Wmm ":" :i :■■!': r.m ■:Xi:\:-i::-..[v::vi : -h ':'{'■ :;:«,■;! IMM'i - >!.;■■ y.:- J '■■: -r - m« (found from published tables), the true density of the liquid, at the same temperature, is W When a lower degree of accuracy suffices (say i in 200) and a good supply of liquid is available, the common hydrometer is more convenient. This instrument consists of a glass bulb (Fig. 6), weighted so as to float upright, and carrying a graduated vertical stem. When placed in a liquid it sinks till the weight of the liquid it displaces just balances its own weight (see §.8). It does not need to sink so far in a dense liquid as in a light one to satisfy this con- dition, and hence the point to which it sinks measures the density of the liquid, according to a scale previously marked on the instrument. Hy- drometers with scales of various ranges may be bought, and their accuracy checked by one or two measurements with the pyknometer. Sometimes it is convenient to test the density of a liquid by means of a set of glass beads of varying density. These, which have their densities marked on them, are dropped into the liquid till one that just floats and one that just sinks are found : the density of the liquid then lies between the two. When only a small quantity of the liquid is available— e. g. Fig. 6. 12 GENERAL PHYSICS blood— the best plan is to prepare a series of liquids of known densities and see in which of these it floats or sinks. The standard liquids may be mixtures of glycerine and water, con- tained in test tubes. The blood is contained in a capillary pipette, and a small drop of it is pressed out into the middle of such a tube ; if it rises, it is lighter than the mixture, and the next lighter mixture must be tried, till it has been found between which pair its density lies. The density of a solid which occurs in good sized pieces may be found by the method of the hydrostatic balance (areometric method) : the specimen is weighed in the usual way (W) : it is then hung by a very fine silk thread or platinum wire fronl an arm of the balance in such a way that the specimen is completely immersed in a beaker of water, the weight of the water being, however, borne by a support independent of the balance : it is then found that the weight appears less (W). The density (see § 8) is W n where D is the density of the water or other liquid in which it is hung. If water attacks or dissolves the substance in ques- tion, another liquid of known density may be used instead. The method is not applicable to solids lighter than the liquid used, except with the complication of a sinker. For light solids, however (i. e. most solids other than the metals), especially when occurring in small pieces, the most accurate and convenient method is to find a liquid mixture in which the pieces neither float nor sink, as described above for blood. The density of gases is found by methods essentially similar to those for liquids : for details, larger works should be consulted. § 4. Momentum : force. The next dynamical quantity to claim attention is momentum, expressively called by Newton the 'quantity of motion.' Amoving body is said to possess a momentum which is measured by the product of its mass and its velocity : thus the effective quantity of motion of a body may be large either on account of its having a large mass (e. g. a wagon) or large velocity (bullet). Momentum, MOMENTUM: FORCE 13 like velocity, is to be considered as having a direction, and con- sequently being positive when the body is moving in one sense, negative in the other. Newton's third law of motion— th& equality of action and re- action — may best be expressed in modern language in terms of momentum. It is that in any action between two bodies, the momentum gained by the one is equal to that lost by the other. In interpreting this, it must be borne in mind that momentum has a positive or negative sign, otherwise the statement will be reduced to nonsense. It may be illustrated by a numerical example ; thus, suppose a tennis racquet of mass 290 gms. moving with a speed of 400 cms. per sec. to collide with a ball of 60 gms. moving in the opposite direction at 700 cms. per sec. We will take the direction of motion of the racquet as positive : then the racquet possesses, before the collision, a momentum = 290 gms. x 400 — '- = 116,000, the ball 60 x ( - 700) = - 42,000. This information is not in itself enough to determine the motion : it would be necessary to know the elastic properties of the racquet and ball as well : if however we find by observation what happens to one of them, the law of motion will show what must happen to the other. Thus, suppose the ball is driven forward at 600 cms. per sec. ; it now possesses a momentum of 60 x ( + 600) = + 36,000. The momentum has been increased by the blow from - 42,000 to + 36,000, a total increase of 78,000 ; hence the momentum of the racquet must have decreased by the same amount, and become 116,000 - 78,000 = 38,000 ; the racquet is therefore still moving forward (its momentum is positive), but its speed has fallen to 38,000-5-290 = 131 cms. per sec. The law of equality of action and reaction really serves as a means of measuring masses. We have, it is true, defined a kilogram as being equal to the platinum standard at Paris : but we have provided no means of testing the equality. We now see that if two equal masses collide or otherwise interact, the gain of velocity by the one must be equal to the loss by the other, since in that way the gain of momentum by the one will be equal to the loss of momentum by the other. A means of testing this experimentally is given by Hicks's ' ballistic balance,' which is a device for projecting two bodies against each other with measured velocities and for measuring the velocities with 14 GENERAL PHYSICS which they rebound 1 . Thus suppose we have a standard kilo and another body supposed to be equal to it in mass (it may differ in material, shape, and size) : project the two in opposite directions with equal velocities, and (for convenience) arrange by means of automatic clips that they do not rebound after meeting. Then if they are really equal they will stop dead after the collision : for beforehand the one possessed a certain positive momentum, the other an equal negative amount. The first body loses all its- positive momentum, and the second therefore necessarily gains the same amount, which just serves to neutralize the negative momentum it had, and bring it to rest. If however the masses are not equal, the greater mass will have the more momentum (since the velocities are equal) and the two bodies after collision will move on together slowly in the direction in which the greater mass was moving. Standards of mass are not actually tested in this way, but by means of their weight : nevertheless the method is important because it shows that the conception of mass is intelligible and complete without introducing that of weight. The conception of force may be approached either from that of momentum or of acceleration. According to the former method, which agrees with the order of reasoning expressed by Newton's laws of motion, force may be defined as the rate of change of momentum : a force can be measured, therefore, by observing the amount of momentum it generates in a measured time, and dividing by that time. Hence as unit of force we have . momentum am. cm. per sec. force = T . = = gm. cm. per sec. per sec. time sec. The expression ' per second ' has to be repeated here, as in the precisely analogous case of acceleration. As this unit is im- portant, and its systematic name is lengthy, a special name has been given to it, viz. the dyne. The conception of force, which may be regarded as a matter of common knowledge, derived in the first place from the sense of muscular effort, is rendered scientifically accurate by con- sidering the first and second laws of motion. The first law is that ' Every body continues in its state of rest or of uniform motion in a straight line except in so far as it is made to change that state 1 For details see W. M. Hicks's Elementary Dynamics, p. 24. MOMENTUM: FORCE 15 by external forces.' So long as the body is at rest, or moving with uniform velocity, it possesses a constant momentum ; hence the effective force on it (measured by the rate of change of momentum) is zero. To say that the effective force on a body is zero, means one of two things, either (1) that there are no forces at all acting on it (this never actually occurs in nature), or (2) that two or more forces act on it, but in such a way as to neutralize each other's effect, i. e. produce equilibrium. A body may therefore be in equilibrium, and yet in motion, provided the motion be uniform and in a straight line. Thus if a heavy body be placed on the floor, it is subject to two forces, viz. the attraction of the earth, downwards (i. e. its weight), and an equal pressure from the floor, upwards : these two neutralize, and the body remains at rest. If, however, the body be placed on the floor of a lift, which is ascending with uniform speed, there is again no change occurring in its momentum, therefore' no effective force on it, so that again the pressure of the floor must be just sufficient to balance its weight. But if the lift be starting to go up, and consequently suffering an acceleration, the case is different : the body is then gaining momentum, so that an effective force is acting on it, which shows that the pressure of the lift floor must be different from (in fact greater than) the weight, and the two forces are no longer in equilibrium. When the several forces acting on a body are not in equilibrium the excess which remains in some direction is called the resultant of the forces, and con- stitutes the effective force producing change of momentum. It is to be noted that since the notion of momentum (or of Velocity) includes that of direction in space, a change of direction is a change of momentum, and therefore requires a force : thus it is only uniform motion in a straight line that implies absence of force : uniform motion, e. g. in a circle (stone tied to a string and swung round uniformly), requires a force constantly acting (tension of the string). The second law of motion may be stated as follows : Rate of change of momentum is equal to tlie force applied, and takes place in the direction in which the force is applied. This, it will be seen, includes the definition of force already given : it also gives information as to the resultant of several forces, as will be shown in detail below. A little consideration of the above statement, or of the nature i6 GENERAL PHYSICS of the unit of force, will show what is the relation between force and acceleration : the unit of force, the dyne, or gm. cm. per sec. per sec. may not only be regarded as (gm. cm. per sec.) per sec, i. e. as momentum divided by time, but as gm. x (cm. per sec. per sec), i.e. as mass' multiplied by acceleration. A force therefore may be considered as measured by the product of the mass it is acting on, into the acceleration which it produces on that mass. The cross relations between these quantities may in fact be simply illustrated by a diagram : velocity acceleration momentum force Thus each of the two lower quantities is the time-rate of change of the quantity standing above it ; while each of the quantities on the right is identical with the corresponding quantity on the left, multiplied by mass. One of the most important deductions to be drawn from the laws of motion is as to the property known as inertia. When a mass at rest is set in motion, an acceleration is produced in it, and therefore a force has to be exerted, independently of any frietional or other resistance to the motion. This inertia, opposing the starting of a heavy body (opposing equally its stoppage), is a familiar fact, evidenced by the behaviour of carts, bicycles, &c. : and it is nothing more than the essential meaning of mass -as shown by the second law of motion. A particular case, important to the c%"l*f physiologist, will serve to render the notion of inertia more definite. To study the contraction of muscles, an arrangement like that of Fig. 7 is used : the muscle, fixed at the upper end, is attached below to a lever, hinged at a, so that when the muscle is stirou- Fig. 7. lated the tracing point at the other end of the lever is lifted and makes a corresponding mark on the chronograph paper which it touches. If a mass of say m grams is hung at p immediately below the muscle, when the contraction occurs, an upward acceleration has to MOMENTUM: FORCE 17 be communicated to it : the tension of the muscle, which when at rest was equal to the weight of m grams, is consequently now somewhat greater. On the other hand, when the lever has risen nearly as high as it will go, and is coming to a stop, its inertia tends to cany it on : in other words, it is now suffering a negative acceleration (its velocity upwards is being lessened) and the tension of the muscle is some- what less than the weight. This fluctuation in tension causes an oscil- lation to be set up, and recorded on the chronograph, as may be seen on comparing Figs. 8 and 9 (taken from Brodie's Experimental Physiology). Each represents a simple muscle twitch, Fig. 8 with a so- called ' light ' lever possessing little inertia : Fig. 9 with a heavy lever ; the latter clearly shows an oscillation of the writing point which is not proper to the muscular movement studied, but introduced by the apparatus. To avoid this misleading effect, it is not however desirable to use a very light lever unloaded, since this would not keep the muscle stretched : the way out of the difficulty was found .by Fick, and is known as the isotonic method. It is illustrated in Fig. 7, and consists in hanging the load much nearer to the axis, and increasing it in proportion : thus instead of m grams at p we may, as in the figure, use 20 m grams at a point N ^ of the distance from- the pivot A : this produces the same tension in the muscle (see § 6 on moments) ; now the movements of if will only be -fa as small as the corresponding movements of p, and the acceleration therefore reduced to -fa that of p. The mass moved is 20 times as great, and since the force exerted is measured by the product of mass and acceleration it is the same in actual amount as when the small mass is hung at p. What we are concerned with however is the relative fluctuation in the tension : and as the weight at n is 20 times as great as that required at p, the same actual force introduced by .inertia will be of far less consequence relatively to it. Of all forces, the most frequent and important in practice is that of gravity, or weight. We have seen that the attraction of the earth produces on all bodies near its surface an acceleration of 981 cm. per sec. per sec. The acceleration varies slightly from one place to another on the earth, but at any one place it is precisely the same on all bodies. Hence at any one spot, a gram of matter has a weight of (about) 981 dynes, and each gram has precisely the same weight as each other gram, regard- less of its quality, size, or shape. It follows that two masses can be compared by comparing their weights, i. e. by weighing one against the other in an ordinary balance, as is commonly done, without having to fall back on the much less accurate and convenient methods depending on inertia, such as that of a g S MOMENTUM: FORCE 19 the ballistic balance. But it must be borne in mind that the fact that the weights of bodies are proportional to their masses is only an observed result, although observed for all known kinds of matter, and is not an essential part of the meaning of mass. So far as we can at present see it would be quite possible to conceive of some kind of matter obeying the second law of motion (possessing 'mass' or 'inertia') and yet not suffering gravitational attraction (i.e. possessing no 'weight'), just as a piece of copper and a piece of iron may have the same mass, yet one is strongly attracted by a magnet, the other not at all. Owing to the frequency with which weights have to be dealt with in practice, it is often convenient to express all forces by means of the weights they are equal to : thus we may not only speak of the ' weight of a gram,' meaning the attraction of the earth for a gram mass, and therefore 981 dynes, but we may describe some other force, such as the tension of a horizontal string, which has nothing to do with gravity at all, as being a ' force equal to the weight of x grams,' or for brevity ' a force of a; grams.' A force so expressed may always be converted into dynes by multiplying by 981. In a. curve such as Fig. 9 distances from the horizontal line express displacement of the end of the muscle from its normal position. Hence to measure the rate at which the muscle is moving at any instant we must take the height above the horizontal line at the beginning and end of some short interval — say one vibration of the tuning-fork. When the muscle is moving at a constant rate, the line traced out rises equal amounts in equal intervals of time, and is consequently straight (but not horizontal) ; and the steeper it is the greater the distance travelled in a given time, i.e. the greater the velocity. If Fig. 9 be examined it will be seen that the muscle is at rest up to a certain point a ; that a little later, at is, it is moving uniformly (the tracing being straight), and from measurement of the curves the velocity then will be found about ao cm. per sec. During this period then the muscle is exerting such a force that its movable end, to which the weight is attached, is accelerated upwards, increasing in speed from o to 20 cm. per sec. in the course of o.oi sec. ; accordingly the measure of the acceleration is ao-J-o-oi = 2000 cm. per sec. per sec. This involves » force which is in addition to the force required to balance the weight : the total tension is therefore 3981 dynes on each gram suspended, and if we say there are x gms., we may write the equation (expressing the second law of motion) thus : — C2 20 GENERAL PHYSICS Tension of muscle (2981 x dynes upwards) — weight supported (981 x dynes downwards) = effective force (2000 x dynes) applied to the lever, and consequently causing an acceleration (2000 cm. per sec. per sec. upwards) in the latter. § 5. Work : energy : power. Mechanical work is done when a body is moved in opposition to a force that resists the motion. The measure of the work done is consequently the product of two factors : (i) the force overcome, (ii) the distance through which the body is moved against it. The simplest case we can consider is that in which the movement takes place in the line of action of the force : e. g. the line of action of weight is vertically downwards, so if a weight be lifted vertically upwards, work is done on it to an amount equal to the product of the weight (force) into the height. The natural measure of work on the C. G. S. system of units, there- fore, is the dyne multiplied by the centimetre, and to this unit, on account of its importance, a special name has been given— the erg. If now the movement do not take place along the line of action of the force, the work is measured by multiplying the force by the distance moved in the direction of the force, e. g. if a weight be lifted slantways, the vertical distance only must be considered in estimating the work : no more work is done in lifting a weight on to a table 80 cm. high slantways, although the space traversed may be a metre or more, than if the weight be lifted vertically up to the table. It may even happen that a force is acting on a body and yet not moving it in its own direction at all : this is the case with a body moving uniformly round a circle (p. 33). Thus if the moon travels in a circle round the earth, it is because of the pull exerted on it, gravita- tionally, by the earth, yet as the pull is always in the direction joining the earth and the moon, and the moon does not move along that line, either towards the earth or away from it, but at right angles across the line, no work is done. Energy is the capacity for doing work, and is therefore of course to be measured in the same unit, the erg. As stated in § 1, the fact that numerous forms of energy are known, and that they can be converted into one another without change in amount, is the leading principle of modern physics. These WORK: ENERGY: POWER 21 various forms will be dealt with in turn later : here we need only consider a certain preliminary distinction into two kinds, and the relation between the two as it occurs in abstract dynamics. A body in motion has a capacity for doing work, in conse- quence of its motion : this is known as its M/netic energy. Also a body (or system of bodies) usually has a capacity for doing work due to its configuration, i.e. the relative position of its parts : this is known as its potential energy, e. g. a spring when compressed has potential energy, since on changing its configuration, viz. on reverting to its normal length, it can do work. When force is exerted on a body, unless it be balanced by other forces, it is spent, we have seen, in giving acceleration to the body : the same fact may be stated in terms of energy, thus : when work is done on a body, unless it be balanced by work done by the body on others, it is spent in imparting velocity, and therefore kinetic energy to the body. Since by the law of conservation the kinetic energy generated must be equal to the work spent, we may learn from this how to measure kinetic energy. For this purpose we may take the most familiar case, that of a falling weight : here the ' system ' consists of the weight and the earth : the change in configuration is the change in distance between them : the system possesses potential energy of gravitation, which is spent as the weight falls towards the earth. If now the fall of the weight is caused, by means of a string, to drive a piece of machinery, such as a clock, work is done by the weight on the clock, so that very little of the work spent by the weight goes to produce kinetic energy in itself, and its rate of descent is very slow. If however the weight fall freely, so that there is no force resisting it, and no work done by it on external bodies, then all the potential energy of gravita- tion that it loses'reappears in the form of kinetic energy. Suppose, for simplicity, that the falling mass be one gram, and that it falls for t seconds : let g be the acceleration of gravity (about 981) ; then it follows from the definition of acceleration that the velocity acquired at the end of the time is v = gt. But since the velocity is increasing uniformly (the acceleration 22 GENERAL PHYSICS is constant) its average value during those t seconds is the mean between o and v, i. e. % v, so that the distance through which the weight has fallen, //, is h=\vt. On the other hand, the force with which the earth is attracting the mass downwards (the ' weight ') is g dynes : and therefore from the definition of work, the work done by the earth on the weight (or potential energy lost by the system: earth + weight) is gh ergs ; this then must be the kinetic energy acquired by the falling mass. But gh = gx%vt = bv i , i. e. the kinetic energy of the moving gram is measured by half the square of its velocity : the kinetic energy of any moving body is measured by half the product of its mass into the square of its velocity (J rnif). We saw, above, that it is common, and in many cases con- venient, to measure forces in terms of weights, instead of in the systematic unit — the dyne : when that is done, it is also con- venient to measure work in a corresponding gravity unit : this is the gram-centimetre, i. e. the work done in lifting a gram weight through a height of one centimetre : since the weight of a gram is 981 dynes, the work done in lifting it one centimetre is 981 ergs. It must be carefully borne in mind in the calculation of kinetic energy given abo\ , e, that the kinetic energy of a body has nothing to do with the attraction of the earth : that its expression (5 mi?) gives the amount of it naturally in the syste- matic unit of energy, and that if some other quantity of energy be given in gravity units it must first be reduced to ergs before a comparison can be made. If the systematic units of force and energy be used throughout, no complication will arise. Even the gram-centimetre is an inconveniently small unit in practice, especially for engineering purposes. The following table will sufficiently explain the other units in use : — Gram-centimetre = 981 ergs. Kilogram-metre = 98,100,000 ergs. Joule = 10,000,000 ergs. Therm (calorie) = 42,000,000 ergs. The first two are gravity units, the second being that commonly WORK: ENERGY: POWER 23 adopted by continental engineers : the third is designed as a convenient multiple of the erg, and is much used for scientific and especially electrical purposes : the fourth is the unit of heat, and since, as stated in § 1, heat is a kind of energy, it is also a unit of energy. In dealing with momentum it was pointed out that that quantity must be understood as being associated with a par- ticular direction in space, and that if a body travelling in one sense along a line is considered to have positive momentum, then in travelling along the same line in the opposite sense its momentum is negative. With this understanding, the third law of motion may be regarded as a law of conservation of momentum, since, one body losing as much momentum as the other gains, the total amount remains unchanged. But the law of the conservation of energy is not to be understood in this sense : energy is not a vector : it Ms no sign : a body cannot have a negative amount of energy, either potential or kinetic: such a statement would have no meaning. Hence the law of conservation implies that the actual arithmetic total of energy possessed by any system of bodies remains unchanged, whatever action may take place amongst those bodies, provided no energy is put into the system from outside. When a force doing work does not remain constant in amount, it becomes necessary to consider what is the right way of averaging the force in order to calculate the work done. This is best done graphically, i. e. by a method of represent- ing the quantities con- sidered, by lines on a dia- gram, somewhat in the fashion already made use of in connexion with the chronograph. In Fig. 10 let the vertical distances (axis of y) represent the magnitude of a force : whilst horizontal distances Fig. 10. (axis of x) represent the distances traversed by the point of application of the force (in the direction in which the force acts). As the most familiar of such dusuuwa 24 GENERAL PHYSICS diagrams are the 'indicator diagrams ' used to show the action of a steam engine, it will help to a precise understanding, if we suppose Fig. 9 to be such. The problem, there, is to find the amount of work done by a piston in its ' stroke,' i. e. its movement along the cylinder of the engine ; hence the horizontal distances in the diagram must represent the length of the cylinder along which the piston travels, while the vertical distances represent the force with which the steam is driving the piston at any moment of its stroke. Now suppose that whilst the piston is moving from j to i the force on the piston remains constant and equal to aj or bi; then from the definition of work (force x distance) it follows that the area abija is in proportion to the work done by the steam during this movement. Similarly, if whilst the piston moves from i to H the force of the steam remains constantly equal to ci or dh, an amount of work measured by the area cdhic is done : and this area, it will be seen, adjoins the preceding area. Now the force does not actually change in this discontinuous manner, but by making the sections ji, ih &c. during which the force is regarded as constant small enough, we may approach as nearly as we please to the actual continuous change such as is represented, e. g. by the curve klm. In this way we see that if the force at any moment be represented by the height of the curve lh, the work done by it is represented by the area enclosed by the curve, two perpendiculars and the axis of x ; e. g. the work done whilst the piston moves from j to a is represented by the area lmgji,. In engi- neering practice the curve is drawn automatically by means of a piece of apparatus called an indicator : in this, as in the chronograph, a pencil is moved vertically whilst a sheet of paper wrapped on a drum is moved across it ; the vertical movement is given by the steam pressure, acting against a spring, and consequently measures the force exerted by the steam ; the horizontal movement is not, as in the chronograph, caused by clockwork, since the intention is, not to record equal times, but equal spaces transversed by the piston : the drum is, therefore, actuated by a cord directly connected to the piston. When the diagram has been drawn, the sheet of paper may be detached and the area measured, in order to calculate the work done in the stroke. The word power (or ' activity ') as used in dynamics means rate of doing work ; it is therefore to be measured by the number of ergs done per second. Two other units are in common use : — Watt = 10,000,000 ergs per sec. = 1 joule per sec. Horse-power = 7,460,000,000 ergs per see. = 746 watts. As an instance of calculations of work and power, we will take the action of the heart. This is estimated to discharge WORK: ENERGY: POWER 25 180 gms. of blood at each beat, against an excess of pressure in the aorta equal to one-third of the atmospheric pressure. This (p. 37) is equivalent to lifting the blood up to such a height as would suffice, in a column of blood, to produce a pres- sure of one-third atmo, or about 325 cms. Hence the work done at each beat is 180 x 325 = 58,500 gm. cm. If there be 72 beats per minute this is 58,500 x 72 = 4,212,000 gm. cms. per min. or 42-12 kilog. metres per min. or 42-12 x 9-81 = 413 joules per minute. The action of the heart is of course not uniform in a second, but this is equivalent to an average activity of 6-9 joules per sec. or 69 watts. This is for the left ventricle only. For comparison it may be mentioned that 40 or 50 watts is as much as a man can do in the way of muscular work, continuously. The relations of energy and power to the other dynamical quantities may be conveniently shown by a diagram thus : — space rate * momentum energy (work) force power (activity) Here each quantity is the rate of change with the time, of the quantity immediately above it : power being rate of change of energy ; .force, rate of change of momentum. Again, the right- hand column is derived from the left, by being its rate of change in space, or the left can be derived from the right by multi- plying it by a length : this is shown in the relations between energy and force ; energy (work) is obtained by multiplying force overcome by the length, through which the point of appli- cation of the force is moved: conversely, force is the rate at which the energy is changed (or work is done) when the point of application is moved by unit amount 1 . 1 The diagram might be extended by putting in the left-hand top corner the quantity known as ' action '—this quantity, however, is never used in elementary dynamics. 26 GENERAL PHYSICS § 6. Vectors. We have already had occasion to remark on the difference in character between energy as a quantity, and certain other quantities such as momentum : they are not to be treated by the same mathematical rules, on account of the association of momentum with a direction in space, which is foreign to the notion of energy. Dealing more explicitly with this distinction we find that there are three kinds, or stages, of quantity, which may be described as (i) arithmetic, (ii) algebraic, (iii) geometric. The first, which includes energy, is the simplest kind of quantity, involving no conceptions of either sign or direction : it is that treated by the ordinary rules of arithmetic, so that to add two quantities of energy together is an operation in the sense of simple addition : if a body contains three ergs of energy, and two ergs are imparted to it, then it possesses five— the sum is necessarily greater than either component. To gain the con- ception of algebraic quantity it is necessary to supplement this by positive and negative sign : examples of such quantities, involving sign but not direction, are to be found in Physics, but for the sake of familiarity we will choose, rather, one from common life, viz. money. If money owned be described with the positive sign, then money owed is negative; and it is not only possible to add together two positive amounts, but also to add up various sums owned and owed, giving a negative sign to the latter, and so arrive at the total value of one's property. In this case of course the sum is not necessarily greater (in the arithmetic sense) than the parts : thus if one has £3 in one's purse, £40 at the bank, and owes one's tailor £15, to arrive at the total value it is necessary to perform the mathe- matical operation represented by + 3 + 40 -15 + 28 The example given on p. 13 with regard to momentum is of that kind : length, velocity, acceleration, momentum, and force, are all quantities which are to be reckoned positive when in VECTORS 27 one sense, negative in the opposite sense, and so possess the characteristics of algebraic quantity. It is true they go beyond this, and being associated with direction in space, are really geometric ; but if for the moment we neglect this fact and con- sider a length, momentum, &c. in one line only, we get a good example of the kind now considered. Thus in Fig. <■ 11, taking o as the starting- _> point, the distance op may Q p be regarded as a step in one Fig. 11. sense along the line (to the right). If we regard this as positive, then any step made in the other sense (towards the left) must be regarded as negative: such e.g. is pq. Hence the result of starting from o and adding together the two steps op (=+2 cm.) and po, (=-3 cm.) is to arrive at q, and the sum of the two steps is 00, (= - 1 cm.). These results may seem almost too obvious to need explicit remark ; but that is certainly not the case as regards the addition of geometric quantities, or vectors. Here again a length, as the simplest of vectors, may best be taken as example, and a length may be looked upon „ 'p.... as a step. In Kg. 12 op is not V\_ " ^--- only a length of two centimetres ; \ ^\^^ ~\ JV it is that length in a direction \ ^^'^ ♦ (which we will call ehe). To this \ S^** W —T—E is to be added the length pq, \ ^s' I which is three centimetres, but }f ,-'" in the north-west direction. The ,,-- result is to arrive at the point q ; Fig. 12. giving the step 00, as the sum of the other two. This is the proposition known as that of the triangle of forces, for all vectors, including of course forces, can be added in this way. If it be desired to find the resultant (i. e. the sum) of two forces acting at a point, this can be done by drawing a diagram in which the forces are represented by lines drawn parallel to the forces and of lengths to represent their magnitude on some fixed scale-. These principles, especially as regards forces, can be illustrated from mechanics, and as well from the mechanics of the human body as any- thing. Thus the radius and the flexor muscles of the fore arm constitute a sort of bracket for supporting the hand. The parts they play are similar to those of the strut A and tie b in Fig. 13. It is easily seen that the strut is in compression, and consequently exerts a thrust x at L (if it 28 GENERAL PHYSICS ■were not strong enough it would crush up), whilst the tie is in tension and exerts a pull t at l (if it were not strong enough it would be pulled apart). These two forces when added together (as vectors) have a resultant g which just balances the JY weight w. This is shown *-._. Y graphically in the tri- X ""* *■ -----* an g' e a * tne B ide. The x resultant of x and t would be the third side of the triangle, directed upwards, and it produces equilibrium with the weight which is repre- sented by that line drawn downwards. It should be noticed that in this example both forces x and t are considerably greater than their resultant — the reason being that they so nearly oppose each other. In the figure as drawn, if the weight were 5 kilos the force x would be about 15, while t is greater still. The same, or even a greater difference, occurs in the arm, the tension of the muscles and the pressure in the bone being much greater than the weight supported. Many problems in mechanics, especially with regard to moving mechanisms, may be most conveniently solved by a direct application of the law of conservation of energy, in a form known in the classical treatises on dynamics as the ' principle of virtual velocitiea' If one end of a machine be caused to travel through a small distance p under the action of a force P, acting in the direction in which the motion takes place, then the force does work on the machine of amount Pp ; if the machine is a perfect one, without friction of any kind, it must give out the same amount of work at the other or driven end : suppose this end moves through the distance q while the former (driving) end moves through p, then the force Q exerted at that end in the direction of motion must be such that Qq the work done = Pp. Pp Hence Q = — and the force exerted at the driven end can be determined by means of the geometrical construction of the machine. If there is no friction, the reasoning just given holds, to whichever end the force is applied : the machine is reversible ; and if worked backwards, Q being a known applied force, VECTORS 29 J> = Ms . B u t if there is friction, the work given out will be less p than the work put in : hence if the end p is driving the other, Pp>Qq, so that Q< — ; whilst if q is the driving end (the machine worked backwards), Qq>Pp, and P< — ■ Thus in any case friction Jr diminishes the force exerted by the machine. As an example, let us take a screw press, such as a copying press ; and apply a force equal to 10 kilos-weight on the handle, the distance of which from the axis of the screw is 15 cms. ; and suppose the pitch of the screw (i. e. the distance it moves forward in one revolution) to be \ cm. Then if we choose for p the movement of the driving during one complete revolution of the handle, this is 2tt x 15 = 94 cms., while the corresponding move- ment q of the driven end is J cm., consequently - = 94 -i- J = 188, and the pressure Q exerted at the driven end would be 1,880 kilos, if there were no friction. Actually the friction is large and reduces this pressure greatly ; but still the magnification of the force applied is very great. The friction in an ordinary screw is usually so great that the machine will not work backwards at all : this is a valuable property, since it prevents the screw from running back when the driving force is removed. The action of machines, as well as of the mechanical contrivances of the animal body, can be discussed by the aid of the method of compounding and resolving forces already described, except in one case, that in which the forces to be dealt with are in parallel lines. When that is so, the lines representing them on a diagram cannot be made to form the sides of a triangle, and consequently some other way of finding their joint effect is required. This is most conveniently done by means of the moment of the forces : if a, body be fixed at one point (say the beam of a balance, at its central support), a force applied elsewhere will in general have a tendency to turn the body round the fixed point, and its tendency to do so, or moment, will be greater, the further off the line of the force is from the fixed point (e. g. the further a weight is put from the centre of the beam, the more it tends to turn the beam round). The exact measure of the moment of a force about a point is the product of the force into the perpendicular distance from the point to the line of the force : e. g. Fig. 14 the force r 3° GENERAL PHYSICS Fig. 15. exerts round p a tendency to turn measured by the product of f into the length pq. That this is so may be verified by considering the work that a force might do, i. e. by the principle of virtual velocities : thus supposing two forces r and a (Fig. 15) to act on the beam of a balance, G being twice as far from the centre as f; then f tends to turn the beam against the hands of a clock, g in the clockwise sense. Suppose the beam to turn a little, say clockwise, then it is clear that f and g will both describe parts of circles round the centre of the beam, and g move twice as far as f. If then f be twice as great a force as g, the work done (force x distance) by G will be equal to that done against f: and this is the condition for equilibrium, since if a gave out more energy than was absorbed by v the difference would go to give velocity (and kinetic energy) to the system, whilst if g gave out less energy than F absorbs the movement would take place in the opposite sense. Hence it is only when the work done on either side is the same that the two forces can balance and leave the beam at rest: and this is the case when the moments of the forces are equal. The moment of a force actually applied to turn a shaft round is usually called the torque on the shaft. The work done by a rotating shaft can be most conveniently expressed by means of the torque exerted in it. For, let n be the number of revo- lutions per second, and suppose the shaft to be driven by a force J" applied at a radius r from the axis of rotation. Now in one revolution the travel of the point of application of the force is awr; hence in one second a-nrn. The work done per second is therefore 2 ir rnF. But rF measures the torque applied ; hence we may write work done = 2irxno. of revolutions x torque. The arrangement just considered is a lever, and the principle of moments has sometimes been known as the principle of the lever. It can be applied to determine the force that can be exerted by such a mechanism ; thus if in' a pair of nut-crackers (two levers hinged together) the hand exerts a pressure of 20 kilos at 12 cms. from the hinge, and a nut be placed 2.5 cms. from it, the moment of the pressure is 20x12 = 240; this must be balanced by the moment of the resistance of the nut ; if the latter is x kilos the moment is x x 2-5 = 240, whence x = 96. This, it will be seen, is practically the same as applying the principle of virtual velocities direct. Another instance of this general principle may be found in a system of pulleys ; thus in Fig. 16 with one movable and one fixed pulley it maybe seen from the geometry of the system that the force F applied to the rope must move 2 cms. for each 1 cm. that the I W Fig. 16. OSCILLATIONS AND WAVES 31 weight w is raised: hence, neglecting friction, the force need only be half the weight. The fixed pulley makes no difference to the magnitude of the force, serving only to change its direction. Since weight always acts downward, i. h. in parallel lines, the resultant of any number of weights, or of the weight of different parts of a body, can be found by the principle of moments. If a single solid body be considered, we must take the weight of each small part separately and compound them together, one after another, till we have the resultant of the whole ; in this way may be found the line along which the resultant or total weight acts. It may be shown that if the body be put successively in any number of different positions the line of the resultant weight will always pass through one point : this is called the centre of gravity of the body. If then a, body is to be supported, the supporting force must be applied vertically under or over the centre of gravity : e.g. if a string be tied to a chair and the chair lifted, it will not in general produce equilibrium, but the chair will swing about, and eventually settle in such a position that its centre of gravity is vertically under the point of attachment of the string. Again, if a body is resting on s. base, the vertical through the centre of gravity must fall within that base ; this is the case in a chair as it usually stands, the base being, here, the quadrilateral formed by the four feet. But if the chair be tipped back till the vertical through the centre of gravity comes to lie outside that quadrilateral, the chair will fall over. § 7. Oscillations and Waves. The most familiar instance of an oscillatory or vibratory- motion is the common pendulum. To study the motion imagine such a pendulum constructed by hanging up a small ball of lead by a long thread, and set swinging, but only to a small extent, in order that we may avoid complications as much as possible. The ball then swings to and fro, in what is really an arc of a circle, drawn round the point of attachment of the thread, but it is so short an arc as to be nearly straight, and we may treat it as such. Then we may note these points about the motion ; first there is a position of rest for the pendulum, viz. whenit is hanging straight down ; next when pulled aside from that position, and let go it tends to go back to its position of rest. This second fact shows that the position of rest is a stable one. The use of this term involves a knowledge of the different kinds of equilibrium ; a body is said to be in equilibrium when the various forces applied to it balance one another, and so do not cause it to move : that is the case e. g. when an egg is 32 GENERAL PHYSICS laid on its side, for the resistance of the table acts in the same vertical line as the weight and balances it ; but it is also the case if the egg be very carefully placed on its end. The resistance of the table is again in the same vertical line as the weight, and balances it, so that the egg will remain at rest. The difference between the two cases is that if some slight accidental dis- turbance moves the egg out of place when laid sideways it will return to its former position : it is then said to be in stable equi- librium ; but, if placed endways, the least disturbance will make it fall over : it is then said to be in unstable equilibrium. When it is neither on end nor on its side, but in an intermediate posi- tion, it is not in equilibrium at all. There is a third case of equilibrium, called neutral, which is exemplified by a ball placed on a horizontal table : the ball is obviously at rest, and, moreover, if moved slightly, it neither tends to fall back into its former position nor to fall away from it, but remains in any position in which it is put. Now if a body be in neutral or unstable equi- librium, and be slightly displaced, it will not tend to rock about that position, since it has no tendency to return : an oscillation, then, is always a movement round a position of stable equilibrium. Returning to the observation of our pendulum we see that, when pulled aside, the ball is slightly raised, and consequently possesses a small amount of potential energy (work stored up in it on lifting) as compared with its position of rest. When the pendulum is let go, this potential energy is gradually converted into kinetic, as the ball acquires more and more velocity, until when the position of rest is reached, and the ball is as low as possible, there is evidently no potential energy left ; and we know, by the law of conservation, that just an equal amount of kinetic must have been generated. The ball will consequently not stop here, for that would mean that a certain amount of energy present in it vanished, which of course is impossible. Accordingly the ball goes past its position of equilibrium, and only stops when it has risen to an equal height on the other side, when all its energy is again in the potential form. This process is repeated an indefinite number of times, and constitutes the vibration : the only limit to it lies in the fact that in any real pendulum there is a small amount of friction (against the air, and at the place of support), so that at each swing a little energy is used up, and the store becomes smaller and smaller, the vibration gradually dying away. In a vibration of the kind just mentioned the most important quantities to record in describing it are (i) the amplitude, by which is meant the ex- treme distance the vibrating body reaches on either side of the position OSCILLATIONS AND WAVES 33 of rest, (ii) the periodic time of vibration, i. e. the time taken to swing com- pletely to and fro, since after that the motion is merely repeated. If a common pendulum be observed with a watch and telescope, it will be found that, although the vibrations gradually diminish in amplitude, the time taken to execute one swing to and fro remains constant : the smaller vibrations are carried out with smajler velocity. This fact is observed with all vibrations of similar kind, so that any oscillation is characterized by a definite periodic time. The reciprocal of the periodic time, i.e. the number of vibrations executed in one second, is called the frequency. Such an oscillation may be regarded as the projection of a uniform motion in a circle : indeed it is easy to see how it might be derived from circular motion. Suppose the ball of the pendulum in the illustration to be pulled aside and then started moving in a small horizontal circle : it will be found that it travels round and round with uniform speed, in- definitely, except that friction slowly brings it to rest as in the previous experiment : in this case it never passes through the position of rest at all in its motion, but behaves like a planet circling round and round the sun. Now go a good distance off, and, keeping the eye on a level with the ball, watch the movement ; we then obtain a ' projection,' for we. are able to see the movement to right and left of the eye, but not the movement •towards and away from it: and it will be found that all the characteristics of the usual pendular movement are reproduced. We may use this as a definition of the kind of movement we are con- cerned with. Any motion that repeats itself at regular intervals is called harmonic ; and the movement of the ordinary pendulum is of the kind called simple harmonic. A simple harmonic motion (S. H. M.) is the projection on a diameter of uniform motion in a circle. Numerous other instances of simple harmonic motions occur, especially when the movement is due to elastic forces. A solid, when in a position of stable equilibrium, may be subjected to a small strain and then released, and it will oscillate to and fro about the equilibrium position until fric- tion causes it to come to rest. To this class belong the motions of tuning-, forks, spring time-markers, and the vibrating parts of musical instru- ments : we have already referred to the tuning-fork in connexion with the chronograph (p. 4), the instrument there described, however, being provided with an artificial arrangement for maintaining the vibrations against friction indefinitely. We will here, therefore, choose as an ex- ample of harmonic motion executed under elastic forces, the time-marker shown in Fig. 3. In this a flat band of steel s, which would normally be horizontal, is bent out of that position by the weight w : it consequently attains a new position of equilibrium, under the joint action of the weight and the elastic reaction set up in it — a position such as that shown in the figure. If now it be depressed a little further the elastic reaction over- 34 GENERAL PHYSICS balances the weight, and on releasing s it will spring upwards under the action of the resultant force : hut when it has recovered its position of rest it has acquired a considerable velocity, and will, of course, not stop suddenly, but rise further ; and as then the spring will be less strained, tho elastic stress in it will be diminished, and will no longer balance the weight, and the resultant force will then be downwards. In this way the motion repeats itself at uniform intervals. The example, moreover, enables us to see what that most important quantity, the periodic time, depends on. The weight w can be shifted to any desired position on the spring, and, of course, the further out it is the wider arc it has to swing through, and so for any given frequency of vibration of the spring the more rapidly it moves : but, as nearly all the mass of the moving body lies in w, this means that when the weight is further out the kinetic energy corresponding to any given time and amplitude of swing will be greater. However the kinetic energy can only be the equivalent of the work done by the spring in recovering its equilibrium position after being bent : if it be bent to a fixed extent, then in recovering it will do a fixed amount of work, and communicate a fixed velocity to w, so that when the weight is put further out the vibration must be executed more slowly. On the other hand, if the spring were made stronger, the weight remaining fixed in magnitude and position, then for a given amount of bending more work will be stored up in the spring, and as this work again spends itself in giving kinetic energy to the weight, the latter will acquire a higher velocity, and the vibration be exerted more quickly. Thus we see that anything that increases the capacity of the system for kinetic energy makes the period longer, anything that increases its capacity for potential energy, when displaced from the position of equilibrium, shortens the period. In the practically important case of the common pendulum, if it be constructed of a small heavy ball suspended by a thread, it may be shown that the time of vibration or period T = 2 it \/- , where I is the length from the suspension to the centre of the ball, and g is the acceleration of gravity. This is the method actually used for measuring g. In addition to the amplitude and period, another piece of information is needed to determine the position of a vibrating body at any time. The body is said to pass through various ptaes in the course of its vibration : the term is one well known in connexion with the periodic changes in appearance of the moon — new moon, full moon, last quarter, &c. : and it is equally applicable to the set of changes cons'ituting any other periodic phenomenon. To express phase quantitatively it may be defined by means of fractions of the whole period : thus, reverting to the case of the pen- dulum, if we call its phase o at the moment when it passes through the centre towards the right, then when it has reached its greatest elongation OSCILLATIONS AND WAVES 35 towards the right its phase is \, when passing through the centre towards the left \, when at the furthest point towards the left f, and when again passing through the centre towards the right 1 or o again, since then the motion repeats. Now clearly if we know the phase at some moment of time, and also the period of vibration, we shall be able to calculate the phase at any future time, and, with the aid of that and the amplitude, shall know the exact position of the vibrating body at that moment. When a vibrating reed or tuning-fork is used in connexion with the chronograph, a wavy curve is obtained, such as that shown in Fig. 8 ; this consists of short pieces similar to each other, each leading into the next : each piece corresponding to one vibration of the reed or fork. The piece of curve corresponding to a single vibration may be of other shapes than that shown in Fig. 8, without interfering with the regular repeti- tion, as may be seen in sphygmographic records. These other, and usually more complicated shapes, correspond to a more complex harmonic motion on the part of the vibrating body : but a reed or tuning-fork vibrates in a manner that is very nearly simple harmonic, and the corresponding tracing on the chronograph is of the shape known as a sine-mrve. The sine-curve, then, is the resultant of two motions, one a S. H. M. in one direction (usually vertical), imparted by the tracing-point of the fork, the other a uniform motion in a straight line at right angles to the former direction, and imparted by the drum of the chronograph. Such a tracing may serve as an illustration of a wave, for a wave con- sists in a harmonic motion propagated in some direction, usually with uniform velocity. As a simple instance of the production of a wave the following experiment may be considered : tie one end of a string to a fixed point on the far side of a room, and holding the other in the hand, give it a vibratory motion; or better, in place of string, take a long piece of rubber tubing, weighted by filling it with water. With this apparatus we may observe that by giving a single to and fro move- ment to the end, a disturbance is produced, which travels down to the other (and even back again by reflection) ; if only a single vibration is executed, this transmitted to a distance constitutes what is called a pidse : if the vibration be kept up, one pulse after another is transmitted, till the whole string is set in vibratory motion, and a regular set of waves is formed. The leading points to be observed— they may most easily be noticed in the case of a single pulse — are (1) each point of the string executes a movement similar to that of the starting-point, (2) the move- ment occurs later and later as the distance from the starting point increases : i. e. the moment at which each point passes through a given phase is later according to the distance of the point from the starting- point. By observing the time when successive points reach the same phase, we may measure the velocity with which the wave is propagated j this will be found uniform in the case chosen for experiment, and in D2 36 GENERAL PHYSICS general ; in fact, the velocity of a wave usually depends only on the nature of the medium in which the wave is transmitted. When a complete wave is set up, the movement of any individual pulse may be watched, and the velocity of the waves determined by means of it. But by the time that one point a (figure, p. 246) has reached a particular phase— say its greatest elongation to the right— another point b, nearer the place of starting, will be found to be in the same phase, being really one whole period in advance of a. The distance ae is called the wave-length ; it is, therefore, the distance that the wave travels forward during the time of one complete vibration, and in the tracing representing the wave (in the figure, p. 246, a curve of sines) it is the length after which the curve repeats itself. It follows then, from this definition, that wave-length = velocity of wave x periodic time ; or since frequency means the reciprocal of the periodic time, we may write this — velocity of wave = wave-length x frequency. The observations made on the vibrating cord may also be made on the ripples set up in a pond by dropping a stone in it, with this difference however, that as the waves spread out into larger and larger rings, the amplitude of vibration set up in the water falls off. This is because the wave only possesses a fixed amount of energy, and as it spreads out it sets more and more water in motion, and conse- quently cannot impart so large a vibration to each point of the water. The same is true of waves — such as those of sound — which spread out in all directions in space - r but not when the waves are propagated in one direction only, as in the experiment with the string. Here there is nothing to make the amplitude fall off except the gradual consumption of energy in overcoming friction. § 8. Hydrostatics. A vessel containing a liquid suffers everywhere a pressure on its walls, tending outwards ; as may easily be