GIFT OF A TEXT-BOOK GENERAL PHYSICS FOR THE USE OF COLLEGES AND SCIENTIFIC SCHOOLS CHARLES S. HASTINGS, Pn.D AND FREDERICK E. BEACH, PH.D. OF YALE UNIVERSITY BOSTON, U.S.A. GINN & COMPANY, PUBLISHERS Cbe &tbenffttm 1901 COPYRIGHT, 1898, BY CHARLES S. HASTINGS AND FREDERICK E. BEACH ALL RIGHTS RESERVED > fo \ PREFACE. THE most marked improvement in the method of recent text-books on physics is the growing tendency to emphasize the essential continuity of the science. It is true that phys- ics, as a matter of convenience, still has to deal with certain phenomena of the sensations of sound and of light, which logically belong to physiology or psychology; but there is comparatively little difficulty in so isolating these portions of the text from those which may be dealt with in a purely objective way that there need remain no reason for the mental confusion which so often arises from the natural tendency to accept a sensation as a just measure of its awakening cause. Thus, if we set aside Chapters XXXIV and XLIV of the following pages, which treat primarily of the relations of sensations to their physical causes, and which will doubtless long be wanting in the scientific precision that characterizes other portions of physics, we find that all the remainder can be described as a strictly quantitative study of various trans- ferences and transformations of energy. An understanding of energy is, therefore, absolutely essential to a satisfactory intellectual grasp of physics. This can only be attained by sustained study of dynamics, whence elementary mechanics must be regarded as the logical basis of the whole science of physics. No pains should be spared on the part of the student in attaining clear notions on this portion of his course. This conviction has prompted the writers to make 96706 iv PREFACE. their treatment of mechanics more complete than is ordi- narily the case, especially in the physical notions which attach to the simplest cases of the action of forces. For the purpose of giving familiarity with these ideas, many problems are appended to the various chapters, and the utility of exercises which involve the solution of these by the students is strongly urged upon teachers. The class of students for which this text-book is designed is supposed to have a useful knowledge of trigonometry, but not of calculus. This is in agreement with the courses of instruction in most of our American colleges ; but it has ordinarily the disadvantage of leaving rather a large inter- val betwee.n the study of the philosophy of physics and the application of its principles to engineering. Especially true is this of thermodynamics and electricity where it is often difficult for the student to recognize the fact that the un- accustomed mathematical processes are simply easier means of attaining an understanding of a physical problem and not an end in themselves. For this reason the subjects men- tioned are developed with somewhat more completeness than usual, so that the engineering student can find the essential notions of his advanced work logically connected with those acquired at an earlier time. Of course such an extension implies an exercise of choice on the part of the teacher as to what may be omitted in first reading with a class of which only a portion expects to pursue the subject farther. In Chapter XLI will be found a treatment, quite ele- mentary in character, of the limiting powers of optical instruments. This is, of course, of great philosophical in- terest, because it is by the means of such instruments that we attain the greatest enlargement of our intellectual horizon. Notwithstanding the simplicity of the exposition, it does not seem to have been done before in unmathematical language. PREFACE. v The book as a whole is designed as an aid to the teacher in presenting a general view of the phenomena and philos- ophy of physics; but it assumes as an essential comple- ment a course of demonstrations in the form of experimental lectures, and, for those students who aspire to acquire a knowledge of physics more than merely sufficient to enable them to follow the growth of science with an intelligent in- terest, a supplementary course in the physical laboratory. CHAKLES S. HASTINGS. FREDERICK E. BEACH. SHEFFIELD SCIENTIFIC SCHOOL OF YALE UNIVERSITY, January, 1899. CONTENTS. PART I. -MECHANICS. CHAPTER PAGES I. PHYSICAL QUANTITIES AND THEIR MEASUREMENT . 1-24 II. SIMPLE TYPES OF MOTION 25-68 III. WORK AND ENERGY 69-78 IV. MECHANICS OF A RIGID BODY 79-97 V. ELASTICITY 98-114 VI. MECHANICS OF FLUIDS 115-137 VII. SURFACE TENSION 138-159 PART II. -HEAT. VIII. THERMOMETRY 160-171 IX. EXPANSION 172-186 X. CALORIMETRY 187-199 XL CHANGE OF STATE 200-225 XII. SOLUTIONS 226-243 XIII. TRANSFERENCE OF HEAT 244-257 XIV. THERMODYNAMICS 258-289 XV. KINETIC THEORY OF GASES . 