity or Permittance of C units. When this body, so insulated, has its potential raised by the amount V, the Charge of Electricity imparted to the conductor is Q = VC units. When a series of conductors whose electrostatic, capacities are C y , C yy , C /y/ , etc., and which are charged to potentials V,, V y/ , V /yy , etc., so that their several charges are respectively C y V y , C yy V yy , C y// V //y , etc. are connected by a wire, the potential thereupon assumed by the whole system is equal to The whole charge = Y y C y + Y y/ C y/ + V^C,,, The whole capacity C y + C /y + C /yy ... XVI.] CAPACITY. 593 The electrostatic capacity of a conductor is the same whether it be solid or hollow : the merest film of gold leaf supported on a wooden ball has as great a capacity as a solid metallic sphere. Electrostatic stress can only persist within the field of force, the die- lectric, which is limited by the surface of the conductor ; beneath this surface it is a matter of indifference what the metallic thickness may be, since within a conductor there can be no permanent difference of potential, no per- manent electrostatic stress. The quantity C is not really a capacity of the conductor for electricity at all, but is a measure of the elastic yielding of the Field of Force involved. The Electrostatic Capacity of a Sphere. A sphere of radius r, within an unlimited air-space containing no other charged bodies, is charged with quantity Q ; this quantity, uni- formly distributed over the surface, acts as if it were gathered at the centre, and therefore at a distance r from the surface. The potential at the surface of the sphere must therefore be V=Q/r. The capacity C = Q/V; this is Q/Q7r=r; the Electrostatic Capacity of a Sphere, insulated in air, is therefore, in C.G.S. electrostatic units, numerically equal to its Radius. The Work spent in charging any body is equal to half the product of Q, the Charge imparted to it, into V, the rise of Potential produced in it. To bring a quantity, Q, of electricity from a place of zero potential to a place of constant potential, V, involves the expenditure of QY units of work, by our definition of Potential. To bring a charge Q by successive instalments into a region whose potential is at first zero, but steadily rises as the successive instalments arrive, work must be done which is equal to half the product of the final Fig.198. rise of potential into the whole charge brought up. In Fig. 198 the small rectangles represent the work done in bringing each successive instalment up to the corresponding potential; these rectangles increase, and their sum, which represents the total work done in bringing up the whole charge OQ to the final poten- tial QP, is represented by the triangle OQP, whose area = OQ x QP = whole quantity x final rise of potential Quan tity The work done in charging a body is equal to the electrical energy stored up in the field of force surrounding that body ; and since this is equal to JQV, we see that the energy of an electrified body depends not only upon the Quantity of 594 ELECTRICITY AND MAGNETISM. [CHAP. electricity borne by that body, but also upon the Potential; just as the potential energy of a mill-pond depends not only upon the Quantity of water contained in it, but also upon the average elevation of that water above surrounding objects. For which reason a mere Quantity of Electricity is not, in itself, a quantity of Energy ; and therefore Electricity, as meas- ured by Quantity of Electricity, is not, like Heat, itself a form of Energy. The energy of a charged conductor of any kind is measured by QV ; but this is equal (since Q = C V, where C is the electrostatic capacity of the conductor) to |CV 2 or to |Q 2 /C. The energy of a system of connected conductors is equal to ^V 2 - 2C, or to ^Q 2 -4- SC, where 3C is the aggregate capacity of the whole system. Suppose now that two conductors, of which the one is charged to poten- tial V while the other is at zero potential, and of which the respective capacities are C, and C 7/ , are placed in metallic communication ; on contact they form a joint conductor whose capacity is (C, + C ;/ ). The energy of the single charged conductor was ^Q 2 /C 7 ; that of both taken together is JQ 2 /C / + C y/ , a smaller quantity. There is therefore an apparent Loss of Energy equal to &QYC, - $QyC, + C,,} - {ttQY^XC,,/^ + C,,)}, or (C/yC, + C /7 ) times the energy of the original charge. If C y = C /y , half the energy of the original charged conductor is apparently lost by partial discharge, being transformed into Heat. Wherever there is a readjustment of electricity in the form of a running-down of electricity from a place of high potential to a place of low potential, there is a loss of energy of electrifi- cation ; just as when a full pond is allowed partly to discharge itself into an empty one, the average level of the whole is lowered, and the energy of position partly disappears, to reap- pear in the form of Heat. In general, where electrified conduc- tors are connected by metallic wires, if there be a current, the potential energy of the system sinks to a minimum ; heat and if a spark pass light, sound, and mechanical effect being pro- duced. Where the components of an electrified and insulated system are allowed to approach or to recede from one another in obedience to the electric forces, the energy of electrification becomes in part converted into mechanical work, and therefore falls in amount; while if they be pulled asunder or made to approach against the electric forces, the mechanical work done upon the insulated system from without is converted into energy of electrification. In the former case the energy left in the sys- tem is that of the same charge at a lower potential; in the latter case it becomes that of the same charge at a higher potential. Electrostatic Induction. When an electrified body or XVI.] ELECTROSTATIC INDUCTION. 595 system is placed within a hollow metallic shell (Fig. 199), with which there is no communication except through non-conduc- tors, the shell becomes charged by 'Indue- ti o 11 ' across the intervening dielectric. If Fi e- 199 - the bodies placed within the shell be posi- tively charged, the inner surface of the shell becomes negatively, the outer positively electrified. The opposite charge thus induced on the inner surface of the shell, the similar charge induced upon its outer surface, and the original inducing charge on the internally-suspended system, are all equal in amount, if the shell completely or, practically, even if it very largely surround the electrified body suspended within it. Thus the positive and the negative charges called into existence by Induction are together algebraically equal to zero. The Difference of Potential between an inducing sphere and an induced spherical surrounding shell is Q/r Q/V, where Q is the charge on the inducing sphere, r its radius, and r' the radius of the hollow spherical shell. The distribution of the induced charge on the interior sur- face of a completely-surrounding shell is such that on external points it produces an effect equal and opposite to that of the interior insulated charge ; the two inner charges therefore pro- duce together no effect upon external bodies, and the induced charge on the outer surface is the only charge which can affect particles situated in the outer air. The two interior charges are bound to each other, for they are of opposite character, and there is a field of force between them ; the outer charge is said to be free. The distribution of electricity over the inner and outer surfaces of the shell is, if the shell be spherical, governed by the law that the superficial density cr at any point E is a- = (CE 2 ~ CM 2 ) . Q/47T- CE- ME 3 , where C is the geometrical centre of the shell, and M the point at which the charge Q is situated. Then the inner charge Q at M, and the 'opposite charge - Q on the surface (the inner surface of the shell), produce together no effect on surrounding particles. The charge + Q on the outer surface acts as if it were all at M. The lines of force or of induction radiating from M are not displaced by the interposition of the insulated shell ; neither are the equipotential sur- faces : but the shell divides the field of force round M into two regions, of which either can be destroyed without affecting the stress or the potential- slope in the other. At page 583 we saw that the number of lines of force from a charge Q Fig.200. 596 ELECTRICITY AND MAGNETISM. [CHAP. was 4 TrQ. Similarly, the total induction in the inductive field is said to be also 4?rQ, or Q/rf 2 per sq. cm.; and every 4?r Lines of Induction passing out of a dielectric into a conductor are said to be able to induce one unit of charge on the bounding surface of that conductor; so that the Induction per sq. cm., i, = 4-7nr, where cr is the superficial density of the in- duced charge. Then, since the Electric Force <}> is also equal to 4^0-, it is said that the Lines of Force and the Lines of Induction coincide in air, the standard medium. If the outer surface of the shell be connected with the sur- face of the earth, the shell and the earth become one extended conductor, and the positive charge on the outer surface of the shell is repelled to the earth's surface ; it now blends with and neutralises the negative charge previously borne by the earth and surrounding objects in consequence of the original positive electrification of the inducing body. As a result of this we have, within the shell, a purely local field of force, restricted to the space between the internally-suspended body and the interior surface of the shell, and giving rise to no phenomena outside that cavity. The potential of the whole system is lowered by this annihilation of the outer region of the original field of force ; but the potential-slope within the interior field remains unchanged. If, on the other hand, the insulated body within be made to touch the enveloping shell, the internal field of force will be destroyed ; but the outer induced charge will remain, distrib- uted over the outer surface of the shell. Any quantity of electricity may thus be wholly transferred to the outer surface of a hollow insulated conductor, if a charged body be made to touch its internal surface. A sheet of tinfoil charged, and separated from a second sheet by an intervening layer of air or glass or mica or waxed paper, will act inductively across the dielectric. The nearer surface of the second sheet is oppositely, the farther surface similarly charged; and if the second sheet of tinfoil be connected to earth, the similar charge escapes and the field of force is now almost wholly limited to the thin space between the two sheets or plates. A layer of dielectric intervening between two conducting surfaces constitutes an Electrostatic Accumulator or Condenser. In this dielectric layer, a limited Field of Force may be set up, the lines of force through which stretch across from one con- ducting surface to the other ; and the Permittance or electro- static Capacity of such a field is greater than that of the field xvi.] CONDENSERS. 597 set up when either of the two conducting surfaces is separately charged in the open air. From the four equations, Q = ACT (i.) ; <|> = (V, V y/ )/c? (ii.) ; <|> = 47r due to the given charge. If the opposed plates be made to approach one another, the potential- difference between the opposed plates falls proportionately, because = 47TO- remains constant if o- be not altered; and in order to maintain a given potential-difference constant, it would be necessary to increase the charge on the insulated plate : or conversely, if the potential-difference be determinate (e.g., where the opposed plates are each connected with one terminal of a galvanic battery), on mutual approach of the plates the charges upon them will proportionately increase. In this, it is assumed that any "free charge " on the farther surface of either conducting film is so small that it may be left out of account. But see p. 630. The nature of the dielectric between the plates of the con- denser is not a matter of indifference. It is found and this proves that in the phenomena of electrical force the dielectric plays an important part that the permittance or capacity of a condenser varies with the nature of the interposed dielectric, and is proportional to a constant special to each substance (or in some instances even to particular directions within the same substance) and called the Specific Inductive Capacity or the Permittivity, K, of that substance. The sp. ind. cap. of air being taken as a standard and equal to unity, that of sulphur is 3-2. Sometimes the sp. ind. cap. of a vacuum, which differs little from that of air, is taken as unity, in which case the 598 ELECTRICITY AND MAGNETISM. [CHAP. dielectric is the Ether itself. The sp. ind. cap. of glass rises slightly when the temperature is increased (between 12 and 83 C. a rise of 2J- per cent). All gases have very nearly the same inductive capacity, whatever their chemical constitution, their temperature, or their density. If, however, their pres- sure be increased or diminished, the minute difference between their sp. ind. cap. and that of a vacuum is also increased or diminished in the same proportion ; and conversely, when a gas is employed as a dielectric, induction across it diminishes its pressure, the gas then adjusting itself so as to become rarer and consequently less inductive. If a given charge Q, will raise a condenser in which air is the dielectric to a potential V, it will only raise a similar condenser whose dielectric has sp. ind. cap. = K to the potential V/K ; and since in the special case of a conducting material used in the place of a dielectric the difference between the inner and outer coats is zero, the sp. ind. cap. of a conductor may, for some purposes, be considered infinite, for = l/oo . Similarly, the greater the sp. ind. cap., the less is the mechanical Force across the dielectric between two given charges at a given distance; F = QQVKcP. This is the most general formula: the formula F = QQ'/d 2 applies only to air as the dielectric. From the air-formula we get the following Equation of Dimensions: [Q] = [Mechanical Force x distance 2 ] i= [ML/T 2 - L 2 ] * = [M*U/T] ; but from the general formula we get [Q] = [KiM*Li/T]. In electrostatic measurements we arbitrarily assume air as a standard and, for air, K = 1, a Number merely, without Dimensions ; but if we approach the subject from any other point of view, we must use the general formula, not the air-formula, in order to ascertain the true Dimensions of electric quan- tities on any other system of measurement. Whatever be the medium, the quantity of induced charge remains the same : this is expressed by saying that the same number of Lines of Induc- tion pass from a given charge whatever be the medium, and if the given charge be Q, measured in the usual air-units, the total number of these lines is 4?rQ = I, the Total Induction ; or, at distance d the lines of induction are Q/d 2 per sq. cm. = i the induction per sq. cm., = 47r, the numerical value of the potential- slope or Electric Force, in a medium of sp. ind. cap. K, varies inversely as VK for charges of given numerical value, measured with reference to that medium, or inversely as VK x VK = K for charges of an equal number of air-units, these units being smaller. Therefore <|>, 47rcr in air, becomes <}> = 47rcr/K in any other medium, when a- is measured in the usual air-units; but i remains equal to 47r0-; hence i = K<|>, and in a medium of sp. ind. cap. K, the lines of induction are more numerous than those of force in the ratio K : 1. The units of quantity vary pari passu in the terms for i and <|> : hence i is always equal to K : but in air i = <|>. xvi.] SPECIFIC INDUCTIVE CAPACITY. 599 Given charges, Q and Q', measured in air-units, thus produce a mutual force F = QQ'/Kd 2 ; the potential is Q/Kd: the potential-slope or electric force varies as 1/K : the equipotential surfaces are at mutual dis- tances K times as great as in air : the capacity of a given conductor varies as K, e.g. , that of a sphere of radius r is Kr: the work done in commu- nicating a given charge to a given conductor varies inversely as K, for If a field of force be made up of layers of different dielectrics, the potential-slope in each is'inversely proportional to K. There is thus a kind of refraction of the potential-slope at the bounding surfaces of the layers. The sp. ind. cap. of a dielectric diminishes with the time, and is there- fore difficult to measure directly ; and when a condenser is discharged by metallic communication set up between its two coatings, its charge does not at once completely vanish, but the condition of the dielectric is apparently very similar to that of a body which, being imperfectly elastic, recovers slowly and irregularly its primitive form and condition after deformation ; and it is curious that the same means vibration, shaking, jarring, etc. which facilitate the return of such a body to its normal condition after a strain, facilitate the prompt and complete discharge of a condenser whose two coatings are put in metallic connection. On sending alternate charges into a condenser, the residual discharge liberates them in the reverse order (Hopkinson) ; a result strikingly like that of Boltzmann with reference to successive torsions. Quartz employed as a dielectric has one-ninth the resi- dual capacity of glass ; Iceland spar seems to have no residual capacity at all, and permits prompt discharge. The dielectric of a condenser may become double-refracting under the influence of Electric Stress, which tends, without altering its total volume, to dilate it at right angles to the lines of force : its optical axis is parallel to the lines of force. Glass and olive oil become like Iceland spar (nega- tive crystals, p. 555) ; bisulphide of carbon, paraffin, resiri, become positive (Kerr). Solids slowly, liquids instantly, acquire or lose this condition of stress : and when an air-condenser is released from its stress by discharge, there is a distinct sound. Since electrostatic capacity or permittance varies as K, the general for- mula for that of a condenser, such as a Leyden jar, is (K/d)- (Surf ace /47r). The form of condenser known as a Leyden jar usually con- sists of a glass vessel lined internally and externally with tinfoil. The inner coating communicates by wire with a smooth metallic knob projecting externally and insulated from the outer coating. By contact between the knob and a charged conductor "the inner coating is charged. By induction through the glass there is produced -an electrical separation in the external tinfoil. The external surface of this is temporarily connected with the earth. Thereafter there remains a Field of Force in the glass between the two tinfoil coatings. This may be discharged by establishing a metallic communication between the two coat- ings, the outer tinfoil being first touched, then the inner. A Leyden jar when charged dilates somewhat, and as it expands its capacity increases ; the potential, to which a given charge is competent to 600 ELECTRICITY AND MAGNETISM. [CHAP. raise the jar, sinks to a corresponding degree. When discharged, the jar makes a dull sound, and the glass glows at the edges of the tinfoil, while the internal air becomes warm. A submarine telegraph cable is, in effect, a very long Leyden jar. The copper core is the inner coating; the guttapercha or other insulator repre- sents the glass ; the outer coating of tinfoil is represented by the protecting iron wire or by the bounding surface of the sea-water. When a charge of electricity is passed into a deep-sea cable, the cable takes some time to become fully charged : it then bears, for a considerable tinfe, an electrostatic charge upon the surface of its copper core. An ordinary aerial telegraph-wire is again, but to a less marked degree, a Leyden jar. The inner coating is the surface of the wire itself; the die- lectric is the air; the outer coating is the surface of the earth. The elec- trostatic capacity of an aerial wire is small in comparison with that of a submarine cable ; but it is not insignificant. If the two coatings of a Leyden jar be slid past one another so as to diminish the opposed surfaces, the capacity diminishes and the potential due to the actual charge of the jar increases : the potential may thus be adjusted (Sliding Condenser). Batteries of Leyden Jars. When a Leyden jar has its inner coating placed in simultaneous metallic communication with the i n n e r coats of a series of uninsulated jars, the whole becomes in effect one great Leyden jar of increased surface, and the jars are said to form a battery connected in Surface. The charge of the first jar, being then distributed throughout an enlarged conductor, brings it to a reduced potential ; and energy is lost in the production of sparks when the battery is charged by the first jar. A series or battery of Leyden jars is said to be charged in Cascade when the outer coat of one jar is connected by metal with the inner coat of the next, and so forth, while a charge is imparted to the inner coating of the first. The difference of potential between the inner coating of the first jar and the outer coating of the last is distributed between the jars of the bat- tery, and thus the risk is diminished of any of the jars being destroyed through an excessive difference of potential in any one jar causing a spark to pass and perforate the glass. The charge of the whole system is only equal to that of a single jar, and the difference of potential in each of n jars is (V, V )/n, where V, and V are the potentials of the first and the last coatings ; whence the energy of the whole ( = half the whole charge x the whole potential-difference) is the same as the energy of a single jar loaded with the same charge as the battery. If the conductors surrounding an inducing charge do not completely enclose it, the charges induced upon them are each numerically less than the inducing charge, and the sum of those of each kind is also numerically less than that charge. In no case can the induced charges exceed the inducing charge. Coefficient of Mutual Induction. The Coefficient of Induction of a conductor A on a conductor B is the ratio of the Charge (or change of charge) developed in B to the Potential (or change of potential) of A. It can be proved that the coefficient of induction of A on B is always equal to the converse coefficient of B on A ; and this reciprocally valid coefficient is xvi.] ELECTROSTATIC INDUCTION. 601 called the Coefficient of Mutual Induction. It depends upon the relative positions of A and B. Inverting the statement, a unit charge on either body will, by induction, alter the potential of the other by an amount equal in both cases. The effect of induction is seen when an electrified body such as a glass rod rubbed, with a dry silk-handkerchief is brought into the neighbourhood of light bodies suspended or floating in the air. Over each of these bodies there is a separa- tion of electricities ; the aspect nearer to the inducing body is charged with electricity of the opposite kind, and is attracted ; the farther aspect is charged with electricity of the same kind, and is repelled, but to a less extent, because it is more distant ; its charge not being " bound " is, besides, more readily dispersed into the surrounding air. On the whole, these light bodies are attracted. If they come in contact with the inducing body, they acquire a part of its charge, and are thereupon repelled. As another effect of induction we find that while two similarly-charged bodies at the same potential within the same field will always repel one another, yet if they be not at precisely the same potential, the one of higher potential will, by its presence, alter the distribution of electricity over the other, the weaker, in such a sense that the weaker one may even become oppositely charged over the nearer aspect, and the attraction of the more highly-charged body for this side of the weaker may prevail over its repul- sion of the farther side ; and on the whole, two such bodies will, if they be placed at a sufficiently small distance and if no spark pass between them, attract one another. In a certain intermediate position there will be unstable equilibrium, and at all greater distances there will be repulsion. When a conducting body is brought into the neighbourhood of a system of insulated and charged conductors, the energy of that system falls, for the interposed body causes by its presence a redistribution of the charge of the system ; and such a redistri- bution of the charge causes a fall of the potential and therefore of the energy of the system. If the body introduced be a die- lectric, the effect produced is similar but smaller. Electric Screens. A conducting sphere surrounding an insulated electrified body and connected with the earth will, as we have seen, shelter an external particle from the inductive action of the enclosed electrified body; and conversely, it will shelter the internal electrified body from the distribution-dis- turbing and potential-lowering influence of the outside particle. A screen of perforated tinfoil or a cage of wire gauze has nearly an equal effect : such screens are used to protect delicate instru- ments from the inductive action of external electrified -bodies. The place of an enveloping sphere may be taken by a plate 602 ELECTRICITY AND MAGNETISM. [CHAP. of metal connected with the earth. If the diameter of this be infinite or practically, if it be very great as compared with the distance between the electrified and the protected particle the screening action will be perfect. In Fig. 201, A is an insulated body positively charged by a galvanic battery or a frictional electric machine ; D is a large Fig.2oi. metallic screen ; B is a metallic body connected 4- > by a wire with the earth ; X N this wire passes round Earth the magnetic needle of > a Galvanometer, G ; the screen D is suddenly removed : there is a sudden separation of electricities in B : a positive charge escapes round the galvanom- eter and deflects its needle by an instantaneous twitch. The phenomena of electricity in equilibrium are very similar in their mathematical aspect to those of the steady flow of Heat; equipotential sur- faces represent isothermal surfaces ; lines of force represent lines of flow of heat ; specific inductive capacity takes the place of thermal conductivity, and potential-slope that of temperature-gradient. Again, the calculation of the variation of the force throughout a die- lectric field resembles very closely that of the distribution of the flow in a steadily-flowing mass of incompressible fluid; just as the stream lines in a field of liquid-flow may be held to exert lateral pressure upon one another, so do the lines of force in an electric field laterally repel one another, as is specially manifest at the surface of a conductor, where the " elements of charge " repel one another ; within the same medium, in each of the tubes of force, or tubes of flow, the product of the force or of the flow into the cross-sectional area is constant (Law of Continuity in Hydrodynamics) : and the Energy per unit of Volume in a field of force or of flow is at each point numerically equal to the electrostatic or hydrostatic Pressure per unit of Area at that point. In Faraday and Maxwell's theory of Ether-stress, the flow of charge across an electrified surface is insisted on. This flow, which takes place whenever a Separation of Electricities occurs, is of the nature of a Displace- ment in the Ether permeating the Field of Force, and is directed along the Lines of Force at every point in that field. One end of a line of force is in a condition which gives rise to what we call positive, the other to what we call negative electrification or charge at the surface of the electrified body. When a thin insulated sheet of tinfoil is exposed to the inductive influence of a charged conductor, there is a separation of positive from negative charge across the conductor influenced, and on each side there is a charge induced whose density is a: This means that the lines of Induction or of Displacement, across which the tinfoil lies, are directed towards the tinfoil on one of its sides, away from it on the other. The Quantity of Electricity thus induced to flow in either direction is, over any given area A, equal to A cr : this quantity is equal to A /4?r, since <}> 47rcr. If any other die- lectric than air intervene between the inducing charge and the conductor xvi.] ETHER-STRESS. 603 acted upon, = 47rcr/K, and the amount of the induced charge, the Quan- tity of Flow, the so-called Electric Displacement, is Q = Acr=AK<}>/47r = A/4:7r x i the Induction per sq. cm. A conductor offers no permanent resistance to the displacement of an electric charge through it, and as long as there is maintained between the extremities of a conductor a permanent difference of potential, so long will the electric displacement produced be continuously relieved by an electric flow or Current; but in a non-conductor or dielectric the extremities may remain under a permanent difference of potential, a permanent stress or state of Polarisation, for the Electric Dis- placement, the flow set up in it during the first instant of exposure to elec- tric stress, is arrested by a certain Electric Elasticity or Elastivity (Heaviside) of the Dielectric, which, being represented by the fraction electric stress electric force acting across each unit of area _ /j'A/4 "I - electric strain ~~ quantity of flow across each unit, of area " v 9/ "V - 4?r/K, is inversely proportional to K, the specific inductive capacity of the dielectric. The Electrostatic Energy of the dielectric is the product of the average displacing electric force, ^ per sq. cm., into the electric displacement, K<|>A/47r over area A, effected along a distance d: the energy of volume Ad is thus 4<|>(K<|)A/47r)^, and that of unit volume is K<|> 2 /87r; while the dynamical Energy-Slope^ or Electric Tension p Q or Traction t, in the direc- tion of the Lines of Force, is also Kf> 2 /S7r dynes across each unit of area of the bounding surface of the dielectric. Dimensions of Electrostatic Measures, in air. The Absolute unit of Quantity of electricity in electrostatic measure is a quantity which, placed at a certain distance, in air, from a similar and equal charge, repels it with a certain mechanical Force. The Force between two quantities at a given distance is therefore equal to (Product of Quantities) + (Distance 2 ). The Dimensions of this expression are [Q] x [Q] -4- [L 2 ] ; but the dimen- sions of a Force are otherwise known to be [ML/T 2 ] ; whence [Q 2 /L 2 ] = [ML/T 2 ] and [Q] = [M*Li/T]. Surface-Density ], in air. Capacity, C: the Quantity necessary to produce a certain rise of Potential: its dimensions are those of (Quantity) -4- (Potential-difference); [C] = [M*L*/T] - [M*L*/T] = [L], a Length simply. The relative capacities of conductors of similar form are simply proportional to their diameters. 604 ELECTRICITY AND MAGNETISM. [CHAP. Specific Inductive Capacity, K: K = Quantity displaced x 477 /"area x $, see equation, p. 603: [K] = [Q] -s- [<}> x area] = [M*jJ/T] -4-{[M5/L5T] x [L 2 ]}; it is therefore simply a Number, when air is taken as the standard medium. Coefficient of Induction; the ratio of a Charge developed to a Potential inducing; Quantity + Potential; [M*L*/T] + [MiL*/T] = [L]. Electrostatic Dimensions in any medium. Let K be the sp. ind. cap. of the medium; what the Dimensions of K may turn out to be we do not know ; but F' = QQ'/Kd 2 , whence [Q] = [MiUKi/T]. Similarly, [>] = [MlKl/L*T]; [E] = [M4LVKiT]; [$] = fa* /K] = [Potential- Slope] = [Mi/L4K*T] ; [i]=[MiK*/L4Tj; [CJ = [KL]; [K] = [K] ; and [Coeffi- cient of Induction] = [KL] . Relations of Electrostatic Quantities, in any medium. Potential- Diiference ; V, - V y/ = E = dVSir-F/AK = dVS-n-f/K = d-^ = ^Tr-d- o-/K = 47r d Q/ AK = d i/K. Electric Force, = E/d = V87r-F/AK = V8w//K = 47TO-/K = 47r Q/ AK = i/K = 2//0-. Surface-Density, /4fl- = VKF/27rA = VKf/^r = KE/4ffrf = Q/A = 1/47T. Induction per sq. cm., i = 47TQ- = K<|> = V87rKF/A = v/87rK/ = KE/d = 47rQ/ A. Quantity, Q = A/47r = ARE /M = Ai / TT = I/4ir. Force F across area A ; F = A KE 2 /87rd 2 = KA| 2 /8ir = 27r Acr/2 = Aio-/2K = Ii/87r = Ai 2 /87rK. Force / per sq. cm. ; /= KE 2 /8rf = K^/8 7r = 2 7 r(7 z /K = 2TrQ J 2 /A 2 K = ^(T/2 = ia/'2K = 4>i/87r = i 2 /87rK. Energy of Field = ^QV = J{A VKF/2irA x d VSirF/AK} = d-F = Ad-f; =%(AK$/4:Tr).d'~4> = Ad'K^/8Tr', = Ad 27ro- 2 / K =. Ad i 2 / STJ-K = AKE 2 / STT^ = Ad . 27rQ 2 / A 2 K = Ad <|>i / STT = Ad 2 /87r = 2iro- 2 /K = KE 2 /87rd 2 = 27rQ 2 /A 2 K = <}>i/87r = |>cr/2 = ol/2K = i 2 /87rK. Capacity or Permittance C = Q/V = KA/47rd. Dielectric Elasticity = /o- = 4v/K. The fundamental equations for the above relations are: = E/d; f= 27r(7 2 /K ; i = K<|> = 47ro- ; where cr is measured in the usual air-units. If in the above expressions we make K = 1, we obtain the ordinary air-equations. OBSERVATION OF DIFFERENCES OF POTENTIAL. Observation of Differences of Potential is effected by means of instruments called Electroscopes and Electrome- ters; the former indicate the nature, the latter measure the amount, of differences of potential. Gold-Leaf Electroscope. A glass flask with a vulcanite stopper : through the stopper passes a metal rod, surmounted by a metallic sphere or plate, and terminated below by a pair of freely-suspended strips of gold leaf. If the metallic part of the electroscope be charged by contact with an electrified body, the gold leaves, becoming similarly charged, repel one another, and diverge, slightly if the charge be feeble, widely if it be great. The electroscope may also be temporarily charged by induction: a 4- electrified body brought into the neighbourhood of the sphere or plate causes that sphere or plate to become negatively, while the more distant gold leaves within the flask are positively charged. If, while the electro- scope is electrified by induction, its upper extremity be momentarily touched by the experimenter, the gold leaves collapse, for their charge escapes to the earth : the plate or sphere, however, retains its charge, and when the indue- xvi.] ELECTROSCOPES. 605 ing body is removed, the opposite charge borne by the sphere or plate becomes free to distribute itself over all the metal of the electroscope, and the leaves again diverge, for the instrument is now permanently charged. If the electroscope be permanently charged, the approach of a body similarly charged will cause a farther divergence of the leaves : the approach of a body oppositely charged will cause the leaves to repel each other with less force : whence the nature of the electrification of a given charged-body can be ascertained. The deficiencies of the electroscope are : that its indications are quali- tative, not accurately quantitative; and that the glass does not thoroughly screen the gold leaf from the direct inductive action of external charged- bodies. In order to obviate the latter defect, the inner surface of the flask is sometimes lined with perforated tinfoil, or the whole is surrounded by a cage of wire gauze. The gold-leaf electroscope is a development of earlier instruments, in which straws, plain balls of elder-pith, or gilt pith-balls, were employed. In the discharging electroscope the gold leaves, when they diverge, come in contact with two metallic uprights which communicate with the earth : they are thus discharged and collapse, again to be charged : the number of oscillations of the gold leaves affords a rough measure of the quantity of electricity borne by a conductor which is discharged to earth through such an electroscope. Peltier's Electroscope. A vertical brass ring, insulated; attached to its inner circumference, at the lowest point, a vertical pointed rod ; on the pointed rod is poised either a magnetic needle or else a metallic rod whose direction is determined by a small magnetic needle attached to it. The whole is turned round a vertical axis until the ring and the poised metallic rod lie in the same plane. If the ring be charged, the charge is shared with the poised metallic mass, and the ring and the poised mass repel one another ; the latter swings round until the force of electrical repulsion is balanced by the tendency of the magnet to point to the magnetic north and south. This instrument may, by imparting to it a series of successive known charges, be so graduated as to act as an electrometer. Bohnenberger's Electroscope. Two vertical dry piles (p. 622), the one with its + pole, the other with its pole upper- most ; between these oppositely-charged uppermost poles there is a field of force, within which a strip of gold leaf is suspended. If uncharged, this strip hangs vertically ; if charged, it is repelled by one pole and at- tracted towards the Fig.202. other. Instead of two piles, the two extremities of one and the same dry pile may be used to make such an electro- scope. In Fig. 202, AB A - -he is a dry pile whose poles are connected with the metallic plates 606 ELECTRICITY AND MAGNETISM. [CHAP. C and D, between which there is thus formed a field of force, in which the gold leaf E is suspended. On the same principle the Quadrant Electrometer of Lord Kelvin is based. In Fig. 203 the two opposite quadrants A and D are connected with one another by wire, but are insulated from B and C. A and D are thus at the same potential, while B and C are also at the same potential, a potential which may differ from that of A and D. A and D may be brought to the potential of the earth by means of a wire connected with gas or water pipes ; B and C may be brought to the potential of any given object by connecting them with it by means of a wire. The quadrants A, D, and B, C, are thus at different potentials, and a metallic needle an aluminium needle of a flat dumb-bell Fig.203. shape will, if it be suspended symmetrically over the quad- rants by means of two threads arranged parallel to one another, and if it be kept charged by constant connection with one coat- ing of a Leyden jar (which may be replenished when necessary), impose a certain amount of torsion upon those two suspending parallel threads ; the amount of this torsion will indicate the nature and approximately the amount of the difference of potential between the two pairs of quadrants, and therefore between the earth and the object whose Potential is to be measured. If the quadrants be made hollow, and the needle suspended within them, the arrangement is better adapted for electrometric purposes. The whole arrangement is well adapted for testing the adjustment to equality of the potentials of two bodies. XVI.] ELECTROMETERS. 607 It would come to the same thing if the potentials really measured were those of the air in the neighbourhood of the quadrants, provided that the quadrants be all of the same metal, or that the potential of the air in the neighbourhood of one uncharged metal be the same as that in the neighbour- hood of another. The amount of deflection of the suspended needle may be observed by connecting with it a very light mirror, upon which a very narrow beam of light shines ; as the needle is deflected, the beam of light reflected from the mirror is deflected through an angle twice as great as that of the deflection of the mirror ; and the beam of light, if received upon a distant scale, thus acts as a weightless pointer. Fig.204. Upon the scale the deflection of the spot of light may be read off ; that deflection is, on a straight scale, proportional to the tangent of twice the angle of de- flection of the mirror : for small angles it is nearly proportional directly to twice the angle (Fig. 161). Coulomb's Torsion Balance. A long, verti- /^~~ cal, slender, hard-wire or silk-fibre AB, Fig. 204, by which there is suspended in a horizontal position a thin counterpoised rod of glass or shellac, CD, which bears at one of its extremities a little gilt sphere D. -L In one position of the suspending wire the gilt sphere >* D comes into contact with a sphere-ended metal rod EF : this rod projects through the walls of the glass case in which the whole is encaged, and is therefore insulated. This metal rod terminates externally in a sphere E, which may be charged by contact with an electrified body, such as a proof -plane. A proof- plane is a small metallic disc provided with an insulating glass or ebonite handle. It is used by laying the disc upon the surface of an electrified body : when the disc is withdrawn, it bears with it a charge proportional to the charge previously borne by that part of the surface of the electrified body with which it had been placed in contact : it is then made to touch the sphere E of the torsion balance. EF being charged, the two spheres F and D, when they come in contact, become charged with electricity of the same kind, and repel one another : they do this until there is equilibrium between the electric repulsion and the torsion of the suspending wire AB. The proof-plane may be used directly in the place of EF ; and instead of a proof- pi a ne a proof-sphere maybe used when the curvature of the body, whose charge is to be examined, is but small. Different charges may be compared by comparing the amounts of torsion necessary to bring the two mutually-repellent bodies, D and F, to equal dis- tances. A preliminary charge is given to the ball D ; a charge Q of the same kind is imparted to F, or brought in by a proof-plane or a proof-sphere. Let the repulsion, between Q and the charge on D, be such that the suspended c \ _. E 608 ELECTRICITY AND MAGNETISM. [CHAP. Fig.205. horizontal fibre makes an angle FED of 10 with that position in which D is in contact with F, while the upper end A is twisted in the contrary direc- tion so as, as it were, to tend to force F and D together through an angle of 410 ; the total torsion of the wire AB is 420. Now remove the charge Q and substitute a charge Q,' ; the index at A indicates 95 of rota- tion there when D is in its former position : the total torsion of the wire is now 105. The charges Q and Q' are proportional to the torsions which their repulsions balance ; and Q : Q' : : 4 : 1 . Coulomb also made use of the method of oscillations (p. 39) : he swung an electrified needle in presence of an electrified ball ; the periods of the oscillations varied as the distance ; but the period varies inversely as the square root of the force acting : therefore the force acting varies inversely as the square of the distance. When the distance is kept fixed, the charges of the needle or ball being varied, the periods of the oscillations vary inversely as the square root of the varied charge. The Absolute Difference of Potential between two bodies may be ascertained by measuring the attraction between two metallic plates which are respectively connected by metallic wires with the two bodies in question. In Fig. 205 AB is a galvanic battery, the extremities of which are permanently at dif- ferent potentials : it is desired to find the dif- ference between these potentials. Connect A and B with the plates C and D. The field of force between C and D is uniform at its centre. D is fixed ; but E, the central part of C, is mov- able. The attraction between E and D may be measured by observing the distortion of a spring which tends to pull E upwards while the electrical attraction tends to pull E down- wards, this observation being made when the distance of D is so adjusted that the lower surface of E is flush with that of C. It is sometimes found advantageous in the use of instruments of this kind to connect D alternately with B and with the earth : the spring tends to become differently distorted in the two cases, and in order to render its distortion equal in both cases the dis- tance of D must be varied. The amount of approximation or retraction of D may be measured by a micrometer-screw. The spring which keeps up E against the attraction of D may be replaced by transforming E into one pan of a delicate balance, of which the other pan may be loaded with known weights. I xvi.] MEASUREMENT OF POTENTIAL. 609 The potential of E is V E ; that of D is V D : the difference of potential to be measured is (V E - V D ). The traction across the field, along the lines of force, or the pull of the field upon the plate E, is 44*" P er S( l- cm -> or ^4>-Ao- upon its whole area A. This is equal to 27ro- 2 -A: and it is bal- anced by F, the stress upon the spring ; F = 27rcr 2 A. Also, kirv = $ = (V E V D )/<7. Hence the Difference of Potential in absolute measurement is (V E V D ) = d V8?rF/A, in which expression d, A, and F can be directly measured ; d being the distance between the plates E and D. Since cr = (V E V D )/47re? per unit area, the charge on the attracted circular disc of radius r is (V E V D )r 2 /4d: the capacity of the system is therefore r 2 /d, and can thus be measured absolutely. Standards of Electrostatic Capacity can thus be constructed. When E and D are connected with A and B, the respective potentials of E and D are V A and V B ; and V A - V B , the difference of potential between the ends of the pile, = o?V87rF/A. When D, instead of being connected with B, is connected with the earth, its potential becomes zero ; and when D (movable in this case) is brought by its micrometer-screw into such a position that the plate E again assumes a position flush with the fixed guard- ring C, the stress F upon the spring is the same as before, and V A V^^ = V A = e^VSTrF/A, where d, is the new distance between the plates E and D. Hence V B = (d, - d) VSTrF/A ; and the Potential of B is easily measurable, for (d t d), the change of distance between E and D, is much more easily measurable than d, the absolute distance between them. Lord Kelvin, to whom the above method is due, has also devised instru- ments by which the difference of potential (or voltage) of electric light- ing currents may be electrostatically measured. This is effected by observing the extent to which an aluminium strip, charged, succeeds in rising up from its position of gravitational equilibrium, in order to place itself immediately between two fixed aluminium plates, oppositely charged (Kelvin's Electro- static Volt-Meters). If the electrifications be reversed, the attraction is the same ; and if they be rapidly alternated, the general action of the instru- ment remains the same. These instruments are graduated so as to measure the potential-difference in Volts, not in electrostatic C.G.S. units. PRODUCTION OF DIFFERENCE OF 'POTENTIAL. The principal source of Difference of Potential is commonly stated to be the Contact of dissimilar surfaces that is, either of different substances or of two pieces of the same substance whose surfaces are in different conditions. A piece of resin and a piece of glass will, upon contact, be more difficult to pull asunder than two pieces of resin or two pieces of glass : and if they be rubbed together, so as to multiply the points of contact, the effect is multiplied. When pulled asunder, two such bodies are found to be charged equally and oppositely : across the sur- face of contact there has been a Separation of positive from negative electricity. The development of electrical condition is thus necessarily a phenomenon of continual recurrence : and 2R 610 ELECTKICITY AND MAGNETISM. [CHAP. it greatly influences the adhesion of one body to another. In all probability, wherever there is friction, the energy ultimately converted into heat is, in the first place, converted into the energy of electrical separation. When two substances have different molecular velocities at their com- mon surface of mutual contact, the molecules hamper one another and energy is lost : this energy, formerly that of molecular motion, now takes the form of the energy of electrical displacement. Within the interior of a homogeneous body the same thing must happen between colliding mole- cules whose velocities are different ; but, all being alike, and the average molecular velocity being the same throughout the mass, there is on the whole no effect. Non-conductors in contact become electrified; but only on their surfaces of actual contact. When they are separated their final discharge is incomplete, and the residual charges their superficial distribution being restricted to those parts of the surfaces which have been most nearly in actual contact are small in quantity but of great density, and there- fore of high potential ; and as these charges are not diffused by conduction over the whole surface, their potentials remain high after separation. When sulphur is melted in a glass test-tube, after cooling the sulphur is found to bear permanently a negative, the glass a positive charge. In the following series, due to Faraday, each member becomes positively charged when rubbed on one following it, negatively when rubbed on one preceding it: Cat and Bearskin Flannel Ivory Feathers Rock Crys- tal Flint Glass Cotton Linen Canvas White Silk the Hand Wood Shellac the Metals ( + Fe, Cu, Brass, Sn, Ag, Pt) Sulphur (Soapstone). There are certain irregularities here to be observed: for example, a feather lightly drawn over a piece of canvas becomes nega- tively electrified, whereas if it be drawn through a pressed fold of canvas it becomes positively charged. The separation of electricities by contact and friction is utilised in the various forms of electric frictional machines, which range in complexity from a simple piece of sealing-wax or a glass rod rubbed with a catskin or a silk handkerchief, or a stout glass tube rubbed with a piece of dry flannel, to a machine in which a glass or vulcanite disc or cylinder, set in rotation, rubs against silk rubbers : these rubbers, whose conductivity is improved by anointing them with a mixture of fat and mercury, communicate with the ground, and their negative electricity is thus carried off to the earth ; the positive charge, borne by the rotating glass or vulcanite, blends with a negative charge developed by induction in the tips of a' comb-like series of sharp metallic points which come almost in contact with the rotating glass ; while the complementary induced positive-charge is conveyed either to a large insulated Conductor connected with these points by a metallic chain or wire, or to the surface of a large insulated hollow conductor which sur- rounds the rubbing parts of the machine, or to the inner coat of a Leyden xvi.] ELECTRIFICATION ON CONTACT. 