shown by making a hole in the wall, when the liquid rushes out in consequence of the pressure. To find the amount of this pressure, suppose a vertical tube, 1 sq. cm. in cross section, to be filled to a height h with a liquid of density d: then the tube contains h cubic centimetres of liquid, or hd grams. The weight of this has to be borne by the base, which consequently suffers a pressure of M grams weight per square centimetre ; but adopting, as before, the systematic unit of force, this amounts to Tidg dynes per sq. cm. (g = 981 = the acceleration of gravity.) HYDROSTATICS 37 Observation shows that however a vessel, or set of com- municating vessels, be shaped, the liquid will stand at the same level throughout it. This means that the pressure at any given depth below the surface must be the same however the vessel may be shaped : if, for instance, the vessel cone outwards from the bottom up, the weight of water it contains is greater than that standing vertically over the base ; but the excess is borne by the sloping sides, leaving the pressure still = hdg on each unit area of the base. So again if it cone inwards the pressure remains the same, although it is now greater than the whole weight of water contained : the difference being accounted for by an upward pressure on the sloping sides. Not only is this the case, but there is an equal pressure on the sides of the vessel whatever direction they have: hence, e.g., if a closed vessel (such as a thermometer bulb) have a tube filled with liquid rising from it, the pressure on its sides may be very considerable, depending as it does on the depth below the free surface of the liquid, and not merely on the weight of liquid contained in the tube. The pressure of the liquid on the base of the vessel must not be confused with the pressure that the vessel exerts on the support bearing it. The latter is, in any case, equal to the whole weight. Thus, taking the example of a vessel coned outwards, some of the weight of water is borne by the sloping sides, and some by the base, but the whole ultimately falls on the table on which the vessel stands. Gases have weight, and consequently produce a pressure in precisely the same way as liquids : hence the great quantity of atmospheric air overhead causes quite a large pressure on any surface exposed to it (including that of the human body, which, however, is so accustomed to the pressure as not to notice it). This can easily be shown by the instrument known as a barometer : if a glass tube some 90 cm. long, ending in a tap, be placed with the tap uppermost, and a reservoir of mercury be attached by flexible tubing to the lower end, then by opening the tap and raising the reservoir, we may drive out the air through the tap and fill the tube completely with mercury ; then if the tap be closed, when the reservoir is lowered the mercury in the tube will no longer be pressed on by air, whilst that in the reservoir is ; and it will be found that the free surface-level 38 GENERAL PHYSICS in the tube is higher than in the reservoir by some 76 cm. Hence the pressure due to the weight of air overhead balances that due to a column of mercury of the height stated. The height varies somewhat from day to day, and from place to place, and a barometer is intended chiefly to measure it ; but as a standard value, 76 cms. of mercury at the temperature of 0° is chosen. Now the density of mercury at that temperature is 13-596, and the mean value of g is 980-61. Hence the normal atmospheric pressure is 76 x 13-596= 1033-3 gms. weight per sq. cm. or 76 x 13-596 x 980-61 = 1,013,200 — Sometimes, however, ' sq. cm. 1,000,000 — : is chosen as the standard pressure. sq. cm. * Gauges to measure fluid pressure are usually made of a column of liquid in a glass tube. The most familiar is the barometer, already mentioned : it measures the actual pressure of the air, because one end of the mercury (that in the reservoir) is pressed on by the air, the other end (under the tap) by nothing, so that the whole pressure of this air has to be balanced by that due to the difference in level of the two mercury surfaces. Barometers are not usually made with a tap and flexible tube, as described above, but either of a simple glass tube, bent into a (J shape, with one end sealed up, the other open to the air ; or of a straight glass tube, sealed at the top, and dipping into a dish full of mercury. Mercury is chosen because (1) it is so heavy that the barometer is shorter than when made with any other liquid ; e. g. a glycerine barometer would need to be about 8 metres high to produce the pressure sufficient to balance the air ; (2) it is almost involatile. If a volatile liquid like water were used the space at the top of the tube would be filled with vapour, and this would produce a considerable downward pres- sure on the liquid, and the height of the liquid column would only represent the difference between the air-pressure and that of water vapour (see p. 96). The chief precautions required in making a barometer are : (1) that the mercury should be pure (preferably distilled in vacuo) • (2) that the tube be completely free from air-bubbles and moisture. To get rid of moisture the tube must be heated strongly : a simple and convenient form for the purpose is that represented in Fig. 17, with an air-trap a : the tube can be filled by merely pouring in mercury ; this is done r IT A J i i -- 1 '£ - to ~ = = 3 -- E 1 \_ - ■ - ; ! HYDROSTATICS 39 partially, and the portion above the trap then freed from air and moisture by boiling repeatedly over a bunsen burner ; it is then completely filled with mercury, the finger put on the open end, and the tube inverted into a dish of mercury. If any air-bubbles remain in the lower part of the tube they collect under the trap, where they do not affect the vacuum at the top. The tube should not be less than 8-10 mm. wide at the top, to avoid capillary effects (p. 112). The most important correction to the readings of the barometer is for temperature : as mercury expands with heating, it is desirable to correct the height of the column to that which it would assume if the mercury were at the standard temperature of the freezing-point : to do this, if a glass scale is used, subtract 0-13 mm. for each degree above zero ; if a brass scale, 0-124 mm. When a gauge is only intended to measure the j- IG ±~ m difference between two pressures, the construction of it is easier ; it may be necessary, e. g., to compare the pressure of the gas in a piece of apparatus with that of the air outside, and this can be done with a simple U-tube of mercury. When gas is enclosed in a vessel, it produces on the walls of the vessel a hydro- static pressure comparable with that existing in the atmosphere on account of its weight. Thus, suppose we have a glass bulb fitted with a tap, and let the tap be open, then obviously the atmospheric pressure, 760 mm. of mercury, prevails through- out. Now let the tap be closed : no difference has been made to the air inside, so that a hydrostatic pressure of 760 mm. still exists in it, although it is no longer directly pressed on by the weight of air oveihead. This fact is explained by the view that a gas consists of small particles (molecules) which are in rapid movement in all directions, and so very frequently collide with the walls of the containing vessel, and tend to drive them out- wards. It is this pressure, due to the velocity of the molecules, which balances the applied pressure the gas has to support, whether that be due to the weight of more gas, or be applied by means of a mercury or other liquid column, or be due to the resistance to expansion of the solid containing vessel. Moreover it is observed that the pressure produced in a given vessel is pro- portional to the mass of gas it contains (Boyle's law, see p. 77) ; 40 GENERAL PHYSICS so that if twice as much air be forced by means of a pump into the bulb in the above experiment, with the tap closed, the pres- sure inside it would be sufficient to balance 2x 760 = 1520 mm. of mercury. Hence we are justified in speaking of the pressure inside a vessel of gas, as well as that in free air : and whilst in liquids it is usually necessary to bear in mind that the pressure is different at different depths, owing to the weight of the liquid, this is hardly ever necessary in the case of a gas, because its weight is so small. Thus if an apparatus containing air were five metres high, the pressure at the bottom of it would be greater than that at the top only by hdg = soo x 0-0012 x 081 = ^8-86 — y ° * ° sq. cm. or about stroTro" atmosphere. To measure the difference between the pressure in two vessels containing gas, or between one vessel and the atmosphere, a simple U-tube of mercury is sufficient, each end of the U being connected to one of the vessels. If the pressure difference is small, a lighter liquid than mercury may be used with advantage, such as sulphuric acid, or oil ; and since these liquids wet glass, no capillary error will be made, provided the two limbs of the gauge are of the same bore, consequently the gauge may be made much narrower : 1 or 2 mm. diameter is sufficient. For registering pressure on the chronograph (i. e. differences of pressure between some apparatus and the air), a float in the open end of the gauge is made to cany the tracing-point. It should be noted that if the gauge is to respond rapidly to changes of pressure, such as those occurring in the vascular system, its moving paits must be made light, for the reasons already discussed in the case of levers (p. 16). But if the mean pressure only is desired a mercury gauge is satisfactory because its considerable inertia prevents its oscillating too easily. Other methods have been adopted in order to avoid, more completely, the disadvantages of inertia in a pressure gauge ; especially the plan of balancing the pressure against the elasticity of a spring of some kind. Thus Pick's ' kymograph,' or manometer, consists of a C-shaped spring, made by uniting two thin strips of metal at the edges, so as to form a flat tube : if pressure be applied to the air in the interior of the tube, the C opens out and so registers the pressure. Aneroid barometers are con- HYDROSTATICS 41 structed on the same principle, the internal space of the tube being evacuated, so that the tube is more or less coiled up according to the pressure of the air acting on its exterior surface. Again, Hiirthle's manometer consists essentially of a drum, or tambour, covered with thick india-rubber ; the interior of the drum is filled with salt solu- tion, which transmits the pressure from the blood-vessels, which it is desired to register : when the pressure increases, it raises the drum- skin, which by a short metal arm moves a very light lever, provided with a tracing-point to record the movements on the chronograph. One of the effects of fluid pressure is that a solid immersed in a fluid is pressed upwards, and consequently loses part of its weight. The upward pressure amounts to the weight of the fluid displaced. The effect is most obvious in liquids on account of their possessing a much greater density than gases, but it occurs also in the latter. Two cases have to be distinguished, according as the solid is denser or less dense than the liquid in which it is immersed. In the latter case the solid floats, a portion of it projecting above the liquid suiface ; that is to say, just enough of it is immersed to make the weight of dis- placed liquid equal to the weight of the solid itself, and the solid is thus practically relieved of weight altogether. Such a case has already been mentioned in the common hydrometer (p. n). If the solid is denser than the liquid, it sinks com- pletely below the surface : it has then displaced a bulk of liquid equal to its own, and is apparently lighter by the weight of that displaced liquid, as may be proved by attaching it, while immersed, to the arm of a balance (p. 12). In the intermediate case, in which the solid and liquid are of precisely the same density, the solid will not necessarily settle either at the top or bottom of the liquid, but will remain freely anywhere, having lost all its weight. This case is sometimes observed in the use of specific gravity heads (p. n). But if a solid lighter than water be forcibly immersed below the level at which it naturally floats, its apparent weight becomes negative, i. e. the upthrust of the liquid more than balances its weight. A boat, e.g., floats freely at a certain level: if a man gets into it, it floats deeper, and the additional water so dis- placed produces an upward pressure just sufficient to balance the weight of the man, which is the external force pressing the boat downwards. 42 GENERAL PHYSICS The hydrostatic support of an immersed solid may be simply proved by the energy principle. For suppose a small piece of solid to be immersed to a depth h below the surface of a liquid, and let m be the mass of liquid displaced by it : then if the solid were caused to rise to the surface, the liquid, flowing down to take its place, coming as it would practically from the surface, would do an amount of work = weight (mg) x height = mgh : this amount of work therefore would contribute towards raising the solid, and the force exerted on the solid is obtained by dividing the work done by the distance through which the force is exerted, or mgh -r h = mg, the weight of the displaced liquid. And this result is clearly true for every small piece of a solid of whatever size or shape, and hence though the large solid may not all be at one depth below the surface, the result is still true. All the bodies we are familiar with are immersed in an ocean of air, and hence lose weight, although very little. This fact must be taken into account in very accurate weighings. Also, though no actual solid is so light as air, an arrangement may be con- structed of a light bag filled with hydrogen or coal gas (balloon) which weighs less than an equal volume of air, and is conse- quently pressed upwards in air, like a piece of wood in water. One of the most important practical problems in connexion with the pressure of gases is that of increasing or decreasing it, i.e. the construction of pumps. Mechanical pumps for compres- sion or rarefaction depend essentially on the working of a piston in a cylinder provided with valves. The ordinary cycle-pump may be taken as an example : it consists of a cylinder a, in which slides a piston b ; the latter carries a flange o of leather or india-rubber attached somewhat aslant, so that when the cylinder is pulled outwards, making the volume under the piston greater, air flows in past the flange; but on reversing the stroke the air in attempting to get out again presses the leather against the wall of the cylinder, and closes the aperture: the flange con- sequently serves as a valve in this apparatus, allow- ing the air to flow one way, but not the reverse. The air, thus compressed in the space between the cylinder Tig. 18. and piston, escapes through the piston-rod, which is hollow. The tube by which this compressed air is delivered into the vessel intended to contain it (e. g. the cycle- tyre) must have another Yalve, to prevent its flowing back HYDROSTATICS 43 Af during the next outstroke of the pump. A similar arrangement with the valves reversed would serve for rarefaction ; but when the vacuum was very good the air-pressure would no longer be sufficient to work the valves : hence to get the best results, they should be worked mechanically, as for instance in the Fleuss pump (of which the essential parts are shown in Fig. 19): it consists of a cylinder m, in which a piston n works: the piston-rod carries the valve g, opening upwards. Both the piston and the valve g are com- pletely covered by a non-volatile oil. The piston on its up-stroke passing the point b leading from the surrounding brass chamber, forces the air in the cylinder up through the valve g, and the oil following the air leaves no waste space: on the down-stroke the oil serves to close the valve g perfectly : a vacuum is consequently produced over the piston, and air flows from any receptacle connected to a through b into the cylinder, and the stroke is repeated. No air-pressure is needed to work the valve g, as it is lifted by the collar o, and replaced by a spring. It is possible to arrange two such pumps 'in series,' i.e. to make the second pump the air out from the barrel of the first. By this means the air-pressure can be reduced to a small fraction of a millimetre, and a vacuum obtained good enough for the manufacture of incandescent lamps and 'vacuum-tubes.' Most mechanical air-pumps are not however capable of reduc- ing the air-pressure below one or two mm. ; for producing much better vacua than this mercury-pumps are mostly used. These may consist of a cylinder with a piston of mercury, instead of a solid : the Topler pump shown in Fig. 20 is actuated by lifting the reservoir a of mercury up and down. As it rises, the mercury fills the cylinder b and forces the air out by the capillary tube c ; when the stroke is reversed, the pressure of the atmosphere drives mercury up c (which must be at least 76 cms. long) and so prevents return of the air ; hence air flows Fig. 19. 44 GENERAL PHYSICS into b from the apparatus to be exhausted, through d, which is usually provided Avith a drying-tube containing phosphorus pentoxide. On the next stroke the air in the cylinder cannot get back into the vessel to be exhausted, because the mercury closes the tube d : the valve e is to prevent mercury being driven up too high into d ; when a is lowered, it falls by its own weight and leaves the pas- sage free. By means of a mercury-pump the air can be all but completely re- moved, the gas remaining in the apparatus being mainly mercury vapour, which has a pressure at ordinary temperatures of about -gV mm. Mercury- pumps are now made which are actuated me- chanically by means of water-pressure, so that it is only necessary to turn a tap and leave the ex- haustion to proceed auto- matically : such have been designed by Kaps, and by Kahlbaum. Another mercury-pump, acting on an entirely diffe- rent principle, is that of Sprengel (Fig. 21). In this the mercury falls in drops down the tube A, and in passing the mouth of the side-tube b each drop carries a small quantity of air with it. The tube a should be of about 1-5 mm. diameter, and long enough for the mercury in it to balance the atmospheric pressure. Fig. 20. Fig. 21. HYDROSTATICS 45 A similar pump can be worked with water (Fig. 22). Here drops falling from the capillary tube b drag down with them air that flows into the chamber a through the side tube 0. The tube d that leads away the water should be long ; if it be only a metre or two, the pump will only diminish the pressure a little, and is chiefly suitable for drawing a slow stream of air through any apparatus, but by using a fall tube of 10 metres a good vacuum can be got. If c be left open, the air delivered at the bottom of d will be under a small ex- cess of pressure and may be led away to an apparatus through which it is desired to drive a slow stream of air, e. g. for arti- ficial respiration. Another water-pump is that shown in Kg. 23, the theory of which is explained below (p. 117). The flow of water through the fine jet a sucks air through the side-tube b. The action of this pump is much more rapid than that of the water-Sprengel, and if the water-pressure be sufficient it may be made to exhaust down to the vapour-pressure of water, i. e. about -^ atmosphere. The air is carried down through the outlet, and may also be separated from the water and delivered under pressure. PiO. 22. CHAPTEE II. HEAT. § i. Temperature. Before dealing with the various properties of matter in turn, it is necessary to consider the very far-reaching influence of temperature, for there appears to be no property except gravita- tion (weight) which is not affected by it. The conception of temperature is originally a physiological one, based on the sensations of hot and cold : to make it physically useful we must make the observation that hot bodies and cold ones left in contact tend to become of the same temperature ; e. g. a vessel of hot water cools when left to itself, i. e. it tends to become of the same temperature as the air with which it is in contact. Generalizing this, we may say that equality of temperature is the state in which two bodies will remain, as regards their heat, if left in contact long enough, and that one body is hotter (at a higher temperature) than another when, on putting them in contact, heat flows from the former to the latter. Now since change of temperature affects nearly all the proper- ties of a body, it is only necessary, in order to measure tempera- ture, to choose some property and record its changes. Almost any property will do, and a good many have actually been used— length of a bar, volume of a liquid, pressure of a gas, electrical resistance of a wire, and so on ; if then we note that a bar of metal is of a certain length when put in a furnace, and another day is of the same length on being put into another furnace, we may conclude that the second furnace is of the same temperature as the first, and we might register a scale of TEMPERATURE 47 temperatures according to the length assumed by the bar in various furnaces. Any instrument thus designed to measure temperatures is called a thermometer ; and the one most in use is the mercury in glass thermometer. This consists of a small cylindrical glass bulb fused on to a glass tube, usually six or seven mm. in diameter externally, hut of very small bore ; the bulb and part of the stem are filled with mercury, and the top sealed, the stem above the mercury being left empty of air. When heated the glass bulb expands slightly, but the mercury expands much more, and consequently has to rise in the tube to find room : the tube is therefore provided with a graduated scale against which the position of the mercury thread may be seen. By the device of the bulb and stem even a very small change in temperature, producing a very small change in volume of the mercury, may be observed. When the instrument has been made, in order to graduate it, it is placed (i) in melting ice, (ii) in steam over boil- ing water. To perform the first operation successfully it is chiefly necessary that the ice should be pure : a test tube full of distilled water should be partly frozen by immersion in a mixture of ice and salt, the water in the tube being stirred to keep the ice formed in a powdery state : the thermometer is then put in the test tube and left, with occasional stirring, for a few minutes : the temperature will then be constant, and will remain so as long as there are both ice and water present in the tube : the level of the mercury is then marked. For the second test a flask with a side tube is taken, partly filled with water, and the thermometer fitted into the neck with a cork, so that its bulb comes a little way above the surface of water in the flask : the water is then boiled for a few minutes, and the level of the mercury thread noted : in the meanwhile the height of the baro- meter should be read. In both operations it is necessary to see that the mercury thread projects as little as possible from the ice or steam, so that the whole of the mercury may be at the desired temperature. The temperature of the ice-point is called 0° (on the centigrade scale), that of the steam-point 100° when under the normal atmospheric pressure of 760 mm. : but the latter varies with pressure, rising 1° for 26-8 mm. rise of pressure, so that the observed position of the mercury must be corrected according to the reading of the barometer. When the ice-point 48 HEAT and (corrected) steam-point have been found the length of stem between them is divided into 100 parts, and the graduation, if desired, is continued above and below *- When so graduated, however, the thermometer will still not read accurately unless the stem is of uniform bore throughout : a very good thermometer must therefore be 'calibrated' to determine any small irregularities in its bore. But as this is a very tedious process, most thermometers are merely compared with a standard, by immersion in baths of various temperatures : this is done for a small fee at Kew Observatory, and at the Berlin Reichsanstalt ; so that the most con- venient way of measuring temperatures accurately is to purchase such a standardized thermometer. A few further points of practical impor- tance must be noted : in the first place, two thermometers made of different materials (e. g. mercury in hard glass, mercury in soft glass), although made to agree at o° and iocr 1 , will usually not quite agree at other temperatures : there is no reason indeed why they should, for that would imply that the expansion of the different materials went proportionally over the whole scale of temperature. The differences on this account, though small (under ^V) between o° and 100°, may amount to several degrees at say 300 : hence it is necessary to choose some pattern of instrument as a standard and reduce the others to it. The standard thermometric scale adopted is that indicated by the pressure of hydrogen gas (or the so-called ' constant- volume hydrogen thermometer'), and in the certificates issued from Berlin and Kew the reduction to this scale is already made. Next, glass when heated and caused to expand does not at once on cooling come back to its original size, but retains a slight change, which slowly disappears r this thermal after-effect is the greatest cause of trouble in accurate thermometry. It is however less in hard glass than soft, and less in thermometers that have been kept for a long time. With a matured thermometer of hard glass it may usually be neglected. The most important correction to be applied is when the thread of mercury is not completely immersed in the body whose temperature is to be measured. To apply this, a. small auxiliary thermometer should be used to take the average temperature of the stem of the main ther- mometer : let t be the reading of the main thermometer, t' of the auxiliary, and n the number of degrees of the mercury thread that are exposed ; then a correction n (t-f) 6400 1 On the Fahrenheit scale the ice-point is called 32 , the steam-point 212° ; consequently a cent, degree = $ Fahr. degrees, e. g. 37 cent. = 37 x f = 66^6° above the freezing-point, or 66-6 + 32 = 98.6° Fahr. TEMPERATURE C49 degrees should be added to the reading of the main thermometer. This is especially important in measuring high temperatures. Thermometers are usually made either with solid stems engraved on the outside, or with hollow glass stems enclosing a scale engraved en milk-glass. The former are stronger, but are more exposed to risk of inaccurate reading due to 'parallax' : this means that as the scale is some way in front of the mercury thread, the reading will appear to differ according as the eye is held high* or low ; for accuracy the eye must be just on a level with the top of the mercury column ; this wUl be the case when the graduations nearest to the top appear to be in line with their reflections in the mercury. In milk-glass scale thermometers this difficulty does not arise, because the scale is almost in contact with the mercury thread. Thermometers should have a small bulb at the top, to leave room for expansion of the mercury in case it is accidentally heated above the range of the instrument, otherwise the expanding liquid will break the glass. Ordinary mercury thermometers have, at most, a range from - 40°, the freezing-point of mercury, to + 350°, at which it boils, but it has recently been found practi- cable to extend the range upwards, by filling the top of the stem with a gas under high pressure : this raises the boiling-point of the mercury and allows of its use up to 550°. Such thermometers are necessarily made of hard gkss in order not to give way at such high tempera- tures and pressures. To make a thermometer extremely sensitive it is necessary either to make the bore very small— and there are practical limits to this— or the bulb large. Ther- mometers with very large bulbs have been much used of late years for Beckmann's freezing-point apparatus and other purposes. Such an instrument is shown in Fig. 24. The bulb is 10 to 12 mm. in diameter : the stem gradu- ated in hundredths of a degree : the instrument there- fore necessarily has a short range— usually some six degrees. But to extend its availability it is provided with a reversed bulb at the top : when it is desired to use the instrument for higher temperatures it is warmed p IG _ 24. until a sufficient amount of mercury has flowed into the upper bulb ; then, with a shake, this is detached, and the l.emainder made use of in the usual way. Of course the scale V 5° HEAT does not in such an instrument indicate any fixed temperature, but serves to measure small differences of temperature, such as that between the freezing-point of water and that of a salt solution. For very low temperatures other liquids must be used instead of mercury. Alcohol, toluene, and petroleum ether are in use : the latter being available down to nearly - 200 . For clinical purposes a thermometer is needed that registers the maximum temperature it has reached. This is usually accomplished by making a constriction in the tube, near the beginning of the scale : the pressure of the mercury drives it past this in expanding ; but on contracting again the mercury thread remains in the tube until it is shaken back. Partly on account of its limited range, and partly for reasons of. a more theoretical kind, the mercury thermometer is not adopted as the standard, but is referred to the scale of a gas thermometer. The various gases, whether their expansion or their rise of pressure be used as indicator of temperature, give very nearly the same thermometric scale : but to be quite exact, the scale due to the increase of pressure of hydrogen, when kept at a constant volume, is chosen as the international or normal scale of temperature. A gas when kept under a constant pressure (as measured by a gauge, like a barometer, communicating with it) expands by 0-367 of its volume on heating from the freezing-point to the boiling-point ; but if by increase of pressure it be prevented from expanding, then its pressure rises by the same fraction ; so that, e. g., if a quantity of hydrogen at o° have a pressure of 1,000 mm. of mercury, then at 100° its pressure will be 1,367 mm. nearly. Accordingly the temperature to which the hydrogen is raised may be expressed by means of the pressure it reaches. The apparatus used is shown, a good deal simplified, in Fig. 25. A bulb a of glass or platinum-iridium contains the hydrogen, and can be raised to the desired temperature by the bath b, in which also mercury thermometers tt may be placed for com- parison. The bulb is connected by a fine capillary tube c to the gauge d : this is a wide tube of glass, containing mercury. The mercury reservoir e, attached by rubber tubing, can be raised or lowered till the mercury in d is brought to a fixed mark ; this ensures that the volume occupied by the hydrogen is always the TEMPERATURE S i same. The other end of the gauge f is a similar wide glass tube in which the mercury can rise and fall, and which serves at the same time as a reservoir for the barometer tube g. Then since- Fig. 25. the mercury in g has a vacuum above it, and that in d is exposed to the pressure of the hydrogen, the difference in level between the two mercury surfaces measures the pressure of the gas, as explained in p. 38. The difference in level is read by means of a scale placed alongside the tubes and a telescope sliding on an upright ; for convenience of reading, g is bent so as to come vertically over d : g and d must be wide tubes at the part where the mercury surface occurs, otherwise the mercury level will be irregularly affected by surface tension (see p. 112). Since the hydrogen increases in pressure by 0-367 of its amount at o° on heating to the boiling-point of water, and this interval is to be described as ioo°, it follows that one degree may be defined by a rise of 000367 = -^^ in pressure : e. g. if the pressure at o° be adjusted to 819 mm., then when on heating at constant volume it rises to 822 mm. the temperature is + 1° : when on cooling it falls to 816 mm., the temperature is - 1°. It is clear that if the temperature were reduced 273° below zero, and the same rule held, the pressure would fall to nothing. As a matter of fact the hydrogen would liquefy- before that point was reached (at about ~ 245°) ; still the supposed point of no pressure is very convenient to reckon from, and temperatures reckoned from it are called 'absolute.' Absolute temperature, then, is equal to the temperature centigrade +273 : it is simply proportional to the pressure in a hydrogen thermometer. The supposed point of no pressure E 2 52 HEAT (- 273° cent.) is called the absolute zero. These results may be put algebraically thus : If_p is the pressure of the hydrogen at 0° cent, (in melting ice), p the pressure at some other temperature f (cent.), then the absolute temperature of melting ice is 273°, the other temperature t + 273, and p a : p = 273 : 273 + t, whence t may be calculated. In connexion with the measurement of temperature arises the equally important practical problem of the maintenance of constant temperature— an extremely important matter for many physical and chemical, as well as physiological' experiments. Two methods are in use : (1) to make a bath of some material that undergoes a change at a fixed temperature, (ii) to use a bath of water, air, or other convenient material, and attach an auto- matic regulator to the heating supply. The changes available for use in (i) may be grouped into (a) fusion, (&) evaporation. Of the former the most familiar is the ice bath, already referred to : so long as both ice and water are present the temperature of the mixture remains constant at o°. Many other temperatures may be obtained by means of other substances : temperatures below o° by means of mixtures of ice and salts in the proper proportion to form ' cryohydrates,' e. g. Kal and ice will fall to — 30 ; high constant temperatures by the melting of metals, such as tin and lead ; and by means of the chemical transforma- tion analogous to fusion suffered by certain hydrated salts, temperatures between o° and 100° may be got, e. g. Glauber's salt NajS0 4 .ioHjO 'melts' in its water of crystallization at 32°-4. Secondly, the boiling-point of a pure liquid depends only on the pressure under which the boiling takes place ; this has already been referred to as a means of testing the ioo° point on a thermo- meter ; other points may be obtained by boiling either carbon disulphide, alcohol, toluene, chlorobenzene, aniline, and other liquids that are easily purified and do not decompose on boiling. The liquid should be boiled in a wide glass tube or other con- venient shaped vessel, inside which can be placed the apparatus whose temperature it is desired to maintain constant, and the boiling tube provided with a reflux condenser. With -a few liquids only a few temperatures are obtainable in this way, but by boiling them under an adjustable pressure any temperature desired may be reached. Fig. 26 shows in outline the apparatus TEMPERATURE 53 required for this purpose. The barometer tube a is to be main- tained at a constant temperature ; it is surrounded by the jacket b, which has a small bulb c' attached in which the liquid is boiled : b is fitted at the bottom by a rubber stopper made gas-tight by a layer of mercury above it ; d is the reflux condenser that keeps the Fig. 26. liquid in the jacket, m a large bulb to keep the pressure more steady', no the gauge by which the pressure is measured. The appropriate liquid is placed in c', the apparatus fitted up air-tight, a burner placed under c', and the pressure reduced by the pump till the liquid in c' boils at the desired temperature. Kamsay and Young 54 HEAT % B have prepared tables showing the pressures under which certain liquids boil at each successive degree of temperature ; with the aid of these any temperature from o° to 350 may be obtained, and when once established in the apparatus may be kept up indefinitely. The alternative method (ii) is to immerse the body whose temperature it is desired to maintain in a bath, usually , of water, or for high temperatures liquid r — ~\ metals, and heat the bath by a gas-burner <; |-LL| controlled by an automatic regulator. Such regulators have been constructed depending on the vapour pressure of a liquid, but more often depending on expansion. A If f very common and useful type, due to Ostwald, is shown in Fig. 27. The large bulb a extends to the whole depth of the bath b and is filled with toluene or a strong solution of calcium chloride; when the temperature rises the expansion of this liquid forces mercury contained in the tube c upwards. In the top of this tube is fitted a glass tube d to carry the gas supply : the gas passes down it, over the surface of the mercury in c and out at e to the burner, but when the temperature rises to a certain point the movement of the mercury up c stops the gas supply : a small hole blown in the side of d allows just enough to flow through to keep the burner alight, so that if the temperature shows a tendency to fall ■-.._--=•.— ._— -..- an( j the mercury contracts away from d Fie. 2". the flame springs up again : / is an iron screw to adjust the level of mercury and so alter the temperature at which the regulator works. With any such device the liquid bath must be kept well stirred by means of a motor. § 2. Quantity of heat : dynamical equivalent of heat. Temperature alone is not sufficient to describe the phenomena of heat. There is needed as well the quantity known as ' quan- tity of heat,' the quantity whose flow is regulated by differences gr-t-- DYNAMICAL EQUIVALENT OF HEAT 55 of temperature. The quantity of heat contained by a body is greater the higher the temperature to which it is raised, but apart from that, is proportional to the mass of the body itself, and moreover varies according to the specific character of the body. Thus a large mass of boiling water will give out heat for a longer time than a small mass of the same substance : and a kilo of hot water will give out more heat than the same mass of iron at the same temperature. Hence to measure quantities of heat it is necessary to choose a unit in which these points are borne in mind. The unit is the quantity of heat required to raise one gram of water through one degree in temperature, and is called a calorie. Other substances differ from water in regard to power of storing heat, hence the need of the term specific heat ; the specific heat of a substance meaning the amount of heat required to raise one gram of it through one degree ; further, it is often convenient to employ the term thermal capacity with regard to an individual body (e. g. a flask, a calorimeter), meaning by that the amount of heat required to raise it through one degree. Then to calculate the heat required to raise any substance through any range of temperature, it is necessary to multiply together the mass of the substance, its specific heat, and the rise of temperature. Or, if a body of m grams be made of a material whose specific heat is s, the thermal capacity of the body = ms ; and to raise it from temperature t' to t^ requires an amount of heat = ms (t 2 - tj. But, it has been remarked in the Introduction, heat is one of the forms of energy : it can be produced by the consumption of mechanical work in friction, as well as by conversion of electrical, chemical, and other energies : and under certain conditions it can, at least partially, be converted into those forms. Hence arises the problem, how much work is equivalent to a given amount of heat ? or to put it concretely, how many ergs of work or other form of energy must disappear when one calorie of heat is produced ? The quantity in question is known as the dyna- mical equivalent of heat, and the experiments of many physicists have been directed to showing that such an equivalent exists, and to measuring it. The solution of the problem owes more to Joule than to any one, since he first clearly stated it, and first 56 HEAT made measurements of an accurate character. The determination he made by the method of friction in water in 1850 was, for long, the standard experiment : besides that, he measured the heat generated by friction in mercury, friction of iron plates, com- pression of air, by an electric current, and in other ways, and in 1878 published a new and still more careful set of experiments- by the water friction method. We shall however describe, instead of his, a research by Eowland, by a similar method, pub- lished in 1879, and generally regarded as having given the most accurate results. Rowland's method, like Joule's, was to rotate a set of paddles vigorously in a vessel of water, hence warming the water : and to measure (i) the work spent in agitating the water, (ii) the heat generated in it, so as to be able to equate the one against the other. His apparatus is shown in Fig. 28. The calorimeter A, or vessel in which the heat was measured, was of brass, nickel plated, and highly polished, so as to enable it to retain heat better : it was surrounded by the double-walled jacket b, which contained water between the two walls, in order that, the surroundings of the calorimeter being perfectly definite, the loss of heat from it during an experiment might be accurately estimated. Inside the calorimeter, which was of eight or nine litres capacity, was an elaborate system of vanes, attached to a spindle, working between another set of vanes fixed to the walls of the calorimeter. The spindle passed through a stuffing- box at the bottom of the calorimeter, and could be driven very rapidly by means of the gear wheels shown in the figure, from an oil-engine. The calorimeter, on the other hand, was attached by a stiff arrangement of wires to a disc, which in turn was fixed to a shaft c, passing through bearings, and suspended by a wire from the head of the apparatus. Consequently, when the engine was started the paddles on the spindle tended to diive the calorimeter round ; but this being prevented from rotating, the water in it was violently churned up, and warmed in consequence. The upper part of the figure shows also the 'frictkn balance ' by which the mechanical power spent was measured. This consisted of an accurately turned pulley d, keyed on to the shaft ; round it passed a pair of tapes, leading over vertical pulleys and carrying weights. The pair of weights were so arranged as to tend to turn the shaft round in the sense opposite to that in which the paddles acted ; hence by adjusting the weights correctly they could be made to balance the turning moment, or torque, of the paddles. When this balance was accurately effected there is no resultant twist on the suspending wire of the apparatus ; and in order to observe whether the condition was DYNAMICAL EQUIVALENT OF HEAT 57 ■=€!3 Fie, 23. 58 HEAT fulfilled or not, a small mirror was attached to the horizontal pulley, and a telescope focussed so as to see the reflection of a lamp in the mirror. If then the pulley be rotated even through a very small angle, the image of the lamp will be shifted in the field of view of the telescope, and by the direction of the shift the observer can tell whether the torque of the engine is overpowering the weights, or the weights are too great for the engine, and can readjust accordingly. The bar e with the heavy slides was for steadying the apparatus, and rendering this adjustment easier. Then the torque exerted by the weights (and therefore by the paddles) = sum of the weight x radius of the horizontal pulley ; and, as explained on p. 30, to measure the work spent by the engine we have work = torque x 2 t x number of revolutions. The measurement of work done therefore requires, in addition to the torque, a knowledge merely of the number of revolutions : this was obtained by means of the chronograph shown in the lower part of the figure, on the drum of which the revolutions of the shaft were registered. The ther- mometer to show the rise in temperature of the water was enclosed in a copper case near the axis of the calorimeter, so that it could be read from outside without stopping the apparatus ; and when the mercury reached each scale division, a mark was made on the chronograph, so that the number of revolutions, and consequently the work required to raise the calorimeter one degree in temperature, could be directly observed. The heat evolved was that required to raise the calorimeter and its con- tents in temperature. The water content was much the more important, amounting to between 8,000 and 9,000 grams, while the brass and other materials of which the vessel was made absorbed as much heat as some 350 grams more of water (i. e. the thermal capacity of the calorimeter was 350). To the amount of heat thus calculated a correction had to be made, as usually in calorimetric experiments, for heat radiated from the calorimeter ; for as soon as the temperature of the calorimeter rises above that of the enclosure in which it is placed, it begins to give off heat ; the amount of this heat was determined by a separate experiment. The power used in the experiments amounted to nearly half a horse- power ; this is to be reckoned an advantage, as by comparison with so large an amount of heat generated, the correction for radiation is com- paratively unimportant. Kowland compared his working mercury thermometers with an air thermometer of his own construction ; they have however since been standardized in a more satisfactory manner by means of the hydrogen thermometer of the International Bureau at Paris, and the values of the dynamicjl equivalent so corrected are probably more trustworthy than that published by Kowland himself. They are given in the following table. Rowland's measurements extended over the range from 6 1 to 40 . and DYNAMICAL EQUIVALENT OF HEAT 59 showed that water (like other substances) has not quite the same specific heat at different temperatures : — Temp. Dynamical equivalent. Specific heat. 6' 4-203 x 10' ergs 1-0007 therms 10° 4.196 „ 09990 „ 15° 4-188 „ 09972 ,, 20° 4- 181 „ 09955 ., 2 5° 4' J 76 „ 0-9943 „ It is evident then that the definition of the unit of heat previously given is not quite precise, for according to the temperature of the water it will take somewhat more or less heat to raise one gram of it through i°. For this reason it has been proposed to adopt as the standard calorie or therm the quantity 4-2 x io T ergs. This is about the amount of energy required to heat a gram of water from 7° to 8°. Of the numerous other methods that have been tried for measuring the dynamical equivalent of heat, only one is comparable in accuracy with the water-friction method, that is, the process in which heat is generated in a wire by the passage of an electric current through it. This method will be referred to again in the chapter on electricity, when the method of calculating the electrical energy spent has been explained. Another experiment, however, deserves to be mentioned, not on account of its accuracy, but for its theoretical interest. Usually mechanical (or electrical) energy has been converted into heat, and the amount of heat generated measured ; Hirn, however, adopted the con- verse plan of converting heat into work, and measuring the amounts of heat spent and work generated. There is an important distinction between the two cases however ; for whereas mechanical (or electrical) energy can be wholly converted into heat, heat can only be partially converted into mechanical and other kinds of energy. Hence in an experiment of the kind now to be described it is necessary to measure three quantities of energy instead of two. Hirn's experiments were made on an actual steam engine, and therefore involved (1) the heat supplied to the engine, (2) the heat rejected by the engine to the con- denser, &c, (3) the work done, which is due to the transformation of the difference between (1) and (a). In the experiments, which were carried out continuously through a working day, the pressure in the boiler was kept as constant as possible, and read every ten minutes by means of a mercury manometer ; the steam from the boiler was passed through a superheater, as otherwise it would have carried with it an amount — very difficult to measure — of liquid water (priming), and the determination of the quantity of heat used would have become uncertain ; as it was, (he temperature of the superheated steam was taken imme- diately before passing into the cylinder— also every ten minutes — and its 6o HEAT pressure being known, Kegnault's measurement s of specific and latent heat sufficed to determine the heat supplied per gram of steam accurately. . The mass of the steam was measured both by means of the boiler feed, and on flowing out of the condenser. The second quantity of heat, that rejected to the condenser, was determined by the quantity of condensing water and its rise in temperature ; the condenser was of the 'jet' type, so that the condensed steam mixed with the water used for the con- densation, and the difference in mass between the inflow and outflow of water from the condenser measured the mass of steam used. The work performed by the steam was carefully determined by means of indicator diagrams. Corrections were applied (i) for the heat radiated from the _ cylinder ; this clearly must be subtracted from the total heat supply to get the amount actually converted into work : (2) the amount of work reconverted into heat and measured as such ; this is the case with the work spent on the injection pump and air-pump of the condenser, and therefore that amount of work must be deducted to get the actual output of the engine. As an example of the results we may take the following experiment : — Steam pressure, 4-4745 atmos. Corresponding saturation temperature, 145 . Actual temperature of the steam, J9°-57". Temperature of boiler-feed, 3o -gi. Mass of steam used per stroke of piston, 198-7 gm. • This, with the known values of the latent heat and specific heat of steam (p. 106), gives as the total supply of heat 198.7 [606 5 + 145 x 0-305 + (195-7 — 145) x 0-5—30-91] = 128120 calories. Temperature of condensation water before use = i6"-i5. ,, „ „ after use = 3o"-9i. Mass of „ ,, per stroke = 77323 gm. Hence the heat rejected was 773 2 -3 (30-91-16-15) = 11413° cals., leaving 13990 as the amount consumed; but the two corrections above- mentioned amounted between them to 1500 cal., leaving 12490 as the equivalent of the work shown by the indicator diagrams. The latter was 5318-8 kilogrammetres. Hence the dynamical equivalent is 5318-8 -^12490 = 0-426 kilogrammetres, or in absolute measure 4 18 x 10 7 ergs per calorie, in very good agreement with the results of Eowland?s and other experi- ments by the direct method. Hirn also was amongst the first to make experiments on the human body, regarded as an appliance for the conversion of heat into work. His experiments in this direction were not successful enough to quote, but others since, using more elaborate appli- DYNAMICAL EQUIVALENT OF HEAT 6t anccs,'have obtained results of value. For this purpose it is necessary to construct a calorimeter large enough for a man to live inside of. A recent experiment by Chauveau made in this ■manner gives a direct verification of the law of conservation of energy as applied to the body. A man inside the calorimeter worked a kind of treadmill, the shaft of which passed outside the calorimetric chamber and was used to drive a wheel against friction. Measurements were made on (i) the heat generated by the body inside the chamber ; (2) the work done on the wheel ; (3) the amount of carbon dioxide and water vapour exhaled by the man. Now if the body be regarded as a heat engine, its fuel is the tissue which, during action, is converted into carbon dioxide and water, so that from the amount of these products it is possible to calculate the heat that would be generated in pro- ducing them. In a certain instance this amounted to 257 calories per hour, while the heat actually measured in the calorimeter was 199 cals. Hence there is a loss of 58 cals. to account for the energy ' exported ' from the calorimeter in the form of work. The work done on reduction by the known value of the dynamical equivalent was found equal to 68 calories, so that, considering the difficulties of the experiment, a very fair agreement was obtained ; and it may be concluded that the conversion of the energy of foodstuffs into mechanical work is the same in its general character as the conversion of the energy of coal in the steam engine. Attention may again be called to the fact that conversion of heat into work is always incomplete. The fraction work given out energy put in for any machine is called its efficiency. In the case of Hirn's steam engine the fraction is 12490 =0-8? 128120 9 /o ' more than nine-tenths of the total heat supply finding its way to the condenser, still in the form of heat. The best modern engines give scarcely more than 15 %. The human body, according to Chauveau's experiment, is a more efficient instrument, for out of 257 cals. of energy supplied to it, it delivered 68 cals. in the form of work, or 26-5 %. 