290-308 PART III. ELECTRICITY. XVI. ELECTRIFICATION 309-318 XVII. THE ELECTRIC FIELD 319-329 XVIII. ELECTROSTATIC INSTRUMENTS 330-346 XIX. THE ELECTRIC DISCHARGE 347-356 XX. MAGNETISM 357-384 XXI. THE ELECTRIC CURRENT 385-405 XXII. THE ELECTROMAGNETIC FIELD 406-414 XXIII. GALVANOMETRY 415-437 XXIV. RELATIONS BETWEEN HEAT AND ELECTRICITY . . . 438-449 XXV. DIMENSIONS AND UNITS OF ELECTRICAL QUANTITIES . 450-460 viii CONTENTS. CHAPTER PAGES XXVI. INDUCTION OP CURRENTS 4C1-485 XXVII. TELEGRAPH AND TELEPHONE 486-494 XXVIII. PASSAGE OF ELECTRICITY THROUGH GASES .... 495-505 XXIX. ELECTRIC WAVES 506-513 PART IV. -SOUND. XXX. WAVES 514-539 XXXI. SOUNDS AND THEIR RELATIONS 540-549 XXXII. PROPAGATION OF SOUND-WAVES 550-554 XXXIII. SONOROUS BODIES 555-575 XXXIV. COMPOUND TONES 576-590 XXXV. MUSICAL INSTRUMENTS . 591-598 PART V.- LIGHT. XXXVI. NATUBE AND PROPAGATION OF LIGHT 599-608 XXXVII. REFLECTION AND REFRACTION 609-626 XXXVIII. ELEMENTARY THEORY OF OPTICAL INSTRUMENTS . . 627-640 XXXIX. INTERFERENCE 641-658 XL. DISPERSION 659-664 XLI. MAXIMUM EFFICIENCY OF OPTICAL INSTRUMENTS . 665-679 XLII. OPTICAL PHENOMENA OF THE ATMOSPHERE .... 680-702 XLIII. RADIATION AND ABSORPTION OF LIGHT-WAVES . . 703-713 XLIV. SENSATIONS OF COLOR 714-726 XLV. POLARIZATION . 727-752 INDEX 753-768 OF THE UNIVERSITY OF GENERAL PHYSICS. PART L MECHANICS. CHAPTER I. PHYSICAL QUANTITIES AND THEIR MEASUREMENT. 1. Measurement of Physical Magnitudes The measure- ment of any physical, i.e. concrete, magnitude, consists in the comparison of the quantity to be measured with a definite portion of the same magnitude selected as a unit. The process of comparison may be either direct or indirect. In practice, the direct method is used only in the comparison of lengths, and the measurement of all other magnitudes has, with an insignificant exception, been reduced to the judging of the coincidence of two lines or the determination of a length, for the reason that the perceptions of the eye are more trustworthy than those of any other sense. The result of the comparison of physical quantities is expressed as so many times the unit chosen. If, for example, a rod is found to be four times as long as a foot-rule, its length is said to be 4 [ft.]. The numerical part of this expression is termed the numeric, and the part enclosed in brackets the physical unit. It is obvious that no estimate of the size of the quantity measured can be formed unless the unit is explicitly stated. MECHANICS. 2. Reduction of Observations. The numbers which arise from the measurement of continuous physical quantities differ in an important respect from those numbers relating to discrete quantities which are the subject of calculation in arithmetic. The latter are complete and exact ; the former will always be incomplete and subject to error either on account of the limitations of the observer or the imperfections of the instruments by whose aid they are obtained. Hence, in order to obtain trustworthy results from a calculation, it is desirable to make a critical inspection of the numbers which enter into every computation. The importance of such examination may be made clear by an example. Sup- pose, for instance, that there are 124 piles of coin on a table, and that each pile contains 48 coins, making a total of 124 X 48 = 5952 coins. This number is complete and cannot be stated with greater accuracy. On the other hand, suppose that it is desired to compute the area of a table top from its measured dimensions, which by the aid of a scale have been found to be the following : Length on one side 96.33 cm. " " opposite side 96.35 cm. Width " one end 68.69 cm. " " opposite end 68.65 cm. Selecting the mean of each of these pairs as the most probable value, the length may be taken as 96.34 cm. and the breadth as 68.67 cm. It is manifestly absurd to express this number to thousandths of a centimeter, for if the table were to be measured at intermediate points the figure in the second decimal place would very likely be changed. To refine these measurements would, in general, be labor thrown away, as far as any practical use of this area is concerned. Accept- ing, then, as satisfactory a measurement which is accurate REDUCTION OF OBSERVATIONS. 