611 jar, or to the inner coat of one of the constituent members of a battery of Leyden jars. A charge of positive electricity may be thus accumulated. If, on the other hand, the positive charge of the glass be conveyed to the earth, while the insulated conductor is metallically connected with the rubbers, a charge of negative electricity may be accumulated in the con- ductor. If the conductor, in which positive electricity is being accumu- lated, be connected by wire with the negatively-charged rubbers, a current of electricity will pass along the connecting wire so long as the machine is worked, and that wire will be heated. If this current be sent through a second electric machine it will tend to cause in it a reversed rotation. It is possible (Gaugain) thus to produce continuous currents by the friction even of dissimilar metals. When two metals come in contact, in air or other gas, they at once become electrified, positively and negatively respectively. The amount and kind of charge on each metal depends upon (1) the nature of the metals, (2) the condition of their sur- faces, (3) their temperatures, and (4) the nature of the sur- rounding or intervening gas, if there be any. In the case of copper and zinc in air, the copper becomes negatively and the zinc positively charged. The older view as to this was, that there was electrical sepa- ration at the surface of contact between the metals, and that each metal was at an equal potential throughout: then the potential of the air in the immediate neighbourhood of the metals was the potential of the metals themselves. The newer view is that the metals are, before contact, by reason of a ten- dency to chemical action (oxidation, etc.) each at a potential different from that of the surrounding air or gas ; that their respective potentials are more or less different from one another ; that when the two metals are brought into contact, the potential throughout the whole conjoint metal becomes uniform, and a momentary current runs in the conjoint metal mass, so as to charge the one metal (zinc) positively and the other (copper) negatively ; that while this is taking place, the surrounding dielectric is being electrically displaced so as to produce a Field of Force ; that the Energy necessary for this is derived from a trifling amount of chemical combination (oxidation, etc.) at the metal-gas surface ; that in the case of a zinc-copper couple, before contact the potential of the air is say V C.G.S. electrostatic units, that of the zinc is V (x -f- 0-0025), and that of the copper is V x ; that after contact the potential of the copper-zinc couple is uniformly V (x + 0-00125), that of the air near the zinc is (V + 0-00125), and that of the air near the copper is (V - 0-00125) ; and that the Field of Force 612 ELECTRICITY AND MAGNETISM. [CHAP. is maintained in the air, through its being a dielectric, the Ether in which offers elastic resistance to further displacement. But there is also another effect. We have spoken of the two metals coming to the same potential when brought into contact. It appears, however, that they cannot perfectly do this, on any view of the facts, even independently of the air, except at a particular temperature. There is almost always a slight difference of potential, a true contact-effect; and this varies so remarkably with the temperature as to give rise to the phenomena of Thermo-electricity, of which later (p. 624). If a metallic disc be composed of four quadrants, soldered together and consisting alternately of zinc and copper respectively, and if the disc be arranged horizontally, a needle suspended horizontally over the centre of the disc will, if that needle be charged with positive electricity, be repelled by the zincs and attracted by the coppers, and it will therefore swing round so as to lie over the copper quadrants ; while if it be charged negatively it will come to lie over the zinc quadrants. The needle may be so suspended by two threads that, when uncharged, it lies along a diameter of the disc, a diameter which coincides with a line of junction between quadrants. Take an electroscope surmounted by a copper plate, varnished on its upper side ; upon this plate lay a zinc plate varnished on its lower side : these plates, separated by the varnish, act as a condenser. Bring a copper and a zinc plate, both of which are unvarnished and insulated, into contact : separate them ; with the zinc touch the zinc, with the copper the copper of the condenser. Repeat this operation several times : then remove the zinc plate of the condenser : the copper is found to be strongly charged with negative electricity, while the zinc plate removed is positively charged. Copper filings falling through an insulated zinc funnel, as they leave that funnel carry with them a negative charge. These experiments may be interpreted in accordance with either of the above views. In Fig. 206 a zinc plate Zn and a copper plate Cu, both in air, are connected by a copper wire. Then, either the zinc is at Fig 206 potential JE = 0-00125 _j_ I electrostatic units, and the copper at potential cu _ | ' -IE =-0-00125; or, according to the newer , x Zn + E . ,, 6 Cu f ' ' view, they are at equal " potentials, while be- tween them there is a Field of Force, the potentials within which present a potential- difference of 0-0025 units. If one of the plates be connected to earth, the potentials of the copper and zinc are altered : but the Field of Force, though its terminal potentials are altered, remains constant in its potential-fall. xvi.] ELECTRIFICATION ON CONTACT. 613 Within the field of force between such plates arranged with an inter- vening dielectric, = 47rcr = const. = (V Zn V Cu )/Krf, where d is the thick- ness of the dielectric, and V Cu , V Zn , the potentials at the copper and the zinc respectively. Hence the superficial density a- = ( V Zn V Cu ) /^irl^d ; but if the numerator of this fraction, the difference of potential, be con- stant, as it is between two metals, cr = const, x (!/ an( i an alloy of 15 parts of antimony and 7 of cadmium), arranged after the fash- ion of Fig. 212. When the face of this pile marked HHH is exposed to heat, the junc- tions H, H, H become warmer than the junc- tions C, C, C, and a current passes through the circuit O from the antimony to the bis- muth terminal. If it be placed opposite a piece of ice, the face HHH will cool itself by radiation, and the current is now in the reverse direction. When a junction of two metals is connected, by a pair of copper wires, with a similar junction at the same temperature, no current passes, whatever be the length of the intervening cable or its local variations of temperature ; but if one of the junctions assume a different temperature from the other, then a current passes, and the temperature of the distant junction may be inferred from the strength of the current which passes, this being measured by a galvanometer, or directly determined by heating or cooling the similar junction situated under the observer's control until its temperature becomes the same as that of the distant one : this is known to have occurred when the current through the galvanometer ceases. A couple of junctions of this Fig.213. kind, with an intervening double wire and galvanometer, form a differen- tial thermometer, the indications of which must be interpreted with refer- ence to the thermo-electric diagram of the two metals iised. As sources of electricity, thermo-electric piles are not much in use. BecquerePs thermo-electric piles, made of thirty pairs of blocks or rods of artificial sulphide of copper (which fuses only at about 1000 C. ) and of xvi.] THERMO-ELECTRIC German-silver, can decompose .water when the differences of temperature employed are from 250 to 300. In Rebicek's form of Noe's thermo-elec- tric pile, twenty-five pairs of plates of German-silver and of an alloy of zinc and antimony are ranged round a Bunsen gas-burner: each such pile maintains an effective difference of potential of from 2 to 2-75 Volts, so long as the Bunsen burner is kept lighted, while the internal resistance is 0-75 Ohms. In (Diamond's pile, about 6000 couples of iron and of bismuth- antimony alloy are ranged round a coke fire, and the E.M.D.P. produced is, if the couples be arranged in file, about 218 Volts. The disadvantages of thermic piles as sources of electricity are that, in general, the E.M.D.P. pro- duced is so extremely small that moderately-slight external resistances make the current extremely weak, and that it is difficult to keep the cold junction cool ; and even in Clamond's pile, which is able to keep a pair of electric arc-lamps in action, about 95 or 96 per cent of the heat of the fire is not converted into the energy of a current, and is thereby practically wasted. For many purposes, such as electroplating on the small scale, ]S T oe's bat- teries, three of which produce an E.M.D.P. nearly equal to that produced by seven DanielFs cells, are very useful, for when they are once built up their current can be produced or arrested at will. An arrangement like that of Fig. 213 has been used as a self-acting source of electrical currents, and therefore of energy, sufficient to maintain in action a self-winding clock. The most important source of electricity is the transform- ation of the energy of work into that of electrical separation by means of magneto-electric and dynamo-electric machines, the action of which will be explained in the sequel. Atmospheric Electricity. The atmosphere in different regions is often found to be at different local potentials, which differ from that of the earth sometimes even by as much as 3000 Volts within 100 feet. This is possibly (Tait) due to a contact-effect between air and aqueous vapour; and it is possibly necessary that there should be at least traces of dust present, as well as water-vapour. A conductor insulated from the earth may be brought to the same potential as any point in the air, by leading to that point a metallic wire, and by furnishing this exploring wire with an extremely fine point, or, better, by fixing at its extremity a sponge dipped in spirit and set on fire, or a little cistern from which a quantity of water is allowed to drop. In the former case the flame continuously con- veys masses of gas away from the end of the exploring wire ; and so long as there is any difference of potential between the region of the air explored and the conducting system of which the exploring wire forms a part, there will be a current along the wire, and finally the whole conducting system will come to the same potential as the air around the flame. Similarly, waterdrops, on falling from an insulated cistern, bring the cistern to the same potential as the air around it: each drop, just before falling off, becomes electrified with a charge opposite, while the nozzle, the cistern, and the main mass of water are electrified with a charge similar to that of the air in the neighbourhood of the falling drop. As the drop is in the act of falling off, it is attracted by the cistern : it is held back as it falls : it falls down with less speed than it would have assumed if it had fallen from an uninsulated cistern; and when it reaches the ground it produces less heat. The energy of the electrification acquired by the cistern is equal to the missing kinetic energy of the falling drops. 630 ELECTRICITY AND MAGNETISM. [CHAP. In an analogous way the air within a room may be strongly electrified ; connect a flame with the conductor of an electrical machine, and work the machine : in one minute a Holtz machine will raise the potential of the air of a room by 2000 Volts. Combustion alone will effect electrification of the air, which is negatively charged by burning coal-gas, positively by burning charcoal. When difference of potential has once been produced, it may be turned to account for the conversion of Work into Electrical Energy. Charge a plate, whose free capacity is C, to potential V ; its charge will be Q = CV. Bring up to it a second plate, parallel and at a distance d; the two plates now form a condenser. The capacity of this condenser is C' = K x surface/47rc? (p. 599). Connect the second plate to earth. Its potential becomes zero, and on its outer surface it has no charge. The capacity of each plate for " free charge " may be reckoned as having been reduced to C by the mutual approach, for lines of force can now only pass from one face of each plate towards surrounding objects. The poten- tial of the inducing plate will now be V = CV/C' + |C, which is less than V. The original charge CV on the inducing plate is now divided into two parts. Of these, one is "free charge " = |C 2 V/C' + --C; the other is "bound charge," C'CV/C' + ^C, which faces an equal and opposite charge on the second plate of the condenser. Now insulate the second plate and remove it to a distance d' : the capacity of the condenser decreases in the ratio d/d' ; then as the capacity of the condenser decreases, the potential of the second plate tends to fall from zero to VC'/C' + -C, while that of the first tends to return to V. In the case of conducting plates in free air, the negative potential imparted to the second plate by this method cannot, therefore, become numerically quite equal to the original potential of the first plate; but in the Electrophorus, next to be considered, the charge cannot travel along the surface of the original charged plate. In that case, the whole charge remains "bound"; and since E = 47rQe?/A, and Q is constant, then on increasing d, the potential-difference will rise. Difference of potential may also be increased by the expen- diture of Work, with the assistance of induction (Holtz, Voss, Wimshurst). The work is done in stretching the Field of Force against the mechanical traction t across the field ; t = E 2 /87re? 2 , in dynes per sq. cm., where E is the rise in potential occasioned by pulling the plates apart through a distance d ; whence E = d V87rt. The Electrophorus consists of a cake of resin or vulcanite and an insulated metallic plate. The former is slightly charged by being rubbed with a catskin or a dry silk-handkerchief : the metallic plate is then laid upon it. The contact between the two can never be perfect at all points ; practically there is an intervening film of air between the resin and the metallic plate, and the latter is charged by induction with an attracted and a repelled charge. The latter charge may be withdrawn by touching the metallic plate with the finger, or by making metallic communication between the metallic plate and the earth ; the former remains, facing and XVI.] ELECTROPHOKUS. 631 attracted by the original charge on the resin. Work is now done from without in pulling the metallic plate away from the resin; as the distance between the metal and the resin increases, the electrostatic capacity of the electrophorus, considered as a condenser, diminishes; the potential there- fore increases, both on the metal and over the resin : the knuckle applied to the edge of the insulated metallic plate may now receive a spark. When the metallic plate is next laid on the resin a new charge is induced in it, which may again be withdrawn in the same way when the plate is removed. Small original charges may thus induce successive charges of high potential. Sir William Thomson's (Lord Kelvin's) Replenishes This instrument, which is used as a means of keeping the Leyden jar connected with the suspended needle of Kelvin's electrometer at a constant poten- tial, is sketched in the accompanying diagram (Fig. 213 a). A, B, two metal half-cylinders, insulated from one another ; C, D, two metallic plates insulated from one another and capable of rotation round the axis O; E, F, an insulated Fig. 213 a. spring capable of touching both C and D when they are in the position shown in the figure ; G, H, two springs connected with A and B and capable of being pressed upon by C and D as they rotate. Start from the position shown in the figure. B is positively charged by contact with one of the plates of the Leyden jar ; C becomes negatively, and D positively, charged. Rotate C to the left, D to the right. Their metallic connection with one another is broken and they remain oppositely charged. As they pass G and H, C's charge escapes to A, and D's + charge to B ; and thereafter D and C respectively acquire and + charges, and stand in the former positions of C and D. The + charge of the -j- plate of the Leyden jar may thus by continuous rotation of CD be con- tinuously increased. If, on the other hand, C be rotated to the right, D to the left, C's negative charge is conveyed to B, and the positive charge of the -f- plate of the con- denser may, by continuous rotation in this sense, be reduced to any desired extent, or even reversed. The potential of the Leyden jar may thus be adjusted to any desired amount, which may be determined by a subsidiary pair of plates, connected with the inner and outer coatings respectively and separated by springs, coming to assume a position at any pre-arranged fixed distance from one another. Sir William Thomson's (Lord Kelvin's) Water-Gravity Electric Machine. In Fig. 214, A and B are two Leyden jars, whose inner coatings consist of sulphuric acid and are connected with the metal tubes C, F and E, D respectively. C and D are co-axial : so are E and F. Water falls in drops from the bifurcated metal-tube G, which, being connected with the ordi- nary water supply, is in communication with the earth, and is therefore at zero potential. A small initial charge, consisting (say) of positive electricity, is imparted to one of the Leyden jars, say A. Water is made te flow from G in streams so thin as to break up into drops within the tubes C and E. Just before these drops break off from the stream, they are by induction 632 ELECTRICITY AND MAGNETISM. [CHAP. Fig.214. within C charged with negative electricity, while the complementary posi- tive charge is conveyed along G to the earth. When the drops have become separate, they fall down charged negatively. They then fall upon a me- tallic funnel placed in the tube D, and charge the ex- terior of that tube nega- tively : this charge is shared with the Ley den jar B. This Ley den jar, thus neg- atively charged, by a cor- responding inductive action causes the drops which fall through E to become posi- tively charged. When these drops fall upon F they in- crease the positive charge of the Leyden jar A. Thus the Leyden jars A and B become more and more highly charged, the one with positive, the other with negative electricity on its inner coat. The energy of their electrification is derived from the work done by gravity upon the falling water ; and thus this contrivance is an electrical machine worked by gravity. Earth Earth- STEADY ELECTRICAL CURRENTS. If the plates of a charged condenser be connected by a wire, the condenser will be discharged ; the quantity Q of electricity disappears in a time , and during that time t certain phenomena occur, with waning vigour, which are spoken of as those of a Current of Electricity. It is as if electricity ran out of a place where it was stored up under a potential-difference E, down to the ordinary potential-level V = ; and as if its path were along the wire. In truth, the phenomenon is one of the release of the Field of Force from constraint, and of transmission of energy through that field, not through the connecting wire. Still, with this reservation, it is convenient to adhere to the terminology according to which such a discharge is spoken of as a Current in the wire. If a very large condenser be discharged through a very long and thin wire, the phenomena of Current may remain fairly steady for any short interval of time ; and if the E.M.D.P. be that between the terminals of a galvanic cell, these phenomena are, apart from Polarisation, etc., continuously uniform or steady, as if there were a practically inexhaustible reservoir of electric quantity to draw upon, so long as the cell holds out. xvi.] STEADY CURRENTS. 633 The Intensity or Strength of a current i.e., the Quantity of electricity which passes any cross-section of the conductor during one second of time depends, on the one hand, upon the effective difference between the potentials at different parts of the conductor, and, on the other, upon the nature of the con- ductor that is to say, upon its size and its substance. A long or thin wire is a worse conductor has less Conductance and offers more Resistance than a thick or short one: a silver wire conducts better than a copper one of the same size. The Density of a Current, A, is the quantity of electricity passing per sq. cm. of cross-section of the conductor. It is therefore equal to Intensity -r- Cross-Section = I/o. The relation between E the electromotive difference of poten- tial, I the Intensity of the current, and R the Resistance of a uniform conductor, is, when the flow is steady, expressed by the equation, I = E/R. When there are several sources of difference of potential within the circuit, or several successive conduc- tors, each of which offers its own resistance to the onward flow of the current, the law assumes the generalised form that ., _ E _ effective sum of all the differences of potential , . SR sum of all the successive resistances is Ohm's Law. The Resistance, as defined by Ohm's Law, is E/I or 2E/I, and it must be specially noted, as an experimental fact, that for any given conductor or set of conductors, this fraction, once found, remains almost absolutely constant, whatever may be the value of E or 2E, provided that the temperature and the structure of the conductor remain unchanged. The Measure- ment of the Resistance of a conductor is the experimental deter- mination of this fraction. Ohm's Law may also be written I = E/Z -4- R/Z; E/Z is the "electro- motive force " or Potential-Slope <|> ; R// is the Resistance per linear centi- metre. Also, if A stand for current-density, A = E// -* R /R = <|>D, where R and D are the Resistivity and Conductivity, as defined below. The C.G.S. Electrostatic Unit of Intensity is the intensity of a current in which one C.G.S. electrdstatic unit of quantity passes a given section of the conductor during one second. It is the current which passes when the difference of potential E = 1 C.G.S. electrostatic unit, and the total resistance is also R = 1 C.G.S. electrostatic unit of resistance. The C.G.S. Electrostatic Unit of Resistance is the resistance offered by a conductor which, when it is interposed between two bodies whose potentials are maintained at a constant difference of one C.G.S. electrostatic unit, allows one C.G.S. Electrostatic unit of Quantity to pass along it, per second. 634 ELECTRICITY AND MAGNETISM. [CHAP. These units are inconvenient for practical purposes, and electricians use as their practical units certain fractional or integral multiples of these. The Resistance of a uniformly-cylindrical conductor, such as a wire, depends upon three things : (1) its length Z, directly ; (2) its cross-section 0, inversely; (3) its Conductivity D, inversely. It is therefore equal to Z/0D = R. The Conductance of a conductor is the reciprocal of its Resistance. In a wire it is therefore equal to ov/L Ohm's La\v may therefore be written I = DE, E = I/D, or D = I/E, where D is the Conductance. The C.G.S. Unit of Conductance is that of a conductor of unit resist- ance. The specific Conductivity, D, of any substance is a constant, special to each substance, and even found to differ from sample to sample of that which is nominally the same substance. It represents the number of units of electricity which can pass, per second, between two bodies kept at a constant potential-difference of one unit, when the conductor interposed between these bodies has a length of 1 cm. and a cross-section of 1 sq. cm. It varies very greatly from one substance to another. The reciprocal of D, (!/D) = R, the Resistivity of a substance. The Resistance of a conductor of length I and cross-section o is therefore equal to IR/O = R. In the following table the first column of figures gives the Resistivities, the next column the Conductivities of a certain number of substances, in electrostatic measure ; while the third column gives the numbers which denote their Relative Conduc- tivities when the conductivity of mercury is taken as a standard and called unity. It is very usual to take the conductivity of silver as a standard = 100. The numbers in the following table have (with the exception of those for the last four substances) been calculated from the data of the authorities named, on the assumption that 144521 grammes of mercury (sp. gr. 13-5955), in the form of a column of uniform cross-section (1 sq. mm.) and 106-3 cm. in length, has a resistance equal to the 900,000,000000th part of an electro- static unit of resistance, that is, equal to one Ohm or practical unit (see p. 711) of Resistance. An Ohm-coil is a coil of wire whose resistance is one Ohm. The conductivity of an Ohm-Coil is called a Mho. If the following table be read without the multiplier or divisor, V 2 , it then expresses the specific resistivities and conductivities in another system the Magnetic or Electromagnetic system of C.G.S. units, from which the Ohm and the Volt are primarily derived, the Ohm being 10 9 electromagnetic units of resistance, and the Volt 10 8 electromagnetic units of potential- difference. This system depends upon the laws of Magnetism, afterwards to be explained. XVI.] RESISTIVITIES AND CONDUCTIVITIES. 635 10 ~ cT 5 o ji* 03 II 3 = M I'd" o s W g I l &: 00 S o o .o '3 6 6 o M ^M. o o '*& l - O ^3 S3 C PH 22.^3 fl 3 d 45 NOPnO 636 ELECTRICITY AND MAGNETISM. [CHAP. The conductivity of metals decreases, that of most bad con- ductors (including carbon) increases, with their temperatures: a heated wire or dynamo-electric machine increases the resist- ance in the circuit of which it forms a part ; while at the tem- perature of the electric arc, carbon appears to offer no resistance. Very roughly, and with well-marked exceptions in the cases of iron and mercury, the resistivity of a pure metal is proportional to its absolute temperature : but pure metals appear (Dewar) to have no resistivity when exceedingly cold, whereas in alloys there is no such result. When metals melt, their conductivities fall suddenly. Alloys are in general worse conductors than the arithmetical considera- tion of their percentage composition and the conductivities of their component metals would lead us to expect; but with changes of temperature, their conductivity varies less than that of pure metals does. There is a broad resemblance between the conductivities of metals for electricity and for heat : the best conductors of the one are in general the best conductors of the other ; and in both cases alloys offer a relatively high resistance. The series are, however, not identical. The velocities of light through films of the metals are also (Kundt) closely related to their conduc- tivities. Variable Conductivity. Conductivity varies not only with varying temperature, but also with varying magnetisation, tension, torsion, or pres- sure. It increases with longitudinal stretching, diminishes with longitu- dinal compression of a wire, and diminishes in iron, but increases in tin and zinc, when the stress, being transverse, tends to widen the wire (Tom- linson). In powders or porous material, such as metal filings, platinum sponge, charcoal, it increases with the pressure ; and if the pressure vary, within small limits, the variations of conductivity follow and are propor- tional to the variations of pressure. This is the principle of the Micro- phone. In such materials Heat raises the internal pressure and therefore the contact, and this modifies the amount of resistance and the heat pro- duced within the conductor : this last itself affects the conductivity, as in the Tasimeter, which detects changes in temperature by the variation of a current passing through a rod of carbon fixed between metallic supports, and exposed to varying temperatures. Selenium, which in the amorphous form is a non-conductor, but in the crystalline form is a conductor, varies in conductivity with its state of aggregation, its temperature, the length of time during which a current has been passing through it; and crystalline selenium, when acted upon by .light (especially the yellow and the red), and to a less extent when acted upon by dark rays, increases in conduc- tivity: in the case of very bright sunlight this increase being sometimes even tenfold. Light of variable intensity produces corresponding and rapidly-responding variations in the conductivity of the crystalline selenium xvi.] VAKIABLE CONDUCTIVITY. 637 upon which it may fall a fact utilised in the construction of the Photo- phone. Metals, unlike selenium, become worse conductors as the temper- ature rises; but (Siemens) at 210 C. selenium changes its character and comes to act like a metal. Reduced Resistance and Reduced Length of a Conductor. This may be explained by a few numerical examples. We suppose the unit of resistance to be the Ohm, as above denned, the resistance of freshly-distilled mercury in a column of 1 sq. mm. section and 1-0(33 metres in length. 1. What length of soft-copper wire of 1 sq. mm. sectional area will give a resistance equal to one Ohm? 1-063 x 61-70 = 65-5871 metres. The figure 61-70 is taken from the table of conductivities above. The Resistance of 65-5871 metres of copper is thus equal to that of 1-063 metres of mercury: the Reduced Length of 65-5871 metres of copper is 1-063 metres of mercury. 2. What will be the resistance of a column of mercury 100 metres long and 1 sq. cm. in section ? It will be equal to that of a column of mercury 1 sq. mm. in section x 1 metre in length multiplied by I = 100, and divided by o = 10 2 . It is therefore 0-940734 Ohm. Its reduced length is 1 metre of standard mercury-column, 1 sq. mm. in cross-section. 3. W T hat will be the absolute resistance, and what the resist- ance in Ohms, of 1000 metres of platinum wire whose diameter is \ millimetre ? Its sectional area o == irr 2 = TT ( V) 2 S( l- cm - = __^_ S q. cm. ; its length I 100,000 cm. ; its resistivity R is 14562-13 -f- V 2 ; the Resistance of the wire is T. I* 1Annnn 14562-13 6400 _ 2,966560,000000 K = - = JU,Ul -^yT~ ~^~ ~^~ C.G.S. electrostatic units, or 2966-6 Ohms. The strength or intensity of a steady current is measured by a Galvanometer (p. 713), round the magnetic needle of which the current is passed : in the Tangent Galvanometer the tangent of the angle of deflection of the needle is proportional to the intensity of the current. The strength- of a current is equal throughout all parts of a circuit in which there is a steady flow. A magnetic needle is equally deflected when brought into the neighbourhood of any part of the circuit, whether the circuit be locally composed of solid, of liquid, or of heated or rarefied gas. The Practical Unit of Intensity is the intensity of that current which is produced in a conductor whose total resistance is 1 Ohm ( = 1^900,000,000000 638 ELECTKICITY AND MAGNETISM. [CHAP. C.G.S. electrostatic unit), when there is kept up between its extremities a potential-difference which constantly amounts to one Volt, or 1/300 C.G.S. electrostatic unit. Since I = = = __ __ C.G.S.E.S.unit^ R 1 Ohrn 1/900,000,000000 C.G.S.E.S. unit the practical unit of intensity, the Ampere, is equal to 3000,000000 C.G.S. electrostatic units of intensity. In a current whose Intensity is one Ampere, the practical unit of quantity, one Coulomb, passes any given section during each second : the Coulomb is thus equal to 3000,000000 C.G.S. electrostatic units of quantity. Electrical engineers have adopted the Ohm, the Volt, etc., as means of practical measurement. The Ohm and the Volt in electrical workshops are not abstract calculations, but standard wires and standard batteries (or multiples or fractions of these), by comparison with which the resistance or the E.M.D.P., the so-called electromotive force, of any given combination of materials may be relatively measured ; e.g., an average pint Daniell cell will deliver a maximum current of about \ Ampere, its D.P. being about 1 Volt, and its internal resistance about 4 Ohms ; an average pint Grove cell will deliver a maximum current of ten Amperes, its D.P. being about 2 Volts, and its internal resistance about i Ohm. Dimensions of Electrostatic Measure in Air. Current-Inten- sity a quantity passing per second: [I] = [Q/T] = [MiLfyT] -j- [T] = Resistance: [R] = [E/I] = [MiLl/T] + [MsU/T 2 ] - [T/L] ; Con- ductance = [1/R]'= [L/T], a Velocity.* Resistivity: [R] = [R] x [o/fj = [T/L] x [L 2 /L] = [T]. Conductivity: [D] = [1/n] = [1/T]. In any medium of sp. ind. cap. K, the Dimensions in E.-S. measure are: Current-strength, [MlL3Ki/T] ; Resistance, [T/LK] ; Conductance, [LK/T] ; Resistivity, [T/KJ ; Conductivity, [K/T]. The above dimensions are based on the assumption that the Quantity of electricity in a current is the same thing as the Quantity of an electro- static charge : they are therefore called Dimensions in Electrostatic Measure. Fall of Potential in a Homogeneous Conductor of uniform thickness. During the maintenance of a steady current one end of a homogeneous conductor is at a higher, the other at a lower potential, and between these points the fall is gradual, so that intermediate points are at intermediate potentials. Fig. 215 shows that if the length of the uniform conductor be repre- * Suppose a sphere of radius r and therefore of capacity C = r to he charged, in air, with quantity Q ; the potential will be V = Q/r ; and Q = Vr. If this sphere be connected with the earth by a wire, whose resistance is R, for a short time t, along that wire a current will run, whose mean intensity is I; the quantity conveyed by that current in time t is It ; and this is lost by the sphere, whose charge sinks to Q'. Hence Q Q' = It. If the potential of the sphere is not to sink, the radius must diminish. If the radius shrink to r' in time t, the velocity of its contraction is (r r')/t: and Q = rV as before; and also, Q' = r'V, V being unchanged. From these we find that (r r') /t = I/V = 1 /R = D, the conductance of the wire. But (r r') /t is a Velocity; whence, in electrostatic measure, the Conductance of a wire is a Velocity. xvi.] FALL OF POTENTIAL ALONG CONDUCTORS. 639 sented by CZ, the end C connected with the positive terminal of the battery is at a potential which differs by (P)P' from the potential of the end Z, connected with the negative terminal. The Fall is steady, and depends (1) upon the difference of poten- tial between the ends of the conductor, arid (2) upon the length of the conductor ; it is meas- ured by the Slope of the line p pcc^-- PP', the amount of fall of 4 | potential per unit of length, o 1 If the battery be con- nected at its midpoint with the earth, the conductor CZ is near C at a positive potential; towards the midpoint of CZ this diminishes; the midpoint of the conductor is a point of zero potential (the potential of the earth); and as we approach Z we find the potential increasingly negative. Necessarily, the Potential-Fall or -Slope along the conductor depends upon the difference between the potentials of its extremities, not upon the values or signs of these in relation to the arbitrary earth-zero of potential. Resistance in a Heterogeneous Conductor. When a con- ductor is made up of a succession of conductors which, on account of their differing materials or conditions or thicknesses, present different resistances to the current, it may become necessary to consider each conductor as reduced to an equiva- lent length of a standard conductor, such as a column of mer- cury 1 sq. mm. in cross-section. For example : a current passes successively along (1) a metre of mercury 1 sq. mm. in section ; (2) 10 metres of mercury 1 sq. cm. in section ; (3) 1 mm. of pure water 1 cm. in section ; (4) 61-70 metres of soft copper wire 4 sq. mm. in cross-section : what is the total resistance of this combination ? We must reduce all to a common term, to Re- duced Lengths of our standard mercury column. Then, above, (1) is equivalent to a metre of such a column, (2) is equivalent to -jig- metre, (3) to 400,000 metres, and (4) to ^ metre of such a mercury column ; and the whole resistance is that of 400,001-35 metres of the standard conductor. Resistance in a Galvanic Circuit. In a galvanic circuit we have to consider two sets of resistances : those internal to the cells, the internal resistance, R f ; those in the conducting media, the external resistance, R e . Then Ohm's Law is I = E/R t + R e . Let n cells be arranged side by side, copper to copper, zinc o zinc ; the E.M.D.P. of the combination is the same as that of one cell, and = E Volts ; 640 ELECTRICITY AND MAGNETISM. [CHAP. the internal resistance (the combination being virtually one cell of w-fold surface) is R t -/n Ohms ; the external resistance is unaltered. The intensity or current-strength is therefore I = (E/(Ri/) + R e } = (nE/R* + nR e } Amperes. If the internal resistance be extremely small in comparison with the external, R, may vanish from this expression; then I = {nE/nR e } = E/R e Amperes, and the current-strength is little increased by the use of many cells ; but if the external resistance be extremely small, the current-intensity becomes {nE/R f }, and the side-by-side arrangement in Surface* secures the highest strength of current. If n cells be arranged in file, copper to zinc, the E.M.D.P. is nE Volts ; the internal resistance is nR f Ohms ; and the external, as before, R e Ohms. The current-intensity is now I = {wE/nRj + R e } Amperes. This arrange- ment of cells behind one another in Indian file or in Seriesf is the best for securing the highest attainable strength of current when the internal resistance is extremely small in comparison with the external; for then, R< vanishing, the current-intensity is {nE/R e } Amperes ; while if the external resistance, on the other hand, be exceedingly small in comparison with the internal, the intensity is {nE/rcR f } = E/R f , which differs but little from -)- R e }, the strength of the current produced by one cell. For extremely great external resistances, then, arrange in Series, if the highest attainable strength of current be aimed at ; for extremely small external resistances, arrange in Surface. When neither the internal resistance nor the external can be considered as vanishingly small the one in comparison with the other, the best arrange- ment, for high intensity, is to unite cells, ab in number, into a series of b each : in each series of 6, the b cells are placed side by side, copper to copper, zinc to zinc ; then a such series are arranged in file, the copper terminal of each series being connected with the zinc of the next. In this way we virtually make up a large cells, each of 6-fold surface, and we arrange these in file or series. In each of these virtual large cells the E.M.D.P. is E Volts ; the resist- ance is 1-frth of r Ohms, the resistance of a single cell. Now couple a such large cells in Series; the E.M.D.P. of the combination is aE Volts; the internal resistance of the whole, R , is equal to a x (r/6) Ohms ; the inten- sity or strength of the current produced is aE aE E 1 = -I - = ^ - = BT Amperes, a'+R e b n n a where n = ab. The denominator of the last fraction is the least possible, and the strength of the current consequently the greatest, when R e /r = a/6. When the current is strongest, R e is thus equal to ar/b, or the external resistance is equal to R f , the internal. If the external resistance be equal to nr, and still more if it be greater than nr, the problem of the most advan- tageous arrangement of the cells in rank and file becomes an insoluble one, and the cells must be arranged in series. * Obsolete synonym " in Quantity." t Obsolete synonym " in Tension." xvi.] RESISTANCE IN GALVANIC CIRCUIT. 641 Problem. Sixty Grove cells, in each of which the resistance is -6 Ohm, are at disposal : a resistance of 10 kilometres of soft copper wire of 4 mm. diameter is to be encountered; what is the best arrangement of the cells ? The external resistance, R e , is that of 1,000,000 cm. of copper wire of cross- section -125664 sq. cm. and relative conductivity 61-70: this is equal to 12-13308 Ohms. Now in the equation R e = ar/b = a 2 r/n, R e = 12-13308, r = 0-6, n = 60 ; whence a = 34-83. The nearest feasible number correspond- ing to this value of a is 30 ; and the best arrangement is the division of the 60 cells into 30 virtual double-surface cells, arranged in Series. If the external resistance be that of one kilometre of such wire, a being found equal to 11-015, the best arrangement is 12 sets of cells, each contain- ing 5 cells joined in surface, and these sets joined to one another in Series. To obtain maximum current-strength is not, however, the most economi- cal way of using a battery; half the energy is wasted in overcoming internal resistance : this internal resistance must be proportionally reduced in order to reduce this waste ; and if this be done, then, though the current is not the maximum obtainable, the amount of zinc consumed is reduced in a still greater ratio. For economical working, therefore, keep the resistance of the cell or battery as low as possible compared with that of the general circuit ; that is, work with high external resistances. The Fall of Potential in a Heterogeneous Conductor. If we draw a diagram, setting out on a base-line and using as abscissse the Reduced Lengths of the several successive conduc- tors which make up a heterogeneous conductor, and if for a moment we let drop from view any local differences of poten- tial set up by contact of different materials, then the line of potentials slopes uniformly down from one end of the hetero- geneous conductor to the other end, and from such a diagram we may find the total fall of potential along each component Fig.216. A Thln T ' hi * ek Water Mercury conductor. Fig. 216 very diagrammatically represents the fall of potential in the composite conductor specified in the preced- ing large-type paragraph, p. 639. A A' is the potential at the junction of the slender and the thicker column of mercury, BB' that at the one surface, CC' that at the other surface of the water, OO'and DD' the terminal potentials. If now we follow this up with another diagram in which the real lengths of the conductors are supposed to be, represented, we find a remarkable appearance presented by it. The poten- 2T 642 ELECTRICITY AND MAGNETISM. [CHAP. tial-line, which indicates the successive falls of potential, is repre- sented by the line O A'B'C'D' in Fig. 217. The fall of potential is exceedingly rapid along the bad conductors, for bad conduct- ors keep up a great difference of potential ; and the whole fall Fig.217. c' D of potential is distributed among the component conductors, to each according to its Reduced Resistance. If there be local differences of potential, these must be added to or subtracted from the total fall of potential for which the conductor has to provide. By way of illustration, let AB be a conductor, of which one-half consists of copper wire, the other half of zinc wire, of an equal thickness, and let its extremities be kept at potentials which differ by 2 AX. In Fig. 218, AC is the reduced length of the copper wire, and CB (= f^Jf AC) the reduced length of the zinc wire. Between the copper and the zinc there is (on the older view discussed on p. 611) a rise of potential represented by DE, which would make the slope of the line of potentials steeper throughout the conductor. To BX' add X r F, which is equal to DE : connect X and F by a Fig.218. dotted line. Of this the portion XD would represent the fall of potential along the copper : the sudden rise of potential at D would bring the line of potentials up to E, whence it would be continued parallel to XF, along the course EX', arriving at the terminal potential X'. From this diagram another might be constructed in which, instead of the reduced lengths AC and CB, the corresponding true lengths would be represented and XVI.] FALL OF POTENTIAL IN CONDUCTORS. 643 the corresponding true slope of the potential-line found for each. In Fig. 219, the line ZAA'CZ shows the slope of potentials in a galvanic circuit^ the various parts being supposed to be brought to their reduced lengths. We mio-ht O ' t> again reduce this diagram to another, in which the reduced Zn cu zn lengths of the different parts of the circuit would be replaced by their true lengths, and the true slope of the potential-line found for each. From this we find that the actual difference of potentials between the plates of a battery in a closed circuit depends upon the relation between the internal and the external resistance. Thus if R* be the internal and R e the external resistance, the difference of potentials between the plates, available for the service of the part of the circuit external to the plates, is to the whole E.M.D.P. of the battery (measured between terminals on open circuit) as R e : R + R e . It is there- fore equal to ER e /(R t -f R e ) and we cannot assume E, the whole E.M.D.P. of the battery, to be available unless R e be so great in comparison with R f that the latter may be neglected. Flow along large Conductors. If a conductor be very wide in comparison with the wires leading to and from it, the current widens out, and no part of the conductor is free from equipotential surfaces and lines of flow. If it be practically infinite, the resistance offered by it depends on the radius of the wires or plates connecting it with the battery, and on the resist- ivity of the conductor itself: not on the distance traversed by the current in the wide conductor. In Fig. 220 the battery B is connected with two galvanometers, Gr and G', by a long telegraphic wire interrupted at A : at K there is a key by Fig.220. which contact may be made or broken. CD is a wire, the^ continuity of which may be broken by another key. When C and D are connected 644 ELECTRICITY AND MAGNETISM. [CHAP. through the wire CD, and connection made at K, both galvanometers are deflected. If, however, the connection CD be broken, and connection be suddenly made at K, the galvanometer G is alone deflected: the earth between E and E' does not simply replace the wire CD between C and D. On the other hand, it is beyond doubt that currents do run in the earth's crust. A telephone, part of whose circuit runs in the straight line joining two telegraph stations, will pick up signals from the earth-currents : and the use of the earth in place of a return wire is familiar in telegraphy. Simultaneous Currents. Any number of currents may co-exist on the same wire, and the resultant at any point is the algebraic sum of the separate currents, positive or negative respectively. Thus two currents in opposite directions and of equal strengths may produce no effect in a single wire which is made a part common to two circuits ; and if this wire be led round a magnetic needle, no deflection will be produced. If the one of these two currents be stronger than the other, the effect will be that corresponding to their difference. Derived Currents. When a steady current finds the con- ductor to divide and then to reunite, it divides into portions which run along the several paths open to it. In Fig. 221 the current arriving at A divides into two moieties ; if the two paths be equal in their resistance, these moieties will be equal. If the resistances be not equal, the current passing along each branch will be inversely proportional to its resistance, for the difference of poten- tial between the extremities is the same for every branch, and in each branch the product of the current-strength into the resistance is equal to the difference of potential. The double path acts like a single conductor whose resistance is equal to 1/{1/R' + 1/R"}, where R' and R" are the resistances of the two branches. The conducting power or Conductance of the double path is the sum of the conductances of the two branches; these are respectively 1/R' and 1/R"; their sum is {1/R' + 1/R"}, and the Resistance of the double path is the reciprocal of this sum. Kirchhoff 's Laws. I. Where a steady current branches, the quantity of electricity arriving by the single wire is equal to the quantity leaving the junction by the branches. The alge- braical sum of the currents passing towards (or passing from) the junction is equal to zero ; 5)1 = 0. II. In a metallic circuit comprising within it a source of permanent difference of potential E, the products of the intensity xvi.] KIRCHHOFF'S LAWS. 645 of the current, within each part of the circuit, into the corre- sponding resistance are, if the elements of current be all taken in cyclical order, together equal to E ; 2 (IR) = E. In a metal- lic circuit in which there is no source of permanent difference of potential, E = 0, and S (IR) = 0. This law applies to each several mesh of a wire network as well as to a single metallic loop, and it holds good even when an extraneous current is passed through the loop. Shunts. If between A and B (Fig. 221) a single wire run whose resistance is R, a certain current I will pass; if a lateral path or Shunt be made available, the resistance in which is ^V^> the current in the shunt is I //} and the current in the original wire will sink to I y , l-100th of its former intensity. This result may be found from the equations I = I y + I y/ and I y R (I /y x ^V^ 1 ) 0- If the original wire contain a galvanometer which would suffer risk of damage if the whole original current were sent through it, the current-intensity I can thus, by the use of shunts, be mod- erated to any desired degree. If the shunt have a very high resistance the current running in it is proportionately very small, and the distribution of potential in the circuit, as well as the intensity of the current in the original path, is very little interfered with by the interposition of this new path. If in this new path, with its known resistance, there be arranged a galvanometer or a sensitive current-measurer of any other kind, the indications of this instrument will measure the intensity of the current, and therefore the difference of poten- tial between A and B. This is realised in Lord Kelvin's Voltmeter. The Resistance of two conductors may be compared, by means of a Voltmeter, by observing the relative differences of potentials between pairs of equidistant points in the two conductors, when these conductors are successively, in the same circuit, traversed by one and the same current. Wheatstone's Bridge. In Fig. 222 there is represented an arrange- ment of conductors known by this name. The respective resistances, inten- sities, and directions of the current are indicated in that figure. Kirchhoif's Laws give us the relations p. between these. Law I. shows B that at A, I (the intensity in the wire h/4?r ergs, where and h are respectively the local electric and magnetic forces per sq. cm. in the field, both at right angles to the direction of transmission of energy. xvi.] EFFECTS OF A STEADY CURRENT. 