62 HEAT § 3. Thermodynamics. The law of the conservation of energy, already so often mentioned, may be put in the form of a denial of the possibility of what is called ' perpetual motion.' Attempts have been made, from time immemorial, to devise a machine that should go on indefinitely turning out useful work without using up any corresponding supply of energy. If the law of conservation of energy broke down anywhere such a machine might be possible, for if a process existed in which one kind of energy was Iransformed into another in a ratio not equivalent, then by carrying out that process in one sense or the other energy would be gained ; by an appropriately designed machine the process in question might be repeated indefinitely. That such a process has not been devised it is hardly necessary to remark : it is of more importance to point out that any new process that professed to be of such a character might safely be con- demned beforehand as false, for it would be in contradiction with the whole of our knowledge of Physics : and indeed that the evidence in favour of the conservation of energy is by now so overwhelming that the impossibility of a perpetual motion may be taken as axiomatic. Heat being a form of energy, the law of conservation is necessarily applicable to it. Nevertheless the law, as applied to heat, has often been stated separately, because taken in conjunction with another law it con- stitutes the basis of the mathematical theory of heat, known as Thermo, dynamics : it is then known as the first law of thermodynamics, and may be stated, in the words of Maxwell, as follows : — ' When work is transformed into heat, or heat into work, the quantity of work is mechanically equivalent to the quantity of heat.' The second law of thermodynamics, which was first clearly stated by Clausius, is : — • 'It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature.' It is, as remarked in § 1, a familiar observation that heat passes of its own accord from hot bodies to cold ones ; this was indeed made the basis of the definition of temperature — temperature being at least qualitatively defined as the criterion according to which heat flows in one direction or the reverse. It might, however, be thought that by some device heat might be caused to flow indirectly from a cold body to a hot one. It can certainly be made so to pass by means of the application of work : e. g. an ice-making machine can be devised by which heat is taken from water already cooler than its surroundings and transferred to the surrounding air, the process being continued till the water freezes ; but no ice-making machine will work without some external power to drive it. The state- ment involved in the second law is indeed not so obvious but that THERMODYNAMICS 63 numerous attempts were made to controvert it. One suggestion was that by collecting the radiation from a large hot body and concentrating it by means of powerful lenses and mirrors it might be made to raise a smaller body to a still higher temperature; this particular case, plausible as it looks, as well as others, was successfully dealt with by Clausius. The law is, of course, a generalization of such observed cases ; but it is more than that, for the evidence in its favour may be immensely strengthened by a consideration of what would happen if it were not true. Just as any single exception to the law of conservation of energy would give the opportunity for a perpetual motion, and so of an indefinitely great departure from the conditions that we recognize as normal, so any exception whatever to the second law of thermodynamics would give opportunity for what has appropriately been called a 'perpetual motion of the second kind.' The atmosphere and all surrounding bodies con- tain vast quantities of heat in them, for we may by artificial means cool parts of them many degrees below their usual temperature, and observe how large a quantity of heat it is necessary to remove from them in order to do so. Nevertheless this energy is not in the least available for conversion into useful work : if the heat be all at the same temperature it is not possible to drive even the smallest heat-engine by means of it ; in actual heat-engines the heat has always to be produced at a high temperature (in a coal furnace, gas flame, &c), in order to be capable of conversion into work. Now if it were possible by means of any exception to the second law to transfer heat from a colder to a hotter body, such means might be employed to take heat from the surrounding atmosphere and store it up at a high temperature (say in a steam boiler) in order to use subsequently for production of work ; and we should have a self- acting appliance for converting the even temperature heat of the atmo- sphere into useful work ; that is what is meant by 3 perpetual motion of the second kind : it involves no creation of energy, and therefore no contradiction of the first law ; but it is equally contrary to experience, and shows that any exception whatever to the second law would also lead to an indefinitely great departure from observed conditions. It is this logical consequence of the laws of thermodynamics that gives to them a validity beyond that attaching to the usual generalizations of Physics, and only comparable to that of the laws of motion. Hence any deduction made by the methods of thermodynamics has behind it all the certainty of the laws themselves. We cannot here give the mathe- matical expression of the laws of thermodynamics, or the deductions to be made from them ; but the general character of the results may be noted as consisting in relations between the numerical values of the various properties of matter. When some property has been numerically determined by measurement it is usually possible to duplicate the result by a thermodynamic argument : e. g. if a certain chemical reaction takes place partially in the cold, then by measuring the evolution or absorption 64 HEAT of heat accompanying it we can foretell whether it will take place to a greater or less extent on rise of temperature. One leading deduction from the laws of thermodynamics must, how- ever, be noted. It refers to the absolute scale of temperature. This scale is that which would be indicated by the expansion or by the pressure of a perfect gas (p. 77) and agrees exceedingly closely with that of the hydrogen thermometer (p. 50). In the first place it may be shown that the absolute zero, on this scale, means the point at which bodies would be deprived of all their heat energy : consequently there is no lower temperature. In the next place the attainment of the absolu'e zero is an impossibility, notwithstanding that a somewhat near approach has already been made in the liquefaction of hydrogen and helium (at about — 250 cent. = 27° absolute). Lastly the absolute scale itself is defined by a certain relation as to the convertibility of heat into work. It may be shown that if heat be contained in a body at temperature T l (absolute) while the surroundings are at the lower temperature T T, the fraction of it available for conversion into work is 1-^. An ideally perfect heat-engine would convert this fraction into work, T leaving the remainder -~ in the form of heat at the same tempera- ture as the surroundings ; any actual engine will, on account of necessary imperfections, convert less. As an example of this we may take Hirn's steam engine mentioned on p. 59. The hot body may here be regarded as the boiler, which is at a temperature 145 cent, or 145 + 273 =418° abs. ; the 'surroundings' mean, effectively, the boiler feed-water at 3o°-9i cent, or 303°-9i abs. Hence the limiting efficiency of conversion is 30^91 418 - 2 7-3/o> whilst the actual efficiency was 98 %. In actual processes of all kinds inefficiency such as the above is observed ; that is to say, a loss of useful work. When heat is converted into work, and anything less than the theoretical maximum of work is obtained (as is always the case in nature), an irreversible loss occurs, for the heat having become diffused at the same temperature as the sur- roundings is no longer available for conversion into any other kind of energy ; still more is this the case when high temperature heat is simply conducted or radiated away to bodies at a low temperature, or when work is converted by friction into heat. None of these processes can be performed in the reverse sense : consequently they all mean a diminu- tion in the amount of available energy. This fact is usually referred to by the name of the dissipation of energy ; i. e. energy, though it cannot be lost, is continually being dissipated into a uniform state of heat out of MOLECULAR THEORY 65 which it can no more be transformed. The second law of thermo- dynamics may, in fact, be looked upon as including two statements ; first, the quantitative relations that would hold for ideal processes that were reversible in character (leading to the definition of the absolute scale of temperature) ; and second, that processes which are irreversible necessarily lead in the sense of the dissipation of energy. § 4. Molecular theory. Whilst the method of thermodynamics allows of co-ordinating the phenomena of heat with certainty, it does nothing towards a com- prehension of the intimate nature of the transformations of energy whose quantitative relationships it allows of formulating ; and the instinct of the physicist is unable to rest in so imperfect an explanation. Attempts have always been made, therefore, to trace out a description of the less obvious facts of Physics — heat, electricity, light, and so on — in terms of the ordinary, visible phenomena of mechanics. These attempts have resulted in the atomic or molecular theory of matter, and this may be regarded as another aspect of the reality, serving to complete the aspect presented by thermodynamic reasoning. The theory was sug- gested, in the first place, by Dalton, as an explanation of the facts of chemical combination : the law of combination in fixed proportions finds a natural explanation in the hypothesis that matter is composed of certain definite particles, atoms, of which there are a good many kinds — hydrogen atoms, chlorine atoms, &c. — but each kind perfectly constant in size and structure, and that these atoms unite in various ways to form molecules, i. e. arrangements of matter which hold together under ordinary circumstances ; o. g. two hj drogen atoms unite to form one hydrogen molecule, and such molecules are the actual constituents of ordinary hydrogen gas : or a single atom may constitute the molecule, as in mercury vapour : or an atom of hydrogen unites with one of chlorine to form a molecule of hydrochloric acid — and it would be impossible to find a compound in whose molecules there was, say, only half an atom of chlorine each. The atomic theory, so started, was rendered much more definite by various physical phenomena which point to a certain measurable scale for the structure of matter. Various lines of argument, due mainly to Lord Kelvin, and to the founders of the so-called kinetic theory of gases, Clausius, Maxwell, and Boltzmann, have led to the con- elusion that matter has a. structure of finite, although very small size, and that just as » piece of plant tissue which looks to the eye uniform throughout, and structureless, is seen under the microscope to consist of cells of approximately equal size and similar character — perhaps a hundred of them to the millimetre in length — so all matter, water, silver, or what not, though uniform in appearance under the most 66 HEAT powerful microscope, is nevertheless made up of parts regularly placed and similar to one another, and that it is possible to estimate the size of these parts or molecules. The estimates of size, which have recently received valuable confirmation from the experiments of J. J. Thomson, lead to the conclusion that a molecule of hydrogen has a mass of about 4-5 x io -23 grams. Since a cubic centimetre of hydrogen at ordinary temperature and pressure weighs 9X10 -5 grams, this means that in 1 cubic centimetre of hydrogen there are 9 x 10- s -r- (4-5 x io _2S ) = 2 x 10" molecules : if these were arranged in cubical order there would be V 2 x I o 18 = 1,300,000 to the centimetre in length. Accepting this view, then, the molecules are not to be regarded as occupying- fixed positions, but as in rapid motion ; and the different states of aggregation in matter are explained as follows : — in gases the molecules are some distance apart, the empty spaces between adjacent molecules being much larger than the molecules themselves. Con- sequently the molecules are for the most part free from one another's influence, and travel, with high velocity, in straight lines. This motion, however, brings them from time to time into collision : actually, in ordinary gases, many million times per second ; the average distance that the molecules travel without a collision, the ' mean free path ' as it is called, is one of the most important quantities to determine on this theory, and a knowledge of it helps to form a picture of the state supposed to exist in a % gas. The molecules of oxygen and nitrogen in the air are estimated to travel at about 50,000 centimetres per second (the speed of a modern rifle bullet), and to move on an average only about centimetres without coming into collision. Notwithstanding 100,000 the smallness of this distance, it is so much larger than the diameter of the molecules themselves that the interference to which their move- ments are subjected by the collision lasts only a small fraction of the whole time. This is still more true when the gas is rarefied, as in the vacuum tubes used for electric discharges : here the number of molecules is so far reduced that they may, on the average, travel as far as a centi- metre without coming into collision. Jt is then possible to regard them as for the most part travelling independently, and the mathematical treat- ment, though difficult, is easier than in the case of liquids and solids. In liquids, of course, the molecules are far more crowded together than in gases, since the same amount of matter goes into a space perhaps a thousand times as small. Now the phenomena of gases show that it is not sufficient to assume molecules moving freely between successive collisions, but that the molecules must be regarded as attracting one another, when not in contact, although the amount of attraction is only of importance when they are very close together. In gases this at- traction only modifies a little the results calculated from the motions of MOLECULAR THEORY 67 the molecules ; but in liquids, since these are always very close together, the attractions between them are always of great importance. The result is to make liquids hold together, and not, like gases, spread out into any space they have access to ; at the same time the flowing of liquids shows that their molecules must be capable of moving over one another with little difficulty. In solids the molecules are usually still closer together than in liquids, but the essential difference is in their keeping on the whole fixed positions relatively to one another ; the fact that a mark can be made on a solid, and will remain unchanged for a thousand years (as in old coins), seems to show that the molecules of the solid, though moving, only move to and fro about certain average positions, whereas those of liquids travel indefinitely through the substance. In gases, then, the molecules, travelling with high velocity in all directions, come frequently into collision with the walls of the vessel containing the gas : and since the number of collisions per second is enormously great, the individual effects they produce cannot be observed, but the general result is a continuous pressure outwards on the wall ; this is the pressure of the gas, measured by a barometer or similar gauge, as explained on p. 37. The amount of pressure depends (1) on the number of molecules producing it, and hence is greater in a dense gas than a rare one, (2) on the velocity of the molecules to whose ' bombard- ment ' it is due. This last point leads to the explanation of heat on the molecular theory. Heat, which is known to be a form of energy, is explained as the kinetic energy of the molecules ; it consists partly of the rectilinear movements (translation) of the molecules, just discussed, which produce the gas pressure, and partly of other kinds of movement, viz. rotations of the molecules, and, when the molecule consists of more than one atom, of relative movements amongst the atoms (intramolecular movements or vibrations of the molecule). The energy of all these movements taken together constitutes the store of heat in the gas, and in a solid or liquid the same is true, though the distribution of energy amongst the various kinds of movement may be very different. Two important theorems may, in the case of gases, be shown to follow from this idea : (1) the pressure of the gas is proportional to the average kinetic energy of the molecules of the gas ; hence when with a hydrogen thermometer we measure temperature by means of the gas-pressure we are really treating the temperatures as proportional to the average kinetic energy of the molecules ; (a) the average kinetic energy of translation of any one gas is equal to that of any other gas at the same temperature : consequently when there is the same number of molecules per cubic centimetre in two gases whose temperature is equal, they possess the same amount of kinetic energy of translation per cubic centimetre, and therefore the same pressure. This is Avogadro's law, stated in terms of the kinetic or molecular theory. F 2 f8 HEAT § 5. Conduction of heat. Heat, as we have seen, has a constant tendency to flow from places of high to places of low temperature till the temperature difference is abolished. This process, which is known as conduc- tion, takes place through all bodies of whatever material, whether solid, liquid, or gaseous, but does not, so far as is known, take place across empty space, in the absence of matter. In order to form a quantitative conception of the process it is necessary to follow out a simple case : imagine, a cubical block of a substance, 1 cm. in side, and let two opposite faces be maintained at different temperatures, one a degree higher than the other. Then at first when the temperature difference is set up, heat will flow into the substance of the block to raise its temperature ; but soon each point of the block will have settled down to a steady temperature, and — the temperature of the two faces being main- tained constant— a steady flow from the hot side to the cold will take place ; this flow measures the process of conduction, and the number of calories per second is called the conductivity. The block may be looked upon as a section of a wall separating a place of high temperature from one of low, and to generalize the case we may suppose the wall to have (1) any area, then the amount of heat flowing will be proportional to the area ; (2) any thickness, then the flow of heat will be inversely as the thickness ; (3) any difference of temperature between its faces, then the flow will be approximately proportional to this difference. Summa- rizing this we have, if a is the area of the wall, d its thickness, /, the temperature on the hot side, t 2 on the cold, and It the con- ductivity, flow of heat per second = q = - — ~ — — ■ The difference of temperature may here be looked upon as of the nature of a force tending to drive a current of heat : the current flowing will depend not only on the magnitude of the force but on the area, thickness, and material of the channel through which it has to flow. To measure the conductivity of any substance, particularly of a solid, it is necessary to realize experimentally the conditions described above. Of arrangements for the purpose the earliest CONDUCTION OF HEAT 69 was that of Peclet ; he experimented on metal plates. Each side of the plate, which was laid horizontally, was brought into contact with water— the lower hot, the upper cold, and both masses of water were vigorously stirred in order to keep the metal surfaces at the temperature of the water and avoid any stagnant layer. The conduction through the plate reduced the temperature of the lower water and raised that of the upper ; so that by observing the rate of change of temperature the amount of heat flowing through could be estimated. The measurements, however, gave values of k that we now know to be too small, and for a reason that is important to note : the two surfaces never really assumed the temperatures indicated by thermometers immersed in the masses of water, despite the most vigorous stirring. Of the whole difference of temperature available, only a part is spent in driving the current of heat through the substance of the metal : the remainder is required to cause the heat to flow from the lower water into the plate, and from the plate into the upper water. There is in fact a resistance to absorption on the one side, and to emission on the other, as well as to conduction through, and though stirring may reduce these resistances it cannot do away with them altogether : indeed, when the plate is made of such good conducting material as the metals, the resistance to conduction through the plate is so small as to be quite obscured by the others. The difficulty is not so great in the case of less good conductors, and the method has been successfully applied in the recent experiments of Lees on stones, glass, ebonite, &c. Discs of the materials, 4 cm. diameter and from 2 mm. and upwards thick, were cut, and pressed between two copper discs : one of the latter was made double, and enclosed a flat coil of silk-covered platinoid wire. An electric current being led through the wire, heat was generated in it, and flowed across the disc of material to the copper on the other side. The disc on the other side may be kept cool by a current of water. In each copper disc two radial holes are bored, inside which are soldered a platinoid and copper wire respectively ; these constitute a thermo-couple (see p. 225) by means of which the temperature of each copper disc may be measured : these are t^ and t 2 in the above equation ; while the flow of heat can be determined, also electrically, from the strength of the electric current flowing through the heating coil. A little of the heat generated escapes from the edge of 70 HEAT the discs instead of flowing across from one face to the other, as the theory supposes, but this can be allowed for. In the case of metals the method was successfully applied by Berget : his experiments on mercury may be taken as typical. The ' wall ' or 'plate,' in order to offer sufficient resistance to the flow of heat, is thickened till it becomes a cylinder considerably longer than it is broad ; this filled a vertical glass tube a (Fig. 29). The supply of heat was from a flask c in contact with the upper face of the mercury, containing alcohol, water, or other liquid, kept boiling ; the current of heat flowed vertically downwards through the mercury to a vessel d containing ice and water in contact with its lower face. This vessel played the part of a Bunsen calori- meter (p. 105), and measured the heat transmitted. But instead of assuming, like Peclet, that the differ- ence of temperature between the boiling liquid at the top and the ice at the bottom was all spent on the mer- cury, the temperature of the latter was determined by a thermo-couple that could be lowered to various measured distances in it. What is needed to know is the quantity l 2 , i. e. the fall of temperature for a given thickness. It is more convenient to regard this as a single factor in the equation — called the tem- perature gradient or slope of temperature ; evidently this gradient, in degrees per centimetre, can be calcu- Fie. 29. lated if the temperature is taken at various depths in the mercury tube. If, further, we fix attention on the flow of heat per square centimetre of cross section of the tube, we may write the equation of p. 68 in a manner that brings out more closely the essential relations : — i = ft x *i=h a d i. e. flow of heat per square centimetre per second = conductivity x temperature gradient. The method of experimentation just described, when the plate of material is elongated into a column or tube, increases the error due to loss of heat from the sides; this error, ho wever, may be avoided by an ingenious device, originally used for a similar electrical problem. The device, as used by Berget, consisted in surrounding the glass tube containing the mercury by a wider concentric tube b, also containing mercury ; then all the heat flowing out of the sides of the apparatus came from b, and the flow in A was left to proceed straight across from the hot to the cold face. Just as A leads below to the working chamber D of the calorimeter, so J C \ < J B A B E D E CONDUCTION OF HEAT 71 the continuation of b formed the ice-jacket e ; consequently only the heat that had flowed through a was measured by the calorimeter. Of other less direct methods that have been employed to measure conductivity it is only necessary to refer to one, that depending on the propagation of waves of heat ; and that not so much for the sake of the measurements made as for the light it throws on the fluctuations of tem- perature near the earth's surface. If one end of a bar of metal be alter- nately heated and cooled — say, by passing a current of steam over it for a few minutes, and then cold water for a few minutes, and so on — a wave of heat is propagated along the bar ; and it may be noticed that (i) every point of the bar suffers fluctuations of temperature similar to those of the end experimented on ; but (ii) it takes time for these fluctuations to travel along the bar, so that the highest temperature is reached later by other points than by the point directly heated, and later by a distant point than a near one ; (iii) the fluctuations in tem- perature become less and less considerable as the distance from the source increases, so that if the bar is long the far end of it may not be appreciably affected. Now the earth's surface is alternately heated and Cooled by the sun, passing through a cyclic change once a year, and also a cycle once a day. Accordingly the material of the earth's crust suffers similar changes ; but the rate of propagation of the heat wave is so slow that only two or three metres below the surface the highest temperature is attained six months later than at the surface, i. e. midsummer occurs in January, and midwinter in July ; also the extent of the fluctuations falls off very rapidly, the daily fluctuations becoming inappieclable at a depth of a metre, the annual at a few metres only. By measuring the rate of propagation and the decrease in intensity Lord Kelvin was enabled to calculate the conductivity of the soil. The following table gives a notion of the results obtained by conductivity measurements : — Copper o-8 Mercury 0-018 Iron 0-16 Water 0-0013 Sandstone o-oi Air 0-000053 Vulcanite 0-0002 Hydrogen 0-00033 This means, e. g., that a temperature gradient of i° per cm. would cause o-8 calories to flow per sec. through each square cm. of a copper plate. The conductivities, however, vary a little with temperature ; in other words, the flow of heat is only approximately proportional to the temperature differences pro- ducing it. The conductivity of liquids is small, that of gases much smaller ; but transfer of heat can take place in them much more rapidly in another way. Rise of temperature reduces the density of fluids 72 -{' HEAT in nearly all cases ; consequently when a fluid is heated from below the parts first heated become lighter and float upwards, being replaced by colder fluid from above, which is in turn heated. This process is termed convection, because heat is con- veyed by means of moving matter ; it may be observed in many common instances, such as the movement of water in a kettle or flask heated over a burner, and the air over a flame, or a stand of hot water pipes. If a fluid be heated from above, convection cannot take place, as the lighter fluid is at the top to start with and stays there : this can be observed by playing on the surface of a beaker of water with a bunsen burner : the lower layers of water are heated by conduction only, and it is found that the surface water can be boiled, while that at the bottom remains cool. In measuring the conductivity of liquids it is obviously necessary to supply the heat from above in order to avoid con- vection. An important exception occurs in the case of water between o° and 4° ; over this range of temperature water on heating becomes more dense, passing through a maximum of density at 4 , after which it behaves like other liquids. Hence between o° and 4 , water, contrary to other liquids, must be heated from above and cooled from below, in order to make use of convection : an important consequence of this in nature is that lakes may be cooled in winter on the surface till the water freezes, and yet the deeper water will not fall below 4°. The same remarks are true of sea water, except that the temperature of maximum density is somewhat lower. Convection is a process as to which it is hardly possible to make any quantitative statements, as it is so greatly affected by accidental circumstances : thus, the loss of heat from the walls of a house, or from the animal body, is much greater when a wind blows over its surface, than in still air at the same temperature ; and convection in both liquids and gases can be greatly promoted by artificial circulation with pumps or fans. § 6. Radiation. In the preceding section we have dealt with the processes by which heat is transferred as such from place to place ; it may, however, also be transferred by means of a transformation into RADIATION >73 another kind of energy and re-transformation, as in the process known as radiation. When heat is radiated from the sun to the earth two transformations take place ; the hot matter of which the sun is composed starts waves in a medium filling the inter- planetary spaces, a medium of which not much else is known than its property of transmitting such waves, and to which the name of the luminiferous ether has been given ; the action, so far, is comparable with that of a ship which, in consequence of the rotation of its screw, starts waves over the surface of the sea, and provides the energy which these waves carry away with them. The waves in the ether, known generally as radiation, are propagated with great velocity in all directions. Some of them falling upon the eye produce the sensation of light, and therefore radiation of appropriate character is itself known as light. But the distinction involved in this term is a purely physiological one : other radiation, differing in no essential features from light, produces no effect on the optic nerve ; but for physical purposes all radiation, whether luminous or not, must be treated together. When radiation falls on a body it is always more or less absorbed : very slightly by gases like air, very strongly by ordinary opaque solids ; since then on absorp- tion the energy of the radiation disappears, it must be by con- version into some other form, usually heat. Hence the more radiation a body absorbs, the more it will be heated ; this is the second transformation above referred to. It may be compared to the action of waves on the sea, produced by the ship, falling on a small boat at a distance ; they will set the boat rocking, i. e., impart kinetic energy to it, and will themselves pass on with a more or less diminished supply of energy. Conceivably an arrangement might be invented that would be so violently rocked by the incoming waves as to absorb the whole of their energy, in which case there would be no wave left to travel further, or, in .other words, the absorption would be complete. The process of communication of heat by radiation differs in its most essential features from the simple flow of heat by conduction. Con- duction depends absolutely on the presence of matter to carry it on : radiation takes place best in free ether, in the absence of matter, and is always to a certain extent obstructed by the presence of matter, even in the form of gas. Conduction is a very slow process, and when occurring in waves it may take, as we have seen, months to travel a few metres : 74 HEAT radiation is enormously rapid, so that it only takes eight minutes to travel from the sun to the earth. These differences correspond to essential differences between the mechanism by which the two processes are carried out. According to the molecular view, conduction consists in the distribution of the kinetic energy of the molecules by their mutual collision ; so that if one set of molecules have originally more average kinetic energy (i. e. are at a higher temperature) than another set with which they are in contact, molecular kinetic energy (i. e. heat) is gradually taken from the hotter and given to the colder ones by collision. Radiation, on the same view, is set up in the ether by the intramolecular movements of vibration, as we have attempted to explain by the analogy of a ship at sea ; and when once such waves of radiation have been established they are propagated to a distance in the ether without further intervention of material molecules. Intensity of radiation depends very largely on temperature, always increasing as the temperature, and therefore the store of heat-energy, increases. When two bodies at the same temperature are placed opposite one another each radiates to the other to the same extent, and conse- quently on the whole no interchange of heat takes place (as might be expected from the second law of thermodynamics). When one of the bodies is at a higher temperature than the other it gives out more radiation than it receives, and so on the whole loses heat. When the difference of temperature between two bodies is small, the net radiation of heat from the hotter to the colder is proportional to the difference of temperature — a statement fre- quently known as Newton's law. The statement, however, may be taken to cover the entire loss of heat, or emission from a hot body : for the proportionality holds, as we have already seen, for conduction, and it is approximately true also for convection. Thus if a body some twenty degrees hotter than the air is exposed, it loses heat by radiation from its surface, by conduction through the air, and its solid supports, and by convection of air. All these processes take place twice as actively as if the excess of temperature were only ten degrees. The rate of emission may be measured by noting the rate at which its temperature falls : if the thermal capacity is known (see p. 55) this will evidently give the rate at which heat is being given off by the body. As the emissivity is a quantity of much practical importance, some results of measurements may be quoted with advantage : Emissivity. Black surface in vacuo . . . 0-00009 Polished silver surface in vacuo 0-00003 Black surface in air .... 00002 RADIATION 75 These are the numbers of calories emitted per square centi- metre per second, when the emitting body is one degree hotter than its surroundings. Generally, the flow of heat per second where h is the emissivity, a the area, t x the temperature of the emitting surface, t* the surrounding temperature. It will be seen from the values of h that the pure radiation (represented by the emission in vacuo) is some three times as great for a black surface as for one highly polished : and that when the surface is exposed to air, the conduction and convection add a loss of heat somewhat greater than that due to radiation from a black surface, raising the coefficient from 0-00009 to 0-0002. When the radiating body is at a much greater temperature than its surroundings the simple law of proportionality does not hold ; the amount of heat radiated increases far more rapidly than the excess of temperature. The conditions are fairly expressed by Stefan's law, that the actual radiation from each body is proportioned to the fourth power of its absolute temperature. The heat actually lost by a hot body is, as we have seen, the difference between what it radiates and what it receives by radiation from surrounding bodies. Hence we may put the loss of heat (per sq. cm. per sec.) as c W - r,«), where T L T 2 are the temperatures of the hot body and its surroundings respectively, and c is a constant : the latter has been found for a black surface to be about 1-28 x io -12 calories. The very rapid way in which the radiation increases with temperature may be illustrated by an example. If we take for T 3 the atmospheric temperature, in round numbers 300 (=27° cent.), then a body at 327° cent. ( = 600 abs.) will radiate sixteen times as much heat as it receives: one at 627° cent. (=900° abs.), a moderate red heat, eighty-one times as much as it receives, and one at 927 cent. ( = 1200 abs.), or verging on white heat, two hundred and fifty-six times as much. It must be noted carefully that Stefan's law and the above calculations refer to radiation only, not to the total emission of heat, which is » complex phenomenon involving conduction and con- vection as well. The latter processes are more nearly proportional to the excess of temperature throughout, so that while at low temperatures they may be the most important, at high temperatures radiation, owing to its very rapid increase with temperature, completely outweighs the other causes of loss of heat. The principles explained above throw light on practical prob- lems of heating as involved in house construction and in clothing. 76 HEAT The materials of which houses are built do not vary greatly in conducting power, with the exception of thatch, walls filled in with loose shavings, and the like. These are a useful resource when heat has to be very carefully husbanded on account of intense cold outside. They are very bad conductors for the same reason that cloth, fur, wool, brown paper and similar substances are, viz. that they are largely composed of air : the air is en- tangled in such small spaces between the fibres of the material that convection currents are practically stopped, so that loss of heat occurs mainly by pure conduction ; the heat transmitted through a layer of such material, when well made, such as fur, may be little more than that which would be calculated for an equivalent layer of air, convection being ignored. For the same reason two or more layers of a woven fabric, even when each is quite thin, keep in heat very efficiently, on account of the layers of stagnant air between them. But as the emissivity of a surface is limited, when a current of heat is flowing through a solid wall, a part of the temperature difference available has to be spent in getting the heat away from the surface of the wall : in other words, this surface effect adds a certain amount to the resistance to conduction of the wall. Hence a wall cannot be indefinitely reduced in its resistance to flow of heat by thinning it down, or by making it of the best conducting material ; it would, on the one hand, be of no appre- ciable advantage to make boiler plates of copper instead of steel : and on the other, a glass window, though so much thinner than a brick wall, does not let very much more heat through per square metre ; and if the window be made double it may even, on account of the layer of stagnant air, serve better than the wall to keep it in. The most efficient protection against loss of heat is a vacuum. Liquid air is now preserved in double-walled vacuum jackets, the space between the two walls being very completely exhausted of air. Here heat can only pass from the atmosphere to the liquid in the inside by radiation, and so little passes in this way as to cause only a very slow evaporation of the liquid air. CHAPTER III. PEOPEETIES OF FLUIDS. § i. Perfect gases. It is convenient to deal first with fluids (under which name both liquids and gases are included) and afterwards with solids, as the structure and properties of the latter are more complicated. In fluids, as we have seen (p. 37), there is a pressure which is everywhere and in all directions the same, except for variations due to the weight of the fluid itself, which may usually be ignored ; this pressure and the temperature are the main factors determining the properties of the fluid. Among fluids, again, the simplest results are obtained in studying gases : with these, then, we shall begin. If a gas be kept at a constant temperature, it is found that the volume it occupies is inversely proportional to the pressure exerted on it. This is usually demonstrated by enclosing the gas in a graduated glass vessel, shut off by mercury : the mercury forms part of a gauge, by means of which the pressure on the gas may be measured. This result, known from the name of its discoverer as Boyle's law, is very closely true for ordinary gases at pressures such as that of the atmosphere ; Ave may express it algebraically as p v = constant (when T is constant), where p is the pressure to which the gas is subject ; if is the volume, which may conveniently be taken as the volume occupied by one gram, the so-called ' specific volume ' ; T is the temperature. Again, if a gas be kept in a vessel of constant volume, and heated, its pressure rises, and it is found that with all ordinary gases the rate of rise is approximately the same, viz. — of the pressure exerted at the freezing-point, per degree : this may be 78 PROPERTIES OF FLUIDS shown experimentally by filling a gas thermometer, such as that shown in Fig. 25, with various gases in turn, and immersing it in ice and steam. The result obtained may, as we have seen, be most conveniently expressed by reckoning temperatures, not from the freezing-point, but from a point 273 below ; using the letter T for temperatures so reckoned (and called ' absolute ') we may say that $ oe T (when if is constant). Finally, if the gas thermometer be so modified that the pressure exerted by the mercury on the gas is always the same, the gas will expand, with rise of temperature ; and if the vessel it expands into be graduated we have a means of observing the relation of volume to temperature under constant pressure. It is again found that the change for one degree is — of the amount at the foregoing point, or, in other words, that now the volume is proportional to the absolute temperature : or i/oc? (when T is constant) (Charles's or G-ay-Lussac's law). These three results may be included in the one equation pv' = E T, where if is again to be taken as the specific volume, and R' is a constant ; thus, for oxygen it is found that at o° cent, and the normal atmospheric pressure of 76 cms. of mercury (= 1013230 dynes/sq. cm.) the density is 0-0014279 gms. per c.c. or v = T = 700-3 c.c. per gram. 0-0014279 tv pxf IOI3230 x 700-3 , m =^7jr = ^-^ - — - = 2,599,000 ergs per degree. 1 273 force (It should be noted that the product pressure x volume = ELVQ2L x (length x area) = force x length = work ; hence the value of p v'/T can be expressed in ergs per degree.) This equation allows of calculating the volume of a gram of oxygen at any temperature or pressure. The result may be extended to all gases in a very convenient way, by the aid of Avogadro's law ; according to that well-known principle of chemistry, in gases at the same tempera- ture and pressure, each cubic centimetre contains the same number of molecules ; hence the mass of a molecule of each gas (molecular weight) is proportional to the mass of a c. c. (density), PERFECT GASES 79 or the molecular weight, taken in grams, of any gas occupies the same volume. As standard of molecular weight is taken that of oxygen as 32 ; the molecular weight in grams, or a gram-molecule, of oxygen occupies, at normal temperature and pressure (0° and 760 mm.), 32 x 700-3 = 22410 c. c, and a gram-molecule of any other gas occupies the same space. Hence it is more convenient to express the relation between volume, temperature, and pressure for a gram-molecule of the gas instead of a gram ; calling v the volume of a gram-molecule (or molecular volume) we have j? v = B T, where B is now a constant common to all gases (the so-called gas-constant) and its value is 32 times that calculated above for Bf in the case of oxygen. B = 83,157,000 ergs/degree, or very nearly 2 therms (p. 59). The equation pv = B T is called the characteristic equation to a gas. The relations of pressure and volume can be most usefully represented by means of diagrams similar to that of Fig. 10, p. 23. If instead of distance and force, as in that case, volume and pressure be represented by lengths drawn horizontally and verti- cally, we may indicate a set of observations relating to Boyle's law, as in Fig. 30. Thus the position of a will represent an observation, if the abscissa on measures, on the scale of the diagram, the volume occupied by the gas experimented on, while ost measures the pressure exerted on it at the same time. Then if b, c, . . . represent in the same way other observa- tions made on the same gas at the same temperature, we may summarize them all by drawing the smooth curve abc through the points, abc will therefore indicate a set of corresponding pressures and volumes, all under a constant temperature, say T x ; the line is called the isothermal of 2\. If another set of observa- 80 PROPERTIES OF FLUIDS tions be taken at some other temperature T a another line will be obtained — say efg if the latter temperature be lower than the former. This line is lower than abc, because for any given volume the pressure exerted is less at the lower temperature : so that if a and e refer to two observations with the gas occupying the same volume, the height en is less than an. The curves which represent a behaviour corresponding to Boyle's law are rectangular hyperbolas, whose asymptotes are the axes of pressure and of volume, op,ov; but obviously the same method of graphi- cal representation is applicable to any substance, whatever the law connecting the pressure and volume may be. The work done by a gas in expanding may be measured with the aid of such a diagram. Since, as remarked on p. 78, the product of a pres- sure and a volume is a quantity of work, the p v diagram may be utilized in a manner precisely similar to the work diagram (Fig. 10), and as there shown, the work is represented by the area bounded by (i) the curve, (ii) two perpendiculars through its end points, (iii) the horizontal axis. Thus the gas in expanding from the condition of volume and pressure represented by a (Fig. 30) to that represented by b, performs an amount of work measured on the scale of the diagram by the area abn'ha. The line joining a to b is sometimes called the path of the expanding substance, because it represents the succession of states through which the substance passes in changing from a to b. This method is by no means confined to the isothermals, whether of a gas or other substance ; if a, b are any two states, differing in temperature as well as in volume and pressure, and the line ab be drawn so as to indicate the intermediate states, the work will still be measured by the area between ab and the base and perpendiculars. Further, it may be possible to proceed from a to b by different intermediate processes, i. e. by different paths ; the work done will then in general be different, since the area of the figure will be different according to the shape of the line ab. E. g. in order to convert ice-cold water into steam at 100° we might either (i) heat the water to ioo° and then boil it ; or (ii) evaporate it in vacuo at o" and then heat the vapour formed up to 100°. The work done in the two processes would not be the same. Another important quantity can be conveniently represented by the diagram, viz. the elasticity, or resistance to compression, of the fluid. A change in volume of the fluid is called a strain : such a change can easily be produced in a gas, but, though the fact is less obvious, liquids also can be compressed to a small extent. To produce a strain a stress is needed — in this case a change of pressure. Since the pressure applied to a fluid is distributed uniformly throughout it, the same pressure will clearly produce the same compression in each cubic centimetre of the PERFECT GASES 81 fluid, and therefore, to measure the strain, the change in Volume of i cubic centimetre must be taken, not the change in the whole quantity of fluid, which is indefinite in amount. Now it is found that when the strains, as so defined, are small, they are proportional to the stresses required to produce them (Hooke's law), and the constant ratio of stress to strain is called the coefficient of elasticity, or small change of pressure stress „ -r: r; — z 1- — ; = —i — — = coefficient of elasticity. corresponding small change of volume strain per c.c. Representing this graphically (Fig. 31), if aa' is a curve showing the relations between pressure and volume, we may look upon the change from a to a' as involving a stress and its corresponding strain, and if the change is small, as the figure roughly indicates, we may apply the above equation to determine the coefficient of elasticity. The stress is the increase of pressure mm'; the strain is the decrease of volume hn' divided by the original volume ots. Hence, if aa' be produced to meet the vertical axis at R, by the properties of the similar triangles aa's and arm we have NN' coefficient of elasticity =JOf -=- — — ■■A'S- AM For a very small change aa' the line ar becomes the tangent to the curve ; hence we may state the rule : ' To determine the elasticity of a fluid in any given condition draw the pv curve for the fluid, construct the tangent to it at the point representing the state con- sidered, and produce it to cut the axis of pressure : the intercept MR on that axis, between the horizontal through the original point and the point of intersection, measures the coefficient of elasti- city, according to the scale of pressure employed.' The pressure- volume curves discussed above were mainly under the condition of constant temperature (isother- mals) ; if measured along them the isothermal elasticity is obtained. But the elasticity may also be measured under other conditions of compression in the same way. For the particular case of a gas obeying Boyle's law, for which there- fore the isothermals are reatangular hyperbolas, it may be shown that the isothermal elasticity is equal to the pressure. This may be shown G 82 PROPERTIES OF FLUIDS algebraically as follows :— Let p be the pressure of the gas, v its volume, and let the pressure be increased by the small amount p', causing a small decrease in volume if 'while the temperature is kept constant ; then according to Boyle's law the product of pressure and volume is the same after the compression as before, or pv = (p+p') (»—'/) - pv+p'v—v'p-v'p'.- Now since p 1 and 1/ are both small quantities, their product will be much smaller still, and we may leave it out of the equation. Then we have p'v = v'p. But the stress (change of pressure) is p' while the strain (fractional 1/ change of volume) is — , so that the coefficient of elasticity to' = — =P, v the pressure of the gas. Thus the isothermal elasticity" of atmospheric dvnes air is one atmosphere, or in C. G. S. units io 6 — . This value may sq. cm. be compared with those for liquids and solids given below. Again, if air be compressed to 20 atmospheres, its coefficient of elasticity becomes 20 atmospheres too, i. e. its resistance to further compression is twenty times as great as when at ordinary pressure. The graphical methods above described are of general application, but the other results, viz. Boyle's and Charles' laws, the characteristic equation pv = RT, and the theorem that the isothermal elasticity is equal to the pressure, apply only to gases ; and really only constitute a limiting or ideal cate to which gases approach in their properties. The term perfect gas is used for an imagined substance which followed these laws exactly : the nearest actual approach is made by hydrogen and helium ; somewhat less close are nitrogen, argon, oxygen, carbon monoxide, nitric oxide, methane, carbon dioxide, sulphur dioxide, ethylene, and so on. §2. Condensation and vapour pressure. When a gas such as air is compressed, it decreases continu- ously in volume as the pressure increases ; it may be made, by sufficient pressure, almost as dense as a liquid, but at no point is there a visible change into the liquid form. During the com- pression the gas does not follow Boyle's law at all closely after the pressure rises to a few atmospheres ; the actual behaviour for air may be described as follows :— At first the compression follows Boyle's law very closely, but the gas is just measurably more compressible (has a lower coefficient of elasticity) than that law indicates, so that under 10 atmospheres the discrepancy CONDENSATION AND VAPOUR PRESSURE 83 amounts to nearly 1 per cent. If we put the product pv = 1 under low pressure, then at 10 atmospheres it has fallen to about 0-992. It continues to fall, the departure from the behaviour of a per- fect gas increasing in amount, till about 65 atmospheres, when pv = o 97 nearly ; after this the compressibility becomes less, so that the value of pv increases again, till, under a few hundred atmospheres, the gas becomes almost as incompressible as a liquid , and the behaviour is entirely different to that of the ideal gas. Certain other gases, N 2 , 2 , CO, NO, and CH 4 , act in a similar way, while hydrogen differs only in the fact that it is from the beginning less compressible than Boyle's law indicates, so that the product pv is always greater than unity. It would be difficult to show these relations on a diagram of pressure and volume, since the behaviour of a perfect gas is there represented by a rectangular hyperbola, and another curve differing very slightly from that, such as would suit the observa- tions on air, would be indistinguishable from it without accurate measurement. But if we represent the product pv vertically as a function of the pressure p (Fig. 33), then the behaviour of a perfect gas will be shown by a horizontal straight line (constant value of pv), and that of air and hydrogen by curves, whose de- pal ture from straight- ness is easily noticed, c :u As a contrast to this, consider the be- haviour of a vapour, such as steam, on com- pression. To be definite, we 200°, which is about the oSAtnws. Fig. 32. will choose the temperature highest used in modern steam- engines. Steam at atmospheric pressure can exist at this tem- perature, if it has space enough ; for if water be boiled in a flask, and the vapour be led through a tube over a flame, it can be raised (' superheated ') to 200° or any higher temperature. If now such steam be passed into a cylinder, closed by a piston, and compressed, its behaviour is very much like that of a perfect gas ; but as the pressure rises, it shows a more considerable a 2 84 PROPERTIES OF FLUIDS departure from Boyle's law than does air, and on the side of becoming more compressible. When, however, the piston is so far lowered that the pressure rises to 15-4 atmospheres, an abrupt change occurs : the steam begins to condense to water. Now if the piston be pressed down lower, it is found that no further rise of pressure is noted ; the steam has suddenly lost its elasti- city, compression being accomplished without any corresponding increase of pressure. This is observed so long as both steam and water (the two ' phases,' as they are called) are present ; as the piston moves on, more and more water is formed at the expense of the steam, but the change in quantity of the phases does not affect the temperature and pressure required for equilibrium between them. Eventually, however, if the piston is continually pushed down, the volume will be so reduced that the whole of the steam is condensed ; after that no reduction of volume can take place except by compression of the liquid water ; then an extremely great increase of pressure is required, in proportion to the compression produced— in other words, the coefficient of elasticity is very great. These changes, which take place while the fluid is maintained at a constant temperature, characterize an isothermal for water (that of 200°), and may consequently be shown with advantage on a pressure-volume diagram (Fig. 33, which is not to scale). The isothermal, which for air was a con- tinuous curve, is here a broken line in three parts ; the first ab, CONDENSATION AND VAPOUR PRESSURE 85 not very different from a rectangular hyperbola, shows the com- pression of the vapour— rise of pressure in proportion to the diminution of volume ; this part stops abruptly when the pressure has risen to 15-4 atmospheres, the volume then being 132 c.c. per gram. The steam is then said to be saturated, or -at its maximum or saturation pressure (for the given temperature). The iso- thermal then proceeds horizontally along bc, indicating that the pressure remains at its saturation value until the volume is reduced to about i-i c.c, that occupied by one gram of liquid water at 200 ; bc therefore refers to the mixture of liquid and vapour : and if the volume has any value intermediate between i-i and 132 c.c. (per gram), the substance will divide itself between the two phases in such proportion as to fill the space available, at 15-4 atmospheres pressure. The third part of the isothermal cd begins at the point c corresponding to complete condensation, and indicates the extremely rapid rise of pressure required to reduce the volume further ; it is not a vertical line, but slopes so little towards the left that the figure does not show the slope. According to the preceding section, the elasticity would be measured by drawing a tangent to cd and measuring the height above c at which this cuts the axis of pressure. For water the coefficient is about 20,003 atmospheres, i. e. by raising the pressure one atmosphere the volume of the water is reduced by about ^-^ part. The behaviour just described is that which is familiar in the case of all liquids; when heated, they are at a certain point (the boiling-point) abruptly converted into vapour— the vapour occupying a space very much greater than the liquid. The phenomenon of boiling in the air is the converse of the conden- sation followed out above, but as it takes place necessarily under the constant pressure of the air, it covers only the horizontal pait of the isothermal. To complete the knowledge of the properties of the fluid it is necessary either (i) to trace out the isothermals for various temperatures other than 200°, or (ii) to observe the conditions of boiling under various pressures. These correspond to two different experimental methods that are in use. (i) To trace an isothermal, with apparatus of a laboratory pattern, we may shut off the substance to be experimented on in a glass tube, by means of mercury. In Fig. 26 (p. 53) a is such a tube, graduated in order to measure the volume of the fluid ; it is surrounded by the jacket b, in which a constant 86 PROPERTIES OF FLUIDS temperature is maintained by the vapour of the liquid boiled in the bulb c, as described on p. 53. The bottom of a fits through an air-tight rubber stopper in the mercury vessel e ; by means of the side tube g air can be pumped into this vessel to any required pressure, and the pressure measured by an attached mercury gauge, a is first filled with mercury and inverted in the mercury of the vessel e ; it then constitutes a barometer. A weighed quantity of the liquid to be experimented on is then introduced into a in a minute stoppered flask ; on rising to the top of the barometer tube the liquid drives out the stopper and evaporates, filling the space above the barometer. If then a small enough quantity of liquid has been taken, it will evaporate completely, and it will be possible to observe the volume occupied at the fixed temperature chosen, and under a pressure equal to that existing in the vessel d, minus that of the mercury column in the tube, i. e. it is possible to observe a point on the part of ab of the isothermal (Fig. 33). The pressure may then be increased by pumping more air intp n, and another reading of pressure and corresponding volume be taken. When, however, the volume has been reduced to a certain extent the vapour will be observed to condense, and thereafter, as the substance is compressed higher up the tube a, no increase of pressure is required ; the pressure then observed is consequently the saturation or 'vapour pressure ' — that of bc (Kg. 33). If it be desired to continue the experi- mentation on the liquid, the tube a must be provided with a capillary ending, into which the substance may be driven as it is condensed, in order to measure the very small changes of volume that occur in the liquid state. If a number of experiments be made on the same substance at different temperatures, it will be found that the general character of the isothermals obtained is the same as that indicated in Fig. 33, but that the saturation pressure increases greatly with rise of temperature; the following short table for water will show this : Vapour pressure of water. o° 50 100° 150 200 o'oo6 o'lai roo 4/71 15-4 atmospheres. In accordance with this, the least volume into which the vapour can be compressed before condensing (represented by on, Fig. 33) is much greater at low temperatures than at high. For this CONDENSATION AND VAPOUR PRESSURE 87 reason the isothermal of 200° was chosen for Fig. 33 ; that of 100° would be still more impracticable to draw to scale, as the volume of steam at the ordinary boiling-point is about 1 ,650 times as great as that of water. The course of the isothermals may be further studied in Fig. 38 (vid. inf.). (ii) The dependence of the saturation pressure on the tempera- ture may equally well be studied by means of an apparatus for boiling a liquid under variable pressure. Such an apparatus has already been described as a means of producing constant tem- peratures (p. 52). If in that apparatus the enclosed barometer- tube be removed and the 'jacket ' with its connexions be regarded as the essential instrument ; if, further, a thermometer be placed in the jacket, we have an arrangement by which the pressure on a liquid can be varied, and the temperature of its vapour under any pressure noted. This is sometimes known as the dynamic method of measuring vapour pres- sures, in contrast to that of the barometer-tube, which is known as static ; the two are found to give identical results, i. e. the temperature at which a liquid boils under a given pressure p is the same as the temperature at which the saturation pressure of the vapour is p, although in the first case the pressure is due mainly to air over the liquid, while in the barometer-tube it is due to the vapour itself. The relation between temperature and saturation pressure, which is of great importance, may advantageously be shown by a diagram in which these two quantities are explicitly repre- sented. Fig. 34 is drawn to scale for water ; it will be noted that the curve is convex to the axis of temperature, indicating that the pressure rises more and more rapidly as the temperature be- comes higher ; as is also shown by the table on the preceding page. ZOO Temp. 83 PROPERTIES OF FLUIDS § 3. Liquids. Turning next to the behaviour of the substance in the liquid state, we have already noted the extremely small influence of pressure ; water, which may be taken as representative in this respect, is compressed only one-twenty-thousandth part per. atmosphere pressure ; mercury only — . Temperature has 250,000 a larger influence than this, although, again, less than its influence on gases ; hence the most important measurement to make is on the change of volume of the liquid under the constant pressure of the atmosphere. The change is in nearly all cases an expansion with rise of temperature, and may be expressed, as in gases, by a coefficient of expansion, which is defined as the increase in volume per degree, divided by the volume at o° ; or, algebraically, if v be the volume at o° and v, that at t° (cent.), coefficient of expansion = tv. but whereas in gases this coefficient is always about — it varies much from one liquid to another- The coefficient of expansion may be measured by any of the processes available for measuring the density of liquids (p. 10), if carried out at various tempera- tures, since it merely expresses the change in density produced by temperature. It may, how- ever, best be accomplished independently by a dilatometer, which is constructed after the principle =>- of the thermometer. The instrument is shown in Fig. 35 ; it is made entirely of glass, consisting of a wide tube a, to which are sealed the capillaries b and c ; b has a number of small bulbs blown on it, and marks etched between each, c is bent as shown, y and drawn out very fine at the end. After being I — ')) cleaned and dried it is filled with mercury, by Fiq. 35. attaching the end of b to a filter-pump and dipping c under mercury in a dish ; the mer- cury is allowed to rise to each of the marks in turn, and the LIQUIDS 89 M weight taken : from this the volume of the main tube and each of the small bulbs can be found. The apparatus is next filled with the liquid to be experimented on (previously freed from dissolved air by boiling) to a convenient level ; the point of c is fused off in the blowpipe, b closed by a cap of rubber tubing, and the whole placed in a bath of water alongside a thermometer. The bath is heated and kept well stirred, and note taken of the temperature at which the liquid reaches each of the marks. The experiment may with advantage be extended to temperatures below that of the air, by means of freezing mixtures, the dilatometer and thermometer being placed in a bath of toluene or paraffin oil. The results may be plotted to form a curve, with temperature and volume represented along the two axes ; from this the volume at the freezing-point v may be found, and the co- efficient calculated according to the formula on the pre- ceding page. It will usually be found that the increase in volume is not quite the same for each degree rise of tempera- ture ; when that is the case the entire result cannot be expressed by means of a single ' coefficient,' for the coefficient will vary according to the tempera- ture chosen for calculation. We must then distinguish between (i) the mean coefficient of expansion over a given range of tern* perature ij to t 2 , or volume at t, - volume at U (t % - t,) x volume at o° and the true coefficient of expansion at a given temperature t, which is thevalue of the coefficient derived from observations very slightly above and below t. This distinction is most clearly shown graphi- cally, as in Fig. 36. If the expansion of a liquid were the same for each degree, i. e. the coefficient of expansion constant, it would be represented by a straight line, and the steepness of slope would 0/ Y^R N^Q L 'P . -"" ' 50 Temp. Fie. 36. 