3 enough for the purpose, let the doubt which attaches to the last figure of each number be denoted by the italic type and the calculation performed in the usual way. Thus : 96.34 cm 68.67 cm 67438 57804 5780^ It will be noted that the doubtful figures have rendered worthless at least four figures of this product. In other words, the percentage error of a product cannot be less than that of the worst factor entering into it. In the example given, the last four figures of the product might better be omitted; for, as they stand, they are liable to mislead the casual reader into supposing the result more accurate than it is. The labor of calculation of useless figures without sacrifice of any precision in the result may be saved by the following abridged method of multiplication. Reverse the order of the figures in the multiplier, writing them directly below those of the multiplicand. The multiplication may now proceed in the usual way except that the first, i.e. the right- hand, figure of each partial product is obtained by multiply- ing the figure of the multiplicand by the one directly below it in the multiplier, mentally adding anything which should be carried from the multiplication of the preceding figure of the multiplicand by this figure of the multiplier. Writing these partial products so that their right-hand figures stand in a vertical column, they may then be added and the deci- mal point determined by inspection thus : MECHANICS. 96.3^ 76.86 5780^ 7707 578 67 6Q15.6 The precision of this method is as great as before, and in the example given there is a saving of six multiplications and the addition of three columns. If the numbers are large, the pointing off will be assisted by expressing them in powers of ten (Art. 3) before perform- ing the multiplication. The corresponding abridgment of division consists in de- leting a figure of the divisor in each partial division after the first, instead of bringing down a cipher. For example : 61803 4353 4120 233 206 27 27 Calculations involving several numbers may be most readily made by the use of logarithms. In this connection it should be noted that four-place tables are sufficient for the ordinary purposes of engineering, navigation, and the work of the chemical and physical laboratory, since in most cases these are all that the accuracy of the data will warrant. Very few measurements in the branches mentioned admit a precision FUNDAMENTAL MAGNITUDES. 5 greater than 1 of one per cent ; on the other hand, the error in a four-place logarithm will never amount to more than Q of this, and is usually far less. 3. Notation in Powers of Ten. The measurements which have to be recorded in Physics differ enormously in their relative magnitudes, though never exceeding in number seven significant figures. Thus, for example, the wave length of yellow light (D^ is \ = 0.00005896 cm., while the velocity is v = 29,990,000,000 cm. per sec. Instead of writing so many ciphers, the following brief and convenient notation in powers of ten is often used : X = 0.5896 (10)- 4 cm. v = 2.999 (10) 10 cm. per sec. It is convenient to place the decimal point before the first figure if it is greater than 5, and after if less, as in above example. 4. Fundamental Magnitudes. Experience has shown that the measurement of all physical quantities may be expressed in terms of three fundamental magnitudes. Those commonly chosen for this purpose are time, length, and mass, or quan- tity of matter. It may be assumed that our ideas of time and space, whether innate -or acquired, are sufficiently exact for all practical purposes. The case of matter, however, requires more particular consideration. Of the three magni- tudes named, matter alone is directly cognizable by the senses, and invested with a variety of interesting properties. From the chemical standpoint the most important of these is its 6 MECHANICS. indestructibility, which may be stated as a formal proposition as follows : In any closed space the total quantity of matter is invariable, irrespective of the changes of form which it may undergo. For present purposes matter may be defined as anything which can be weighed, and the quantity of matter as propor- tional to its weight, i.e. its attraction toward the earth. 