649 EFFECTS OF A STEADY CURRENT. Production of Heat. If a galvanic circuit be completed and allowed, as it were, steadily to run to waste, no external work being done by it, heat is developed within the cell and in the conducting wire. The Heat produced represents the total Energy of the steady current, and is equal, like that energy, to PR ergs per second (Joule's Law), or to E 2 /R or El ergs per second, where R is the total resistance of the circuit, and I the intensity of the current actually passing. The heat per cub. cm. of the wire is EI/vol. = E// x I/o = A. Problem. A uniform copper-wire whose cross-section is 4 sq. mm., and whose length is 106-3 metres, connects the poles of a cell whose effective difference of potential is one Volt, and whose internal resistance is 4 Ohms. How much heat will be developed during one minute ? E is one Volt = ?fa C.G.S. electrostatic unit of E.M.D.P. The total resistance, R, is 4 Ohms internal + x i x - Ohms external = 4-405 Ohms = - ^^ - C.G.S. electrostatic units. The 900000,000000 Heat = Energy = ^ per second = + - ^5 - = 2,270148 ergs R 300 2 900000,000000 per second = 136,208880 ergs per minute = about 3-3 ca per minute. If a current be maintained in a wire, the temperature rises until a point is reached at which the loss by radiation and convection, on the one hand, and the heat supplied at the expense of the energy of the current, on the other, exactly balance one another. Thereafter the temperature remains constant. The wire, being warm, expands. This expansion may be meas- ured. It varies as the intensity varies. If the current-intensity do not materially vary, the amount of expansion may be utilised as a means of ascertaining E, the difference of potential between the ends of the wire. This principle is applied in Major Cardew's and Ayrton and Perry's Volt- meters. When a current is made to pass through a heterogeneous conductor composed of different metals, between which there is developed a difference of electric potential (true contact- effect, p. 624), the energy, which is wholly converted into heat when no work is done by the current, is divided into two parts. Of these one part obeys Joule's Law, and is equal, per second, to PR, the product of the total resistance into the square of the actual intensity: the other, which may be positive or negative, goes to produce Peltier's effect, which is the follow- ing: Consider a junction of metals, A and B, such that when this junction is made the hot junction of a thermo-electric cir- cuit a current passes through it from A to B : let a current be 650 ELECTRICITY AND MAGNETISM. [CHAP. made to run ab externo through that junction in the same direc- tion, A to B ; that junction will under such circumstances be cooled, while if, on the other hand, the current be made to flow from B to A, the junction will be heated. In Fig. 224, OL is a conductor composed partly of iron, partly of cop- per ; a steady current is made to flow through the conductor from O to L, V, Fig.224. (Iron, R,) (Copper, R,,) between the potentials V, and V IV . At the junction J there is a sudden fall of potential (V,, - V,,,). In OJ the intensity = 1 of potential = V.-V, Resistance of OJ R, in JL the intensity is y _ T = V, - V, R Resistance of OJ In both it is equal : hence ( V, ~ Viv) ~ (V,, ~ where E OL is the total fall of potential, Ej the fall at J, and R the total resistance. Hence E OL = RI + Ej. The Energy of the current is E OL x I = Rl 2 + EjL The first part of this expression, RI 2 , represents heat distributed over the whole conductor ; the second part, Ejl, represents heat locally devel- oped at J, and proportional to the fail of potential there. If the current be made to pass from copper to iron there will be a rise, a negative fall : the heat developed at the junction will be a negative quantity, and the junction will be cooled.* In a thermo-electric circuit of copper and iron, the current flows from the copper to the iron across the hot junction. At the hot junction the current passes through a rise of potential (copper-iron) ; the current there- fore tends to cool the hot junction. At the cold junction the current passes through a fall of potential (iron-copper) ; it therefore tends also to heat the cool junction. This cooling of the hot and heating of the cool junction is Peltier's Effect. When a current is passed ab externo through iron, copper, iron succes- sively, it again heats the iron-copper and cools the copper-iron junction. The main current is weakened by the reverse thermo-electric current sec- ondarily produced, and when the main current is cut off, the latter acts alone until the junctions come to the same temperature. Thomson's Effect (Lord Kelvin's). The same thing may occur even within a single metal. Hot iron is negative to colder iron ; a current, made to pass within a mass of iron from a hotter region to a colder, travels against * Prof. O. J. Lodge points out that there is no such phenomenon at a junction of copper and zinc: whence he concludes that there is at such a junction no real fall of potential, and that the apparent D.P. of copper-zinc is really the sum of a copper-air and a zinc-air contact-difference, XVI.] THOMSON'S EFFECT. 651 progressively rising potentials and cools the iron in the cooler region ; made to pass from cold to hot iron, it heats the iron in the hotter region. It thus tends to exaggerate the existing differences of temperature. These effects are reversed in copper or brass. The convection of heat by a current of electricity in unequally heated iron is " negative," that is, it is opposed to that convection of heat which would be brought about by the flow of water through an unequally heated tube. In copper, on the other hand, the electric convection of heat is " positive." In a thermo-electric circuit, therefore, the current, as it travels in the iron from hot to cold, absorbs heat and gains energy ; in the copper, travel- ling from cold to hot it again absorbs heat. The Thermo-electric Diagram may be made to represent the Thomson and Peltier effects. Let Fig. 225 be a diagram for iron and copper between the temperatures t, and t ir The area marked " Peltier hot junction " repre- sents the amount of energy absorbed at the hot junction when a unit- current passes; the area marked "Thomson-Fe" represents the energy Fig. 225. THOMSON - FE THOMSON - CU ' ? Abs. absorbed from the iron when a unit-current passes in it, from hot to cold ; the area marked " Thomson-Cu " in the same way represents the Thomson effect in the copper, the amount of energy absorbed from the copper when a unit-current passes in it from cold to hot. The whole shaded area thus represents the energy absorbed, by the cooling of the hot junction and of the unequally heated iron and copper, when the unit-current runs in the direction indicated by the arrows. Plainly, if the hotter junction be heated to T, the Xeutral Point, we shall have the two Thomson effects, and, at the hot junction, no Peltier effect. There is at that temperature no D.P. between the two metals. Now turn to the energy evolved. This takes two forms: (1) Heat liberated at the colder junction (Peltier effect) ; and (2) the Energy of Electric Current. The latter, when the current-intensity is unity, is equal to the E.M.D.P. ; and we have already seen that this E.M.D.P. is repre- sented by the area between two metal-lines and the ordinates corresponding to the two temperatures. Hence the accompanying diagram (Fig. 226) needs little explanation. If the colder junction be at a temperature of T, 652 ELECTRICITY AND MAGNETISM. [CHAP. the Neutral Point, there will, at that junction, be no Peltier effect, no libera- tion of energy as Heat. The student may now exercise himself in showing that when the colder junction is at temperature T the effect is the reverse of that obtained when the hotter junction is at T; that when one junction is as far below T as Fig. 226. the other is above T, the area representing the Current-Energy vanishes ; and that when the hotter junction is at a temperature farther above T c .than that of the colder is below it, the current is reversed. In these figures the energy supplied is equal to the energy accounted for. The whole arrangement is a kind of thermic engine, in which Heat is absorbed from a Source, partly restored to a Condenser or Sink, and partly converted into the Energy of an Electric Current. The Thomson effects are themselves reversed in iron at a low red heat, and probably again at a higher temperature, so as to make one if not two new neutral points. The same phenomena occur in nickel at comparatively low temperatures. When a circuit is composed of various conductors which successively offer different resistances to the current, the Heat produced is distributed among them, to each according to its resistance, or its total potential-fall. Numerical Example : A circuit consisting of one cell whose E.M.D.P. is 1-8 Volts, and whose internal resistance is 0-7313 Ohm, and of an external conductor composed of 6-170 metres of soft copper-wire 4 sq. mm. in cross- section, in which is interpolated a piece of platinum wire ^ mm. in diar. and 4 cm. in length, will have a total resistance amounting to Battery 0-7333 Ohm, copper wire ^ Ohm, and platinum wire (equivalent to a f - x , , mercury column f - x - J metres long and -007854 sq. mm. in section) 7417 Ohm; or on the whole 1-500 Ohms, or 900000,000000 static units. We assume that a steady current can be set up and maintained for a second within such a circuit, and further, that radiation and convec- xvi.] HEAT IN CIRCUIT. 653 rt )-roqo tion of heat may be set aside. Of the total heat produced, is developed 1*500 in the battery, " J in the copper wire, and ' in the small piece of .1/ouU JL*oUU platinum wire. The total heat produced in a second is E2 = \ ( l ' S Y+ 1>5 I R < V300/ ' 900000,000000 I C.G.S. units or ergs; this is 21,600000 ergs or (21,600000-^-41,593000) ca. The heat evolved in the battery 4889 of the whole would be, if we sup- pose the battery to contain 1 kilogramme of material of a mean specific heat of 0-8, sufficient to raise its temperature by about -000314 C. in a second ; that evolved in the copper wire (whose weight is about 217 grammes and sp. heat = 0-095) by about -0004 C. ; while that liberated in the platinum wire (whose weight is about 0-0276 grammes, and whose sp. heat = 0-0325) would be competent to raise it in a second to the temperature of 289 C. In electric welding, the two pieces of metal to be welded together are made terminals for a powerful current. They are brought into contact : the current runs : the point of contact offers resistance and becomes very hot : and the hotter it is the worse is its conductivity, and therefore all the greater is its resistance. . Heat is thus locally developed : and the metal pieces may, by this means, even be fused together. In electric blasting and the electric cautery the current is made to flow through a very thin piece of platinum wire, which locally becomes red-hot or white-hot ; and in electric fuses an excessive current heats a specially fusible part of the circuit so far that it melts, and thus breaks the circuit or " cuts-off " the current. When a steady current is divided into derived currents (Fig. 221, et seq.), the division is such as to correspond to the least possible value of 2 (I 2 R) in the branches ; that is, to the minimum aggregate production of Heat. Production of Light. When one part of a circuit presents a relatively-great resistance, the greater part of the heat devel- oped within the circuit is concentrated within that part. When the local resistance is due to a thin platinum wire or a thin fila- ment of carbon or of carbonised paper or vegetable fibre or paste, that bad conductor is so far heated as to emit a considerable amount of light. This is illustrated by the various forms of incandescent lamps or electric " glow-lamps." Those in which the carbon filament is arranged within a vacuum give out, according to the type of lamp, the number in circuit, and the intensity of the current employed, a light equal to that of from 1 to 1000 candles each (usually 8 or 16). The ordinary data are : 2^ candle-power, 5 Volts x 1-9 Amperes to 25 Volts x 0-4 Amperes ; 8 c.-p., 15 V x 1-9 A to 55 V x 0-6 A ; 16 c.-p., 30 V x 1-85 A to 105 V x 0-58 A ; 32 c.-p., 55 V x 2-0 A to 105 V x 1-05 A ; 100 c.-p., 80 V x 4-4 A to 105 V x 3-3 A ; 200 c.-p., 80 V x 8-5 A to 105 V x 6-5 A ; 1000 c.-p., 80 V x 43-5 A to 105 V x 33 A. The consumpt of energy is, per candle-power, from 3 to 4 Ampere- Volts per second, or from 0-0047 to 0-0054 horse-power, absorbed from' the energy of the current. 654 ELECTRICITY AND MAGNETISM. [CHAP. If the carbon filament be so constructed as to present the form of a hollow tube, of relatively- great surface and small actual cross-section, the luminous efficiency of the lamp is greatly increased (Bernstein, Cruto). When too strong a current is driven through such a lamp, the superficial particles of the heated carbon are scattered throughout the vacuum ; the carbon is volatilised, and condenses on the wall of the lamp ; so with platinum and iridium heated above 1700 C. When a strong electric current is driven through a carbon rod to a thicker piece of carbon, the thin rod becomes heated , when this takes place in air the carbon burns away rapidly; but if the rod rest loosely by one end upon the thicker mass, the contact is always maintained, and the light is fairly steady so long as any carbon remains. When the interposed resistance is that of a certain thick- ness of air, the current will not pass unless the interval be so small or the difference of potential on its two sides be so great that a spark can fly across it. When this is the case the current is established across the interval. If the poles be of carbon, their extremities become intensely hot and wear away by oxida- tion in the air. The intervening air is so good a conductor when intensely heated that, when the arc has once been estab- lished, the poles may be separated to a distance greater than the striking distance in cold air ; still, the resistance of the hot air will not alone explain the resistance offered by the voltaic arc to the transmission of the current. This is due to a kind of thermo-electric effect. The positive pole is hotter (4000 C.) than the negative (3000 - 3500 C.) ; and the greater part of the fall of potential is at the positive carbon, while there is another potential-fall at the negative carbon. These sudden falls of potential are equivalent to a reverse E.M.D.P. ; and the resist- ance of the arc presents two terms, the one constant, the other increasing with the length of the arc. The air in the arc is dark, not bright. The temperature attained seems to be that of the volatilisation of carbon. In small arc-lamps, this temperature is attained over a small area ; in large, with more powerful currents, a larger area attains it; but the brightness of the incandescent carbon per unit of luminous area is the same in both large and small. The positive pole, being the hotter, wears away about twice as fast as the negative, and becomes hollowed. The problem of electric lighting is to keep the arc in the same place, the carbons xvi.] PRODUCTION OF LIGHT. 655 being allowed to approach one another as far as, and only as far as, is necessary in order to make up for their wear. In JablochkofPs and Jamin's candles the two carbons were rods, parallel to one another and of equal length ; the arc passed between their apices. If the current passed in one direction only, one carbon would wear away faster than the other ; the carbons would thus cease to be of the same length. The currents used must rapidly alternate in their direction ; both carbons are thus equally worn away, and the length of the arc is constant. The usual fall of potential in Jablochkoff's lamps was from 42 to 43 Volts ; the inten- sity of the current producing the light from 8 to 9 Amperes, and the candles per horse-power about 400. In arc-lamps two carbon points are placed opposite to one another, and it is the part of a special regulatory mechanism to keep the carbons at a constant distance (3 to 4J mm.) apart. Such regulating mechanisms depend for their action (1) upon variations of intensity of the current traversing the lamp, or (2) upon variations in the differences of potential between the two ends of the arc, or (3) upon departures from a predeter- mined relation between this difference of potential and the intensity of the current, or (4) upon variations in the amount of heat developed in the arc. The light given generally varies very much according to the angle from which the lamp is viewed. The resistance of the voltaic arc is 4 to 10 Ohms ; the fall of potential is from 32 to 58 Volts ; a 12,000-candle lamp consumes about 7 engine horse- power, an 875-candle lamp about 1 engine h.-p., or f h.-p. electrical (say 10 Amperes x 50 Volts). The heat developed in the arc has been utilised by Messrs. Siemens and Huntington, who produced the electric arc within the interior of a crucible, and by its means fused very refractory metals with considerable expedition. M. Moissan has recently succeeded in effecting many extraordinary chemi- cal reductions by the temperature of an arc (450 A and 70 V) enclosed between two blocks of burned lime. At the temperatures obtained, he distilled platinum and evaporated silicium and carbon. When the electric arc is produced between carbons in vacua a beautiful glow is obtained, the negative pole being surrounded by a blue aureole, and the positive by a stratified pale-blue light. The carbon evaporates, the vessel becomes filled with a blue vapour which darkens to indigo, and this condenses and renders the whole opaque. If a very little vapour of bisulphide of carbon be introduced into the vacuum, the light becomes insupportably bright, and of an extremely brill- iant green. Its spectrum presents channelled regions in the red, yellow, green, and violet, which look like duplicates of one another, Reproduced in different colours (Jamin). 656 ELECTRICITY AND MAGNETISM. [CHAP. Geissler's Vacuum Tubes. When a discharge of high D.P., as from an electric frictional-machine, or an "induction coil," or a battery of 400 Groves, is sent through a mass of rarefied gas (about yoVo atmos.) contained within a so-called vacuum tube, that gas glows with a bright light, characteristic, as regards its spectrum, of the gas exposed to this operation. The positive pole is surrounded by a bright glow, the negative by a dark space and a set of striae, and in this case the negative pole is the hotter. If the vacuum be very good and the tube containing the rarefied gas be somewhat narrow at its middle, the glow breaks up into striae, which flow and flicker if the current which pro- duces them be in the slightest degree variable. (See p. 662.) The discharge through a vacuum is shown to be disruptive by the fact that the fall of potential in the vacuum tube is inde- pendent of the E.M.D.P. of the circuit, provided that this be sufficient to produce such a discharge at all; and there is a large fall of potential at the negative pole. The approach of external conductors repels the internal glow and causes its deflection ; and the glow is deflected by a magnet in the same way as a wire bearing a current would be. In very high vacua the discharges from the two poles of a vacuum tube appear to be independent of one another : each pole discharges itself with- out, as it were, feeling the condition of the opposite pole of the tube. Even where both are positive they may discharge towards one another into the same space. Electrification of Radiant Matter. When the rarefac- tion of a gas is extreme (one-millionth) its matter becomes radiant. The movement of its molecules may be guided and rendered manifest by electrification. In a Geissler's tube, the mattey filling which is radiant, the molecules which come in contact with the negative pole are at once repelled from it in lines at right angles to its surface. Energy is imparted to these molecules by the electrified negative-pole, and where these mole- cules strike each other or other molecules they produce an internal glow ; where they strike glass (or diamond or ruby) they produce light and cause phosphorescence ; they also pro- duce heat, so that when they are directed from a concave nega- tive-pole upon a piece of platinum, the energy conveyed by them brings that piece of metal to its melting point ; and when they strike a movable body they produce obvious mechanical effect (Crookes). xvi.] RADIANT MATTER. 657 Two streams of molecules proceeding from a forked negative-pole repel one another like two similarly-electrified gold leaves, and the negatively- electrified particles which constitute such a stream are attracted and deflected from their course by a magnet. Chemical Effects of a Steady Current Electrolysis. In most cases, if a liquid permit the passage of a steady current through it, different chemical elements or groups of elements, contained within it, travel in opposite directions along the lines of force, the result being apparent decomposition; in other words, most liquids which possess conductivity are Electro- lytes. A few liquids, such as alcohol and ether, though not absolutely non-conductive, are not decomposed by the passage of a current. As a rule, substances which conduct when melted, but insulate when solid and cold, are electrolytes, e.g., glass. Let us take as an example the effect of a current upon a solution of hydrochloric acid in water. In such a solution insert two platinum plates, the one, the positive-electrode, connected by wire with the copper terminal of a sufficient battery; the other, the negative-electrode, with the zinc or negative terminal. Hydrogen is liberated on the surface of the negative-electrode ; chlorine is liberated upon the positive- electrode, which by a secondary reaction it attacks and dissolves, with the formation of PtCl 4 . According to recent researches, the mechanism of Electroly- sis seems to be somewhat the following : If the liquid lying between the positive and negative electrodes were pure water, these electrodes would be brought by the outside battery to a definite difference of potentials, and there would thereafter be no current, but a condition of electrostatic equilibrium would result, in which the water constituted a field of force ; for pure water is a non-conductor. If, however, the water contain, say, HC1 in solution, then some hydrogen and some chlorine are already dissociated from one another, and exist equally disseminated throughout the solvent as separate atoms or ions ; the hydrogen- atoms are positively and the chlorine-atoms negatively charged with definite and equal quantities of electricity. The positive hydrogen-atoms are attracted by the negative electrode, the negative chlorine by the positive. Each atom, as it comes up to its electrode, discharges its electricity into the general cir- cuit ; it is then free to combine with another similar atom, similarly discharged, to form a molecule of ordinary free non- electrified chlorine, or of hydrogen. The phenomena of Elec- 2u 658 ELECTRICITY AND MAGNETISM. [CHAP. trolysis are, therefore, not phenomena of decomposition, but of discharge of already-dissociated ions upon the electrodes ; and the number of free charged ions which can come up to the electrodes in a given time limits the quantity of electricity which they can bring to the electrodes in that time, and thus determines the Conductance (or, inversely, the Resistance) of the electrolyte. Any molecules which are not decomposed by dissociation appear to play no part in the electrolytic conduc- tion: and the electrolytic conductivity of a solution, after mak- ing due allowance for dilution, thus measures the extent to which dissociation has taken place in it. The ions cannot travel with indefinite velocity through the electrolyte ; each kind of ion has its own specific velocity under the electromotive action of a given slope of potential. This causes differences in the number of free ions which reach the electrodes, and corresponding differences in the quantities of electricity dis- charged upon the electrodes in a given time ; hence the total Conductivity depends upon the number of free ions of each kind and the specific velocity of each : and it increases with temperature so long as dissociation remains incomplete, but no further. In the electrolysis of a solution of hydrochloric acid in water, since the hydrogen travels five times as fast as the chlorine, the solution becomes more weakened towards the positive than towards the negative electrode : while in the electrolysis of a solution of potassium chloride, since the velocities of the ions happen to be nearly equal, the solution is weakened almost equally at the two electrodes. In copper sulphate, CuSO 4 , the copper plays the part of the hydrogen of the previous example, while the part of the atom of chlorine is taken by the atom-group or salt-radicle SO 4 . The copper liberated at the negative-electrode forms a film upon that electrode ; the SO 4 liberated at the positive-electrode reacts secondarily upon the water present; SO 4 + H 2 O = H 2 SO 4 + O ; the positive-electrode is surrounded by sulphuric acid, and oxy- gen is liberated on its surface. If a solution of sulphate of potash be electrolysed, the ions liberated are K, K, at the negative and SO 4 at the positive elec- trode ; the SO 4 causes the evolution of oxygen as a secondary product at the positive-electrode ; the potassium, by the reac- tion K + K + 2H 2 O = H 2 4- 2KHO, causes .the evolution of hydrogen at the negative-electrode. Here the water seems to have been decomposed; the apparent decomposition of the water is, however, a secondary result of the liberation of the potassium and SO 4 ions. xvi.] ELECTROLYSIS. 659 Alcohol, oil, and bisulphide of carbon are practically like water, non- conductors when in a pure state ; but if metallic salts be dissolved in them, the solutions are electrolytes. The secondary reactions met with in electrolysis depend upon the time which is allowed for them, and are therefore favoured by currents of small intensity. If copper chloride be electrolysed between copper elec- trodes, the one, the negative-electrode, is thickened by a deposit of copper, while the other is worn away, being attacked by the chlorine ; and the intervening solution of copper chloride is, if the electrolysing current be feeble, maintained in its state of saturation; but if the electrolysis be very rapid, the solution of the copper electrode does not keep pace with the evolution of chlorine upon it, and the solution becomes weaker in copper and acid in its reaction. The secondary reactions observed are sometimes very peculiar. When hydrochloric acid is electrolysed, the chlorine evolved at the positive-electrode attacks the water present and liberates oxygen, which in its turn attacks some of the hydrochloric acid present and produces chloric and perchloric acids. When Na 2 CO 3 is electrolysed, Na appears at the negative pole (pro- ducing a secondary evolution of hydrogen) and CNaO 3 at the positive pole ; this last group reacts upon water and forms oxygen and NaHCO 3 ; 2CNaO 3 + H O = O + 2 HNaCO 3 . When NaHCO 3 is electrolysed, it produces Na and CHO 3 , and then 2 CHO 3 = 2 CO 2 + H 2 O + O. When formic acid (H.COOH) is electrolysed it breaks up into H and COOH ; then 2 COOH + H 2 O = 2H.COOH + O, or formic acid and oxygen; but the oxygen reacts upon some of the formic acid present, and then H.COOH + O = H 2 O + CO 2 . When fused caustic-potash is electrolysed, K appears at the negative, HO at the positive ; this coalesces into H 2 O 2 , and breaks up into H 2 O and O ; but if the action be slow the K acts upon some of the water, and hydrogen is evolved. The nascent hydrogen evolved at the negative pole will attack aldehydes, forming alcohols, and thus certain ill-tasted rough alcohols may be greatly improved. Faraday's Laws of Electrolysis. First Law. The quan- tity of material liberated at the electrodes from a given elec- trolyte in a given time is directly proportional to the intensity or strength of the actual current; or, in other words, the quantity of electricity which passes in a given time, in a circuit of which a given electrolyte forms part, is propor- tional to the quantity of the ions actually liberated at the elec- trodes in that time. Faraday's Second Law. This Law of Electrochemical Equivalence may be divided into the following propositions, of which the fourth may be regarded as a paraphrase, of the law itself : 660 ELECTRICITY AND MAGNETISM. [CHAP. 1. The gramme-equivalent of a metal is that quantity which will chemically replace one gramme of hydrogen. For example : in comparing HC1 (H = 1, Cl = 35-5) with AgCl (Ag = 108, Cl = 35-5), we find Ag (= 108) to be equivalent to H(= 1) ; the gramme-equivalent of silver is 108 grammes. In comparing CuSO 4 with H 2 SO 4 we find Cu (= 6346) equiva- lent to H 2 (=2); the gramme-equivalent of copper is 31-73 grammes. 2. The gramme-equivalent of a salt-radicle or halogen is that quantity which will combine with one gramme of hydrogen. In HC1, 35-5 grammes of chlorine unite with 1 gramme of hydro- gen ; the gramme-equivalent of chlorine is 35'5 grammes. In HNO 3 (H = 1, NO 3 = 62), 62 grammes of NO 3 unite with 1 gramme of H ; the gramme-equivalent of NO 3 is 62. In H 2 SO 4 (H 2 = 2, SO 4 = 96), the gramme-equivalent of SO 4 is48 grammes. In H 3 PO 4 (H 3 = 3, PO 4 = 95) the gramme-equivalent of PO 4 is 31f. 3. The gramme-equivalent of a salt or acid is that quantity which contains 1 gramme-equivalent of the halogen or salt- radicle. 4. When a current whose intensity, after the current has become steady, is equal to A Amperes passes in a solution of a salt, 0*000, 010352A gramme-equivalents of the salt are electro- lysed during each second ; 0-000,010352J. gramme-equivalents of the salt-radicle or halogen being liberated at the positive elec- trode and a corresponding quantity of the metal at the negative. If both ions be monovalent, as in Ag | N"O 3 , the electrolysis of the solu- tion causes the liberation or deposition of 0-000,010352/1 gramme-equiva- lents of the metal as well as of the negative ion, where A is the number of Amperes. A current of 1 Ampere deposits from a solution of pure nitrate of silver 0-001,118 gramme of silver per second, or 4-025 grammes per hour; this corresponds to 0-000,010352 gramme of H per second. If the ions differ in chemical valency, the rule holds good that the amount of the negative ion, the halogen or the salt-radicle, liberated at the positive-electrode, is equal to 0-000,010352^4 gramme-equivalents. Thus let a current pass simultaneously, in " series," through a solution of cupric chloride and a solution of cuprous chloride, and continue to pass through both solutions for five minutes ; its intensity is 18 Amperes ; com- pare the amounts of copper deposited on the negative-electrodes in the two solutions. In each the amount of the halogen the chlorine liberated is (-030010352 x 18 Amp. x 300 sec.) gramme-equivalents,' or (-000010352 x 18 x 300 x 35-5) grammes. In CuCl 9 every 71 parts of chlorine are combined with 63-46 of copper ; the copper deposited from the CuCl 2 is therefore (-000010352 x 18 x 300 x 35-5) x grammes. xvi.] ELECTROLYSIS. 661 In Cu 2 Cl 2 every 71 parts of chlorine are combined with 126-92 of copper; the copper deposited from the Cu 2 Cl 2 solution is therefore (-000010352 x 18 x 300 x 35-5) x grammes, double the quantity deposited by the same current from a solution of cupric chloride. When acidulated water is electrolysed, 0-0000103524 grm.-equivts. of salt-radicle are set free: these take up the hydrogen of 0-0000103524 grm.- equivts. of water, and set free 0-0000103524 grm.-equivts. of oxygen, that is, (0-0000103524 x 8) grammes. Correspondingly, 0-0000103524 grammes of hydrogen are set free at the negative-electrode. Thus it is said that a current whose intensity is 4 Amperes will decompose 0-000,092,9614 grammes of water per second : though a current will not decompose water at all, except by secondary reactions such as the above. A Voltameter is an instrument in which a current is made to pass through acidulated water between platinum electrodes ; the hydrogen and oxygen liberated at the electrodes are collected, either together or separately, in a graduated tube or tubes, and measured ; a simple calculation gives the strength of the current actually passing through the voltameter. Instead of sending the whole current through the voltameter, we may send a known fractional part of it by arranging the instrument in a Shunt. Since each Ampere will liberate 0-000010352 grm.-equivt. of salt-radicle or halogen per second, each Coulomb of electricity will liberate that quan- tity independently of the time; and each C.G.S. electrostatic unit will lib- erate one 3000,000000th part of this quantity, i.e., 0-000000,000000,0034506 gramme-equivalents. This last quantity is otherwise known as the electro- static Electrochemical Equivalent of the salt-radicle or halogen liberated or salt electrolysed. The Energy of a current passing a quantity Q down a constant poten- tial-fall E is equal to EQ; when Q = 1 this energy is, numerically, E ergs; but it is also, if no energy be lost or gained collaterally, equal to the work done by the unit quantity in electrolysing one electrochemical equivalent of a salt. Hence, conversely, the chemical energy liberated by the production of one electrochemical equivalent of a salt may be measured in terms of a potential-fall or an electromotive difference of potential. The E.M.D.P. in a galvanic cell may thus be computed in terms of the chemical energy set free in it. Let us enquire what is the value of the E.M.D.P. of a Daniell's cell. Here we have a chemical action going on which liberates electric energy : this energy, if not utilised as the energy of a current of electricity, may wholly appear as heat ; this heat has been measured in various ways, and the mean result of several observations is that (between extreme values 714 and 805) the amount of heat liberated when one gramme of zinc is dissolved in the cell amounts to 760 ca or 31,610,680000 ergs. The gramme-equivalent of zinc is 32-645 grammes (Marignac); the electrostatic electrochemical equivalent of zinc is (0-000000,000000,0034506 x 32-645) = 0-000000,000000,112645 grammes. The amount of energy liberated on the solution of one electrostatic electrochemi- cal equivalent of zinc is therefore (0-000000,000000,112645 x 31,610,680000) ergs or 0-00356 ergs. This being equal to the amount of energy liberated during the solution of one electrostatic electrochemical equivalent of zinc in a Daniell's cell and the production of a current of one electrostatic unit- intensity for one second, is necessarily equal to the energy which would be spent by a current of the same intensity, and enduring for the same time, 662 ELECTRICITY AND MAGNETISM. [CHAP. on reversing the whole chemical process which results in the production of an electrochemical equivalent of zinc sulphate, that is, on electrolysing one electrochemical equivalent of that salt. This we have seen to be numerically equal to E, the difference of potential under which the unit current passes. The E.M.D.P. of a Daniell's cell is therefore E = 0-00356 electrostatic units of difference of potential. This is equal to 1-068 Volts; a theoretical result which does not depart widely from the experimental values, which range between 1 Volt and 1-124: Volt. This mode of computation is due to Lord Kelvin. Electrolysis in Gases. Cold gas will not conduct, if dustless : hot gas will : it seems that some amount of dissociation is necessary for con- duction. Those gases which are most readily dissociated are the best con- ductors. Mass for mass, highly rarefied gases have high conductivities : and dissociated molecules appear to be concerned in conduction. These molecules appear to be disposed in chains here and there along the path of the current, those of each kind acting together, and the average length of the chains being the same as the distance between the strice. When a cur- rent goes through rarefied steam, oxygen and hydrogen are liberated, in approximately the same amount as in a voltameter traversed by the same current (Perrot and J. J. Thomson). A current once started in a gas can be kept up with comparative ease ; the molecules have been sufficiently dis- sociated to act as carriers. When oxygen is sent through a space across which a silent discharge is passing, it becomes in part converted into ozone, especially if there be rapid variations in the potential-difference. Prof. Schuster has shown reason for believing that in highly rarefied gases there is considerable dissociation of molecules into atoms around the negative electrode. In mercury-vapour, which is monatomic, the phenomena of glow are the same round both terminals. In Electrolysis, Work is done by the current ; its Energy is spent in separation of the charged ions of the electrolyte. The energy absorbed in the electrolytic production of, say, 9 grammes of oxy-hydrogen gas from acidulated water, is approximately equal to the Heat liberated by chemical action and change of physical state when 1 gramme of hydrogen and 8 of oxygen are exploded together and condensed into water. Work done on electrolysis, if such work be other than chemical, causes divergences from Faraday's second law. Such work is done when the pro- ducts of electrolysis of a liquid become gaseous, or when the metal of an electrolyte is lifted up towards the negative-electrode ; while the energy of the current is increased when these circumstances are reversed. The relation of the energy liberated by the chemical action of the battery to the heat produced in the battery and the energy spent in doing electrolytic work is represented by the equation Battery-energy = Heat evolved in battery + Electrolytic work done. The last term cannot be greater than the first; if it were, the heat developed in the battery would be a negative quan- xvi.] ELECTROLYSIS. 663 tity, and the battery would cool itself as well as all surrounding objects. That this should go on indefinitely is impossible : and in such cases the E.M.D.P. of the battery falls as the temperature falls : and, further, heat- ing the battery causes a rise of E.M.D.P. Similarly, if the battery become heated in action, heating it causes a fall in its E.M.D.P. If the battery become neither heated nor cooled, as in the case of Daniell's cell, heating or cooling it from without causes no change in its E.M.D.P. ; and it is only in such cases that the E.M.D.P. can be directly calculated from the chemical energy alone. In a Daniell cell the replacement of one grm.-equivt. of copper by one grm.-equivt. of zinc is attended with the evolution of 24,200 ca of heat ; in a Grove the energy evolved is 47,000 ca for every grm.-equivt. of Zn dis- solved. When 1 grm. H and 8 grm. O unite to form one grm.-equivt. of water, 34,462 ca of heat are evolved. A single Grove cell can therefore electrolyse acidulated water; a cell with a potential-difference equal to f If H times that of a Daniell would just be able to do so; a single Daniell cannot. Two such cells are, however, able to effect electrolysis of water ; the energy supplied by two cells arranged in series is double that supplied by one, even though the resistances be so adjusted that the current pro- duced is of the same intensity : for Energy per second = El, and if E be doubled while I remains the same, the energy is doubled. A single cell will, however, electrolyse water if the positive-electrode be of a substance such as copper, which will combine with oxygen, and thereby liberate energy sufficient to make up the deficiency in that supplied by the cell. The process of electrolysis is turned to practical account in the arts of electroplating, etc. ; the article to be covered with a metallic film is made the negative-electrode in a suitable solution of the metal. The positive- electrode is often itself made of the metal to be deposited from the solution : as the metal is deposited from the solution upon the negative-electrode, the positive-electrode is attacked and its substance dissolved in the solution, which is thus kept saturated if the action be not too rapid. In the electrolysis of mixtures, the reaction which requires the least energy is the first to be completed; and thus, by regulating the potential- difference, the different metals may be successively deposited from a mixed solution. When a mixed solution of acetates of lead and copper (Nobili), or a solution of litharge in caustic potash (Becquerel), is electrolysed between two electrodes, of which the negative is a sharp-pointed platinum wire or steel needle, while the positive is a large plate of silver, german-silver, sil- vered copper (spangle metal), or even thin sheet-iron, the current being one of relatively-great E.M.D.P. (15 to 20 very small Bunsen cells mounted in file), there is formed on the positive-electrode a series of rings concentric with the point of the negative-electrode, Nobili's rings, produced by deposition of PbO 9 . If the positive-electrode be complex and consist of two or more points, or if currents be made to run through some of these points towards the plate, while in others the direction is reversed, the rings are modified into representations, in iridescent hues, of complete systems of equipotential surfaces (Guebhard). , 664 ELECTRICITY AND MAGNETISM. [CHAP. Polarisation of Electrodes. When platinum electrodes have been used in the electrolysis of water, the negative one is found to contain a certain quantity of hydrogen not only adherent to its surface, but also occluded within its substance ; while the positive one carries similarly a certain quantity of oxygen. These oxygen and hydrogen films tend to produce a reverse current, which weakens the main current. When the main current ceases, the reverse current continues for some time and dies away gradually. The smallest potential-difference between the electrodes is sufficient to direct the ions so as to set up this state of polarisation ; and the reverse cur- rent tending to be produced by this is equal and opposite to the main current, so that between the two there is no effect, until the latter reaches a certain limit. When it exceeds this limit, a current passes; but it is only the excess of the main current above the virtual reverse current which can pass through the electrolyte. The occluded hydrogen is very gradually reduced by the oxygen brought to it by a reverse current, and a corresponding quantity of hydrogen is liberated on the positive pole, where it is occluded by the electrode or oxi- dised by the occluded oxygen or dissolved by the water. Thus, though a reverse current passes when the main current ceases, no products visibly appear. In Grove's gas-battery a number of such electrodes are arranged so as to give a current, or which amounts to the same thing a circuit is arranged in which a current passes through: (1) water; (2) a plate of platinum standing partly in water, partly in hydrogen gas ; (3) conducting wire ; (4) a plate of platinum standing partly in an atmosphere of ozonif- erous or electrolytic oxygen, partly in the water. If electrodes of palladium be used in the decomposition of water, the negative one absorbs as much as 936 vols. of hydrogen, with which it forms an alloy, greater in size than the original electrode. The electrodes in such a case are said to be polarised, and their polarisation within a given substance is (if the resistance of that substance be not so great as to prevent any perceptible current from passing under any potential-difference) the best test as to whether that substance is really an electrolyte. Warm glass is thus found to be actually an electrolyte. The capacity of electrodes so polarised is very great. Two square inches of platinum electrode immersed in dilute sulphuric acid have (Varley) when the E.M.D.P. is one-fiftieth that of a Daniell's cell, a capacity equal to that of an electrostatic condenser whose plates have an area of 80,000000 square inches separated by | inch of air ; i.e., a capacity of 175 microfarads : while as the E.M.D.P. increases, the capacity increases also. Electrodes of amalgamated zinc are not at all polarised when used to transmit a current through a perfectly neutral solution of ZnSO 4 (Beetz). When a powerful current is sent through a metal immersed in dilute sulphuric acid, hydrogen evolved on its surface offers local resistance, and xvi.] POLARISATION OF ELECTRODES. 665 heat is there evolved; the metal may rapidly become even white-hot (Lagrange and Hoho's process for welding). Secondary Cells and Batteries. When acidulated water is electrolysed between electrodes of lead, the negative electrode remains bright, but the positive becomes covered with a film of PbO 2 . When the current ceases, if the positive and negative electrodes be connected by a conducting wire, a reverse current a polarisation-current, or Secondary Current passes ; the film of PbO 2 is electro-negative like the copper in an ordinary cell, the lead electro-positive, like the zinc ; and the current passes in the conducting wire from PbO 2 to Pb; for which reason the PbO 2 pole of the secondary cell or battery is called the positive pole. The reverse action appears to be (Gladstone and Tribe) in the main the following : Pb0 2 H 2 S0 4 I H 2 S0 4 I Pb = PbO H 2 H 2 S0 4 I PbSO 4 which becomes PbSO 4 | H 2 O | H 2 O | PbSO 4 . On passing in a charging current, the reaction is, for thin layers, PbSO 4 , H 2 O, O; H 2 , SO 4 Pb = PbO 2 , SO 4 H 2 ; H 2 SO 4 , Pb. An arrangement of this kind, into which a current of elec- tricity can be passed and a reverse or secondary current obtained at will, is a Secondary Cell; and secondary cells may be grouped into Secondary Batteries. Planters original form of secondary cell consisted of two large lead electrodes, separated by a sheet of felt, rolled up into a spiral and immersed in 10% dilute sulphuric acid. Faure improved this by covering both electrodes with a layer of red-lead of about 10 kilogrs. per sq. metre, held on by layers of felt and parchment between the opposed plates. When a current has been passed into a so-called ' Faure-accumulator ' for some time, the red-lead on the surface of the negative-electrode is converted into spongy lead, while that on the positive-electrode is oxidised to PbO 2 . This improvement greatly abridged the tedious process pre- viously necessary for Charging Planters 4 accumulators.' In the Faure-Sellon-Volckmar accumulators ' there is no felt ; the plates of lead are pierced or cast with holes, into which there is compressed a quan- tity of red-lead, of reduced lead, or of a salt of lead, or a mixture of PbO, Pb 8 O 4 , and PbSO 4 . One of these, weighing 140 kgr., and composed of 43 plates, gave (Hospitalier) a current of 120 Amp. for 6 hours. The efficiency of these batteries is in part due to the fact that their porous metal or oxide is in close contact with the lead plate, and is, on 666 ELECTRICITY AND MAGNETISM. [CHAP. account of its porosity, able. to retain large quantities of the oxygen or hydrogen which is evolved when an external current is passed through the accumulator. When the secondary current has passed for some time, both films become mainly converted into sulphate of lead, and the apparatus is ready for a renewed charge. When a secondary battery is charged by two or three Grove cells and disconnected from them, it will be rapidly discharged if connection be established between its poles by means of a thick wire. From sixty to seventy per cent, or, in the newer forms, a still greater proportion of the energy actually sunk in it can be recovered in the form of the energy of the secondary current, especially if the battery be not allowed to run down too far. When the cells of a secondary battery, charged side by side, are disconnected from the source, and then connected in series and discharged, the electric current produced is one of "high ten- sion" or great fall of potential. The D.P. of a single cell is about 2-25 Volts, and hence a hundred such cells, first arranged in surface and charged by prolonged connection with a few cells, can be made, if arranged in series, to pass for a short time a current of E.M.D.P. = 225 Volts; such a current produces a high temperature, together with vibration and crumpling of the conducting wire. The internal resistance of these cells is very small, being only 0-006 Ohm in a single cell whose surface is about 300 sq. cm. A secondary cell of about 18 Ibs. weight, the plates in which are each about 2 sq. feet in area, will, when charged by three Leclanches, keep at a white heat during 5 to 10 minutes a platinum wire of ^ inch diar. and 8 inches long. A pile of this weight, kept constantly connected with three or four Leclanche elements (which require little attention beyond that of keep- ing them moist), is a very convenient means of heating such a thing as a galvano-cautery wire, which must be raised to a high temperature for a short time. The rate of discharge should not exceed 1 Ampere per sq. decimetre of plates. Electrical Storage of Energy. When energy is stored up in bent steel springs, about 3924 megergs per kilogramme can be stored up i.e., 40 kilogrs. can be lifted through 1 metre by the elasticity of a spring weighing one kilogramme. When it is stored up in compressed air, 1. kilogr. of air com- pressed to one-sixth contains 2,250,000 megergs, of which about 450,000 can be recovered in the form of work. When it is stored in secondary batteries, about 500,000 megergs are stored up per kilogr. of secondary battery. Of these, xvi.] ELECTRICAL STORAGE OF ENERGY. 