96 PROPERTIES OF FLUIDS be a measure of the coefficient ; if, for instance, the volume at o" and 50 be known, the difference between them (vertical distance between e and r) must be divided by the difference of temperature (horizontal distance between e and r) to give the rise per i", so that the quotient, i. e. the tangent of the angle of slope, is pro- portional to the coefficient. Next, in the actual case, to get the mean coefficient between 20 and 50° we must join l and m, and the tangent of the slope of this line will be proportional to the quantity required ; or if the mean coefficient between 30° and 40 be desired, it will be given by the slope of the chord no, while in the limit the chord becomes a tangent, and the true coefficient of expansion at 35° is indicated by the slope of the tangent at that point, pqe. The coefficient of expansion obtained by means of a dilatometer (whether true or mean coefficient) requires a correction on account of the expansion of the glass vessel itself; since this expands with rise of temperature, the contained liquid will not rise so far up the stem as if it did not ; hence the coefficient of expansion, as observed, will be too low. The correction amounts to about 0-000025 for ordinary glass, so that e. g. the coefficient of expansion of benzene as observed is 0-001176 per 1° ; the true value 0-001201. The coefficients for different liquids vary pretty regularly according to their boiling-points, being smaller the higher the boiling-point ; the value just given for benzene may be taken as an average for ordinary liquids ; mercury, in accordance with its much higher boiling-point, has a coefficient as low as 0-000180 per degree. The behaviour of water is peculiar. From o° to 4° it contracts with rise of temperature ; its volume is then a minimum, and at 8° is about the same as at 4 . The expansion becomes more rapid as the temperature rises, but even at ioo° is abnormally small ; the curve representing its behaviour consequently slopes much less steeply than that of a normal liquid, such as benzene. The physiographical effects of this peculiarity were referred to in the preceding chapter. Of methods other than the dilatometer for measuring • the expansion of liquids, we may mention the following : — (i) The weight thermometer : this is essentially similar to the specific gravity bottle in principle ; it consists of a bulb — say 10 c.c. in capacity — drawn out to a fine point. It is weighed empty, filled by alternate Liquids 91 W- ^B heating and cooling, in much the same way as a thermometer, with the liquid to be studied, and when completely full at a known temperature (that of a water bath or ice bath) weighed again ; it is then heated to some higher temperature, when the expansion drives out a little of the liquid, and on. cooling weighed again. Regarding the volume of the bulb as unchanged in this process, it is clear that the weight of liquid contained will be proportional to the density, therefore inversely as the specific volume of the liquid. If the latter be i> at the freezing-point, and the coefficient of expansion a, Weight of liquid contained at < ( sp. vol. at t 2 11, (1 + o (J i+al, Weight of liquid contained at t 2 sp. vol. at t L ~ v„ ( 1 + a t L ) ~ 1 + a «, whence a can be calculated. Here again the coefficient a obtained is that relative to glass, and must therefore be increased by 0000025 to give the true value. (ii) The areometrio method : this consists in weighing a solid in air, and when immersed in the liquid as described on p. 12, and repeating the process at different temperatures. The apparent loss of weight of the solid is propor- tional to the density of the liquid in which it is immersed ; hence this method, like the last, is essentially a method of comparing the density at different temperatures, and of course gives re- sults relative to the solid used, and must be corrected by adding the coefficient of expansion in volume of the solid, as before. (iii) Just as the densities of two liquids may be compared by arranging a column of each to produce the same hydrostatic pressure, so the density of the same liquid at different tem- peratures may be studied. Kegnault in this way measured the coefficient of expansion ; his apparatus consisted of a pair of vertical tubes aa, eb (Fig. 37) to contain the mercury, one immersed in a bath of tap water, the other in a liquid bath that could be heated : the two tubes were connected at the top by u cross-tube of fine bore ab, which served to equalize the pressure on the two sides, but allowed so little liquid to diffuse through it as to cause no appreciable transfer of heat from the hot to the cold side. At the bottom each of the main tubes was connected to a short upright glass tube, c, d ; these were connected at the top to one another and to a supply of compressed air. c and n served as a gauge to measure the difference in pressure produced by the two columns a and B of mercury, which were of the same length but different densities ; since (p. 36) the pressure produced by a liquid column is proportional to its density, this gives the means of determining the effect of temperature on the density Fig. 37. 92 PROPERTIES OF FLUIDS (and therefore volume) of mercury ; and since {he pressure produced at a given depth is independent of the size or nature of the containing vessel, this method, unlike the preceding ones, requires no correction for the expansion of glass, but gives directly and accurately the coefficient of expansion of the liquid studied. § 4. Critical point. The marked difference, noted in § 2, between the behaviour of air and steam on being compressed, is not characteristic of those substances, but according to the temperature either set of pheno- mena can be obtained ; steam at a very high temperature (over 400 ) would behave on compression like air under ordinary condi- tions, while air at - 160 would show the phenomena described above in the case of steam. Neither is a convenient material on which to show the transition between the two states on account of the extreme tem- peratures required : carbon dioxide is more suitable, and was in- deed the substance with which Andrews made such experi- ments and discovered the existence of the critical point. It has already been remarked in the case of steam, that with rise of temperature the saturation pressure rises greatly, and so the volume occu- pied by a gram of saturated vapour becomes less and less (Fig. 38). This is well brought out by the behaviour of carbon dioxide : the gas was compressed into a strong capillary glass tube, and in an experiment made at i3°-i found to condense under about 50 atmos. pressure; this then is the saturation J4C.C. CRITICAL POINT 93 pressure of the substance at that temperature, the behaviour being similar to that previously described for steam. But on account of the high pressure involved the specific volume of the saturated vapour was only 12 c.c. whilst that of the liquid formed was 2 c.c. But at 2i°-i, the next temperature used in the experi- ments, the approximation between vapour and liquid had gone still further : the vapour pressure was 60 atmos., the volume of the vapour before condensation 9 c. c. , that of the liquid formed about 2-2. Finally at 31 the distinction into vapour and liquid ceased : at no time during the compression were two forms of the substance visible in the experimental tube together, as was the case at lower temperatures, and at higher temperatures the behaviour was analogous to that described for air. The tempera- ture limit between these two conditions is called the critical temperature— it may be looked upon as the limit beyond which the phenomena of evaporation and condensation cease to exist, or beyond which liquid and vapour cease to be distinguishable. The saturation, or vapour pressure, therefore, is a quantity relating only to temperatures below the critical, and the limiting value corre- sponding to that temperature is called the critical pressure. The state of the fluid marked by the critical temperature and pressure is called the critical point (c, Fig. 38), and the volume the fluid then occupies, the critical volume ; the latter is, clearly, at the same time the least volume of the saturated vapour and the greatest volume of the liquid. There are only a few substances whose critical points lie below the atmospheric temperature, and consequently cannot be con- densed by pressure alone : they include hydrogen, oxygen, nitrogen (and consequently air), carbon monoxide, methane, nitric oxide, argon and helium. Two or three distinct methods have been successfully employed to cool these gases sufficiently to permit of their being condensed. That of Pictet was to proceed by steps : sulphur dioxide was liquefied by compression, and the liquid rapidly evaporated by means of a pump, the liquid thus fell in temperature to its boiling-point under the low pressure maintained by the pump, viz. about - 70 ; the liquid and vapour rising from it were used as a jacket to maintain this low temperature, on the same principle as the jackets described, p. 76 ; by means of it carbon dioxide, under a pressure of a few atmospheres, was condensed and allowed to collect in a separate 94 PROPERTIES OF FLUIDS reservoir, from which it was rapidly evaporated by a second pump. The boiling-point of the carbon dioxide was thus reduced to - 130 : it was used as a jacket for the tube in which oxygen was to be condensed, and on forcing oxygen at high pressure into this the liquefaction was accomplished, showing that the critical point of oxygen is not so low as -r 130 . Cailletet liquefied oxygen at about the same time as Pictet, with simpler apparatus, by adopting a novel principle ; the gas was raised to as high a pres- sure as possible in an apparatus of the same essential character as that referred to on p. 86, but very strongly constructed, so as to withstand pressures of 1,000 atmospheres : the glass tube containing the gas was enclosed in a steel block, from which only the capillary end projected ; the gas when compressed was cooled as far as could conveniently be done by a bath of liquid sulphur dioxide, still it did not condense. Then, by opening a valve, the pressure was suddenly reduced to a few atmospheres only : the gas in expanding does work in driving the mercury or water compressing it out through the valve ; this work is done at the expense of the store of heat-energy in the gas itself, which conse- quently falls in temperature. The fall in temperature can be made great enough to condense oxygen or nitrogen, which appear in small drops on the inside of the capillary tube : but the quantity obtained is very small and does not allow of any practical use being made of the liquid. A further modification of principle, introduced quite recently by Linde, allows of the liquefaction being carried out continuously, so that any quantity can be obtained : liquid air can by that means be prepared in a few minutes, and at a cost of one or two pence per litre. When air under pressure escapes through a fine hole, e. g. through a throttle- valve, the work it does in expanding is spent in overcoming friction in the valve, and consequently the heat is generated there and returned to the expanding gas : its temperature there- fore does not fall like that of a gas expanding in such a way as to do work on an external system, as in Cailletet's experiment, or by means of a cylinder and piston ; indeed if the gas were a 'perfect' one it would issue from the valve at precisely the temperature it went in at. In air, however, the two effects do not quite balance : the heat lost by the gas on account of the expansion being somewhat greater than that returned to it by friction (as the equivalent of the work done by the expanding MIXTURES 95 gas) ; hence there is a small fall of temperature, about a quarter of a degree per atmosphere pressure. In machines constructed after this principle a pressure of some 200 atmos. is used : this would not in itself cool the air sufficiently to condense it, but the cooled air coming from the valve is passed through a pipe called an ' interchange^ ' which jackets the air approaching the valve and cools that, so that the process intensifies itself, till eventually a part of the air liquefies. The method consists, there- fore, in compressing air by a machine pump to about 200 atmos., passing the compressed air through a spiral tube cooled by tap- water, to remove the heat generated in the compression, freeing it from carbon dioxide and water vapour by means of soda-lime, and letting it flow through the interchanger to the throttle-valve and back ; when the apparatus has been working for a few minutes a portion of the air liquefies at the valve and drops into a receiver. The latter is a double-walled glass vessel with a vacuous space between the two walls (cf. p. 76). Air liquefies under the atmospheric pressure at - 194". Hydrogen can be liquefied in the same way if previously cooled by liquid air : its liquef ying-point (or boiling-point) is about - 235 . § 5. Mixtures. When two or more gases are mixed, they each produce almost the same effect as if they separately occupied the entire space. In particular they each produce a partial pressure equal to that which would be produced if the other gases were not present (Dalton's law) ; the total pressure of the gas, as measured by a manometer, being the sum of these partial pressures. Thus air is a mixture of oxygen (21 % by volume) and nitrogen (79 %) ; if the two pure gases were taken in these proportions, and made to occupy in turn a glass vessel of fixed volume, and their pressures measured by an attached gauge, and then the two gases mixed and compressed into the same vessel, the pressure would then be found to be equal to the other two measured pressures added together. Further, a vapour, whether saturated or unsaturated, in presence of gases, produces its own partial pressure, just as if the gases were not there. We have already had an instance of this in the agreement between the static and dynamic methods for measuring vapour pressures (p. 85), since the former measures 96 PROPERTIES OF FLUIDS the vapour pressure of a liquid in the absence of any other sub- stance, while the latter does so in presence of air. This principle is of importance in finding the mass of a specimen of gas. Gas is usually collected for measuring in graduated glass vessels over mercury, water, dilute sulphuric acid or other liquid. Mercury has no appreciable vapour pressure at ordinary tempera- tures, but water and dilute acid have ; hence the pressure as measured is that of the gas saturated with water vapour, and to get the pressure of the gas alone, that of the vapour must be deducted. Thus, suppose b to be the height of the barometer, and that the liquid over which the gas is collected stands at a height h higher in the collecting vessel than in the trough outside, while d is the density of the liquid : then the column of liquid inside the collecting vessel produces a pressure equal to hd that of a mercury column — ^ high (13-6= density of mercury) ; hd so that the actual pressure inside the collecting vessel is 6 -. If at the temperature of the experiment (f cent.) the liquid has a vapour pressure=s, the partial pressure of the gas is b y - s (s may be taken from published tables ; it is less for dilute acid than for pure water). Now the density of a gas is by Boyle's law proportional to its pressure, and by Charles's law inversely pro- portional to the absolute temperature T ( = t + 273) ; hence if p is the density of the gas at o° and 76 cm. pressure, its density under the observed condition / hd ^3-6 ; ~ " v 273 ~ pX 76 " 273+r and the mass of gas is obtained by multiplying this quantity by the observed volume. When the gas is collected over mercury or other non-volatile liquid, such as strong sulphuric acid, the term s can be neglected. The air expired from the lungs is also saturated with water vapour at the temperature of the body or nearly so ; the vapour pressure of water at that temperature (37 ) is 47 mm. Hence the water vapour in the breath amounts to 7% of the whole (by volume). Carbon dioxide is found to constitute about 2-2 % or 17 MIXTURES 97 parts in 760, and consequently exerts a partial pressure of 17 mm. It may be desired, by ventilation, to' reduce the partial pressure of the carbon dioxide to 1 mm. (in fresh air it amounts to 0-3 mm., so that this may be regarded as reasonable) ; to do so it would be necessary to make the volume of the air 17 times as great, or rather more so, seeing that the ventilating air itself contains a little. If the volume of air expired be taken at 8 litres per minute per head, it may be reckoned that about 150 litres a minute is a sufficient supply. Atmospheric air always contains some moisture, and for meteor- ological purposes it is desirable to estimate the amount of it. The most direct process is to draw a measured volume of air through drying-tubes and weigh the moisture absorbed ; this, sometimes known as the chemical method, is represented in Fig. 38a. The aspirator a serves to draw a measured volume of air through the tube d into the drying bulbs c : these may contain cal- cium chloride, strong sul- phuric acid, or phos- phorus pentoxide, of which the last is the most efficient ; the dry- ing bulb b serves to keep any vapour from the water in the aspirator from diffusing back into the bulbs c. From the increase in weight of c the concen- tration of the vapour may be calculated, and thence, if the tem- perature of the air be noted, the pressure obtained. The results may be expressed either as (i) actual partial pressure (incorrectly called ' tension ') of the water vapour ; (ii) the ratio which that pressure bears to the saturation pressures at the same tempera- ture (called the relative humidity) ; (hi) the concentration of the vapour (weight in unit volume) ; (iv) the drying power, i. e. the additional weight that unit volume of the air could take up .before becoming saturated. All these are recorded in the daily weather reports as published in the Times. A more rapid process to determine the hygrometric state of the air consists in determining the dew point. Dew is deposited on any surface that is cooled till the air in its immediate neighbour- H r r "\ SBa : p-^^^ ~====i &^=l BRUE BUB Fia. 98 PROPERTIES OF FLUIDS hood becomes saturated with vapour. To determine the dew point a polished metallic surface (usually a small silver cup) is cooled, by evaporating ether inside it, and watched till dew forms on it ; it is then left to itself till the dew disappears by evapora- tion, and the temperature at that moment noted ; we may then find from the tables of vapour pressure of water what the pressure corresponding to the observed dew-point temperature is ; that will be the actual pressure of the water vapour in the air. Another process is that of the wet and dry bulbs, i. e. of reading the temperature of the air by an ordinary (dry) thermometer, and reading at the same time another thermometer whose bulb is kept wet by a wrapping of cotton wool dipping into a vessel of water. The process is not reliable, however. A more useful instrument is the hair hygrometer : this consists of a long hair, previously freed from grease by means of ether, which is kept stretched by a small weight ; it contracts in moist air and expands in dry, and may be made to record its changes in length by means of a magnifying lever working over a scale. The instrument is only roughly accurate, but is very useful for certain observations, because it is direct reading. Its scale must be calibrated by comparison with determinations of humidity made by the other methods. When two liquids both give off vapour to the same space, it is still true that the pressure produced by each vapour is that that would be calculated from the quantity present (the concentration) independently of the other. But it is not in general true that the saturation pressure of either remains unaffected ; if the two liquids do not mix at all, then each has the same vapour pressure as if the other were not present, and the total vapour pressure is the sum of the two ; this is nearly the case for benzene and water. If the two liquids are completely miscible, the vapour pressure of any mixture usually lies between the pressures of the two pure substances, approximating to that of either component according as that component preponderates in the liquid ; this is true, e.g., of benzene and carbon tetrachloride. Sometimes, however, a mixture of two liquids in certain proportions may possess greater vapour pressure than either separately (benzene and alcohol), or less than either separately (water and formic acid) ; the latter case indicates a considerable attrac- tion between the molecules of the two substances, and the mixture may be regarded as approaching a chemical combination in its properties. When a solid is dissolved in a liquid, the solid being practically non- volatile, the only vapour pressure to be considered is that of the solvent, MIXTURES ' 99 and that is diminished. The lowering of vapour pressure produced can be calculated according to a rule first given empirically by Raoult, and subsequently explained on theoretical grounds by van 't Hoff. If the solution contains n molecules of dissolved substance for one of solvent, the vapour pressure is i — n times that of the pure solvent. This law, how- ever, like other generalizations with regard to solutions (vid. inf. p. 127), applies only when the solution is sufficiently dilute, just as Boyle's law is applicable to gases only when their concentration (or density) is small enough. As an example we may take a 1 % aqueous solution of cane sugar CiaH^O,! ; the molecular weight of sugar is (12 x 12 + 22 x 1 + 11 x 16 =) 342 ; that of water H 2 is 18. Hence 1 gram molecule (18 grams) of water contain ■££$ x -jj^ = Y&n; gram molecule of sugar ; therefore at ioo°, while the vapour pressure of pure water is 760 mm., that of the sugar solution is less by I ^ nr part, and amounts to 759-6 mm. Raoult's rule has been used to determine the molecular weight of bodies in solution ; it leads in some cases to apparently anomalous results, the most important being for solutions of salts, acids, and bases in water. Here the vapour pressure is lowered more than would be expected according to the rule, and these observations, with other evidence, have led to the ' electrolytic dissociation theory,' according to which salts in solution are dissociated ; the number of dissolved particles is therefore greater than would be the case if the salt were not disso- ciated, and the change of vapour pressure correspondingly increased. To say that, at a given temperature, the vapour pressure of a solution is less than that of the pure solvent is equivalent to saying that, under a given pressure, the boiling-point of a solution is higher than that of the pure solvent, since in all cases the vapour pressure increases with the temperature, and the boiling-point is the temperature at which the vapour pressure becomes equal to the external pressure on the liquid. It is convenient in practice to measure the rise in boiling-point, rather than the diminution in vapour pressure ; the measurement is usually made in an apparatus such as that shown in Fig. 39, the design of which is due to Beckmann. It consists of a boiling tube a provided with two side tubes t L , t 2 ; ), the two processes leading to an identical result, viz. the solution, saturated with silver nitrate, D. If, on the other hand, we have a solution containing very little salt, and it be cooled, then at a temperature somewhat below o° it begins to freeze. But now it is saturated not with salt, but with the other con- stituent, and ice crystallizes out. This corresponds to the line bo, or to mixtures of a little naphthalene with much paratoluidine. Thus we see that the freezing-point of water is lowered by addition of salt, and that there must exist a cryohydric point (different for each kind of dissolved substance) : common salt mixed in the right proportion with ice forms a cryohydrate melting at about —20° and is cooled to that temperature, forming a freezing mixture. The difference between the pair of organic solids quoted, and the pair water-silver nitrate, is chiefly in the fact that the melting-points of the latter lie so much further apart, and with other salts the difference is even greater. The lowering of the freezing-point of a liquid, due to the presence of .144 PROPERTIES OF SOLIDS another substance in solution, may be looked at in quite a different light. We have seen (p. 126) that the dissolved substance produces an osmotic pressure, which may show itself in drawing the solvent in through a semi- permeable partition, and would resist the converse process of removing the solvent in that manner. The osmotic pressure, in fact, shows itself in a tendency to resist removal of the dissolved substance in any manner, for by reducing the quantity of solvent one compresses the dissolved sub- stance into a smaller space, and so increases the pressure which it exerts. It has already been noted that solution of a solid in a liquid raises its boiling-point (p. 99), i. e. adds a resistance to separation of the solvent in the form of vapour. Now, if a dilute solution — say of sugar in water — is cooled, the water begins to crystallize out, and the dissolved sugar is com- pelled to occupy a smaller volume : it follows, from the general principle referred to, that the freezing will only occur at a lower temperature than that of pure water. This depression of the freezing-point is, for dilute solu- tions, proportional to the osmotic pressure, and therefore to the concen- tration of the dissolved body (expressed in gram molecules per litre). It is, in fact, by measurements of the freezing-point that the osmotic pressure of solutions is mostly determined ; and since the molecular weight (in grams) of any substance dissolved in a given volume of water produces the same lowering of the freezing-point, if we have a substance of unknown molecular weight, we may find this quantity by the effect produced on the freezing-point, e.g. molecular amounts of alcohol (C a H 6 = 46 grams) and urea (CO(NH 2 ) a = 6o grams) produce the same lowering ; if, then, we did not know the molecular weight of glycerine, but found that 92 grams of it produced the same effect as 46 of alcohol; we might conclude that the molecular weight of glycerine (C 3 H s 3 ) was 92. The same remarks apply to other solvents, except that the depression of the freezing-point due to a given concentration of dissolved substance varies from one solvent to another. But it must be borne in mind that the lowering is only pro- portional to the concentration for dilute solutions, because it is only to -them that the laws of ideal gases are applicable (p. 127). As was remarked in considering osmotic pressure, there is a large group of bodies which become electrolytically dissociated in aqueous solution, and so give anomalous results. Just as they give greater osmotic pres- sures than those calculated from their usual molecular formulae, so they give greater depressions of the freezing-point. Further explanation of $his phenomenon must be postponed to the chapter on electricity. CHAPTER V. SOUND. § i. Sources of sound. Sound consists of waves propagated through the air, and received by the sensory apparatus of the ear ; it is, therefore, to be treated in accordance with the general principles of wave motion, considered in § 7 of Chap. I. As in that section simple oscillations were considered before waves, so here it will be convenient to treat first, for simplicity, the vibrations of bodies such as reeds and strings that generate sound-waves, making as few assumptions as possible as to the nature of the mechanism by which their motion is transmitted to the ear ; and leave for the subsequent sections those properties of sound which are more immediately connected with the behaviour of the medium through which it passes. The vibrations to which sound-waves are due usually occur in solid bodies, and are nearly always due to elastic forces. In the preceding section we considered in some detail the motion of one vibrating elastic solid— a chronograph reed— chosen as a speci- men of its class ; it may suffice here, therefore, to give a catalogue of the vibrating bodies that have most commonly to be dealt with : the motion of each is substantially the same in its har- monic or periodic character. Now, the most important kinds of strain of which a solid is capable are longitudinal strain (extension or compression), flexure, and torsion ; accordingly vibrating bodies may be classified as suffering these various strains. A bar or wire of metal when stretched tends to recover its former length, the tendency to do so being measured by one of the coefficients of elasticity, Young's modulus : the tendency to recover produces vibrations ; but on account of the great magnitude of Young's modulus for ordinary solids, these are usually too rapid to be used in music. Thus, a glass or steel L 146 SOUND rod, clamped in the middle, or a steel wire, stretched between two pegs and stroked lengthways with a resined cloth, will emit a shrieking sound, the vibrations of which may be many thousand to the second. Such vibrations are longitudinal, i. e. each particle of the bar or wire vibrates in the direction of its length. But stretched wires or strings may also be made to vibrate transversely ; this is the case which is of the most importance musically, including both piano and violin strings. A piano wire is struck by the hammer, and pushed sideways out of place, the ends remaining the while fixed on their pegs; consequently the wire becomes slightly longer, for the moment, and tends to contract again till it resumes its straight position, In this way vibrations are set up, each point of the string moving to and fro at right angles to the direction of its length. The frequency of these transverse vibrations is increased by making the wire more taut, so as to increase the force tending to restore it to its equilibrium position ; the frequency is decreased, if the length or weight be increased so as to give it greater inertia. The exact formula for frequency of vibration is w=t/v/— where n = number of vibrations per second, I = length (in cm.), w = tension of string (in dynes), m = mass of one cm. length of string (in grams). All transverse vibrations of strings, whether bowed as in the violin, or plucked as in the guitar, may be reckoned from this formula. The flexural rigidity of a bar may be made use of to produce sound. The reed of Fig. 3 vibrates in that way, and really sets up sound-waves in air, but the vibrations are so slow as to produce no impression on the nerves of the human ear. If they be made more rapid by removing the weight w altogether, the spring will produce a faint, dull, booming sound, while a thicker bar would, on account of its greater stiffness, give more rapid and consequently more easily audible vibrations. A tuning- fork is essentially a pah- of such flat bars, cast on the same base ; its vibrations are more rapid the stiffer the prongs. They can be adjusted slightly in frequency by filing the prongs near the base (to make them slower) or near the point (to make them faster). The vibrations of flat and rounded plates (cymbals, gongs, bells) belong to the same class, though they are more complex in character. SOURCES OF SOUND 147 Vibrations can also be set up by twisting rods and wires, but they are of no practical importance. There is another class of musical sounds not fully accounted for in the description given above, viz. the sounds produced by wind instruments. In this class may be included the human voice and organ-pipes, as well as the orchestral instruments that are actuated by blowing. In all such cases the vibration is started by an elastic solid body — in the voice by the vocal cords, in organ-pipes by the reed, in the oboe or trumpet by the lips of the player ; but the vibration is greatly intensified and modi- fied in character by the air enclosed in the instrument. The way in which this air takes part in the motion may best be dealt with later, while considering the phenomenon of 'resonance.' (i) Musical sounds produced in any of these ways possess certain characteristics corresponding to those of the oscillations which generate them. These are, firstly, loudness: this is pro- portional to the total energy of the oscillation, and therefore depends on the amplitude. Now, by doubling the amplitude we double the force tending to restore an elastic body to its normal shape (Hooke's law), as well as double the distance through which that force has to act ; accordingly the work done by the force is quadrupled. We see, then, in general that the energy or loudness of a sound is proportional to the square of the amplitude of vibration of the sounding body. (ii) Secondly, the pitch of the sound depends on the frequency of vibration. Notes of high pitch are those due to rapid oscilla- tions, notes of low pitch to slow ones ; while a noise, which is not a musical sound at all, is a sound corresponding to oscillations of so irregular a character that it is not possible to assign any definite pitch to it. Pitch is reckoned by intervals from some fundamental note— a process of a purely musical character— and it is found that a constant interval in the musical sense corresponds to a fixed ratio, not a fixed difference, in frequency of oscillation. Thus, the interval known as an octave corresponds to the ratio 2:1. If a certain low note makes 80 vibrations per second, the note an octave above it makes 160 ; if a certain high note makes 1,000 vibrations per second, the note an octave above that makes 2,000, although the difference in frequency is so much greater in this case. L2 148 SOUND The intervals of most importance in music are those expressible by the simplest numerical relations, viz. the octave 2 : 1, the fifth 3 : 2, and the major third 5 : 4. Out of these intervals, taken in various combinations, is made up the scale or succession of notes required in connexion with a starting or key-note arbitrarily chosen. The following table gives the diatonic major scale, which may be regarded as the most natural one for singing when not accompanied by any instrument whose mechanical construction gives a bias in favour of any more artificial arrangement. The key-note chosen is C, and the vibration numbers in the last column are those that would be obtained by starting from the middle note of a modern piano. Ratio of Actual frequency Note. Interval. frequencies. (Just intonation). C Key-note 1 : 1 264 D Second 9:8 297 E Major third 5:4 33o F Fourth 4 = 3 352 G Fifth 3:2 396 A Major sixth 5:3 440 B Major seventh 15:8 495 C Octave 2 : 1 528 Modern music requires that the key-note may be changed to any point of the scale originally adopted (modulation), and hence a larger number of notes within the range of an octave is indispensable. A compromise between all the notes that might possibly be needed, consisting of twelve notes, has been adopted in practice ; and in modern instruments with fixed pitches (piano, organ) the arrangement known as equal temperament is aimed at. This consists in making the interval (i. e. ratio of frequen- cies) between each two successive notes the same. Hence, each interval — called an equal semitone — is represented by the ratio 2^: 1, and, starting from C as above, the frequencies are 264, 264 x 2^ = 279-7, 264 x 2^ = 296-3, and so on. In contradistinction, the notes of the natural scale given above are said to be in just intonation. (iii) Sounds of identical pitch differ also in quality, as for instance the same note played on a piano, a violin, and a horn. This is a distinction which cannot be accounted for by any of the properties of simple harmonic motion, and indeed it is found experimentally that most musical sounds, while harmonic, are not simple. A tuning-fork gives vibrations of practically simple harmonic character, and accordingly when a tracing of them is obtained by means of the chronograph it is found to be a curve of sines (Fig. 8). If some other source of sound, such as SOURCES OF SOUND 149 a violin string, be made to record its movements in an analogous way, the curve obtained will be harmonic, i. e. will repeat itself at regular intervals, but will not be of the same shape as for a tuning-fork. Innumerable different shapes can be obtained, some of them differing as much from a curve of sines as the sphygmograph curves already referred to (p. 35). These differ- ences in harmonic character may be explained by saying that the vibrations of a body like a violin string are compounded of several S.H.M.'s arranged in a particular sequence : thus the string may vibrate as a whole — this gives the so-called funda- mental tone, by which its pitch is determined. But the string might divide in two at the middle point, and the two halves vibrate in opposite phases ; this can actually be done by holding the finger very lightly on the middle of the string while it is being bowed. Now each half of the string will emit separately a tone of twice the frequency of the fundamental (for the frequency of a string is inversely as its length, p. 146), i. e. a tone one octave higher; this is called the second partial tone or harmonic. Again, the string will break up into three, if the finger be put lightly at one-third of the length ; the vibrations of each segment have then three times the frequency of the whole string, and their pitch is an octave and a fifth above the fundamental tone ; this constitutes the third harmonic, and so on. Now when a violin string is bowed in the ordinary way, its motion is not that due to the fundamental S. H. M. alone, but the various other partial tones occur at the same time, and the actual course of each point of the string is a complex harmonic motion. The analysis of a complex periodic motion into a succession of S.H.M.'s with frequencies in the ratio 123, &c, was first given by Fourier, and the series is often known by his name. We can now see that the different relative strengths of these partials will produce differences in quality between two sounds, although the fundamental tones of the two may be the same in pitch. Actually the flute pipes of an organ possess only the second, and faintly, the third harmonics ; violin strings the first eight or ten, in regularly decreasing amplitude ; piano strings only the first six, but the second, and even the third, sometimes more strongly than the first ; while in the note of a trumpet even the twelfth or fourteenth may be detected, and so on. 150 SOUND The mor.e important musical instruments, from their construction, are capable of producing partial tones whose frequency, as just shown, is an exact multiple of that of the fundamental tone : these are called harmonies. But vibrating bodies of more complex shape, such as bells, possess, it is true, more than one way of vibrating, but the frequencies corresponding to the various modes do not bear this simple relation to one another. The partial tones, i. e. the simple harmonic motions into which the actual vibration can be analysed, are said in this case to be anharmonic. § 2. Sound-waves. A vibrating body, such as those described in the preceding section, acts as a source of waves in the medium— usually air- around it. Thus, to take a tuning-fork as an example, when its prongs move outwards it pushes against the surrounding air, tending to compress it ; the air cannot move outwards instan- taneously, although it does so very rapidly, consequently the air in the immediate neighbourhood of the fork is actually com- pressed for the moment. But air has elasticity of volume, i.e. tends to recover its size after compression ; so, as the compressed air is at a slightly higher pressure than the undisturbed air outside, it will tend to recover its normal volume by compressing in turn the next layer of air. Thus the compression is passed onwards from each layer to the next. But very soon the prongs of the fork are moving in again, leaving room for the ah in immediate contact with them to recover its volume, and even expand. This change is in a similar way communicated outwards to one layer of the air after another ; and so it comes about that the complete oscillatory motion of the air — conden- sation and rarefaction alternately — is propagated outwards from the fork, and each part of the air suffers in turn changes similar to those of the air first acted upon : this constitutes the wave. Waves may be either longititdmal or transverse, i. e. the direction in which the vibrating particles move to and fro may be lengthways or crossways to the direction in which the wave is proceeding through the medium ; but in any particular case it is only possible to generate the type of wave for which there is the necessary elasticity in the medium. Thus, in the experi- ment with a rope (p. 35) the motion of each point is obviously transverse to the direction of the rope itself, and the latter constitutes the medium ; this is possible because of the degree SOUND-WAVES 151 of rigid connexion that exists between successive portions of the rope ; although flexible, a shear in any section of it tends to cause a shear in the next section, otherwise the rope would go to pieces. If instead of a rope we had a cylinder of sand, this rigidity would not exist, and if one section of the sand were displaced sideways, it would have no tendency to drag the next section after it. Now, air has no rigidity, but it has elasticity of volume; consequently transverse waves are not possible in it, and the waves of sound are longitudinal. These longitudinal waves may perhaps be more easily imagined, if their propagation along a straight tube of air be considered ; it is then obvious that rarefaction and condensation of the air in the tube will be produced as the air moves to and fro. When waves are propagated in one direction only (as along a tube), they do not diminish in amplitude (and therefore in intensity) except so far as absorbed by frictional resistance. But when propagated in space, as the wave spreads outwards the energy it carries is spread out over a larger and larger spherical surface, in proportion to the square of the distance from the source, and consequently the energy at any one point (or intensity of the sound) is inversely proportioned to the square of the distance from the origin. In a wave the frequency of vibration at any point is neces- sarily the same as that of the source which is producing the wave; but there is another characteristic quantity— the wave velocity — which depends on the nature of the medium through which the sound is passing, and, in the case of sound-waves at any rate, depends on that only. We may form a clearer notion of the velocity of a wave and the associated quantity, the wave-length, by considering in more detail the mode of propagation. The five curves (Fig. 49) are intended to show this. No. 1 may he taken as representing the condition at a certain moment of a portion of air (say in a tube) through which a sound-wave is passing. The air at the point A is at that moment at its normal (equilibrium) position, but moving towards the right ; that of the point B, less distant from the source of the wave, is in a subsequent phase of its motion, viz. as far towards the right as it can go ; for the condition that B is in now is that towards which a point further on such as A is tending. Again, c is in a more advanced phase still, that of passing through the centre 152 SOUND towards the left ; d is at the extreme left ; while e is in the same condi- tion as a, the cycle of phases being complete. Next consider No. s ; this is drawn for the same points, but at a moment later by one quarter period. Consequently the air at a, which was moving through the centre towards the right, has now reached the extreme right — the phase that B was at in the earlier diagram ; similarly, b is in the phase formerly held by c, and so on. So that the total effect is as if the arrangement No. i were shifted forwards through the distance ba, but otherwise unchanged. Similarly, Nos. 3, 4, 5 represent the same wave at moments later than No. i by \, |, and i period, and each one shows the wave advanced by the same amount, as compared with the preceding figure. We have, then, a picture of the propagation of the wave, and see that the distance between two points in the same phase, such as a and e — the wave-length — is the distance that the wave travels in the interval between Nos. i and 5, that is in one period. D B » 1 »- _• • , 3 5 < 1 • •— . , _ .... _ ,, Fig. 49. Hence the relation already given (p. 36) that the wave-length = wave velocity x periodic time. We have so far described the sound-wave in air by means of the displace- ments of the air particles which it produces. It may also be looked upon as a wave of compression and expansion, i. e. of change in density • but it is to be noted that the phases are different according to these two ways of looking at the phenomenon. Thus, looking at No. 1 in the foregoing figure, it appears that at a, c, and e there is no displacement of the air ; at b and i> the maximum amount of displacement. But the air particles are crowding in on a and e from both sides, spreading out from c on both sides, consequently a and e are points of maximum compression, c of maximum expansion, while it is at b and d, the points where the greatest change of position occurs, that the density possesses its normal value. Fig. 49, it will be seen, if read horizontally, indicates the state of all the air at an instant, i. e. the nature of the wave at that instant ; but if SOUND-WAVES 153 read vertically it shows the changes of position or oscillation of a single air particle in time. The velocity with which a wave is propagated in air or other medium depends on the density and elasticity of the medium. It is easy to see in a general way that increased elasticity would cause each layer of air to recover more quickly from the com- pression it suffers, and consequently transmit the motion more rapidly to the adjacent layer ; while increased density would mean a greater mass to move, and consequently the elastic forces would be slower in producing their effect, and the rate of propagation would be less. Exact treatment of the problem shows that the , . , „ , /coefficient of elasticity of the medium velocity 01 sound = teristics of the sound-wave in the air outside. Beyond this, however, it is probable that some process of selective vibration or' resonance takes place. The most probable organ for this purpose- is the basilar membrane ; this consists of parallel' fibres, only' loosely connected together, so that the membrane is tense in the direction of its breadth, but not in that of its length. In such a structure each fibre would vibrate more or less independently of its neighbours, and as the fibres vary greatly in length, and also possibly in tension, they would possess a great, variety of natural periods of vibration, and would constitute a set of some thousands of resonators. These resonators would each pick out its own tone from a complex sound-wave, and communicate an- impression to the corresponding nerve-ending : so that each component S. H. M. in the sound-wave would find a separate means of conveying a sensation to the brain. It is natural to, conclude that this process of mechanical analysis takes place to some extent at least, although no doubt a direct action of the brain in distinguishing sounds, of different characters is not to be excluded. M 2 CHAPTER VI. CHEMISTRY. § i. Law of mass action. The branch of Physics known as Chemistry has long been recognized, owing to its great extent and mass of detail, as needing independent treatment ; and, apart from its detailed application to individual sub- stances, the general principles of chemical reaction, which are of more interest from the physical point of view, have also received attention from the writers of Chemistry ; but whilst such of those principles as have long been known — the theory of atomic weights, of combining pro- portions, of valency, and so on — are adequately treated in all books on the subject, there are others, of more recent discovery, which are not yet thoroughly incorporated in the teaching of systematic chemistry. For that reason a chapter will be devoted to the most important of them here : the subject of osmotic pressure and its bearing on solutions having been touched upon already, there remains of leading importance the law of mass action, and the phase rule — regulating between them the con- ditions of chemical equilibrium — and the relations of energy changes to chemical action. A system may be called homogeneous when, whatever its chemical character, it is incapable of separation by mechanical means into dis- similar parts. Thus a mixture of gases is always homogeneous, a liquid usually so, but many instances occur of liquids which separate into two layers ; e. g. ether and water : in this case each layer is homogeneous. Confining our attention for the present to a homogeneous fluid, if there be substances present capable of reacting chemically with one another, a certain state of equilibrium will be reached in which the various sub- stances are present in calculable proportions. The conditions regulating the reaction will be more easily followed by the aid of a concrete example — one that has been much studied — the equilibrium between alcohol and acetic acid. These substances are capable of reacting as follows : — C 2 H 5 OH + CH 3 COOH«-C s H 5 OOCCH 3 + H 2 ethyl alcohol acetic acid ethyl acetate water We will call the pair of substances on the left of the equation, for brevity, LAW OF MASS ACTION 165 group A, that on the right, group B. Groups A and B, then, contain the same atoms, only differently arranged ; and the conversion of the one group into the other belongs to the class known as reversible reaction, i. e. if group A be produced by mixing appropriate quantities of its com- ponents, it will gradually suffer a change, but will not bo wholly con- verted into group B ; and similarly, if we have group B to start with, it will suffer a change, but will not be completely converted into A. Thus the reaction is capable of proceeding in either sense, hut in each case not com- pletely. These facts are intended to be indicated by the symbol ^ which is used instead of the ordinary symbol of equality. Observation shows that in this particular case, when no further change takes place, i. e. when the system has reached its state of equilibrium, one-third of the mixture is in the form A, two-thirds in the form B. Now, this state of equilibrium may best be understood by aid of molecular considerations. Suppose a mixture A of alcohol and acetic acid made up ; then the mole- cules of the components come into frequent collision with one another. It is only at the time of collision that such a rearrangement of atoms as is needed to produce molecules of ethyl acetate and water instead can take place ; but this rearrangement does not by any means take place every time a molecule of alcohol collides with one of acetic acid : if that were the case the whole action would be over in a fraction of a second, whereas it takes hours or days to complete. We must suppose, then, that the molecules of alcohol and acetic acid possess a considerable degree of stability, and that it is only under exceptional circumstances — pre- sumably when they collide with exceptionally great velocity— that the collision breaks up these molecules so far that they are capable of re- arrangement in the form of the group B. We may put it, then, that a certain number of collisions occur every second between a molecule of alcohol and one of acetic acid, and that in a minute, but definite, fraction of these, the atoms come out of the collision rearranged, con- stituting a molecule of ethyl acetate and one of water. In precisely the same way, however, collisions are continually occurring between a mole- cule of ethyl acetate and one of water, and in a certain fraction of cases they lead to a break up of these molecules and a rearrangement in the form A. We must suppose these two processes to go on at the same time, equilibrium being reached when the amount of each is equal. This con- ception of a kinetic equilibrium is familiar in other departments of Physics; thus the equilibrium between a liquid and its vapour is, from the molecular point of view, regarded as such : it is not assumed that no interchange takes place between the two phases, hut that some liquid molecules escape and become vapour, only that on the average an equal number of vaporous molecules become entangled in the liquid and stay there. Now, the number of collisions of any particular kind of molecules per lU CHEMISTRY second is proportional to the number of such molecules per unit of volume ; so that the collisions between a molecule of alcohol and one of acid are proportional (i) to the number of alcohol molecules per c.c. ; (ii) to the number of acid molecules per c.c. The number of molecules per c.c. is in any case proportional to the concentration, i. e. the number of gram-molecular-weights (or mols.) in i c.c. Calling this quantity C we may put — Quantity of A converted intoB per c.c. per second = /exC a lcohol x C ac id. Similarly — Quantity of B converted into A per cc] per second = k' x C es ter x C wa t er . The quantity (in mols.) of any substances suffering reaction per c.c. per second is called the velocity of reaction, and the constant factors k k' are called the velocity constants for the reactions in question. Suppose, then, only alcohol and acid to be present to start with : the reaction from left to i-right will go comparatively fast, that from right to left will not take place at all, for there is no ester and water to suffer conversion. -As the .process continues, the quantity of acid and alcohol present will constantly decrease, and so, according to the equation, the velocity of the reaction from left to right; at the same time the quantity of ester and of water present will constantly increase, and so the velocity of the reaction from right to left, until eventually the two partial re- actions become equal in velocity, and equilibrium is reached. If, instead, we have only water and -ethyl acetate present at -first, similar changes will ensue, until the same state of equilibrium as before is reached. This gtate of equilibrium is evidently governed by the condition — & Calcohol • Cacid = ft C e s(er * ^vvater 7c and the ratio — is called the reaction constant : we may then write — k k Cester • C W ater fc' Calcohol • C ac i{i In this case the reaction constant = 4 approximately, since for equi- librium % of the mixture consists of ethyl acetate and water, i. e. 2 x 2 K =■ * 2 = 4 ; and the value in question is practically unaltered by i x i temperature changes. In any reaction there is a corresponding constant regulating the state of equilibrium arrived at ; but in calculating the constant it must be noted that when two molecules of the same kind are involved in the reaction (alone or with others) the square of their con- centration occurs in the equation for K, when three of the same kind, the cube, and so on. This is because the number of collisions involving two molecules of the same kind will be quadrupled by doubling the number of such molecules, and so on. As an example of a reaction that LAW OF MASS ACTION 167 lias been much studied, we may take the partial decomposition of gaseous hydriodic acid that occurs at high temperatures. The reaction is — Hence, the law of mass action gives for the velocity from left to right (decomposition of HI) = Zc C 2 H ij for that from right to left (formation of HI) = k' C H2 Ci £ , and equilibrium ensues when — * = K = 0fti ' Cla k G'ai K has been found to be 0-01984 at 448° cent. Substituting this number in the equation it will be found that about 22 per cent, of the acid is ■decomposed at that temperature. The constant is not independent of temperature in this case, as a somewhat greater fraction is decomposed at higher temperatures. So far we have only considered mixtures of the reagents in molecular proportions, i.e. in the proportions required for the reaction. But the law of mass action gives important information as to the effect of an excess of one of the reagents. It follows from the reasoning .on jrchich the constancy of K is based, that whatever be the proportions in which the reagents are mixed, they must eventually reach concentrations such that the fraction given above is equal to the value of K for the reaction in question. Thus, reverting to the example of ethyl acetate, suppose that we have only ethyl acetate and water to start with, but that instead of these being in molecular proportion, the water is present in great excess — forming, in fact, a weak aqueous solution of ethyl acetate. Then in the equation — Cester ■ C wa ter Calcohol • Cacid :the factor C w ater will be very much larger than before ; .consequently, for equilibrium, Cester must be much smaller. Accordingly the addition of a large excess of water causes the partial reaction in which it takes part (ester + water -9- alcohol + acid) to go much further than it otherwise would, and the ethyl acetate is nearly all broken up into its components. The reversible reaction, such as has been discussed, is really the general type of reaction, although in the usual treatment of chemistry it appears secondary. A reaction which appears to go one way only, such as the formation of water — 2H 2 +0 2 = 2H 2 may be regarded as one in which the reaction constant is exceedingly great, i. e. the velocity of the change from left to right enormously exceeds that from right to left. If a mixture of hydrogen and oxygen be kept at 500° it will not explode, but will react slowly with formation of water — there is therefore a finite velocity of reaction ; and i68 CHEMISTRY we must suppose that when two molecules of steam collide there is always a chance, though an extremely small one, that they may effect the converse change, i. e. rearrange themselves into a molecule of oxygen and two of hydrogen. This actually takes place at higher temperatures : it is observed that at 2,000° or thereabouts there is an appreciable dis- sociation of steam, i. e. when