5. Unit of Time. The unit of time used for scientific purposes is the second. It is defined as ^g J^ part of the mean solar day, by which is meant the average time between successive passages of the sun across the meridian through- out the year. The period of the rotation of the earth on its axis, as determined by successive transits of a star, is an interval of nearly perfect constancy, but of such length that it does not conveniently fit civil affairs, and hence is used only for astronomical purposes. It is known as the sidereal day. j'j'o'f^-:*. <. vT* Zst^*+y 6. Instruments for Measurement of Time. Clock. The subdivision of time may be effected by any regularly recurring phenomenon. The pendulum, which consists of a heavy body supported at one point and vibrating under the influ- ence of weight, is one of the most familiar instruments for securing equal intervals of time. When supplied with a driving train to overcome the resistances which would bring it to rest, and a pointer to record the number of beats, the mechanism forms a clock. A perfect clock would be one which had a constant daily gain or loss, technically known as the rate. The error of a clock is determined from the observation of a transit of some fixed star. Chronometer. The chronometer is a timepiece designed with special view to its portability. The isochronous ele- INSTRUMENTS FOR MEASUREMENT OF TIME. 7 ment is a small spiral spring and balance wheel (Fig. 1 44;, mounted so as to oscillate with the utmost freedom about a vertical axis with the bending and unbending of the spring. The accuracy of a chronometer does not equal that of the pendulum clock at its best ; but, on the other hand, the chro- nometer is capable of giving reliable results where, as at sea, the clock would be perfectly useless. A watch does not differ essentially from the chronometer, except that it is smaller, and the mechanism of the escape- ment is altered to adapt it to the somewhat rougher handling it is likely to receive. Chronograph. The chronograph (Fig. 1) is an instrument designed to record intervals of time. Its essential parts are ' ' iuui mnmlu.ii FIG. l. a cylinder carrying a sheet of paper and revolved uniformly by clock-work once a minute. Supported above the paper, and resting upon it so as to leave a helical trace as the cylinder turns, is a pen electrically connected with a clock, and so arranged that each time the clock beats it receives a slight and momentary displacement lengthwise of the cylin- der. It is also capable of a similar movement at the will of the observer, by the depression of a telegraphic key. A copy of the record thus made is shown in Fig. 2. MECHANICS. The V-shaped notches mark the beginning of each second, while the square indentations record the epoch of two events at A and B observed by the operator. The interval which 20-' FIG. 2. elapsed is found by measuring the distance from A to B and comparing it with the scale of seconds. In another form of chronograph the record is made on a sheet of smoked paper by a style fastened to the prong of a tuning fork of known period. The trace in this case is a sinuous curve on which the beginning and end of an event are marked by minute perforations, as at A and J9, which are produced by an electric spark made to pass from the FIG. 3. style to the cylinder when the circuit is closed. Since each wave corresponds to a definite interval of time, the duration of the phenomenon may be readily deduced. In the first form of instnjn^ent the limit of precision may be regarded as not far from -f^th of a second, while in the second it may be less than -j^Vo^h of a second. 7. Unit of Length. The unit of length employed in science is the centimeter. It is defined as one hundredth part of a certain platinum bar in the possession of the French Government, measured at a temperature of C. This bar, the standard meter, was intended to represent a ten-millionth of the earth quadrant measured on the meridian of Paris from the equator to the pole ; but as more recent measurements have revealed a slight deviation in the length of the bar from INSTRUMENTS FOR THE MEASUREMENT OF LENGTH. 9 this dimension of the earth, the meter is to be regarded in reality as an arbitrary rather than a natural length. The relation of the centimeter to the inch is given very exactly by the equation 1 in. = 2.5400 cm. 8. Instruments for the Measurement of Length. * Scales. The determination of a length is commonly effected by the direct comparison of the required distance with a scale or copy of the standard of length, which is divided into equal parts by lines drawn on its face. The scale being FIG, 4. placed alongside the given length, the whole number of units may be read off and the fraction of a unit estimated to one tenth of a millimeter under favorable conditions. When greater accuracy is desired, various special instruments are used, differing somewhat with the manner in which the length to be measured is denned. Micrometer Microscope. When a length is determined by the distance between two carefully ruled lines, the frac- tional part of a scale division may be found by the use of a micrometer microscope (Fig. 4). This consists of an ordi- nary microscope furnished with a pair of cross hairs, which can be moved across the field by a screw of fine pitch, and 10 MECHANICS. BG