667 from 250,000 to 330,000 may be recovered if the batteries be used within a day or two after charging. The great fault of these " accumulators " in their present form is their want of durability. Equalisation of a Current. A current passed through one plate of an Electrostatic Condenser will be apparently absorbed when the current is increased, and will be given out equably when the current falls off ; such a condenser is therefore com- petent to play the part taken by a flywheel in the mechanical transmission of power. Similarly, Secondary Cells may be made to serve as regu- lators of a current if they be fitted up in the course of the conductor in the manner indicated in Fig. 227. The direction of Fig. 227. Z- the secondary current is indicated by the dotted lines and arrows connected with the secondary cell ; it opposes the main current in CZnCuD, but aids it in DEC. THE DYNAMICAL PROPERTIES OF A STEADY CURRENT. If a straight conducting wire, forming part of a wide circuit, bearing a steady current, be passed vertically through a hole in a piece of card or of silver-paper adjusted to a horizontal position, and if iron filings be then sprinkled upon the card, and if the card be gently tapped downwards so that the filings may leap into positions spontaneously assumed by them, they will be found to range themselves in concentric circles round the current, while each filing becomes, for the time being, a little magnet. The space round the current is therefore an Electromag- netic Field of Force, permeated by concentric circular Lines of Magnetic Force or of Magnetic Induction and by Magnetic Equipotential Surfaces, which ,are at right angles to these. The magnetic equipotential surfaces all have 'U-II7IRSIT7' At* 668 ELECTRICITY AND MAGNETISM. [CHAP. the line of the current for their common edge or boundary. If the current be straight, these equipotential surfaces are planes ; and if they were visible, and if the current could be looked at end-on, so as to appear a mere point, these surfaces would seem to radiate from it like equidistant radii from a centre. The lines of Magnetic Force mark the direction in which an ordinary magnet, such as a small compass-needle, when placed within the field, tends to place itself. The one end of the magnet is driven in one direction, the other end equally in the opposite direction, along these lines of force: the magnet is acted upon by a couple, which acts upon the two extremities or Poles like the hands on the handles of a copying press one pole being pushed or repelled, the other being pulled or attracted, until the magnet lies along a line of force. The moment of the couple gradually diminishes as this position is being assumed, and the couple ultimately ceases to produce farther rotation; and further, since one pole is attracted as much as the other is repelled, the magnet as a whole undergoes in such a field no movement of translation. The direction in which a current tends to throw the positive or north-seeking pole of a magnet placed in its neighbourhood is shown by Fig. 228. This direction is called the Positive Direction of the lines of magnetic Fig.228. force. The Current in the figure passes verti- cally Upwards ; the Positive pole is thrown to the Left Hand of the Current. This expression, left hand of the current, is obtained by suppos- ing the current to be replaced by a person whose head is at B and feet at A, and who turns so as constantly to keep the magnet-pole in full view. The relation between the direction of the current and the positive direction of the lines of magnetic force is always the same as that between the propulsion of the point and the twist of the hand in the ordinary use of a European corkscrew. Conversely, action and reaction being opposite, stationary positive magnet-poles tend to throw movable conduct- ors, bearing upward currents, to their left. For negative mag- net-poles the directions given are reversed. Proposed Mnemonic Rule. If a pen be held in the right hand in the usual way, the penholder may represent the wire, and the direction of flow of ink (towards the point) the direc- tion of the current ; if then the thumb be stretched across the xvi.] FIELD SURROUNDING A CURRENT. penholder it will represent the magnet, and the thumb-nail its marked or north-seeking pole. The same relation may be still more simply brought to mind by laying the thumb across the forefinger of the right hand ; either of these will then represent the current (flowing towards the finger-tip or the thumb-tip, as the case may be), the other the magnet. A magnet-pole may be made to rotate round a current by keeping the other pole in the axis of rotation. In general, magnet-poles tend to rotate round currents so long as these are maintained ; and, conversely, currents tend to rotate round magnet-poles: and the deflection of a magnet by a current is only a particular case of this. The action between a current and a magnet-pole is thus at right angles to the line joining them. There is a magnetic field even within the wire ; if a magnet be made itself a conductor, the steady current being led from its midpoint to one end, it will spin. Further, Linear Currents act upon other Linear Cur- rents ; but they do not throw them to the right or left ; they attract or repel them directly. In Fig. 229, AB is a steady current ; round it there are lines of magnetic force, the number of which per sq. cm., at any point, varies inversely as the mean distance from the axis of the wire. Let another current, parallel and in the same direction, be brought to CD : then, be- Fig.229 tween AB and CD the lines B Fj of force of the two currents ,^"~" are opposed, but beyond ^'.^.--si"--^--.-----. , c * -"-*- CD they concur in their direction. The result is that the medium beyond t ..~- CD is in a state of greater v *-. constraint than that be- tween AB and CD ; so is that to the left side of AB ; and the two conductors are impelled towards one another. If the current in AB be opposed to that in CD, the directions of the lines of force coincide between AB and CD, but are opposed beyond AB or CD, and the stress is such as to drive AB and CD asunder. If the currents in AB and CD be at right angles, approaching one another (Fig. 230), the current in AB has lines of force whose direction is upward on the left side of AB, downward on its right. The current CD has lines which would depress a posi- tive pole placed in the upper, and raise a positive pole placed 670 ELECTRICITY AND MAGNETISM. [CHAP. in the lower part of the figure. Fig.230. Up Up Down The concurrence of lines of force is in the upper part of I the diagram (Fig. 230): if the current AB be fixed, the conductor CD is repelled in ^ a direction contrary to that | of the course of the current in AB. If the current in CD pass from D towards C, the conductor CD tends to move in the same direction as the current in AB. Down Fig. 231. 1 ft "A 1 ( / / ' 1 These results were first summarised in Ampere's formula. The mechanical force of Attraction between very small elements of two linear circuits is equal, in dynes, to ii' II' (2 sin sin 0' cos w cos cos O f ) /d 2 , where i and i' are, in mag- netic measurement (p. 707), the in- tensities of the currents passing in the two wires ; I and /' the lengths of the elements ; d the distance between their midpoints ; and 0, 0', w are the angles which deter- mine their relative direction as follows : Each element makes an angle 6 or 0' with the line AB ; but fur- ther, these elements are situated in planes which make an angle co with one another. We may take some particular cases. Let both currents lie in the plane of the paper ; the angle , of the Shell ; it is equal to the magnetic quantity per unit of area x the thickness of the shell. AB as the central point C is; i.e., at distance d from the midpoint AB, at distance VcF 2 -\- ?/ 2 from either A or B. The pole D is attracted by the one end of AB and repelled by the other; in each case with a force m-ffi/l- AU 2 , or mffi/1- (d 2 + ^) 2 - The resultant force on D is parallel to AB, and is therefore equal to the whole force acting X i '/ \A? 2 + i/ 2 ; i.e., it is equal to mfH / (eZ 2 +i/ 2 )f. The resultant on E is equal and opposed in direction. The couple on DE is therefore mfH/ (d 2 + i/ 2 )i X DE X cos 6 = ffl, fflL-cosfl./Ccff + i/ 2 )^. This is equal to the terrestrial couple t-Jft-sintf; whence tan xvi.] MAGNETIC SHELL. 683 The quantity of magnetism per unit of area is the Mag- netic Superficial Density, 9, =

?'//*] If the shell be fixed, a positive pole would tend to move away to regions of less potential, and thus to travel round to the negative side of the shell. If the unit pole be fixed, a shell would tend to move in such a way as to diminish its apparent area, and even to present what is equivalent to a negative area, namely, its negative side, to the positive pole ; it would therefore tend to rotate. In the immediate neighbourhood of a magnetic shell the angle subtended by it is 2?r; the potential near the positive surface is therefore 27rcp, where > is the strength of the shell ; near the negative surface it is 2Tr = the Magnetic Moment of the equivalent shell ; but if the circuit be coiled, so as not to be a single loop, but to surround its own axis n times, the magnetic moment of the circuit = Am; and if the medium constituting the field be any other than air, the magnetic moment Ay of the equivalent shell = /x An - i, or for a single loop,

]. Magnetic Rotatory Polarisation of Light. If a plane- polarised beam of light or of radiant heat be sent through a magnetic field occupied by a transparent medium, its plane will, by the retardation or acceleration in phase of one of its circular components, be rotated. The sense of this rotation depends upon the direction of the lines of force and upon the nature and chemical constitution of the medium ; its amount upon the thickness and the nature and physical state of the medium and upon the intensity of the magnetic field, resolved in the direction of the ray ; but it does not occur in free Ether. This has, in the hands of Becquerel and Lord Rayleigh, been made the basis of a method of measurement of the intensity of a current ; the current, when passed through a solenoid, produces within this an electromagnetic field, the intensity of which is proportional to the strength of the current ; a plane-polarised beam sent through glass along the axis of the solenoid is rotated to an extent proportionate to the current-strength. The direction of rotation of the plane of polarisation is in most cases, including flint glass and thin films of iron, cobalt, Fig.244. and nickel, positive, being that shown by Fig. 244, in which AB represents a line of magnetic force, and the arrowed circle represents the direction of rotation of the plane. Whether the ray travel in the direction AB or in the direction BA, the absolute rotation imparted to it on its transmission through the magnetic field remains the same ; whence, if it be re- A fleeted from a mirror and sent back through the field, the rotation of its plane will be doubled. In uniaxial crystals, the rotation is most marked along the axis. In a concentrated solution of perchloride of iron it is negatiye. Hall's Experiment. A film of metal, in the form of a cross, laid upon glass. A current from a battery passes through two opposite arms of the cross, and does not affect a galvanometer connected with the other two arms. When the cross is made to face the lines of force of a strong mag- netic field, a small constant current is indicated by the galvanometer (Fig. XVI.] MAGNETIC ROTATORY POLARISATION. 695 Fig. 245. 245). The strength of this current depends upon the intensity of the mag- netic field and the strength of the primary current ; and also upon the kind of metal of which the film consists. Its direction depends upon the direc- tion of the field and that of the primary current, and on the metal of the film. Kerr's Experiment. Polar- ised light reflected from the polished face of a magnet undergoes rotation of the plane of its polarisation; when reflected from the north-seeking pole, the rotation is negative. When reflected from the sides of a magnet, it also undergoes rotation, the sense of which varies with the plane of polarisation and the angle of incidence. Kundt has also found a variety of phenomena of rotation of the plane of polarisation of light transmitted through thin magnetised metallic films. Thus far we have dealt with Steady Currents and Steady Fields, electrostatic, magnetic, or electromagnetic. We have now to consider the properties of Varying Currents and Fields. Fig. 246. THE VARIABLE PERIOD. When an open circuit is abruptly closed for an instant, and an instantaneous current is produced in the wire, this current is not felt simultaneously over the whole circuit. In the case (a) of Fig. 246 the galvanometers B and B f twitch first when the interrupted circuit is momentarily completed ; in case (5) of that figure, under similar circum- stances, the galvanometers A and A' twitch first. The dis- tance between A and B must be great in order to show this effect. By the time a state has been arrived at, in which a steady current passes, a certain amount of energy has been elastically accumulated in the dielectric ; but between the instant at which the current begins to flow and that at which it has assumed its steady state, there is a period of adjustment, the variable period. During this period the field has Kinetic Energy, and there^is a Displace- ment- or Polarisation-Current. 696 ELECTKICITY AND MAGNETISM. [CHAP. When, as in Fig. 247, a battery of which one pole is con- nected to earth has its other pole suddenly brought into commu- Fig.247. nication with a long wire whose other extremity is con- nected with the earth, the time which elapses be- fore the current through the wire becomes steady is found to vary as the square of the length of that wire. In long lines, the time spent in acquiring at any point of the wire a certain definite intensity of current is approximately proportional to RC/ 2 /E, where E is the D.P. employed, R the resistance, and C the electrostatic capacity of the wire per cm., and I its length. The time which elapses before a certain proportion of the ultimate intensity is attained varies approximately as RC/ 2 ; it is also, in practice, not independent of the effec- tive D.P. set up by the galvanic cell employed. The product RC/ 2 , which measures the Electrostatic Retardation, is thus of great importance in long lines ; but in short lines, the effects of electrostatic retardation are masked by those of self-induction and the induction of other circuits. A lightning discharge through a lightning conductor is so brief that the laws of steady flow do not hold good : it is of advantage, in order to diminish the risk of lateral divergence, to render the current more uniform, or, in other words, to retard it ; for this purpose the capacity of the conductor should be increased, and therefore its surface; and lightning conductors should be broad flat plates of metal rather than compact rods. The effect of self-induction of the current also aids in bringing about this result ; cur- rents running parallel and in the same direction retard one another. The circumstance that the flow is too brief to affect the interior of the wire to any considerable extent also aids in making it more important to increase the relative surface of the conductor than to increase its cross-section ; for the phenomena of so abrupt a discharge are practically restricted to the field of force, the dielectric, surrounding the wire. In a uniform wire, OL, between whose extremities a dif- ference of potential is maintained equal to OP (Fig. 248), the ultimate Line of Potentials is PL ; and when such a distribu- tion of potentials has once been produced along the conductor, Ohm's law is obeyed; L but at various instants during the preliminary variable period, the distribution of poten- Fig.248. XVI.] THE VARIABLE PERIOD. 697 Fig.249. tials along the wire is such as is indicated by the curved lines, 1, 2, 3, etc., sketched in Fig. 248. The momentary and local intensity is always the momentary and local E/R(= Potential-Slope -4- Resistance per linear cm.), but during the vari- able period it varies from point to point and from instant to instant. When the extremity of a long wire is momentarily charged by contact with a charged conductor or with one pole of a bat- tery, its home end suddenly acquires a high potential, which is immediately thereupon reduced by communication of the charge acquired by the ex- tremity of the wire to the rest of the wire. In Fig. 249 the end O of the conductor OL is suddenly raised to the potential OP. A point such as A is found, as it were, to leap up to a high potential and then to descend. A wave of sudden increase of potential thus travels along the conductor, but falls off pro- gressively, both in abruptness and in height, the farther it travels. At the distant end, for a short interval after the circuit has been actually completed, no effect is perceived; the current then begins to become sensible : and it then, if the contact be kept up at the home end, appears to increase in intensity after the man- ner indicated by the Arrival-Curve represented in Fig. 250. A current, even though Fig. 250. it be constantly main- tained at the home end, would take an infinite time to acquire its maxi- mum value at the distant 5 /" end of such a conductor - / as an Atlantic cable, if / that conductor had, when the current commenced Time to traverse it, been uncharged ; it would, however, require only about 108 seconds to attain -f-$ of its maximum value, and about the fifth part of a second to attain T ^ of its maximum value. The apparent velocity of transmission of signals in a given con- ductor is thus seen to be mainly an affair of the dejicacy of the instruments which detect the current on its arrival at the distant 698 ELECTRICITY AND MAGNETISM. [CHAP. end, and is perfectly distinct from the velocity of propagation of an electromagnetic disturbance ; and it depends on the capacity of the conductor, for the transmission is greatly delayed in con- ductors whose capacity is great, such as submarine cables, appre- ciably so in long air-lines, inappreciably so in short air-lines. The attainment of the steady state is greatly facilitated, though the currents produced are weakened, by leakage. The signals produced are thus rendered clear (Oliver Heaviside). When the current suddenly stops after having acquired a steady flow, its cessation at the distant end presents a similar deliberation. When a wire is momentarily connected with a charged body and then connected with the earth, or " put to earth," the arrival-curve at its distant end is a curve due to the superposition of two arrival-curves ; the first of these is the arrival-curve, resembling that of Fig. 250, due to the contact with the charged body ; the second is curved in the opposite sense, and is due to the sudden discharge of the conductor. The dotted curve of Fig. 250 is the result of the superposition of two such opposed arrival-curves. This curve indicates that there is an abrupt and brief variation of potential at the end of the wire distant from the galvanic cell. More effective still than this in producing an abrupt and brief current is the process of following up each positive charge, immediately after putting to earth, with a negative one, after which the wire is again put to earth. The disadvantage of this is that the potential, while it abruptly ceases to be positive, sinks at once to a negative condition, as in Fig. Fig.25l. 251; for which reason it is X.. customary so to arrange the mechanism at the signalling station that each apparently simple making of contact is in reality a complex operation, in which an odd number of cur- rents of opposite kinds are sent in rapid succession into the wire, the wire being, after each, put to earth ; each of these currents being briefer than its predecessor, and correcting it. The arrival-curve for such a combination indicates an abrupt rise of potential, an abrupt fall, and then a slightly-wavy line, which at no point diverges to any material extent from the base-line. Even these methods are increased in effect, the arrival-curve being ren- dered still more abrupt, by the use of Condensers. Each condenser is com- posed of a large number of plates of tinfoil separated by waxed paper and paraffin : the alternate plates are in metallic communication with one another. One series of alternate plates in each condenser is in communica- tion with the cable ; the other set is in communication with the galvanic battery or with the galvanometer G (Fig. 252). Any sudden variation in the potential of the landward plates of the home condenser is immediately followed by an equally-sudden flow of electricity XVI.] THE VARIABLE PERIOD. 699 into or from the cableward plates of that condenser : this flow takes place either from or into the cable itself ; this disturbance is propagated along the cable ; the potential of the cableward plates of the condenser at the receiving station is affected ; by induction the distribution of electricity in the landward plates of that condenser is affected, and a current passes through the galvanometer, either from the condenser to the earth or in the Fig. 252. reverse direction. On connection of the home condenser with the positive pole of the battery employed, a positive current runs through the distant galvanometer Gr to the earth ; and on putting the home condenser to earth a reverse current passes through G, which may be corrected as before. Even if the key be kept permanently pressed down at the transmitting station the current passing through G is but momentary, for both condensers quickly assume a condition of electrostatic equilibrium. During the Variable Period, the Lines of Force are slipping along the wires with the velocity of Light. They travel with their ends on the wires, and approximately lie at right angles to these, until the condition approaches that of the steady state. During the variable state, they accumulate (or thin away) in the field ; when the steady state has been attained, there is no accumulation of them in the field, but only transit, while their direction becomes approximately parallel to that of the wires. At the same time, during the variable period, the Lines of Transmission of Energy through the field are themselves in motion ; and the axis of the wire is the last thing to be affected. During this period there may be production of induced currents in neighbouring conductors. ELECTROMAGNETIC CURRENT-INDUCTION. If there be a closed current-bearing circuit, with its positive face facing a positive magnetic pole, there will be mutual repul- sion between that circuit and pole ; and if the current and the magnetic pole be brought nearer one another, then, since work must be done in order to bring about this approach in the face of mutual repulsion, the potential energy of the system is increased by a fixed amount: a portion of this energy takes the form of a temporary increase of the current in the closed circuit; while the remainder may, by induction, produce an increased magnetic condition in the magnet. Now replace the magnet by an equivalent closed circuit 700 ELECTRICITY AND MAGNETISM. [CHAP. (" circuit B "), the positive aspect of which faces the positive aspect of the original closed circuit ("circuit A"). These two circuits, again, repel one another : and if work be done in forc- ing them together, the energy appears in the form of a tempo- rary increase in the intensities of both currents, and is presently converted into heat in the circuits. Conversely, when the magnet or the equivalent circuit is withdrawn, there is a corresponding temporary diminution in the corresponding current-intensities, and possibly in the corre- sponding magnet-strength. These increases and diminutions in current-intensities are equivalent to the Induction of New Currents. The duration of these new induced currents is limited to the Variable Period, the time spent in changing the relative positions of the mutually- inducing magnets or currents. The lines of electric force accumulate, or else fall off, upon and parallel to the circuit-wire. The steady state is thus interfered with, and an electrical effect is produced, analogous to mechanical acceleration. If a circuit bearing a current be brought towards a circuit capable of bearing a current, and if the former, the inducing cur- rent, have its positive face turned towards the circuit approached, there will be two effects produced: (1) an increase in the inten- sity of the inducing current, and (2) a new current developed by induction in the circuit approached, which had previously appeared to bear no current. This current has its positive face turned towards the approaching positive face of the inducing current, and is therefore opposed to it in its direction. If two wires be laid alongside one another (Fig. 253), and if one of these wires be connected with the two poles of a battery, and thus form part of a Primary or Battery Circuit; while the other wire is merely a part of a complete metallic circuit, a so-called Secon- dary Circuit ; then, when contact is suddenly made in the primary cir- cuit, a current of brief duration a duration not exceeding in time the variable state -of the primary current is produced in the secon- dary circuit and is known as the Secondary Current. The primary current and the secondary current are, in the wires laid alongside one another, opposed in their direction. Fig. 253. xvi.] ELECTROMAGNETIC CURRENT-INDUCTION. 7Q1 So long as the intensity of the primary current remains constant, the secondary circuit has in it no current ; but any increase is accompanied by a brief opposed secondary current. When the primary current is diminished, the primary cir- cuit again presents a variable state ; and so long as that variable state lasts, there is again a current in the secondary circuit, which is on this occasion in the same direction as the waning primary current. When the primary current stops abruptly, there is a very abrupt secondary current, parallel to the ceasing current. These secondary currents represent a definite amount of energy subtracted from the energy of the primary current, an amount which depends only on the initial and final states or intensities of that current. Being of extremely short duration, they are of correspondingly great intensity and high potential. The secondary current produced on breaking the primary cur- rent is briefer, and therefore more intense than that produced on making it. When a magnet is thrust into the axis of a bobbin which forms part of a closed circuit, there is a current produced in that circuit. The current is opposed in direction to the mag- netic molecular-currents, the Ampere-currents, of the pole which is introduced first. If a long magnet be drawn wholly through such a coil, there is at first a current in one direction as the one pole approaches ; then, as its midpoint passes the midpoint of the coil, the current is ra7, but is reversed as the opposite pole passes out. The current is at first opposed to the Ampere- currents of the approaching pole; and as all parts of a bar- magnet, looked at end-on, have their currents in the same direction in space, the induced current changes in its direction as the magnet passes through. These statements may be generalised by saying that wher- ever a closed circuit, capable of bearing an electric current, lies wholly or in part in a Magnetic or Electromagnetic Field of Force, any disturbance in the Intensity of the Field of Force will induce a Current in the circuit ; and the direction of the induced current is determined by the rule (Lenz's Law) that the new current will increase the already-existing resistances, or develope new resistance to that disturbance of the field which is the cause of induction. A telephone circuit passing through a disturbed field of force will pick up signals : for example, at every lightning flash the instrument is heard to roar; and in order to prevent such effects of induction, no part of the current 702 ELECTRICITY AND MAGNETISM. [CHAP. is entrusted to earth, but the double wire necessary is coiled round itself so as to form a strand composed of two insulated wires. The effect of induc- tion on one wire is then equal to the opposite effect of induction on the other wire. During thunderstorms military mine-fuses have been known to explode through induction in the wires controlling them. We have seen that a closed current, A, whose positive aspect faces a positive magnetic pole or face of a magnetic shell or equivalent electric cur- rent, B, sends towards the latter, from within its own contour, a number of positive lines of force or of induction, which radiate from its positive face. If we change our standpoint, and regard .the current first mentioned a current borne by circuit A as placed within the magnetic field of the shell B or the electromagnetic field of the equivalent circuit, then the positive lines proceeding from the latter are, as regards the circuit A, negative, for they trend not from but towards its positive aspect. Circuit A, as we have seen, tends to move by translation to a greater dis- tance from circuit B. It will also tend to rotate until its negative aspect faces the positive side of B ; it is then attracted towards B. In the former case, as A moves away, the number of negative lines which pass towards its area diminishes. In the latter case that of rota- tion A tends, as it turns, first to set itself edge-on to B's lines of induction, and then so to place itself (its negative face opposite to B's positive face) that the lines of induction which emerge from B positively also emerge from A's positive face positively, and are positive with respect to A. In either of these cases, translation or rotation, the number of negative lines met by the area of A is diminished as far as possible, or the number of positive lines embraced by its contour attains a maximum. A little movable circuit may be made De la Rive's floating battery by thrusting a strip of copper and a strip of zinc through a cork, and con- necting them by an arch of copper wire : when the whole is floated in water, the arch tends to lay itself at right angles to the magnetic meridian, copper to the west, zinc to the east ; in this position the positive face of the arch is to the north, and the magnetic lines of force or induction which trend towards the north are embraced by the arch in the greatest possible number. The general statement of the phenomenon is : A movable circuit tends so to place itself as to meet as few negative or to have as many positive lines of induction passing through it as possible ; a line passing through a circuit being held positive when, after passing through, it emerges from the positive face in a positive direction ; a line being held to be negative when its direction is towards the positive face. The position thus assumed is the position of least potential energy, that into which the whole system tends, as it were, to sink. A circuit in this position of least potential energy embraces as great a number as possible of positive lines of induction. Mutual Attraction and Repulsion of Currents. Suppose two cur- rents in the plane of the paper, similar in their directions and having in con- sequence their nearer portions opposed in direction, as in Fig. 243. Let their directions be the same as those of the two currents in that figure. The left- hand current in that figure has lines of induction which ascend from the plane of the paper and tend to descend through the contour of the right-hand circuit, meeting its ascending lines. These descending lines are therefore negative to the right-hand circuit, and that circuit tends to move away so as to meet as few of them as possible. The portions of the currents which are nearest one another, running in opposite directions, thus seem to repel one xvi.] MUTUAL ACTION OF CURRENTS. 703 another. The area in which downward lines meet upward lines is thus diminished as far as possible, and this enables us now to understand the propositions illustrated by Figs. 229, 230. If a circuit embracing the greatest possible number of positive lines of induction, and therefore occupying the position of least potential energy, be pulled or turned into any other position, work must be done upon it ; and this work is done against mutual attractions. This doing of work is associ- ated with diminution of the number of positive lines of induction embraced by the movable circuit. As the circuit moves in the field, lines of induction must be cut through by it. AU cutting through lines of induction, when the number of lines enclosed by the circuit is diminished by the opera- tion, is effected by the expenditure of work. The process attains its maximum when the movable circuit has been swung round through 180. If it be still farther rotated, it comes to meet fewer negative lines, then to enclose an increasing number of positive lines, until it regains its original position . The work done takes the form of the energy of induced currents, which always increase the resistance to the actual movement ; if A and B repel one another, their intensities are increased when they are urged together, dimin- ished when they are drawn asunder ; if they attract one another these actions are reversed. This may be otherwise expressed by saying that when a circuit is made to meet a greater number of negative lines of induction, or to enclose a smaller number of positive lines, its current is increased in intensity, or a new induced current set up in it ; while if it be made to meet a smaller number of negative or to enclose a greater number of positive lines, the inten- sity of its current is diminished, or a new reverse current is set up in it. The result is the same whether it move so as to enclose more or fewer lines in an existing magnetic field, or whether the magnetic field itself vary so that its lines either open out and become fewer, or become more numerous and approach one another a smaller or a greater number of them conse- quently passing through the given circuit. If a part of a circuit of total resistance r be movable in a magnetic field which presents b lines of magnetic induction per sq. cm., it will cut through all the magnetic lines in a certain area A in the course of time t ; it will therefore cut through Ab/t B/f lines of induction per second. The cur- rent set up is, firstly, such that if the movable part of the circuit uniformly diminish the area of the circuit as it moves in the terrestrial magnetic field, the current will run in the circuit (which is supposed to be set in a plane at right angles to the magnetic meridian) in a direction which seems from the standpoint of an observer stationed to the south to be the same as that of the hands of a watch : and, secondly, its intensity is proportional to B / rt. In Electromagnetic measure, the units are so adjusted that the intensity of the induced current is equal to B / rt = Ab / rt. If B be the additional number of lines which the circuit comes to enclose, the intensity of the induced current is i = B/rt = Ab/rt. This relation is the same, whatever be the permeability //,. When a block of copper is whirled within a magnetic field, currents are set up in it, which produce resistance to the motion; the motion of the block very rapidly ceases, as if the magnetic field were highly viscous, and the block becomes hot. When a magnet-needle is suspended immediately above a copper plate, any oscillation in the magnet developes^retarding cur- rents in the copper, and the magnet almost immediately comes to rest. 704 ELECTRICITY AND MAGNETISM. [CHAP. Self-induction. A current suddenly formed in a spiral wire is retarded by the mutual action of the different turns ; it does not flow on, and its intensity is, at first, less than it would have been in a straight wire : when suddenly broken it is pro- longed and is as it were piled up, so that the so-called Extra- Current can force its way through greater resistance than the steady current can. In fact, a single Daniell cell can be made to electrolyse water by delivering a part of the energy of its cur- rent, at high potential, in the form of the so-called Extra-current. These phenomena closely remind us of the phenomena of momentum in a water-pipe, already discussed under the Hydrau- lic Ram ; and they can be explained as phenomena of momen- tum of the Ether in the electromagnetic field. Two wires bearing currents in opposite directions, and twisted round one another, present no phenomena of self-induc- tion ; for which reason the wires leading to and from a galva- nometer should be twisted together for some distance from the needle. Coefficient of Mutual Induction of two Currents. The Ether sur- rounding a pair of current-loops of intensities i y and i n must possess Energy, which Clerk Maxwell showed to be proportional to squares and products of the intensities, and which may be written thus : { 1 L^ / 2 -f Mz/'^ + ^L'i,/ 2 }. The second term vanishes when the currents are at an infinite distance from one another; it is at its greatest practical when the two circuits almost touch one another, its greatest theoretical when they absolutely coincide : at intermediate points it has intermediate values. It can be shown that Miyi^ is, in any given position of A and B, numerically equal (1) to the Mutual Potential Energy of the two circuits and (2) to the Number of Lines of Induction which, being due to A, pass from A through B or, equally, being due to B, pass from B through A; and M is styled the Mutual Inductance or the Coefficient of Mutual Induction. M varies with the relative position of the two circuits. The maximum value of M is its value when the two currents are made to run in the same circuit ; let this be called M . The number of lines ( = MO^,) due to i, and the number due to i n are to be added together for the conjoined current (i t + i /y ) ; for they all pass through the same circuit ; hence the number actually threading the circuit will be 2M i / i // . In this case ^Ji i l i ll is equal to the part of the energy of the field which is due to the approximation of the currents i, and i n from an infinite distance and their coincidence ; and M is equal, numerically, to half the number of lines of induction which pass through the circuit itself when i t and i tl are both unity, that is, when the conjoined current has an intensity = 2. It is there- fore equal to the number of lines of induction passing through when i = 1. Coefficient of Self-induction, or Inductance. We next see that M = L. If the intensity of the second current i n be 0, the energy of the field is ^Li, 2 only. A second current of the same intensity in a circuit of the same size, etc., at an infinite distance will have energy also equal to xvi.] SELF-INDUCTION. 705 iL*, 2 . Together the energy will be Li, 2 . Now bring the two currents together and blend them ; the energy is -^-L(2i / ) 2 = 2Li y 2 . The system pos- sesses energy equal to Li, 2 , due to the approximation ; but this is also M^ 3 if both currents be equal to i t ; whence L = M . L is the Coefficient of Self-induction; and the coefficient of self-induction of a circuit is equal, numerically, to the number of Lines of Magnetic Induction which thread that circuit when it bears a current whose intensity is unity in electromag- netic measure. Within the contour of a circuit, B = Li. In a solenoid of n turns, and length / cm., the number of lines of induc- tion for unit current is kirn- p,/ 1 per sq. cm., or 4irn- A- p./ 1 across- area A. If a second solenoid, of n' turns, surround the first, each turn of it embraces 4r7rn- A n/l lines once ; and its n' turns embrace 4?rn n'- Ap/l lines. Hence for two such solenoids, M = 4-Tr nn'- Ap/l ; and for a single solenoid, L = 47rn 2 A^/Z. Hence the self-induction of a coil of many turns is very great, Extra-Currents. If the energy of a current traversing a single circuit be derived from any external source, such as a battery, which is independent of induction, the energy supplied from that source during a very short time 8t will be equal to ei - &, where e is the E.M.D.P. and f the current-intensity. (All our measurements in these paragraphs are supposed to be made in electromagnetic measure.) This energy is divided into three parts. (1.) Heat in the circuit. This is equal to ri 2 &, where r is the resistance. (2.) External work, mechanical, chemical or other. This we shall sup- pose = 0. (3.) Work spent in imparting energy to the electromagnetic field. This is equal to |L{(i + Si) 2 i 2 }, where Si is the small change in the intensity produced during the time 8t. We thus have the equation ei - & = ri* & + 4L {(t + Si) 2 - *} ; (i.) an equation which can be dealt with by integration, the effect being that we find the intensity, at any time t after the introduction of a new E.M.D.P. = e into the circuit, to be i t = e /r - e/r (2-718281 -*/!). (ii.) The intensity never comes fully up to the value e/r; but it approaches it indefinitely nearly as the time t increases. If, however, the coefficient L be large, as it is in a coil of wire, the second term on the right-hand side is not immeasurably small, and it represents what is equivalent to a reverse cur- rent lasting for an appreciable time, and delaying the development of a current of full intensity e/r. This reverse current is called the Reverse Extra-Current or the Extra-Current of Closure or of Making. When a circuit is suddenly broken, the intensity at a time t after the current has been stopped is + (e/r} (2-718281-*/ L ). This indicates that there is still an onflow, a" Direct Extra-Current or Extra-Current of Opening or Breaking; an onflow which results in a high potential at the broken extremities of the wire, and, since the capacities of these extremities are small, in a high value of a at these extremities. These extra-currents are thus associated with absorption of energy by the electromagnetic field while currents, the energy of which is derived from extraneous sources, are being produced or increased, and with libera- tion of energy by that field when such currents are brokers or while they are being diminished. 2z 706 ELECTRICITY AND MAGNETISM. [CHAP. They may also be looked at as phenomena of Induction. When the intensity of a current is increased, the circuit is made to embrace more lines of induction: if it embrace B more lines of force in time &, an E.M.D.P. is set up, equal in electromagnetic measure, to B/&. Where L is the self-induction of a circuit, the establishment of current i in it causes the development of Li lines of induction embraced by it ; and this causes an E.M.D.P. e, =. Li/Bt; whence ',, the mean intensity, = Li/r- 8t ; this being the mean intensity of an induced current opposed in its direction to the originating current. When the main-current ceases, the induced cur- rent now produced on the disappearance of lines of induction is direct, the Direct Extra-Current. The steady intensity i = e/r ; whence the new " electromotive force " e / = Le/r St. Since 8t is very small this may greatly exceed e ; and, for a given length of wire, it is greatest when the current passes through con- ductors of a spiral form, in wliich the value of L is great. The extra-cur- rent may thus be able to spark across striking distances beyond the power of the main-current. The quantity of electricity in the extra-current is in each case q = i, $t = eL/r 2 . These relations are the same, whatever be the permeability ju,. Measurement of Inductance L. Take a Wheatstone Bridge (Fig. 222), with arms AB, BC, AD, DC, with a battery as in the figure, and with a galvanometer between B and D; and then lengthen the wire of the bat- tery circuit AC, and of the galvanometer branch BD, so as to bring them both up to a rotating commutator or contrivance which shall, in repeated succession, perform the following cycle of operations, viz. : (1) disconnect the battery, (2) disconnect the galvanometer, (3) connect up the battery, and (4) bring in the galvanometer. Then, if the resistances in the arms AB, BC, CD, and DA be respectively r', r", r'", and a resistance r"" which comprises that of the loop or coil which is to be tested ; and if these be so adjusted that there is no current in the galvanometer; then, on setting the commutator in action, the balance of resistance will appear to be disturbed, for the " impedance " of the coil (p. 722) is not the same as its Resistance to steady currents ; and the galvanometer-needle will be deflected. The re- sistance r"" will have to be reduced by a certain amount Br to restore the balance ; and Ayrton and Perry have shown that if n be the number of complete cycles per second, the inductance L is equal to 8r/n. (Ayrton and Perry's Secohm-meter.) Induction Coils. The effect of induction is multiplied when the two wires, that of the primary and that of the secon- dary circuit, though kept insulated from one another, are wound together round the same axis. The secondary current is then proportional in its intensity to the product of the number of turns in the two wires, provided that the resistances introduced by multiplying the coils, or the differences between the mutual distances of the different turns, be not too considerable. In Induction Coils the wires of the two circuits are wound round separate bobbins, which are then slipped the one over the other to a greater or less extent. On this extent depends the inten- xvi.] INDUCTION-COILS. 707 sity of the secondary current. The primary current is made and broken with great frequency by means of a Contact- breaker. This may be a mere mechanical contrivance, or it may be automatic. In the latter case there lies a bar of soft iron in the axis of the inner, the primary, bobbin. When the primary current passes, this bar or core becomes an electromagnet. This electromagnet pulls towards itself an armature, a mass of soft iron, which is arranged near one of its extremi- ties ; this mass of soft iron is an integral part of the circuit of the primary current, and by its movement the primary current is suddenly broken. The electromagnet now loses its magnetic condition; it ceases to attract the armature ; the latter, under the pressure of a spring, returns to its former position, and again completes the primary circuit ; the electromagnet is again made, and the armature again displaced. The soft iron armature is thus caused to oscillate and to impart to the primary current an iiitermit- tence, whose frequency depends upon the intensity of the current and upon the pressure of the spring. MAGNETIC OR ELECTROMAGNETIC MEASURE. A current of given intensity, I in electrostatic units, must be represented by smaller numbers when magnetic units are used ; a current of I = 60000,000000 is a current of i = 2 ; for the C.G.S. magnetic unit of current-intensity or -strength is 30000,000000 times as great as the C.G.S. electrostatic unit. The basis of the Magnetic or Electromagnetic system of measurement is the identity of effect, in air, between a mag- netic shell of strength qp and a closed current of the same con- tour and of a particular intensity i ; the units of current-intensity are so adjusted that i becomes, in air, numerically equal to y. If i = l, the current is equivalent in magnetic effect to a magnetic shell whose strength q> is unity and whose area and outline are the .same as that of the circuit; and this is the Magnetic C.G.S. Unit of Current-Intensity (Definition i.). [qp = fjd.