oxygen and hydrogen are mixed at that temperature they suffer a reversible reaction, and » state of equilibrium is reached in which some of the constituents are in the form of steam, some in the form of the elementary gases, in other words, a state similar to that of hydrogen and iodine at 400°. One special case of chemical equilibrium is of so much importance as to deserve separate mention here. When an electrolyte, i.e. an acid, base, or salt, is dissolved in water, it suffers, according to modern views, a dissociation of a peculiar kind ; it forms so-called ions or atomic groups possessed of positive (cation) or negative (anion) electric charges (p. 188). Thus, common salt NaCl is dissociated into one cation, viz. the sodium atom with a positive charge (indicated by a •), and one anion, viz. the chlorine atom with a negative charge (indicated by a ' ) ; we may regard this process as a chemical reaction, and write it — NaCl $ Na- + CI' The electric charges carried by the atoms are of constant magnitude, whether positive or negative ; but one ion may carry two or more charges, e. g. ZnGl 2 dissociates as follows — ZnCl 3 ^ Zn- • + a 01' the zinc ion carrying twice the charge that the sodium ion did in the preceding case. Indeed the number of charges of electricity is a measure of the valency of the atoms in an electrolyte.. The number of positive charges is always equal to the number of negative ones, and it is not possible to obtain a cation without an equivalent quantity of anion, or vice versa. The chemical equations above have been written as if referring to a reversible reaction, for when an electrolyte is dissolved in water a part of it dissociates in the manner described, but a part remains in its molecular state, the electric charges neutralizing one another; and a definite state of equilibrium between the dissociated and undissociated parts is arrived at. An acid is an electrolyte of which the cation is hydrogen. Thus, acetic acid in solution dissociates — CH 3 COOH ^ CH 3 COO' + H- but the extent to which the dissociation proceeds varies very greatly from one acid to another, and consequently the amount of hydrogen ions formed ; e. g. nitric acid of ordinary concentration — say 1 mol. in 10 litres of water (decinormal)— is almost completely dissociated, acetic of the same LAW OF MASS ACTION 169 concentration only to the extent of one or two per cent. Now, the acid properties of the liquid are due exclusively to the presence of hydrogen ions, so that the strength or avidity of an acid depends solely on the extent to which it is dissociated into ions. Similarly, a base is an electrolyte of which the anion is hydroxyl ; e. g. baryta in solution gives — Ba(OH) 2 ^ Ba- + 2OH' and the basic properties depend solely upon the amount of hydroxyl ions present. In all cases the dissociation is favoured by dilution, so that when a salt, acid, or base is dissolved in a very great amount of water it may be regarded as completely broken up into its ions. Water itself dissociates to a small extent, yielding both hydrogen and hydroxyl ions — H 2 $ H- + OH' but the amount of them is very small, only about 1 part in 10,000,000.000 being dissociated at ordinary temperature. It may, however, on this account, be looked upon either as a very weak acid or a very weak base. The velocity of all reactions is increased by rise of temperature, and that very rapidly ; the rates of increase per i° in cases that have been measured vary from 2 % to 13 %, and are mostly nearer the higher than the lower limit. This increase is in geometrical, not arithmetical pro- gression, i. e. each degree raises the velocity by the same fraction of its then value, not by the same actual amount as in the expansion of gas. Hence, the increase over a wide range of temperature is enormous : if the rate per t° is 10 %, that for ioo° is to be found by reckoning com- pound interest on the amount at o° one hundred times over at 10 %. Thus, the rate of decomposition of dibromsuccinic acid in aqueous solution lias actually been measured at 15 and at 101 , and proves to be 3,000 times as rapid at the latter temperature as at the former. In consequence of this the range of temperature over which any particular reaction is observable is very restricted, e. g. the combination of oxygen and hydrogen goes on at a moderate and measurable rate at the boiling-point of sulphur (448 ) ; but at atmospheric temperature, if it takes place at all, the reaction is so slow that it is not possible to observe it ; whilst at 700 or there- abouts it is so rapid as to constitute an explosion. In a reversible reaction both partial reactions are accelerated by rise of temperature, but not necessarily at the same rate. If they are the state of equilibrium will not be affected ; this is the case for the forma- tion of ethyl acetate. But usually one reaction gains on the other as temperature rises ; thus, the decomposition 2HI -» H 2 + I 2 gains on the formation H 2 + I 2 ->2HI, and consequently at high temperatures more hydriodic acid is decomposed than at low. 110 CHEMISTRY Certain changes in organisms have heen studied for the influence of temperature, ;:nd appear to behave like chemical reactions ; thus, the respiration of plants has been found to increase nearly 10 % per i 9 between o° and 25°, and seems to go on increasing (not reaching a true optimum point) until the excessive rate of action destroys the plant. The action of enzymes has been found to increase at about the same rate, and the excessive action of the human organism during fever may probably be classed with these effects. The changes occurring in the fluids of the digestive organs, the blood, &c, are essentially chemical changes, and consequently follow essentially the same laws as ordinary chemical reactions that have been followed out in the laboratory. § 2. The phase rule. In contradistinction to the definition given at the beginning of the last section, a system is called heterogeneous when it can be separated mechanically into parts ; and each of the parls is called a phase. Thus, if water and its vapour be present together, as in the boiler of a steam Engine, we have two phases ; if a solid salt lie at the bottom of a vessel containing its aqueous solution, we have again two ; and if, further, some ■water vapour exist above the liquid there are three phases to consider. Generally gases form only one phase, since they all mix together ; liquids may form one or two or even more according to their miscibility, e.g. if phenol and water be mixed in equal proportions they will form two Jayers, one of phenol, which however contains some dissolved water, and one of water, which likewise contains some dissolved phenol : by warming the mixture to 60° or 70° the two liquids come to dissolve more and more of each other till they completely mix, and then there is only cne phase. Solids, on the other hand, usually constitute a jphase each, since they do not mix at all. The number of possible phases in a system is associated with the number of chemical components that go to form it. In reckoning these it is not necessary to push the analysis as far as it will go ; it is sufficient to find what is the least number of components by which the analytical composition may be expressed : the meaning of this will become clear on considering some particular cases. Thus, if we have ice, water, and steam together (which is possible under certain conditions), there is no need to take into account the fact that these substances are formed by the combination of oxygen and hydrogen, nor to enter into discussions as to the molecular weight of the water in its various phases, or its electrolytic dissociation. It is sufficient to note that the analytical composition of all three is identical, and to say that there is only one component. Suppose, however, some hydrochloric acid added to the water ; tr.en the THE PHASE RULE 171 phases will not all have the same composition, for the ice will be un- affected, remaining pure H a O, the liquid phase will contain a certain percentage of HC1, and the vapour will also contain a certain percentage of HC1, but not in general the same as the liquid. However, to express the composition of these three phases, it is not necessary to write down the percentage of hydrogen, oxygen, and chlorine in each (three sub- stances), for the oxygen is throughout combined with a fixed proportion of hydrogen, and the chlorine with another fixed proportion of hydrogen ; hence, we shall know all about the composition by stating the percentage of water and hydrochloric acid in each phase (two substances). The number of components therefore is two. Now, considering first, for simplicity, a system of one component, say H 2 0, let us find in how many ways it may vary independently. For this purpose we will start by supposing that there is only one phase, steam. The steam can obviously not be varied in composition, but it may be varied in pressure while the temperature is kept constant (provided we do not increase its pressure so far. as to make it condense), and it may be varied in temperature while .kept under constant pressure (with the same proviso as to condensation brought about by cooling). There are, consequently, two independent variations possible without passing out of the given phase, or, as it is -stated, two degrees of freedom. We have here left out of account a mere change in the quantity of steam, apart from any change in its condition, for, as we shall see throughout, the quantity of any phase present is without influence on the state of equilibrium reached (in marked contrast to the effect of the quantity of the different constituents within one phase, as considered in the preceding section). But suppose that the steam is brought into a state in which some of it condenses to water : there are then two phases. We have seen already (p. 86) that steam in order to be saturated, i. e. to exist in equilibrium along with liquid water, at any given temperature, must be at a definite pressure corresponding to that temperature — known as the saturation pressure. It is, consequently, not now possible to effect two independent variations in the state of the system without the dis- appearance of one of the phases ; for if the temperature of the system be raised the pressure must be raised too, or it will all be converted into vapour, and if the temperature be lowered, the pressure must be lowered too, or it will all be condensed into liquid. The change in pressure is therefore no longer independent of the change in temperature, so that there is only one degree of freedom. Next suppose the mixture of water and steam to Le cooled, taking care to adjust the pressures so that neither phase disappears, till ice begins to form; there are now three phases. It will be found that there are no degrees of freedom left, i. e. no change is possible in either temperature or pressure without one of the phases .disappearing ; if the temperature is raised the ice will melt, if it is 172 CHEMISTRY lowered the water will all freeze, unless that is prevented by an appro- priate rise of pressure, and then the vapour phase would disappear. The condition of temperature and pressure under which all three phases can coexist is called the triple point. It is slightly above o° in tem- perature and about 5 mm. in pressure.. These relations are perfectly general ; the appearance of a new phase always implies the loss of a degree of freedom, so that if the system be of one component only, it has two degrees of freedom when one phase is present, one when two phases are present, forms a triple point when three are present, and can in no case contain more than three phases. There are instances in which four or more phases of a single substance are known, e. g. sulphur in the forms of (i) vapour, (ii) liquid, (iii) mono- symmetric crystals, (iv) rhombic crystals ; but it is impossible to have more than three of these in equilibrium together. The triple point and associated phenomena may be clearly represented by a diagram with temperature and pressure as ordinates, such as has already been used to explain the properties of the melting-point (Fig. 46). The lines on that figure have already been referred to, with the exception of de ; this is the boundary between the solid and liquid states, in other words the melting-point line, d, as already explained, is the melting- point of the solid when under the pressure of its own vapour ; it is consequently the triple point, for the solid and liquid can there not only exist alongside of one another, but in presence of their vapour too : for benzene, to which Fig. 46 primarily refers, the triple point is at about 36 mm. pressure and 5°-58. Suppose then a mixture of solid, liquid, and vaporous benzene kept in a cylinder closed by a piston ; the pressure being adjusted to the value just mentioned and the correct temperature being maintained by a bath; now push in the piston. The pressure will not at first rise, for as the volume gets less, more and more vapour will condense ; when no vapour is left, however, and there are conse- quently only two phases present, the system will possess one degree of freedom— it can be changed in pressure, but only on condition that the temperature changes in a definite manner too, if both phases are to be preserved. This definite condition of change is shown by the line de ; in the case of benzene, as the pressure rises, the temperature must rise too, i. e. the melting-point is raised by pressure. This is because solid benzene is a little denser than liquid, and consequently the increasing pressure would squeeze the whole substance into the solid form, were it not counteracted by a rise in temperature. The line de accordingly slopes to the right, upwards; the slope is very slight however (much exaggerated in the figure), amounting only to one- or two-hundredths of a degree per atmosphere of increased pressure : consequently the melting- point under atmospheric pressure is practically identical in temperature with the triple point. For water the line de slopes a little to the left, THE PHASE RULE 173 because ice is less dense than water; pressure consequently tends to convert it all into the liquid form, and has to be counteracted by lowering of temperature. From the triple point d therefore there start out three lines,, »B, dc, de, dividing the diagram into three areas, cde, edb, bdc. The point » represents the simultaneous existence of three phases ; each line two phases ; each area one phase only. Further it is easily seen that the point allows of no movement at all, each line allows of movement in one direction (viz. along its length \ in other words one degree of freedom ; while within the spaces, independent movement in two directions is possible, i.e. changes in pressure and temperature can be made separately. If a system has two components, e. g. water and hydrochloric acid, then it may possess three degrees of freedom, for in addition to tempera- ture and pressure, we may vary the percentage, say, of hydrochloric acid in it. Again, by introducing one new phase, one degree of freedom is lost, so that starting from a single phase we may add three more, and reach a maximum of four. Similarly if there be three components, five phases may coexist, and so on. The phase rule which is due to Gibbs may then be stated generally as follows : — 'The number of degrees of freedom of a chemical system is equal to the number of components increased by two, and diminished by the number of phases.' The melting-point is really only one of a class called transition points at which one phase passes over into another. One of the most familiar instances of such is the transition between the two crystalline forms of sulphur ; the rhombic form, in which that substance is usually found, is stable at ordinary temperatures, but at 95° it is converted into another form, that of crystals belonging to the monosymmetric system. The conversion is a much slower process than that of fusion, but if rhombic crystals be kept at ioo° for some hours they will be found partially con-, verted into the other form, and the conversion will proceed steadily until the one form is completely replaced by the other. If then the product is maintained a long time at 90 it will suffer the converse change, and with equal completeness. The transition point is therefore a point of interchange of stability, like the melting-point : and when the form stable at lower temperatures (rhombic) passes over into that stable at higher temperatures (monosymmetric) there is always an absorption of heat, the latent heat of transition, exactly comparable with the latent heat of fusion. In the case of sulphur the transition is accompanied by an expansion, but that is not necessarily the case ; the rule as to absorption of heat is universal. Another example is the transition between the ordinary form of iron, which is magnetic, at about 700° into another form which possesses no magnetic properties. Again, it has lately been found by Cohen that T74 r CHEMISTRY" - ordinary tin is not stable below 20°,, but is converted into a grey' crystalline form ; this sometimes takes place spontaneously in very cold winters, a sort of ' tin-pest ' breaking out on the surface of the metal, and gradually spreading. Ordinary tin can be 'infected' by means of specimens of the grey form, and thereby converted, more or less rapidly,, into the latter form, which is the more stable at low temperatures. Innumerable cases of transition amongst compounds are known ; e. g, Glauber's salt melts in its own water of crystallization at 3a . This process is really the transition : Na 2 S0 4 • ioH.O ^ Na 2 S0 4 + ioH 2 (solid crystals of hydrate) (anhydrous solid + saturated solution) , All transition points, like the melting-point, are but little affected by pressure,, because the change in volume accompanying the transition is small ; accordingly it makes little difference whether the temperature of transition be measured at atmospheric pressure, or at the saturation pressure of the substance concerned. What little difference there is, however, follows the same rules as in the case of the melting-point. The two leading results as to transition may then be stated formally as follows : — ' The system which is formed from the other with absorption of heat, is the more stable at temperatures above the transition point, and vice versa.' ' The transition temperature is raised or lowered by pressure according as the system expands or contracts in passing from the form stable below the transition point to that stable above it.*' The importance of the phase rule as a guiding principle among the phenomena of chemical equilibrium may be illustrated by the following (out of very many possible) examples. Amongst drying agents, anhydrous copper sulphate and strong sulphuric acid are commonly used : their action however is different in principle. Copper sulphate is usually obtained in crystals containing five molecules of water ; these on drying (either by heating or in a vacuum desiccator) dissociate, leaving a salt with three molecules of water ; on further drying this breaks up, yielding a monohydrate, and the latter finally dissociates in a similar manner, giving the anhydrous salt. We need only consider the last reaction, which may be written — CuS0 4 ■ H 2 ^ CuSO t + H ? 0, Now if some of this hydrate be placed in a tube and the pump applied; to evacuate it, so that it gives off some, water, we have a system con- sisting of three phases, viz. two solids, the hydrate, and the anhydrous, salt,, and one gas, steam. The system is made up of two components, clearly, for it is only necessary to analyse any one of the phases into its percentage of CuSOj and of H 2 to state its composition. According to. THE PHASE RULE 175 the phase rule,' therefore, the degrees of freedom possessed by the system amount to 2 + 2 — 3 = 1. This means that if the temperature be changed the pressure must be changed in a definite manner : there is a fixed pressure (called the dissociation or saturation pressure) corresponding to each temperature. The case is in fact similar to that of a liquid and its vapour. If a vessel containing water and water vapour, kept at say itf cent., be connected to a pump, when the pressure is reduced to about 13 mm. the water will boil, and it will not be possible to reduce the pressure further, so long as any water is left, for if water vapour be pumped out, it will merely be replaced by further evaporation ,' and if steam at higher pressure be let in, it will condense. In the same Way the hydrated salt CuS0 4 -H 2 possesses at rs° a definite dissociation- pressure of about 0.6 mm. Consequently if the pressure be reduced below this point it will dissociate and give off water vapour,, while on the other hand if damp air be passed over the anhydrous salt, the reaction will proceed in the opposite sense, and water will be removed from the air till the pressure (of water vapour) has fallen to o.6 mm. ; hence anhydrous eopper sulphate is a fairly effective drying agent, removing some •£$ of the moisture in the air, but it is incapable of going beyond that point. The drying action of strong sulphuric acid is different : sulphuric acid being a liquid, it, with the water it absorbs, constitutes only one phase, and the vapour over it another. Now in a system of two components with two phases there are 2 + 2 — 2 = 2 degrees of freedom ; i.e. there is not in this case a definite pressure corresponding to any particular temperature ; but, the temperature being kept constant, it is still possible to vary the vapour pressure by varying the composition of the acid ; e. g„ if a bulb containing 10 gms. of pure H 2 S0 4 be used and some air con- taining o-i gm. of moisture be passed over it, most of this will be absorbed, and the strength of the sulphuric acid thereby reduced to. about 99 % ; such acid has a vapour pressure say x, and consequently will not absorb more moisture from the air passed' through it if the pressure of water vapour in the air be lower than x. If however another bulb be put behind it, containing another charge of pure H 2 S0 4 , this will absorb more moisture from the air which has passed through the first : thus the drying power depends essentially on the quantity of sulphuric acid used, whereas with copper sulphate the quantity is immaterial, so long as there, is any of the anhydrous substance left. Another example of the phase rule may be found in the aeration of the blood. Here the equilibrium is between the two forms of haemoglobin, oxidized and reduced, and the oxygen of the air ; there are thus twq solids and a gas, and the case is similar to that of copper sulphate and water vapour. The oxyhaemoglobin gives off oxygen — dissociates — at a definite pressure, so that it can be alternately formed and dissociated by 176 CHEMISTRY exposure to oxygen at a comparatively high pressure (in free air} and at low pressure (in the tissues). This at least is the general outline of the phenomenon: it is really more complicated and less perfectly under- stood than the dissociation of a hydrated salt. § 3. Thermochemistry. Changes of state, both physical and chemical, are in general accom- panied either by an absorption or evolution of heat ; we have seen, for instance, that when ice is converted into- water or water into steam, a considerable absorption of heat takes place ; again, that any transition, such as that from rhombic into monosymmetric sulphur, involves absorp- tion of heat ; while it is a well-known fact that many chemical processes involve a large solution of heat, e. g. when carbon combines with oxygen of the air and burns, when acid is mixed with alkali, and so on. Examining more closely these thermal changes, we find that in accor- dance with the law of conservation of energy, a distinction must be made between cases in which the substance suffering the change remains of the same volume as before or not. As there is no essential difference between physical and chemical instances we will take one of the former, for simplicity. When a gram of water at ioo° is boiled, some 536 calories of heat have to be put into it. This implies that the steam formed contains a very much larger amount of energy than the water it was formed from ; at the same time, not all the heat absorbed is spent in adding to the stock of energy in the substance; for whilst the water occupies only 1 c.c, the steam occupies about 1,654 e -°- ! therefore during the process of evapora- tion the atmospheric pressure has been overcome through a volume equal to the difference of these, and work has been done by the expanding steam to the extent = (1654 — 1) 1013200 ergs, i. e. the increase in volume 1653 x 1013200 multiplied by the pressure. In calories this is = 41. 42000000 Since then the substance we are dealing with gives out during the process mechanical work equivalent to 41 calories, while it absorbs 536, there can only remain 536 — 41 =495 as the increase in internal energy of the water on conversion into steam. This is sometimes called the internal latent heat. For chemical purposes it is more convenient to refer results to a gram- molecule of material than to a gram ; the molecular weight of steam being 18, we may put the molecular internal latent heat = 495 x 18 = 8910 cal., and to express the heat change it is customary to write down the chemical equation with the amount of heat evoked in it at the end. The reaction we have just considered is then — H.fi (liquid) = H 2 (vapour) - 8910 the negative sign before the quantity of heat implying that heat is THERMOCHEMISTR Y 177 absorbed during the change indicated. In each case it should be carefully specified whether the internal or total heat change is meant. If however the conversion is from one solid or liquid into another, the change in volume is so small that the external work done may be neglected by com- parison with the absorption of heat, and the distinction ceases to be of consequence. Thus ice in melting absorbs 80 calories per gram, while the amount of work done (in this case done on it by the atmosphere, since the ice contracts) does not amount to the equivalent of T fo calorie. Turning now to a chemical case we note precisely the same distinctions. If only solids and liquids are concerned we may ignore any changes in volume ; thus iron and sulphur combine with evolution of heat, the reaction only needing to be started by a local high temperature ; it may be written — Fe + S = FeS + 23800 This statement implies evolution of heat (since the sign is + ), and that when 56 grams of iron combine with 32 of sulphur 23,800 calories of heat are given off. In other words, the internal energy possessed by 88 grams of ferrous sulphide is less by 23,800 calories than that of the elements it was formed from. This quantity is called the heat of reaction. If, on the contrary, gases are concerned, the external work done by the reacting substance if it expand, or done on it if it contract, is considerable enough to take into account. Practically a reaction is performed either under the constant pressure of the atmosphere, free contraction or ex- pansion being allowed, or else in a space of constant yolume (such as Berthelot's calorimetric bomb). In the latter case evidently no external work is done, and the true or internal heat of reaction is directly measured. When the volume is allowed to change the circumstances are slightly more complicated. As example we will suppose four grams of hydrogen to be burnt under ordinary conditions. It is then observed that 136,720 calories are evolved : the equation may be written — 2H 2 + 2 = 2H 2 + 136720 (at constant pressure) the water formed being collected as liquid. Now three mols. of gas disappear in the equation, while only a liquid is formed, so that work is done by the atmospheric pressure on the system — e. g. if the com- bustion takes place at the top of a eudiometer tube, mercury is driven in by the atmospheric pressure to take the place of the gas formerly there. The work done in this way can easily be calculated, as in the preceding example, and since a mol. of any gas occupies the same space when under the same conditions of temperature and pressure, the same amount of work is done for each mol., the amount being (at atmospheric tem- perature) about the equivalent of 580 cals. Hence 3 x 580 = 1740 cals. of heat is produced by means of this work, and of the observed generation of heat so much is accounted for, and there remains only 136720 — 1 740 = 134980 cals. really due to the combination. This is the H 178 CHEMISTRY heat that would be evolved if the combustion took place in a closed space such as the ealorimetric bomb, when no question of external work comes in ; it is therefore the true decrease in energy of the materials : four grams of hydrogen and 32 of oxygen uncombined possess 134,980 cals, more energy than 36 grams of water. It follows from the law of the conservation of energy, that if a chemical system be transformed from a certain initial state A to a certain final state B, it gains or loses a fixed amount of energy, however the trans^ formation be effected ; for if we transformed it by a certain process and it gained energy, and we then transform it back by a different process, it must lose precisely the same amount of energy, otherwise it would not in the end have the same content of energy as at starting, but being in the same state at the end as at the beginning it must have the same content of energy. The important results that this principle leads to may best be under- stood by the aid of an example. Suppose it is desired to know how much heat is evolved or absorbed in forming benzene (C 6 H 6 ) from its elements. It is not possible to synthesize benzene directly, so to determine the amount of heat we must have recourse to indirect methods. Now carbon and hydrogen can be burnt in a calorimeter, and the heat evolved measured ; so too can benzene ; and it is found that the heat of com- bustion of six grams of hydrogen is 202,470 cals., of 72 grams of carbon 581,760, making a total of 784,230, while 78 grams of benzene on com- bustion give out 797,900 cals. Hence if we could synthesize benzene from carbon and hydrogen, and the heat evolved in doing so be x, we may compare the following processes : — (i) C 6 + 3H 2 = C 6 H 6 + x and C 6 H 6 + 7£0 2 = 6C0 2 + 3H 2 + 797900 (ii) C 6 + 3 H 2 + 7i0 2 = 6C0 2 + 3H 2 + 784230 (i) and (ii) lead from the same initial to the same final conditions, so C - that the total change in energy must be the same in each ; it follows that x + 797900 = 784330, or x = — 13670. Accordingly the requisite quantities of carbon and hydrogen on combining to form benzene gain 13,670 cals. of energy. Fig. 55 may serve as a simple illustration of the principle here involved, a represents the initial state of elementary carbon, hydrogen, and oxygen ; B the final state of carbon dioxide and water ; in passing from a to e the energy-content falls by 784,230 cals. represents the intermediate condition of benzene and oxygen, and it is found that in passing from to b the energy falls by 797,900 cals. ; hence it is obvious that c is 13,670 cals. higher in energy than a. B 1 In all the above reasoning it must be carefully remembered Fig. 55. that we are concerned with changes of internal energy only, not with work done by the reacting system on other bodies, so that the true heat of reaction, i.e. at constant volume, must always be employed. A THERMOCHEMISTRY 179 A reaction such as the formation of benzene from its elements, in which the heat of reaction is negative, is called endothermic : the more fre- quent class in which the heat of reaction is positive is called exothermic. A convenient abbreviated notation for heats of reaction is to write within square brackets the substances out of which the compound is formed with a comma between each, and put the heat evolved on the opposite side, h. g. — [C„ H 6 ] = - 13670 [H, CI, Aq] = + 39300 The symbol Aq represents an indefinite quantity of water, used as a solvent, H 2 being retained for the molecular quantity, such as enters into reaction ; consequently the latter equation means that when one gram of hydrogen and 35-4 grams of chlorine, both in the gaseous state, combine to form hydrochloric acid, and dissolve in water, 39,300 calories are evolved. This is a strongly exothermic reaction ; as a rule the reactions which take place most readily are those in which a, good deal of heat is liberated ; but for more exact information as to the bearing of heat of reaction on chemical equilibrium, see below. The most important thermochemical data, so far as medicine is con- cerned, are the heats of oxidation of the elements forming organic compounds. These are — [H a 0] = + 67490 or hydrogen . . 33745 cals. per gm. [C, 0] = + 26300 carbon to monoxide . 2192 „ „ [C, 2 ] ■= + 94640 carbon to dioxide . 7887 „ ,, [S, 3 ] = + 71000 sulphur . . . 2219 „ ,, Nitrogen, in the decomposition of organic compounds, is usually given off in the elementary state. The thermal efficacy of food-stuffs (and coal) may be roughly calculated by means of the above numbers ; only roughly, for we have seen, in the instance of benzene, that the energy of the elements when combined is not quite the same as when free. In reckoning the thermal value in this way, if there is oxygen in the compound it should be regarded as combined with the hydrogen, and so reducing from the total supply available. As an instance of the mode of calculation we will take a proteid of composition C 53 % , H 7 %, N 16 %, 22 %, S 1 %, ash 1 %. Then in 1 gram we have 0.22 gm. of oxygen : this may be regarded as com- bined with I of its weight in hydrogen, i. e. 0-0275 gm. ; deducting this from the 0-07 gm. of hydrogen we have 0-0425 left for combustion. The heat generated will therefore be, approximately — 0-53 gm. of carbon @ 7887 = 4179 calories 00425 „ hydrogen @ 33745 = 1434 „ 001 „ sulphur @ 2219 = 22 ,, 5635 u «2 i8o CHEMISTRY An estimate of average human diet gives the following as the total supply of energy to the body per day : — ioo gm. proteid = 550000 cals. 100 „ fat = 950000 „ 240 ,, carbohydrate = 960000 „ 2460000 ,, This in mechanical units is 2460000 x 4-2 = 10,330,000 joules per day = 120 joules per sep. (watts). This is the average total activity of a full- grown man throughout the day and night : the average of mechanical work done by those engaged in muscular occupations may be taken at 20 watts (or say 48 per hour during ten working hours). Hence the efficiency of the human machine, averaged throughout the twenty-four hours, may be taken as one-sixth. § 4. Heat and chemical equilibrium. In Mechanics one is familiar with the principle that the potential energy of a system tends to run down to a minimum ; this was partly taken into account in considering the various kinds of equilibrium on p. 31. But as the principle is applicable throughout Physics, and has a special bearing on chemical equilibrium, it will be reconsidered here. Beginning with a simple mechanical example, suppose a ball to be placed on an incline ; it has a certain amount of potential energy, due to its height above the earth, the amount being measured by its weight multiplied by the height above whatever is chosen as the standard level. If the ball be moved higher up it will there- fore possess more energy ; if lower, less ; and accordingly if left to itself it will roll down- wards, i.e. in the direction of diminishing potential energy. If it be at the bottom of an incline, say at A (Fig; 56), it will be in equilibrium, and in particular in stable equilibrium ; this is because the level rises in all possible directions from a, and such a point is called a point of minimum level. This does not mean that it is the lowest point of all, for e is lower ; but merely the fact that the level rises in all directions from A. Again, if the ball be at the top of an incline, as at c, it is in equilibrium, but this time unstable ; its height, and therefore its potential energy, is a maximum, i. e. the level slopes down from that point in all directions, although c is not the highest point of all, for t is higher. We see, then, that Fig. 56. HEAT AND CHEMICAL EQUILIBRIUM 181 stable equilibrium corresponds to a minimum of potential energy, un- stable to a maximum ; while a point which is neither a minimum nor a maximum, like B and r, is not a point of equilibrium at all. Further, the equilibrium at E is more stable than that at A, nevertheless it is not possible to transform A into e without giving the body sufficient energy to climb the intervening maximum. The term metastable is some- times used to distinguish a condition like a, which is stable, and yet not the most stable condition possible. A closer analogy to the chemical case is supplied by the action of a siphon. Suppose a and e connected through c by a siphon ; then water could not flow of its own accord from a to o, but if the siphon were once started it would enable an indefinite quantity of water to pass from the metastable condition a to the stable e ; and in order to start the siphon it is only necessary to impart a small impulse from without, just to raise enough water to fill the siphon. Applying these considerations to the case of chemical reaction, we may say that a mixture of benzene vapour and oxygen is a system in metastable equilibrium : it possesses a very considerable degree of stability, and indeed may be kept for an indefinite time without change ; yet it is not so stable as the water and carbon dioxide that may be formed from it, only an intermediate condition of greater potential energy has to be passed through to transform the one into the other. This can be accomplished by means of local heating — say by passing an electric spark through some point of the mixture ; this causes an explosion, and the whole is transformed into the stable form of water vapour and carbon dioxide ; we may pass sparks repeatedly through this without getting it back into the metastable form again. A similar process constitutes the metabolism of the animal organism. The food-stuffs possess a considerable degree of stability— sugar, starch, fat, and so on, can be kept indefinitely ; but they possess a large amount of potential energy that can be given out by combining them with oxygen of the air to form carbon dioxide and water. This is con- tinuously accomplished in the organism by a mechanism that builds up the materials into the form of protoplasm, which if not actually unstable, no doubt constitutes a metastable form possessing so much energy that it easily breaks down ; the breaking down of this by oxida- tion then supplies the energy by which the organism is kept in action. It was at one time thought that the potential energy of a chemical system, according to which its transformations are determined, could be measured by means of heats of reaction. It is true indeed that by means of the heat evolved in a reaction we measure the change in the total energy of the reacting system ; but it does not follow that this total energy is what must be taken into account to decide upon the stability or in- stability. As a matter of fact it is a part only of the energy, which is called 182 CHEMISTRY the free energy, that determines that. To continue the mechanical analogy, we may suppose the ball in Fig. 56 to be set spinning about a vertical axis : then it possesses energy of two kinds, the potential energy we have already considered, and kinetic energy of rotation ; now the ball might be spinning at a so rapidly as to possess more energy on the whole than if it were at rest at u. Yet, though the circumstances would be more complex than in the previous state, it would still be true that a was the position of stable equilibrium, and the ball would not pass of its own accord from A to c ; it is therefore in this case only a part of the total energy which is decisive as to the equilibrium of the ball. With regard to the chemical case, something similar is true ; though it is not possible within the scope of this book to explain fully the method of determining this part — the so-called free energy, the conclusion is that the change of total energy, as measured by heat of reaction, is an unsafe guide. True, in the majority of cases a reaction will proceed in the sense that causes heat to be evolved ; but the opposite case, that of an endothermic re- action, is by no means rare ; thus among processes of a more physical character the formation of a freezing mixture out of salt and snow is a striking instance ; the mixture takes place of its own accord, although accompanied by so much absorption of heat that the product falls many degrees below zero. Again, to take a purely chemical example, hydro- chloric acid, being a stronger acid than hydrofluoric, decomposes sodium fluoride in aqueous solution, although some 2,300 calories are absorbed in the process. The rule then is that a reaction will proceed so that the free energy of the system is reduced to a minimum. In some cases changes of free energy can be determined by means of electromotive force ; but unfortunately this method is not general, as most substances are not electrolytes, and in the majority of reactions the changes in free energy are as yet unknown. Nevertheless certain deductions from this principle can be made. In the first place, the value of a food-stuff in supplying energy to the organism is not to be reckoned solely by its heat of combustion ; this, as we have seen, is a rough guide, showing in fact what is the total energy the substance can give up when converted by the processes of the organism into water, carbon dioxide (and urea) ; but it is not this total energy which we are concerned with. In » steam engine it would be, and if the body were a heat engine it would be so in that ; in a heat engine all the chemical energy of the fuel is converted into heat, by burning it, and that heat used for the production of work (by means of steam or otherwise). Now a heat engine can only convert into work at most a definite fraction of the heat it consumes, viz. — — . where T. is the absolute temperature at which the heat is supplied, and T 2 that at HEAT AND CHEMICAL EQUILIBRIUM 183 which the waste is withdrawn (p. 64). If the body were a heat engine it could only work between its own temperature — say 37 cent. — and that of the surrounding air, say on the average 15 cent. The efficiency of such an engine could- at most be only — 37 - I5 =7%' 273 + 37 whereas the efficiency of the human body during actual work has been found to be about 26-5 % (p. 61). Hence the mechanism of production of work in the organism must be quite different : the energy of food is not first transformed into heat and then used, but is transformed directly into mechanical energy ; it is therefore incorrect to speak of food as the fuel of the animal body — the metaphor is no doubt strengthened by the observation that much heat is given off from the organism in its working, but the analogy between it and a heat engine is really misleading, the two mechanisms being fundamentally different. Consequently in order to estimate the value of a food-stuff as a means of producing the work necessary to the organism, it would be necessary to know the amount of its free energy ; but that quantity has not yet been determined. In the second place, it has been shown (by van 't Hoff ) as a deduction from the principle of free energy that in any reversible reaction by which one system of bodies is converted into another — ' Rise of temperature favours the system formed with absorption of heat? We have already seen that this principle is applicable to cases of tran- sition from one phase to another. To take the simplest physical instance, water and steam are two phases that can be reversibly converted into one another, and steam is formed out of water with absorption of heat ; consequently rise of temperature favours the formation of steam, fall of temperature, water. This is in fact only another aspect of the principle stated at the end of § 2, that in passing from the form stable at low temperatures to that stable at high temperatures, heat is absorbed. But van 't Hoff's statement was intended to apply rather to equilibrium in homogeneous systems. This, as we saw, follows the law of mass action, and when any reversible reaction A^.B can take place a certain state of balance is arrived at, in which some of A and some of B are present ; such an equilibrium however holds only for a particular tem- perature ; when the temperature is altered the equilibrium will be dis- placed, the percentage of A being increased or decreased. The rule stated above shows which way the displacement will take place ; thus carbon dioxide is formed from carbon monoxide and oxygen with a large evolution of heat. At low temperatures the reaction is carried out with practical completeness in one sense — nothing but carbon dioxide is formed when a mixture of the gases is sparked or otherwise made to react. But rise 184 CHEMISTRY of temperature favours the system formed with absorption of heat, i.e. the carbon monoxide and oxygen, so that at 2,000° the reaction aCO + 2 i» 2C0 2 is decidedly a reversible one, quite a considerable amount of the system on the left of the equation existing in equilibrium ; this has been noted in the gases of a blast furnace — the carbon dioxide is said to be partly dissociated at so high a temperature. Conversely, as we have seen that hydriodic acid is dissociated to a. somewhat greater extent at high tem- peratures than at low, we may conclude that the system H 2 +I 2 is formed from the system 2HI with absorption of heat. Now the atmospheric temperature is a low one, being only about 290° above the absolute zero, whereas we have to deal with temperatures up. to 2,000°, and even higher. Hence it is natural that in most cases of equilibrium at low temperatures the system formed with evolution of heat should prevail ; and it is found in practice that reactions which take place energetically at ordinary temperature are accompanied by a large generation of heat ; this led at first to the opinion that chemical reaction necessarily took place in the direction to evolve heat. That opinion is not correct, since endothermic reactions also are known ; but at the absolute zero it would be true. CHAPTER VII. ELECTEIC CUEEENTS. § i. General properties of currents. An electric current is a phenomenon to the production of which two things are necessary : (i) a source of electrical energy ; (ii) a conducting circuit. What the current really is, in itself, is by no means completely known at present, nevertheless the properties it possesses and actions it exerts have been investi- gated with a thoroughness and exactitude that are not excelled in any other department of Physics. What we have to do, therefore, is to describe these properties, and in order to do so we shall begin by leaving out of consideration the source of electrical energy, which may be a voltaic cell, a dynamo, &c, and confining attention for the moment to the effects observed in and near the conducting current. To begin with, then, substances may be classed into those which' do and those which do not conduct electric current, or to be more accurate, may be classed according to the facility they possess for conducting it. In the first class come all the metals ; these are good conductors : in the second class a group of bodies whose relation to the electric current is peculiar, since they suffer chemical decomposition when the current is led through them ; these are the so-called electrolytes, and include salts, acids, and bases, in aqueous solution, and also, in many cases, when fused. A third class may be constituted of all the remaining bodies, their conducting power being extremely small, and in the case of gases perhaps nil ; amongst these bodies, called insulators, some that are important for electrical purposes, from the very fact that they strongly resist the flow of current through them, are mica, glass, ebonite, paraffin. 186 ELECTRIC CURRENTS Next, in order to allow an electric current to flow, it is not sufficient to provide a path of conductors, whether metallic or electrolytic, but the path must be a completely closed circuit. For it is found that electric currents flow only in closed, circuital paths ; any break in the chain of conductors, made by interposing an air space or a piece of ebonite, &c, stops the current. Hence it is easy to construct electrical Jceys or switches, by means of which the current may be turned on or off as desired^ Such a key is shown in Fig. 57 ; it consists of two brass blocks, mounted on an ebonite base. Each block carries a ' binding screw ' for conveniently making metallic con- nexion by means of wires with other pieces of apparatus. The space be- tween the blocks can be filled up Fig. 57. at will by a tightly fitting conical brass plug, provided with an ebonite handle. When the plug is inserted the current flows freely through the apparatus, when it is removed the metallic circuit is broken and the current stops. Many other patterns of keys are in use. The principal effects produced by an electric current are (i) thermal, (ii) magnetic, (iii) chemical ; we will consider these briefly, in the order stated. (i) When an electric current flows through any substance whatever it meets with some resistance, and in overcoming this it generates heat ; the energy of the current is in fact converted into heat — a process of dissipation analogous to that which friction causes in the case of ordinary mechanical motion. Now it will be necessary, as we shall see immediately, to distinguish which way a current may be regarded as flowing through a con- ductor, and also to.measure the strength of it ; this can be done by means of the effects which the current produces, but the heating effect can only be used for the latter purpose, for it is found that if a current be sent first one and then the other way through a given conductor, just the same amount of heat is generated by it per second, i. e. the heating effect is irreversible. Further, if the current be measured in the manner described below, -by means of its chemical or magnetic effect, it will be GENERAL PROPERTIES OF CURRENTS 187 found that the heat generated per second is proportional to the square of the current strength. A current may be measured by means of the heat it produces. Any instrument for measuring current is called an ammeter (from ampere, the name of the unit of current, vid. inf.), and there are in use ' hot wire ammeters ' whose construction depends on this principle. These consist of a fine wire, usually stretched horizontally, the middle point of which is attached to a pointer ; when a current is passed through the wire, it becomes hot and sags ; this causes the pointer to move round a dial, and registers the strength of a current. Such instruments are only adapted for strong currents, and, from what was said above, it follows that they are not capable of showing which way the current is flowing. (ii) An electric current may easily be shown to produce mag- netic effects in its neighbourhood. Thus if an insulated copper wire be wrapped spirally round a bar of soft iron, and a current be caused to flow through the wire, the iron will become mag- netic. And again if a magnet, pivoted so as to be free to move in a horizontal plane (e. g. a compass needle), be held near a wire through which current is passing, it will in general be turned out of its normal N. and S. direction. In distinction from the thermal effects it is to be noted that the magnetic effect of a current is produced near to, but outside the conductor ; and that it is reversible, i. e. when the current is made to flow the other way, the magnetic effect, whatever it is, is inverted ; in the case of magnetizing an iron bar, what was the north pole becomes the south ; in the case of deflecting a compass needle the deflec- tion will take place in the opposite sense. The magnetic effect is made use of for the definition and measurement of the strength of the current. The actual definition of unit current must be deferred till later, since it involves magnetic quantities which have not yet been considered ; but we may at least state the qualitative rule, since that is of use as a means of detecting the sense in which a current flows. If there be a single straight wire carrying a current, then a magnet placed near to it tends to set itself in the direction of the tangent to a circle drawn round the wire and passing through the centre of the magnet. Further, if one imagines oneself swimming in the wire, in the direction of the current, and facing the magnet, then flie north pole of 188 ELECTRIC CURRENTS the magnet will he deflected towards the left. This is known as Ampere's rule. The unit of electric current in practical use is called the ampere, from the name of the great French physicist just mentioned. The magnetic effect is that most conveniently employed to measure currents by, and instruments for the purpose are called galvanometers. These instruments take many forms, but the principle involved is usually to place a small pivoted or sus- pended magnet at the centre of a coil of wire ; the coil being arranged first parallel to the natural direction of the magnet needle, when a current is passed through it, the needle is turned out of its position to right or left according to the direction of the current, and to an extent that measures the amount of it. (iii) An electric current produces a chemical action only when it is passed through a conductor belonging to the group already referred to as electrolytes. These consist, mainly, of aqueous solutions, and the peculiar chemical action produced may best be considered by means of a concrete example— say a strong solution of hydrochloric acid (HC1). Hydrochloric acid gas, on solution in water, suffers, like other electrolytic substances, a dissociation into parts called ions. These consist of either atoms or groups of atoms, associated with an electrical charge, and are consequently of two kinds, according as their charge is positive or negative ; the former being called cations, the latter anions. In the case of hydrochloric acid, the dissociation is as simple as possible, since there are only two atoms in the molecule of that compound ; of these the hydrogen takes the positive charge and the chlorine the negative. The two charges are equal in amount, so that the water containing hydrogen ions and chlorine ions possesses no electric charge as a whole. The last statement is always true, consequently the total (positive) charge on the cations in any electrolyte must be equal to the total (negative) charge on the anions. Moreover it is found that a very simple relation holds between the charges on the various ions : for these are always either equal to the charge on a hydrogen atom, or a whole number of times that quantity ; the number in question in fact expresses the valency of the atom or group. Thus, the cation is usually a metal : H, K, Na, Tl, Ag carry one unit charge each ; Ca, Ba, Zn, two each ; Cu, Hg may carry one or two ; Fe, either two or three, and so on. As a conve- GENERAL PROPERTIES OF CURRENTS 183 nient notation the positive electric charge may be indicated by a dot after the symbol, so that Fe * * represents ferrous iron, Fe • • " ferric iron. Again, the anion usually forms all the rest of the molecule, so that some of the most common are the hydroxyl-ion OH, the chlorion CI, nitrion N0 3 , acetyl-ion CH 3 COO, all with a single charge ; sulphion S0 4 with a double charge. The negative charge is indicated by a dash, e. g. OH'. The ions in a solution are so far free from each other's influence that when an electric force is applied to the solution, it drags them in opposite directions. If, therefore, we pass a current through the supposed strong hydrochloric acid, we shall separate the components, carrying the cations with the current, the anions against it. The current is led into and out of the liquid by means of metallic conductors — say small sheets of platinum foil, to which wires have been welded. These are called electrodes, and in particular that by which the current is led in, and consequently to which the anions are attracted, is called the anode, the other the cathode. The current in the liquid is actually constituted by the movement of the charged particles ; negative charges moving in the negative direction count as well as positive charges moving in the positive direction, and produce identical effects. The ions are led by the electric force as far as the electrodes, but there their progress is evidently stopped ; and unless the current is to be stopped too, they must give up their electric charges to the electrodes, to flow round the metallic circuit. In doing so the ions become changed into ordinary matter, and will appear as such : in the example chosen the chlorions at the anode become ordinary gaseous chlorine, which is given off in bubbles ; similarly the hydrogen ions give up their charge to the cathode, and become gaseous hydrogen, which is evolved from the solution. Thus happens the remarkable fact which dis- tinguishes an electrochemical reaction from an ordinary chemical one, that the products of decomposition appear only at the electrodes, and separately ; whereas in ordinary chemical cases they are produced together, and indifferently, in any part of the reacting substance. Since the charge of electricity conveyed by an atom of any univalent substance is the same, that conveyed by the atomic weight taken in grams must be ; and, further, in the case of multivalent substances, the equivalent weight in grams will 19° ELECTRIC CURRENTS convey the same charge. This, then, is a fundamental quantity of great importance with regard to the chemical action of the current. The experimental laws that have led to the conclusions just mentioned as to atomic charges were first investigated by Faraday, and his results may be expressed as follows : — (a) The amount of chemical action produced is proportional to the amount of electricity flowing, and is independent of the rate of flow (i. e. the current). Thus a current of one ampere flowing for five seconds will convey as much electricity as one of five amperes flowing for one second, and therefore liberate the same amount of any chemical product. The unit in which quantity of electricity is measured is that conveyed by one ampere in one second, and is called a coulomb. (&) The amounts of various substances liberated by the same quantity of electricity are proportional to their chemical equiva- lents. Thus if the same current be led in turn through solutions of sulphuric acid, copper sulphate, and silver nitrate, hydrogen, copper, and silver will be the cations in these three liquids, and the weights liberated at the electrodes will be in the ratio i : 31 • 8 : 108, those being the equivalent weights of the elements in question. Both results are included, along with the numerical observation, in the statement — One gram equivalent of any ion conveys 96,610 coulombs of elec- tricity. Chemical action accordingly gives a means of measuring quanti- ties of electricity, and so, indirectly, current. An instrument for this purpose is called a voltameter : those in most frequent use depend on the production of hydrogen, copper, or silver. The latter is the most accurate, but involves some trouble in use. The electrolysis is usually carried out in a platinum bowl which serves at the same time as cathode : in this is placed a fairly strong solution of silver nitrate ; an anode of pure silver wire is arranged to dip into the liquid, and should be