A the number of revolutions of the screw read from a disc attached to its head and divided into equal parts. By caus- ing the cross hairs to traverse the small length which it is desired to measure, viewed through the microscope, the whole aud frac- tional number of turns of the screw may be counted and reduced to the unit of length by means of the de- PIG 5 termined value of one revolution. Vernier Caliper. When the length to be measured is defined by the bounding surfaces of a solid, use is often made of the instrument shown in Fig. 5, called a vernier caliper. It consists of a fixed jaw, A, and a movable one, B, sliding on the bar D and arranged so as to be brought into contact with the piece Gr to be measured. The opening of the jaws is then read off the v graduated scale from the posi- tion of the point 0. To assist in the accurate deter- mination of this point an auxiliary sliding scale, 0, called a vernier, is added. Its principle is as follows : A dis- tance, US, is laid off on the sliding scale equal to nine divisions of the fixed one and divided into ten parts. If now the zeros of both scales be placed in coincidence, it is evident that 1 on the vernier will be one tenth of a scale division behind division 1 on the fixed scale. Similarly the divisions marked 2 on each scale will be two tenths apart, divisions 3, three tenths, and so on. Accordingly, to bring any mark, say 6, 5 10 R 1 i i 1 i i 1 1 I S I" ' ; Mil i D PIG. 6. 1 1 I M UNIT OF MASS. 11 A B of the vernier in coincidence with the corresponding one on the scale, the vernier must be moved six tenths of a division beyond the zero, or in general the number of the mark on the vernier in coincidence with one on the fixed scale gives the position of the vernier zero in tenths. Thus, in the figure ,*he reading is 3.2. The precision of reading of the vernier is hardly more than twice that with which the eye will readily estimate the position of the line R on the scale I), but the convenience of its use is considerably in its favor. Micrometer Oaliper. The most accurate measurement of the linear dimensions of solid bodies of moderate size may be made by means of a micrometer caliper (Fig. 7). A is a fixed pin, and B the end of a screw which may be approached or with- drawn by turning the milled head of the sleeve E. The separa- tion of A and B in any position is read by the aid of the scale on D and the graduated edge of the sleeve F. The precision of reading in such an instrument considerably exceeds the accuracy of setting. 9. Unit of Mass. The unit of mass used in science is called the gram. It is defined as an amount of matter equivalent to one-thousandth part of that of a certain piece of platinum, known as the standard kilogram, in the posses- sion of the French Government. When this standard was constructed it was intended that the gram should be an amount of matter equivalent -to that contained in a cubic FIG. 7. 12 MECHANICS. centimeter of distilled water at its maximum density, i.e. at a temperature of 4 C. More recent measurements show that this intention was not exactly realized, though the dis- crepancy is so slight that the two definitions may for prac- tical purposes be regarded as identical. The relation of the kilogram to the pound is given with considerable precision by the equation lkil. = 2.20 Ibs. 10. Measurement of Mass. - - The usual method of determination of the mass of a body depends upon the prin- ciple, first demonstrated by Sir Isaac Newton, that every body is attracted toward the earth with a force proportional to the quantity of matter it contains, or, in other words, that the mass of a body is proportional to its weight. The instrument employed for comparing masses by deter- mining the equality of their weights is called a balance. It consists of a light rigid beamsupported at its center by a knife edge on which it freely turns. From the ends of the beam are suspended two pans so adjusted that when they are without load the beam remains in a horizontal position, which is accurately indicated by a long pointer moving over a graduated scale. If exactly equal masses are placed in each pan, it is obvious that the equilibrium will be undisturbed, for the mass of each pan with its contents is equally attracted toward the earth. If, however, one of the masses is greater than the other, then that pan in which it is placed, on account of its greater weight, will descend. Thus, by means of a set of standard masses it is possible to determine the quantity of matter in a body of moderate size, to a degree limited only by the sensitiveness of the balance. The standard masses composing the set are often called SYSTEMS OF UNITS. 