~\ If we suppose a wire bearing a current, and one cm. in length, to be bent into a circular arc whose radius is one cm., and if we suppose a unit mag- netic pole to be placed at the centre of the circle of which the circular arc forms a part ; and if we further suppose that the mechanical force exerted by the current upon the unit magnet-pole is equal to one dyne ; then the current is one whose intensity is, in magnetic measure, equal to unity. In such a case i = 1 ; and this may be taken as Definition ii. of the Magnetic C.G.S. Unit of Current-Intensity. The general formula is F = tn t//r 2 , where F is the force exerted upon a magnet-pole placed at the centre of such an arc, m the strength of that pole, t the intensity of the current, I the length and r the radius of the circular arc into which the wire is bent. [F does not depend upon /x.] If the wire be bent into a complete single circular loop pf radius r, I 2irr, and F = tn i 2ir/r. 708 ELECTRICITY AND MAGNETISM. [CHAP. If the pole ttt be in the axis of the single loop, but not necessarily in its plane, F = /-i-27r-r 2 /w 3 , where u is the distance, more or less oblique to the axis, between the pole and the wire. When u is reduced to u = r ; that is, when the pole comes into the plane of the loop, F = m i - 2ir - r 2 /r 3 = m i 2-jr / r ; which agrees with the expression above. If a current, = i in magnetic units, pass along a wire across a uniform field of magnetic force or strength h, in air, the wire is acted upon, trans- versely, by a force F = hil dynes, if the length of the wire be I cm. ; and this gives us another definition (Definition iii.) of the Magnetic C.G.S. Unit of Current-Intensity. This also enables us to measure the intensity h of an intense magnetic or electromagnetic field ; a current is led through it ; the wire is forced in one or other direction ; this force can be balanced by a known weight. If the current be sent through a column of mercury in a known magnetic field, there is a difference between the manometric pressures at the two sides of the column (Lippmann). [F = /x,-hi-/.] (Definition iv.) : a current of intensity t, traversing a straight wire of indefinite length, acts upon a magnetic pole m, at a distance d from the wire, with a force F = nth = m 2i/d. (Compare p. 189, prop. 7.) A unit current would therefore act upon a pole m with a force F = 2m/ d. [Does not depend upon /x,.] When a current i passes into a long solenoid, of n turns and length I cm., a magnetic pole m anywhere within the solenoid is acted upon with a force F = h-m = ^ir-m-i-n/l. Whence a unit current will act with a force F = m 4-rr n/l, or (4?r m) X the number of turns in the solenoid per unit of length. (Definition v.) [Does not depend upon /A.] The Intensity of a Current, magnetically measured, and the Magnetic Strength of a Shell must accordingly have the same dimensions. The Magnetic Strength of a magnetic shell is (p. 682) numerically equal to the product of the Magnetic Quantity per unit of area into the Thickness of the shell ; its dimensions must hence be those of Magnetic Quantity, divided by an Area, and multiplied by a linear Thickness ; but the dimensions of Magnetic Quantity must (since the imaginary magnetic matter obeys laws resembling those of repelling or attracting electric matter, similarly imaginary) be like those of electric quantity, [M*L*/T] ; the dimensions of Magnet-Strength are accordingly [M 4 L*/T] -*- [L 2 ] x [L] = [M*L*/T] ; the Mag- netic Measure of Current-Intensity has the same dimensions, and therefore differs from the electrostatic measure, whose dimensions are [M*L*/T 2 ], by the term [L/T], which represents a Velocity : the numerical value of this ratio must be found by experiment, which shows it to be 30,000,000000 nearly, = V. Measurement of V. The ratio between the magnetic and electro- static measures may be determined by several methods, of which two may be taken as examples. Weber's method. Charge a Ley den jar with a known quantity of electricity, Q ; discharge the jar through a wire, which passes round a gal- xvi.] MEASUREMENT OF V. 709 vanometer-needle. The quantity of Electricity passing through the galva- nometer may be measured in terms of the deviation undergone by the needle in consequence of being thrown by the instantaneous current. This gives the quantity, q, in magnetic measure. These separate measurements of the numerical values, Q and q, of the one quantity of electricity give the ratio between the electrostatic and the magnetic unit. Lord Kelvin's method. The two ends of a wire of great resist- ance, R, are kept at a constant potential-difference, E ; a constant current runs through the wire; this current is found to have an intensity i C.G.S. magnetic units ; the difference of potential is (by Ohm's law) E = IR or e = ir. E and e, or i and I, bear to one another the relation of 1 : V ; whence V may be found numerically. The meaning of this ratio between the Electrostatic and the Magnetic or electromagnetic units is frequently found to be puzzling. Its real basis is the following. Both these systems of units are based on independent arbitrary conventions ; and neither of them can absolutely represent phys- ical truth, though all calculations will work out accurately if we adhere to either system. In the Electrostatic system the units are so adjusted that equal charges at unit distance apart, repelling or attracting one another through air with unit force, are called " unit charges : " but more generally, the mutual action of two charges depends upon K, the sp. ind. cap. of the medium between them; F = QQ'/IW 2 ; whence [Q] = [Ktal*I*t/T] ; and it is only by assuming the sp. ind. cap. of air to be unity, and the sp. ind. caps, of other media to be Numbers merely, that we arrive at the air-equa- tion F = QQ'/^ 2 ' an d the corresponding Equation of Dimensions [Q] = [M^U/T]. Similarly, the Force F between two currents i and i' varies as //, it', where p is the magnetic permeability of the medium : that is, [Force] = [>-Q 2 /T 2 ] and [Q] = [MiLV/n*] ; but by similarly assuming the mag- netic permeability of air to be unity, and the magn. perms, of other media to be numbers merely, we arrive at that air-equation of Dimensions, [Q] = [MW], which lies at the basis of the Magnetic system of measure- ment. The conventional units of electric Quantity thus bear to one another the ratio of [M4U/T] to [M*L*], or [L/T] to 1; but this apparent want of equality arises from these conventions themselves. On a natural system, the same quantity ought to have the same Dimensions, whether looked at from an electrostatic or a magnetic point of view ; and therefore, throw- ing aside these conventions, we must have [MWKs/T] = [M^L*///,*], or [L/T- VK/x] = a Number merely, and that number = 1. When we assume, in accordance with our conventions, that K and /u, are both numerically = 1, assumptions which cannot possibly both be true, we find that L/T, the term of difference between the conventional e.-s. and in. units of electric Quantity, is experimentally determinable as numerically equal to 3 x 10 10 ; and as from its Dimensions it is a Velocity, it is said to be a Velocity of 3 x 10 10 cm. per second. What the last equation of dimensions more truly shows, however, is that [K/x] = [T 2 /L 2 ], and that K/x = 9 x 10*>. We do not know the absolute numerical values of either K or /w, separately ; neither do we know their Dimensions separately. See, however, p. 746. Dimensions in the Conventional Magnetic Air-system. Current Intensity or Strength, f. Page 670, no. 1; attraction or repulsion (= mechanical Force) a ii' x ll'/d 2 ; i.e., [ML/T 2 ] = [intensity 2 ]; .-. [i] = [MlLi/T], Quantity, q, = Intensity x Time; [?] = {[MW/T] x fT]} = [MIL*]. 710 ELECTRICITY AND MAGNETISM. [CHAP. Potential, or Difference of Potential, e, = work done -f- quantity of electricity upon which work is done ; \_e] = {[ML 2 /T 2 ] -* [M*Li]} = Electrical Force, the mechanical force acting on magnetic unit of electric quantity. Its dimensions are those of mechanical force -H- electric quantity ; [f] = [F/j = {[ML/T 2 ] - [M*Li]} = [MilJ/T 2 ]. Otherwise, this is Potential-Slope, [WLt/T*] * [L] = [MlLl/T^. Resistance = difference of potential -f- current-intensity ; {[M^Li/T 2 ] -i- [MAL5/T]} = [L/T], a Velocity ; * whence the resistance of an Ohm wire (10 9 magnetic C.G.S. units of resistance) is said to be 1C 9 cm. per sec. ; and so on in proportion. Capacity is quantity of electricity stored up per unit potential-differ- ence produced by it ; its dimensions are {[M*L4] -5- [M^Li/T 2 ]} = [T 2 /L]. Conductivity: the intensity of current passing across unit area under the action of unit electrical force. Its dimensions are those of cur- rent-intensity -T- (electrical force x area), viz., {[M*L*/T] + [(MiLi/T 2 x Resistivity, the reciprocal of the Conductivity ; [L 2 /T]. Coefficients of Self-Induction and Mutual Induction of Currents: Ratios between E.M.D.P. produced and the rate of change of current-intensity producing it : the dimensions of the former are [MsU/T 2 ] ; those of the latter are [Intensity -=- Time] = [MiJ>/T] -i- [T] = [M*U/T 2 ] ; the ratio therefore has the dimensions [M*lJ/T 2 ] -*- [MiJJ/T 2 ] = [L]. Magnetic Dimensions in any medium. Current-intensity, [M*Li/ T/A*]; Electric Quantity, [M^Li//* 4 ] 5 Electric Potential, j>*M4U/T]; Elec- tric Force, [/^MsLs/T 2 ] ; Resistance, |>L/T] ; Conductance, [T/L/x] ; Resistivity, [>L 2 /T]; Conductivity, [T/L 2 /jJ] ; Capacity, [T 2 /L/x] ; Coeffi- cients of Induction, [/xL]. From these dimensions we find that that which is measured electrostatically as a current of intensity I e.-s. units is magneti- cally a current of intensity i = (I/V) magnetic units. Similarly, by comparison with the electrostatic measures, we find that electrostatic quantity, Q e.-s. units, is numerically expressible as 2=(Q/V) magnetic units; potential-difference, E in electrostatic measure, as e = (EV) in magnetic ; resistance, R electrostatic units, as r = (RV 2 ) magnetic units; capacity, C, as (C/V 2 ) *Resistance a Velocit y. In the Tangent-Galvanometer, p. 712, and by defini- tion (ii.) , p. 707, h = fy tan e = il/rad. 2 ; .-. i = fj . tan rad.^/l. By definition, p. 703, i = A lo/rt, where r is the resistance ; let (Ab/i) lines of induction per second be cut by a vertical slider, connecting two parallel horizontal rails which lie East and West, one vertically above the other, at a mutual distance of d cm., and which, with the aid of the slider, form part of a circuit; then, in order to cut this exact number of lines per second, the slider must travel with a particular mean velocity v. The horizontal component of intensity of the terrestrial magnetic field is jjj ; the number of lines of induction cut per second is thus Ab/ = fy v d', whence i, the mean intensity of the current induced in the circuit, = JjueZ/r. Then {j v d/r = i = fy -tan0 -rad.z/l; whence r = v-d'l/(rad. 2 t&n0). Now impose two conditions; first, the wire coiled in the galvanometer is to be of length I = rad. 2 /d; and second, the velocity is to be such as to produce a deflection = 45. Then r = v ; the Resist- ance is, in this case, numerically equal to the Velocity of the slider ; and it is always some merely numerical multiple of the slider-velocity. xvi.] MAGNETIC MEASURE. magnetic units ; conductivity and resistivity, equal to D and R electrostatic units, as respectively equal to (DV 2 ) and to (R/V 2 ) magnetic units. Practical Units. Some of the units of the C.G.S. Mag- netic System are inconveniently large or small. It is therefore the practice not to use the C.G.S. magnetic units of electrical quantity, intensity, resistance, etc., but to build up a magnetic system based on new units of length, Z, and of mass, m. These are respectively 1000,000000 cm. (the earth's quadrant) and the 100,000,000000th part of a gramme. The unit of current- intensity is then [w/T] = [(M/ 100,000,000000)* . (L x 1000,000000) */TJ = JQ [M 4 L*/T]. The new unit of intensity, the Ampere, is thus equal to -fa C.G.S. Magnetic Unit; and the new unit of quantity, the Coulomb, is similarly equal to -^Q C.G.S. Magnetic Unit. In the same way we find the unit of resistance, the Ohm, = 10 9 C.G.S. Magnetic Units. The Megohm = 1 million Ohms ; the Microhm = one-millionth Ohm. The unit of difference of potential,, the Volt, = 10 8 C.G.S. Magnetic Units ; the Megavolt = 1 million Volts ; the Micro- volt = one-millionth Volt. The unit of capacity [T 2 //] = [T 2 / 1000,000000 L] = {[T2/L] *- 1000,000000| = 10~ 9 C.G.S. Magnetic Unit = 1 Farad. The Farad = 10~ 9 x one C.G.S. Magnetic Unit of Capacity ; but the latter unit is equal to the electrostatic unit x V 2 , or to 9 x 10 20 Electrostatic Units; the Farad is therefore equal to 10~ 9 x (9 x 10 20 ) = (9 x 10 11 ) Elec- trostatic Units of Capacity. The electrostatic capacity of a sphere is equal to its radius ; a Farad is therefore the electro- static capacity of a sphere of (9 x 10 11 ) cm. radius; and for convenience the standard in use is the Microfarad, the millionth of a Farad. The coefficient of self-induction, the Henry or Secohm or Quadrant, is [1000,OOOOOOL]= 10 9 Magnetic C.G.S. units. In this system the quantity i n, which so often occurs in our equations, is known as the Ampere-turns. The heat developed in a wire, per second, by a steady current of A Amperes, under a potential-difference of V Volts, is (V x 10 8 ) x (A x 10- 1 ) ergs = 10 7 - VA ergs = 0-24 FJ. ca. In electric lighting a certain unit is commonly made use of as a con- ventional basis for estimating the sum due by the consumer. This unit represents 1000 A rape re- Volt-Hours, and is equivalent to the Energy con- veyed by a current of one Ampere intensity, passing down a fall of poten- tial of one Volt, and sustained for 1000 hours. This amount of Energy = 712 ELECTRICITY AND MAGNETISM. [CHAP. 1 Ampere- Volt or Watt x 3,600,000 sec., and is therefore equal to {10,000000 Ergs per sec. x 3,600000 sec.} = 36,000000,000000 Ergs or 2,654,340 foot- pounds, or about 865,000 ca, an amount of heat which would convert 2-95 Ibs. of ice-cold water into steam at 100 C. ; and the commercial unit of current is a current of any intensity continued until this quantity of energy has been transmitted through the consumer's apparatus. The accompanying table, pp. 714, 715, gives a conspectus of the relations between the quantities dealt with in this chap- ter, and of the numerical data which are requisite in order to transform a quantity, numerically stated in terms of one of the three systems of conventional units described, into the same quantity numerically stated in terms of either of the other two systems. Magnetic Measurement of Current-Intensity. The mag- netic units of measurement have all been derived in theory from the magnetic measurement of intensity of a steady current: the magnetic measurement of a steady current is therefore a fundamental measurement. It is effected by the use of galva- nometers and electrodynamometers. A magnet surrounded by a coil of wire will, when a current is passed through the wire, tend to place itself at right angles to the plane of the current, that is, to place its axis along the Lines of Induction. If the coil be placed vertically in the plane of the magnetic meridian, and if the needle be suspended horizontally at its centre, so that it can swing round a vertical axis of rotation, then, on the supposition that both poles of the magnet are at the centre of the coil an ideal approximated to when the coil has an extremely large diameter, or when the needle is extremely short the deflection of the needle from the magnetic meridian is such that its tangent is proportional to the current passing. Such an arrangement is called a Tangent-Galvanometer. In a tangent-galvanometer in which the coil consists of only one turn, of radius r, the force acting upon a pole tn very near the centre is F = hm = mil/r 2 = mi - 2irr/r 2 = mi - 27r/r, when i is measured in C.G.S. Magnetic Units, and where h is the force acting on a unit-pole. The mechanical force exerted by the horizontal component of the earth's magnetism is fj on a unit-pole, fym on a pole m. The deflection of the needle is 0. The mag- netic couple is ij ml- sin if the moment of the magnet be / m. From the " Equilibrium of Couples," page 160, prop. 2, we learn that F : fntt : : tan : 1. Therefore i Zir/r = fj tan 0, or i = fj tan - r/2-Tr. fj can be found as on page 681, or turned up in observational tables of local magnetic intensities ; 6 can be observed ; r can be measured ; whence { can be found numerically in mag- netic C.G.S. measure : and it will be observed that it is independent of vari- ations in the strength m of either pole of the magnet. It is also the same whatever the value of /x. If the coil consist of n turns, whose mean radius is r', the force h acting on unit-pole (the ' intensity of the electromagnetic field ') at the centre is i - n - 27r/r / . If it be a coil of rectangular section with inner and outer radii r t and r lfl and of length I cm., with n turns in it on the whole, the intensity of field is xvi.] GALVANOMETERS. 713 The Tangent-Galvanometer is most sensitive when is 45. If the coil be placed parallel to the deflected needle, the sine of the deflection becomes proportional to the current-intensity : we then have the Sine Galvanometer, in which a long needle is used. (Compare Figs. 83 a and 83 &.) In Galvanometers, in which a passing current produces an electromagnetic field in which a magnetic needle is deflected, the amount of this deflection indicates the strength of the cur- rent. It is well in all cases to produce as uniform a field of force as possible. This is effected by arranging a number of coils so as to surround the field, not wholly but in outline. In von Helmholtz's Galvanometer, for example, there are two parallel coils, between which the needle is placed, at a mean distance from each equal to half the mean radius of either. For sensi- tiveness, each winding of the wire is made to come as near the magnet as is practicable. Galvanometer-Constant. When a current passes through the wire of a galvanometer, the needle is in a magnetic field of a certain intensity or strength, measured, as usual, by h, the force locally acting upon a unit mag- netic-pole. If i = 1, h has a certain numerical value which involves only measurements derived from the construction of the galvanometer itself : this is known as the Galvanometer -Constant, and gives the numerical value of the strength of the field when the current traversing the instrument is of unit intensity. It is distinctively represented by the symbol F, and the force h acting on a unit-pole, when the intensity of the current is i, is equal to Ti ; acting on a pole tn, the mechanical force is F = htn = Fmt- For exam- ple, in a tangent-galvanometer of one turn the force acting is, as above, F = mi'2ir/r; whence F = 2?r/r. The dimensions of F are [1/L]. Ballistic Galvanometer. If a tangent-galvanometer be constructed with a short heavy needle of length I, and if a very brief current, enduring- only for the exceedingly small interval 8t, be passed through it, the needle will receive a twitch and, after the current has passed, will swing through an angle 0. The last equations under Ballistic Pendulum (p. 215) were I N = 2VHij/X.sin072 which represents the angular velocity imparted to it. 714 TABLE OF ELECTRICAL DIMENSIONS OF UNITS. ELECTROSTATIC. MAGNETIC. Derivation. Dimensions. Derivation. Dimensions. 1 Electric Quantity, Q, q ..... 2 Electric Surface-Density, a . . . 3 Electric Force on Unit Quantity; Electromotive Intensity ; Inten- sity of Electrostatic Field ; Im- pressed Electromotive Force Potential-Slope, <}>,.... 4 Total Electrostatic Induction, I 5 Induction per sq. cm., i . . . . 6 Sp. Ind . Capacity, or Permittivity, K 7 Electrostatic Potential V ; Differ- ence of Potential, E, e . . . 8 Electrostatic Capacity or Permit- tance, C VForce-cPK Q/Area Force/ Q 47rQ 47T(r 47r Work/Q Q/V Q/Time I/Area I/E V4 1/D 1/D F=mil/d 2 K*M*L*/T K*M*/L*T M*/L*TK* K*M*L*/T K*MS/L*T K M*L*/TK* KL K*M*L*/T 2 K5M*/L*T2 KL/T K/T T/LK T/K M*L 4 /K* RiMiLi/T 2 M*L*/K* M*/L*R* K*M*L*/T2 M^/L f K^ M*/L*K* M5/L*K* M*L*/K* T2/L2R T2/L 2 K T2/LK TV^K K^M^LVT 2 KL/T 2 KLVT 2 i x Time *=9VV M'L^/Ai* M/LM ^M^L^/T 2 M^L*//x* M*/LV TVL 2 ^ ^MW/T 2 TYI^ M*L*/Tf** MVL^T/i* T/L/, T/L 2 ^ M L/T A*L 2 /T M*M*L*/T M*/L*TAI* ^M^/T /JMS/I^T M*L4/T/t* ffotf/IJT At*MiL*/T ^M^/L^T ^ M J L f/ T /x pi M L ML M*L*/T/MJ I/ML VA; 9 Current-Intensity or Strength, I, i 10 Current-Density A . 11 Conductance, D 12 Conductivity, D . 13 Resistance, K, r 14 Resistivity, B 15 Magnetic Quantity, m . 16 Magnetic Force ; Mechanical Force on Unit Quantity ; Intensity of Magnetic Field; Potential-Slope; Number of Lines of Force per SQ cm ' h VForce ^ /* Force /m m x Length /vol. Work/m tn/Area s thickness 47rm/Area 47rtn i/fe e+M/dt e+di/dt 4:Tri n I/Aft (I/A^)>(I/A) 17 Magnetic Moment, fSi .... 18 Intensity of Magnetisation, K . . 19 Magnetic [scalar] Potential, ft . 20 Magnetic Surface-Density, s . . 21 Strength of Magnetic Shell, ) x (9 x 10 20 ) x (9 x 10 20 ) x (3 x 101) -4- (3 x 10 9 ) - (3 x 10- 9 ) - (9 x ion) - (9 x 10 2 ) x (9xlOU) x (9 x 10 2 ) x (3 x 10 2 ) x (9 x 1020) x (3 x 101) x (3 x 101) x (9 x 10) x (9 x 10 20 ) -(9x1020) - (3 x 101) xlO 9 xlO xlOi 9 xlO 9 xlOi 8 -10 9 -lOi 8 -10 8 Farads, x(9xlOH) Amperes, x (3 x 10 9 ) x (3 x 10~ 9 ) Mhos, x(9xlOU) x(9x!0 2 ) Ohms, -(9xlOU) -(9 xlO 2 ) H-(3xl0 2 ) -id 9 -10 -10i 9 -10 9 xlO 9 xlO 8 8 9 10 11 18 14 15 M*L*/T 2 M*/L f lf*L*/T pt/L* M*/L* M*/L*T M*/L*T M*/L*T - (3 x 101) x (3 x 101) x (3x101) -(3x101) x (3 x 101) x (3 x 101) -3 x(3xlO- 7 ) x(3x!0 23 ) - (3 x 10 9 ) x (3 x 10 2) ) x (3 x ion) x (3 x 101) -(3xlOio) - (3 x IQio) x (3 x 10i) - (3 x 101) -(3 xlO 10 ) xlOio xlOio xlO xlOio xlO Gausses, x3 - (3 x 10~ 7 ) ^ (3 x 1020) x (3 x 10 9 ) -(SxlO 20 ) ^-(3xlOU) xlO" -10 -10 in 17 18 19 20 21 MVL* M*/L*T x (3xlOio) x(SxlO-) - (3 x 101) xlOio - (3 x 10 20 ) -101 22 Hit* T 2 /L 2 T 2 /L 2 M*L ? /T (No.) (No.) x (3 x 101) x (9 x 10 20 ) x (9 x 1020) x (3 x 10 2 ) x (9 x 10 20 ) - (3 x lOio) - (9 x 10 20 ) -(9x1020) -10 8 Same Same Webers, -(3xl0 2 ) -(9x1020) H-(9xl0 20 ) xlO 8 Same Same L>3 24 26 T 2 /L L x (9 x 10 20 ) x (9xlOH) -(OxIO 20 ) -10 9 Henries, -(9x10") xlO 9 26 T2/L L/T 2 L M*L*/T 1/L x (9 x 10 20 ) - (3 x lOio) - (9 X 10 20 ) x (9 x 10 11 ) - (3 x 10) - (9 x 10") - (9 x 10 2 0) x (3 x 101) x (9 x 1020) -10 9 xlO XlO 9 + (9 x ion) Gilberts, x(3x!0 9 ) Oersteds, x (9 x 10 11 ) xlO 9 -10 -f-10 9 27 28 L2/T 2 (No.) - (9 x 10 20 ) - (9 x 10 20 ) X (9 X 1020) Same x (9-x 1020) Same 80 C =(K/ where w is the angular velocity imparted to the needle, and Bt the time during which the current lasts. The work done is therefore %Tim - 1 - $ti>8t = ^T-ml-iBt-^ = ?r|BQ i 2 . Now equating these two values of |Nco 2 the energy, we have ^FfHQ to = ffilfr.2siii 2 (0/2); or Q = 4fr/r.2 sin 2 ((9/2). /o> = 4fr/r2 sin 2 (0/2) -s- [2>/iflfHj/N".siii (0/2)] = 4ij/r- VN/jmij.sin0/2. But T, the time of a com- plete to-and-fro oscillation of a needle swinging freely in the terrestrial field, is T = 2uVN/jafj; whence Q = 2Jj/F T/TT- sin (0/2), in which aU the terms are measurable. When the current ceases, the magnet tends to oscillate for some time, like a pendulum ; but if it oscillate in a strong magnetic field of force as, for example, in the neighbourhood of a strong magnet its oscillations will be very rapid and of small amplitude. If masses of metal be so arranged that any oscillations of the magnet tend to produce retarding induced-cur- rents in these masses, then, especially if the needle be light, the oscillations of the magnet rapidly cease, as if it were immersed in a viscous medium, and the magnet is, without further oscillation, restored to its position of repose. A galvanometer arranged on this principle is a Dead-Beat gal- vanometer. The same dead-beat effect is mechanically produced by making the magnet move in a small closed chamber of air which it nearly fills ; it thus moves against air-resistance. In Differential Galvanometers, two equal and separate wires are similarly coiled round the same needle ; through these wires currents may be sent in opposite directions ; if the two currents be equal, the needle remains at rest ; if either predominate, the needle moves. In Electrodynamometers the current is passed through a coil which is suspended within a strong and uniform magnetic field, such as that produced by powerful electromagnets actuated by a second current, or again by fixed coils through which a second current is passing. The deflection of the suspended coil depends upon the strength of the current passing through it, and also upon the strength of the magnetic field surrounding it. If the same current traverse both the fixed and the suspended coils, the rotating couple is proportional to z 2 , and therefore to the energy of the current ; and it is independent of its direction. The two coils have the respective mean radii r and r,, r the greater, r, the less ; the respective numbers of turns are n and n t ; when the plane of the suspended coil makes with the plane of the larger coil an angle 0, the couple, tending to bring the two coils into the same plane with their currents opposed, is />u 2 2?r 2 r, 2 rm^sin 0. /r. If I and I, be the lengths of wire in the two coils respectively, this expression may be written as ^u'2. //,./- sin 0./r 2 . If the current to be measured be a rapidly alternating one, the result is the same as if it were constant ; it is reversed in both coils at the same time, and the algebraic sign of i 2 is always positive. xvi.] ELECTRODYNAMOMETERS. 717 In the Siemens electrodynamometer, used for measuring electric- lighting currents, the same current is made to pass in succession through two thick-wire loops. These tend to place themselves in the same plane ; but by the torsion of a spring, they are forced into a standard position at right angles to one another. This torsion is measured by the angle of rotation of a pointer connected with the spring; and it is proportional to the square of the intensity of the current. If a movable coil be free to slip up and down the axis of a fixed coil in which a current is passing, the inner coil may be sucked in or repelled with a force which may be balanced and measured by known weights or elastic tensions or torsions ; or if the current in the suspended coil be variable, the tension tending to draw it in to the fixed coil may be made to act against a spring, and graphically to record its own variations upon a uniformly-moving piece of paper. If, instead of a movable coil free to slip up and down the axis of a fixed coil, we have a bar of soft iron, it will also be sucked in or repelled ; but in this case the magnetisation of the bar is not strictly proportional to the intensity of the current : whence the law, that the force of suction or of repulsion is proportional to the square of the current-intensity, fails us. If, however, the bar be reduced to a very thin soft-iron tube, it rapidly becomes saturated and soon becomes practically constant in strength. When this limit has been reached, the force is directly proportional to the intensity of the current. This is the principle of Ayrton and Perry's Ammeter ( = " Ampere-meter "). A slender piece of soft iron tends to move towards the centre of a current-bearing coil; it may thus be made to rotate against gravity round a fixed pivot: a pointer attached to it will indicate the amount of rotation: and this is the principle ofSchuckert's Ammeter. A short thin bar of soft iron tends to be pulled so as to lie in the strongest part of the field between two electromagnet-poles (Ever shed's Amme- ter). The tendency towards suction of a suitable electromagnet into a coil can also be measured by balancing it against weights, as in the Electric Power Storage Co.'s Steelyard Ammeter. Ampere-meters, as their name indicates, are graduated in Amperes, not in magnetic C.G.S. units. The principle of the differential galvanometer may be here applied, as in Prof . Langley's Thermic Balance or Bolometer. The suspended coil is composed of two separate wires wound together, but insulated from one another : a single current is divided into two equal moieties which run in opposite directions through the two wires of the coil; there is no effect. The least variation in one of these moieties, as when the conductivity of its path is affected by the local application of heat, causes imperfect compensa- tion, and practically a small uncompensated current passes : however feeble this may be, it can be rendered manifest and measurable by increasing the strength of the magnetic field within which the double coil is suspended. The part of the divided circuit to which heat may be locally applied may be an exceedingly thin strip of platinum. This may be moved up and down, say, in the dark region of the spectrum. In some places it is heated, in others dark lines it is not. By thus groping in the dark it discovers 718 ELECTRICITY AND MAGNETISM. [CHAP. the dark lines and the specially " bright " lines of the heat-spectrum. An instrument of this kind is sensitive to differences of temperature of TT^TFTF F- Magnetic Measurement of Resistance. If a circle of wire, radius r, stand at right angles to the magnetic meridian it will embrace /xfy lines of terrestrial magnetic induction (yx = 1) per sq. crn. or yuij rrr 2 lines over its whole area; if it be turned round a vertical axis through 180, it will come to embrace /xij -jrr 2 lines oppositely directed with reference to it : the number of lines of induction passing through it has therefore been increased or decreased by 2/xfj Trr 2 . The circle of wire thus rotated (E a r t h - i n d u c t o r) becomes the seat of a current whose mean E.M.D.P. is numerically equal to 2/xfy 7rr' 2 /t in magnetic units, and whose mean intensity is i = 2/xfj 7rr 2 /R, where R is the resistance (also measured in magnetic units) and t the time occupied in the rotation through 180. If a small needle be suspended at the centre of this rotating circle, that needle will be deflected ; the rotating circle acts somewhat as if it were its own Tangent-galvanometer; but instead of a deflection such that tan = i-27rn/rfy, where n is the number of coils and r their mean radius, we have (approximately) tan 20 = i-ir 2 - n 2 /rty. But i = 2/4 7rr*/Rt ; whence tan 20 = 2/x ^n^r/Rt. If t be the 2Nth part of a second, the coil will make N complete turns per second, and tan 20 = fyNTr^V/R. Therefore R, = 4/xN7r 8 n 2 >-/tan 20. Of these quantities, n the number of coils and r their mean radius are obtained by measurement ; tan 20 is the ratio between ' the scale-reading (straight scale) and the dis- tance of the scale from the mirror fixed to the centre of the deflecting needle ; N can be read off on a speed-indicator ; and /z = 1 in air. When the resistance is so adjusted that to a speed N there corresponds a deflection such that the product above (with due corrections) is numerically equal to 1000,000000, the resistance employed is equal to one Ohm. This is the principle of the method by which the British Association Committee on Electrical Standards constructed the original standard Ohm. Measurement of the Capacity of a Conductor or Condenser. The capacity is C = Q/V, and therefore we can find the value of C if we find, in terms of units of the same system, the quantity Q with which a body is charged, and the potential V to which this charge raises it. This potential V may be the difference between the potential of the body charged and that of the earth, or it may be the difference between the potentials of the opposed plates of a condenser. This is, however, not a convenient method ; and the practical method is first to construct standard condensers of known capacity, and then to compare the capacity of the body examined with that of these standards. Standard Condensers. Suppose a conductor of capacity C and bearing a charge Q to be discharged through a known resistance which includes a galvanometer ; the resistance being so considerable that the dis- charge is far from instantaneous. The initial potential of the conductor is V = Q/C. A current will pass through the galvanometer, but will con- tinuously diminish in intensity. At the end of time t let the potential have sunk to V y which is the nth part of V, and the charge to Q, ; and at the end of a very small further interval &, let these have farther sunk by the amounts V y and Q, respectively. Then the quantity which has escaped in time & is Q,, which is necessarily equal to C V, ; it is also equal to the instan- taneous intensity I multiplied by the time /, and this product is equal, by Ohm's law, to (V y /R) x &. Hence (V,/R) & = CV,, or & = CR- V,/V y . From this we find, by means of the Integral Calculus, that the time which XVI.] MEASUREMENT OF CAPACITY. 719 must have elapsed between the initial instant at which the potential had been V and that instant at which the potential had sunk to V, is equal to CR log (V/V,). But by our supposition this time is t, and the ratio V/V y is equal to n. Whence t = CR log n, or C = t/R log n. If we observe the successive values of the current-intensity at equal intervals of time we can find the value of n, and then, knowing the value of R in electrostatic, in magnetic, or in practical units, we can find the corresponding value of C, the capacity, measured in units of the corresponding system. Comparison of Capacities. 1. De Sauty's Bridge-method, appli- cable to small capacities. Two condensers of capacities C, and C 7/ , if charged to the same potential V, must be charged with the respective quantities C/V and C y/ V. If one of these be discharged through a resist- ance R; for time &, the current which is set up is of mean intensity I,, and the quantity passing in time & is I, &. But this is equal to the fall in the value of Q during the time &; that is, to Q. But Q = C/V",, where V, is the fall of potential in the condenser C y . Whence I, = C/^/&; and R,, which varies inversely as I /5 is proportional to Similarly, if C /y be discharged by a current of intensity I y/ through a resistance R /y , that resist- ance must bear the same proportion to &/,/$" ; and if V y/ , the fall of potential in C /y , be the same as in condenser C,, that is, if V /y = V, ; then the equation shows that the Resistances through which the two condensers charged to equal potentials must be discharged, in order that the potentials of the two condensers may fall concurrently and remain persistently equal to one another, must be inversely proportional to the respective Capacities of the bodies discharged through them. This being postulated, the arrangement of the apparatus is indicated by Fig. 254. A, a battery; K, a key with which the wire B may be at will con- nected either with the battery A or directly with the earth, or else, as in the figure, isolated from these. When the battery is connected with B, the condensers C y and C 7/ are charged through the resistances R, and R /y . Connect, then, A with B for a cer- tain time; disconnect. The two condensers, if not already at equal K " potentials, soon become so, for an equalising current traverses EGF; when equalisation is complete, the needle of the galvanometer G re- turns to rest. Now put B to earth. The charges of C, and C 7/ escape through R, and R /y respectively. If the resistance R/ be disproportion- ately great, the outflow through it is disproportionately small, and the potential at F sinks faster than that at E ; a current therefore passes from E to F, and the galvanometer-needle in G is deflected. If R, : R,, : : C /y : C y , the galvanometer-needle regains at rest, for the potentials at E and F as they sink, sink together and are concur- EARTH EARTH 720 ELECTRICITY AND MAGNETISM. [CHAP. rently equal to one another. If therefore we adjust the resistances R y and R yy until we find that on effecting the three operations (1) connecting B with the battery A ; (2) isolating B from A until the galvanometer-needle comes to rest ; (3) putting B to earth the last of these is followed by no deflec- tion of the galvanometer-needle, then, since we know the relative values of the resistances R y and R //? we know, inversely, the relative values of the capacities C /7 and C,; and as one of these capacities is a standard, we are thus enabled to state absolutely the actual value of the capacity to be measured. 2. Compensation-method. If a wire Cu Zn (Fig. 255), connecting two poles of a battery, be connected at any one point with the earth, the potential Fig. 255. O WIRE Zn, Cu. EARTH t of that point must become equal to zero ; but the difference of potential be- tween the extremities of the wire remains unaffected. The positive potential at Cu (Fig. 255) bears to the negative potential at Zn a numerical ratio, the same as that between CuO and OZn ; for obviously CuA : ZnB : : CuO : OZn. Let now between the points Cu and O a resistance of reduced length R ; be placed, and between the points O and Zn a resistance of reduced length R y/ . The potential at a point just between R y and Cu, and the potential at a point just between R /y and Zn, bear to one another the ratio of R y : R /7 . If these potentials be + V y and V,, respectively, we have V y : V y/ : : R y : R /y . Let these two points, at potentials V y and V /y respectively, be connected with two condensers of which the one has standard capacity C y , the other the capacity C /y to be determined. The two condensers will become charged to the respective potentials V, and V y/ ; but the aim is so to adjust the poten- Fig. 266. +X Viff w&m tials that these condensers shall become charged with equal but opposite quan- tities of electricity. Sup- pose this adjustment to have been effected. Then Fig. 256 illustrates the successive operations. (i.) Connect at D and E. The condensers become charged to po- tentials + V, and - V /y respectively. They are therefore charged with quantities + C y V y and C // V // . Disconnect at D and E. (ii.) Connect at F. The two charges + C,V, and C y/ V y/ blend, and there remains in the conjoined condensers a resid- ual charge of (C y V y - C y/ V /y ), which, if C y V y = C y/ V y/ , is equal to zero. EARTH EARTH xvi.] MEASUREMENT OF CAPACITY. 721 (iii.) Connect at H. The residual charge, if any, runs to earth and deflects the needle of the galvanometer; if none, there is no deflection. There is no deflection when C,V, = C,,V /y . But V, : V ;/ : : R, : R /y . There- fore, when there is no deflection on making contact at H, C / R / = C^R,,; and the capacities C, and C y/ are inversely as the corresponding resistances. Adjust therefore the resistances R y and R y/ until there comes to be no de- flection of the galvanometer-needle after operation (iii.), and from the known ratio of the resistances we find that of the capacities, for R / /R // = C y/ /C y . OSCILLATING OR ALTERNATING CURRENTS. If the potential, which tends to give rise to a current, itself fluctuate between positive and negative values, its variation being simple-harmonic, we have an oscillating or alternating current produced in the circuit. Oscillating currents differ in some important respects from the steady currents hitherto discussed. If their frequency be small, they approximate to steady currents in their character, and merely fluctuate in their intensity and direction : if it be great, they present, as it were, nothing but the Variable Period, and never arrive at the Steady State. We shall mention the main characteristics of those cur- rents in which the frequency is great, say a million oscillations per second. How such currents are produced we shall learn later. The phenomena of the current tend, as the frequency increases, to be entirely confined to the dielectric, so that only a thin skin of the wire is concerned in the cur- rent : but the less the magnetic permeability of the conducting wire, the thicker is that conducting skin ; and the smaller the frequency of alternation, again the thicker is that skin. Even when the frequency is as low as 100 per second, the skin, in the case of iron, is practically not more than 0*3 cm. thick, while with a Leyden-jar discharge it is less than 0-001 cm. in thickness. In the dielectric there are Waves of Propagation of Lines of Force ; these lines travel back and fore with the velocity of Light, with their ends on the conducting wires, and approximately at right angles to these.' In the dielectric or the electromagnetic field surrounding the wire, the lines of magnetic induction have the same direction as in the case of a steady current, but the field-intensity fades away very rapidly as the wire is receded from. The Electrostatic Attraction between the two sides of the circuit tends, as the frequency increases, towards equality with the Electromagnetic Repulsion between them. The Transmis- sion of Energy through the dielectric is approximately parallel 3 A 722 ELECTRICITY AND MAGNETISM. [CHAP. to the wire. The Apparent Resistance, or Impedance, r, of the wire tends towards a value inversely proportional to the cir- cumference, instead of to the cross-section of the wire ; and it is greater the greater the magnetic permeability of that wire, for the conducting skin is in that case all the thinner. As a wave of potential passes along the wire, its amplitude wanes ; and the effect of this is more marked the greater the frequency ; so that after passing over a long wire a complex harmonic dis- turbance may have its higher harmonics considerably more atten- uated than the lower components. Besides this, disturbances of different wave-lengths produce their maximum effect, at a given distant point, at different times. The conducting skin screens the interior of the wire from the inductions which would otherwise be set up in it by the alternating field ; and generally, a film of metal acts as a screen, or is opaque, to the alternating condition of the field, while an insulator is not. An electrolyte is not quite opaque in this sense, but has a high absorptive coefficient, and a thickness of some millimetres acts like a film of metal. Suppose that the E.M.D.P. is itself subject to variation which follows the S.H.M. law; that is, with a frequency n times per second, it oscillates between the extremes + e and e, and is at any given time t equal (assum- ing it to be zero at the initial instant when t = 0) to e-sin ^imt. Then, instead of e in equation i., p. 705, we have (e - sin 27rnt) ; and that equation, when so modified, upon being integrated, gives i t = (e/ Vr' 2 + 4LV 2 7r 2 ) ; sin (lirnt tan" 1 27rnL/r), a S.H. function : that is, the intensity of the cur- rent varies as time goes on, according to the simple harmonic law, and its maximum alternating value is e/^/r 2 + 4L 2 nV 2 . Observe from this expres- sion that if the coefficient L be comparatively great, as in a coil of wire, the maximum values of the current may fall considerably short of the value e/r ultimately attained by a steady current, but will tend to be little affected by changes in the resistance of the circuit ; and that as the frequency n of oscillation increases, the maximum intensities fall off. The apparent resist- ance, or Impedance, is thus vV 2 -f- 4LW = r. Hence, when an alter- nating current of high frequency is sent through a coil of many turns round a soft-iron core, the impedance may have a very high value, and the current round the coil may be reduced to a minimum. The current may thus be "throttled," or choked down. Observe also that in the equation arrived at above, the S.H. variation of the current-intensity does not keep step with the variations of the E.M.D.P., as it would do if the last term were simply sin27rw; there is a lag; the variations of current fall behind those of E.M.D.P. by an amount represented by an angle whose tangent is ZirnL/r. This Lag is greater the greater the frequency n of the variations of the applied E.M.D.P., or the greater the self-induction L of the circuit or coil, or the less its resistance r; but the angle cannot exceed 90 ; hence the zero current is delayed after the zero E.M.D.P., by an interval of time corresponding to not more than a quarter of a revolution in the circle of reference (see S.H.M., Fig. 29) ; that is, it is in arrear by an interval of time not greater than a quarter period, T/4, or l/4n. xvi.] ALTERNATING CURRENTS. 723 Since, during any S.H. variation of potential or of current-strength, etc., the average value, taken throughout the whole of a positive or a negative phase, is 2/?r x the maximum value (see p. 85), the actual mean intensity of an alternating current is 2/7r or 0-6368 times the maximum intensity. Alternating currents produce Heat, and therefore also Light, as in incandescent and arc lamps; and they can light up Geissler- tubes. A Cardew's Ammeter, or an Electrodynamometer, produces Heat, or a Torque, proportional to the square of the current at any instant; such instruments therefore indicate the mean value of i 2 , not of i; and in S.H. variations of i, the mean value of i 2 is half the square of the maximum inten- sity ; whence the mean jalue of , as given by such instruments, is the maximum intensity x V^. This differs from the true arithmetical mean in the ratio V : 2/Tr, or -707/-637 ; and it is called the effective or virtual mean intensity. Similarly for the voltages. Hence the Heat produced by an alternating current whose maximum intensity is i / is \i?r. If an object of some capacity a small porcelain ball in a vacuum chamber be connected with one terminal only of the secondary coil of an induction coil subjected to alternations of extreme frequency, it may itself become very hot ; and if it be itself surrounded by air, it may subject the molecules of that air to collisions and shock, so that the air glows with a phospho- rescent light. A Geissler-tube connected in the same way will light up. The human body may be charged in this extremely- rapidly-alternating manner by being connected in the same way ; when this is done, a Geissler-tube held in the hand will light up ; while the current then passing in the body, though of enormous voltage, appears to do no more harm than the impact of light-waves does. Mr. Nicola Tesla has recently devised a series of extraordinary experiments on these lines. He has con- structed lamps connected only by one wire with the terminals of secondary coils; and he has obtained, at these terminals, brushes in the form of veritable flames, consisting merely of air-molecules subjected to collision and shock. Since the current is alternating, there can be no electrolytic effect, except, apparently, a small residual decomposition in some cases, which is possibly due to greater ease of charging particular elements with one kind of electricity than with another. When an alternating current is sent through a loop, as in Fig. 221, the derived currents are not so distributed as to produce the minimum amount of Heat, as is the case in steady currents ; but they are so distributed as to keep the kinetic energy of the field down to a minimum, and to neutralise each'other's pro- 724 ELECTRICITY AND MAGNETISM. [CHAP. duction of an electromagnetic field as far as possible. The currents in the two branches of a loop may thus be opposed, and even be each much greater than the leading current ; but their mean difference will be equal to the mean intensity of that current. The preponderating branch-current goes in the direction of the leading current along that branch which has the less self-induction ; that is, along that branch the passage of a current along which would give rise to the smaller amount of total magnetic induction in the part of the field affected by that branch; but if the electrostatic capacity of the other branch be increased, this may neutralise the effect of its superior self-induction. When an alternating current is used to excite an electro- magnet, there is a strong tendency to the formation of Eddy- Currents, parallel to the wire of the inducing coils. This is combated by building up the core, of laminae or of thin wires. If an alternating current be used to excite a long electromagnet, the alternating magnetising effect falls off very rapidly at a distance from the exciting coil, and does so the more rapidly the greater the frequency; the lines of magnetic induction leak out laterally from the iron, and find closed return-paths through the air. In a non-uniform magnetic or electromagnetic field, in a case where a conductor, say a coil, bearing a steady current, would act like a magnet, one bearing an alternating current acts like a diamagnetic body ; it is repelled into the weakest part of the field. Two alternating currents in the same direction and in the same phase attract one another ; if in opposite phases, they repel one another ; if their phases differ by ?r/2, they have no effect upon one another. Hence the mutual attraction or repulsion of two alternating currents will depend upon their relative amounts of Lag. When an alternating current acts by induction on a coil laid parallel to the inducing coil, in the absence of self-induction the induced current would be strongest when the inducing current passed through its zero value, for the strength of that current would then vary most rapidly. But, when the secondary coil is excited by an electromagnet, itself excited by an alter- nating current, there is Lag in the secondary coil ; there is now repulsion between the two coils ; and the secondary coil tends to fly off the electro- magnet. Similarly, an alternate-current electromagnet may repel copper by its action on the currents induced in that copper. Transformers. An alternating current of high voltage and few Amperes can be sent to a great distance, for there is comparatively little loss by transformation into heat. But for use in houses, etc., its voltage must be reduced. This is effected xvi.] TRANSFORMERS. 