wrapped up in filter paper, so that no disintegrated fragments of it that may be formed during the electrolysis drop into the dish. With an 8 cm. dish and 30 per cent, solution currents up to 2 amperes may be used : the silver is deposited in a coherent crystalline form on the platinum. It must then be thoroughly washed and dried, and then weighed. Afterwards the silver is dissolved off ELECTROMOTIVE FORCE 191 with nitric acid and the bowl is ready for use again, i coulomb deposits 0-0011172 gm. of silver. For copper electrolysis a solution of the cupric sulphate (CuSOJ containing about 100 gms. of the crystalline salt per litre is the best. A platinum dish is unnecessary, as the anode and cathode may both be made of copper, and fixed vertically side by side in the solution. About 100 sq. cms. of depositing surface should be used per ampere ; the weight of copper deposited on the cathode for each coulomb is 0-0003292 gm. The water voltameter is most commonly used in Hofmann's form, as shown in Fig. 58. Dilute sulphuric acid, or dilute caustic soda, is used as electro- lyte, and is decomposed between the platinum plates aa ; oxygen is formed at the anode, hydrogen at the cathode, and these are col- lected separately in the graduated tubes bb. The volume of hydrogen should be double that of oxygen, but owing to solubility of the oxygen, and also to formation of ozone, the volume of oxygen is usually somewhat less, so that it is best to calculate the current from the hydrogen alone. The volume of hydrogen must be corrected for temperature, pressure, and presence of aqueous vapour, as described in Chap. III. "When reduced in that way to 0° and 760 mm., the volume of hydrogen per coulomb is 0-1160 c.c. The hydrogen voltameter is particularly useful for measuring small quantities of electricity. Fig. 58. § 2. Electromotive force. Turning our attention now to the causes that make a current of electricity flow in a conducting circuit, we find that the pheno» mena cannot be described with the aid only of the conception ' quantity of electricity,' or the rate at which that flows, which is the ' current.' For in order to make a current flow it is neces- sary to provide a source of energy, either mechanical, as in a dynamo driven by a steam engine ; or chemical, as in a voltaic cell ; or thermal, as in a thermopile ; and in any case the current 192 ELECTRIC CURRENTS produced is not in itself a measure of the rate at which the supply of energy is used up— a second quantity, the electromotive force (E. M. F.) or voltage, must be measured as well. We may form a preliminary notion of this quantity, as measuring the tendency to produce current ; a voltaic cell, e. g., is an arrangement always ready to produce a current, but the amount produced at any moment depends on the nature of the circuit provided for it to flow round ; the electromotive force, i. e. the strength of the tendency, on the other hand, depends only on the cell itself. Thus electromotive force is comparable to pressure (in a water supply), while the electric current corresponds to the flow (of water) produced by it. As a simple example of an arrangement for producing electric current we may take the Daniell cell. This may be constructed as shown in Kg. 59. A glass or earthenware pot a contains a sheet of zinc (Zn) bent into a cylinder for convenience ; inside this stands a pot of porous earthenware b, and in it a rod of copper (Cu). The outer pot is filled up with dilute sulphuric acid (say 1 to 5 of water) or else a solution of zinc sulphate, the inner pot with a solution of copper sulphate. Wires attached to the copper and zinc plates serve to convey the current from and to the cell. The essential is merely the chemical materials used, the arrangement is otherwise a matter of We may describe the cell, Fie. 59. :Cu the chemical action is convenience, then, in its essential features as follows — Zn : H 2 SO, : CuS0 4 : Cu or Zn : ZnS0 4 : CuS0 4 according to the way it is made up ; practically the same in the two cases. We have then two metals serving as electrodes, one (zinc) with a very strong tendency to pass into the state of ions, i e. to acquire a (positive) electric charge and dissolve in the liquid, the other (copper) with a much weaker tendency to do so. If then a path be provided by joining Cu to Zn outside the cell by a wire a current will flow ; it is carried into the liquid by zinc ELECTROMOTIVE FORCE 193 which dissolves— the zinc electrode is therefore the anode— and out of it by an equivalent amount of copper which is deposited on the copper plate (consequently the cathode) being driven out by the stronger tendency of the zinc to go in. Hence the actual chemical effect is replacement of copper by zinc, and may be represented by the equation— Zn + CuS0 1 = Cu + ZnS0 4 How the energy liberated in such a reaction can easily be measured, for if the reaction be conducted in an ordinary chemical Way all that energy is converted into heat. It has been measured and found to amount to 50,110 calories = 209,910 joules (for the quantities represented in the equation, i.e. 65 grams of zinc). This then is the store of energy available for producing a current ; but we saw in the preceding section that, according to Faraday's law, 65 grams of zinc (a equivalents) carry with them 2 x 96610 coulombs of electricity. Hence the energy available is 209910 -=-193220=1-087 joules per coulomb. We have then arrived at a quantitatively exact method of expressing the tendency of the Daniell cell to produce current, i. e. of its electromotive force ; it may be put in the form of a definition as follows : — ' The electromotive force (of a cell, dynamo, &c.) is the electrical energy supplied by it per unit quantity of electricity flowing from it.' ' The unit of E. M. F. adopted in practice is the joule -f coulomb, and is called a volt. Hence the Daniell cell has an E. M. F. of 1-087 volts. It should be noted that in the above reasoning we have assumed that all the chemical energy spent by the materials of the cell becomes electrical energy ; this is nearly the case in the Daniell cell, though in many other cases it is not so. The defini- tion of electromotive force, however, remains unaffected ; it is in any case calculable if the electrical energy available is known. It is sometimes more convenient to take instead of the energy the rate at which energy is spent (i. e. the power), and instead of the quantity of electricity the rate at which electricity flows (i. e. the current), and say that the E. M. F. is measured by dividing the power by the current, or in practical units that the volt = watt ■¥ ampere. Proceeding now to describe the commonest forms of voltaic cell, we may note first, that the essential constitution of all of them is the contact of two different metals with an electrolyte (or o 194 ELECTRIC CURRENTS sometimes two electrolytes). The original cell, as constructed by Volta, consisted of zinc and silver plates dipped into dilute sulphuric acid ; the scheme being, therefore, Zn : H 2 S0 4 : Ag. This cell is not satisfactory in its working, and is not now used, but a consideration of its defects will serve to explain the other cells that have been adapted from it. When the Volta cell is freshly made up it has an E. M. F. of about two volts. But even before any circuit is made with it, it will be found that zinc dissolves in the acid — at least if ordinary (impure) zinc be used. This is due to what is called local action : small particles of other metals present in the zinc constitute, with that metal, little local voltaic cells all over the plate. Now to get current out of a cell it is necessary to connect the electrodes by a conductor : if this conductor is very short and thick the greatest possible current will be obtained— the cell is said to be short-circuited : that would be the case with the Daniell cell described above, if Cu and Zn were connected by a thick wire ; still better if the two electrodes touched one another. But when an impure zinc plate is put into acid, the little local cells formed are all short-circuited, since all their electrodes form part of one metal plate, and accordingly electrolytic action will take place, as may be seen by the bubbles of hydrogen given off all over the plate. The current so pro- duced will of course all be wasted in the zinc plate and serve no useful purpose. If a plate of pure redistilled zinc be used the local action will not take place ; and the same thing may be accom- plished more economically in practice by cleaning the zinc with acid and rubbing it over well with mercury ; by this means the surface of the plate is covered by a uniform layer of zint amalgam, and the local differences of composition which cause local action avoided. Hence voltaic cells in general should be made with amalgamated zinc plates, unless chemically pure zinc is used. Next it will be found that if the zinc and silver plates be con- nected so as to cause a current to flow, this current will rapidly fall off in strength, and if the voltage of the cell be tested after a few minutes' working it will be found much less than two— the cell is said to be polarized. The cause lies in the production of hydrogen on the cathode— the silverplate. Current in an electro- lyte necessarily consists in the movement either of positively charged ions in the direction of the current, or negative ones in ELECTROMOTIVE FORCE 195 the contrary direction : so in this case the current is conveyed into the electrolyte by the zinc which dissolves ; but to convey it out at the cathode there is no negative ion to go into solution, for the silver plate contains no materials for such, conse- quently a positive ion must go out of solution, and the only kind that is to hand is the hydrogen of the acid. Accord- ingly hydrogen will be liberated on the cathode, as in a water voltameter. The effect of the hydrogen is twofold. In the first place, gases are all non-conductors of electricity, so that wherever a bubble of hydrogen appears the conducting circuit will be interrupted, and if much hydrogen is produced the channel for the current to flow through will be much obstructed, and the flow of current correspondingly diminished : in the language of the next section we may say that the internal resistance of the cell is increased. But even before any visible amount of hydrogen appears on the plate, its chemical influence becomes apparent ; the hydrogen may in 'the first instance be regarded as dissolving in the silver — this phenomenon is well known in the case of platinum and palladium, the latter of which will absorb many times its own volume of hydrogen, but even in the case of silver a minute amount is probably absorbed. Now the dissolved hydrogen practically con- verts the silver into a hydrogen plate, and alters the character of the cell altogether ; this is the phenomenon known aspolarimtion. The dissolved hydrogen has a considerable tendency to go back into the ionic form, and indeed hydrogen combining with the elements sulphur and oxygen to form H 2 S0 4 would evolve not much less heat than zinc in forming the corresponding com- pound ZnS0 4 ; so that instead of the energy of combination of the zinc being available for producing electromotive force, as at the moment when the cell first comes into action, the energy of recombination of the hydrogen must be deducted, and the E. M. F. corresponds merely to the difference in the tendencies of zinc and hydrogen to ionize : instead of two volts it is a fraction of a volt. Thus in order to construct a workable cell it is necessary to avoid polarization. This can be accomplished by mechanical means, such as brushing away the hydrogen, but only very imperfectly ; it can be better done by chemical action, oxidizing away the hydrogen as it is formed. Nitric acid, potassium. 02 .196 ELECTRIC CURRENTS bichromate, and manganese dioxide are the oxidizing agents mostly used. The cells depending on their action are- Grove's. Zn : H 2 S0 4 (dilute) : HN0 3 (strong) : Pt The sulphuric acid may be about i to 5 or 1 to 7 of water, the nitric acid not diluted, in order to get the highest electromotive force. The two liquids must be separated, most conveniently by a jar of porous earthenware ; for if the nitric acid came in contact with the zinc it would dissolve it even when the circuit was not completed. Silver cannot be used for cathode as it is attacked by nitric acid ; platinum is therefore used instead. The Grove cell has an E. M. F. of about 1-9 volts, is capable of giving fairly strong currents, and is steady in working for two or three hours ; but it does not last long, and must be taken to pieces after using; moreover it gives off unpleasant nitrous fumes. Bunsen's. Zn : H 2 S0 4 (dilute) : HN0 3 (strong) : C Identical with Grove's, except that a stick of hard carbon is substituted for the expensive platinum ; its efficiency is practi- cally the same. The bichromate. . Zn : H 2 S0 4 + K 2 Cr 2 7 (dilute) : The solution may consist of 200 grams of sulphuric acid and 80 grams of potassium bichromate, made up to a litre of water. No porous pot is used, because the solution does not act much on the zinc when the cell is not in use ; but it is desirable to have an arrangement for drawing the zinc up out of the liquid when the cell is done with, to prevent waste. Its electromotive force is from i-8 to 2 Yolts, and is not quite so steady as that of a Grove ; the cell is used for similar purposes. Leclanche's. Zn : NH 4 C1 solution : Mn0 2 : C The solution not being an acid does not attack the zinc at all when the circuit is. not completed. Hence the cell may be left fitted up for any length of time, until the solution needs renew- ing. It is not suited for giving strong currents, as it polarizes a good deal ; the manganese dioxide, used as depolarizer, is a solid, and its action consequently slow, but if the cell is only used for a short time and then left-to itself, the polarization is removed, and the E. M. P. recovers to its original value ; hence it is particu- larly useful for electric bells. The Leclanche may be con- veniently used for most electrical measurements : its E. M. F. is «bout 1 -4 volts. ELECTROMOTIVE FORCE 197 Dry cells are Leclanche's made up with sawdust or some other substance to prevent the liquid getting spilt, and sealed up. They are very convenient where small currents are required, as they need no attention, produce no fumes, and give no opportunity for mess. Daniell's. This has already been described. In it polarization is not so much counteracted as avoided altogether. It gives stronger currents if made up with sulphuric acid, but is steadier in action if made with zinc sulphate solution. In the latter case hydrogen ions have practically nothing to do with its action, for the current is carried into the solution at the anode by zinc ions and out at the cathode by copper ions. It is necessary to keep the copper sulphate solution away from the zinc to prevent direct deposition of copper on it; this can be done by a porous pot as above described, or by making the zinc sulphate solution much denser than the copper sulphate and pouring the latter carefully on top of the former. The cell always breaks down in the end by diffusion, and so needs to be set up afresh from time to time. Clark's standard cell. Zn : ZnS0 4 (saturated) : Hgj30 4 : Hg The constitution of this is very similar to that of the Daniell cell, but the depolarizer, the mercurous sulphate, is an almost insoluble substance. This has as one result that the cell polarizes if at all strong currents are taken from it, and requires some time to depolarize ; but it is not intended for giving appreciable cur- rents, but only as a standard of electromotive force ; while the insolubility of the mercurous sulphate keeps it from action on the zinc. The- Clark cell, consequently, if made up with suffi- cient care, will keep indefinitely, and forms a perfectly reliable standard of E. M. F. The practical construction is shown in Fig. 60. A short wide test tube contains the mercury m for cathode ; connexion with this is made by means of a platinum wire p sealed through the glass. Over the mercury is poured a neutral saturated solution of zinc sulphate, to which mer- curous sulphate has been added to form a thick paste. This should be so full of crystals as to set firmly and so prevent any chance of 198 ELECTRIC CURRENTS spilling the mercury if the cell be upset ; the anode of pure zinc z is held in the solution by a cork e, a copper wire c soldered on to it serving to make connexion ; and the whole is closed air- tight by marine glue g. The E. M. F. is 1433 volts at 15 , and falls off by 0-0012 volt per 1° rise of temperature. The above are so-called primary cells, i. e. arrangements for producing electric current by the expenditure of chemical mate- rials : when the zinc, or acid, or other material in them is used up, it must be replaced by fresh. There are also, however, secondary cells or accumulators, in which the chemical materials after use are reproduced by running a current in the opposite sense through the cell. The most important of these is the ordinary lead accumulator. Its action, in outline, is as follows : When the cell is fully charged, the anode plate is of lead, the cathode of lead peroxide (held together by means of a grid of metallic lead), the electrolyte dilute sulphuric acid (about 1 to 5 of water), i. e. Pb : H 2 S0 4 : Pb0 2 . The action of the cell is to oxidize the anode and reduce the cathode, so that both tend to become PbO, or rather, in presence of sulphuric acid, PbS0 4 ; when the two plates are alike the E. M. F. of course vanishes ; but by running current from an outside source in the opposite sense, Pb and Pb0 2 are reformed— this is known as charging the cell— and it is then ready for a further discharge. The accumulator possesses the advantages of all the other cells combined ; it is capable of providing very strong currents, its E. M. F. remains for a long time constant ; it does not waste when not in use, and is consequently always ready for use, and does not require much attention. The disadvantage is of course the necessity for charging ; this is usually done by means of a dynamo. When con- tinuous current from a public supply is available, it may be used very conveniently, and the cells should be charged once a week, or thereabouts. Their E. M. F. is normally 205 volts ; during charging 20 per cent, more is required, and the cells should not be discharged any more when their voltage has dropped some 10 per cent, below the normal amount. § 3. Resistance. It appeared in the preceding section that the electromotive force in a circuit, being located where there is a source of electri- cal energy (cell, dynamo, &c), could be calculated by means of RESISTANCE 199 the amount of energy available ; but the strength of the current which flows through the circuit depends not only on the electro- motive force in it, but also on the nature of the circuit itself. The latter influence may be described by means of the resistance offered by the various conductors to the flow. Thus if in a given circuit it is found that x volts are required to make one ampere flow, x maybe regarded as a measure of the resistance offered to the flow ; for if in some other circuit it was found that 2 x volts were needed to maintain the same current, we should naturally say that the second circuit offered twice as much resistance. Now it is found that, confining attention to a single circuit, the amount of current produced in it is proportional to the electromotive force applied. This most important result, which is known as Ohm's law, is verified by experiment quite strictly, both in the case of metallic and electrolytic conductors ; at the same time it must not be supposed that it is an a priori law involved in the nature of current and electromotive force ; that that is not so may be clearly seen by considering the analogous case of the flow of water through a pipe —the rate of flow is here not pro- portional to the pressure driving it. The law is, as a matter' of fact, found to be true not only for complete circuits, but for every separate conductor and material. So it may be put in the form — ' The current produced in any conductor is proportional to the E. M. F: applied to it.' In accordance with this law, therefore, the resistance of the conductor may be defined as the (constant) ratio of E. M. F. applied to current produced : or in symbols — where E = electromotive force, C = current, B — resistance. The resistance of a wire is therefore to be measured in volts per ampere ; this unit is called the ohm. Sometimes it is more convenient to use, instead of resistance, its reciprocal, i. e. the ratio of current to E. M. F. This is called the conductance, and is measured in amperes per volt, or mhos. For example, a sixteen-candle power electric lamp adapted for use with a public supply of electricity at no volts may take 0-5 ampere. Accordingly its resistance is no -r 05 = 220 ohms, or its conductance 5J5 mho. 20O ELECTRIC CURRENTS The resistance of an ordinary cylindrical conductor, or wire, is found experimentally to be proportional to the length, inversely to the area of cross section, and to depend very much on the nature of the material it is made of. The latter influence may be described by taking as standard a length of i cm. of a con- ductor whose cross section is i sq. cm. ; the resistance of this is called the resistivity (or specific resistance) of the material. Hence we may put— a Here p = resistivity, I = length, a = area of cross section ; e. g. if it be known that the resistivity of pure copper is i-6 x io- 6 ohms, it is desired to find the resistance of a No. 24 copper wire 20 metres long. Here I = 2,000 cms., a = 0-00245 sq. cms., according to the wire gauge table, so that E = i-6 x io -6 x 2000 -f- 0-00245 = Il 3^ ohm. Similarly the conductance of a specimen 1 cm. long and 1 sq. cm. in cross section is called the conductivity, so that the con- ductance of a wire (or tube of electrolyte) = conductivity x cross section -5- length. The resistivity of all substances depends on temperature. Accordingly in the following table the temperature coefficient is given ; this means the increase of resistance per 1° expressed as a fraction of tJie resistance at 0°. Resistivity at Temperature o° (ohms). coefficient. Silver . . 1-47 x io -8 -1- 0-00400 Copper , . 1-56 x io -6 00428 Aluminium , . . 2-66 x io -5 00043s Platinum . , . . 10-92 X IO -5 0-00367 0-00625 German silver a Hoy . . 20. X IO - ' 0-00044 Platinum silver alloy . 24- X IO -8 0-00031 Manganin alloy . . 42- X IO -0 negligible Mercury . . 94-07 X IO" 8 0-00088 Graphite and electric \ 1 2400 x io -8 light carbons j and upwards — 0*0005 It will be observed that all the metals increase in resistance with rise of temperature, and further that the various pure metals have mostly temperature coefficients of about 0-00366 = ?$%, which is the coefficient of expansion of a gas. That is, the resis- tance of a pure metal varies in nearly the same way as the volume of a gas when the temperature is altered, and accordingly RESISTANCE 201 would vanish, or at least become very small at the absolute zero. Carbon, on the other hand, diminishes in resistance when the temperature rises, so that the resistance of an incandescent lamp is much less when hot than when cold. Electrolytes have all much greater resistance than metals ; one of the best conducting is strong nitric acid, but the resistivity even of this is of the order of 1 ohm, i. e. roughly a million times greater than copper. As a general rule the conductivity of solu- tions goes hand in hand with the amount of dissolved substance they contain, so that the weaker a solution is made the greater its resistance is. 06% salt solution, which is commonly taken to correspond in salt content with the blood, has a resistivity of about 80 ohms. Most of the tissues have probably a higher resistance than this. Absolutely pure water is almost a non-conductor, having a resistivity of about 25,000,000 ohms. Almost all the electro- lytes conduct better hot than cold, the rate of change being very rapid, usually 0-02 per degree. Even the value for water is however far exceeded by the actual insulators— glass, silk, gutta-percha, and so on— e.g. the resis- tivity of mica has been estimated at io u ohms. The difference between this value and those for metals is so enormous that a No. 24 copper wire stretching all round the earth would have less resistance than a thin mica wad, so that the current would rather travel by that route, 40,000 kilometres long, than from one face of the mica to the other.- It is this fact that renders elec- trical energy so superior to any other kind in practical availability, for the current conveying the energy can be guided by wires to any point desired, however distant, without excessive leakage ; it is thus possible to drive an electromotor by means of current generated a hundred kilometres away ; and the small currents needed for telegraphic signalling can even be carried in practice across the ocean. It is easy to construct permanent standards of electrical resis- tance, since the resistance of a wire (at least when properly annealed) remains quite constant but for temperature variations. To avoid the influence of these last, such standards are made of some alloy whose temperature coefficient is small, usually now- adays of manganin. Coils of resistance equal to 1, 2, 3, 10, 100, 1,000, or any desired number of ohms are made ; and usually these 202 ELECTRIC CURRENTS are made up into resistance boxes, arranged as shown in Fig. 6i (a plan, 6 elevation). Here pqbst are brass blocks, to which are soldered the ends of a coil, a well-fitting conical brass plug fills up the space between the blocks, and the current flows through this with no appreciable resistance ; but when the plug is removed the current can only flow through the coil, and so much resistance is thrown into the circuit. The coils are made in sets which can be combined so as to give any desired total, like a set of weights. For very large resistances it is more convenient to use. either carbon or an electrolyte. Megohm (1,000,000 ohms) standards are made consisting of a streak of graphite (i. e. a lead-pencil mark) on an ebonite plate. Liquid resistances of 1,000 ohms or more otcz)cz^czz>a W (b) Fig. 61. can easily be constructed by filling a glass tube with zinc sulphate solution, and fitting it with electrodes of pure amal- gamated zinc, since such electrodes do not polarize appre- ciably ; a liquid resistance would however be inconstant if it polarized, for the reasons discussed in the last section, unless it were used with an alternating current (i. e. one that oscillates to and fro instead of flowing always in the same direction). Besides standards of resistance for purposes of measurement it is desirable to have variable resistances, by which to adjust the strength of the current in a circuit. For weak currents (fa ampere and less) resistance boxes may be used, but it is simpler to have a wire of variable length, such as that of the apparatus (rheostat) RESISTANCE 20$ Fig. 62. shown in Fig. 62. Here the wire is wound spirally on a drum, the ends being con- nected to the bind- ing screws a, e. The third screw c is con- nected to a brass spring d, which slides along a bar. Accord- ing to the position of the sliding contact, more or less of the wire is interposed electrically between a and c. Another piece of apparatus for this purpose is the "Varley rheostat (Fig. 62 a), consisting of a pile of loose carbon discs ; when these are pressed closer together by a screw their resistance diminishes. For very small currents, such as those occurring in nerve-muscle work, it is convenient to use liquid resistances made as described above, but with one of the electrodes fixed on a rod so that it can be pushed in and out of the tube. The longer the dis- tance between the electrodes is made the greater, of course, is the resistance offered by the liquid. An actual electric circuit is made up of various conductors— battery, connect- ing wires, galvanometer, resistance coils, and what not, and in the simplest case the current flows through all these in turn. Since Ohm's law is applicable to each of them, it is applicable to the circuit as a whole, and to get the current flowing we have only to divide the electromotive force by the total resistance in the circuit. Further, in accordance with Ohm's law the electromotive force is spent partly on each conductor, and in proportion to the resistance overcome. This may be elu- cidated by an actual example. Thus suppose (Fig. 63) an accu- mulator bb' of 2-05 volts, joined up to a small lamp l of 3-6 ohms resistance, an ammeter a of 0-3 ohm, and let the internal resistance of the cell (i. e. the resistance from terminal to ter- minal, mainly made up of that of the electrolyte between the plates) be 02 ohm, that of the conducting wires negligible. The Fig. 62 a. 204 ELECTRIC CURRENTS total resistance is 3-6 + 0-3 + 02 = 4-1, hence the current is 2 05 -7- 4-1 = 0-5 ampere. If now a voltmeter be attached to the ends of the lamp, it will indicate that part of the E. M. F. used to drive the current through the lamp alone, viz. 0-5 ampere * 3-6 ohms=i-8 volts ; if it be attached to the ends of the ammeter it will read, similarly, 0-5 x 0-3=0-15 volt. If how- 3?J S! ever it be attached to the terminals of the cell it / ; \ will Tead 1-95 volts, for the current in flowing through the cell would require 05 x 0-2=0-1 volt to drive it, and on that account the terminal e' (the lead plate) where the current enters would be o-i volt above b (the peroxide plate) ; but the cell is itself the source of electromotive force, so that in passing from the lead to the peroxide plate there would on account of chemical action be a rise of 2-05 volts ; hence on the whole b is higher than b' by 1-95 volts, and b is the positive terminal. Of course if no current were flowing from the cell a voltmeter attached to it would read 205 volts. The point here involved is very clearly brought out by com- paring a voltaic cell (Cu Zn, Fig. 64) with an electrolytic cell (Cu Cu, Fig. 65). When currents flow through these there is in Li Fig. 63. - - ' v ■ — Fig. 64. Fig. 65. each a drop in volts in passing from the point where the current enters (anode) to that at which it leaves (cathode) on account of resistance overcome ; in the electrolytic cell there is nothing to compensate this, so the anode is + Te . But in the voltaic cell there is an electromotive force acting in the direction in which the current is flowing (from Zn to Cu through the cell), which causes the cathode (Cu) to be the + ve . To drive a current through an electrolytic cell needs a certain RESISTANCE 205 voltage on account of the resistance of the solution, and when there is no chemical work to be done that is all ; i. e. when copper sulphate is electrolyzed between copper plates, just as much copper is dissolved at one pole as is deposited at the other, so that no chemical work is done. But when that condi- tion does not hold a further voltage is required. Whenever polarization occurs this is the case, e. g. when dilute sulphuric acid is decomposed between platinum electrodes ; here work is done in decomposing the solution and generating oxygen and hydrogen, and it is found that about 1-7 volts has to be spent to do this, in addition to anything required to merely overcome the resistance of this electrolytic cell. The amount thus needed, which may be called the back electromotive force of polarization, can be approximately determined, like the E. M. F. of a voltaic cell, by dividing the chemical energy involved in the reaction by the quantity of electricity which, according to Faraday's laws, is con- veyed : only here the chemical work is done by the current, whereas in the voltaic cell the current is produced at the expense of chemical energy. An electrolytic cell is therefore, in a sense, the converse of a voltaic cell. Indeed the same piece of apparatus may serve for both. Thus a lead accumulator when charged is a voltaic cell, and is capable of giving current at the expense of its store of chemical energy ; but when run down it has to be treated like an electrolytic cell, and a current run against its own E. M. F. so as to ' charge ' it again, i. e. restore the chemical energy which it has lost. Here the peroxide plate is in both cases positive to the other, but it acts as cathode during the discharge, anode during the charge. We arrive then at the following results as to distribution of energy in electric circuits : — (i) The total electromotive force E in a circuit is spent (a) in overcoming any back electromotive force E' due to polarization that may exist in the circuit, and (&) the remainder of it in driving a current C against the resistance B of the circuit, so that E=E' +CB If there is no back electromotive force E becomes — CB. (ii) Bearing in mind that electromotive force means power -5- current (p. 193) it follows that the total electrical power (or activity) A in a circuit is spent (a) in effecting chemical decom- 206 ELECTRIC CURRENTS position against the electromotive force of polarization, the power spent being E'G ; (b) in producing heat in the circuit at the rate CR x C, so that a - EC + cm Here the power will be given in watts if the ampere, volt, and ohm are the units employed. (hi) The work W done is of course the power multiplied by the time U), so that W=E'Ct + C 2 Rt where the first term on the right-hand side is the work done in .chemical decomposition, and the second the heat produced in the circuit ; if the time is given in seconds both these will be given in joules. To calculate the heat in calories the number must be divided by 4-2. It may be added that an electromotor (vid. inf. p. 238) behaves like an electrolytic cell in giving a back electromotive force. Hence in the preceding equations E' may also be taken to mean that, and E'G will be the amount of electrical power spent in doing mechanical work by means of the motor. Example : A motor whose resistance is 6 ohms is connected to a supply of electricity at no volts, and it is observed that 2 amperes flow through it. Consequently the voltage used in overcoming resistance is (CR) 2 x 6=12, leaving 110-12=98 volts for actually doing work by the motor (E'). Hence further, the power obtained from the motor (neglecting friction and other mechanical imperfections) is 98 x 2=196 watts ; the power spent in heating is (C 2 R) 2 x 2 x 6=24 watts, and the total power taken from the supply (A) is 196 + 24 = 220 watts. Again, the heat pro- duced per hour (3,600 seconds) is 24 x 3600 = 86400 joules or 86400 -j- 4-2 = 20570 calories. § 4. Distribution of currents. We have so far considered only the simplest kind of circuit, one in which all the current flows in turn through all the con- ductors. But it often happens in practice that branches and networks of conductors exist, so that the simplest formulation of Ohm's law is not applicable. We have to consider in this case how the current in each branch may be calculated. All such cases may be reduced, so far as the resistance is concerned, to Fig. DISTRIBUTION OF CURRENTS 207 two : the conductors are arranged either in series or in parallel, or it may be some combination of the two. Series means that the same current flows through each of them in turn (Fig. 66, a), (a,) — ^-*www^— vwwwwJL- parallel that the conductors offer alternative paths to the ^ vwwwwa ^ current (Fig. 66, I). In order W '^T^^^^S^ - ^ to determine the joint resis- tance, that is the total resis- tance from a to b or from c to d in the figure, we have the following rules : — (a) ' The resistance of any number of conductors in series is the sum of their separate resistances.' This almost obvious rule was assumed in the course of the preceding section. (&) ' The conductance of any number of conductors in parallel is the sum of their separate conductances.' Thus ifp q r be the resistances of the separate conductors, their conductances are - - — ; the total conductance is therefore p q r 111 1 « + 7. + z.i an d the total resistance p q r 111 - + -+- p q r One of the most familiar cases of putting conductors in parallel is in the ordinary way of wiring a house for electric lighting. This is to place each lamp independently across the mains, i. e. from the main conductor leading from the positive end of the dynamo to the other main conductor leading to the negative end. All these lamps are in parallel to one another, therefore ; suppose a supply at no volts, and that 18 lamps each of 200 ohms, one large lamp of 50 ohms, and a heater of 20 ohms are put in. Then the conductances are, each small lamp ^5=0-005 mho, hence the 18 small lamps are 18 x 0-005=0-09 mho, the large lamp 0-02 mho, the heater 0-05 mho, making a total of 0-09 + 0-02 + 005 = 0-16 mho. The total resistance when all are switched on is thus 1 -r- 0-16 = 6-66 ohms, and the total current =E. M. F. -?- resistance, or =E. M. F. x conductance, i. e. no x 0-16 = 17-6 amperes. Another common case is a galvanometer and its shunt. When a galvanometer is too sensitive for the purpose in hand a wire 208 ELECTRIC CURRENTS resistance is arranged in parallel with it, as shown in Pig. 67 ; then only part of the current flows through it. If g and s be the resistance of the galvanometer and shunt respectively, their I SGI- combined resistance is - — - = — — • Usually g is very much II S + G S G greater than s ; in that case the combined resistance is practically that of the shunt alone (i.e. the conductance of the galvanometer -_ can be neglected by comparison with that l—\ G of the shunt). When conductors are put in parallel there is evidently the same voltage applied to each, seeing that they connect the same points (e. g. c to d in Fig. 66) ; hence in Fig. 67. accordance with Ohm's law the current flowing through each must be proportional to its conductance. Thus the currents through galvanometer and shunt respectively, and the total, will be galv. current : shunt current : total current ::—:-:- + -. G S G S Hence the fraction of the total current flowing through the galvanometer is — -t-(- + -) = — — Usually the shunt is made to have a resistance \, or jfo, or -^ of that of the galvanometer, i. e. 9, 99, or 999 times its conductance, and accordingly -fa, xJ^, or j^jtj of the total current flows through the galvanometer according to the shunt used. A ' short circuit ' is a shunt of very low resistance, so that when a piece of apparatus is short-circuited almost none of the current flows through it, and it is practically cut out of use, except in the case of the battery. The current of course all comes from it in any ease, and a short circuit merely makes it very large (and probably damages the battery). It sometimes happens that the voltaic cells themselves have to be arranged in parallel or in series ; then their internal resis- tances can be combined according to the preceding rules just like any other resistances ; but the electromotive forces require separate treatment. Three arrangements may be considered, (a) Cells are placed in series, i. e. the zinc of the first is connected to the copper or carbon of the second, the zinc of that to the DISTRIBUTION OF CURRENTS 209 copper or carbon of the third, and so on ; then the electromotive force is simply the sum of the separate electromotive forces. Example : one Daniell cell of 1-08 volts is found incapable of electrolyzing water ; 1 1— 1 1 two such are put in series, so giving 2-16 volts, and the electrolysis is found Fig. 68, to take place (Fig. 68). (&) Two cells are said to be in opposition when they are arranged as in series, except that one of them is turned the other way round. The total electromotive force is now the difference between the two (Fig. 69). Example : in order to charge a battery of 40 accumulators they are connected in series, and then put in opposition to a — 1 — 1 1 7 — dynamo of no volts. If during charging each cell has an E. M. F. of 2-45 volts, Fig. 69. the total E. M. F. of the battery is 40 x 2-45 = 98 ; hence the total E. M. F. in the circuit is 110 — 98 = 12 volts. It is this that is effective in driving the current. (c) Cells are arranged in parallel (Fig. 70) by joining together all their positive terminals and all their negative terminals, and attaching these joint electrodes to the external circuit. This should only be done when all the cells are alike, and so have the same electromotive force. In this case the combined E. M. F. is merely that of a single cell ; in fact the group practically forms one cell with bigger j> IQ _ 70, electrodes. The only effect, then, as com- pared with a single cell, is to reduce the internal resistance ; but sometimes that ia the most effective way of increasing the current in a circuit. So far we have paid no special attention to the internal resis- tance of a cell, as it is constituted in the same way as the resistance of any other conductor, and depends on the same factors. But it plays such an important part in determining the efficiency of a battery as to deserve some separate mention. To take a practical example, let us suppose two dry cells, each of 1-4 volts, and 3 ohms internal resistance. Either cell by itself is capable of giving as a maximum 1-4 -J- 3 = 0-47 amperes, for if it were short-circuited, i. e. the poles connected by a wire of negligible p 210 ELECTRIC CURRENTS resistance, then the total resistance in the circuit would be the internal resistance only, and the current is found, according to Ohm's law, by dividing this into the electromotive force. If the two cells were put in series and short-circuited, the total E. M. F. would be 2-8 volts, and total resistance 6 ohms, and the current would be 2-8 -=- 6 = 0-47, the same as before ; but if the two were put in parallel, while the E. M. F. would remain 1-4 volts, the joint resistance would be reduced to 1-5 ohms, and the current 1-4 -s- 1-5 = P-94 ampere, would be doubled. Hence we see that if it is desired to get the largest current through a very small external resistance it is best to put the cells in parallel. On the other hand, if the external resistance is very large, the rule is reversed. Suppose it to be 1,000 ohms, then we may practically neglect the internal resistance by comparison with this, and we find that one cell would give 1-4 -e- 1000 = 0-0014 ampere, and any number of cells in parallel would only give the same ; but the two cells in series would give 2-8 -=- 1000 = 0-0028, L-er-4;wice as much current. Thus, when the external resistance is very large, the cells should be put in series. Accumulators have very low internal resistance : a good-sized one may have jfoj ohm or less ; consequently they are capable of giving very strong currents. The extreme contrary case is to be found in a nerve-muscle preparation ; this acts as a voltaic cell, giving an electromotive force of about o-i volt; but it has a very high internal resistance— 100,000 ohms possibly. Hence, any ordinary electric circuit, even a galvanometer of 10,000 ohms, acts practically as a short circuit to it, and all that has been said of short circuits is applicable ; it makes, in fact, very little difference what path is provided for the muscle-current, the amount obtained will be practically the same if any ordinary wire circuit at all is used. § 5. Methods of measurement. Current. Of the quantities we have so far discussed, one, the intensity of an electric current, is commonly measured directly by an appropriate instrument based on one of the effects it pro- duces; whereas electromotive force and resistance are usually found by means of more complex and indirect processes, essen- tially involving some current-measuring device ; it is necessary, therefore, first to deal with current measurement. METHODS OF MEASUREMENT 211 The principles on which such instruments can be con- structed were referred to in general terms in § 1, and it appears that they fall under two main heads— thermal and magnetic effects. The use of the former is, however, restricted to a few electrical engineering instruments, so that we may say the only current measures with which we have to deal belong to the class of galvanometers, using that word to cover any instrument in which magnetic effects are used for the purpose in question. To give the precise theory of any galvanometer would be anticipating the subsequent chapter on electromagnetics : it will be sufficient here to mention points of theory only so far as they are indis- pensable in going over the practical construction of galvano- meters. Three groups of instruments may be made, according as the instrument contains (i) a fixed coil of wire through which the current flows, and in consequence tends to rotate a magnet, Fig. 71. either suspended or pivoted so as to be free to move ; (ii) a sus- pended or pivoted coil through which the current flows, and which, placed between the poles of a magnet, tends in consequence to rotate ; (iii) a fixed coil and a movable coil, through both of which the current is led. For measuring currents of intensity down to about one milli- ampere (njfor ampere) direct reading instruments based on these various plans are in common use. Designed primarily for en- gineering purposes, such ' ammeters ' usually possess an accuracy of the order of one per cent., and can be used for physical experi- ments except when the highest order of exactness is required ;' in physiological experiments always, when such large currents are to be measured. There are, moreover, instruments of a more exact construction, intended mainly for standardizing others, . among which may be mentioned Lord Kelvin's ' current balances '; these (Fig. 71) consist essentially of four fixed coils aaaa, and P2 213 ELECTRIC CURRENTS two movable ones bb, the current being led through all in turn. The two movable coils are attached to a frame supported at its centre so as to be free to turn like a balance. Of the two fixed coils on the right one attracts the movable coil near it, the other repels, so that both tend to raise the movable coil which lies between them ; similarly, the two at the left both tend to lower the movable coil which lies between them. Hence, all four coils tend to tilt the beam in the same direction ; to balance this a small weight is hung on the beam and slid along it (as in a steel- yard) till the beam lies exactly horizontal. The moment of the weight then serves to measure the current flowing. This instru- ment, though depending on what may by analogy be called the magnetic attractions and repulsions of electric currents, does not involve any actual steel magnets, and is consequently not exposed to errors due to accidental weakening of the magnet, like most direct reading ammeters. It must be noted that, like all other instruments in which the current passes through both a fixed and a movable coil, the scale reading is proportional to the square of the current, for doubling the current doubles the force exerted on account of the fixed and movable coils separately, making it on the whole four times as great. In such an instrument the motion is always one way, whichever way the current flows, but that is for some purposes an advantage, because it allows of measuring an alternating current, i. e. not a steady flow of electricity in one direction, but an oscillation of electricity to and fro along a conductor. Amongst instruments for measuring comparatively large currents may be mentioned, on account of its theoretical simpli- city, the tangent galvanometer. This consists of a circular coil of wire fixed on a frame. At the centre of the coil, which is commonly made from 10 to 20 cms. in diameter, is fixed a small horizontal graduated circle ; here a short compass needle, provided with a long, light pointer of aluminium or glass, is pivoted. The needle being free to move in a horizontal plane sets north and south ; but when a current is led through the coil there is a tendency for the needle to set itself at right angles to the coil. Accordingly, to use the instrument the coil is rotated till it is parallel to the needle (i. e. in the magnetic north and south line) ; then the current exerts a couple on the needle, tending to make it lie east and west, and this, together with the couple due to the METHODS OF MEASUREMENT 213 earth's magnetic action, will cause the needle to rest in an inter- mediate position, say at an angle 8 with the north and south line. Then it may be shown that the current C = Jc tan 8, where h is a constant depending on the construction of the instrument and the strength of the earth's magnetic action. Such galvanometers have been much used, but they are liable to disturbance by magnets and masses of iron (gas pipes, steel girders, &c.) in their neighbourhood ; and as moreover they have to be adjusted to the N. and S. position, and require some arithmetic to deduce the strength of current from their readings, they are neither so convenient nor so accurate as the commercial ammeters referred to above. Tor currents from about a milliampere downwards reflecting galvanometers are always used. The essential point here is that in order to obtain very great sensitiveness, a beam of light is used instead of an ordinary pointer. The usual indicating arrange- ment is shown in Fig. 72 (plan). The moving part of the galvanometer a, whether magnet or coil, carries a very light mirror b ; a lamp c placed behind a narrow slit in a screen Fig. 72. throws a beam of light on to the mirror, from which it is reflected to form an image of the slit at d on a scale usually fixed to the screen. The image may be formed either by using a concave mirror, or by causing the light to converge from a convex lens. When electric light is available the filament of an incandescent lamp may itself be focussed on the scale, a more brilliant image being so obtained. "When no current is flowing through the instrument, and the mirror is therefore at its normal position, the light is reflected to the middle point of the scale d ; if now a current produces a small deflection of the moving part of the instrument so that the mirror makes an angle 8 with its former position, the light will be reflected in a direction be, making twice that angle with the incident beam bc. The scale may be of paper, observed — j.— E ®c *«q 214 ELECTRIC CURRENTS from the side of the galvanometer, or of ground glass or other translucent material, and read from behind. For the most exact observations it is preferable to replace the lamp and slit by a telescope, and, taking care that the scale is in a good light, observe the point on the scale which appears by reflection in the mirror. The electrical construction of a mirror galvanometer is essentially the same as that of the instruments already referred to, and con- sists either of a fixed coil and movable magnet on the plan of the tangent galvanometer (Thomson pattern) or of a coil suspended between the poles of a strong magnet (D'Arsonval pattern). In the former, which until recently at any rate was regarded as the customary type, the magnet is a little bit of steel, perhaps a centi- metre long, pasted on the back of the mirror, the whole being suspended by a fine fibre of silk or quartz, so as to offer as little resistance as possible to turning. Two coils are commonly used, placed close together in front and behind the magnet ; but in order to increase the sensitiveness still further, an astatic magnet system can be used. This consists of a pair of magnetic needles fixed horizontally at a convenient distance apart on a stiff vertical axis (an aluminium wire) ; the two needles are parallel, but their poles point in opposite directions, and as they are made of about equal strength, the actions of the earth on the two nearly neutralize, and the combined system has hardly any tendency to set itself one way or the other. The two needles are arranged so as each to lie at the centres of a pair of coils ; the astatic system is consequently very easily turned a measurable amount from its position of rest by a current flowing through the coils. As it would be inconvenient to rotate the coils of such a galvano- meter in order to bring them parallel to the needle, it is customary to provide a control magnet, or a pair of such, fixed on the stem of the instrument, so that by adjusting them the suspended needle may be turned to suit the position of the coils instead. By combining the various artifices that contribute to the sensitiveness of the instrument— large number of turns of wire, astatic needle, delicate suspension, mirror method of observation — it has been found possible to make galvanometers that will indicate a current as small as io -10 ampere, or even less. Such a current would have to flow for three centuries to transfer one METHODS OF MEASUREMENT 215 coulomb of electricity, and so (according to Faraday's laws) suffice to liberate about a tenth of a c.c. of hydrogen. For physiological observations a galvanometer of this type is the most suitable, and it may with advantage be wound with very fine wire so as to have as many turns as possible in the coils. This increases the sensitiveness and so allows of measuring smaller currents-; it also increases the resistance of the instru- ment, but, as we have seen, the internal resistance of the muscle or nerve producing the current is so large that the addition of even io,ood ohms in the galvanometer does not much matter. The D'Arsonval pattern consists of a coil of fine wire, some- times circular, sometimes elongated ; in either case suspended between the poles of a strong horse-shoe, or c-shaped, magnet. The suspension must be as delicate as possible ; it is not so easy to arrange as in the other pattern, since the current has to be led into and out of the coil by the suspending wires ; but satisfactory results have been obtained by using a pair of very fine strips of phosphor-bronze close side by side. The coil then offers very little resistance to rotation about the axis of suspension. When a current flows through it, its tendency is to lie with the plane of the coils crossways to the line joining the north and south poles of the magnet ; it is therefore necessary to set up the galvanometer with its coils parallel to that line, in order that the least tendency to turn may be observed. A mirror is attached to the coil, and the motion observed with a lamp and scale or telescope and scale in the usual way. A sensitiveness of about 10- 8 amperes per scale division is usual. The ad- vantages of such galvanometers depend on the fact that the moving part is placed in a strong magnetic field ; magnetic changes in the neighbourhood are therefore practically without influence on them, and the instrument may be used even in a dynamo-room, where a Thomson galvanometer would be set in continual fluctuation by the moving masses of iron. As a consequence of this, the sensitiveness, i. e. the current required to produce one scale division deflection, can be determined once for all, and the galvanometer may afterwards be used to measure the actual strength of current in fractions of an ampere. The current is sensibly proportional to the deflection produced on the scale. 216 ELECTRIC CURRENTS Electromotive force. There exist, also, instruments for the direct measurement of electromotive force— those of an engineer- ing type being usually called voltmeters. When an electromotive force of moderate amount (say T V volt and upwards) produced by a battery or dynamo is _to_ be measured with a moderate degree of accuracy a commercial voltmeter with a direct reading scale is appropriate. Such instruments are usually ammeters provided with a resistance coil in series with the working coil ; e.g. suppose an instrument be provided of which each scale division corresponds to a current of o-ooi ampere, the resistance of the working coil will probably be a few ohms merely. Let a coil of manganin or other suitable material be put in series with it, making the total resistance 100 ohms ; then ae- cording to Ohm's law the voltage applied to the whole will be O'ooi x ioo = o-i volt per scale division. The instrument with the series coil thus added constitutes a voltmeter, which may be used, e. g., to test the voltage of an accumulator. But such an instrument is only available when the internal resistance of the arrangement producing the electromotive force is negligible. Even a Leclanche cell could not be tested very well with it, for if the resistance of the cell were, say, 3 ohms, the current produced through the voltmeter would not be E. M. F. -=- 100, but E. M. F. -7- 103 ohms, i. e. 3 % less ; in other words, while the instrument would indicate correctly the voltage applied to it, that will not be the same as the total voltage of the cell, as explained at the end of the last section. If an arrangement with a high internal resistance, such as a muscle-nerve preparation, were tested in this way, the method would break down altogether and the voltmeter indicate practically nothing. If therefore a direct reading instrument is to be used as a voltmeter when the in- ternal resistance is high, it must be one that does not consume any current. These are known by the name of electrometers, and are of two types : the ' static ' electrometer, usually of what is known as the ' quadrant ' (see p. 255) construction, and the capillary electrometer. The capillary electrometer, due originally to Lippmann, depends on the fact that the surface tension between mercury and an electrolyte such as dilute sulphuric acid, is altered when there is a difference of potential between the two. The usual form of the instrument is shown in Fig. 73. A glass tube of METHODS OF MEASUREMENT 217 about 1 mm. internal diameter is drawn out to a very fine bore ; this is supported vertically and surmounted by a long glass tube (30-40 cms.), the two being connected by rubber tubing provided with a screw clip. The finely drawn tube dips into a glass cylinder. Mercury is poured on to the bottom of this, and also fills the tube down to a point in the capillary part ; the rest of the capillary and the cylinder being occupied by 10 % sulphuric acid. The mercury cannot however flow out of the tube on account of the resistance offered by its surface tension in the very fine bore ; the pressure has in fact to be adjusted by altering the height of mercury in the tube (for which purpose the tap at the side is convenient) till the acid-mercury surface is brought to the required point. Electrodes are sealed through the glass to make contact with the mercury in the cylinder and tube respectively; if now an electro- motive force be applied by means of these the mercury will move up or down in the capillary. The instrument is mostly used only for null purposes^ i. e. to indicate when there is no potential difference be- tween two points, by the absence of movement of the mercury meniscus. The movements are observed by means of a microscope, and o-oooi volt can be detected. When- it is desired to measure an electromotive force with accuracy it is best to use an indirect method, that known as the potentiometer, in which a galvanometer or electrometer is em- ployed, but only as a null instrument, not to give direct readings by means of its deflection. The connexions are shown in Pig. 74 diagrammatically ; 1m is a thin wire of uniform gauge, commonly a metre long, stretched over a scale ; since it is exposed to air, and it is necessary to make contact with it at any point, a hard platinum alloy is the best material to use. Through Fig. 73. 