13 weights, and the operation of determining the mass, weigh- ing ; but it should be noted that this process tells nothing as to the weight of a body in the strict sense of the term, i.e. the absolute value of the earth's attraction for it. A common form of the balance enclosed in a glass case is shown in Fig. 8. The large knob at the front actuates a mechanism by which the beam may be lifted off its supporting knife edge when not in use. In order to effect a final adjustment of the masses without opening the case or disturbing the pans, a FIG. s. rod is inserted through the upper right-hand portion of the case, by means of which a small bent wire, called a rider, may be placed astride the balance arm at any point. The effective influence of this added weigrit is read at once from a graduated scale on the beam. 11. Systems of Units. As has already been pointed out, any three independent units are sufficient to express the measurement of all physical quantities. When these meas- ures are so expressed they form what is known as an abso- lute system, and all units other than the fundamental are termed derived units. The system in which the centimeter, the gram, and the second are selected as the fundamental units is usually desig- nated as the C. G. S. system. Other systems are rarely used in scientific works, but occasional reference may be found to the English units, or the Foot, Pound, Second system. 14 MECHANICS. 12. Dimensions. When it is desired to make a general statement of a concrete magnitude without explicit mention of any unit, the numeric will be denoted by a small letter and the unit by a bracketed capital. Thus, any length may be written l[L], The most general expression for any derived unit is of the form where the numbers p, q, and r are termed the dimensions of the derived unit in terms of the fundamental units. Numer- ous examples of dimensional formulae will be found in the articles following. Since in any equation involving concrete magnitudes equality can subsist only between quantities of the same kind, it follows that every true equation must be homo- geneous with respect to the physical units entering it. In this way dimensions furnish a valuable check upon the equations resulting from a train of reasoning. They are also of great assistance in changing from one set of units to another. 13. Derived Units. Area. The area of any figure is proportional to the product of two of its linear dimensions. The dimensional formula of area is accordingly [i 2 ], and the unit one square centimeter. This unit may be symbolized by 1 cm. 2 Volume. The volume of any solid is proportional to the product of three of its linear dimensions. The dimensional formula of volume is [Z 8 ], and the unit one cubic centimeter. This unit is usually written 1 cm. 3 or 1 cc. Density. The density of a body, or the concentration of matter, is defined as the mass per unit volume. The dimen- DERIVED yNTp. 15 sions of density are ML' 3 . The unit is the density of one gram per cubic centimeter. It is symbolized by - : , but JL CC. has received no name. The ratio of the density of any substance to the density of water is called the specific gravity of the substance. Since in the C. G. S. system the density of water is sensibly unity, there will be no occasion to use the term specific gravity unless the measures of mass or volume should be given in some other system. Angle. The measure of an angle at the center of a circle is defined as the quotient of the subtending arc by the radius. The dimensions of angle are zero ; that is to say, its measure is a pure number. A similar remark applies to all the trigonometric functions. The unit angle is an angle sub- tended by an arc equal to the radius. It is called the radian. Velocity. When the position of a point changes continu- ously as time goes on, the point is said to possess a velocity, which is measured by the distance traversed per second, or the time rate at which its path is described. To illustrate, let s (Fig. 9) be the position of the point measured along the path at the beginning of any time, and s t its position at the end of t seconds. Then FlG 9 if equal distances are described in equal intervals of time, the velocity is constant and equal to the distance traversed divided by the time, or If the velocity is variable, its value at any instant will be the limit which this expression approaches as t is made smaller and smaller, or, in mathematical symbols, 16 MECHANICS. (2) v = limit t ~ S - The term speed is commonly used to designate the mag- nitude of a velocity considered apart from its direction. The dimensions of velocity