725 by Transformers, which are, in effect, Induction Coils reversed in their action. The current of high voltage is sent through the coil of many turns (inside the other coil), and an induced alternating current of lower voltage and correspondingly greater quantity is induced in the coil of fewer turns. The soft-iron wire or laminated core may form either a closed or an open " magnetic circuit." An interrupter or contact-breaker is not necessary. When the core is large and the alternations rapid, the effective current-intensities are inversely proportional to the number of turns in the respective coils. The induced currents are opposite in phase to the inducing currents ; they thus tend to demagnetise the soft-iron core when the house-circuit is closed. They thus tend to diminish the Impedance of the main circuit ; and they do this the more completely, the less the resist- ance in the house-circuit. When the house-circuit is broken, little or no current passes through the main coil, on account of the impedance of that coil, with its core ; but as the resistance in the house-circuit is reduced by increasing the number of paths along which the house-current may pass, that house-cur- rent is allowed to gain in strength in proportion to the reduc- tion, that is, in proportion to the work to be done. From 3 to 6 per cent of the energy supplied is usually lost in eddy-currents and hysteresis. PRODUCTION OF ALTERNATING CURRENTS. There are two main methods of producing these ; (1) by the action of dynamo-electric machinery, the frequencies pro- duced by which range, say, from 40 to 150 per second ; and (2) by means of the discharge of a Leyden-jar or other electrostatic condenser, the frequencies produced by which may amount to, say, 10,000000 per second. We shall deal with the latter first. Oscillation in Leyden-jar Discharge. Assume the source of elec- tricity in a circuit, which circuit may have an air-gap in it equivalent to an interposed resistance, to be a Ley den jar charged with a quantity Q; the E.M.D.P. is E = Q/C, where C is the Capacity of the jar. The equa- tion (i.), p. 705, is easily reduced neglecting squares of SI, which is a proper omission in a calculation involving subsequent integration to the form E = RI +' L 81 ; or, since I is itself equal to - Q, we have Q/C = _ (RQ + LQ) ; R being the total Resistance of the circuit. This equation is a Differential Equation ; and, starting from a charge Q and no current at the initial instant, this equation is reduced, by appropriate mathematical treatment, to the statement that at the end of any given time-interval f, the charge Q, left in the Leyden jar is Q, = Q e-^/ 2L {((2La '+ R)/4La) <* 726 ELECTRICITY AND MAGNETISM. [CHAP. -f ((2La R)/4La) e~ a *}, in which expression the letter a is made to do duty for V(R/2L) 2 - 1/CL, and c is equal to 2-718281. On interpreting this statement we find that there are two cases, in which the consequences are different; (1) when L is less than R 2 C/4, a is a possible positive quan- tity, and the charge gradually diminishes as t increases, that is, as time goes on, so that the Leyden jar steadily discharges itself; but (2) when L is greater than R 2 C/4, a becomes the square root of a minus quantity, and being itself therefore impossible, renders the expression an unintelligible one as it stands. But if we transform it by making aV 1 = a', we find the result to be that the expression takes the form Q, =Q -R/2L. ( C os a 't (R/2Lo/-sina/)). That is to say, the charge in the jar oscillates in amount, being nothing when (sin a't /cos a't) = 2La'/R> and attaining a succession of rapidly-decreasing maxima (alternately positive and negative), which occur whenever a't == mr, where n is any whole number. The current- intensity is a maximum, positive or negative, when the charge in the jar is passing through a zero value ; when this is the case, the intensity is equal to Q t = Q/CLo/- e -R*/ 2 L. sin a't; and it thus oscillates in value, rapidly diminishing, with a period T of complete oscillation = 2-jr/a! = 27T/ Vl/CL (R/2L) 2 . This period T, when R is very small in compari- son with L, is approximately T = 27rVCL. Whether, therefore, a steady or an oscillating discharge will be obtained depends on the relations between the capacity C of the jar, the self-induction L of the circuit, and its resist- ance R : diminish C or R or increase L sufficiently, and an oscillating dis- charge may be obtained ; while by increasing L very much, as by interposing very large coils in the circuit, the rate of oscillation may be greatly reduced. Since a maximal positive value of the current-intensity occurs once in each period, the time-interval between any two such positive maxima is T = 2-7T VCL. The successive maximal positive values of the current-intensity are thus Q/CLa'-e-^/ 2L = Q/CLa' c~ BrVc/L f or the first; Q/CLa' -R2,i>/c7L for the second, and so on ; the value being Q/CLa' e- R '* 7rV ' c / L for the nth positive maximum. Numerical Example. In a solenoid of n turns and of length I cm., L = 47m 2 A//,/. Let n = 100, and I = 20 ; and let r = 1 cm., and, as the medium is air, /* = 1. Then L = 19,739, the number of lines of induction, in magnetic measure, which thread the solenoid when unit-current passes. The Resistance of this wire may be taken as 0-0045 Ohm per metre, or 45000 magnetic units per cm. ; R = 45000 '2-nr n = (nearly) 28,275,000 magnetic units. Next, suppose a Leyden jar to have opposed surfaces of 250 sq. cm. each, at a mutual distance 0-3 cm. across glass whose sp. ind. cap. is, say, K = 2-5 : then its Capacity C will be K/47T surface/ d = 165-78 C.G.S. electro- static units, or (165-78 -r- 9.10 20 ) magnetic units. Now discharge the Leyden jar through this solenoid, it being assumed that there is no other part of the circuit to be taken into consideration. Then a' Vl/CL (R/2L) 2 = 16,583500, in magnetic units ; and the period of complete oscillation T = 27r/a'=- 27p ^ second. _ To ascertain the progressive decrease of the successive maxima of cur- rent-intensity, put these numerical values of C, R, and L in the expressions given above for that intensity : then we find that the current-intensity is, at the first, the 10th, the 100th, the 1000th, and the 10000th maxima respec- tively, in the ratios of 0-99973 : 0-99742 : 0-97323 : 0-76237 : 0-06632 ; and the maxima of current-intensity are reduced to a millionth in 0-02428 second. xvi.] LEYDEN-JAR DISCHARGE. 727 The discharge of a Leyden jar is thus practically instantaneous ; and in order that it may keep up a continued discharge, the jar must be fed from a machine or an induction-coil. If the apparatus of the above example be reduced to half the size, line- arly, the time of oscillation will be reduced to one-half; and so on in pro- portion. A Leyden jar of molecular size would give an oscillating discharge whose frequency is of the same order as that of light-waves ; but if the mole- cule be simple in its structure, the frequency, thus calculated, will lie beyond the violet. It has, however, been assumed in the above that the current is uniform all along the wire ; that is, that the wave-length is great in comparison with the length of the circuit ; and that the current-density is uniform across the cross-section of the wire. If we take into account that these things are not true at high frequencies, we find that we have to replace the Resistance by the Impedance, and that the value of L is affected by the frequency ; which modifies the numerical results. If the condenser be reduced to the mere tips of the wire of a loop or coil, the to-and-fro reflexion of the disturbances in the wire will occur at a definite frequency, which depends on the size of that loop or coil. Similar phe- nomena of to-and-fro reflexion occur whenever abrupt signals are sent over a short circuit, but on a long one they die out ; they are due to the electro- static charge on the wire. The second -method involves the use of Magneto-Electric and Dynamo-Electric Machines. The former are now seldom seen, except in small apparatus such as that used for medical purposes : the latter have assumed great importance in Electric Lighting, Electrolysis, Transmission of Power and, to a smaller extent, in Heating. Given an existing magnetic field : then, if a loop of wire be moved in this field, so as to embrace more or fewer lines of mag- netic induction, a current will be set up in that wire. In the former case, Work has to be done upon the loop ; in the latter, the field does work upon the loop ; and both these amounts of Work appear as the Energy of Electric Current in the loop. The direction of that current depends upon whether work is being done by or against the field at the moment. We have seen that if the lines of magnetic induction point eastwards, and the direction of motion of the lines of force in the field be northward, the corresponding lines of Electric Force will themselves point upwards. Here we have the converse case : if any small part of the wire stand vertical and move broadside-on, towards the south, across lines of magnetic induction whose trend is towards the east, there will be set up in that part of the wire a current whose direction is upward. In the former case, the lines moved up to the wire ; in the latter the wire moves in the opposite direction towards them. If a loop, or a coil, be flashed past the two poles of a per- manent magnet, so as alternately to embrace the magnetic lines 728 ELECTRICITY AND MAGNETISM. [CHAP. radiating from these two poles, it will, as it approaches the one pole, travel into a field whose strength it finds to increase to a maximum, and then to fall away as it recedes from that pole. As the coil goes farther, it goes through a region of zero and then into one of opposite potential, which in its turn again reaches a maximum and then falls away. The currents produced in any given part of the wire are thus alternately in opposite directions. In Pixii's machine, permanent magnets were themselves flashed past the coils, of which there were two, parallel and wound on bobbins, with a soft-iron core in each. These cores assumed, in rapid alternation, opposite magnetic characters, and exposed the wire to a more intense alternating magnetic field than they would have been exposed to had there been no cores. In other cases Clarke's, etc. the magnets were fixed, and the bobbins were made to pass their poles. This kind of machine, by multipli- cation of the magnets and of the rotating bobbins, has been made in very large sizes ; and with the substitution of electromagnets for permanent magnets, the principle is still applied in some disc dynamos. It is more common, however, instead of dragging a loop or a coil sideways past the polar face of a magnet, to provide for it an axis of rotation traversing the middle of a nearly uni- form magnetic field, at right angles to the magnetic lines of induction of that field ; and so to fit up the loop, that at two points in its rotation round that axis, it may be so placed as to embrace the greatest possible number of lines of induction. This may be done in two ways : first, by making the axis of rotation lie across the loop itself, which is spun in the field round its own diameter ; or second, by making the axis of rota- tion lie outside the loop itself, and parallel to it, so that the loop is swung in a circle in the field. In both these cases, the loop, at one part of its rotation, will embrace a maximum number of magnetic lines, to which its own axis is then parallel ; when it has turned round through 90, the axis of the loop is at right angles to these lines, and the loop itself embraces none of them. When it is in this latter position, it is most rapidly altering the number of lines which pass through it, and the induced current in that loop is then a maximum. The number of lines of induction embraced by it when its axis is parallel to these lines is B = its area A x b ; when it is at an angle 6 ( = STrnt) to that position, the induction through it is reduced to B cos = B . But the current-strength i e in any position is, omitting self-induction, &B 9 /r8t, r being the resistance ; and this, on giving B its value, and differentiating, leads to the result that i$ = (B/r) . 2irn - sin ; and the maximum value of this is i = (B/r) 27rn, when 6 = 90 ; that is, when the axis of the coil is at right angles to the lines of induction. Hence the maximum value xvi.] DYNAMO-ELECTRIC MACHINES. 729 of e = 27rnB ; and at any position 0, e& e sin = e - sin 27rnt; the condition required for finding the Lag and the Impedance (p. 722), when self-induc- tion is taken into account. The average value of e is 2/7r x the maximum value (p. 85), and is therefore, for a single loop, equal to 4nB ; n being the number of complete revolutions of the loop, through 360. In the latter of the two methods of rotation referred to, the whole loop, and in the former, any part of it, is carried round in the course of its rotation from a positive into a negative part of the field, and the same operation, with negative sign, is repeated there ; the current now produced is, as regards the loop or the part of it referred to, now in the opposite sense, and passes through a maximum in the same way. In any given part of the loop, therefore, the current produced passes through alternating positive and negative maximal values ; and its varia- tion between these extremes is simple-harmonic, one complete alternation to each complete revolution of the loop. The extremities of the loop may be connected with two separate rings, which rotate along with the loop round the axis of rotation ; and if an external circuit terminate in flexible metallic " brushes," which rest upon these collecting rings, the alternating currents developed in the loop will be propagated round that circuit. A single loop is, however, an illustrative rather than a practical apparatus. A coil would produce a greater current, for each turn in it would be acted upon, practically, as if it were a loop ; and in Siemens' Inductor a coil, with a soft-iron core, was rotated round an axis passing through its own centre, and was so shaped as to lie as close to the magnetic pole-faces as possible, and at the same time to have a minimum moment of inertia. But in the course of each revolution there is a period during which such a coil or loop is very nearly idle: that is, when its own plane is at right angles to the lines of induction : and since there is only one alternation per revolution, the speed for rapidly-alternating currents would have to be excessive. It is therefore the practice in alternating-current machines, or Alternators, to multiply the opposed magnetic fields through which the coil has to travel; and this is done by multiplying the opposite pole-faces past which the coil is driven. Such machines are said to be multipolar: and the frequency of alternation is correspondingly increased by this device, though the alternations are now, initially, not so nearly simple-harmonic in their character as when a single loop or coil is employed in 730 ELECTRICITY AND MAGNETISM. [CHAP. a single uniform field. But further, instead of a single coil passing one magnet-pole at a time, a number are put into simul- taneous action, all similarly situated with respect to their several magnetic fields ; and great ingenuity has been displayed in con- triving the machine so that these may act in concert. These coils may move past the magnet-poles, or the magnet-poles past them : or again, both coils and magnets may be stationary, the strength of the magnetic fields being alternately increased and diminished by masses of soft iron moving in or near those fields. For some purposes it is convenient so to arrange the coils and their metallic connections as to send along, say, three wires three equal currents differing in their phase of alternation by equal amounts ; alternators of this kind are called multiphase alternators. The congeries of coils is borne by a drum, by a ring, or on the periphery of a disc; and each coil has a laminated soft-iron core, which forms a part of the magnetic circuit, and greatly intensifies the inductive effect on the wire of the coils. The whole arrangement of coils, with their core or cores, is called the Armature. The magnetic field is, in all modern machines of any size, that of an electromagnet; it has to be intense, while at the same time there must be plenty of iron in the magnetic circuit. The electromagnets are excited, when the machine is at work, by a part of the current from the machine itself ; but as this is alternating, it would not excite an electromagnet so as to impart to it a uniformly-directed magnetic polarity, unless its alternate phases had been made to go, not in opposite, but in the same directions. This is accomplished by a Commutator: the two "brushes," which take current off for the electromag- nets, do not each continuously touch one of the collecting rings ; but each comes in contact, first with a projecting tooth of the one, and then with a projecting tooth of the other collecting ring. Thus, in step with the alternations of the current yielded by the machine, there is an alternation of the directions into which it is guided ; and the result is a current not uniform in strength, but constant in direction, and useful for the electro- magnet. Two alternators put in series tend to assume opposition of phase and to deliver, jointly, no current; but they will work in parallel. They then go into step, and tend to keep step, co-phasally. xvi.] DYNAMO-ELECTRIC MACHINES. 731 Direct-Current Dynamos. In these the whole cur- rent of the machine is led through a Commutator. The current from a single loop or coil would vary from zero to a maximum and back to zero twice in each revolution ; and as currents, merely in different states of variation of positive or of negative value on one side only of zero, do not tend to neutralise one another when sent into the same wire, but tend, by their summation, to render the aggregate current more uniform in character, the direct-current dynamo generally has its several coils or groups of coils so arranged that each sends its own current into the general circuit, in whatever phase it may happen to be, and is cut out of that circuit only during such time as there may be danger of other coils or groups of coils being short-circuited through it during its own comparatively idle period. The arma- ture, by multiplication of loops or coils lying across the field or towards its periphery, usually takes the form of a drum or of a ring ; and its soft-iron core is laminated, to prevent the forma- tion of eddy-currents. Both in ring and drum armatures, the armature-core tends to become magnetised transversely to the main magnetic field. The actual magnetic field is thus the resultant of the main field and a cross-field. If it had not been for this, the proper posi- tion for the brushes would have been at right angles to the field, so as to lead off the current through those coils which, at the moment, are least engaged in the actual production of cur- rent ; but the effect of the resultant obliquity of the field is that the brushes must also lie obliquely, to an equal extent; and thus, as is said, the brushes must be given a certain lead. The amount of this lead is, further, somewhat increased by the necessity, in order to prevent sparking at the brushes, of letting each loop or coil get a little way into the opposing field before being cut-out ; by which means the extra current is neutralised. This conduces, however, to demagnetisation of the magnetic cir- cuit as a whole. The mean current produced is proportional to the speed, less a certain number'of revolutions per second, called the Dead Turns; and is also proportional to the number of coils in the field and to the strength of that field. At speeds less than the so-called Dead Turns the machine will not deliver any current at all. As to the mode of excitation of the Field Magnets, that is, of the electromagnets which produce the magnetic field within 732 ELECTRICITY AND MAGNETISM. [CHAP. which the armature rotates : these are feebly excited by an ordinary magnet, or there may be sufficient residual magnetism in them to serve the purpose, or they may be feebly magnetic under the induction of the earth's magnetic field; then, when the armature is set in rotation, an extremely feeble current is generated. This feeble current is not permitted at once to pass away, but is sent, either wholly (in "Series " dynamos) or partly (in " Shunt " dynamos, by means of a shunt always kept in action), round the soft-iron magnet, and thereby increases its magnetisation. The soft-iron electromagnet, thus strengthened, induces a still stronger current in the rotating armature ; and thus, the current-intensity attains in a short time a maximum, the potential of which depends upon the speed of rotation and upon the product of the intensity of the current actually pass- ing round the field-magnets into the number of turns which it takes round them, as well as upon the number of turns within the armature, effectively connected in Series. In "Series-Shunt" machines the current is divided into two parts; one part runs in a shunt round the electromagnet : the other runs both round the electromagnet and through the external circuit. In a Series machine, as the Amperes increase, the total voltage in the circuit also increases, rapidly at first, but then more slowly, as the permea- bility of the iron begins to fall off, and the cross magnetic field to become more intense ; and at extreme ampereages, the total voltage even tends to droop away for these reasons. As the ampereage increases, the available voltage at the terminals differs by a steadily-increasing amount from the total voltage in the circuit, because of the increased voltage consumed in the passage of a greater current through the armature : hence, as the cur- rent increases, the available voltage at the terminals reaches a maximum and then falls off. Within this limit, however, when the resistance of the circuit is increased, the electromagnet is enfeebled, and the voltage at the same time falls. In a Shunt machine, on the other hand, as this resistance increases, the tendency is for a larger proportion of the total current to pass through the shunt-winding round the electromagnet, and thus to strengthen it and raise the voltage. When the voltage is so increased, the Amperes rise to a maxi- mum, and then fall off to nothing, while the voltage goes on rising to a maximum, the potential-difference on open circuit. In Series-Shunt machines, the electromagnet is wound with shunt-coils, and the main current is also sent round it. When the resistance is increased, the opposite variations of potential, due to the shunt and to the series-wind- ing, may partly compensate one another ; when they 'are so adjusted that, for a particular speed of running, the machine gives a constant voltage whatever be the resistance, the machine is said to be " Compound-Wound." In another class of machines, there is separate excitation of the electromagnet by particular coils driven on the same axis as the coils supply- ing the general working circuit, or by a separate subsidiary machine. ELECTRICITY AND MAGNETISM. 733 TRANSMISSION OF ENERGY TO A DISTANCE. All current and electromagnetic phenomena are, as we have seen, associated with the transmission of Energy to a distance, across the dielectric. We have already considered the action of Galvanometers ; and also of Ballistic Galvanometers, in which the throw of the needle renders manifest the passage of a very brief current ; just as the position of equilibrium assumed by the needle, as it lies more or less completely across the current, with its axis directed along the lines of force, indicates the persistence of a steady current. As often as a momentary-current is sent round the magnet of a galvanometer, so often will the twitch of the suspended magnet be repeated, and at intervals of time equal to those between the successive momentary-currents. This action which is the simplest form of transmission of energy to a dis- tance, for work is done in displacing the magnet within the field is the basis of telegraphic signalling. Longer and shorter currents produce longer throws and shorter twitches of the galvanometer-needle. These form the basis of a signal alphabet the Morse code. The following is the alphabet, the upper line, where there are two, being the European or " International," the lower the American form : A B c - A D B - F -I" G H - I " j K L M ' N - I~A~- p Q R IT-.'. s -- T U v w -~ x IIIT Y A Z ..TV! A - 6 -- u --- N [CH- -] E- i--ir-_r:-v2v.zi.T7--3: : : = 4- - - 5_imi^ 6 -.-.-.-. 7= = ::- 8= 91 - - I 734 ELECTRICITY AND MAGNETISM. [CHAP. Full stop I I _I_~_I_~_ . Stroke Semicolon (Amer.) Apostrophe Comma I ZZ I ZZ ~ ~ Parenthesis mi ITJZT " Exclamation ZZ ZZ 1I_Z~ Repeat or ? 1_1 7Z ZZ I " Paragraph (Amer.) Hyphen Italics " " (Amer.) & - A (Amer.) In some of the American forms it will be observed that a period of time, represented by A, intervenes in the midst of a set of signals represent- ing one letter. The American form (U.S. and Canada) is the original, as devised by Prof. Morse : the European is an improved version (International Congress, 1851). In submarine telegraphy the signals used are not long and short, but right and left deflections that is, positive and negative momentary-cur- rents. By means of differential galvanometers two messages may be sent along the same telegraphic wire at the same time (Duplex Telegraphy). Station A has a single wire leading from the positive pole of his battery : the current running in this he divides into two moieties, which he sends in opposite directions round the needle of his differential galvanometer : these two moieties are then sent on, the one to the distant station B, the other to A's own earth-plate. In the course of the one or the other branch-current the operator at A interposes resistances until the intensities of the opposed currents round his galvanometer-needle are equal ; then, in whatever way A may make or break circuit, his galvanometer-needle will remain steady; but the needle at B will respond. Similarly, B sends signals to A, to which his own instrument is mute. The two stations may thus signal simulta- neously, two operators being employed at each end, one to transmit, the other to receive ; and the variations of electric condition produced in the single connecting-wire run through one another in a manner analogous to that in which waves meeting on a cord traverse one another. Bridge- Method in Duplex Telegraphy. Suppose a triangle ABC ; the current enters at A ; B is connected with the distant station D ; C is connected to earth through a resistance equal to that of the line BD ; between B and C is the recording instrument of the home station. One moiety of the current which enters at A will run to earth, the other will travel to D. If the resistance in AB be so adjusted as to be equal to that in AC, B and C will be at equal potentials ; no current will run through BC ; the home instrument stands motionless. At the receiving station the apparatus may be precisely similar ; it will then indicate the arrival of sig- nals from A, but will be insensible to the movements of its own key. Quadruplex Telegraphy. A small current always runs in the circuit. There are two transmitting keys. The one reverses the direction of the current ; this causes a needle within a magnetic field at the receiving station to swing to left or right ; an effect which depends upon change of direction of the current within the circuit. The other key, when depressed, introduces a new battery into the circuit; the strength of the current is thereby increased, and the current is now enabled to make a certain soft- xvi.] TELEGRAPHY. 735 iron electromagnet move at the receiving station ; an effect which depends upon the strength, but not upon the direction of the current in the circuit. The one receiving instrument thus records reversals, the other the enhance- ments of current-intensity. Two sets of signals may thus be sent in the same direction at the same time; and this arrangement when duplexed, preferably by the bridge-method above described, becomes quadruplex. This is the ground-principle of Prescott and Edison's system, which is described at length in Prescott's Telephone. The practical details are extremely ingenious ; there may, for instance, be a critical instant at which the intensity-receiver is liable to be interfered with and to fail, through the current supplied to it fading away while being reversed by the reversing- key; a condenser then acts as a reservoir, and its discharge keeps up a current which tides over the critical instant ; a result which is aided by a subsidiary local battery then brought into action by means of a relay. In Multiplex Telegraphy, each operator gets the use of the circuit several times a second ; his signals are like cyclostyle writing, broken up, but practically continuous. When at the distant end of a circuit the conducting wire is passed round a soft-iron core, that soft-iron core becomes an electromagnet just as often, and remains an electromagnet just as long, as the circuit is or remains completed by a key at the home station. This electromagnet may govern the movements of a neighbouring mass of iron, and do work upon it : and the movements of this second mass may be utilised in an endless variety of ways for the repetition of movements similar to those executed at the home station by the hand of the operator, or by any mechanical contrivance adjusted so as to make and break contact in any pre-arranged manner. The mass of iron moved at the distant station may itself, by its movement, make and break a second electric circuit, and may thus control the move- ment of metallic masses at still more distant stations, as in the case of telegraphic relays. Electromagnetic Interrupter for Tuning-Forks. Atuning- fork of known pitch is set in vibration. As it vibrates, it alternately makes and breaks a current which traverses the tuning-fork itself. This current is passed, in its course, round a little electromagnet, which is alternately made and unmade. This electromagnet is so arranged as alternately to attract and release one of the prongs of the tuning-fork, which is thus kept in continuous action. The intermittent current produced is sent round a second electromagnet, which rhythmically attracts and releases a second tuning-fork; this is thus kept vibrating in unison with the first, even although it be not precisely in tune with it. Signalling by Alternating Currents. The Pho-nophore. In this there are two wires, simply coiled together : their farther ends are both free : the nearer end of one is connected with the line-wire. When a brief current is sent, the other wire is acted upon by induction, and signal&rmay be heard in a telephone connected with it and also to earth. This instrument is 736 ELECTRICITY AND MAGNETISM. [CHAP. worked by alternating currents, which do not affect the ordinary telegraphic instruments, and will not pass through their coils, being throttled by them. These signals may thus be made independently of the ordinary direct-cur- rent signals, and both systems may be duplexed. If a current which has 300 maxima of intensity per second, and another of say 800 maxima per second, be sent along the same wire, the conjoined current will present variations of intensity such as might be represented by the curves of Fig. 45. Suppose a current presenting such variations of intensity to be passed round a soft-iron core, near the end of which is a steel reed tuned to vibrate 300 times a second, and so adjusted as to be attracted in the sense of its vibrations when the soft-iron core attains its maximum intensity. The steel reed would, among other impulses to which it would not respond, receive a set of 300 maximum attractions per second, which would set it in vibration. The same current may be also passed round a core, opposite the extremity of which is placed a reed tuned to 800 vibra- tions per second ; that reed will pick out and will respond to the more rapid component of variation of intensity of the current, and will respond to it only. Further, suppose that the several components of variation are each not continuous, but interrupted : the corresponding vibrations of each reed will be similarly interrupted, and one telegraph clerk may be occupied with listening to each. A current whose variation of intensity is as complex as the sum of eight distinct S.H.M.'s may be practically resolved by as many distinct receiving-reeds into distinct signals ; and since the duplex method of working may be applied to this plan, as many as sixteen distinct mes- sages may travel along a single wire at the same time. This is the principle of Mr. Elisha Grey's Harmonic Telegraph. The Telephone, in its simplest form, presents a plate of iron, P, placed in the magnetic field of a magnet, M : the plate is caused, by being spoken at, to enter into certain vibrations ; the vibrating plate P, by induction, acts upon the magnetism of the F . 257 magnet M; the latter is alter- nately strengthened and weak- ened in accordance with the varying position of the vibrating plate : as M varies in strength it causes variations in the strength of a current passing through a coil, C, wound round its pole, or else, if there be no appreciable current passing in that wire, it causes a current to be formed in that wire whose intensity varies continuously on either side of zero-value, being now in the one direction, now in the other. This induced current reproduces in the mode of its variation the complex-harmonic curve which might have been recorded by a delicate writing-point attached to the vibrating plate. The variable current thus produced passes at the receiving station through the coil of a similar telephone. It there causes, by induction, variations in the strength of the xvi.] TELEPHONE. 737 magnet, which attracts the plate with varying degrees of force. That plate is either bent as a mass towards and from the magnet, or its molecules are disturbed by the varying induction : or these actions may be combined ; in any case, the plate exerts varying pressure upon the surrounding air and produces in it Sound- Waves, which approximately reproduce in their complexity the sound-waves produced by the original voice. There are many causes of distortion of the signals sent, both in the instrument and in the line. In the latter, the higher harmonics tend to thin away more rapidly than the graver components, and they are propagated at different speeds: but the resulting distortion can be reduced to a minimum by lowering the line-resistance, and would also be reduced through increas- ing the inductance L by hanging the wires far apart, or through increasing the leakage (Heaviside), though this would weaken the current reaching the receiving instrument, or through reducing the electrostatic capacity of the line. Mr. Heaviside has shown that the distortion would be zero, if R/L = D/C, where D is the leakage-conductance, all per unit of length. It is a matter of indifference to the receiving telephone by what means the variations of current-intensity which it reveals have been produced. These may be due to variations of electromotive D.P. (vibrations of one of the plates of an electrostatic condenser or oscillatory variations in its charge, variations of the potential of a mass of mercury vibrating, while in contact with water, up and down a conical capillary tube), or to varia- tions in the total resistance (length, cross-section, conductivity) of the con- ducting wire. The conductance of the circuit may be caused to vary by squeezing the wire, by causing a certain length of it to vibrate ; or again by interposing a certain length of a conductor whose conductivity varies with varying pressure (microphone) or with varying illumination (photophone). According to Prof. Tait, the variations of current in an ordinary tele- phone are equivalent to actual currents whose intensity is one-thousand millionth part of the current ordinarily used in telegraphic work. This telegraphic current may, on long lines, be stated to be about one-sixtieth Ampere. Page-Effect. A telephone will work feebly even without any plate P; the varying constraint of the particles of the magnet M causes them to exert varying pressure upon the air. If a plate of any substance be con- nected with the extremity of M, that plate will act as a sounding-board, and will enhance the sound produced. Reversed Action. A reverse current of high potential sent through a frictional machine may maintain rotation in it, so that a stronger machine in circuit with a weaker one may drive it backwards. If a dynamo deliver all its current in one direction, an extraneous current sent through the machine in the same sense causes a reversed rotation of its armature. In consequence of this, if we couple two direct-current dynamo- electric machines by connecting wires, so that both dynamos (so-called for the sake of brevity) are on the same metallic cir- 3s 738 ELECTRICITY AND MAGNETISM. [CHAP. cuit, and if we force the armature of the one into rotation, the armature of the other rotates in a reverse sense (that is, against its brushes unless these be reversed or the armature-connections reversed) as soon as the current transmitted attains a certain intensity. The distant dynamo, which bears under such cir- cumstances the name of Electromotor, may be of any size, and the simple use of a key or commutator arrests or reverses its action at will. The intensity of the current passing round the circuit is diminished by the reversed rotation of the electro- motor: this is equivalent to the production of a reverse cur- rent by the electromotor. The usefulness of the arrangement, the proportion of the Energy Absorbed by the electromotor, in rotating against resistances, to the Total Energy imparted by water-wheel or steam-engine to the driving dynamo, is equal to the ratio between the intensity of the virtual reverse-current produced by the electromotor, while running, and the intensity of the current produced by the dynamo when the motor is kept from rotating. This Utility or Efficiency is not to be measured by the relative rapidities of rotation of the electro- motor and dynamo, on account of the so-called dead turns; the rotation of the dynamo must exceed a certain speed before any current will be produced, and the current produced must exceed a certain strength before the electromotor will turn. The Activity of the arrangement (i.e. the rate at which the electromotor can do external work, the amount of energy trans- mitted per second) is theoretically greatest ( Jacobi's Law) when the virtual reverse-current is half that produced by the dynamo when the motor is stopped that is, when the exterior circuit, including the running motor, acts as if it were a wire-circuit whose resistance is equal to that within the dynamo itself. In practice it is better to make the whole resistance about *- times the internal resistance. Jacobi's Law is arrived at thus: During each second the external work done is w ergs ; w also represents numerically the Activity in ques- tion ; the energy converted into heat in the whole circuit is PR, and the energy provided by the dynamo is El, ergs per sec. Then El = I 2 R + w ; a quadratic ; whence I = (E \/E 2 - 4Rw)/2R. The quantity (E 2 - 4Rw) cannot have, physically, a negative sign, for its square root would then become an impossible quantity. (E 2 4Rw) cannot be less than zero : whence w cannot be greater than E 2 /4R: when it is equal to E 2 /4R, I = E/2R, and the intensity has been diminished from E/R to E/2R, that is, to one half, while the total resistance must have been doubled. The Efficiency-relation may be thus arrived at: If the dynamo run while the motor is held fast, the E.M.D.P. and current-intensity will be xvi.] ELECTROMOTORS. 739 E D and I D . If the motor were to run at its actual speed while the dynamo was held fast, the reverse-current produced by it would be at E M and I M . When the two are coupled, the actual current is (I D I M ) : the energy supplied by the dynamo during each second is w ergs ; that taken up and transmitted by the motor is M? M : the resistance of the whole circuit is R : and (T D - I M ) 2 R is the Heat developed in the whole circuit : then the Energy supplied by the dynamo is W D = E D (I D - I M ) = E D (E D - E M ) /R = {, + (I D - I M )2R) = (w M + (E D - E M ) / R} ; whence the Efficiency W M /W = E M / E D = I M / I D . The ratio E M /E D , and therefore the Efficiency, may be raised by raising the value of E M : and this may be done by giving the motor a small load, so that it may rotate rapidly, or by making its magnetic field a comparatively strong one ; so that efficiencies of 86 per cent have been attained at such distances as 600 metres (f mile) ; 44-8 per cent at 36 miles (Creil-Paris), with 6000 Volts. As to the thickness of conducting wire necessary, there is no limit other than that imposed by the necessity of very good insulation. An ordinary telegraph wire could convey the whole energy of Niagara Falls, and convey it to any distance ; but the wire would be at so high a potential that sparks would fly from it into the surrounding air. In the same way, if the amount of onflow of a fluid in a pipe were found to vary directly as the motive difference of pressure, any amount of energy might be transferred from one place to another by the smallest flow of water, for any water allowed to flow out of the pipe might be made to escape with any assignable velocity ; provided always that the tube were strong enough at all points to sustain at all intermediate points the necessary pressure. If a dynamo of resistance 5 Ohms, and producing a difference of poten- tial of 1000 Volts, be the source, and a similar machine be the electromotor, while the connecting wire offers a resistance of R Ohms, the intensity of the current produced is / - j Amperes. If 500 such dynamos be coupled in file, their joint E.D.P. will be 500,000 Volts, and their resistances 2500 Ohms ; if the receiving electromotors be also multiplied five-hundredfold, their resistances will be 2500 Ohms ; if the connecting wire be unaltered, the 500 000 intensity of the current passing will be ' Amperes ; but if the connecting wire be also 500 times as long as at first, the intensity is _ 500 ' 000 _ = ( 100 } Amperes, the same as in the former case. 2500 + 2500 + 500R (lO + R / Though the intensity of the current passing is the same, the energy trans- mitted per second is not the same : it is 500 times as great. In the former case it is Intensity x E.D.P. = ( * 000 *} x 1000 Ampere-Volts or Watts: in \ 10 -4- J*/ the latter it is -1M-. Amperes x 500,000 Volts = 500 ' 00 Q 00 Watts. 10 + R iy + J* p When the total resistance is the internal resistance R f , I => ' ; when it is F ? f (R f -j- R e ), I, = =3 These two distinct sets of circumstances are linked 740 ELECTRICITY AND MAGNETISM. [CHAP. together by (1) The criterion of maximum activity, R t - +e = -W Rfl and (2) The energy imparted to the dynamos is a constant, = EJ, = w ergs per sec- ond. From these equations we find I,= V T % 8 ff w/R i? and E ; = Ri+el, = (-^YR-O (V^w/R^m V -Vg - RiW = VR He w. We must now choose numerical values for Ri +e , the total resistance internal and external, and w, the energy im- parted to the system per second. We shall use C.G.S. Electrostatic units. Let the resistance be that of 4000 kilometres of copper wire of 1 sq. cm. in cross-section, and that of x dynamos and x electromotors. The dynamos are each supposed to generate an E.D.P. of 1000 Volts, or 3 C.G.S. Electro- static units. Their number must be (E / -=- 3i). Let the joint resistance of each dynamo and motor be 10 Ohms, or 9oo.ooo.oooooo C.G.S. Electrostatic unit of resistance. The resistance of the (E,-*-3&) pairs of machines will be {(E, -s- 3J) x 90 o,ooo.oooooi)}> or L C.G.S. Electrostatic units. 300,000,000000 The wire (4000 kilometres) will offer a resistance of about 648 Ohms, or Electrostatic C.G.S. units. 3 oo,ooo.oooooo The total resistance R t+e = goo.ooo'.oooooo * E / + 216 } C.G.S.E.S. units. Let w be the energy of the Falls of Niagara, per second, also in C.G.S. units or ergs. About 100,000000,000000 grammes of water fall per hour through a height of about 4830 cm. The potential energy lost by the water is about 132,000,000000,000000 ergs per second = w. The equation E, = VR f+e w; is now E, - V airo>000 i 000000 {, + 216} x 132,000,000000,000000, a quadratic ; whence E, = 440,020 C.G.S.E.S. units or 132,006000 Volts. If iron telegraphic wire 4 mm. in diar. were used, its resistance would be (the resistance of iron being fff that of copper) 31680 Ohms; the total resistance would be , 00 .ooo 1 .oooooo (E, + 10560); and E, = 450,320 C.G.S.E.S. units, or 135,096000 Volts. No practicable insulation could be set up, adequate to sustain perma- nently so great a stress ; and no possible dynamo-electric machines could be ranged in file to the number necessary, for the insulation of their coils would be broken down by sparks from the wire to the outer air. It is practicable, however, with ordinary telegraphic wires insulated in the ordinary way, and with a 16-horse-power dynamo, to drive a 6-horse power electromotor at a distance of 30 miles. The wire must also, by possessing sufficient thickness, offer so little resistance that it is not so far heated as to deteriorate in conductivity. Alternating Current Motors. If two alternating current machines be coupled in circuit, and if they be once in synchro- nous motion, they will tend to assume and to maintain uniform- ity of phase ; while if the mechanical load on the motor be increased, within appropriate limits, the motor will present a less complete opposition of phase, and a stronger current will run, so that the mechanical forces upon the motor-armature will be correspondingly increased. The magnetic field of such a motor must be kept constant in its direction. The practical difficulty connected with such motors is that they are not self- starting. xvi.] ELECTROMOTORS. 741 At pages 90-91 we learned that two S.H.M.'s, differing by | period, pro- duce an ellipse. Similarly if we have two S.-H.-varying currents, each tend- ing to produce an alternating magnetic field in its own particular direction, but differing from one another in phase, the result will be a continuous rotation of the direction of the resultant magnetic field. In such a field any mass of metal will tend to rotate ; and this is the basis of rotary-field alternating current motors (Tesla, Dobrowolski). In Schallenberger's alter- nate-current meter, a vertical coil receives the alternating current to be measured ; within it is fixed another vertical coil, closed, and standing at an adjustable angle with the preceding. Inside the latter is a soft-iron disc, horizontal, pivoted on a vertical axis of rotation. The outer coil tends, alternating! y, to magnetise the disc along a certain line ; the inner presents induced currents, nearly opposite in phase, which tend to magnetise the disc, alternatingly, along another line, which makes an angle with the pre- ceding : the joint action of the two coils (or sets of coils) on the disc is to magnetise it in a direction which itself continuously rotates; the disc tends to rotate with a velocity proportional to the square of the current-strength. The tendency is for the disc to maintain a position in which the retarding eddy-currents in the iron are a minimum. In such apparatus the result is, as regards the intensity of the induced magnetisation (which tends, if the component inducing forces be not equal and at right angles to one another, and at phases differing by exactly ^ period, to present maxima and minima), more uniform if three or more S.H. variations be simultaneously induced in directions making equal angles with each other, as in Brown's three- phase alternating-current motor. Multiple-phase motors are self -starting, and gain in speed until synchronism is attained. The Lauffen-Frankfurt experiments of 1891 gave an efficiency of 72 per cent in the transmission of 108 horse-power over 110 miles, at 30,000 Volts. OSCILLATORY ELECTROMAGNETIC DISTURBANCES IN FREE ETHER. Herz's Experiments. Two metallic plates, each say 16 cm. square, are suspended in the same plane, and are connected each with one terminal of an induction-coil. They are also almost con- nected with one another by means of wires terminated at their free extremities by polished knobs, between which there is a small air-gap. When the induction-coil is at work, a stream of sparks runs across this air-gap, from knob to knob. The electric displacements in the region of the spark are of an oscillatory character, and are parallel to the length of the air-gap, from knob to knob. The lines of force are accordingly parallel to the length of that gap. This apparatus is called Herz's Vibrator. Next we have his Resonator, which is a single circle of wire, broken by an air-gap between two knobs. This resonator has its own fundamental period of electric oscillation. The vibrator sends out a mixture of oscillatory ether-disturbances'of various 742 ELECTRICITY AND MAGNETISM. [CHAP. periods. Now assume that the vibrator air-gap, in its length, runs East and West, horizontally. Lay the resonator hori- zontally, with its air-gap also East and West, and facing the vibrator centrally. Then sparks pass in the resonator air-gap. Sparks will continue to pass in this air-gap, though with a diminished striking distance, when the resonator is turned round in its own plane into any position, so long as it is kept horizontal. Now turn the resonator into a vertical plane, which plane lies parallel to the length of the vibrator air-gap, that is, East and West; some sparks will pass should the resonator air-gap be also parallel to the vibrator air-gap, but when it is at right angles to the same no sparks will pass. Again, turn the resonator round so that its vertical plane lies North and South, or at right angles to the length of the vibrator air-gap; no sparks will pass at all, whatever be the direction of its air-gap. It is (J. J. Thomson) as if the lines of electric force in the Leyden-jar discharge through the vibrator air-gap were parallel to the length of that gap ; and as if when travel- ling broadside-on, outwards from that gap, they produced some sparks when they struck the resonator air-gap in such a way that their length coincided with its length, and produced a maximum effect when they struck the wire of the resonator longitudinally, so that their lengths coincide with its length, while at the same time the magnetic induction was directed along the axis of the resonator. In the last case there would be reflexion, from end to end of the resonator, of these lines of force ; and these lines of force would oscillate to-and-f ro along that resonator, with a result analogous to Resonance in Acous- tics. Those disturbances, radiated from the vibrator, which had been in tune with the resonator would be taken up and piled up by it, until sparks passed in the resonator air-gap, or until a Geissler-tube held in or near that gap would light up. By this means the Resonator can be used to detect the existence and the direction of electric oscillatory disturbances in the Ether, such as have periods corresponding to the rate of propagation of electric disturbance along the wire of the reso- nator, and to the length of its wire ; but the resonator will respond to disturbances of a considerable range of frequencies above and below this limit. The waves produced by this electric method traverse brick walls with ease, but they are reflected by metallic mirrors. If the resonator be used to explore the path of the reflected waves, xvi.] HERZ'S EXPERIMENTS. 