2l3 ELECTRIC CURRENTS this wire a constant current flows from the battery a ; this should consist of an accumulator, as it yields more steady currents than any primary cell ; the E.M.F. of A must be greater than that to be measured, so that if the latter is higher than 2 volts, two or more accumulators must be put in series. Since a current is flowing through Im and the resistance of this wire is uniform along its length, the potential falls off uniformly from I (connected to the + ve terminal of a) towards m, the latter point being about two volts lower than the former. If then the source of electromotive force b to be tested be connected with its positive pole to I, there must be some point be- tween I and m which is at the same potential as its negative pole ; e. g. sup- pose b to be i -a volts, then the point in ques- tion, x, will lie about three-fifths of the way from I to m. If therefore the negative pole of b be connected to x no current will flow through it ; if it be connected to a point on the graduated wire nearer to I current will flow from b, if to a point nearer to m current will flow into b from the accumulator. Thus if x is a sliding contact, and the galvanometer or Lippmann electro- meter c be inserted in the circuit, the point of balance can easily be found. The length Ix then measures the E. M. F. of b. This, however, is on an arbitrary scale ; in order to know the actual voltage the experiment must be repeated on a cell of known voltage, such as a Clark c. The key Jc allows of putting either b or c into circuit at will ; if the balancing point for the latter be x', we have E. M. F. of b _ length 7a? E.M.F. ofc length Ix" Measurement of a resistance, like that of an electromotive force, is essentially a comparison with a standard, and is always effected by an indirect method ; numerous arrangements of apparatus have been designed for the purpose, but we need only describe two. (i) It follows immediately from the definition METHODS OF MEASUREMENT 219 of resistance, that if the "E. M. F. applied to a conductor be measured by a voltmeter, and the current by an ammeter, the resistance will be given by dividing the reading of the former by that of the latter. The arrangement is shown in Fig. 75, where a is the ammeter, v the voltmeter, and e the resistance to be tested, b the source of current, k a key ; such a method is suitable for determin- ing the resistance of an incandescent lamp, or any object through which a moderately large current can be led. It must be remarked however that, according to the arrangement shown, any current flowing through the volt- meter must also flow through the am- meter, and so ought to be deducted from the total reading of the ammeter. The voltmeter current is often small enough to neglect, and is zero in a static instrument ; but if not very small it should be determined independently in order to apply the above-mentioned correction. Then the volts applied -7- current (corrected) produced = resistance in ohms. When a veiy high resistance is to be measured— such as a million ohms or more— a modification of this method is available, in which an ordinary mirror galvanometer is sub- stituted for the ammeter, on account of the smallness of the currents involved, and the voltmeter is dispensed with alto- gether, the current being taken from a battery whose voltage is approximately known. The most important method for measuring resistances is however the process of compari- son known as Wheatstone's bridge. A diagram of this is given in Fig. 76. The current from the battery b (usually a single Le- clanche cell) divides at a into two branches, pq and es, to join again at c and return to the negative pole. Two points bd in these branches are connected together through the galvanometer G. If then p bears the same ratio to // the accumulator will suffer from the excessive current taken out of it on short-circuiting. It is often desirable to have a rapid series of induction shocks. This is accomplished by an automatic interrupter, analogous to that of the electromagnetic tuning- fork (p. 4). This some- times forms a separate piece of apparatus, such as Neef's hammer, shown in Fig. 89. The primary current is led in by a ter- minal on the brass pillar a which carries a steel spring, goes thence through the screw Sj to the induction V£v^_ jll Jv ^y K coil pc ; s x is screwed down till it just touches the spring, and to ensure the contacts against rusting they are made of platinum. From the coil, the current passes round a small electromagnet e and thence by the terminal back to the battery. The electromagnet is arranged so as to attract the Fig. 89. 246 ELECTRO-MAGNETISM armature of the spring v. It does so as soon as the current passes, but in doing so it breaks the circuit and the current stops : the electromagnet then ceases to act, the spring rises again, and the circuit is remade. Thus make and break succeed each other automatically, with a rapidity that depends only on the frequency of vibration of the spring. For Helmholtz's modification it is only necessary to raise the screw s, out of the way, and raise the screw s,, so as to make the vibrating contact through it ; and at the same time make a permanent wire connexion from Si to the battery as shown. Instead of an independent electromagnet the iron armature of the induction coil itself may be used to actuate an interrupter. § 5. Alternating currents. If an earth coil is rotated at a uniform rate, the magnetic induction through it will alternately increase and decrease, coming back every revolution to the same value. Consequently currents 'will be induced in it, first one way, then the other, in the course of one revolution, and so on indefinitely. Such currents are called alternating ; they consist in a surging to and fro of electricity, instead of a continuous flow in one direction. In the particular case considered, the alternations, or oscilla- tions, of the electricity would be of the kind known as simple harmonic Fig. 89 a (i). (P- 33)- Fig. 89 a (i) may serve as a geometrical representation of an alternating current, the horizontal distances being times, the vertical current strengths, the current in one sense being called positive, in the other negative. The earth coil is essentially similar to a single armature segment in a dynamo, although, to increase the efficiency, a magnetic field far stronger than that of the earth is used in the latter. Hence in each segment of the armature an alternating current naturally arises. If the ends of the segment are led to two separate metal rings, and brushes be allowed to rest on these, ALTERNATING CURRENTS £247 so that they make a conducting connexion with the armature throughout the rotation, the alternating current may be led away to any external circuit : it is in this way that an alternating dynamo is constructed. Current from such dynamos is supplied in many places for house lighting, instead of direct current : the frequency is commonly 100 complete alternations or cycles per second. In a dynamo designed to supply continuous current/ instead of the collector rings, it is necessary to provide a commutator, as mentioned in the last section, the function of which is to rectify the current, i. e. to automatically reverse the connexion between the armature and the external circuit every half period. The result of such a commuta- tion on a simple harmonic current is indicated by Tig. 89 a (ii), where alternate halves of the S. H. curves have been inverted. This makes the current flow always in the same sense, although it is far from uniform. In a practical direct-current dynamo there are many sections of the armature, the effects of which are superposed in such a way that when one is giving its maximum effect, another is giving the minimum, and so the total current taken from the brushes is approximately uniform in strength. If alternating current be supplied to the primary of an induction coil, an interrupter is needless, for, as the magnetic induotion due to that cur- Fig. 89 a (ii). rent will be constantly changing, and will pass through the secondary, alternating currents will be induced in the secondary too. Here there is no ' make ' and ' break,' but a continuous rise and fall : such induced currents are not, therefore, so effective in stimulation as the more abrupt ones produced by the ordinary method ; but the principle involved is extensively used for another purpose. The transformer is essentially an induction coil without an interrupter ; but the primary and secondary are interwound, in order to make sure that there is no 'magnetic leakage,' i. e. that all the magnetic induction proceeding from the one flows through the other — a condition only obtained in an induction coil of the pattern described above when the secondary is pushed home. Transformers are used to convert low voltages into high, and vice versa : approximately, the E. M. F.'s in the primary and secondary circuits are in the same ratio as the number of turns of wire in them. It is economical in practice to generate alternating current, for domestic use, at a high voltage— say 3,000— and then, as it would be dangerous to introduce so high a voltage into houses, transform it down to, say, 200 volts, by a trans- former placed near to each house supplied. In this case, the ratio of transformation being 10 : 1, there must be ten times as many turns in 248 ELECTRO-MAGNETISM the primary coil as in the secondary — a state of things the converse of that in the ordinary induction coil, where the secondary has by far the larger number, and consequently the instrument transforms ' up.' It must be remembered, however, that a transformer, like any other machine, is not capable of giving out more power than is put into it ; hence the power, measured by the product of the volts into the amperes, must be (ignoring the imperfections of the apparatus) the same on the primary and on the secondary side, whether the appliance is for trans- forming up or down. Thus, if a transformer receive on the primary side 2 amperes at 2,000 volts, i. e. 2 x 2000 = 4000 watts, and the ratio of transformation be 10 : 1, it should give off the secondary 20 amperes at 200 volts. A transformer is an obviously reversible appliance ; either side may be made the primary, at will. Hence the same transformer may be used to transform up or down. It has already been mentioned that alternating currents are required in order to measure the resistance of electrolytes (p. 221), the arrangement being a Wheatstone bridge, in which the secondary of an induction coil is substituted for the battery, and a telephone for the galvanometer. The coil used for this purpose should be an extremely small one, capable of being driven by a single Leclanche 1 cell, and should have a very high rate of interruption, 500 a second or thereabouts, as that is about the pitch of the human voice, for which telephones are adapted : this is secured by having a very light contact spring v, Fig. 88, not loaded. The core of the coil itself is used as electromagnet. CHAPTEE IX. ELECTEOSTATICS. § i. Forces between electric charges. If the poles of a battery be connected by wires to conductors of any shape— say a pair of parallel metal plates— near to one another, but not touching, of course no continuous current will flow, since the circuit is broken by an air-space ; but a momentary current will flow, producing a charge on the plates, i. e. a quantity of positive electricity on one, an equal quantity of negative on the other. An electrostatic stress is set up in the air-space between the plates, and, when the amount of this is suflicient to balance the electromotive force of the battery, the action stops. The process may be illustrated by the parallel of a circuit made up of materials of varying conductivity : suppose an accumulator connected by thick copper wires to an incandescent lamp : then the electromotive force of the accumulator is distributed in proportion to the resistance to be overcome, and, as that of the lamp is far greater than that of the wires, it is practically all thrown on to the lamp. Now for the lamp substitute an air- space, as above : then all the electromotive force is spent on the air-space, or in other words there is a difference of potential between the plates equal to that of the battery. This difference of potential produces a strain in the space ' between, which shows itself in the phenomenon of a charge on the plates. Just as in the magnetic case we had to consider the existence of a field of magnetic force between magnet poles, and were able to trace it out by means of lines of magnetic force, so here there is afield 1 It is not the air that is strained, for the phenomena are just as well observed in a vacuum : it is rather the ' ether,' the medium that is assumed to fill all space, and transmit light as well as electrical actions. 250 ELECTROSTATICS of electric force (or stress) between the charged plates, and we can trace it out by means of lines of force leading from the one to the other. These lines indicate the direction in which the strain lies, and necessarily reach across from the one conductor to the other, because a conductor is a material that cannot support an electric stress ; consequently where the line of force reaches a conductor it ends — the state of strain ceases— and it follows also that each positive charge of electricity must be associated with an equal. negative charge, for that is only to say that one cannot have one end to a line without having the other end. If the conductors on which the charge is located have any definite size and shape, the quantity of electricity stored up / / / / c±: / Fig. 90. on them will be proportional to the difference of potential applied, and the ratio rrs f — i — r^r is called the capacity rr difference of potential of those conductors. When the conductors are arranged so as to have a large capacity, they form what is called a condenser ; this is best attained by making the metallic surfaces large, and the air-space between them small, the capacity being propor- tional to the area, and inversely to the thickness of the layer of non-conductor. The stress that exists in the non-conductor, or dielectric as it is usually called in this connexion, is a tension along the lines of force, combined with a pressure across them. Thus in Fig. 90, which represents the electric field between a pair of FORCES BETWEEN ELECTRIC CHARGES plates, and Kg. 91, for a pair of charged balls, and similar figures, we are enabled by drawing the lines of force 1 to see at a glance what forces will act on the charged bodies. We must imagine that each line of force has a tendency to shorten up, like an elastic thread, but that ad- jacent lines push each other apart sideways. It is evi- dent that the posi- tive and negative charges on the plates or balls attract each other. On the other hand, if two conduc- tors, both charged with the same kind of electricity (e. g. two balls connected to the same pole of a battery), be placed near together, there is a repulsion between them (Fig. 92). The precise law of force is identical with that of magnetism, viz. the force exerted between two charged bodies is equal to the product of their charges divided by the square of the distance between Fig. 91 Fig. 92. them : when the charges are of the same sign the force is a repulsion; when of different signs, an attraction. This law 1 The lines of force are identical in arrangement with those of magnetic forces which would be produced if each quantity of electricity were replaced by an equal quantity of magnetism. 25 2 ELECTROSTATICS gives a definition of unit quantity of electricity, viz. that quantity which placed i cm. from a similar and equal quantity repels it with a force of i dyne. This is the C. Gr. S. electrostatic unit of quantity : that which we have previously used is either the electromagnetic unit of the C. Gr. S. system or the practical unit, the coulomb, which is ^ of the electromagnetic unit. The electrostatic unit is far smaller— 30,000,000,000 times smaller— than the electromagnetic unit. The quantities of electricity practically occurring in electrostatic experiments, e. g. in charging a condenser, are in fact excessively small compared with those that flow in currents of ordinary magnitude. On the other hand, the differences of potential required for electrostatic experiments, such as demonstrating the attractions and repulsions of charged bodies, are usually great— hundreds or thousands of volts. The connexion between the potential difference applied and the state of electric force or stress set up in a non-conducting space may be most conveniently illus- trated by means of the parallel plate condenser already referred to. It must be borne in mind that difference of potential or electromotive force (see the definition, p. 193) means the work done in carrying unit quantity of electricity from one point to another : if now the plates be put at a fixed distance apart and a fixed E. M. F. be applied to them, an electric force, or stress, is produced in the intervening space measured by the applied E. M. F. -T- distance apart. But the work done in moving from one point to another 4- distance is the measure of the mechanical force overcome. Hence, just as E. M. F. is the expression of the work done in transporting a unit of electricity, so the electric force (or stress, or intensity, or strength of field) is the expression of the mechanical force exerted on a unit of electricity. It is therefore a quantity precisely comparable with the magnetic force, which is the mechanical force exerted at a point on a unit of magnetism. Now, suppose, the distance between the plates to be doubled while the applied E. M. F. is kept the same : then the electric force between the plates is halved. But, as we have seen, the ' charges ' on the plates are merely the ends of the lines of force ; if the force is halved in intensity, that means that there are only half as many bines of force, and the charge must be halved in amount. Thus we arrive at the result that the charge on a parallel plate condenser, to which a given E. M. F. is applied, FORCES BETWEEN ELECTRIC CHARGES 253 varies inversely as its thickness. But the charge for unit E. M. F. is the quantity we have defined as the capacity of the condenser. So we may say that the capacity varies inversely as the thickness. Capacity in electrostatics plays a part analogous to that of conductance in current electricity : for an electromotive force applied to a dielectric produces a charge, applied to a conductor produces a current ; and the ratio of charge (resp. current) to E. M. F. measures the capacity (resp. conductance). Hence all the rules given previously for the combination of conductances are applicable to the combination of capacities. The main difference between the laws of electric charges and magnetism is that whilst oh breaking a magnet each fragment becomes a complete magnet, so that it is impossible to separate the north and south poles, that is not true in the electrical case. If a pair of plates be charged by means of a battery and the plates then insulated from the poles of the battery, by removing or breaking the connecting wires, the plates will remain charged, and we shall have an isolated positive and an isolated negative charge, which may be carried about independently. It is found that merely rubbing two bodies together sets up an electromotive force between them, so that on separating the bodies they are found to be charged. And as the electro- motive force thus produced is exceedingly large (thousands of volts), although the quantity of electricity generated is very small, this process is most conveniently adapted for performing electrostatic experiments. A glass or ebonite rod rubbed with a catskin gives good results, provided both are sedulously dried beforehand ; the glass or ebonite becomes negative, the fur positive. It is therefore convenient, in demonstrating the forces between electric charges, properties of condensers, &c, to use such a generator of electricity, instead of the enormously large voltaic battery that would be needed. Another important means of obtaining electric charges is by what is known as electrostatic induction (not to be confused with induction of currents, described above). If a charged body, say a metal knob, supported by an insulating stand, charged positively by means of a glass rod and catskin, be placed near another conductor, it will attract, or induce negative electricity on the part of the other conductor nearest to it, and drive away 254 ELECTROSTATICS an equal amount of positive to the far end of the second con- ductor. If the latter be insulated, these two charges remain in equilibrium so long as the original charge is in place, but if that be removed, they reunite, and the second conductor is again in the neutral condition. But if while the original or inducing charge is still there, the conductor be connected to the earth, say by touching it with the finger, the positive induced charge will be driven further away, now that a conducting path is open to it, on to the earth ; then if the finger be removed the negative induced charge will be left by itself on the second conductor, and when the inducing charge is taken away, the second conductor will remain negatively charged. The simplest machine constructed on this principle is the electrophorus. This consists of an ebonite disc, usually provided with a metal backing ; and a metal disc of the same size, fixed on a long glass handle. The ebonite disc is laid on a table and rubbed with fur or silk to give it a negative charge. The metal disc, held by the insulating handle, is then placed on it : the charge does not leak on to the metal quickly because ebonite is an exceedingly bad conductor, and also there is usually contact between the ebonite and metal only at a few points, and consequently a thin film of air between the two over the greater part of the surface. The action consequently is to induce positive electricity on the lower, negative on the upper. surface of the metal. Now touch the upper surface for a moment with the finger : the negative charge flows away, leaving only positive electricity on the metal plate. If the metal plate be now lifted up by the insulating handle so as to avoid leakage, it will be found to be charged ; moreover, as the metal and ebonite plates, being very close together, form a condenser of large capacity, there will be a somewhat large charge, and on separating them the capacity is much reduced, so that the potential is correspondingly increased. The negative charge remains on the ebonite, and can be used over and over again, till it disappears through leakage. The more elaborate electrical machines, such as those of Holtz and Wimshurst, are developments of the principles used in the electrophorus. In making electrostatic experiments an electroscope is made use of, i. e. an instrument for observing the potential of charged bodies. The simplest and most useful FORCES BETWEEN ELECTRIC CHARGES 255 pattern is the gold-leaf electroscope ; this consists essentially of a pair of gold leaves hung from a metallic support, which in turn is insulated. It may easily be made as follows :— a conical flask of good insulating glass (shellacked to keep the surface dry, and so improve the insulation further) is fitted with a rubber stopper ; through this passes a stiff brass wire ; a brass knob or plate is soldered to the top of the wire, while below it is bent so that a pair of strips of gold-leaf may conveniently be attached to it, and allowed to hang down side by side. If now the knob be connected to a source of high potential, positive or negative, or charged by means of a rubbed ebonite rod, a charge will be imparted to each gold-leaf, and these repelling one another, the leaves will stand apart at an angle. Gold-leaf is chosen on account of its extreme lightness. Such an electroscope may be provided with a scale, by which to measure roughly the potential to which the leaves are raised : it then becomes an electrometer. But a better instrument is that known as the ' quadrant. ' This, as usually constructed, consists of four quadrants, seen in plan in Fig. 93, forming a nearly closed cylindrical box, but with air-spaces between : cen- trally to them, the ' needle,' a flat sheet of aluminium shown by the dotted line, is sus- pended, so as to lie between the upper and under surfaces of the box. The quadrants are rigidly supported and insu- lated: one pair, diagonally opposite to each other, are to be connected to the positive, the other pair to the negative end of the source of electromotive force to be measured. The needle is supported by two parallel silk fibres, which serve at the same time to insulate it. A wire attached to its lower side dips in a dish of strong sulphuric acid ; this serves three pur r poses, as a means of conveying a charge to the needle, as a dash- pot to prevent oscillations of the needle, and as a desiccator to maintain good insulation in the instrument. The needle is charged by a rubbed ebonite rod or otherwise, and the quadrants connected, as shown : then the (negative) charge on the needle Fig. 93. 256 ELECTROSTATICS is attracted by the two positive quadrants, repelled by the two negative, with the result of giving a small rotation to the needle. This is measured by mirror and scale as in a galvano- meter. The sensitiveness is usually about 100 scale divisions per volt. We have so far tacitly assumed that the dielectric, in which the phenomenon of electric stress shows itself, is air, or if not so a vacuum. There are, however, many solids and liquids that can be used instead, and which are comparable, as non-conductors, with gases. The chief difference produced by substituting one of these for air, is that the strain corresponding to a given stress is increased ; this is analogous to the differences that exist between different solids in their ordinary elastic properties. Up to the present we have identified stress and strain in the electric case, and spoken indifferently of the stress or electric force, as measured by the difference of potential per unit length in the dielectric, and the effect it produces, or strain, showing itself in the form of a charge on the conductors bounding the dielectric. If now we have to do with solids or liquids we must not overlook this distinction. A given stress (volts per cm.) produces a greater strain or charge (coulombs per sq. cm.), and the ratio of the charge produced to that which would be produced if there were air present is called the dielectric constant. Since charge -J- potential is defined as capacity, we may express the same idea by saying that the capacity of a condenser is greater when it has a solid or liquid dielectric than when there is air between the plates, and that the ratio of the capacities is called the dielectric constant : for this reason the term specific inductive capacity has been given to the same quantity. Thus, e. g., a con- denser formed of two brass plates in air has a certain capacity : if the space between the plates be filled with benzene the capacity is about 2-25 times greater. A very common form of condenser is the Leyden jar, which is a glass cylinder, coated inside and out with tin-foil : the glass here forms the dielectric, and its dielectric constant is from 6 to 8. § 2. Electric discharges. There is a limit to the electric stress that any material can stand. This limit for ordinary air may be put roughly at 30,000 ELECTRIC DISCHARGES -257 volts per cm. ; if this be exceeded, a discharge, usually in the form of a spark, takes place. Thus the sparks given by an induction coil when the wires leading from the secondary are held near together, show the discharge current, and the spark length is roughly a measure of the E. M. F. that the coil can give. If a Leyden jar be charged, and the two coatings be touched by a wire, of course a current flows through the wire, and the jar is discharged. But the sudden rush of electricity that takes place when a conducting channel is provided, behaves like a pendulum pulled aside and then released : it does not merely settle down to the equilibrium state, but overpasses it and swings to and fro. Thus, there is an alternating current in the wire, which only gradually dies away, leaving the two coatings of the jar at the same potential. This at least is what happens if there is not too much resistance offered to the dis- charge ; but if the connecting wire were very long and thin, the current might be so much damped that no oscillations would take place ; just as a pendulum might be so damped, by immersing the bob in glycerine, that it would only slowly settle down to the equilibrium position without swinging past it. The discharge from a Leyden jar may take from l0 ^ o0 to l oaiooo of a second to execute one complete oscillation : in any case the frequency is far greater than that of the alternating currents derived from an ordinary dynamo. But far more rapid alternations than this were obtained by Hertz. He connected the terminals of an induction coil to a condenser, consisting of two large metal balls ; these were placed a metre or so apart, and a pair of thick wires arranged so as nearly to connect them, leaving however a small air-gap in the middle (Fig. 94). The Fig. 94. action of the coil charged the balls up to a high difference of potential, and then a spark crossed the gap : but now it has been observed that though air is normally a perfect non-conductor, when a spark is passed through it it becomes for a short time a fairly good o — ^ — o 258 ELECTROSTATICS conductor. Consequently, when the resistance of the air-gap Was once broken down, the electric current set up could surge backwards and forwards between the balls a good many times before it was frittered away. The frequency of the oscillations thus produced was about a hundred million per second, or even more. Such oscillations set up waves in the surrounding medium (ether), just as a vibrating tuning-fork sets up waves of sound in air. Hertz was able to detect these waves, investigate their properties, and in particular to show that they are propagated with a velocity of 3 x io 10 cms. per sec. This is the velocity of light, and in fact the electromagnetic radiation turned out to have all the properties of light, except that it was on an enormously greater scale. Hence we may conclude that light is really an electromagnetic radiation, set up by the vibrations of molecules, which play a part similar to that of Hertz's apparatus : and as the molecules are very much smaller, they send out waves that are shorter, and consequently of far higher frequency even than the Hertzian waves. When air is reduced in pressure, its resistance to electric stress is reduced too ; therefore vacuum tubes are made, with which many phenomena of discharge can be obtained which are not possible in air at atmospheric pressure. Such a tube consists of a glass vessel, provided with electrodes of platinum or aluminium fused through the glass ; it is then evacuated by a mercury pump, and hermetically sealed. Air in small traces that the pump will not remove may be left in it, or the air may be replaced by traces of some other gas or vapour. When an induction coil is connected to the electrodes of such a tube, discharges can be produced^ through it, varying very much in shape and character, according to circumstances, but usually attended by an appreciable amount of light. Of all the numerous phenomena that have been observed in this way, we can only refer to one— the production of Eontgen rays. If a very good vacuum be produced in the tube, the negative electrode (called, as in electrolytic cells, the cathode) sends out a discharge which proceeds in straight lines, as if consisting of fine particles shot out from the electrode with a high velocity ; these, striking the glass of the tube, produce on it a characteristic green phos- phorescence, and there generate Eontgen rays, which also travel ELECTRIC DISCHARGES 259 in straight lines, and penetrate all substances to an extent inversely proportional to their density. Thus bone, being comparatively dense, obstructs the rays and produces a shadow, which the lighter tissues do in much less degree, while a metal object, such as a needle inside the hand, throws a deep shadow to the Bontgen rays. An ordinary induction coil can be used to excite the vacuum tube, but one of much larger size and much stronger than the ordinary physiological pattern is desirable. But if used with an ordinary spring interruptor, the number of discharges given by the coil per second makes the process of photographing by BOntgen rays slow. Wehnelt's electrolytic interruptor has therefore been generally adopted instead. This consists of a pair of electrodes in a jar of dilute sulphuric acid : the one, of lead, is of large size, the other, platinum, is very small, being only a few millimetres of thick wire projecting into the liquid. When a current passes, the small electrode polarizes far more rapidly than the big one, and a bubble of gas forming there, stops the current ; this escapes, and the current is renewed. By putting such an interruptor in the primary of the induction coil, discharges at the rate of 1,000 and more per second are obtained, and the vacuum tube for generating EOntgen rays becomes proportionately more effective. s 2 CHAPTER X. LIGHT. § i. Production of light. We saw, at the end of the last chapter, that it is possible to produce waves of electromagnetic disturbance, which possess all the characters of ordinary radiation, except that they are of very much less frequency, and consequently greater wave- length. Light, or more generally speaking radiation, is now known to consist of electromagnetic waves, set up by vibrations of the molecules, and transmitted through the all-pervading medium known as the ether. Since light consists of waves, much that was stated of sound is applicable here also, and it is well to keep the statements of that chapter in view. In the first place, then, the vibrations "producing light can differ in (i) amplitude, (ii) frequency. As in sound, the energy or intensity of a vibration depends on the amplitude, being proportional to the square of it. It is a matter of common observation, that by raising the temperature of any body it may be made to emit light, and the hotter it is the more light it emits ; thus an incandescent lamp is made to glow by running a current through it, so as to make it hot : if the current is increased in strength, the lamp becomes hotter, and gives out more light. This means, then, that the vibrations executed by the molecules of the lamp filament become greater in amplitude and so possess more energy, enabling the molecules to radiate energy in the form of light more vigorously. Varying frequency of vibration, which in sound produces the sensation of pitch, in light produces that of colour. If a vacuum tube be made with a trace of hydrogen in it, and an electric discharge be passed through it, it glows, and if examined by means of a spectroscope (vid. inf.), it will be found that it is PRODUCTION OF LIGHT 261 giving out three distinct kinds of light, one red, one blue, one violet, just as a bell when struck might give out two or three partial tones of different pitch. Of the three, the red has the lowest frequency, violet the highest. It is however customary to speak of the wave-length rather than the frequency in the case of light : and it must be remembered that the wave-length = velocity of wave -j- frequency, consequently the red has the longest wave-length, violet the shortest. The actual wave- lengths are — Hydrogen, red (C) 6.563 x io -5 cm. „ blue (F) 4.861 „ violet 4.341 (The letters C and F are names by which the particular kinds of light are sometimes known, see p. 276.) It is not, however, generally true in light as in sound, that a substance gives out only one or a few vibrations of definite frequency. When a solid is heated to whiteness, the radiation from it is found to be of all possible wave-lengths between ceitain limits ; it is as if one played all the notes of a piano at once. The range of wave-length observed is from 7-6 x 10 ~ 5 cm. (deepest red), through orange, yellow, green, blue, violet, to about 3-8 x io -5 . This series of colours is called the spectrum, and we may speak of going ' up ' the spectrum from red to violet, by analogy with going up the piano from notes of low frequency to those of high. But the limits just mentioned are really in no way inherent in the physical phenomenon of radiation : they are purely physiological. Just as the ear is incapable of appreciating aerial waves above and below certain limits of frequency, so with the perception of ethereal waves by the eye. Waves of length greater than 7-6 and less than 3-8 exist, and can be observed by appropriate instruments, consequently the spectrum really extends beyond the visible portion, and includes an 'infra-red' region (of long wave-length), and an ' ultra-violet' (of short wave-length). When a solid is heated, it first gives out radiation only of very great wave-length ; as the temperature rises the spectrum extends further up, coming to include shorter waves, till at between 400° and 500° cent, the limit of visibility is reached, and the solid appears a dull red. As the temperature is further raised, other colours of shorter wave-length are produced, and the body appears, as a consequence of their blending, first yellow, 262 LIGHT then white, and, even in an extreme case, slightly bluish, owing to preponderance of the blue and violet in the spectrum. It is only at very high temperatures that an appreciable amount of ultra-violet is produced. § 2. Light-waves. When radiation proceeds from some source, such as an incan- descent solid, it may sometimes be made to travel in one direction only, and not spread out. That may be accomplished by a properly placed concave reflector, or by a lens, as is used in light- houses ; it also occurs when light is generated at one end of a glass rod, especially if the rod be silvered : the light is led along the length of the rod, with little loss at the sides, even if it be somewhat bent— a contrivance sometimes used in microscopy and in surgery. In either case, since the light does not spread out, it remains of the same intensity, however far it travels, except in so far as the substance conveying it absorbs some of the energy of the waves, i. e. is not perfectly trans- parent. But unless some precaution is taken, light spreads out in all directions, so that the energy of the waves has to be spread out over larger and larger surfaces, as the light travels further from its origin, and becomes attenuated. For the same reason as in sound, the intensity of the light is inversely as the square of the distance of the source, disregarding any further diminution of intensity due to want of perfect trans- parency. This fact is made use of in order to measure the strength, or ' candle-power ' of lamps. Any apparatus for this purpose is called a plwtometer. It is not possible to estimate directly by eye the ratio between the intensities of illumination pro- duced by two lights, but it is possible to tell when the two illuminations are the same. Hence, in a photometer, the brighter light is taken gradually further away, till it produces the same intensity of illumination as the other : the candle- powers of the two lights are then in proportion to the squares of their distances. The most familiar arrangement for com- parison is the Bunsen photometer : this consists of a sheet of white paper with a spot of grease in the middle. When illuminated from the front the grease spot looks darker than LIGHT-WAVES 263 the rest of the paper, when from behind it looks lighter ; if the illumination back and front be equal, the grease spot is hardly visible. Now suppose a standard candle to be fixed 50 cm. from one side of the paper, and an incandescent lamp moved to and fro on a graduated bar, extending on the opposite side of the paper, and suppose that at a distance of 200 cm. the grease spot disappears ; then the ratio of distances is 4:1, and the candle-power of the lamp is 4 2 = 16. The velocity with which light is propagated is enormously great : it is found to be the same for light of all wave-lengths (including artificial electromagnetic waves) when traversing cms empty space. This common velocity is about 2-999 x IO ° ' • Kemembering then the relation between velocity, wave-length, and frequency, we are able to calculate the latter. In the case of red hydrogen light it is 2-999 x io ,0 -f6-563x io -0 = 4-57 x io' 4 per second : this is, of course, far beyond anything that can be produced artificially ; it is, in fact, about a hundred thousand times more rapid than the most rapid Hertzian waves that have been produced. In media other than empty space, the velocity of light is less, and moreover it varies slightly according to the kind of light considered. These relations are most conveniently expressed by means of what is known as the refractive index of the medium ; this , , „ , ,. ,. velocity of light in vacuo may be defined as the ratio — = — ;- — = — fr ^ rs 3* J velocity in the medium considered The refractive index for any given substance, say water, varies for fight of varying wave-length : the general rule is that it is greater for shorter waves, i.e. greater for blue than for red. The variation is called the dispersion of the medium, since it is the means practically adopted for separating or dispersing light of different kinds. As the refractive index and dispersion play a most important part in the construction of optical instruments, we subjoin a short table of numerical values. It will be noticed that the refractive index of air is so nearly unity that it is rarely necessary to take into account either it or the dispersion of air, and the same is true of other gases ; so the refractive indices of solids and liquids are often measured by comparison with air instead of with a vacuum. 264 LIGHT Refractive index Dispersion from hy- for sodium light. drogen blue to : red. fp-i P-D Hf-Po Pt-Pc Air at 0° and 760 mm. 1 .00029 1 00000288 ioi- Water at 20° 13330 000600 55-5 Silicate crown glass (Schott & Co. 40) 1 5166 00849 60-9 Ordinary silicate flint glass (Schott & Co. O118) 1.6129 00 r 660 369 Carbon disulphide at 20^ 1-628 00342 18-4 The second column gives the refractive index for the yellow light of incandescent sodium vapour of a wave-length inter- mediate between hydrogen blue and red. The third column gives the difference between the refractive indices for blue and red light ; the fourth column gives the ratio (mean refractive index - 1) -=- difference for blue and red ; the usefulness of this will appear later. A difficulty was at first experienced in accepting the view that sound and light are both due to wave motions, in the fact that sound travels round corners with very little difficulty, whereas light, when it meets an opaque obstacle, casts' a sharp shadow. The difference, however, is not a real objection ; it is all a matter of scale. Sound-waves are ordinarily a metre or so long, light-waves about ^^inr of a centimetre ; hence, to perform comparable experiments in the two cases, it is necessary to make the apparatus enormously large in the one case, or very small in the other. If a church bell be taken as a source of sound, and a house as obstacle, it will be found that quite a marked sound shadow is formed— the sound is much less intense at points where the house stands between the observer and the bell. Still more marked is the shadow thrown by a hill to the sound-waves from a large explosion of gunpowder. If, on the other hand, the shadow thrown by a small sharp-edged object to a source of light be examined minutely, it will be found that a little of the light does get round corners. The experiment may be made as follows : paste tin-foil on a sheet of glass, and cut a narrow slit in the foil with a knife ; mount this slit in front of a very brilliant source of light ; some 20 or 30 cms. away mount a sharply cut straight edge of metal, taking great care that the edge is parallel to the slit ; then examine the shadow behind the edge with a fairly high power eye-piece, holding this so as to look in the direc- tion of the light, and to have half its field of view in the light, half in the shadow. It will be found that there is no sharp line of demarcation between light and dark, but that the light fades away quite gradually towards the dark side, and that on the light side a series of alternations of light and dark occur, getting fainter as one looks further out into the LIGHT-WAVES 265 illuminated part of the field. These alternations are called diffraction fringes, and it is found that a complete explanation of them can be given on the theory that light consists of waves, so that, far from being a diffi- culty, the phenomena of shadow-formation constitute the strongest evidence in favour of that theory. Accordingly we may say that light produced at a source in the air travels outwards in spherical waves, with uniform velocity. We may trace the course of the light by means of the wave-front, i. e. the surface including all the points that are in the same phase at the same moment (see figures, pp. 155 to 157) : the wave-front is a sphere round the source of light, but, if the source is far away, and we are only considering a small piece of the wave-front (e. g. the piece of a wave-front from a star, that enters a telescope), we may regard that piece as plane. Further, we may speak of rays of light, lines at right angles to the wave-front, and consequently in the direction in which the light is going : we may trace out the course of the light by means of the rays, if more convenient ; and, if the wave-front meets an opaque obstacle, that part of it may be considered extinguished, and, very approximately, a shadow will be formed which is marked out by the rays that pass the edge of the obstacle. The space marked out by these rays is called the geometrical shadow, and, as we have seen, light really bends into this space, but only to a minute extent : practically the light may be regarded as travelling in straight lines, i. e. along the rays. When a wave of light strikes against a surface constituting the boundary between two media it is in general partly re- flected, and partly passes on into the second medium. The amount reflected varies very much— if the surface be of polished silver it may be more than 90% of the whole. The geometry of the reflection is shown in Fig. 95. ab is the reflecting surface ; ef is a small portion of a wave-front, such as may be regarded as plane (the diagram is to be regarded as a section at right angles to the plane of the reflecting surface, or of the wave- 266 LIGHT front), ef is travelling in the direction shown by the arrow, and at the point r has reached the reflecting surface; the light from e only does so later, viz. when it has reached the point g. Now from f as centre draw a circle with radius equal to eg, and from g draw the tangent gh to this circle. Then, by the time the light from e has reached g, that from f reflected has necessarily covered the distance fh, and gh is the new wave-front, i. e. surface, all points of which are reached simultaneously by the light. This is travelling in the direction of the arrow, along fhk or gi. Hence, we see (by symmetry) that the reflected wave-front makes the same angle with the reflecting surface that the incident wave does : efg is called the angle of incidence, hgf the angle of reflection : these two are equal. This result may also be expressed by means of rays ; e. g. df is a ray of the incident light (perpendicular to the incident wave-front), fk a ray of the reflected light. These, then, make equal angles with the normal fn to the surface, and since dfn = efg, and hfn = hgf, we may equally well define the angles of incidence and reflection as the angles which the incident and reflected rays make with the normal to the surface, and state the law of reflection as follows :—Thc incident a>id reflected rays are in the same plane with the normal to the surface, and make equal angles with it, but on opposite sides. Light- waves that strike a boundary between two media break up in general into a reflected part and a transmitted part, but, as many substances are opaque, the latter is often rapidly absorbed and cannot be ob- served. If, however, the second medium be trans- parent, e. g. glass, the trans- mitted beam of light can be observed, and is usually much the stronger of the two ; it does not in general proceed straight, but is said to suffer refraction. The cause of this is the unequal velocity of light in the two media, as may be explained with the aid of Fig. 96. Hence, as before, ab is the trace of the bounding surface, ef that of a plane Fig. 96. LIGHT-WAVES 267 wave-front falling on it. But during the time that the incident light takes to travel from e to g in air the light from f travelling in the glass will only have gone a shorter distance. Let the refractive index ^ = 1-5, then the light in the glass will travel only f as fast, and we must draw round e a circle of radius equal to f of eg ; then the tangent from G on this circle, viz. gh, is the refracted wave-front. It is evident that this makes a smaller angle with the refracting surface than the incident wave-front does. If we call efg the angle of incidence, Zj and fgh the angle of refraction Z 2 , then sin L x = eg 4- fg, while sin Z 2 = fh t fg, so that sin Z t _ eg sin Z 2 ~~ fh Putting this in terms of the rays, we may define Z 1; the angle of incidence, as the angle between the incident ray df and fn normal to the refracting surface, while Z 2 , the angle of refraction, is the angle between the refracted ray fk and fn 1 the normal, and say : — The incident and refracted rays are in the same plane with the normal to the surface, and the angles of incidence and refraction are connected by the relation sin A = /* si n A- Here, in whichever sense the light travels, Zj is to be taken as the angle in air, Z 2 in the other medium, as may be seen from the nature of the above proof. The consequences of the laws of reflection and refraction will be considered in detail below in dealing with optical instruments. It was mentioned, in dealing with sound, that vibrations in air are necessarily longitudinal. Those of light, on the other hand, are trans- verse, that is to say, the movements which constitute the light take place in directions at right angles to the direction in which the light is travel- ling. This difference entails an entirely new set of phenomena, for whereas there is only one direction in which the wave is travelling, there are an infinite number at right angles to it. Consequently, the behaviour of light- waves must be associated with some direction in space at right angles to their line of propagation (i. e. in the wave-front), whereas, in longitudinal waves, there is nothing to distinguish any one direction in the wave-front from any other. This fact may be readily shown in the case of the long artificial electro- magnetic waves. If a grid of fine wires, wound parallel to one another on a wooden frame, be placed in front of a straight Hertzian vibrator, it is found that it stops the waves when the wires are parallel to the vibrator, 268 LIGHT lets them freely through when the wires are at right angles to the vibrator. For convenience of description, let us suppose that the vibratoi is placed horizontal in a N. and S. line, and that the grid is placed to the east of it, and on the same level : now the electromotive force generated in the vibrator is along its length, i.e. N. and S., and this is propagated through the surrounding space, keeping the same direction ; consequently, when the waves fall on the grid, their direction of propagation is towards the east, and their direction of vibration is (so far as electromotive force is concerned) N. and S. If the vibrator be not moved, the direction of vibration remains the same for any length of time ; when that is so the radiation is said to be polarized, and, to specify it precisely, a certain plane is chosen : this, by a convention based on quite other phenomena than those just described, is the plane at right angles to the electromotive force, i. e. the vertical plane. Accordingly, when radiation is stated to be polarized in a certain plane, this means that the electric oscillations it consists of take place at right angles to that plane. Now ordinary light is not polarized, because the molecular vibrators from which it is derived are in all sorts of positions, and continually moving. It is, however, possible to sort out the waves by a device analogous to the metal grid, and any such device is called a polarizer : the simplest is a plate of tourmaline crystal. Such a plate allows light-waves to pass through it when their vibrations are in a certain direction relatively to the structure of the crystal, but, if they are at right angles to that direc- tion, absorbs them ; hence, whatever be the nature of the light before reaching the tourmaline, it is plane polarized afterwards. Then, if a second plate of tourmaline be put after the first, it will treat this polarized light just as the metal grid treats the radiation from the Hert- zian vibrator ; that is, if the second crystal plate is placed similarly to the first ;so that the definite direction in the crystal, known as the optic axis, is the same in the two), it lets through the light coming from the first ; but, if the axis of the second be put at right angles to that of the first, it is in a position to stop just that light which gets through the first, so that no light at all gets through the combination. The second plate is called an analyser, and, in the latter case, is said to be crossed to the polarizer. Various other devices for polarizing have been invented, and it may be mentioned here that it has even been found possible to construct a grid of such exceedingly fine wires that natural radiation of the longest wave- length (infra red) from hot bodies could be polarized with it. § 3. Phenomena of emission and absorption. Light is usually produced by raising a solid to a high tem- perature ; but, in order to examine precisely the transformations of energy accompanying this process, it is necessary to have PHENOMENA OF EMISSION AND ABSORPTION 269 some instrument whose sensitiveness to radiation is not limited, as that of the eye is for physiological reasons, but is capable of appreciating indifferently radiation of all wave-lengths. Such an instrument is the thermometer, provided its working part be covered with a substance like lampblack, which absorbs all the radiation that falls on it, for then all the radiation is converted into heat, and the amount of it is registered by the rise in temperature. Thermometers far more delicate than the ordinary mercury in glass one have been invented for the purpose, into the details of which it is not necessary to go here. But again, the observations to be made on radiation would be very incomplete if it were not possible to examine separately that of different wave-lengths ; to accomplish this a spectroscope must be used. Practical details of this instrument are given below (p._294) ; here it may suffice to say that a narrow vertical slit is illuminated strongly by the light to be examined, and the slit viewed through a short telescope ; but before reaching the telescope, the light is caused to pass through a triangular glass prism with its edge vertical. Such a prism bends the rays of light falling on it, and as the amount of bending depends on the refractive index of the glass, it is different for the various kinds of light, and these are dispersed. Thus, instead of seeing in the telescope a sharp white image of the slit, one sees a long coloured band, of the same vertical height as the slit, but changing laterally from red through the various colours of the spectrum to violet, the red rays being less bent than the blue. If an appropriate kind of thermometer be substituted for the eye-piece of the telescope, it can be moved about in the spectrum, and the latter can be traced into regions beyond the red. It should also be possible to trace it into regions beyond the violet at the other end of the visible band, but the radiation there is so faint that no thermometer sufficiently delicate has yet been constructed. Fortunately photography is of assistance here, so that our knowledge of the ultra-violet spectrum has mostly been obtained by putting a photographic plate in the place otherwise occupied by the eye-piece in a spectroscope. Armed with these means of investigation, then, it is possible to determine the way in which radiation from a hot body depends on the nature and temperature of the latter. 