are and the unit, a velocity of one centimeter per second. This unit is symbolized by , but has received no generally accepted name. x sec* * Angular Velocity. When any line in a body changes its direction continuously with the time, it is said to possess an angular velocity, which is measured by the time rate at which the angle is described, or, in other words, by the angle swept over per second. When the angular velocity is constant, its value is found by dividing the angle traversed, by the time. As the dimensions of angle are zero, those of angular velocity will be T~ l and its unit one radian per second. Acceleration. When the velocity of a point varies con- tinuously with the time, the point is said to possess an acceleration which is measured by the time rate of change of the velocity. If v denote the velocity at the beginning, and v t the velocity at the end of any time, , the acceleration f may be written (3) /= limit ^^- If the acceleration is constant, this reduces to the change of velocity divided by the time. The dimensions of accelera- tion zxtf LT~ 2 and the unit, an acceleration in which the velocity changes one centimeter per second, per second. This unit is symbolized by - ^ but nas no accepted name. -L sec. DERIVED UNITS. 17 Momentum. The quantity of motion, or momentum of a moving body, is denned as the product of the mass of the body by its velocity. Its dimensions are MLT~\ and the unit, the momentum of a gram moving with a velocity of one centimeter per second. This unit is symbolized by sjr- '-> but has received no name. This important 1 sec. property of a moving body, which varies as both the velocity and the mass, may be illustrated by suspending two spheres of the same size but of different material, say wood and lead, by strings of equal length. If the spheres be set swinging together, it will be observed that an obstacle, such as a piece of moderately stiff paper, which largely affects the velocity of one, will produce but little change in that of the other. It is thus shown that, although the velocities of the two bodies appeared to be the same, the quantity of motion was greater in the greater mass. Force. As a body has no power within itself to change its motion, we conclude in every case of changing motion that the body is acted upon by some external cause. This cause is called a force, and its measure is defined as the time rate of change of the motion. Thus, if F represents the force, mv the motion at the beginning of the time, and mv t the motion at the end of the time, , the expression for the forpe may be written . , mv f mv (4) F= limit- ~- -t,^> t Since m is constant, this may be written as the mass multi- plied by the time rate of change of velocity, or (5) F=mf. The dimensions of force are MLT~ 2 . The unit is that force which would change the velocity of one gram by one 18 MECHANICS. centimeter per second, in a second. It is called a dyne, and is symbolized by 8 -'. Experiment shows that a gram falling freely in a vacuum receives an acceleration of 980 -^1 ; therefore, the attraction of the earth for the unit sec. 2 mass, or the weight of one gram, is 980 dynes. The accelera- tion due to weight is usually denoted by the letter g, thus : = 980=82.2 ft sec. 2 sec.2 The weight of any mass, m, is accordingly mg. An arbitrary unit of force much used in some branches of science is the attraction of the earth for the unit of mass. In this sense it is customary to speak of a gram force, or a pound force. The use of such a unit is open to two objec- tions: 1 the application of the term gram weight to two things so distinct as a mass and a force is a source of serious confusion; and 2 the weight of a gram is not a perfectly definite force even on the surface of the earth, since it varies slightly from place to place. (See Art. 30.) 14. Vectors. A vector, or directed quantity, is one which requires a number and a direction to define it completely. A scalar quantity is one which is capable of complete definition by a single numerical specification, and whose value does not in any way involve a direction. The displacement of a point, represented by a straight line drawn from the initial to the final position of the point, is the simplest illustration of a vector. A velocity, a momentum, an acceleration, a force, a flow, an electric cur- rent, the magnetization of iron, an angular velocity, are other examples of vectors. SUM OF TWO VECTORS. 