743 it is found that there is interference between the direct and the reflected waves, exactly as in the case of Sound or of Light; the resonator gives maxima of sparking or of illumination at distances equal to half a wave-length from one another ; and the wave-length, thus determined, is consistent with the velocity of Light, together with the frequency of vibration as calculated from the dimensions of the resonator. If a large prism of pitch be employed, it is found that the waves are refracted by the pitch ; and a large lens of pitch acts in an analogous way. If a vibrator-gap be adjusted in the focus of a mirror which consists of a sheet of metal bent into a parabolic form, then a suitable resonator-gap, placed in the focus of a similar mirror opposite to the first, may give sparks. These waves, like Light polarised at right angles to the plane of incidence and reflexion, fail at a particular angle of incidence to be reflected from a metal mirror, provided that the vibrator air-gap be in the plane of incidence. If therefore these electromagnetic waves are like waves of Light or Radiant Heat, or Actinic Radiation, and differ from these in wave-length only, the electric oscillations, as distinguished from the magnetic inductions, are at right angles to the " Plane of Polarisation." The leading phenomena of Light including Reflexion, Refraction, the Angle of Polarisation, and Polarisa- tion itself, Scattering by Haze, and, to a large extent, Metallic Reflexion and the change of phase on transmission through thin films of metal, along with Newton's Rings and the black region of a thin soap-bubble, as well as Diffraction have been imitated on the large scale by means of these electro- magnetic waves ; while the effect of the Magnetic Field on Light can be largely explained if it be admitted that Light con- sists of such Electromagnetic Waves of small wave-length, in which the electric oscillations, at right angles to the direction of propagation, are also at right angles to the plane of polari- sation, while the magnetic inductions are in that plane. The field in the immediate neighbourhood of the vibrator is in a singular condition of alternate withdrawal and emergence of Lines of Force : there are various peculiarities in the amount of the electric force and in the velocity of propagation at that place ; but the upshot is, that after inter- ference between lines passing outward and lines re-absorbed has had full swing, the ether-waves emerge as if from wave-centres about half a wave- length from the vibrator-gap, and the electric forces are thereafter exactly at right angles to the direction of propagation, and vary inversely as the square of the distance. If an electrostatically-charged body could be whirled roun^a magnetic needle at the rate of 30,000,000000 cm. per second, it ought to act upon it 744 ELECTRICITY AND MAGNETISM. [CHAP. much in the same way as a circulating electric current. At very high speeds, such as are physically within our reach, such an effect should be observed in small degree; and Prof. Rowland of Baltimore has succeeded in making it manifest. Maxwell's Theory of Light. These results, obtained by Herz and others, furnish a verification of Clerk Maxwell's Theory of Light. According to this theory, the Electric Displacements, parallel to the wave-front, and at right angles to the plane of polarisation, are the cause of Optical Phenomena. The Magnetic Inductions or Displacements, at right angles to the preceding, and parallel to the plane of polarisation, but also parallel to the wave-front, produce no effect on the eye. The electric displacements will be propagated, in an seolotropic medium, in the same way and with the same velocity as a light-wave; and the magnetic dis- turbance, whose energy is equal to that of the electric, is propagated with the same velocity. The intensity of Light, its average energy per cub. cm., is 27r/xo- 2 v 2 , where v is the velocity of propagation, cr the maximum electric displacement per sq. cm. at the ends of the lines of electric force, and p the permeability of the medium. In a doubly-refracting medium, the electric displacement and the magnetic induction are propagated according to Fresnel's wave-surface. There is no dilatational wave possible : and this removes a difficulty in optical theories. Since in non-conductors an electric Displacement produces an Electric Restitution-Force which varies as the Displacement a criterion of vibratory movement propagated with a definite velocity ; but in conductors no such force is manifest, and the energy of electric disturbance is continuously dissipated by transformation into Heat : then Light-vibrations ought not to be possible in Conductors, which should be always opaque, while non- conductors ought, if homogeneous, to be transparent. With few exceptions this is the rule. The velocity of propagation of an electromagnetic disturb- ance is the same as the ratio of the electromagnetic to the electrostatic unit of current-intensity or quantity. This ratio is experimentally found to be, on the C.G.S. system, in round numbers, equal to 30,000,000000 ; which, in cm. per second, coincides sufficiently with the velocity of light. On comparing the formulae for the transverse variations of an elastic solid with those worked out to represent the stresses in an Ether concerned in electromagnetic phenomena, it is found (Clerk Maxwell, Elect, and Magn., vol. ii., chap, xx.) that in the former a term V, the velocity of propagation of transverse disturbances, occupies the same place as 1/VK in the latter. By our electrostatic convention, in vacuo K = 1 and /x = l/V 2 ; .-. V = (K/x)"*; but electromagnetically, K = 1/V 2 and /A = 1 ; whence V = K~* = V; i.e. the velocity in vacuo is equal to the ratio-number V., With other media than air, (K/z)~5 has other values ; but /A is nearly unity in most non- conductors; roughly, V = Vl/K; but in Light, v = V, and varies inversely as the Refractive Index; whence the Specific Inductive Capacity K of a dielectric ought (Maxwell) to be equal to /3 2 , where fi is the refractive index xvi.] MAXWELL'S THEORY OF LIGHT. 745 for waves of infinite length.* In some substances this is the case; it is so in sulphur (Romich, Nowak, Boltzmann), and in turpentine, petroleum, and benzol (Silow) ; but in vegetable and animal oils (Hopkinson) and in glass, Iceland spar, fluor spar, and quartz (Romich and Nowak), and generally in substances not simple in chemical constitution or homogeneous in structure, the sp. ind. cap. is too great. We ought not, however, to ex- pect more than a general agreement ; even the oscillations produced by a Leyden-jar discharge are millions of times less frequent than those electro- magnetic alternations which we call Light ; and it appears that the sp. ind. caps, for these already approach the values required by the theory. The theory explains most optical phenomena as a part of electrical science ; but it is still weak in its treatment of Dispersion, both ordinary and anomalous, of Metallic Reflexion, and of the rotation of the plane of polarisation by magnets. It does not, however, profess yet to explain the interaction of Ether and ordinary Matter. THE ETHER. The properties of the Ether which are involved in the phe- nomena of Electricity, Magnetism, Electromagnetism, Light, Radiant Heat, and Actinic Radiation have been referred to under these several heads, where required. We can now merely note a few remaining points. In terms of Maxwell's Theory, it is sup- posed that there is some kind of Rotation round each Line of Force, upon which rotation the Elasticity of the Ether may depend : and the facts of Electrolysis or of the Galvanic Cell, in which the charge that can be liberated upon an electrode or plate is limited to a definite quantity per free atom, seem to show that the Lines of Force are not indefinite but definite in number, so that we may perhaps have (J. J. Thomson) one line of force between each pair of free atoms. But these lines of force, on this view, need not all have free ends situated upon matter: there may be, in the Ether, closed lines, like vortex- rings : and these may, in a Magnetic Field, be so directed as to take up a position parallel to one another. Again, if Ether be a form of Matter, it ought to have Iner- tia. In Self-induction it appears to have inertia ; but in a Steady Current it appears to have absolutely none. The only way, and mathematically an easy way, out of the difficulty seems to be to assume that the Ether is really double in its consti- tution, so that when we say, for example, that positive lines of *If j3, j3,, be the refractive indices corresponding to the respective wave-lengths, X and X,, we know that to a rough approximation ]8 = A + B/X, where A and B are constants, found by experiment. From these, knowing the numerical values of j8, j8 ( , X, X,, we can find that of A, which is the approximate value of j3 when X = . 746 ELECTRICITY AND MAGNETISM. [CHAP, xvi.] force move in a certain direction, we ought to add that an equal number of negative lines move in an opposite direction, shearing past the former: the apparent inertia would then be zero in a steady current. This would make a steady current really a double current, positive one way and negative the other ; but in Electrolysis we actually have a double transfer of atoms, to which this would precisely correspond ; and the phenomena of the electric spark point in the same direction. The sp. ind. cap. of a dielectric would, on this view (Lodge), correspond to a Shearability, while the permeability p at the same time represented an Inertia or Density of the Ether. Mr. W. Williams has shown (Proc. Phys. Soc. Lond., xi. 357) that we are practically restricted either to this view or to the opposite, that K is a Density, or inertia per unit volume, and /x a Shearability. On the former view, the magnetic energy of the field is kinetic : and if we take care to keep in view the proper Direction of each element of Length which enters into the Dimensional Equations, these equations themselves bring out the closest analogies between magnetic and electric phenomena and those of vortex- motion and transverse stresses in an incompressible fluid. The Dimensions of fj. in this view would be M/L 8 , where the three L's are at right angles to one another, so that their product truly represents a volume ; and that of K would be T 2 L/M or more properly T 2 L 2 /LM; where the three directions have again to be distinguished. Such an expression as [h] [Mi/LsT/n*] then becomes [h] = [L/T], and h is a Linear Velocity along the Lines of Force ; and so forth. This tends to elucidate the purely mechanical aspect of magnetic and electrical phenomena in the Ether : and the paper should be consulted. But the subject is still more obscure when we consider the relation of Ether to ordinary Matter. Why the magnetic in- duction under a given magnetising force should be 300 times as great in a particular sample of iron as in a corresponding amount of air or of copper is still a mystery ; and even if it were estab- lished that the density of the Ether was greater in iron, that would itself have to be explained. Meantime, it will be kept in view that all our statements as to positive and negative quantities and currents are based upon the purely arbitrary convention that vitreous electricity is positive, and resinous electricity negative. This convention happens to harmonise with that which regards the north- seeking end of a magnet as its positive pole; and thus uniformity of language throughout the subject-matter of this chapter happens to have been readily attained. APPENDIX. Notation. In adjusting the notation used in this volume, the pur- poses kept in view have been to provide, as far as might be, for the whole subject-matter of the book, but to depart as little as possible from the symbols ordinarily in use, while the letters employed should be distinctive and at the same time typographically suitable. One guiding principle has been to separate physical quantities in general from the same quantities per unit of area, where such a distinction seemed needful, by using, in order to represent these, capital and small letters respectively. Then Mr. Oliver Heaviside's suggestion as to the use of blackf aced type for directed quantities was found to promise to work well, and to enable useful distinctions to be made. Thus F is a force acting, in general; f is a force per unit of area; and f is a force acting per unit of area in some given direction. Again, B is a total magnetic induction, b is a magnetic induction per sq. cm.; and both these, being blackfaced, call to mind the Lines or directions in which the induction acts. Other typographical devices have had to be employed for the sake mainly of obtaining a larger number of distinctive characters; but it is hoped that none of these will offend the eye, and that the grouping is reasonably consistent. Perfect symmetry seems hardly attainable, on account of the varying demand for the different letters in different parts of the subject. Still, even with the notation as it stands, it has been interest- ing to the author to note in how many instances the mere necessity of ascertaining which symbol ought to be employed has enabled him to set matters forth with greater definiteness than in the former editions. In its issue of Aug. 25, 1894, the Electrical World of New York has published the recommendations of the Committee on Notation of the Chamber of Delegates of the International Electrical Congress, Chicago, 1893. These recommendations as to notation are at present the subject of a good deal of discussion, and it remains to be seen to what extent they will be generally adopted. In the meantime they are recommendations only, and will have to ,be further considered when an International Electrical Congress next takes place ; but the following conspectus shows their rela- tions to the notation employed in this volume. It is not explained what distinctions the respective large and small letters denote, if any; the same Dimensions apply to both the large and the small letters. The measurement in the fourth and sixth columns is in electromagnetic, not in electrostatic, units. The manuscript type in the last column is that known as the French Script of Messrs. Damon & Peets, New York. 747 748 APPENDIX. THIS VOLUME. COMMITTEE. THIS VOLUME. COMMITTEE. THIS VOLUME. COMMITTEE. I , g Q,q tn m m ir e E, e for so-called h 5C t r,< E.M.F. H (P A #, s J7,M for difference of potential m 3E SB v it 3 Angle 5, 0, 0,0 (C) i (7, c I,* b V V (T>} 6? K /c H 0) * ' ' ^ /* A (T\*\ 7 d $ ' ' L z, z F F,f (R) , r Magnetomotive - f - J ? ' W W (R) P Force ff Activity Power, P W Activity TF Power, P Reluctance of Circuit ' (R P P N K Reluctivity " =1/ " Dimensions. In the Dimensions given in this volume from time to time, it will be observed that, for example, a Torque has Dimensions ML 2 /T 2 , while those of Energy are the same. There are other instances of the same kind. So long as we use the Dimensions only for checking our equations numerically, or for translating from one system of units to another, this identity of expression between physical quantities which, like the two in question, are truly dissimilar, is of no importance ; but if we wish the Dimensional Equations to convey to us an idea as to the physical real- ity lying behind them, we must find some means of introducing into them a representation of the Directions involved. Now the symbol V 1 signifies a Rotation through 90, for the operation which it represents would, if effected twice, convert a directed quantity x into x, that is, would turn its direction round through 180 ; and it has been proposed to distinguish the dimensions of a Torque from those of Energy or Work, by introducing the factor V 1 into it. The expressions would then be ML/T 2 - LA/ 1 and ML 2 /T 2 respectively ; and the former of these would show that the second L was at right angles to the first, while in the other case both the L's are in the same direction. Mr. Williams, in the paper referred to on p. 746, has shown that this idea is capable of great extension by means of keeping the three rectangular axes of direction, X, Y, and Z, entirely separate, so that the corresponding L's do not cancel on division or multiplication unless they be in the same direction. The Dimensional Equations thus acquire a deeper significance and an enhanced utility. 4ir. In the formulae of Chap, xvi., the factor 4?r, or some multiple or sub-multiple or power thereof, appears with painful -frequency. Mr. Oliver Heaviside has pointed out that this is due to the total flux of force, or rather of induction, I or B, round a quantity Q or m, being taken as equal to 4?rQ or 47rin, as the case may be. This is itself a necessary consequence of so choos- ing our units that the force, F dynes, between two equal quantities Q (or m) is Q 2 /Kd 2 (or m 2 //^ 2 )- Mr - Heaviside proposes that we should, while APPENDIX. 749 retaining the dyne as our unit of force, so alter our units of electrical and magnetic quantity as to make the force F = Q 2 /47rKd 2 or m 2 /47r/W 2 . In other words, he proposes to make the units such that not 4?rQ or 47rtn, but Q or tn lines radiate from each quantity Q or m, as measured in the new units. Then the new numerical jralue Q r or tn^ which stands in the place of the old Q or m, is equal to Q V4?r or to m V4^, as the case may be. Whence the new units, which Mr. Heaviside calls rational units, of quantity ai*e smaller than the present air-units in the ratio of 1 to V^TT. If this change were effected, most of the units used in electricity and magnetism would have to be changed at the same time. If we make the suffix r signify that the number indicated by the letter is now to be a number expressing the same physical quantity in terms of the new units, we find that Q r /Q = ov//j, r = V4TT ; uir/m - h/h r = b/b r = Sr / = G/a = Vi^ ; and further, R/R r - C r /C = L/L r = M/M r = 4?r; while K and /x remain the same. The fundamental electrostatic equations would then become <|> r = or r , i r = K<|> r , /= o> 2 /2K, r = E r /d; or, dropping the suffixes and con- fining ourselves to air, $ = o- = i ; /= cr 2 /2 ; <|> = E/rf; whence V, V y/ = E /= E 2 /2 2 /2 = o- 2 /2 = i 2 /2 = 4r/2 = icr/2 = <|>i/2 = Energy of Field per cub. cm. ; and C = KA /d, or, for a unit cube condenser, C = K, while the Dielectric Elasticity = 1/K. In the magnetic and electromagnetic parts of the subject there would be corresponding simplifications. The factor 4?r would, however, make its appearance in other formulae where it does not at present occur, but only where the conditions of the problem are truly spheri- cal and not merely superficial or linear ; for example, instead of b = tn/r 2 per sq. cm. at a distance r from a central pole m, we would have b r nv/47rr 2 = B r /area, which is a better representation of the fact. This change of the fundamental units of electrical and magnetic measurement would involve the study of an additional system of units. But it is not clear that, admitting the reasoning, the proposal goes far enough. The relation be- tween the Unit of Mass and the Unit of Force is equally based upon neglect of the central nature of gravitational forces, and of the Field of Force, with its Lines of Force and Equipotential Surfaces, round an attracting mass. If the same reasoning were here applied, either the unit of mass or the unit of force, or both, would have to be changed. If the gramme were retained as our unit of mass, and if the Force of Gravitation between two equal masses, m grammes each, at distance d, were written G s = ra 2 /47rfZ 2 , the new unit of force would be equal to 4?ry dynes. Then, between two equal electrical quan- tities Q, as measured in the present C.G.S. units, the force would be F R ( = F / 47ry) = Q B 2 / 4fl-Krf 2 ; and Q R 2 = F R 4arKd* = F/ 4-jry x 4rrK 640; Ar- rangement of, 640 ; Discharge of,, 632 ,- E.M.D.P. of measured, 646 ; internal Resistance of measured, 647 ; standard , 622, 646 Centigrade Thermometer, 402 Centimetre, 11 Centre of Figure or Mass, 73, 146, 158, 165, 206, 446 ; of Gravity, 74, 206 ; of Inertia, 146 ; Optical , 534, 541; of Oscillation, 164, 214; of Percussion, 165, 214 ; of Suspen- sion, 165, 214 "Centrifugal Force," 165-169 C.G.S. (Centimetre-Gramme-Second) or Absolute System, 13 et passim Change of Direction, 18, 68 Change of Motion, 6 Change of State, 235, 354-358 Change of Velocity, 18, 68 "Character" of Sound, 415 Charge, 52, 578, 591 ; Division of, 592 ; ''free" and "bound," 591, 595, 601, 630 ; Static Charge of Conductor, 579, 600 ; on Ellipsoid, 580 ; on point, 580 ; do. in motion, 743. Charging a Condenser, 648 Charles's Law, 251, 253 Chemical Action as a source of Elec- trical Energy, 611, 614; Affinity, 52, 256, 328, 354, 356, 358, 614, 615 ; Analysis, 21 7 ; Atoms, 238-245 ; Combination, 236 ; Decomposi- tion by Light, 481, 482, 486, 488, 503 569 ; by Radiant Heat, 482 ; Effect of Electric Current, 657-667 ; Ele- ment, 217, 219, 494 ; Energy, 356 ; Equivalence, 220, 239 ; Equiv- alents, 52, 239 ; - - Forces, 355 ; Formulae, 239, 240, 244 ; Rays, 481 ; _ Work, 355, 359 Chemistry, 1, 217, 219, 24* Cheval-vapeur, 42, 647 760 INDEX. Chevreul's Black, 576 Chimney roaring, 477 Chirp of insects, 414 Chlorophane, 507 Chlorophyll, 49, 482, 499, 504, 505, 569. Chromatic Aberration, 541, 571, 573. Cigar smoke, 503 Circle, Osculating, 79 Circle of Reference in S.H.M., 81, 82, 185 Circuit, Electric, 613-616, 639, 643; open, 613, 614, 617 ; closed, 613, 643 ; Positive side of, 672 Circuit, Magnetic, 691, 725, 730, 731 Circuits, Primary and Secondary, 700, 724 Circuit, Thermo-electric, 624, 651 Circular Motion, 8, 57, 78, 161-169; = 2 S.H.M.'s, 87, 101 ; Friction in, 185 Circular Polarisation, 514, 560, 562, 564 ; detected, 563 Clamond's Thermo-electric Pile, 629 Clarionet, 451 Clepsydra, 34 Clerk Maxwell ; see Maxwell Clock, 8 ; Energy in, 45 ; Pendulum, 9, 34, 212, 380 ; Wheelwork in, 34 Closed Galvanic Circuit, 613, 643; closed Magnetic Circuit, 691 Cloud, 203 Cobbler's Wax, 226 Cochlea, 468 Coefficient of Absorption of Gases, 328 ; of Cubical Compressibility, 259, 301 ; of Linear do., 262 ; of Electric Con- ductivity, 634; s of Thermal do., 407 ; of Thermometric do., 407, 408 ; of Diffusibility, 283 ; of Elas- ticity, 264, 322 ; of Elastic Restitution, 264 ; of Electrostatic Induction, 600, 604 ; of Expansion by Heat, 378, 380 ; of Extensibility, 260 ; of Induced Magnetisations, 686, 693; of Inertia, 147, 166; of Kinetic Friction, 179; of Statical Friction, 176 ; of Magnetic Induction, 685, 692, 694 ; of Mutual Induction of Charged Bodies, 600, of Currents, 658, 704 ; of Resistance to Compression, 259, to Extension, 261, to Shear, 260, to Twist, 263 ; of Res- titution, in Elasticity, 264, in Impact, 151 ; of Rigidity, 226, 260 ; of Self- induction, 705, 710; of Solubility, 280, 328 ; of Transmission of Light, etc., 499 ; of Transpiration, 331 ; of Viscosity, 227, 307, 316; Kine- matical - of do., 228 Coercitive Force, 684, 690, 691 Cohesion, 256 ; solids, 257 ; liquids, 254, 270, 345 Cohesion figures, 279 Coil: Induction , 646,656,706, 725; Resistance-, 634, 646 Collecting Rings, 729, 730 Collodion Films, 273 Colloids, 284 Colour, 281, 483-488, 501; by trans- mitted light, 498 ; by reflected light, 501 ; produced by interposed doubly- refracting lamina, 559; Complemen- tary s, 487, 563, 574 Colour-Blindness, 575 Coloured Light : Simple, 483 ; Com- pound, 486 Colours : Analysis of, 486 ; Matching of, 575 ; Mixture of, 574 ; Primary , 575 Colours of Metals, 502, 503 Colours of Thin Films, 545 Combinational Tones, 473 Combining Proportions, 239 Combustion-equivalent, 356 Combustion, Heat of, 49, 236, 354, 357. Comet, 80, 203 Comma, 423, 424 Commensurable, Commensurate, 93, 103 Common Light, 515, 563 Communicating Vases, 293 Commutator, 730, 738 Compensation-Method of Comparison of Electrostatic Capacities, 720 Compensation-Pendulum, 380 Compensator in Saccharimeter, 568 ; Babinet's Compensator, 564 Complementary Colours, 487, 563, 564, 565, 574, 575 Complementary Distribution of Elec- tricity, 580, 591, 592 Complex HarmoniCjMotion ; see Fourier- motion, 103 Complex Sound- Waves, 432 Components of a Force, 143 ; of a Fourier-motion, 416 ; of a Velocity, 60, 63 Component Tones of a Note, 417, 429 Composition of Forces, 143; of Rota- tions, 74; of S.H.M.'s, 86-103; of Transversal Vibrations, 107 ; of Uni- form with Accelerated Motion, 71 ; of two Velocities, 60 ; of more than two, 65 Compound Coloured Light, 486 Compound Pendulum, 213 Compound Winding, 732 Compressibility, Cubical Solids, 259; gases, 325 ; water, 462 ; coefficient of , 259 Compressibility, Linear, 262 ; coeffi- cient, 262 Compression, adiabatic, 324, 358, 373 Concert Pitch, 421 Concertina, 412, 442, 451 Condensation of gases, 231, 375, 377; condensation-temperature, 392 INDEX. 761 Condenser, 596, 630 ; charging a , 648 ; discharge of a , 632 ; Sliding , 600 ; Standard s, 609, 718 ; s in Submarine Telegraphy, 698 ; vibra- tion of, 737 Condenser and Source: in Galvanic Cell, 615, 648 ; in Heat-engine, 384, 396 ; in Thermodynamic Circuit, 652 Conductance, 633, 634, 638,644; " a Velocity," 638 Conduction-Current, 586 Conduction of Electricity, 588, 603, 692 ; Surface , 696, 721 ; Electrolytic- , 588, 622 Conduction of Heat, 406-410 ; 602, 692, 710, 740 ; in crystals, 409 ; in gases, 236, 251, 409 Conductivity, Electrical^ 281, 634, 635, 638; of electrolytes, 646, 658, 722; Variable, 636, 737 ; Magnetic, 686 Conductivity, Thermal, 407, 602, 636 ; three coefficients of , 497 ; Ther- mometric , 407, 408 Conductor of Frictional Electric Ma- chine, 610 Conductors of Electricity, 588, 603 ; kinds of, 590 ; currents in homoge- neous , 638 ; in heterogeneous , 639 ; in wide , 643 ; Charge of , 579, 600. Conductors of Heat, 406 Conical Pendulum, 80 ; its Energy, 142 ; in Viscous Medium, 185 Conical Kefraction, 557 Conjugate points, 130, 526, 535 Conservation of Electricity, 581 Conservation of Energy, 7, 8, 47, 52, 127, 144, 230, 517 et passim " Conservation of Force," 52, 144, 230, 292 Conservation of Vires vivce, 517 Conservative System, 44, 186, 255 Constancy of Nature, 2 Constitution of Matter, 238 et seq. Contact-Effect, true, 612, 616, 624,627, 649 Contact of Metals (electricity), 611 ; of Non-conductors, 610 Contact-Breaker, 707, 725 Contact in Friction, 179 Continuity between Liquids and Gases, 232, 254, 375, 496 Continuity, Law of, 300, 334, 602 Continuous Spectrum, 495 . Contraction on Heating, 379 ; on Pull- ing, 260 Convection of Heat, 410 ; Convection- currents, 410 ; Electric Convection, 651 Convections : as to Attraction and Re- pulsion + , 189 ; Standard Pitch, 421 ; as to Electric and Magnetic Formulae in Air, 578, 598, 604, 675, 707 ; as to Gravitation-Potential, 192 ; North- seeking Pole of Magnet, Positive, 675, 746 ; as to Plane of Polarisation, 521 ; Vitreous Electricity Positive, 579, 746 Convergent Lenses, 533, 535 Convergent Rays or Beam, 117 Cooling, during evaporation, 389 ; during expansion, 358 ; in still air, 410 ; Dulong and Petit's Law of, 410 ; Newton's Law of, 410 Copper electroneagtive to zinc, positive in the battery, 611, 615, 617, 643; cast under water, 364 ; colour of , 498, 502 ; repulsion of, 724 Cord, Vibrations of, 134, 413 Core, soft-iron, 687, 728, 731 Cornet, 453; cornet of guttapercha, 454 Cornopean, 453 Corona, 551 Corresponding Points of Eyes, 576 Corti, 469 Coulomb, 638, 711, 715 Coulomb's Torsion-Balance, 607 Counterpoising of Forces, 37 Couple, 158 ; equilibrium of s, 160 ; examples of, 160 ; Moment of, 159, 166, 677 ; Magnetic, 676, 681, 693 Crank, 89 Creeping, 380 Crest and Trough, 104 Critical Angle in Kinetic Friction, 180 ; Density, 376 ; Pressure (Car- nelley's), 236 ; Pressure (Van der Waals's), 233, 376 ; State of Mat- ter, 232, 376 ; Temperature, 232, 237, 255, 376, 390, (magnetic) 689 ; Villari's Value, 690 ; Volume, 233, 376 Crookes's Layer, 363 Crooks of Trumpet, 453 Crossed (Nicol's) prisms, 559 Cross-field (dynamo), 731 Crushing, 262 Crust of Earth, 228 Cryophorus, 389 Crystalline form, 255 Crystalloids, 284 Crystals, Axes of, 551, 556, 623 ; Bi- naxial, 556 ; Conduction of Heat in, 409 ; Expansion in, 378 ; Positive and Negative doubly-refracting crystals, 555 ; electrified on heating, 624, on pressure, 623 ; Principal Section, 551 Cubical Compressibility, 259, 325, 462 Currents, 586, 603, 611, 632-674 ; Con- duction , 586 ; Displacement, 586, 611, 648, 695 ; Eddy- s, 703 ; En- ergy of 49, 615, 647, 651; Oscillat- ing , 721-733, 735, 740 ; Steady 632-674, 745 ; Direction of , 615 ; Mutual Action of , 670, 702, 724 ; 762 INDEX. Secondary , 687, 700 ; Simultane- ous s, 644 ; in Telephones, 737 Current-Density, 633, 649 Current-Induction, 699-707 Current-Intensity or Strength, 633, 637, 638, 659, 684, 694, 703, 707, 709 Current-Sheet, 693 Curvature, 79 ; Radius of, 79, 80, 165 Curves: see Adiabatic, Arrival, Gau- gain, Harmonic, Isentropic, Isother- mal, Periodic, Sines Curved path, 57 ; Acceleration in, 79 ; Velocity in, 59 Curved Surface, Refraction at, 130 Curved Wave-Front : Reflexion of, 120 ; Refraction of, 128 Cyanite, 674 Cycle, 394 ; Carnot's, 395 Cyclical order, 63, 66, 645 DALTON'S Atomic Theory, 239.; D.'s Law of Gaseous Pressures, 250, 253 Damp air as an insulator, 589 Damping, 186, 299, 3U7, 701, 703, 716 Daniell's Cell, 620, 638, 661, 663, 704 ; its various forms, 620; its C.M.D.P. computed, 661 Dark-Heat Waves, 481 Dark Lines in Spectrum, 494 ; in Heat- Spectrum, 495, 717 Dead-Beat, 703, 716 Dead Points of a Crank, 89 Dead Turns in Dynamo, 731, 738 Decay, 357, 507 Decimal Candle, 513 Declination, 678 Decomposition, Chemical, 236 ; of Water by Electrolysis, 658 ; by a Grove Cell, not by a Daniell, 663 ; by Light, 481, 482, 486, 488, 503, 569; by Radiant Heat, 482. See Dissocia- tion Deferred Restitution-force, 266 Deflection-methods, 680 Deformation, Resistance to, 259 Degradation of Energy, 399 Degrees of Freedom of a Particle, 72, 357 ; of a Rigid Body, 76 Demagnetisation, 692, 725, 731 Demon, Clerk Maxwell's, 52, 398 Denser Medium, Wave entering, 124; wave leaving, 125 Densimeter, 223 Densities of gases, 241, 250, 324 Density, measurement of, 221-224,294, 295, 382 ; measurement of Vapour- density, 391 Density of Air, 324 ; of the Earth, 202 ; of the Ether, 235, 746 ; of Hydrogen, 224, 324 ; of Matter, 220 ; of Solu- tions, 281 ; of Water, 13, 220, 224 ; Maximum Density of Water, 13, 359 Density, Critical, 376 ; Optical , 509 ; Specific, 221 ; Superficial, of Matter, 188 ; Electrical, 579, 583,603, 604 ; Magnetic, 683, 686, 692, 693 ; Vol- ume-density, 188 Density of Current, 633, 649 Derived Currents : steady, 644, 653 ; oscillatory, 723 Dermis, 287 De Sauty's Comparison of Capacities, 719 Deviation : minimum, 529 ; without dispersion, 531; dispersion without deviation, 532 Dew, 393 ; dewpoint, 392 Dextro-rotatory, 567 Dialysis, 286 Diagram, Indicator, 55, 593 ; Thermo- electric, 626, 651 Diamagnetic, 685, (oscillating currents) 724 Diaphragm in Lenses, 537 Diathermancy, 497 Diatonic Scale, 422 Dichroism, Dichromatism, 499 Dicrotic Pulse, 322 Dielectric, 587, 589, 590, 591, 593, 595, 597, 602, 603, 611, 648, 695, 744 ; - in Oscillating Currents, 721 ; Elas- ticity, 603, 604 ; Energy of the , 602, 603, 604, 695 Difference of Potential, 193, 584, 593. 597, 603, 604, 609, 612, 623, 624 709; Actual D.P. in closed circuit, 643; Electromotive I). P., 587, 608, 609, 616, 625, 646 ; do., measured 608, 646 ; observed, 604 ; produced, 609 Differential Equations, 83, 185, 186, 395, 705, 718, 722, 725, 728 Differential Galvanometer, 713, 734 Differential Tones, 473 Diffraction, 139; of Sound- Waves, 458 ; of Light, 548 ; of e.-m. waves, 743 Diffraction-grating, 141, 486, 549 Divisibility, Coefficient of, 283 Diffusion of gases, 247, 251, 330; of liquids, 247, 283, 288 ; of solids, 257 ; of gases through membranes, 332 Diffusivity, Thermal, 407 Dilatancy, 288 Dimensions, Theory of, 15, 746 ; of Physical Quantities, 59, 75, 166, 224, 263, 308, 598, 603, 604, 638, 693, 694, 708, 709, 714 Dimensions of Space, 9, 10 Dip, 678 Direct Extra-Current, 705 Direct-Vision Spectroscope, 532 Direction, 57 ; Change of , 18, 68 ; of Current, 615 ; of Electromag- netic Lines of Force, 668 ; of Mag- netic Lines, 676 ; of Movement, 191 ; of Sound, 471 INDEX. 763 Discharge, 580, 599, 632, 696, 725, 742, 744 ; residual , 599 ; in a vacu- um, 656, 662 Discharging Electroscope, 605 Discord, 471 Discs, vibration of, 137, 443, 479 Dispersion, 245, 509, 530, 745 ; Abnor- mal, 532 ; Irrationality of, 531 ; De- viation without , 531 ; without Deviation, 532 Displacement, 41, 82, 166 ; angular , 166 ; electric, 602, 603, 611 Displacement-Current, 586, 611, 648, 695 Disruptive Tension, 582 Dissipation of Energy ; see Availability (50) and Degradation (399) of En- ergy Dissipation of Sound in Air, 460, 463 Dissociation, 241, 243, 248, 355, 367, 615, 662; in Solutions, 248, 280, 386, 590, 614 Dissonance, 471 Distance, Action at a, 577 Distorted Image (lenses), 538 Distortion of Air-waves, 477 ; of elec- tric waves, 722, 737 Divergent Lenses, 533-537 ; Rays, 117 Dividing Engine, 30 Divisibility of Matter, 220, 238/245, 246 Division of Charge, 592 Doppler's Principle, 465, 484 Double Bass, 450 Double Refraction, 228, 551-566, 575, 599, 744 Double-Refracting Lamina, 559; Power, detection of, 565 ; Double- Refraction Dynamometer, 565 Drum, 412, 444,448; of Ear (mem- brana tympani), 457, 466, 473 Dry bulb, 392 Dry pile, 605, 622 Ductility, 259 Dulong's Water Calorimeter, 405 Dulong and Petit's Law of Atomic Heat, 366, 367 ; of Cooling, 410 Duplex Telegraphy, 733 Dust, 583 ; dust-haze, 503, 522, 523 Dynamical Coefficient of Thermal Con- ductivity, 407 Dynamo-Electric Machines ("dyna- mos"), 353, 637, 692, 725, 737; Direct-Current , 731 ; 'series, 732 ; shunt, 732 ; series-shunt, 732 ; sep- arately excited, 732 ; Activity, 000 ; Efficiency, ooO 5 Alternators, 729-730 ; Disc-dynamos, 728 ; drum , 730 ; ring , 730 Dynamometer (Force), 38; (Energy), 53; Double - Refraction , 665; Friction , 184 Dyne, 21 EAR, the Human, 448, 458, 465-471 ; the Ear-trumpet, 428, 461 Earth : Attraction by, 6, 201 ; Crust of, 228 ; Currents, 644 ; Density, 202 ; 's Drag on the Ether, 510 ; In- ductor, 687, 718 ; Loss of Heat by , 407 ; Mass, 202 ; as a Magnet, 679, 693 ; 's Permeability to the Ether, 570 ; 's Positive Pole the Southern, 693; its Potential Zero, 588 ; Pressure of Sunlight on , 570 ; Putting to , 698, 719, 720; Radii of, 205 ; Rotation of, 164, 168, 205 ; in Telegraphy, 644 ; Variations of g over , 22, 41, 205 Ebullition, 386 Echo, 460 Ecliptic, 81 Eddies, 306, 313 Eddy Currents, 703, 724, 725 Effect, Cause and, 3 Effective Mean Intensity, 723 Efficiency of Electromotor, 738 ; of Heat-engine, 397; of Carnot's ideal do., 397 Effusion of gases, 330 Egg, spinning of, 76, 164 Elastic Bodies, their Impact, 117, 151, 251, 266; E. intermediary, 56, 268; Restitution, 264 ; Toughness, 265 ; Tubes, Flow through, 320- 323 Elasticity : coefficient of , 264, 322 ; in Solids, 256, 263-267 ; in Liquids, 000 5 in Gases, 229, 267, 324 ; Perfect and Imperfect in Solids, 265, 267 ; Fatigue of, 267 ; physiological exam- ples, 268; Limits of, 256, 265; Mechanical advantages, 56, 268 ; Vibrations due to E., 151, 251, 266 ; Electric of Dielectric, 603, 604 ; E. of the Ether, 235, 745 ; E. of Volume, 229, 259 Elastivity, 603, 604 Electric Arc, 654 ; Attraction, 234, 577, 581, 602, 721 ; capacity, 592, 597, 599, 603, 604 ; Charge, 578 ; free, 591 ; bound, 591 ; Circuit, 613-616, 672 ; Conduction, 588, 603, 692 ; Conductivity, 634, 658 ; Convec- tion of Heat, 651 ; Cautery, 653, 666 ; Currents, 586, 603, 611, 632- 674, 721-733, 735, 740 ; Density, 579 ; Displacement of Dielectric, 603, 611, 744 ; Elasticity of Dielec- tric, 603 ; Energy, 49, 52, 593, 603, 604, 648 ; Equipotential Surfaces, 582-588, 592, 595, 602, 648, 672 ; Force, 581, 583, 587, 599, 603, 604, 609, 648, 710 ; lines of Force, 583, 591-595, 604, 648, 667, 671, 692, 699, 700, 742, 745 ; Motion of these lines, 591, 648, 699 ; in oscillating field", 721 ; Fur- 764 INDEX. nace, 655 ; Fuses, 653 ; Light, 484, 636, 653, 723, 727 ; Machines (frictional), 610, 656, 737 ; Matter (imaginary), 578, 68.1, 583 ; Pres- sure, 587 ; Quantity, 578, 594, 604, 638 ; Screen, 601 ; Storage of Energy, 666 ; Stress, 234, 577, 693, 599, 603, 744; Tension, 582, 603 ; Welding, 653 ; al Wind, 580, 602 Electricity, 7 ; not a form of Energy, 577, 594 ; Conduction of, 588, 603, 692 ; electrolytic, 588, 622 ; Conservation of , 581 ; Separation of Electrici- ties, 602, 609 ; in the Universe = 0, 581 Electrocapillarity, 624 Electrochemical Equivalence, 659; Equivalent, 661 Electrodes, 657, 745 ; Capacity of, 664 ; Polarization of, 664 Electrodynamic Units, 670 Electrodynamometer, 716, 723 Electrokinetic, 234, 591 Electrolysis, 49, 327, 388, 590, 657-667, 723, 727, 745, 746 ; in Gases, 662 ; of Mixtures, 663 ; under Oscillat- ing Currents, 723 Electrolytes, 386, 590, 615, 657, 722, 664 (glass) ; conductivity of, 646, 658 ; in oscillating currents, 722 Electrolytic Conduction, 588, 622 ; Field of Force, 615, 657 Electromagnet, 38, 689, 692, 707, (al- ternate current) 724 Electromagnetic Field of Force, 667, 683, 689, 695, 701 ; Current-Induc- tion, 699-707 ; Interrupter for Tuning Forks, 267, 447, 735 ; Lines of Force, 667, 671, 699, 702 ; their Direction, 668 ; Measure, 625, 634, 670, 684, 703, 707, 714 ; Units, see Magnetic Units ; Unit of Heat (= Joule), 41, 353, 647; Waves, 479-481, 498, 741 Electrometer, 245, 604, 606 Electromotive Force, 587, 588, 633, 706 Electromotive Intensity, 587, 603 Electromotor, 738 ; Alternating Cur- rent , 740; triphase do., 741; Efficiency in , QOO Electronegative, 611, 617, 643 Electrophorus, 630 Electroplating, 663 Electropositive, 611, 617, 643 Electroscopes, 579, 604, 612 Electrostatic, 234, 591; Capacity, 592, 597, 599, 603, 612, 624, 664, 696, 710; Energy, 593, 603, 604,648; Electromagnetic Ratio V, 708, 709, 744 ; Field, Intensity of, 583 ; Retardation, 696 ; Units, see Units Element, Chemical, 217, 219, 494 ; s of Charge, 602; Galvanic , 616; Geometrical, 58 Elliptical Motion = 2 S.H.M.'s, 90 Elliptically Polarized Light, 515, 560 ; detected, 563 Elongation in S.H.M., 82 Elongation under Traction, 260 E.M.D.P., Electromotive Difference of Potential ( = "E.M.F., Electromo- tive Force "), 587, 608, 609, 616, 625, 646, 709 ; Measured, 608, 645 End-on Method, 681 Endosmotic Equivalent, 287 Energy, 2, 42, 354 ; Potential , 43, 45, 190, 354, 684, 699, 702 ; Kinetic , 46 ; Availability of , 50 ; Con- servation of, 7, 47, 52, 125 et passim ; Degradation of, 399 ; Flow of, 648, 684, 695 ; Fluctuation of , 164 ; Graphic representation of (areas}, 52, 53-56, 393 ; Indestructibility of, 7, 47 ; Measurement of , 5, 3 ; Slope, 42, 60, 603; Storage of , in the Ether, 586, 593, 603, 648, 721 ; Transformations of, 47 et passim Energy absorbed or evolved during Change of State, 237, 356 ; do. do. in Galvanic Cell, 614, 615 ; in Charg- ing a Condenser, 648 ; Chemical , 356 ; of Conical Pendulum, 142 ; of Dielectric, 603, 604, 695 ; of Electrified Body, 593 ; of Electric Current, 49, 615, 647, 651 ; in Elec- tric Field of Force, 603, 604, 648 ; Flow of in Current-Field, 648, 684, 695 ; in Impact, 151, 152 ; Intrin- sic , 48, 247, 356 ; of Jet, 303 ; of Magnetic Field, 686 ; of Mole- cules, 351 ; of Niagara F.alls, 739 ; in case of Repulsion, 190 ; of Rotation of a Particle, 162 ; of a Mass, 162 ; in Secondary Batteries, 666 ; of S. H.M., 141 ; of Sound-waves, 142, 414, 476 ; of Sun, 50, 478 ; of Sunlight, 479 ; in Superposed Waves, 458 ; Transmission of , 182, 383, 230; by Steady Current, 632, 648 ; in Vision, 573 ; of Wave- motion, 142 Engine : doing and not doing Work, 49, 352 ; Carnot's , 397 ; Dividing , 30 ; Harmonic , 476 ; Marine , 89 ; Perfect , 374 ; Railway, 181 ; Reciprocating, 397 Engineer's Unit of Force, 23 Entropy, 395, 398, 399 Epoch in S.H.M., 83, 103 Equal Temperament, 425 Equalisation of a Current, 667 Equation to a Curve, 72 Equator, Magnetic, 679 INDEX. 765 Equilibrium, 187 ; stable, unstable, and neutral, 210 ; E. of Couples, 160 ; Electrostatic , 590, 613, 615, 657 ; of Forces, 145 ; of Liquids, 289 " Equilibrium-position," 39, 263 Equipotential SucfaS^i03-199, 301, 409, 5&>r&8% 585, 588, 592, 595, 599, 602, 648, 667, 672, 676, 682, 683 Equivalence, Chemical, 220, 239; equiv- alents, 52, 239 Equivalence of Forces, 3 Equivalence of Shell and Circuit, 683, 707 Equivalent, Electro-chemical, 661 ; En- dosmotic , 287 ; Gramme , 367 ; Water , 405 Equivalent Lens, 538 Ereinacausis, 507 Erg, 41 ; Ergten, 41 Ericsson's Sun-motor, 491 Essential Properties of Matter, 216 Ether, the, 234, 361, 411, 478, 479, 480, 481, 488, 497, 504, 508, 509, 510, 511, 512, 513, 514, 521, 522, 570, 577, 582, 583, 586, 589, 591, 593, 598, 602, 612, 648, 669, 671, 679, 694, 700, 704, 721, 727, 742-746 ; its Constitution, 745 ; its Density, 235, 513 ; its Elasticity, 235, 513, 593, 612, 745 ; Energy stored in, 586, 593, 603, 648; Transverse Vibrations of, 479, 509 Ether-Shear, 746 Ether-Stress, 234, 577, 582, 586, 599, 602, 603 Ether- Waves, 234, 350, 478, 508, 741 ; their Length measured, 542, 544, 550 ; their Pressure, 570 ; their Velocity, " 480, 510, 512, 637, 743 Ethylene, 389 Euphonium, 453 Evaporation, 237, 386 ; Cooling during, 389 ; Electricity in , 623 ; of Ice, 390 ; Latent Heat of , 390 Evolution or absorption of Energy dur- ing Change of State, 237, 356 Exchange of Radiations, 489 Excitation of Field Magnets, 731 Expansion of Gases, indefinite, 229, 250, 326, 338 ; explained, 250 Expansion by Heat, Coefficient of, 378 ; in Crystals, 378 ; Measurement of, 380-382 ; examples of , 379 ; in hollow bodies, 379 Experimentation, 5 Exploration of air-potential, 629 ; of Fluid Pressure, 296 Explosives, 4, 44, 148 Exposure in Photography, 482 Extensibility, 260 ; Coefficient of, 260 Extension of Matter in Three Dimen- sions, 218 Exterior Work done by Heat, 359, 370 External Conical Refraction, 658 Extra-Current, 704, 705, 731 ; Direct, 705 ; Reverse, 705 Extraordinary Ray, 553 Eye, 536, 570 FAHRENHEIT Areometer, 223 ; Thermoneter, 364, 402 ; F. Zero, 386, 402 Fall of Electric Potential, 638, 641, 650, 654, 656 Fall under Gravitation, 21, 202 ; down inclined plane, 173 Farad, 715 Faraday's Laws of Electrolysis, 659 Fatigue of Elasticity, 267 Faure's Accumulators, 665 Federmanometer, 297 Format's Law, 133 Ferromagnetic, 685 Field of Force, 198 ; Electrostatic, 582- 585, 588, 590, 591, 593, 595-599, 605, 609, 612, 615, 632, 685 ; discharge of, 632 ; intensity of, 583, 676 ; Electro- lytic, 615, 657 ; Electromagnetic, 667, 683, 689, 695, 701, 708; Magnetic, 669, 676, 682, 686, 701, 745 ; Rota- tory, 741 Field Magnets, 731 Fife, 453 Figure, Centre of, 73, 146, 158, 165, 206, 446 Films, Colours of, 545 Filtration, 340 Fish, air-bladder of, 325 ; luminosity of, 507 Fixity of proportions in chemical com- pounds, 238 Fizeau's method, 512 Flageolet, 453 Flame, 355, 506, 576, 723 ; as a Reflec- tor, 459 ; Singing and Sensitive, 454 Flexibility, 262 Flexible Lenses, 537 ; Mirrors, 527 Flexure of a Rod, 263 Flow of Electrical Charge, 602-603 Flow of Energy in Current-Field, 648, 684, 695 Flow of Gas, 334, 336; measurement, 336 Flow of Heat, 407, 409, 602 ; in a Bar, 408 Flow of Liquids, 288, 299-323 ; Forces producing , 300 ; in Suspended Loops, 301 ; through rigid Tubes, 309 ; in Capillary Tubes, 315 ; in Elastic Tubes, 320 ; Lines of , 300; Measured, 317; Steadiness of , 300 Flow of Magnetism, 692 Flow of Temperature, 409 Fluctuation of Energy, 164 Fluid, 218, 225 ; Friction, 184 ; 766 INDEX. Pressure, 296 ; Pressure in Vibrat- ing , 335 Fluorescence, 484, 486, 504 Flute, 453 Flux, Magnetic, 685, 686, 691 Fluxion-notation, 59 Fly-wheel, 89, 164, 168, 169, 667 Focal Distance in Eefraction, 130 ; Principal , 130, 539 Focal Length of Minor ( = principal focal distance), 525 ; of Lens, 534, 535, 536 Focus, 117 ; Approximate , 129, 525 ; of Mirror, 525 ; Principal , 525 ; Conjugate, 526 ; Virtual, 525, 527 ; of Lens, 533 ; real, 533 ; virtual, 533; principal, 533, 534; Conjugate, 535; Heat F. and Photographic F., 541, 542 ; F. of Eye, 538 Fogg, 411 ; Clearing , 583 ; Reflexion of Light from, 503 ; Sound through, 460 Foot, 10 Foot-pound, 40, 41 ; Foot-poundal, 41 Force (1) Any Cause of Motion, 4, 20 ; (2) Action = ma, 20, 166 ; (3) Time- rate of Change of Momentum, 20, 60, 248 ; (4) Space-rate of Change of Energy, Energy- Slope, 42, 60, 603; (5) Rate of Variation of Change of Configuration, 47 ; said to Act upon bodies, 20, 40 ; to do Work, 40, or to have Work done against it, 40 ; " Centrifugal ," 165-169 ; Chemical , 355 ; Components of a , 143 ; Composition of s, 143 ; Conserva- tion of , 52, 144, 230, 292 ; Elec- tromotive , 587, 588, 633, 706; Equilibrium of s, 145 ; Equiva- lence of s, 3; Field of , 198; electrostatic, 582-585, 587, 588, 590, 591, 593, 595-599, 603-605, 609, 611, 612, 615, 632, 648, 685, 710, 742; electromagnetic, 667-674, 683, 689, 695, 701 ; magnetic, 618, 667, 676, 682, 683, 701 ; electrolytic, 615, 657 ; uni- form, 198, 583, 608, 673 ; produc- ing Flow, 300; Graphic representation of (lines), 52 ; Lines of , 196, 582, 591, 592, 595, 604, 648, 667, 692, 745 ; electromagnetic, 667, 672, 699, 702, direction of, 668 ; magnetic, 676, their direction, 676 ; Magnetic, , 676, 693 ; Magnetomotive , 691 ; Molecular s, 253-256, measurement of, 271 ; Measurement ,20, 35-39 ; Moment of , 155, 166; Parallelo- pipedon of s, 145 ; Parallelogram of s, 143 ; Polygon of s, 145 ; Resolution of s, 143 ; Resultant Electric , 581, 58'3, 603 ; Simul- taneous s, 143, 145; Triangle of s, 145 ; Tubes of , 197, 583, 602 ; Unit of , 21 ; Unit Tubes of , 197 Forced Vibrations, 447-449, 479 Force-pump, 345, 587 Forceps, 170 Form, of Matter, 218; Perception of ,576 Formulas, Chemical, 239, 244 ; Graphic, 244 ; Mathematical, 15 Foucault's Principle, 512 ; his Prism, 556 Fourier-motion (a periodic motion com- pounded of commensurate S.H.M.'s), (Fourier's Theorem), 103, 266, 412, 414, 416, 417, 434 Fourier's Theorem, 103 ; applied to vibrating strings, 135 Fragility, 258 Fraunhofer lines, 484, 485, 494, 550 Free Charge, 591, 595, 597, 630 Free Path of Molecules, 251, 252, 253, 464 Free Vibration, 434-445, 479 Freedom, Degrees of, 72, 76, 351 Freezing Mixture, 235; Point, 281, 402, lowered by molecules in solu- tion, 281, 386 French Horn, 453 Frequency, 83, 112, 412, 414, 435, 479 ; of S.H.M., 83; of Alternating Cur- rents, 725 ; Ether-Waves, 480 Fresnel, 379, 517, 519, 521, 543 ; Fres- nel's Rhomb, 562 Friction, 7, 176, 203, 352, 393, 436 ; at Axles, 184 ; Coefficient of Stati- cal between Solids, 176 ; Coeffi- cient of Kinetical between Solids, 179 ; variations therein, 179, 184 ; in special cases, 179-184 ; in air, 335 ; -Dynamometer, 184 ; factor, 184 ; between Fluids and Solids, 184 ; within Fluids, 306 ; in the Mechanical Powers, 180; a Resistance, 176, 180; Roll- ing , 182 ; in S.H.M., 185 ; in Violin, 435-436 ; -Wheels, 182 Frictional Electric Machines, 610, 656, 737 ; vapour-friction do. do., 623 Fringes, 140 Frothing, 278 Function, 15, 82 ; Carnot's-, 397 ; Periodic, 82 ; Thermodynamic, 395 Furnace, Electric, 655 Fuses, 653 Fusing point affected by Pressure, 237, 384 Fusion, 360, 384 GALILEO'S Principle, 4 ; his Doublet, 572 ; and Pendulum, 34, 202 Gallon, 13 Galvanic Cell, 353; Battery, Cell, Pile, 608, 616, 745 ; coupled up, 617, 640 ; discharge of, 632 ; economical use of, 641 ; and ends, 615, 617 ; INDEX. 767 Circuit, 616, 639, 643 ; Constants in, measured, 646 Galvanometer, 602, 637, 643, 645, 647, 695, 710, 712, 733 ; Ballistic , 687, 713 ; Constant, V& ; Dead-beat s, 716 ; Differential , 716, 733 ; von Helmholtz's ,713 ; Marine , 691 ; Resistance in, 647 ; Sensitive- ness of , 713 ; Sine , 713 ; Tan- gent , 637, 710, 712, 718 Gases, 229, 241, 250, 324-349, 375, 377|; perfect, 369-374 Gas-battery, 664 Gate on its hinges, 170, 216 Gaugain's Curves, 627 Gauges, 30 ; Steam Gauge, 297 Gauss's Lens-sy stem-method, 539 Gausses, 677, 715 Geissler's Vacuum-Tubes, 656, 723 General Properties of Matter, 219 Gerhardt, 240 Gilberts, 715 Glacier-Flow: (1) Creeping, 380; (2) Plasticity, 385 ; (3) Regelation, 385 Glow-lamps, 653 Gradient, Potential, 191, 585, 587, 597, 598, 602, 633, 639, 648, 710; Pressure , 301 ; Temperature , 407, 602 Gramme, 13 ; gramme-atom, gramme- molecule, 367 ; gramme-equivalent, 660 Graphic Formulse (chem.), 244 Graphic Representation of Energy, 52, 53-56, 393 ; of Force, 52 Grating, Diffraction, 141, 486, 549 Gravitation, 41, 44, 146, 201, 247 ; Constant, 189, 202 ; Fall under , 21, 173, 202 ; Law of , 201 ; Po- tential, 192, 195, 584 ; Universal , 203 Gravity, acceleration g due to, 22, 202 ; Variations of g, 22, 41, 205 ; Meas- urement of g, 36, 206 ; Centre of , 74, 206 ; Effect of on flow, 301 ; Intensity of , 22, 713; Specific , 221 ; do. bulbs, 223 Gravity Electric Machine, 631 Grays, 576 <: Grove's Cell, 621, 638 Guebhard, 663 Guitar, 449 Gun, 6, 46 Gunpowder, 44. 580 Gyration, Radius of, 162 HAEMODROMOGRAPH, 319 Haemodromometre, 319 Haemotachymetre, 318 Hall's Experiment, 694 Halo, 530 Hammock, 149 Hardness, 258 Harmonic Curve, 85, 07, 106, 111, 736 ; Engine, 476 ; Telegraphy, 736 Harmonic Motion, 80 ; Acceleration in, proportional to Displacement, 83, 434; Amplitude in, 82, xv ? , 114; Angular Velocity in, 82 ; Circle of Reference in, 81, 185 ; Composition of, 86-103 ; H.M.'s compounded by Blackburn's Pendulum, 211, by vi- brating reeds, 442, by vibrating strings, 434 ; due to Elasticity, 266 ; Elongation in, 82 ; Energy in (x am- plitude 2 ), 141 ; Epoch in, 83, 103 ; Frequency of, 83 ; Friction in, 185 ; Isochronous, 83 ; Pendulum-move- ment, 86, 211 ; Period in, 82, 414, 680 ; Phase in, 82 ; Projection of, 84 ; Resolution of, 96, 102 ; Viscosity in, 185 Harmonic Variations of Potential and Current, 721, 722, 729 Harmonicon, 442 Harmonics^ 416, 429,. 430, 472, 475 Harmonium, 412, 425, 442, 451,. 472, 473 Harmony, 417,. 