270 LIGHT Now it is a matter of common observation that a hot body, placed inside a cooler enclosure, eventually becomes of the same temperature as that enclosure. The equalization of temperature is accomplished partly or entirely by radiation : so long as the inside body is hotter than the enclosure it radiates heat to it ; when equality of temperature is reached, the process apparently stops. That, however, is not a complete account of the phenomenon, for to suppose the radiation to stop would imply that whether a body radiates or not is regulated by the condition of some other body elsewhere— viz. the en- closure. In the words of Stokes 'the molecules of the body cannot prophesy what is ultimately to become of the motion they may communicate to the ether, and regulate the com- munication accordingly.' The true explanation is, therefore, that the body and the envelope both radiate to each other, under all circumstances, but when their temperatures are equal the amount of radiation is the same on both sides, and so no change of temperature results, but if the temperatures are unequal, the hotter body radiates more than it receives from the other, and consequently falls in temperature. This view is known as the theory of exchanges. Experiments on the amount of radiation from hot bodies are best satisfied by Stefan's formula (p. 75), according to which the radiation is proportional to the fourth power of the absolute temperature. This means that if two surfaces opposite one another are at different temperatures, say T 1} T % (absolute), the energy radiated by the first to the second is equal to, say, hT*, that radiated by the second to the first kT 2 *, and consequently the heat transferred by radiation from the hot to the cold body is the difference h (Tj* - T 2 4 ). When, however, the difference of temperature is only a few degrees (e. g. between the human body and the walls of houses), Stefan's law leads to the same result as Newton's law of cooling (p. 74), that the loss of heat from a body by radiation is proportional to its excess of temperature above the surroundings. The constant Jc for a perfectly black surface (e. g. lampblack) is about 1-28 x 10- 12 calories per sq. cm. per sec. If this number be used to calculate the radiation from the outer to the inner surface of a vacuum jacket such as is used to contain liquid air,' it will be found that the heat received by the liquid air is so PHENOMENA OF EMISSION AND ABSORPTION 271 small that there is nothing anomalous in the fact that it evapo- rates very slowly. Actually, if the surfaces are made of glass, the radiation is a good deal less than the amount just mentioned. To revert to the theory of exchanges, suppose a polished metal ball enclosed in a chamber with lampblacked walls ; then we know that eventually the temperature of the ball will be the same as that of the enclosure. But now the radiation from the enclosure falling on the ball would balance that proceeding from the ball, if the latter were lampblacked ; as it is the same amount falls on the ball, but some of it is reflected, and conse- quently only the remainder— say the fraction x of the whole- is absorbed ; yet this suffices to balance the radiation from the polished ball. It follows, then, that the radiation from the polished ball can only be x of that from the same ball lamp- blacked, and at the same temperature. This result as to the relation between the powers of radiation and absorption is due to Kirchhoff. We may state it formally thus.:— If a surface absorbs the fraction x of the radiation falling on it, then it will radiate x times as much as a black surface at the same temperature. Accordingly, the constant k given above for lampblack must not be taken as correct for surfaces that reflect any of the rays that fall on it. For polished silver the radiation (at low temperatures) has been found to be only about 30% that of lamp- black, so that k = 0-3 x 1-28 x io -12 = 3-84 x io -13 cal. per sq. cm. per sec. Whilst, however, the total radiation from, a solid when heated increases in proportion, to the fourth power of the ab- solute temperature, it is not true that all the kinds of light increase in the same proportion. On the contrary, as already remarked, a solid at first gives out only radiation of very long wave-length, far too long to see ; as the temperature rises the intensity of this infra-red radiation is increased, but other kinds higher up the scale are added, until at a sufficiently high temperature the solid becomes luminous, i. e. radiation of wave- length shorter than 7-6 x io -6 cm. is produced, so that the eye is affected by it. When the temperature is still further raised, the phenomena proceed in a similar manner : the intensity of the red as well as of the infra red is increased, but other colours, yellow, green, blue, &c, are added ; and always the increase in intensity is greater in the higher regions of the spectrum than 272 LIGHT in the lower. The distribution of energy in the spectrum will be more easily grasped from the curves in Fig. 97. The height of the curve there represents the intensity of the radiation of the particular wave-length, as shown by the abscissae, and as measured by putting a sensitive thermo- meter in the corre- sponding part of the actual spectrum. Curve I represents ap- proximately, accord- ing to Langley's mea- surements, the solar radiation ; curve II, that from a source at low temperature ; and it will be noticed that not only is I much more intense than II in all parts, but that the difference becomes greater as we pass from the long wave- lengths to short, and that consequently the wave-length (oa) for which the maximum of radiation occurs is shorter in sunlight than that (ob) for the other source. Further, as a consequence of this, the fraction of the total radiation which occurs between the limits (vr) of the visible spectrum (re- presented by shading), is much greater for sunlight than for the low temperature source. Even for sunlight it is only about ,36 per cent., so that regarding the sun as an illuminating appliance, more than half its energy is useless ; while for such a source as an incandescent lamp only 2 or 3 per cent, of the total radiation lies within the limits violet to red, and all the rest is wasted. Fig. 97. What precedes is an account of the normal production of radiation as a consequence of temperature — what may conveniently be known by the name of incandescence. But it very often happens that substances, for one cause or another, give out radiation of a kind very different from that which their temperatures would indicate, becoming luminous, for example, at temperatures far below a red-heat ; these cases are included under the general term luminescence. The most important causes which give rise to this phenomena are (i) chemical, (ii) electrical, (iii) action of light. Chemi-luminescence occurs in many instances, of which the phosphor- PHENOMENA OF EMISSION AND ABSORPTION S273 escence of phosphorus may be taken as example. It is probable that this is due to a slow oxidation, but it is at any rate certain that the phosphorus, while giving a whitish glimmer, remains cold. Probably the luminous phenomena of the glow-worm, and other animal organisms, come under the same head. Electro-luminescence is very familiar in the discharge of electricity through gases : vacuum tubes can be made to glow with con- siderable intensity by a discharge, but the temperature reached by their contents is by no means high. Fhoto-luminescence may be taken as including all the cases in which substances exposed to light become phos- phorescent, i. e. capable of giving off light on their own account afterwards (e. g. calcium sulphide, or Balmain's luminous paint) ; phosphorescence is really a very common phenomenon, but in a much slighter degree, that requires special appliances to detect it. Finally, the important case of fluorescence comes under the same head : there are substances (e. g. fluo- rescein) which send out light on their own account, so long as they are illuminated by an independent source, but not afterwards ; thus, if a solu- tion of sulphate of quinine be illuminated by violet light, it will be seen, looked at sideways to the light, to be of a pale blue colour, a3 if self- luminous. A solution of chlorophyll in alcohol, similarly treated with green light, appears red. The fluorescent substance absorbs some par- ticular kind of light falling on it — violet in the one case, green in the other, and, in consequence, becomes luminous, giving out in all directions light, not of the same kind, but always lower down in the spectrum — blue and red respectively in the instances quoted. It appears to be in somewhat the same way that BOntgen rays, falling on a fluorescent screen, make it self-luminous, so that if an organ be placed between the source of the rays and the screen, those parts of the organ which most obstruct the rays, such as bones, appear in shadow on the screen. Solids and liquids, when made incandescent, all give a con- tinuous spectrum, i. e. one including all possible kinds of light within certain limits of wave-length ; so that their spectra are not characteristic of the substances producing them, but merely of the temperatures. Gases, on the other hand, when not under too high pressure, give out spectra consisting of light of a few definite kinds only, and therefore appearing in the spectroscope as a series of bright lines (images of the slit) in definite positions, with dark spaces between. The explanation of this difference on the molecular theory, is that the molecules of a gas, being widely spaced, are able to travel quite appreciable distances without coming into collision, or being affected by neighbouring molecules ; hence, after a collision has set a mole- cule in vigorous vibration, it may execute a thousand, or even T 274 LIGHT a million, oscillations before it is again interfered with by another collision. During this interval it gives out waves of its own natural period to the ether, just as a tuning-fork struck, and then after five minutes struck again, would give out its natural sound to the air, the time occupied in striking it being negligibly small compared to the intervals in which it is left alone. Whereas in solids and liquids a molecule never escapes from the influence of its neighbours, so that it has no opportunity of vibrating in its own natural period. The transition between the two may be seen by observing the spectrum of a gas (produced by an electric discharge through it), and gradually increasing the density of the gas. The ' lines ' in the spectrum widen out into bands, these spread more and more, till they fill up all the intervening dark spaces, and a continuous spectrum is formed. Since the spectrum of each gas consists only of a few bright "Jines (or in some cases bands), occupying fixed positions in the scale of wave-lengths, and answering to free periods of vibration of the gas molecules, that of each gas is different to all others, and the spectroscope affords a means— highly delicate and certain —of detecting particular gases. Thus, hydrogen gives the three bright lines referred to on p. 261, the position of which can be measured with an accuracy approaching one part in a million ; sodium vapour gives two lines in the yellow, called D x and D 2 , of wave-length 5-596 x 10 -5 and 5-890 x 10- 6 cms. respectively. These two appear as one in a spectroscope of small dispersion, and are known as the D line ; and since they are very bright and easy to get, by merely placing a sodium salt in a bunsen burner, refractive indices and other optical constants are very commonly measured with the aid of it. Thus, the standard refractive indices given in the table, p. 264, and called pp, are the values of n for this particular kind of light. In order to obtain a gaseous spectrum for examination, the substance to be studied may be volatilized in a bunsen flame, or if that is not hot enough, in an arc lamp ; or, if a metal, electrodes may be made of it, and a spark from an induction coil passed between them, when it will volatilize some of the metal, and take a characteristic colour accordingly ; or a vacuum tube may be prepared, containing a little of the substance, and a discharge passed through it, rendering it luminous. The 'latter method PHENOMENA OF EMISSION AND ABSORPTION 275 lias the advantage that, the gas being rarefied, the molecules are more completely free to execute undisturbed vibrations than when at atmospheric pressure, and the lines of the spectrum are therefore sharper. By examining the light from nebular comets, and the outer fringe of the sun as it appears in total eclipses, it has been shown that these heavenly bodies are gaseous, and the actual gases they consist of have been determined. Kirchoff's law as to the equality between emission and ab- sorption is true not merely for the radiation as a whole, but for each kind separately; consequently, substances which possess special powers of emitting particular rays of light, possess special powers of absorbing the same rays. Hence arise what are known as absorption spectra, which may be as characteristic of a sub- stance as the emission spectra. If white light from a very hot source, such as an arc lamp, be allowed to pass through a bunsen burner made yellow with sodium, and the light be then examined with a spectroscope, it is found that the continuous spectrum of the arc is interrupted by two dark lines, precisely in the positions in which the bright yellow lines would be seen, if the sodium burner were looked at alone. These lines are not really dark, but appear so because they are less luminous than the neighbouring parts of the spectrum. They arise in this way : sodium vapour is transparent to most kinds of light, so that the arc spectrum as a whole is unchanged by the flame ; but when waves of the particular frequency of the D lines fall upon sodium molecules, they set those molecules vibrating in precisely the same way that an acoustical resonator is set vibrating, when waves in unison with its own natural note pass over it. All that has been said (p. 157) about resonance in sound is applic- able here ; the energy of the incoming light-waves of the right frequency is absorbed, i. e. it is spent in setting the molecules of the absorbing substance (sodium vapour) in vibration, and consequently does not pass on and reach the spectroscope. However, the sodium molecules themselves in this way become a source of waves (in addition to the effect produced by the heat of the gas flame), and give out light, not only, of course, in the direction of the spectroscope, but in all directions. Some light of the given frequency therefore reaches the spectroscope, and it is a mere question of temperature whether the D lines 12 276 LIGHT HYDROGEN VIOLET HYDROGEN BLUE appear dark on a bright spectrum, or brighter than the rest of the spectrum. With an arc as source of light, the radiation is so intense that the flame absorbs more than it gives out on its own account ; but with an incandescent lamp as source, the radiation is weaker, and the flame supplies more yellow light than it absorbs, so that the lines appear bright. The spectra of the sun and stars show a continuous band of light interrupted by fine lines (the Fraunhofer lines), which are now known to be due to the white light from the core of the sun or star being partly absorbed by gases in the atmosphere above it — partly too by the earth's atmosphere. Very many of the lines have been identified in position with bright lines in the emission spectra of known bodies, and in this way the existence of many elements known on earth has been proved in the sun and stars. Even an element called helium was known for many years in the sun, being characterized by a line that did not agree with those of any known sub- stance, and, subsequently, that element was discovered by Ramsay in certain terrestrial minerals. A few of the most prominent of the Fraunhofer lines have received letters as a designation, and they form a most convenient way of describing the different parts of the spectrum, Their position is sufficiently indi- cated by the accompanying chart (Fig. 97 a) ; as already mentioned, C and F are due to hydrogen, D to sodium. Solids and liquids, though never emitting light of sharply defined character like gases, sometimes emit light belonging to one region of the spectrum more strongly than another ; the corresponding fact, absorption of one part and not another, is familiar— it is in fact the SODIUM b C z HYDROGEN REO ^ Fie. 97 a. REFLECTION AND REFRACTION 277 cause of the colour of ordinary objects. Thus, if a cell containing bichromate solution be placed in front of the slit of a spectro- scope, it will be found that the spectrum of a source of white light is reduced to the red end only ; the solution is transparent to red light, but opaque to (i. e. absorbs) other colours. As a rule, in solids and liquids, such selective absorption extends in a somewhat indefinite manner to large tracts of the spectrum,' but in a few cases it consists in bands of fairly definite position. Thus, haemoglobin produces two dark bands lying between J> and E in the yellow and green. Hence, the appearance of these bands may be used as a delicate test for the presence of blood. It should be noted that selective absorption may occur just as well in the ultra violet or infra red : a body in which this occurs appears colourless, but in the physical sense it is similar to a coloured body, e. g. glass absorbs the ultra violet strongly ; hence, if we obtained visual sensations from the ultra violet, it would not appear perfectly transparent and colourless. Quartz, on the other hand, transmits ultra-violet light freely. Water is opaque for radiation of wave-length of more than about 14x10-" cm. The limitation of the eye to radiation between 7-5 and 3-8 x io -5 cm. may be due either to a physical or a physiological cause : the radiation may be absorbed before it reaches the retina, or it may be without effect on the retina when it does reach it. The former appears to be the cause of the limit at the violet end, but- is certainly not so at the red end, as it has been shown that longer waves can traverse the media of the eye. § 4. Reflection and refraction. Very many optical phenomena, and especially those involved in the action of mirrors, lenses, and other optical instruments, can be determined with sufficient approximation by the assump- tion that light consists of rays which, when travelling in a uni- form medium, are straight, and which, at the boundary of two media, suffer reflection and refraction according to the laws stated on pp. 266, 267. This method of treatment yields what is known as geometrical optics ; being based only on rules which give an approximation to the facts, it is occasionally insufficient, e. g. the theory of the resolving power of the microscope, due to Abbe, 27a LIGHT though of great practical importance in the construction and use of that instrument, lies outside the domain of geometrical optics. But we shall not attempt any such refinements here, it being sufficient for most purposes to treat optical instruments, includ- ing the eye, by means of the rules of reflection and refraction. The application of these rules consists largely in determining the position of ' images ' formed by mirrors and refracting surfaces. The precise meaning of this term will be best understood after considering the simplest instance in detail. Reflection at a plane surface. Let a (Kg. 98) be a point source of light, bcd a plane reflecting surface j draw the perpendicular ab and produce it to a'. Then a ray of light from a, falling on the mirror at b, since it is normal to the surface, will be reflected back along the same line. Draw another ray ac, and at c erect the normal cm ; draw ce on the opposite side of the normal, so that ca and ce make the same angle with the normal ; there- fore ce is the course of the ray reflected at c. Produce ce backwards to meet the production of ab at a' ; then a'b is equal to ab. For the triangles abc, a'bc have equal angles and a common side ; therefore they are equal in all respects. Now the lines of the two reflected rays ba and ce intersect in A' ; and, to an eye placed in the direction of a and e, they will appear to come from a'. That, however, will not be true of the whole beam, unless all the rays, after reflection, are in lines passing through a'. Consider, then, some other ray, ad, and in a similar way draw the normal dn and the reflected ray df ; then, by a similar argument, it follows that df, produced back- REFLECTION AND REFRACTION 279. wards, passes through a', and so on for all the rays. Hence, to an eye placed, say, in the direction of e and f, all the light after reflection appears to come from a', and it is immaterial whether- it com.es from a source at a' through a window bcd, or from a source at A by means of a mirror bod. a' is called the image of a, and we may give the following formal definition — When, after reflection or refraction, all the light proceeding from one point proceeds to, or appears to proceed from, a second point, the latter is called an image of the former ; if the light, actually passes through the second point it is called a real image, otherwise a virtual one. A. plane mirror, therefore, forms a virtual image. This fact may easily be verified by holding a candle in front of a looking- glass. Refraction at a plane surface. Let a (Fig. 99) be a point-source of light, bcd the bounding surface between air and a second medium. The course of any ray ac may be found by drawing the normal mcm', then the angle ecm', such that sin acm = \i sin ecu', where ^ is the refractive index of the second me- dium ; then ce is the direc» tion of the refracted ray- It will be found that the refracted rays ce, dp, &c, diverge very approximately from the point a', which is on the normal ab, and at a distance from the surface at B such that ab = /* a'b. The image formed in this case is only approximate, whereas in the former case the lines of all the rays passed exactly through a'. Departure from exactness in the formation of images is called aberration, and occurs in nearly all optical in- struments to a greater or smaller extent. 28a. LIGHT Reflection at a spherical surface. In this and the following cases also the formation of an image is only approximate, but is suffi- ciently close for most purposes, when the surface does not form too large an arc of a sphere, e. g. an ophthalmoscope mirror con- sists of a section of a sphere not more than a few degrees across ; if it were a hemisphere, or even a quarter sphere, the aberration would cause great distortion in the appearance of the objects looked at. Two problems arise, (i) to find the position of the image of a point, (ii) to find the size of the image of an object of finite size— this being made up of the images of all the points that compose the object. With regard to the former, it will be sufficient to consider points on the axis, i. e. the normal to the mirror through its middle point. A geometrical construction for this is given in Fig. 100 for the case of a concave mirror. Let A be the source of light, on the axis acb; since the axis is a normal to the mirror, it must pass through the centre of the sphere, let this be c. Draw a ray ap and the normal cp, and make the angle cpa' = cpa ; then pa' is the course of the ray reflected at p. It may be shown that, all the rays after reflection cut the axis at the same point a', which is consequently the image, and in this case a real one. The most convenient way of finding the position of a' is, however, an arithmetical one that will be explained below. The position of the image varies with that of the object, cer- tain cases being especially important. If the object is at the centre of the sphere the image coincides with it. If the object is very far away (mathematically speaking at infinity) the image is at f halfway between c and the mirror at b— this position is known as the principal focus ; and if the object is at the principal focus the image is at infinity. Further, if the object is nearer REFLECTION AND REFRACTION 281 to the mirror than F, the image is virtual and on the opposite side of the mirror. The distance bp is called the focal lengthy and, in the case of a mirror— concave or convex (but not of a refracting surface)— is clearly half the radius of curvature. All these points should be verified by drawing the corre- sponding diagrams, and the convex mirror should be treated similarly. Assuming then the proof, for which there is not space here, that a spherical mirror does bring to a point-image all the rays falling on it from a point-source, it is very easy to find the posi- tion of the image of a point off the axis. For to do so we have only to trace the course of two rays : where they intersect we know that the other rays must intersect too. The geometrical method is given in Fig. 101. From h draw two rays, the first, hp, through the centre c of the mirror ; then, since it falls Fig. 101. normally on the mirror, it must be reflected along the same line ; the other, ho,, parallel to the axis acb ; then, since this might be part of the rays coming from an object indefinitely far away, it must be reflected through the principal focus f, so that its direction after reflection can be found with a pair of dividers and a ruler. The intersection h' is the image of h. Further, by dropping perpendiculars from h and h' on to the axis at a and a', the size of the object and image may be com- pared ; for if there be an object of finite size ah, then a'h' will be the image of it, and we see that when object and image are on the same side of the centre, the image is upright, when (as in the figure), they are on opposite sides, the image is inverted ; and that (seeing the triangles ahc and a'h'c are similar) a'h' : ah : : a'c: ac, or size of image _ distance of image from centre size of object "~ distance of object from centre 282 LIGHT This ratio is called the magnification produced by the mirror. It may be (as in the figure) less than unity, but is still called magnification. The student should draw corresponding figures with the object in other positions, and also for a convex mirror. Refraction at a single spherical surface. A lens has two refracting surfaces, but the case of a single surface is not only important as leading up to the other, but it gives a good first approximation to the action of the eye. The ' reduced eye ' of Listing, which reproduces fairly the really complex optical structure of the eye, consists of a single surface, between air and a medium such as the aqueous humour, convex towards the air, and of radius 5-1248 mm. "We shall take this as example. In Fig. 102 ep is the curved surface, ab the axis, a a luminous point on the axis, c the centre of curvature, cpn consequently the normal N A Fig. 102. at p. An incident ray ap makes the angle of incidence apn in air ; the angle of refraction in the aqueous humour cpa' is such that sin apn = /* sin cpa', where /*, the refractive index, is 1-3376. The refracted ray is pa' cutting the axis at a', so that this point is the image (real) of a. If light comes from infinitely far away on the axis, on the air side, it is brought to an image at p', which, since it corresponds to an infinitely distant source, is called the principal focus. Again, if light started from a point indefinitely far away in the aqueous humour, it would form an image at p ; this is, therefore, also a principal focus, and to distinguish the two, f is called anterior, f' posterior. The positions of f and f' can be found geometrically, but we shall subsequently give a convenient method of finding them arithmetically. Now suppose that the source of light is a small object ah REFLECTION AND REFRACTION 283 (Fig. 103), so that it includes points off the axis. To find the image of h two rays must be traced out, and the same rays as before are the most convenient ; one hch' passing through the centre, since it falls normally on the refracting surface, is undeviated ; another, hp, parallel to the axis is refracted through Fia. 103. the posterior principal focus f', so that its course is pf'h' ; the intersection h' is the image of h. Consequently a'h' is the image of the object ah, and the same rules hold for magnification as in the foregoing case. Numerical treatment of Tenses and mirrors. It remains to show how to deal in a convenient numerical manner with the instances of reflection and refraction described above. For this certain definitions must be introduced. We have already referred to the radius of curvature of a surface, i. e. the radius of the sphere of which the surface forms part ; but it is clear that the larger the radius is, the flatter the surface will be. Accordingly, it is more useful to state the curvature, defining this as the reciprocal of the radius of curvature. To measure this a unit is required, which is the reciprocal of a length : the unit in use is the reciprocal of a metre, and is called a diopter (written D), e. g. the surface of the reduced eye has a radius of 0-0051248 metres, so that its curvature is 1 -f 0-0051248 = 195-1 D. Again, a beam of light is in general convergent or divergent, and clearly— to take the former case— it is more strongly con- vergent when converging towards a point near at hand than one far away. It is best to take as a measure of the convergence the refractive index of the medium in which the light is travelling H- distance of tlie point to which the light is converging, and to treat divergence as negative convergence; these quantities can also, then, be measured in diopters.. The reason for this definition lies in its convenience, as will best be understood from what follows. By way of example,- in Fig. 102 the light is supposed to 284: LIGHT start from the point a at a distance of 01 metre from the refracting surface ; consequently, when it reaches that surface, it possesses a divergence of 1 -r o-i = 10 diopters, or a con- vergence of -10D. Now it can be shown that a spherical surface, whatever be the divergence or convergence of the light falling on it, alters that divergence or convergence by a fixed amount. We therefore give the following definition -.—The strength of a spherical reflecting or refracting surface is the amount by which it increases the convergence of light falling upon it. By the aid of the quantities here defined it is extremely easy to trace out the changes produced in a beam of light by any number of surfaces. The actual strengths of the various surfaces are as follows : — Plane reflecting or refracting surface . o Concave mirror 2 /> Convex mirror — 2/1 Spherical refracting surface convex towards the air . (/t— 1) p Do. concave towards the air (1— /i) p Here p is the curvature of the surface, p the refractive index of the medium on one side of the refracting surface, it being supposed that there is air on the other side 1 . The strength of the reduced eye is consequently 0-3376 x 195-1 = 65-8 diopters. In Kg. 102 the beam falling on the eye has a convergence, as shown above, of — 10 D ; add to this the strength of the surface and we arrive at +55-8D as the conver- gence of the refracted beam on leaving the surface. Hence, by the definition of convergence, it is converging towards a point (image) at a distance p -5- 55-8 = 0-02398 metres from the surface : this is the point a'. It is still simpler to find the principal focal points v and f'. If a beam of light come from a very distant object on the air side, its convergence on reaching the refracting surface is o ; consequently, on leaving that surface it is 658 D, and it will form a focus at 1-3376 -5- 65-8 = 0-02030 metres or 20-30 mm. 1 When there are two media other than air separated by the surface, the strength is (/«, — fi 3 )/>, /i x being the refractive index on the concave side, H 2 on the convex side of the surface. This formula is not used in the examples given in the text, but is of great practical importance, e. g. in the case of a water immersion objective. OPTICAL INSTRUMENTS 285 from b. The posterior focal length Br* is, therefore, 20-30 mm. The retina is situated at f' so that objects at a great distance off form images on the retina. The object in Fig. 103 would form, as we have seen, its image 23-98 mm. from b and consequently 3-68 mm. behind the retina. The actual eye possesses a power of accommodation to bring the images of near objects on to the retina. If a beam of light could start from indefinitely far away on the side of the aqueous humour in the reduced eye, it would fall on the refracting surface with zero convergence and come out into the air with convergence 65-8. But as it is now travelling in a medium of refractive index unity, the point to which it converges is at a distance 1 -j- 65-8 = 0-01518 metres or 15-18 mm, away. This is the anterior focal length bf. § 5. Optical instruments. Thin lens. A lens is a combination of two refracting surfaces, either of which may be convex, plane, or concave. Usually the two surfaces are but a short -distance apart ; this alone, however, does not constitute a thin lens in the technical sense of the word : it is necessary that the distance between the two faces should be small compared with their radii. This condition is commonly, but not always, satisfied by the lenses in actual use ; the higher power objectives of a microscope contain lenses which, though very small actually, approach a hemisphere in shape, and consequently cannot be treated as thin. The action of a thin lens can be represented by its total strength, which is the algebraic sum of the strengths of the two refracting faces. Thus in a bi-convex lens, i. e. one in which both surfaces curve outwards, if p and o- are the curvatures, the strength is (p. - 1) (p + o) ; in a lens convex one side, with curva- ture p, concave the other, with curvature o-, the strength would be(p.-i)(p- h i- ©■ sin i 2 > ~ • The value of i 2 which has a sine equal to - is called the critical angle. A spectroscope is an arrangement for separating the various 294 LIGHT kinds of light as far as possible from one another. It consists of the following parts (Fig. no) : A slit formed by a pair of Fig. 110. metal jaws moved by a screw to be a convenient distance apart. The length of the slit must be placed accurately vertical, and the breadth made as small as the intensity of the light employed will permit. The slit is carried by a tube called the collimator, and at the other end of this is placed the collimating lens, which is an achromatic telescope objective. The slit is moved to and fro in the tube till it is precisely in the principal focus of this lens, so that the light after traversing it falls in a parallel beam on the prism. The latter is usually carried on a rotating table, which can be accurately levelled, so that the refracting edge of the prism is vertical, i. e. parallel to the slit. (The figure shows two prisms on such a table.) Considering now one particular wave-length of light— say that from a sodium flame— since all the beam incident on the prism is parallel to itself, all that emerging from the prism is parallel to itself ; it is allowed to fall on an achromatic telescope, focussed to receive parallel rays (i. e. to view distant objects). The telescope for this purpose must be capable of rotation round a vertical axis passing through the prism. If it be turned in the direction of the deviated light, a sharp yellow image of the slit will be seen (or two images side by side, if the spectroscope is good enough DISPERSION 295 to separate the two sodium rays Dj and D 2 ). Now by turning the prism to and fro a position may be found for it in which the deviated image approaches nearer to the line of the collimator than in any other. This is accordingly the position of minimum deviation, or symmetrical position ; if the telescope be provided with a divided circle by which to measure the angle of deviation, the refractive index of the prism can be calculated by means of the formula on p. 292. The angle of the prism can be measured on the same instrument ; and for liquids, a hollow glass prism is made, closed by plane glass sides ; if this be filled up with the liquid in question, it can be used like the glass prism, and the refractive index of the liquid determined. If instead of light of one or two definite kinds, such as is given by an incandescent gas, a source of white light is used, containing radiation of all possible wave-lengths, there will be an infinite number of images of the slit side by side— i. e. actually a continuous horizontal band of light, of vertical height equal to that of the slit. It is this coloured band which, when derived from sunlight, is interrupted by narrow vertical dark- spaces— the Fraunhofer lines. Of course the images of the slit formed by light of neighbouring wave-lengths overlap ; the arrangement of apparatus described is designed to make the overlapping as small as possible — to obtain a so-called 'pure' spectrum. The instrument in the above form is known as the spectro- meter ; in a spectroscope for examining the character of light from various sources, observing absorption bands, &c, it is not necessary that the prism should be placed on a rotating table, nor that the telescope should be capable of more than a small angular motion so as to pass from one end of the spectrum to the other. On the other hand, a single prism gives a short spectrum : more dispersion may be obtained by using two or more prisms in succession. Again, for minute examination of the spectrum, photography is better than a subjective method, so that a spectroscope may conveniently consist of a collimator as described, a chain of prisms permanently set up, so as to be in the position of least deviation for some standard kind of light, say the D ray, and a photographic camera arranged so as to receive the light from the prisms and throw it on a horizontal strip of sensitized film. 296 LIGHT To compare the spectra of two lights, one is placed directly in front of the collimator, the other at the side in such a position that a totally reflecting prism can throw the beam from it on to the slit, this prism arranged so as to cover only; say, the upper half of the slit. Consequently, the upper half of the field of view of the telescope or camera receives only light from one of the sources, the lower only from the other. If one of the spectra is formed by sunlight, the Fraunhofer lines serve as a convenient means of comparison by which to determine the position of any lines seen in the other spectrum. An instrument, such as that described above, cannot be easily moved ; for that reason portable hand spectroscopes have been devised, in which all the parts are enclosed in one tube. The instrument can therefore be directed at any object like a telescope. The usual construction of the prism in such an instrument is of alternate crown and flint prisms, turned with their edges oppo- site ways (Fig. in). The deviation produced by the flint may thus be neutralized by the crown, without neu- tralizing the dispersion. Fig. 111. Taking the approximate formula for thin prisms, d = (ji— 1) 1, the theory of this may be given simply. Using, as before, suffixes cdf to indicate the kind of light employed, we may take as the mean deviation (/i D — 1) 1, and choose the angles of the prisms so that this quantity has the same value for the flint prism (or sum of the prisms) as for the sum of the crown prisms ; hence the D (yellow) ray will emerge from the set of prisms in the same direction as it entered. But the angular extent of spectrum (between two given rays, say C and F) produced by a single prism is the difference between the deviation of the C ray {n — 1) ' and that of the F ray (p F —i) 1, that is (jJ-F—fo)'; or, as we may write it, or, in words, the mean deviation, divided by the ratio given in the last column of the table, p. 264. As the mean deviation is to be the same for flint and crown, but the ratio is 36-9 for flint, and 60-9 for crown, the dispersion of the flint will overpower that of the crown, and a spectrum will be produced in the same direction, though not so long as that which the flint prisms alone would give. The achromatization of objectives involves the converse problem, that THE EYE 297 of producing deviation (change of convergence) in the light falling on them, but 'without dispersion, and can accordingly be treated by the same method. We have (jjl — i)r for the strength; consequently, as before, 1 f — Po) r for the difference in strengths for the blue and red rays ; but the difference in strength of the crown lens must be neutralized by an equal difference in the opposite sense produced by the (concave) flint lens. So that (mj— Mo) •" being the same for each, their respective strengths for the mean (yellow) ray will be in proportion to the numbers given in the last column of the table, i. e. as 609 : 36-9. If, for example, the crown lens is of 6.09 D, and the flint of —3-69, they will achromatize and yield a compound lens of 2-4 D. § 7. The eye. The optical structure of the eye is essentially as follows — (i) The cornea, approximately spherical in shape. The radius of curvature of the anterior surface is about 8 mm., the curva- ture, therefore, 10004-8 = 125 diopters. The medium that the cornea is composed of has a refractive index of 1-3376, slightly greater than that of water. The surface, it is easily seen, pro- duces a converging effect on light incident on it (see Fig. 112), and its strength is 125 x (1-3376 - i)=42-2 D ; it accounts, therefore, for about two-thirds of the whole refracting power of the eye (p. 284) : the so-called ' reduced eye ' may be regarded as the cornea, some- what strengthened so as to include the effect of the crystalline lens. Behind the cornea comes the vitreous humour, but as this has practically the same refractive index, the distinction has.no optical effect. (ii) 4 mm. behind the anterior face of the cornea comes the anterior face of the lens. The lens is 4 mm. thick, and more sharply curved behind than in front, its curvatures (in repose) being 100 D (10 mm.) in front and 167 D (6 mm.) behind. It is not, like a glass lens, made of homogeneous material, but grows denser (and more strongly refracting) towards the centre. The mean refractive index is 1-4545. According to the rule given in the footnote to p. 284, both faces of the lens produce a converging effect, which therefore augments that of the cornea. (iii) 13 mm. behind the posterior face of the lens is the retina, the screen on which the optical image is to be formed. The space between the two is filled by the vitreous humour, which has the same refractive index as the aqueous. 29a LIGHT The eye caiinot be treated as a ' thin lens,' in which the effects of the two faces need merely he added to get the effect of the whole, for the thicknesses involved are of the same order of magnitude as the radii of curvature. For a complete optical treatment we must refer to larger treatises. The result is to show that it has a strength of about 66 D, and is so arranged that the image of a verydistant object is formed on the retina, the course of the rays being that shown in Fig. 112. Here a is the Fig. 112. anterior surface of the cornea, bc the lens. The ray pq is bent by the cornea into the direction qeg; hence the image of a distant object if formed by the cornea alone would fall at g. The ray, however, is bent again at k, and again at s, and reaches the axis at f, which is, consequently, the principal focus for the whole eye. It is here that the retina lies in the normal eye. "When the eye has to view an object near at hand, it is accommo- dated for the purpose by a peripheral compression of the lens, which increases its curvature, and consequently its strength. The amount of accommodation that can be effected varies with age, being greatest for children ; it is usually stated at 5 diopters for adults, i. e. a distance of 20 cm. ; but with effort the eye can usually see objects distinctly at shorter distances than this ; this means an increase in strength of the eye from 66 to 71 D. The course of the rays in dealing with a near object may be gathered by a comparison of Fig. 102. Here the single refracting surface (representing a simplified eye) has its principal focus at f', and forms an image at a' : in the eye the adjustment would be such that a' falls on the retina, and consequently the principal focus (which, for a distant object, is on the retina) now lies in front of it. The chief optical defects of the eye are — ma. The eye is too strongly curved, consequently the THE EYE 299 image of a distant object is brought to a focus in front of the retina, and is not clearly perceived. .If e. g. the eye be myopic by 2 diopters, then when in repose it is adapted to light of that divergence, that is to light from an object half a metre away ; that is the ' far point of vision ' for such an eye. The accommo- dation may be normal : in this case, instead of being able to deal with light of 5 D divergence, as the normal eye can, it would be able to deal with 7 D, i. e. with light coming from about 14 cms. distance ; such an eye would, therefore, be able to see more detail in a small object than the normal eye without a magnify- ing glass. The remedy for myopia is to use divergent spectacles such as will reduce the eye to the normal strength ; in the case supposed, 2 D. Hypermetropia. The eye is not sufficiently curved. Hence the ' near point of vision ' is abnormally far off. Thus, if an eye be hypermetropic by 2 D, and has the normal accommodation, it will at most be able to deal with 5-2=3 D divergence, and an object must be placed 33 cms. away to be seen. The remedy is the use of convergent spectacles. Presbyopia is loss of accommodation, usually due to age. Its optical effect is the same as that of hyper mel^opia. Astigmatism is unequal strength in different directions, usually due to the cornea being more curved in one meridian than another. It can be remedied by the use of astigmatic lenses, i. e. lenses made with cylindrical surfaces, instead of spherical. Such lenses have a refracting effect at right angles to the axis of the cylinder, but parallel to the axis have none, i. e. they behave like plane glass. INDEX Abbe, 277. Aberration, 279. Absolute temperature, 51, 64. Absorption and radiation, 271. — of wave, 73. — spectra, 277. Acceleration, 8. — of gravity, 17. Accommodation, 298. Accumulator, 198. Achromatism, 296. Aeration of blood, 175. Air-pump, 42. Alternating currents, 221. — dynamo, 247. Amalgamation of zinc, 194. Ammeter, 187. Ampere, 235. Ampere's rule, 233. Amplitude, 32. Andrews, 92. Anode, 189. Aperture of a lens, ego. Areometric method, 12. Arm, mechanics of, 28. Artery, flow of blood in, 119. Astatic galvanometer, 214. Atmospheric pressure, 38. Atomic theory, 3. Automatic interrupter, 243. Avidity, 169. Avogadro's law, 67, 78. Back electromotive force, 205. Ballistic balance, 13. — galvanometer, 241. Barlow's wheel, 237. Barometer, 37. Barometric correction of thermo- meter, 47. Beckmann, 99. Beckmann thermometer, 49. Berget, 70. Bekthelot, 104. Bichromate cell, 196. Blood, density of, in. Boiling, air bubbles in, in. Boltzmann, 65. Boyle, 39. Boyle's law, 77. Breaking point, 136. Brittleness, 136. Bunsen, 105. Bunsen cell, 196. Caelletet, 93. Calorie, 55. Calorimeter, 103. Camera, 291. Capacity, 250. — and conductance, 253. Capillary electrometer, 216. — error of barometer, 112. — tube, 112. Carbon megohm, 202. Cathode, 103. Centre of gravity, 31. Characteristic equation, 79. Charles's law, 78. Chauveau, 61. Chemical equilibrium and tem- perature, 183. Chronograph, 5. Circulatory system, 118. Clark cell, 197. Clausius, 65. Clinical thermometer, 50. Clothing, efficiency of, 76. Coefficient of expansion, 88. Cohen, 173. Colloids, 128. Concentration, 166. Condensation of gases, 93. Condenser, 250. Conductance, 199. Conductivity, electric, 200. — of the soil, 71. — thermal, 68. Conductors and insulators, 185. Constant volume thermometer, 50. Contracted vein of liquid, 114. Convection, 72. Creeping of salt solutions, 112. Critical angle, 293. — point, 92. Cryohydrates, 143. INDEX 301 Current balance, an. Curvature, 283. Dalton, 6s. Dalton's law, 95. Damping of galvanometer, 242. Daniell cell, 192, 197. D'Araonval galvanometer, 213, 239. Decimation, 230. Degrees of freedom, 171. Density, 9. — of solids, 12. Dew point, 97. Dielectric, 250. — constant, 256. Diffraction, 265. Diffusion of dissolved substances, 122. Diffusivity, 121. Dilatometer, 88. Diopter, 283. Dip, 230. Direct vision spectroscope, 296. Dispersion, 263, 292. Dispersive power, 296. Dissipation of energy, 64. Dissociation of water, 169. Distribution of energy in spectrum, 272. Drying agents, 175. Dulong and Petit's law, 141. Dynamical equivalent of heat, 55. Dynamic vapour pressure measure- ment, 87. Dynamo, 241. Dyne, 16. Ear, perception of sound by, 161. Earth coil, 240. Effective force, 15. Efficiency of an engine, 61, 64. — of human body, 180, 183. — of lights, 272. Elasticity, 80. — of a liquid, 85. — of solids, 129. Elastic limit, 136. — after-effect, 138. Electrical equivalent of heat, 224. — power, 193. Electric heating, 223. Electrolysis, 189. Electrolyte resistance, 22-1. Electrolytic dissociation theory, 99, 108, 168. Electromagnet, 233, 236. Electro-magnetic radiation, 258, 268. Electrometer, 216. Electromotive force, distribution of, 204. of cell, 193. Electromotor, 238. Electrophorus, 253. Electroscope, 254. Electrostatic induction, 253. — stress, 249. Emissivity, 74. Endosmotic equivalent, 125. Endothermic reaction, 179. Energetics, 3. Energy, 1. — units, 22. Equilibrium, 15. Erg, 20. Ether, 249. Evaporation from the skin, 102. Expansion of solids, 138. Expired air, composition of, 96. Eye, 297. Eye-piece, 288. Fakaday, 190. Fick, 17, 40. Filter pump, 117. Fleuss pump, 43. Flexure, 134. Flow of liquids, 131. — of solids, 137. Fluid pressure, 36. Fluorescence, 273. Focal length, 281, 284. Food stuffs, thermal value of, 179. Force, 14. Fraunhofer lines, 276, 295. Free energy, 183. — expansion of gases, 108. Frictional electricity, 253. Galvanometer, 188. Gas constant, 79. Gases, spectra of, 273. — weight of, 96. Gibbs, 173. Glauber salt, transition of, 174. Gradient of concentration, 120. ;o2 INDEX Grove cell, 196. Hardness, 137. Harmonics, 149. Heart, work of the, 25. Heat in electric circuit, 216. Heating effect of an electric cur- rent, 186. Heat.of reaction, 177. Helium, 276. Helmholtz, 162, 244. Hertz, 237. Hicks, 13. Hirn, 2, 59. Homogeneous system, 164. Hooke's law, 81, 130. Horse-power, 24. Human body, efficiency of, 61. Hurthle, 41. Hydrates, equilibrium of, 174. Hydriodic acid, equilibrium of, 167. Hydrogen and oxygen, combination of, 167. Hydrometer, n. Hygrometry, 97. Hysteresis, 231. Image, 279. Incandescence, 272. Index notation, 4. Indicator diagram, 24. Induction coil, 243. Inertia, 16. Infra red, 261. Insulation resistance, 201. Intensity of magnetization, 229. — of sound, 162. Internal energy, 176. — energy of fluids, 101. — latent heat, 107. Ion, 168, 188. Isothermal, 79. — of a liquid, 84. Isotonic method, 17. Joly, 105. Joule, 22. Joule, 2, 56, 108. Kelvin, 65, 71, 108, 211. Kinetic energy, 21. Kirchhoee, 271. Kymograph, 40. Laplace, 153. Latent heat, 102. of fusion, 141. of transition, 173. Law of inverse squares, 251. Laws of solution, 127. Lees, 69. Length, unit of, 3. Lenz's law, 240. Lever, 30. ■ Light and sound waves, 264. Linde, 94, 108. Lines of force, 251. magnetic force, 227. Line spectra, 274. Lippmann, 216. Liquefication of air, 95. Liquid mixtures, vapour pressure of, 98. Listing, 282. Litre, 4. Longitudinal and transverse vibra- tions, 150. Luminescence, 272. Magnetic effect of current, 187. — field, 229. — induction, 240. — moment, 228. Magnetism of iron, 173. Magnetization of iron, 231. Magnetizing coil, 235. Magneto, 241. Magnetometer, 231. Magnification of image, 282. Magnifying glass, 287. Manganin, 201. Mass, standards of, 9. Maximum and minimum, 180. — density of water, 72. Maxwell, 63. Mean free path, 66. Melting point, 139. and pressure, 140. Mercury pump, 44. Metabolism, 181. Metastable equilibrium, 181. Mho, 199. Michelson, 3. Microscope, 290. Minimum deviation, 292. — potential energy, 180. Mirror galvanometer, 213. INDEX 3°3 Mixtures, melting point of, 14a. Molecular volume, 79. Molecule, 65. Moment of a force, 29. Momentum, 12. Musical scale, 148. Neef's hammer, 245. Nerve-muscle electromotive force, 210. Newton, 270. Normal salt solution, conductivity of, 201. — scale of temperature, 50. Ohm's law, 199. Optical centre, 286. Organ pipes, 147. Oscillations, 32. Oscillatory discharge, 257. Osmotic pressure, 126. at freezing point, 144. Peclet, 69. Pendulum, 34. Perfect gas, deviation from, 83. — gases, 77. Permanent set, 136. Perpetual motion, 62. Phase, 34. — of chemical system, 170. Phosphorescence, 273. Photometer, 262. Physiological calorimeter, 61, Pictet, 93. Pitch of sounds, 147. Plants, respiration of, 170. Plasmolysis, 128. Platinum thermometer, 223. Polarization of electrodes, 194. — of light, 268. Positive and negative poles, 204. Post office box, 220. Potential energy, 21. Potentiometer, 217. Power, 24. — in electric circuit, 207. Pressure gradient, 115. — gauge, 40. — in blood vessels, 114. — in branch tubes, 118. — in soap bubble, no. Principal focus, 280. Prism, 291. Pulley system, 30. Pulse, 119. Quadrant electrometer, 255. Quality of sounds, 148. Quantity of magnetism, 228. Eamsay, 276. Raotjlt, 99. Ratio of static and magnetic units, 252. Rayleigh, 162. Reaction constant, 166. Reduced eye, 282. Reflection and refraction of sound, 154- — law of, 266, 267. Refractive index, 263. Regnault, 91, 104, 153. Resistivity, 200. Resonance, 157. — of the ear, 163. Respiration, 121. Resultant of forces, 15. Reversal of spectral lines, 275. Reversible reaction, 165. Rigidity, 131. ROntgen rays, 258. Rowland, 56. Rumfokd, 2. Salt solutions,freezing point of, 143. Sap, rise of, 112, 128. Saturation pressure, 85. Screw, 29. Second, 4. Semi-permeable membrane, 125. Sensitiveness of the eye, 277. Series and parallel arrangement, 207. Shear, 130. Short circuit, 194, 208. Shunt, 208. Simple harmonic motion, 33. Sine curve, 35. Size of atoms, 66. — of drops, 1 10. Soft tissues, elasticity of, 132. Solutions, vapour pressure of, 99. — boiling point of, 99. Specific gravity bottle, 10. — heat, 55. 3°4 INDEX Specific heat of gases, 102, 107. Spectacle lenses, 299. Spectroscope, 269, 294. Sprengel pyknometer, 10. Stability and temperature, 174. — of solid and liquid, 141. Stable equilibrium, 31. Static vapour pressure measure- ment, 86. Steam engine test, 60. Stefan, 270. Stefan's law, 75. Stokes, 270. Strain, 80. Sulphur, transformation of, 172. Tangent galvanometer, 212, 235. Telescope, 289. Temperament, 148. Temperature, effect of reaction, 169. — gradient, 70. — and molecular theory, 67. — of visibility, 261. Terrestrial magnetism, 230. Theory of exchanges, 270. Therm, 59. Thermal capacity, 55. — aftereffect, 139. — stress on solids, 139. Thermo-couple, 225. Thermometer, 46. Thermostat, 52. Thin lens, 285. Third law of motion, 13. Thomson, J. J., 66. Time marker, 7. Tin, transition of, 174. Torque, 30. Total reflection, 293. Transformer, 247. Transition point, 173. Triple point, 172. True and mean coefficient, 89. True specific heat, 102. Tuning-fork (electro-magnetic), 4. Twist, 132. Ultra violet, 261. spectrum, 269. Under cooled liquid, 141. Unit of current, 235. Vacuum jacket, 76. — tube, 258. Van 't Hoff, 99. Van 't Hoff s analogy, 127. Vectors, 8, 26. Velocity, 8. — head, 113, 116. — of light, 263. — of reaction, 166. — of sound, 153. Ventilation, 97. Vibrations in wires, 146. Virtual velocities, principle of, 28. Viscosity of liquids, 115, 116. Visible spectrum, 261. Volt, 193. Voltage, 192. Voltameter, 190. Voltmeter, 216. Vowel-sounds, 160. Water pump, 45. Water, resistance of pure^2oi. Watt, 24. Wave, propagation of, 152 — front, 265. — length, 36. — length and temperature, 271. *«-" — length of light, 261. ^~ — — length standard, 3. — motion, 35. Wehnelt interrupter, 259. Weight thermometer, 90. Wet and dry bulb thermometer, 98. Wheatstone's bridge, 219. Wimshurst machine, 254. Work, 20. ;ram, 23. Young's modulus, 133. OXFORD: HORACE HART PRINTER TO THE UNIVERSITY Arnold's School Series. THE LONDON SCHOOL ATLAS. Edited by H. O. Abnold-Eoestbb, M.P., Author of "The Citiaen Reader," "This World of Ours," etc. A magnificent Atlas, including 48 pages of Coloured Maps, several of them double-page, and Pictorial Diagrams. 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FORUM LATINUM. A First Latin Book. By E. Vernon Arnold, Litt.D., Professor of Latin at the University College of North Wales. In three parts. Is. 4d. each. Complete, 3s. 6d. G/ESAR S GALLIC WAR. Books I. and II. Edited by T. W. H addon, M.A., Second Classical Master at the City of London School, and G. C. Harrison, M.A., Assistant-Master of Fettes College. With Notes, Maps, Plans, Illustrations, Helps for Composition, and Vocabulary. Cloth, Is. 6d. Books Hl.-V. Edited for the use of Schools by M. T. Tatham, M.A. Uniform with Books I. and II. Crown 8vo., cloth, Is. 6d. Books VI. and VII. By M. T. Tatham, M.A. Uniform with Books III.-V. Is. 6d. A LATIN TRANSLATION PRIMER. With Grammatical Hints, Exercises and Vocabulary. By George B. Gardiner, Assistant-Master at the Edinburgh Academy, and Andrew Gardiner, M.A. Grown 8vo., cloth, Is. LONDON : EDWARD ARNOLD. ( 3 ) Arnold's School Series. GERMAN. EASY GERMAN TEXTS. For pupils who have acquired a simple vocabulary and the elements of German. Edited by Walter Rippmann, M.A., Professor of German at Queen's College, London. With exercises on the text and a list of the strong and irregular verbs. Small crown 8vo., cloth, Is. 3d. each. ANDEESEN'S BILDERBUCH OHNE BILDER (What the Moon Saw). PRINZESSIN ILSE. By Makie Petersen. GERMAN WITHOUT TEARS. By Mrs. Hugh Bell. A version in German of the author's very popular "French Without Tears." With the original illustrations. Crown 8vo., cloth, Part I., 9d. Part II., Is. Part III., Is. 3d. LESSONS IN GERMAN. A graduated German Course, with Exercises and Vocabulary, by L. Ihhes Ldmsden, late Warden of University Hall, St. Andrews. Crown 8vo., 8s. EXERCISES IN GERMAN COMPOSITION. By Kiohabd Kaiseb, Teacher of Modern Languages in the High School of Glasgow. Including care- fully graded Exercises, Idiomatic Phrases, and Vocabulary. Crown 8vo., cloth, Is. 6d. ELEINES HATTSTHEATER. 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Crown 8vo., cloth, Is. 6d. POEMES CHOISIS. Selected and Edited by R. L. A. Du Pontet, MA. Cloth, Is. 6d. LES FRANCAIS EN MANAGE. By Jetta S. Wolff. With Illustrations. Crown 8vo., cloth, Is. 6d. An entirely original book, teaching the ordinary conversation of family life in France by a series of bright and entertaining scenes. LES FRANCAIS EN VOYAGE. By Jetta S. Wolff. A com- panion volume to the preceding, Riving a lively account of travelling on the continent. Cleverly illustrated. Crown 8vo., cloth, Is. 6d. FRANCAIS POUR LES TOUT PETITS. By Jetta S. Wolff. With Illustrations by W.|Fosteb. Cloth, ls..8d. FRENCH DRAMATIC SCENES. By C. Abel Musgrave. With Notes and Vocabulary. Crown 8vo., cloth, 2s. LONDON: EDWARD ARNOLD. ( 4 ) Arnold's School Series. FRENCH. FRENCH WITHOUT TEARS. A graduated Series of French Reading Books, carefully arranged to suit the requirements of quite young chil- dren beginning French. With Humorous Illustrations, Notea, and Vocabulary. By Mrs. Hugh Bill, author of " Le Petit Theatre Francais." drown 8vo., eloth, Book I., 9d. Book II., Is. Book III., Is. Sd. A FIRST FRENCH COURSE. Complete, with Grammar, Exercises and Vocabulary. By James BoSelle, B.A. (Univ. Gall.), Senior Frenoh Master at Dvilwich College, etc. drown 8vo., oloth, Is. 6d. A FIRST FRENCH READER. With Exercises for Re-translation. Edited by W. J. Greenstreet, M. A., Head Master of the Marling School, Stroud, drown 8vo., cloth, Is. FRENCH REVOLUTION READINGS. Edited, with Notes, Introduction, Helps for Composition and Exercises. By A. Jamson Smith, M. A., Head Master of King Edward's School, Camp Hill, Birmingham ; and C. M. Dix, M.A., Assistant Master at the Oratory School, Birmingham. Crown 8vo., cloth, 2s. SIMPLE FRENCH STORIES. An entirely new series of easy textB, with Notes, Vocabulary, and Table of Irregular Verbs, prepared under the General Editorship of Mr. L. Von Glehn, Assistant Master at Merchant Taylors' School. About 80 pages in each volume. Lamp cloth, 9d. T/n Drame dans les Airs. By Jules Verne. Edited by I. G. Lloyd- Jones, B.A., Assistant Master at Cheltenham College. Pif-Paf. By Edouabd Laboulaye. Edited by W. M. Poole, M. A., Assistant Master at Merchant Taylors' School. La Petite Souris Orise; andHistoire de Rosette. By Madame de Sequr. Edited by Blanche Daly Cocking. T/n Anniversaire a Londres, and two other stories. By P. J. Stahl. Edited by d. E. B. Hewitt, M.A., Assistant Master at Marlborough College, Monsieur le Vent et Madame la Plule. By Paul de Musset. Edited by Miss Leaky, Assistant Mistress at the Girls' High School, Sheffield. Poucinet, and two other tales. By Edouard Laboulaye. Edited by "W.M. Poole, M.A., Assistant Master at Mer- chant Taylors' School. GIL BLAS IN THE DEN OF THIEVES. Arranged from Le Sage. With Notes and Vocabulary by R. de Blanchaud, B.A., Assistant Master at George Watson's Ladies College. Limp cloth, crown 8vo., 9d. [ Uniform with Vie above series. 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Lehpeldt, Professor of Physics at the East London Technical College. 2s. 6d. The Standard Course of Elementary Chemistry. By E. J. Cox, E.C.S., Headmaster of the Technical School, Birmingham. In Five Parts, issued separately, bound in cloth and illustrated. Parts I. -IV., 7d. each ; Part V., Is. The complete work in one vol., 3s. Physical Determinations. Laboratory Instructions for the Determination of Physical Quantities connected with General Physics, Heat, Electricity and Magnetism, Light and Sound. By W. E. Kelsey, B.So., A.I.E.E. Crown 8vo., cloth, 4s. 6d. A Text-Book of Physical Chemistry. By Dr. E. A. Lehpeldt, Professor of Physics at the East London Technical College. With 40 Illustrations. Crown 8vo., cloth, 7s. 6d. Chemistry for Agricultural Students. ByT. S. Dymond, F.I.C., Lecturer in the County Technical Laboratories, Chelmsford. With a Preface by Professor Meldola, F.R.S. Cloth, 2s. 6d. A Text-Book of Zoology. By G. P. Mudge, A.R.C.Sc. 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