471 Harp, 449 Harpsichord, 436, 449 Haze, 503, 522, 523, 743 Head of Liquid, 302, 587 ; of a Gas, 330, 334; Pressure , 310, 334; Velocity- , 310, 334 Heart, 311 ; Valves of, 346, 431 ; Work done by, 320 Heat, 7, 48, 2o4, 350-411 ; Atomic , 366 ; of Combustion, 49, 236, 354, 357 ; Conduction of, 406-410, in gases, 236, 409; Convection of , 410; produced by Electric Current, 649, 652, 711, 723 ; a Form of Energy, 351 ; Engine, 49, 352, 397 ; Ex- pansion bv, 378-382 ; Flow of , 407, 409, 602 ; Focus, 541, 542 ; Latent , 235, 361, 364; Material Theory of Heat, 351 ; Molecular , 367 ; not Motion, 351 ; not by Pressure alone, 351 ; Radiant , 48, 234, 350, 410, 481, 541, 743 ; Rays, 481 ; Sensible , 350 ; Specific , 353, 365, 370, 404, 405 ; -Spectrum, 486, 495, 500, 533, 717 ; Transference of _, 406 ; Transport of, 410 Heavy, 216 ; Heavy liquids, 291 Helium, 218, 495 Henries, 711, 715 Henry's Law, 329 Herz's Experiments, 741 ; his Resona- tor, 741 ; his Vibrator, 741 Heterogeneous Conductors, 639 Hollow Body, Expansion of, 379 , Homogeneous Atmosphere, 347 Homogeneous Conductors^485, 638 Homogeneous Light, 485 768 INDEX. Homogeneous Strain, 78 Hooke's Law, 256, 265, 473 Hopkinson, 599, 689 Horizontal Component of Terrestrial Magnetic Force, 679, 680, 712 Horn-bands, Russian, 449 Horsepower, 42, 647, 653, 655 ; effective, 42 ; nominal, 42 Humidity, 392, 463 Huyghens, 34 Hydraulic Press, 21, 52, 290, 333 ; Ram, 148, 312, 704 ; Tourniquet, 306 Hydrogen ; a metal, 327 ; alloy with palladium, 327, 664; Critical Tem- perature, etc., 376 ; Density of, 224, 324 ; Flame of , 496 ; Heat evolved on Combustion of, 354, 356, 358; Molecular Velocity of, 250, 465 ; Self- repulsion of, 253, 375, 377 ; Sound- waves in, 413, 428 ; in Organ-pipe, 452 ; Specific Heat of, 366, 369 ; Thermal capacity of, 366 Hydrometer, 222, 277 Hydrostatic Paradox, 292 ; Pendu- lum, 318, 336 ; Pressure, 25, 290, 301, 310, 325, 602; Stress, 225, 257, 300 Hydrostatics, 287-299 Hygrometer, 392 Hypertrophy of Heart, 311, 323 Hypothesis, 8 Hysteresis, 690, 725 ICE : Evaporation of, 390 ; Fusion of, 351, 360, 384, 402 ; affected by Pres- sure, 384 ; in hot Crucible, 363 ; Latent Heat of ,361, 390 ; Plas- ticity of , 385 ; Spectrum of , 495 Iceland Spar, 551, 553, 555 ; as a Di- electric, 599 Image : of Lens, real, 533, 535 ; virtual, 533, 535 ; distorted, 538 ; in Mir- ror, 526 ; real, 526 ; virtual, 527 Imaginary Electric Matter, 57 8, 581, 583 Imaginary Magnetic Matter, 675 Imbibition, 282 Impact, 117, 150-152, 251 ; Energy in, 151, 152 ; Vibration on , 151, 251, 266 Impedance, 706, 722, 727, 729 ; in Transformers, 725 Impenetrability of Matter, 218 Imperfect Elasticity, 265, 267 Impulse, 20, 166 ; Moment of , 166 Incandescence, 654 Incandescent Lamps, 653, 723 Incidence, Angle of, 120, 126, 518 j Plane of , 517, 519, 521 Inclination (magnetic), 678 Inclined Plane, 172 ; Fall down , 173 ; Pull up an , 181 Incus, 467 Indiarubber, 266, 360, 379, 570, 58 9 Indestructibility of Energy, 7, 47 ; of Matter, 7, 219 Index of Refraction, 127, 509, 518, 522, 528, 534, 540, 744 ; (for X = co ), 745 ; measured, 528, 529 Indicator Diagram, 55, 393 Induced Magnetic Poles, 685 Induced Magnetisation, Coefficient of, 686, 693 Inductance, 705, 722, 737 ; Mutual , 704 Induction : -Coil, 646, 656, 706, 725 ; Electromagnetic , 667, 699-707, 727-729 ; Electrostatic , 594, 604 ; coefficient of, 601, 604, 692 ; lines of, 595, 596, 598, 602 ; total, 596, 598, 604 ; per sq. cm., 596, 598, 603, 604 ; induc- tive capacity, 597, 598, 603, 604, 692 ; Magnetic , 684, 693 ; coefficient of, 685, 692, 693 ; lines of, 667, 671, 685, 702, 705, 710, 718 ; general problem of magnetic , 692 ; Mutual , 600, 704, 710 ; Self- , 646, 704, 710, 722, 724, 745, 746 Inductive Capacity, Specific, 597, 602, 603, 604, 612, 692, 709, 744 Inductivity, 685, 714 Inductor, Earth-, 687 Inertia, 5, 146, 219 ; Centre of, 146 ; Coefficient of, 147, 166 ; examples of , 147 ; in Ether discussed, 745 ; Moment of , 162, 166, 263, 680 ; Radius of , 162, 166 Infinite (oo ), 58 Instantaneous Axis, 75 ; Current, 695 Insulation, 582, 583, 589 Integration, 162, 188, 395 Intensity of Alternating Current, 723, 726 ; mean, 723 ; virtual mean, 723 ; of Steady Current, 633, 637, 638, 659, 684, 694, 703, 707, 709; meas- ured, 637, 694; Electromotive , 587, 603; of Electrostatic Field, 583 ; of wave-motion at a Focus, 117; of Gravity, 22, 713; of Light, 513, 744; of Magnetic Field, 676, 685, 693, 701, 708 ; of Magnetisation, 677, 691, 693; of Pressure, 25 ; of Radiation, 513 ; Slope, 200 ; of Sound, 414 ; of Stress, 24 ; of Tension, 26 Intensity of S.H.M. (cc Energy or oc Amplitude 2 ), 141, 414 Interference-Bands, 140, 142 Interference of Waves, 137 ; of Ether- waves, 542 ; of electro-magnetic waves, 743; of Sound-waves, 461, 463 Intermolecular, 359 Internal Conical Refraction, 558 INDEX. 769 Internal Work, 359, 369-371 ; in Air, etc., 345, 347, 375 Interrupter, Electromagnetic, 735 Intervals (musical), 422 Intramolecular, 351, 356, 359 Intrinsic Energy, 48, 247, 356 Inverse Squares, Law of, 187, 582, 585, 675 Inversion (thermo-elect.), 628, 651 Iodine, 498, 499, 500 Ions, 280, 288, 386, 590, 614, 658, 659 ; Direction of, 664; Travel of, 658, 664 ; Velocity of, 658 Iridescence, 547 Irradiation, 576 Irrationality of Dispersion, 531 Isentropic Curves, 395, 398 Isochronous Oscillations of Elastic Body, 267 ; of Pendulum, 34, 212 ; S.H.M.'s, 83 Isodynamic Surfaces, 200 Isothermal Lines, 394, 400 ; Sur- faces, 408, 602 Isotropic, 113, 138, 547, 551; (mag- netic) 687 JACOBI'S Law, 738 Jelly, a Solid, 226 ; the Ether analo- gous to a , 235 Jet, 72, 277, 302, 303-305, 579 ; Energy of, 303 Joule, 41, 48, 352 ; 's Equivalent, 48, 352, 353 ; his experiments on Gases, 372-374 ; his Law of the Heat pro- duced by a Current, 649 "Joule" = 10 7 ergs, 41, 353, 647 Jupiter's Satellites, 81, 511 Just Intonation, 422, 475 KALEIDOSCOPE, 524 Kepler's Laws, 203 Kerr's Experiment, 695 Kettle singing, 477 Kilogramme, 13 ; Kilogramme-metre, 23,41 Kilowatt, 42 Kine (C.G.S. unit of Velocity, one cm. per sec.), 14 Kinematics, 57 ; Kinetics, 143, 434 Kinetic Energy, 46 ; Friction, 179 ; Theory of Gases, 247, 361 ; applied to Radiometer, 361 ; to Spheroidal State, 363 ; to Velocity of Sound, 464 Kirchhoff's Laws, 644 Knee, 175 Koenig's Manometric Capsule, 431, 444, 452 LAEVO-ROTATORT, 567 Lag, 722, 724, 729 ; Angle of, 722 Lampblack, 455, 498 Lamp wickholder, 408 Laryngoscope, 524 Larynx, 148, 433, 475, 476, 524 Latent Heat, 235, 361, 364, 384, 411, 689 ; of Expansion, 370, 372, 377 ; of Ice, 361, 390 ; Methods of Calorimetry, 406 Latimer Clark's Cell, 622, 646 Lattice girders, 269 Law of Causality, 3 ; of Continuity, 300, 334, 602 ; Law of Electric At- traction and Repulsion, 579, 581 ; Law of Gravitation. 201 ; Law of Inverse Squares, 187, 582, 585, 675 ; Law of Magnetic Attraction and Repulsion, 675 ; Laws of Motion, 5 ; Laws of Thermodynamics, 353, 398. See Avvogadro, Boyle, Briot, Charles, Dbppler, Dulong and Petit, Earaday, Eermat, Eoucault, Fourier, Galileo, Henry, Hooke, Jacobi, Joule, Kepler, Kirchhoff,Lenz, Newton, Ohm, Pascal, Poiseuille, Prevost, Raoult, Stokes, Torricelli, Van der Waals Laws of Nature, 1, 5 Lead of Brushes, 731 Leakage : Electric, 618, 698, 737 ; Mag- netic, 692 Leclanche' Cell, 619, 666 Left hand of the Current, 668, 689 Left-handed Polarised, 515, 517, 560 Length, Focal, 130, 525, 534, 535, 536 ( = Principal Focal Distance) Length, Measurement of, 27 ; Units of -10 Length Reduced, of Conductor, 637, 641, 642 Length of Simple Pendulum, 206, 212, 214 ; of Compound, 213 Lens : Equivalent, 538 ; Power of, 534 ; Reversibility of, 534 Lenses, 533, 743 ; achromatic, 541 ; convergent, 534, 535 ; divergent, 534, 536 ; flexible, 537 Lens-method, Gauss's, 539 Lenz's Law, 701 Level of Water, 294 ; Potential analo- gous to, 193, 195, 586 Lever, 169-71 Leyden jar, 599, 600, 606 ; dis- charge, steady, 726 ; oscillatory, 721, 725, 742, 744 Light, 7, 48, 234, 481, 694, 723, 743 ; production of, 506; by electric current, 653 ; monochromatic, 485, 542, 543, 545; coloured, 483, 486; polarised, 514 ; common or natural, 515, 563 ; Maxwell's Theory, 234, 510, 743, 744 Light (adjective), 216 Lightning Conductor, 586, 696 ; flash, distance of, 463 Limiting Angle, in statical friction, 177 Limits of Elasticity, 256 V 265 ; of Hear- ing, 471 ; of Magnetisation, 688 770 INDEX. Line, 9 Linear Compressibility, 262 Linear Currents, their Mutual Forces, 669-671 Linear Waves, 104 ; Reflexion of, 117 Line-Measurement, 27 Line, Neutral, 262 ; Nodal, 137, 443 Line of No Pressure (Indicator Dia- gram), 55 Line of Potential, 639, 641, 696, 720 Lines, Adiabatic, '394, 398, 400 ; Iso- thermal, 394, 400 Lines of Electrostatic Induction, 595, 596, 598, 602 Lines, Fraunhofer, 484, 485, 495, 550 Lines of Flow, 300, 602 ; of Heat, 409, 602 ; Stream , 300, 602 Lines of Force, 196, 301, 692 ; Electric, 582, 591, 592, 595, 602, 671, 700, 742, 745 ; closed, 745 ; whether limited in number-, 746$ Electromagnetic, 667, 671, 699, 702 ; direction of, 671 ; cut across, 703 ; travel of, 648, 671, 699, 721 ; Magnetic, 667-670, 702 ; direc- tion of, 668 Lines of Induction : Electric, 595, 596, 598, 602, 603, 604, 671; Magnetic, 667, 671, 685, 702, 705, 710, 718 Lines of Propagation of Heat, 409, 602 Lines of Slope, 200, 687 Lines of Transmission of Energy, 671, 699 Line-Spectrum, 495, 496 Liquefaction, 235, 358, 360, 384 ; lique- fied air, 389 ; marsh-gas, 224 Liquids, 254, 271-323; friction against, 184 ; mobile and viscous, 226 Lissajous's Figures; e.g., figs. 35-41, 211, 442 Litre, 12 Loading a Vibrating Body, 445 Locomotive, 6 Locomotive Pulse, 313 Logarithmic Decrement (viscosity), 185, 308 Logarithmic Increment (musical pitch), 424, 426 Logarithmic Spiral, 185 Longitudinal Vibration, 110, 135, 441 ; Compression and Displacement in, 111 Loops and Nodes, 134, 137, 438, 452 Loops, suspended, liquid in, 293, 301, 344 Loss of Energy of Electrification, 594 Loss of Half a Wave-Length, 125, 518, 545 Loudness, 413, 415, 425, 471 Lubricants, 176, 184 Luminiferous Ether, 234 (see Ether) Luminosity, 507 Luminous Radiations, 489 Lungs, 330, 337, 339, 340 Lustre, 576 Lyre, 449 MACHINES : Electric, 610, 656, 727 ; Simple, 169 Magdeburg Hemispheres 339 Magnet, 674, 693 ; Solenoidal, 675 ; how made, 689 ; Spherical, 678 Magnetic Axis, 674, 679 ; Circuit, 691, 725, 730,731; [Conductivity, 686 ; Declination, 678 Dip, 678 ; Displacement, 744 ; Equator, 679 ; Equipotential Surfaces, 667, 672 ; Ether-vortices, 234, 745 ; Field, 669, 676, 682, 683, 701, 708, 745 ; Energy of, 686, 744; its [nature, 745 ; Flux, 685, 686, 691 ; Force, 648, 667, 676, 682, 683, 693, 701, 746 ; lines of, 667-674, 676 ; motion of these, 671 ; in oscillating currents, 721 ; Inclination, 678 ; Induction, 684, 693, 744; Coefficient of magnetic induction, 685, 692, 693; lines of, 667, 671, 685,702, 705, 710, 718 ; general problem of magnetic induction, 692 ; Intensity, 676 685, 693, 701 ; Lines of Force, 676, of Induction, 667, 671 ; imaginary Matter, 675 ; Measure, 625, 634, 670, 684, 703, 707; Meridian, 679, 680 ; Moment, 677, 680, 693 ; of Shell, 682 ; North, 678 ; Par- allels, 679 ; Permeability, 671, 676, 682, 683, 684, 685, 686, 691, 703, 706, 707, 708, 709, 722, 744, 746 ; Pole (terrestrial), 679, of magnet \ 668, 675, 685, 691 ; Potential, 682, 693 ; Quantity, 675, 693 ; Rotatory Po- larisation, 694, 745 ; Screen, 691 ; Shell, 682, 707 ; Storms, 679, strength of, 682, 684, 693, 707, po- tential of, 683; Strength (pole), 675, (field) 676,693, 701 ; Superfi- cial Density, 683, 686, 692, 693 ; Susceptibility, 686, 690 ; Tick, 690 ; Twist, 688 ; Units, 625, 634, 670, 684, 703, 707, 714 Magnetisation : Induced , 684 ; Coeff t. of, 686, 693; Intensity of , 677, 691, 693 ; Limit of , 688 ; Residual , 684, 690; Weber's Theory of, 688 Magnetism, 49, 234, 257, 674-695 ; Flow of, 692 ; nature of, 674, 692 ; separation of, 686, 689, 692 ; terres- trial, 679, 691 Magneto-electric Machines, 629, 687 Magnetometry, 680 Magnetomotive Force, 691, 714 Malleability, 259 Malleus, 466, 467 Manometer, 47, 296, 325, 333 ; ma- nometre metallique inscripteur, 298 ; INDEX. 771 m. compensateur, 299; manometric capsule, 431, 444, 452 Marie-Davy's Cell, 622 Marine Engine, 89 Marsh-Gas liquefied, 224 Mass, 12, 166, 216, 247 ; Centre of , 12, 146 ; Measurement of , 35 Massive, 216 Matching Colours, 575 Material Theory of Heat, 351 Materials, Strength of, 255 Mathematical Formula?, 15 Matter, 2, 7, 216, 577, 745 ; its Consti- tution : chemical views, 238 ; phy- sical views, 245 ; Kinetic Theory of , 247 ; Essential Properties of , 216; Extension of ,218; Impene- trability of , 218 ; Indestructibility of , 7, 219 ; States of , 225-237 ; General Properties of 5, 219 ; Iner- tia, 5, 146, 219 ; Weight, 12, 21, 201, 219, 333 ; Divisibility, 220, 238, 245, 246 ; Porosity, 220 ; Density, 220 ; Perception of Matter, 216, 246 ; Ra- diant , 234, 252, 656 Matter, Imaginary : Electric, 578, 581, 583 ; Magnetic, 675 Maximum Density of Water, 13, 359 Maxwell on Coercitive Force, 690 ; 's " Demon," 52, 398 ; 's Discs, 574 ; "Electromotive Intensity," 587, 603 ; Theory of Light,234, 510,744 ; Theory of the Magnetic Field, 745 Mean Free Path, 251, 278 Measure : Electrodynamic, 670 ; Elec- tromagnetic or Magnetic, 625, 634, 770, 684, 703, 707; Electrostatic, 638 ; Numerical, 16 Measurement, 1, 27 ; of Adhesion, 37 ; Calorimetric , 404, 513 ; of Density, 221-224, 294, 295, 382; End , 30 ; of Energy, 53 ; of E.M.D.P., 608, 646 ; of Expansion, 380-382 ; of Flow of Gases, 336 ; of Flow of Heat, 408 ; of Flow of Liquids, 317; of Force, 20, 35-39; of g, the acceleration of Gravity, 36, 206 ; of Heat, 404 ; of Inductance, 706 ; of Intensity of Current, 637, 694, 707 ; of Intensity of Magnetic Field, 708 ; of Length, 27 ; Line- _, 27 ; of Liquid Pressure, 295- 299 ; of Mass, 35 ; of Magnetic Moment, 680 ; of Magnetic Permea- bility, 687 ; of Molecular Forces, 270 ; of n, the Atmospheric Pressure, 347 ; of Pressure in a Stream, 317 ; of Re- fractive Indices, 528, 529 ; of Resis- tance of a Battery, 647 ; of Resistance to Electric Current, 633, 645, 647 ; of Resistance of a Galvanometer, 647 ; of Surface, 32; of Surface-Tension, 37, 275 ; Temperature, 400 ; of Time, 34 ; of V, 708, 709 ; of Vapour- density, 391 ; of Velocity of Liquid Stream, 317 ; of Velocity of Sound, 461 ; of Velocity of Wave-motion, 136 ; of Volume, 33; of Wave-lengths in Ether, 544, 550 ; of Weight (spring- balance), 37 Mechanical Equivalent of Heat, 353 Mechanical Powers, 169-176; with Friction, 180 Medium, 44 ; air as a standard , 578, 598, 604, 675, 707 Megadyne, 21 Megalerg, 41 Mega volt, 711 Megohm, 711 Melde's Experiments, 439 Melting Point of Ice, 360, 384, 402 Membrana Basilaris, 469 ; membrana Tympani, 466, 473 Membrane, Diffusion through, 285, 332 ; Vibration of, free, 137, 444, forced, 448 Mercury, Expansion of, 382 ; Fusing point of, 384 ; Thermometer, 401 ; Vapour, 243, 369, 465, 662 Meridian, 9 ; Magnetic , 678, 680 Metallic Reflexion, 502, 521, 561; in electromagnetic waves, 743, 745 Metals, Colours of, 502, 503 ; Conduc- tivity of, 636 ; Contact of, 611 Meteoric Dust, 50 ; Meteorites, 50 Method of Mixtures, 404 Method of Oscillations, 39, 213 Method of Vibrations, 680 Metre, 10 Metric System, 10 Mho, 634, 715 Mica, 564 Microfarad, 711 Microhm, 711 Microphone, 636, 737 Microscope, 571 Microvolt, 625, 711 Minimum Angular Velocity, 164 Minimum Deviation, 529 Mirrors, 459, 523 ; concave, 525 ; con- vex, 527 ; flexible, 527 ; rotated, 512, 525, 607 ; spherical , 123, 525, 527 Miscibility of Liquids, 282 Mixture of Colours, 574 Mobile Liquid, 226 Modulus of Superficial Tension, 276 ; Young's , 261, 321, 441, 443, 461 Molecule, 240 ; Gramme , 367 Molecules, 218, 240, 242, 244, 246, 272, 675, 685, 688, 692, 723, 727; Nature of, 246 ; Number of, 251 ; Energy of, 351 ; Free Path of, 251, 252, 464 ; Size of, 246, 251 ; Velocity of , 250, 464, 610, 613 ; in Vibration/ 247,251, 351,479, 500 Molecular Asymmetry, 244, 479, 568 ; 772 INDEX. Distance of Action, 278, 612; Forces, 253-256, measurement of, 271 ; Heat, 366 ; Kinetic Energy, 361 ; Refractive Power, 529 Moment : of a Couple, 159, 166, 677 ; of a Force, 156, 166 ; of Impulse, 166 ; of Inertia, 162, 166, 263, 680 ; Magnetic , 677, 680, 682, 693; measurement of, 680 ; of momen- tum, 166 ; Principle of Moments, 155 ; Twisting , 158, 160, 166, 263, 676, 681, 691, 716, 723 Momentum, 6, 14, 19, 20, 149, 150, 166, 351 ; Angular , 163, 166 ; Moment of , 166; of a System, 149; in Impact, 150-152 ; in the Elec- tromagnetic Field, 704 Monochord, 437, 449 Monochromatic Light, 485, 542, 543, 545 Moon, 6, 76, 146, 201, 204, 679 Morse-code, 733 Motion, 14 ; accelerated , 69, 152 ; Change of , 6 ; in a Circle, 8, 57, 78, 161-169; in Curved paths, 57 ; Laws of , 5 ; The Perpetual , 7, 193, 357, 385; parallel to or across Equipotential Surfaces, 192, 194; of Lines of Force, 699; Quantity of , 19 ; Reciprocating, 88, 89; Simple Harmonic , 80 et seq. ; Simultaneous s, 4, 60 " Motion" = Momentum, 6, 14, 19, 149, 166, 351 Motivity, 51 Motors : electro ; thermo-magnetic , 689 Multiphase, 730 Multiple Proportions, Laws of, 239 Multiplex Telegraphy, 735 Multipolar, 729 Muscle : Breaking Weight of, 261 ; Dif- fraction-Grating, 550 ; Extensibility of, 261 ; Mechanical Disadvantage, 171 ; _ -Sound, 431, 471 Musical Box, 451 ; Intervals, 422 ; No- tation, 423 ; Pitch, 420 ; Scale, 420, 422 Mutual Action at a Distance, 577 Mutual Attraction and Repulsion of Currents : Alternating, 724 ; Steady, 670, 702 Mutual Inductance, 704 Mutual Induction of two Charged Bodies, 600, 601 ; of two Currents, 701, 704 Myopic, 572 NATURE, Constancy of, 2 ; Laws of, 5 Negative Crystals, 555 Negative Electric Charge (resinous), 579, 602 Negative Electrode, 657 Negative Pole of Magnet (South-seek- ing), 675 Negative Pressure in Thorax, 340 Negative Waves, reflected, 125, 458, 463, 518, 542, 545 Nerve-ends, 350, 469, 575, 576 Neumann and MacCullagh, 522 Neutral Line, 262 Neutral Point (thermo-elect.), 626, 651 Neutral State (electric), 579, 580 Newton's Laws of Motion, 5, 146, 167 ; his Law of Cooling, 410 ; Law of Universal Gravitation, 203 ; Newton's Rings, 546, in electromagnetic waves, 743 ; on Sound, 412 ; 's Law of Velocity of Wave-motion, 267, 368 Niagara Falls, 352 ; Energy of, 739 Nicol's Prism, 555, 559, 568 Nobert, 220 Nobili's Rings, 663 Nodal Lines, 137, 443; Nodal Points, 108, 134 ; (Gauss's) 539 Nodes and Loops, 134, 137 ; in a Membrane, 137 ; on Monochord, 438 ; in an Organ-pipe, 452 ; on a vibrating String, 438, 439 Noe's Thermo-electric Pile, 629 Noise, 416 Non-commensurable, 93, 100, 103 Non-conductors, 588, 603; Contact of , 610 Non-conservative System, 45 Non-polarisable Electrodes, 664 Normal to Wave-front, 113; Propaga- tion along , 113, 117, 131, 139 Normal Spectrum, 141, 550 Note, 417, 429 Numerical Measure, 16 Nutation, 76 OBOE, 451 Oersteds, 715 Ohm, 353, 634, 637, 640, 711, 715, 718 Ohm's Law, 633, 638, 640 et passim Oil in painting, 501 ; on waves, 279 Opacity, 497, 722, 744 Opalescence, 503, 523 Open Circuit, 613, 614, 617 Opera Glass, 572 Ophicleide, 453 Ophthalmoscope, 572 Opposition of Phase in Resonance, 446 Optic Axes, 557, 599 Optical Centre, 534, 541 ; Density, 509 ; measurement of Length, 32 Ordinary Ray, 553, 555 Ordinate, 11 Organ : pipe, 412, 446 ; open, 451, 452, 457, 462 ; stopped, 454 ; -reed- pipe, 429, 451 Oscillating Currents, 721-733, 735, 740 Oscillation, 39, 414 ; as a means of measuring Force, 39, 213, 608, 680 ; INDEX. 773 Centre of , 164, 214 ; of Elastic Body, 267 ; Frequency, 83, 112, 412, 414, 435, 479 ; s of a mercury- column, 147 ; of Molecules, 351, 479 : of Pendulum, 34, 21? Oscillatory movements of Systems of Particles, 103-142 Osculating Circle, 79 Osmosis, 285 : Osmotic Pressure, 281, 288 Ounce, 13 Outflow, 302, 303, 334; from elastic tubes, 323 Overcooling, 281 Overturning, 207-210 ; Angle of, 209 Oxygen, Condensation of, 224, 232; magnetic properties, 232, 685 ; spe- cific heat, 369 Ozone, Condensation of, 232 ; forma- tion of, 662 PAGE-EFFECT, 737 Palladium, 327, 664 Parabolic Mirrors, 122, 525; Path, 72 ; Surface of rotating liquid, 169, 294 Paradox, Hydrostatic, 292 Parallel Beam, 117 Parallel, Cells arranged in, 617 ; Dyna- mos in , 730 Parallelepipedon of Accelerations, 68 ; of Forces, 145 ; of Velocities, 67 Parallelogram of Accelerations, 68 ; of Forces, 143; experimental proof of, 144 ; of Velocities, 61 Paramagnetic, 685, 688 Partially-polarised, 516, 521, 522 ; de- tected, 563 Partials (i.e., Harmonics, ^.v.) Pascal's Principle, 290, 333 Peltier's Effect, 649, 650 ; 's Electro- scope, 605 Pendulum, 9, 34, 36, 86, 149, 206, 210 ; Ballistic , 214, 713 ; Blackburn's, 211; Compensation , 380; Com- pound, 213 ; Conical, 80, 185 ; energy of, 142 ; Formula, 212, 680 ; Gali- leo and , 34, 202 ; Hydrostatic , 318, 336; -- Oscillations Isochro- nous, 34, 212 ; movement Har- monic, 86, 211 ; Length of Simple , 206, 212, 214; of Compound, 213; Simple , 206, 210-213 ; Work done in moving a Simple , 213 Penumbra, 547 Perception of Colour, 573, 575 ; of Form, 576 ; of Matter, 216, 246 Percussion, Centre of, 165 Perfect Conductors and Non-conduc- tors, 588 Perfect Elasticity, 265, 267 Perfect Engine, 397, 399 Perfect Gas: denned, 369; Internal Work in, 372; No Physical Gas Per- fect, 374 Perfect Solid, 225 Perfectly Conducting Molecules, 692 Period of S.H.M., 82, 414, 680; of Wave-Motion, 112 Periodic Curve, 103 ; Function, 82 Permanent Magnet, 674, 689, 692 Permeability, Magnetic, 671, 676, 682- 684, 685, 686, 691, 694, 703, 706-709, 722, 744, 746 ; measured, 687 Permittance, 592, 597, 599, 603, 604, 612, 692, 709, 744, 746 Permittivity, 597 Perpetual Motion, The, 7, 193, 195, 357, 385 Phase in S.H.M., 82; perceived by Ear, 459 Phonautograph, 433 Phonograph, 433, 448 Phonomotor, 457 Phonophore, 735 Phosphorescence, 504, 542, 656 Phosphoroscope, 505 Phosphorus, 282 ; Vapour, 243 Photichthys, 507 Photographic Focus, 541 Photographing Vibrations, 433 Photography, 433, 481, 503, 569; Ex- posure in , 482 ; Pinhole , 548 Photometry, 513 ; Unit in, 513 Photophone, 637, 737 Physiology, 1 Pianoforte, 416, 420, 425, 437, 450 Piccolo, 453 Piezometer-tube, 295, 310, 334 Pigments, 575 Pile, Dry, 605, 622; Galvanic, 608, 616 ; Thermo-electric, 542, 628 ; Vol- ta's, 617, 618, 623 Pint, 13 Pipette, 343 Piston, 40, 55, 229, 237, 356, 399 Pitch of a Screw, 173 Pitch of a Sound, 415, 483; musical Specification, 420 ; physical, 418 ; standard , 421 ; of a Vibrating String, 435, 441 ; Variations of in an Organ-pipe, 452 ; Variations of with Temperature, 452 Pitot's Tubes, 319, 336 Plane in Space, Pendulum Swinging in fixed, 148 ; Wheel rotating, 149. Plane of Incidence, 517, 619, 521 Plane, Inclined, 172, 173, 181 Plane of Polarisation, 521, 522, 555, 559, 566, 694 Plane-polarised, 514, 521, 555, 558, 560, 666 ; detected, 563 Plane Wave-Front, 115 ; reflected, 119 Plant6, 665 Plasticity, 259 ; of Ice, 585 Plates, Vibration of, 443 774 INDEX. Pneumatometer, 336 Pneumothorax, 337 Points : Conjugate, 130, 526, 535 ; Cor- responding, 576 ; Critical, 233, 325 ; Dead, 89 ; Neutral, 626, 651 ; Nodal, 108, 134, (Jauss') 539 Poiseuille's Law, 308, 315 Poisson's Ratio, 260 Polarisation of Light, etc., 514 ; Plane, 514; Circular, 514, 560, 562-564; Elliptical, 515, 560; Partial, 516; Right- and Left-Handed, 515, 517, 560; Rotatory, 517, 566; Angle of Complete , 521 ; Plane of , 521, 555, 559, 566, 694; Magnetic Rota- tory , 694 ; in Electromagnetic Waves, 743, 745 Polarisation of Dielectric, 603; of Electrodes, 664 ; of Galvanic Cell, 619, 632 ; -Current, 664 Polarised Light, 514 ; to detect , 563, 564 Polariser, 516, 555 Pole : Magnetic, of Earth, 679 ; of Magnet, 668, 675, 685, 691 ; Secon- dary, 677 Polygon of Accelerations, 68 ; of Forces, 145; of Velocities, 66, 68; Skew- polygon, 68, 145 Porosity, 220 Position, Change of, 14 Positive Crystals, 555 ; Direction up- wards or to the right, 11 ; Direc- tion of Electromagnetic Lines of Force, 668; of Magnetic, 676; Electrical charge (vitreous), 579, 602, 746 ; Electrodes, 657 ; Mutual Action Repulsive, 189 ; Pole of the Earth (Antarctic), 693 ; Pole of a Magnet (north-seeking), 668, 675, 746 ; Side of a Circuit, 672 ; Thermo-electrically , 625 Potential, 52, 191, 301, 583, 608, 692 ; Analogy of Sea-Level, 193, of Water- Level, 193, 195, 586 ; Analogy of Temperature, 588 ; Absolute , 192, 584 ; Absolute Zero of , 192 ; Ar- bitrary Zero, 193, 588 ; Continuity through Zero Value, 193, 588 ; Difference, 193, 584, 586, 587, 595, 597, 604, 609, 612, 623, 624, 709 ; meas- ured, 608, 646 ; produced, 609- ; electromotive difference of , 587, 608, 609, 616, 625, 646 ; of Air, 607, 611 ; of Double Sheet, 200 ; Elec- trical , 583 ; Fall of , 638, 641, 650, 654, 656 ; Gravitation , 193, 195, 492, 584 ; Gradient, 191, 585, 587, 597, 598, 602, 633, 639, 648, 710 ; Line, 639, 641, 696, 720, , ; Magnetic , 682, 693 ; of Mag- netic Shell, 683; Mutual , 191; Positive and Negative , 192; Slope, 191, 585, 587, 597, 598, 602, 604, 633, 639, 648, 710 ; The , 588, 609 ; Zero , 192, 193 Potential Energy, 43, 45, 354 ; in cur- rent-field, 684, 699, 702 ; in case of Repulsion, 190 Potentiometer, 646 Pound, 12 Poundal, 21 Powdered Transparent Substance, 497 Power, - Activity, 42, 59 Power, Horse-, 42, 647, 653, 655 Power of Lens, 534 Power, Molecular Refractive, 579 Powers, the Mechanical, 169-176, 180 Practical Electrical Units, 618, 634, 637, 711 Precession, 75 ; Angle of, 75 Pressure, 24 ; Atmospheric, 229, 293, 336-349; Critical (Carnelley's), 236; Critical (Andrews'), 233, 376; Electric , 587 ; Electricity on , 623; an Energy-Slope, 42, 603; of Ether-Waves, 570 ; affecting Fusing- Point, 237, 384 ; in a Gas, = coefft. of elast., 324, 368 ; in Gases, 229, 333, Dalton's Law, 250, 253, by Kinetic Theory, 250; pres- sure and volume in gases, 230 ; produced by heated solids, 377 ; di- minished in Fluid in Motion, 335 ; -Gradient, 301 ; Head, 310, 334 ; Hydrostatic , 25, 290, 301, 310, 325, 602 ; Intensity of , 25 ; Line, 311, 334 ; in Liquids, 289, 602 ; in Heavy Liquids, 291 ; measure- ment, 295-299 ; Osmotic or Solu- tion , 281, 288; Restitution , 256, 264, 266 ; Saturation , 390 ; Slope, 301 ; in Streams, 309- 317 ; of Sunlight, 570 ; Total , 24 ; Vapour , 387, of a Solution, 281, 387 ; Vibrating Fluid, 335 Prevost's Law, 491 Primary Circuit, 700, 724 Primary Colours, 575 Principal Focal Distance (= Focal Length), 130, 525, 534, 535, 536 Principal Focus of Mirror, 525 ; of Lens, 534 Principal Section of a Crystal, 551 Principle of Moments, 155 Prism, 485, 493, 528, 743 ; achromatic , 531; Foucault's, 556; Nicol's, 555, 559, 568 ; Rochon's, 556 Projectile, Path of, 203 Projection, 84 ; of S.H.M., 84 Proof-plane, 607 ;' proof -sphere, 607 Propagation of Elasticity- Waves, 267 ; of Electromagnetic Disturbance, 479, 480, 481, 498 ; of Groups of Waves, 142 ; of Heat, 409, 602 ; of Sound, 428, 456-465 ; of Temperature, INDEX. 775 409 ; of Waves along Normals, 113, 116, 131, 139 Properties of Matter : Contingent, 220 ; Essential, 216 ; General, 219 Proportion, fixity of (chem.), 238 ; Multiple proportions, 239 Ptolemy's Law, 133 Puissance, 42 Pull, 23 Pulleys, 174 Pulse : dicrotic, 322 ; locomotive, 313 ; wave, 321 Pump, 342, 345, 587 Putting to Earth, 698, 719, 720 Pyrometry, 404 Pythagorean Intervals, 425 QUADRANT, 711 ; of Earth, 10 Quadrant Electrometer, 606 Quadruplex Telegraphy, 734 Quality of Sound, 415, 428 Quantity : Cells arranged in , 641 ; Electric , 578, 594, 603, 638 ; unit of electric (C.G.S. electrostatic), 578, 603, 604 (C.G.S. electromag- netic), 675, 693, 714 (practical), 711, 714 ; of Magnetism, 675, 193 ; - of Matter = Mass, 12, 166, 216, 247 ; of Motion (= Momentum j, 6, 19, 149, 150, 166, 351 Quarter-undulation Plate, 561, 563 Quartz as a Dielectric, 599 ; Prisms and Lenses, 486, 498, 504; in Rotatory Polarisation, 666 RACEMIC Acid, 568 Radian, 75 Radiant Heat, 48, 234, 350, 410, 481, 541, 743 ; chemical decomposition by, 482 Radiant Matter, 234, 252, 656 Radiation, 410, 478 ; from a hot body, 488 ; from gas and vapour, 495 ; _ from liquids and solids, 495 ; exchange of radiations, 489 ; and absorption, 491 ; Intensity of , 513 Radii of Earth, 205 Radiometer, 361 Radiophony, 455 Radius of Curvature, 79, 165 ; of Gyration, 162 ; of Inertia, 162, 166 Railway : Force of Engine, 181 ; Super- elevation, 209 ; wheels, 182 Rain, 348 Rainbow, 530 Raindrop, 255, 272, 530, 579; friction on , 185 Ram, Hydraulic, 148, 312, 704 Raoult's Laws of Freezing Point, 386, Osmotic Pressure, 288, and Vapour Pressure, 387, in Solutions Rarefied Air, Sound in, 413 Rate of Change of Momentum = Force, 19, 154, 248 Ratio, Electrostatic-Electromagnetic, 708, 709 Ray, 116, 131 ; Kinds of Radiation, 480 ; Convergent s, 117 ; Divergent, 117 ; Ordinary and Extraordinary Rays, 553 Reaction, 6 ; Action and , 6, 23 Real Image, 526, 535, 536 Reciprocal or Reciprocating Motion, 88, 89 Reciprocating Engine, 397 Recoil, 306, 334 Reduced Length, 637, 641, 642 ; Re- sistance, 637, 641, 642 Reduplication, Principle of (in Pulleys),. 174 Reeds, Vibrating, 442, 443, 451,. 736- Reference to Axes, 11,. 66 Reference, Circle of r 81, 82, 185 Reflexion, 117-124 ; Angle of , 120, 518 ; Caustic by , 123, 525 ; of Ether- Waves, 517, 743; Metallic , 502, 521, 561, 743, 745 ; of Negative Waves, 125, 458, 463, 518, 542, 545 ; in electromagnetic waves, 727, 743 ; of Sound- Waves, 459 ; Total , 519 Refraction, 124-131, 217 ; Angle of , 126, 518, 528 ; Caustic by , 129, 130, 537 ; Conical , 557 ; Double , 228, 551-566, 575, 599 ; of Elec- tromagnetic Waves, 743 ; of Ether- Waves, 517, 527 ; Index of , 127, 509, 518, 522, 528, 534, 540; of liquid, 529, of solid, 528 ; for X = oo, of Sound- Waves, 461 ; Total , 520 Refractions, Atomic, 529 Refractive Index, 127,509, 518, 522, 528, 534, 540, 744 Refractive Power, Molecular, 529 Regelation, 385 Relaxation, Time of, 228 Relay, 735 Reluctance, 691, 714 Reluctivity, 714 Replenisher, Lord Kelvin's, 631 Repose, Angle of, 178 Repulsion Conventionally Positive, 189 ; Direction of, 584, 585 ; in Field of Force or Flow, 602, 721 ; Potential Energy in case of, 190 ; Work done by, 190 ; Repulsion of Resonator, 431 ; Self (elect.), 582 Research, 8 Residual Discharge, 599 Residual Magnetisation, 684, 690 Residual Restitution, 266 Resinous (negative), 579, 746 Resistance to Electric Current, 615, 616, 633, 634, 638, 639, 644,^54, 710, 725 ; 776 INDEX. measured, 633, 645, 647, 718 ; reduced do., 637, 641, 642 ; Coil, 634, 646 to Deformation, 259, 264, Compres- sion, 259, Extension, 261, Shear, 260 Twist, 263 ; of Electrolytes, 646 ; to Flow, 306, 310 ; Friction a , 176, 180 1 ; to Traction, 181 ; a Ve- locity, 710 ; Viscosity s, 185 Resistivity, 634, 638, 689, 710 Resolution, of Forces, 143 ; of S.H.M.'s, 96, 102 ; of Velocities, 63 ; of Vibrations, longitudinal, 110, transversal, 107, 108 Resonance, 430, 445, 742 Resonators, 417, 430, 431, 432, 476; Herz's , 741 ; Repulsion of , 431 Restitution-Pressure, 256, 264 ; de- ferred , 266; electric, 744; Co- efficient of R. (impact), 151, (elas- ticity), 264, of deferred R., 266 Resultant Electric Force, 581, 583, 603 ; Force, 142 ; Harmonic Motion', 102 ; Motion, 60 Retardation, Electrostatic, 596 Retina, 570, 573, 575, 576 Reversal in Thermo-Electricity, 627, 651 ; Temperature of , 627 Reversed Action, 611, 737 Reverse Current, 650, 664 Reverse Extra-Current, 705 Reverse E.M.D.P., 654, (electromo- tor), 738 Reversibility of Carnot's Ideal Engine, 397 ; of Lenses, 534 Rheochord, Rheostat, 647 Rhomb, Fresnel's, 562 Right-handed Polarised Light, 515, 560 Rigid- Body, 73 ; degrees of Freedom of, 76 ; Solid, 225 Rigidity, 225, 227, 260, 415 ; coefficient of, 226, 260 ; through Rotation, 226 ;. affecting Vibrations, 415, 435 Rings, Newton's, 546, 743 ; Nobili's, 663 Rochon's Prism, 556 Rods, Flexure of, 263 ; Vibrations of, 135, 441, 443. Rolling down a Curve, 173 Rolling Friction, 182 Rope and Post, Friction between, 178 ; wetted rope, 282 Rotating Mirror, 512, 525, 607 Rotation, 74, 161 ;< Axis of , 74, 76, 162; Composition of s, 74; of the Earth,, 164 r , 205 ; Energy of of a particle, 162, of a Mass, 163; Force causing Constant in Direc- tion, 158 ; Instantaneous Axis of , 75 ; round Lines of Force, 745 ; of a Liquid, 168 r 294; of Magnet-Poles round Currents, 669, of Molecules, 351, 465; of Plane of Polarisation, 244, 517, 566 ; of Plane of S.H.M., 101 ; Simple , 74 ; Axis of Spontaneous , 165 Rotatory Field, 741 ; Polarisation, 517, 566 ; Magnetic , 694, 745 ; Power, 567 ; Vibration, 443 Rowland, 352, 550, 744 Ruhelage, 39, 263 Rupert's Drops, 255, 388 SACCHARIMETER, 567 ; Soleil's, 568 Safety-tube, 340 Salt- radicle, 658, 660 Sand-blast, 258 Saponine, 278 Sassafras, 547 Saturated Solution, 280 ; Steam, 371, 390 ; Vapour, 231, spec, heat of, 371 Saturation-Pressure, 390 Savart's Wheel, 418 Scale (measuring), 27, (musical), 420, 422, (therometric), 364, 402. Scattering by Haze, 503 ; in electro- magnetic waves, 743 Schallenberger's Alternating Current Meter, 741 Screen, Electric, 601 ; Magnetic, 691, to Alternating Field, 722 ; effect of S. on Waves, 138, 139 Screw, 29, 173, 181; male and female, 30 Sea-Level, Analogy in Potential, 193 Sealing-wax a fluid, 226 Secohm, 711 ; Secohm-Meter, 706 Second, 9 Secondary Batteries, Cells, 665-667 ; Circuit, 700 ; Current (induction), 687, 700, 724, (polarisation), 665 ; electrolytic reactions, 659 ; Poles, 679 Selective Absorption, 498 Selenite, 563 Selenium, 570, 636 Self-induction, 646, 704, 710, 722, 724, 727, 745 ; Coefficient of, 705, 710 Self- Repulsion : electric, 582 ; of Hy- drogen, 254, 375, 377 Semitone, true (if), 422 ; so-called (Iff). 423 Sensible Heat, 350 Sensitive Flames, 454 Sensitiveness of Galvanometers, 713 ; of Thermometers, 402 Separately Excited Dynamos, 730. Separation of Electricities, 602, 609 ; of Magnetisms, 686, 689, 692 Series, Cells coupled in, 618, 640; Alternators in Series, 730 ; Series Dynamos, 732 Series-Shunt Dynamos, 732 Sextant- Vernier, 28 Shadow, 140, 547 S.H.M. (Simple Harmonic Motion) 80. See Harmonic Motion. INDEX. 777 Shear, 78, 227, 260 ; Ether- , 746 Shearability, 260 ; of Ether, 746 Sheet, Current, 693 Shell, Momentum of explosive, 149. Shell, Magnetic, 682, 707 ; Equivalence of and Circuit, 683, 707 ; Poten- tial of, 683 ; Strength of, 682, 693 Shunts, 645, 661 ; Shunt Dynamos, 732 Siderial Time, 9 Siemens's Electrodynamometer, 717 ; Governor, 168 ; Inductor, 729 Simple Harmonic Motion, 80 et seq. ; Variations of Current, etc., 721, 722, 729 Simple Machines, 169 Simple Pendulum, 206, 210-213 Simple Rotation, 74 Simple Translation, 73 Simultaneous Causes, 4; Currents, 644 ; Forces, 143, 145 ; Motions, 4, 60 Sine-Galvanometer, 713 Sines, Curve of, 85, 97, 106, 111 Singing Flames, 454 Siphon, 345 Skew-polygon, 68, 145 Sky, 503 Sliding, 176, 177, 210 Sliding Condenser, 600 Slope: Energy-Slope, 42, 603; Inten- sity-Slope, 200 ; Lines of Slope, 200, 687 ; Potential-Slope, 191, 585, 587, 597, 598, 602, 604, 633, 639, 648, 710 ; 5. of Potential-Line, 639, 641, Pres- sure-Slope, 301 Soap-bubble, 37, 330, 582 ; soap-film, 37, 245, 273, 330, 498, 545 ; electro- magnetic waves, 743 Sodium-flame, 483, 493, 502 Soft Iron, 684, 689 Softness, 258 Soft Solid, 226 Solar Time, 9 Soleil's Saccharimeter, 568 Solenoids, 673, 674, 705, 726; Sole- noidal Magnet, 675. Solid, 12, 225, 256-270; rigid, 225; soft, 226 ; perfect , 225 Solubility, 328 ; coefficient of, 280, 328 Solution': Saturated , 280; super- saturated , 281 ; pressure, 281, 288 ; Density of a , 281 ; Dissocia- tion upon, 248, 280, 386, 590, 614 ; s as conductors, 590 Solution in Liquids : of Solids, 279, 358 ; of Gases, 328, 358 ; Coefficient of Solubility, 280, 328. Solution of Gases in Solids, 327 ; in gases, 330 ; of Solids in gases, 330 Sonorescence, 570 Sound, 7. 48, 412-477 ; Analysis of, 429 ; Direction of, 471 ; Propagation of, 428, 456-465; Velocity in Steel, 267; Waves in Air, 413, 432, 456, 464, 737 Sounding-Board, 413, 737 Source and Condenser, 384, 396, 615, 648, 652 Space, 9; Dimensions of, 10; tra- versed under Uniform Acceleration, 70 Spark, 580, 582, 589, 590, 612, 706, 741 Speaking-Trumpet, 414 Specific Conductivity, 634 ; Density, 220 ; Measurement of, 221-224, 294, 295, 382 ; Gravity, 221 ; bulbs, 223 ; Heat, 353, 365, 370, 404, 405, 653 ; at const, vol., 367 ; at const, pressure, 367 ; ratio of Heats, 368 ; Differences in this Ratio, 369 ; inductive capacity, 597, 602-604, 612, 692, 709, 744, 746 ; of dielectric = /3 2 , 744 ; Resistivity, 634, 638, 689 ; Thermal Capacities, 324, 365, 367, 372 Spectrum, 484, 486, 494, 530; Abnor- mal, 532 ; Band , 496 ; Continuous , 495 ; Dark Lines in, 494, 495 ; Diffraction , 550 ; Heat- , 486, 500, 533 ; of Ice, 495 ; Line , 495, 496 ; Normal , 141, 550 Spectrum Analysis, 217, 494 Speed, 15 Sphere, Capacity of, 593, 599 Spherical Aberration of Lens, 537 ; of Mirror, 525 ; Form, 254 ; Mag- net, 678 ; Mirror, 123, 525, 527 ; Wave, 128, 130, 514 Spheroidal State, 363, 623 Spherometer, 31 Sphygmoscope, 298 Spinning-Top, 75 Spirometer, 336 Spontaneous Rotation, 165 Spoud, (C.G.S. Unit of Acceleration, one kine per second), 18 Sprengel-pump, 252, 326 Spring Balance, 37 Squares, Law of Inverse, 187, 582, 585, 675 Stability, 215 Standards: See Units; Cell, 622, 646; Condensers, 609, 718 ; Ohm, 718 ; Pitch, 421 ; Resistance- Coil, 634, 646, 718 Standard Atmospheric Pressure, 348, 349 Stapes, 467 Stars, Twinkling of, 550 State, Critical, 232, 376 States of Matter, 225; Change of , 235, 354-358 Statical Friction, 176 Stationary Vibrations, 134 Steady Currents, 632-674, 721, 745 Steady Flow of Heat, 407 : of Liquids, 299 778 INDEX. Steadiness of Flow, 300 Steam : Steam-engine, 40, 49, 352, 397, 399 ; Steam-gauge, 297 ; Jet issuing into air, 373, 374 ; latent heat of, 390, 411 ; specific heat of, 371 ; saturated , 371, 390 ; thermal capacity of , 371 Stereoscope, 576 Stethoscope, 428, 456 Stokes's Law, 492 Stopping Component Vibrations, 135 Storage of Energy, 666 ; in the Ether, 586, 593, 603, 648 Straight Path, 5, 57 'Strain, 23, 77; Homogeneous , 78; Strain-Ellipsoid, 78 Streams, 308-323; Stream-Lines, 300, 602 ; Streams of Liquid, 299 ; pres- sure in, 309-317 ; velocity of, 317- 319 ; viscosity in, 301, 307 ; work done in keeping up, 319 ; of Gas, 334 Strength of Current, 633, 637, 638, 659, 684, 694, 703, 707, 709 ; alternat- ing, 728, 726 ; of Magnet, 675 ; of Magnetic Field, 676, 693, 701 ; of Magnetic Shell, 682, 684, 693, 710 ; of Materials, 255; of Structures, 269 Stress, 23, 43, 145, 187, 579 ; Intensity of , 24 ; Measurement of, 37 ; , Electric, 234, 577, 582, 586, 593, 599, 602, 603 ; Hydrostatic, 225, 257, 300 ; Effect of Eepeated Variations of, 267 Stria?, 656, 662 Striking Distance, 580, 589, 590, 613 String Organ, 451 String, Tension in Stretched, 26 Strings, Vibration of, 423, 434-441, 446 ; when bowed, 435 ; plucked, 434, 436 ; struck, 436 Stroboscopic Disc, 305 Stromuhr, 318 Structures, Strength of, 269 Sublimation, 386 Submarine Cable, 600, 696 ; Teleg- raphy, 600, 696-699, 734 Sucker, 339 Suction, 305, 339 Sulphurous Acid, 325, 363, 389 Summational Tones, 473, 474 Sun's Atmosphere j 218, 486, 494, 496 ; 's Attraction, 6, 76, 204 ; En- ergy, 50, 478 ; Sunlight, 50, 478, 486, 488, 491, 496 ; Energy of, 479 ; Pres- sure of, 570 ; Sun-motor, 491 ; Sun- set, 527 ; Sun's disc, 491 ; Sun's magnetic effect, 679 ; Sun's Tempera- ture, 489, 491 Superelevation of Rails, 209 Superficial Charge of Conductor, 579, 583, 600; Density (elect.), 579, 583, 603, 604, 612 (magn.), 683, 686, 692, 693 ; Film, 254 ; Tension, modulus of, 276 ; Viscosity, 278 Supersaturated Solution, 281 Surf, 279 Surface, 11 ; Adhesion, 306 ; At- traction of a Gas, 327 ; Cells coupled in , 617, 640; Conduction of Currents, 696, 721 ; Density, 188, 579, 583, 603, 604 ; Equipotential s, 193-199, 301, 409, 582, 583, 585, 588, 592, 595, 599, 602, 648, 672, 676, 682, 683; of Falling Bodies, 203; Friction, 306 ; in Gases, no Free , 229 ; Isodynamic s,200; Isothermal s, 408, 602 ; Ley den jars coupled in , 600 ; of Liquids, 169, 272, 294 ; Measurement of, 32 ; Tension, 26, 254, 272, 294, 305, 579, 603, 624 ; measurement of, 37, 275 Susceptibility, Magnetic, 686, 690 Suspended Body, 164, 209 Suspension, Bifilar, 215, 606 Swing-back, 538 Synthesis of Sound, 432 ; of Vowel- Sound, 476 Syren, 418 Syringe, 339 TANGENT, 57 ; Tangent-Galvanometer, 637, 710, 713, 718 ; -Scale, 607 ; Screw, 29 Tangential Velocity, 59, 165 Tasimeter, 551, 636 Teinte de passage, 568 Telegraphic Code (Morse) 733 ; Re- lays, 735 Telegraphy, 644, 679, 697, 727, 733; Deep Sea , 600, 696-699, 734 ; Du- plex , 734 ; Harmonic , 736 ; Mul- tiplex, 735 ; Quadruplex, 734 Telegraph Wire, 600, 643, 739, 740 Telephone, 471, 644, 646, 701, 736 ; -Currents, 737 ; Mechanical Pulsion , 456 ; Wire , 456 Telescope, 549, 572 Temperament, Equal, 425 Temperature, 249, 250, 359, 364, 399, 588 ; Absolute , 251, 364, 397, 399 ; true C.G.S. unit of , 365 ; of Condensation of Vapour, 392 ; Crit- ical , 232, 237, 253, 376,390, (mag- netic) 689; Flow of , 409; -Gradient, 407, 602 ; Measurement of , 400 ; Propagation of , 408, 409 ; of Reversal, 627 ; Waves of , 409 T:nacity of a Liquid Stream, 254, 345 Tension, 25, 167, 183, 184, 260, 415 ; Intensity of, 26 ; Electric Surface , 582, 603 ; Cells coupled in , 640, 666 ; of String in circular motion, 167 ; Surface , 26, 252, 272, 294, 305, 579, 603, 624 Tenth-metre, 480, 483, 575 Terrestrial Magnetism, 679, 691 Tesla's Experiments, 723 INDEX. 779 Theorem, Fourier's, 103, 135, 266, 412, 414, 416, 417, 434 Theory, 8 Theory of Dimensions, 16 Thermal Capacities, 324, 365, 367, 370, 372 Thermal Conductivity, Coefficients of, 407, 602, 636 Thermal Diffusivity, 407 Thermic Balance, 542, 551, 717 ] Thermodynamic Constant, 370 Thermodynamic Function, 395 Thermodynamics : First Law, 353 ; Second Law, 398 Thermo-electricity, 612, 624, 649-652 Thermo-electric Circuit, 624, 651 ; Diagram, 626, 651 ; Effect in Elec- tric Arc, 654 ; Pile, 542, 628 ; Power, 625, 689; Eeversal, 627, 651 ; Series, 625 ; Thermome- ter, 404, 628 Thermo-electrically positive and nega- tive, 625 Thermolysis, 243, 247, 248, 355, 367 Thermo-magnetic Motors, 689 Thermometers : Air, 401 ; BrSguet's, 401 ; Mercury, 401 ; special forms, 403 ; sensitiveness of s, 402 ; thermo-electric, 404, 628 ; Bolometer, 717 , Thermometric Conductivity, 407, 408 Thermometric Scales, 364, 402 Thermopile. See Thermo-electric Pile Thermoscope, 401 Thomson's Effect, 650 Thoracic Duct, 309 Throttling, 722, 736 Tide, 6, 204, 205, 220 Tide-calculating Machine, 103 Timbre of a Sound (Quality or Charac- ter), 415, 428 Time, 9 ; measurement of, 34 ; Sidereal , 9 ; Solar , 9 ; unit of, 9 ; Time of Relaxation in Canada Balsam, 228 Tone, 418 ; Combinational s, 473 ; Differential, 473 ; Summational, 473, 474 Torque, 158, 160, 166, 263, 676, 681, 691, 716, 723 Torricelli's Law, 302, 330, 334; Vacuum, 342, 344, 349 Torsibility, 263 Torsion, 39, 263, 606, 607, 680 Torsion Balance, 607 Total Induction : electric, 596, 598, 604 Total Number of Lines of Force : elec- tric, 583 ; magnetic, 685 Total Pressure, 24 Total Reflexion, 519 Total Refraction, 520 Total Tension, 26, 167, 260 Toughners, 262, 265 ; Elastic , 265 Tourmaline, 556, 623 Tourniquet, Hydraulic, 306 Traction, 26, 181, 260, 435, 444, 582, 583; Electricity on , 623; Fric- tional Resistance to , 181 Transference of Heat, 406 Transformations of Energy, 47 et passim Transformers, 724 ; Impedance in, 725 Transition in Music, 423 Translation of Molecules, 351 ; of a Particle, 73; of a Rigid Body, 74, 76 Translucency, 497 Transmissibility of Fluid Pressures, 290, 333 Transmission of Energy, 182, 183, 230 ; 739; Lines of, 671, 699; by Steady Current, 632, 648, 739 ; by Oscillat- ing Current, 721, 735 ; by Intermit- tent Current, 733 Transmission of Light, Coefficient of, 499 Transmutation of Elements, 218, 219 Transparence, 497, 509, 744 Transpiration of Gases, 331 ; Coefficient of, 331 Transport of Heat, 411 Transversal Vibrations, 106-110, 412, 424, 440 ; of Ether, 479, 509 Travelling of Electric Condition, 685 ; of Molecules of Solids, 257 ; of Wave-Form, 104, 111 Trevelyan's Rocker, 455 Triangle of Accelerations, 68 ; of Forces, 145 ; of Velocities, 63 Triphase Electromotors, 741 Trombone, 422, 453 Trough and Crest, 104 True Contact-effect, 612, 616, 624, 627, 649 Trumpet, 453 Tubes, Capillary, 315 ; Elastic , 320- 323 ; Rigid , 309 ; Geissler's s, 656, 723; Piezometer s, 295, 310, 334 ; Pilot's s, 319, 336 ; Safety- ,340 Tubes of Flow, 602 ; of Force, 197, 583, 602 Tuning-Fork : 35, 48, 267, 335, 412, 414, 419, 421, 429, 430, 442, 446, 447, 456, 458, 471, 475, 477, 735 ; Electromag- netic Interrupter for , 267, 447, 735 Twinkling of Stars, 550 Twist in a Magnet, 688 Twisting Moment, 158, 160, 166, 263, 716, 723 ULTRA-GASEOUS Matter, 233, 252, 656 Ultra-red Rays, 486 Ultra-violet Rays, 482, 484, 486, 589 Umbra, 547 Uniaxial, 551, 694 Uniform Field, 198, 583, 608, 673 780 INDEX. Units : of Acceleration, 18 ; Angle, 75 ; Area, 12 ; Astronomical Units, 202 ; Density, 220. Electrical : Electro- dynamic, 670 ; Electromagnetic or Magnetic , 625, 634, 670, 684, 703, 707, 709 ; Magnetic Field, Intensity of, 677 ; Magnetic Force, 675 ; Mag- netic Pole, 676. Electrostatic: Capacity, 592, 603; Conductivity, 634; Density, 579; Difference of Potential, 584, 585, 603 ; Force, 578, 602; Inductive Capacity, 597, 604; Intensity of Current, 633 ; Quantity, 578, 579, 709 ; Resistance, 633 ; Re- sistivity, 634. Practical : Capacity (Farad), 711 ; Difference of Poten- tial (Volt), 587, 618, 711 ; Intensity of Current (Ampere), 711 ; Quantity (Coulomb), 638, 711 ; Resistance (Ohm), 353, 634, 637, 640, 711 ; Self- induction (Henry), 711 ; Elongation, 260 ; Force, 21 ; Heat, 353, 404 ; Length, 10; Light, 513; Mass, 12; Astronomical, 202 ; Space, 10 ; Time, 9 ; Velocity, 14 ; Volume, 11 ; Work, 41 Universal Gravitation, 203 Universe, Electricity in the, = 0, 581 ; Energy in the a constant, 47 Unsaturated Vapour, 231 Utility of Electromotor, 738 U-tube, 294 VACUUM, 235, 237, 360, 480 ; Discharge through , 656, 662 ; as an In- sulator, 589 ; Torricelli's , 342, 344, 349 ; _ -Tubes, 656 Valves, 346 ; of Heart, 346, 431 Van der Waals's Law, 375 Vapour (1) in presence of Liquid, ready to condense, 231 ; (2) saturated or unsaturated, 231 ; (3) a gas con- densible by pressure alone, 232; Boyle's Law in s, 390 ; saturated , 231, sp. heat of, 371 ; unsaturated, 231 Vapour-Density, measurement of, 391 Vapour- Friction, electrification on, 623 Vapour-Pressure of Solution, 281, 387 Variable Period, 695-699, 721 Variation, Magnetic, 678 Variations in Barometric Pressure, 348 ; in Difference of Potential, 199, 737 ; of Conductivity, 737 ; in the Earth's Magnetic Force, 679; of Gravity, 22, 41, 205 ; of Stress, 267 Varley's Condenser, 664 Velocity, 14, 58, 59, 166 ; Absolute, 17 ; Average under uniform accelera- tion, 70 ; mean, 17, 79 ; Relative, 17 ; Uniform , 15, 58 ; Variable, 17, 58 ; Angular ,75, 166 ; Change of , 18, 68 ; in curved paths, 58 ; Mini- mum angular, 164 ; Resolution of into Components, 60, 63 ; Tangential , 58, 167 ; Conductance (elec- trostatic) a , 638 ; Electrostatic- Electromagnetic Ratio a , 708, 709 ; Magnetic Force possibly a , 746 ; Resistance (electromagnetic), a , 710 Velocities : Composition of two , 60 ; of more than two, 65 ; Parallele- pipedon of , 67 ; Parallelogram of, 61 ; Polygon of, 66, 67 ; Skew-Poly- gon of, 68 ; Triangle of , 63 Velocity of Ether- Waves, 480, 510, 512, 743 ; in metals, 637 ; of Propagation of an Electromagnetic Disturbance, 234, 635, 698, 743, 744; Measure- ment of do., 708 ; Head, 310, 334 ; of Irons, 658 ; of Molecules, 250, 465, 610, 613 ; of Outflow, 302, 305, 334 ; of Sound, 267, 461-465 ; of a Stream, 317-319 ; of Trans- mission of Telegraph Signals, 697 ; of Wave- Motion, 105, 112 ; in a Gas, 268, 370, 464 ; in an Elastic Solid, 267 ; in a Liquid, 461 Vena Contracta, 304 Venturi's Water-Meters, 314 Verniers, 28 Vibrating Body, Loading a, 445 , Vibrating Fluid, Pressure in, 335 Vibrations : Free , 434^45, 479 ; Forced , 445-449, 479 ; of Bells, 444 ; of Condenser, 737 ; of Cords, 133, 413; Transverse , 106-110, 412, 424, 435-441, 479, 509 ; Longi- tudinal, 110, 135, 441 ; of Discs, 443, 479 ; the Result of Elasticity, 151, 251, 266 ; as affecting Electric Discharge, 599 ; of the Ether, 479, 508, 509, 513, 593, 612 ; on Impact, 151, 251, 266; of Membranes, 137, 444, 448 ; of Molecules, 247, 251, 351, 479, 506 ; as affecting Mag- netic Permeability, 684, 687 ; photographed, 433 ; of Plates, 443 ; of Reeds, 442, 443, 451, 736 ; Resolution of, 107, 110 ; affected by Rigidity, 415, 435; of Rods, 135, 441, 443 ; Stationary , 134 ; Stopping Component , 135 ; of Stretched Strings, 413, 434-441, 446. Vibrations, Method of (magnetometry), (580 " Vibrations Simples," 414 Vibrator, Herz's, 741 Villari's Critical Value, 690 Viola, 450 Violin, 412, 415, 422, 425, 435, 436, 438, 449, 450 Violoncello, 450 Virial, 249 INDEX. 781 Virtual Image, 525, 527, 535, 537 ; Focus, 527, 535 Viscosity, 185, 186, 226, 227, 579, 703 ; Coefficient of , 226, 307, 316; Kinematical Coefficient of , 227 ; in Gases, 251, 404, 428, 477 ; in Liquids, 254, 300, 307, 315; in Liquid Streams, 301, 307 ; Magnetic , 716 ; Resistances, 185 ; in S.H.M., 185; of Elastic Solids, 267 ; of a Sounding Body, 414, 415 ; Superficial in Liquids, 278 ; in Gaseous Streams, 331, 334 Vis viva, 517 Vision, Energy in, 573 Vitreous (positive), 579, 746 Voice, 475 Volatilisation, 236 ; of carbon, 654, 655 ; of Pt and Ir, 654 ; of Snow, 236, 390 Volt, 587, 618, 647, 711, 715 Voltage, 687, 609, 724, 732 Volt-meter, 609, 645, 649 Voltaic : see Galvanic Voltaic Balance, 617 Voltameter, 661, 662 Volta's Pile, 617, 618, 623 Volume, 12, 218; Critical , 233, 375 ; Density, 188 ; Elasticity of, 229, 259 ; Measurement of, 33 ; and Pressure in Gases, 230 Vortex-atom, Vortex-ring, 246, 745 Vortices, Magnetic, 234 Vowel, 475 WALKING, 6, 175, 181, 209, 212 Water : Conductivity of , 635 ; Den- sity of, 13, 220, 224 ; maximum density (at 3-9 C.), 13, 402; Elec- trolysis of , 658 ; Equivalent, 405 ; Level, 294 ; Meters, 314 ; Viscosity of, 308 Watt, 42, 647 Wave : in Air, see Sound ; in Ether, (chap, xv.) ; originated, 506, 741 ; pro- pagated, 508, 742 ; measured, 542, 544, 550 ; relation to Fringes, 138 ; Compression s in the Ether, 510, 744 ; Form, Travelling of, 104, 111 ; Velocity of Propagation of , 105, 112, 269, 370, 461, 464 ; of groups of s, 142 ; Front, 111, 112 ; Di- rection of, 115 ; Normal to, 115 ; -Length, 104, 112 ; Crest and Trough, 104 ; Loss of half wave-length, 125, 518, 545; -Motion: Energy of, 142, 414, 476 ; Velocity of, in Elastic gas, 267, 370 ; liquid, 461, 462 ; solid, 267 Waves : bidimensional, 112 ; concen- tric, 114; flat, 115; distorted, 116; linear, 104 ; tridimensional, 114 ; Ef- fect of Screen on waves, 139 ; Fre- quency of , 112 ; Interference of , 137 ; Propagated along Nor- mals, 113, 131, 139; Reflected, 117-124; reflected in Elastic Tubes, 321; Refracted, 124; Spherical, 128, 130, 514; -Surface, 557, 744 ; Traversing an Aperture, 116, 117, 131, 140 Waves of Potential, 697 ; of Propa- gation of Lines of Electric Force, 721 ; of Temperature, 409 ; Elec- tromagnetic , 479, 480, 481, 498 Webers, 715 Weber's measurement of v, 708; his Theory of Induced Magnetisation, Wedge, 174 Weighing, 333 Weight, 12, 21, 201, 219, 333; atomic , 217 ; measured, 37 Welding, 257, 258 ; Electric , 653 Well, depth of, 463 Wheatstone's Bridge, 645, 706 Wheel and Axle, 171 Wheels of Railway Train, 182 ; Fly 89, 164, 168, 169, 667; Friction , 182 ; Savart's , 418 Wheel work, 172 ; in a Clock, 34 Whispering Galleries, 460 White Light, 484, 485, 487 ; de- composed, 485, 530 ; recom- pounded, 531 Whitworth's Measuring Machine. 29, 31 Wickholder, 408 Winch, 172 Wind, 116, 348, 463 ; Work done against , 40 : Electrical , 580 Wire Telephone, 456 Work, 40 ; Unit of, 41 Work done, 7, 40 ; Mean Rate of doing , 42 ; done against Attraction, 192 ; in Charging a Conductor, 593, 599 ; in producing Com- pression, 260 ; by Ether- Waves, 670 ; by Expanding Substance, 54, 359, 360, 394, 396 ; in producing Exten- sion, 260 ; in Electrolysis, 662 ; in moving across Equipotential Surfaces, 194, 685 ; by or against Force, 40 ; against Friction, 181, 185 ; by the Heart, 320 ; by Heat, 359, 394 ; dar- ing Overturning, 209; in moving a Pendulum, 213 ; by Repulsion, 190 ; in producing Shear, 260 ; in keeping up a Stream of Liquid, 319 ; in pro- ducing Twist, 263; Internal in Gases, 359, 369-371, 375 Worm-wheel, 29 YELLOW Spot (in eye), 573 Young's Experiment, 547 Young's Modulus, 261, 321, 441, 443, 461 782 INDEX. 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