19 Mass, volume, density, speed, and quantity of work are examples of scalars. Any vector magnitude may be represented by a straight line drawn in the proper direction, and having as many units of length as there are units of quantity to be represented. In the case of an angular velocity, the vector direction is taken as that of the advance of a right-handed screw, when revolved the same way as the body. 15. Sum of Two Vectors. It is a matter of experience that if p and q (Fig. 10) represent two vectors without limi- tation as to position, then the line AC, in the tri- angle ABC, drawn with the side AB equal and /^ P parallel to p, and BC equal -and parallel to q, will represent in magni- tude and direction the sum or combined effect of p and q. This prop- osition may be most readily verified in the case of velocities, where it is easily shown that if a body mov- ing with an assigned velocity have another velocity impressed upon it, the movement in the first direction takes place as if that in the second did not exist. An illustration of such combination of velocities may be found in Art. 474. The magnitude of the sum of two vectors may be calcu- lated as follows : From the point C (Fig. 10) drop a perpen- dicular on AB produced. Call the angle between the vectors, i.e. the angle through which q must be turned to coincide in direction with p, , and denote the length of AC\)jr. Then P FIG. 10. D 20 MECHANICS. or = (p + q cos $) 2 + # 2 sin cos (>. P (6) (7) =p*- 16. Difference of Two Vectors. If a minus sign before a vector be understood as reversing its direction, then the differ- ence of two vectors, AB CD, may be written AB + DO; that is to say, the difference is found by adding them after the reversal of the direction of the second vector. If the difference of two vectors, p and q (Fig. 11), be denoted by s, and the angle between them by <, the magnitude of s is given by the relation FIG. 11. (8) s 2 = 17. Resolution of a Vector into Components. The reso- lution of a vector into components is not determinate unless the directions of the components required, or the P/ A FIG. 12. magnitude of one component and the direction of one, are given. For instance, let R (Fig. 12) be a vector which it is SUM OF ANY NUMBER OF VECTORS. 21 desired to resolve into two components in the directions F and 6r. Through the extremities of R draw lines parallel to F and Gr which intersect at A, then P and Q are the components required, for P + Q = R. The most important case is where the given directions are at right angles. If a be the angle between the given vector and one of the directions, the component \r \$ parallel to this direction would be R cos a, and the one at right angles R sin a. /Q 18. Sum of any Number of Vectors. The resultant of a number of vectors, P, Q, S, T, U, may be found by first adding any two of them, then add- ing a third to this sum, and so on, as in Fig. 13. It is easy to show by trial that the final sum is independent of the order in which the vectors are taken; whence, as appears from Fig. 13, if the beginning of the second vector be applied to the end of the first, the beginning of the third to the end of the second, and so on, the line R drawn from the beginning of the first to the end of the last will represent in magnitude and direction the sum of the vectors P, Q, 8, T, U. R FIG. 13. 22 MECHANICS. The magnitude of the resultant of any number of vectors is most easily calculated by referring the vectors to coordi- nate axes. Thus, let the lengths of the given vectors be denoted by p v PV - p n , and the angles which they respectively make with the axis of abscissas, by a v 2 , a n . Call the horizontal components of _ x the vectors x v x 2 # n , and the vertical components y^ y^ - y * Flo 14 Then the sum of the x components is *i+^2 + " '+ x n=Pi cos Oj + ^a cos <% + - (9) = 2 p cos a = JT, say. Similarly, ( 10 ) y\ + y* + ' ' 'y n = ^ P sina= r,say. The magnitude of the resultant R is and the angle < which it makes with the axis of abscissas given by (12) tan $ = ^, or cos = ~ -A 1 The preceding results may without difficulty be extent! a to space of three dimensions. It is obvious that the differ- ence of any number of vectors does not call for explanation since it may be reduced to the case of successive addition. EXAMPLES. 23 EXAMPLES. 1. A train runs 45.6 miles in 38.2 min. What is the velocity in C. G. S. units ? v = 3200 cm. / sec. 2. The mean distance of the earth from the sun is 0.929(10) 8 miles. What is the orbital velocity of the earth ? ^ = 18.5 miles/ sec. = 2.97(10) 6 cm. /sec. 3. What is the velocity of a point on the equator, due to the earth's rotation, taking the sidereal day as four minutes shorter than the mean solar day ? v = 4.64(10) 4 cm. / sec. 4. What is the angular velocity of the earth's rotation ? Ans. 0.729 (10)~ 4 radian /sec. 5. What quantity of motion is possessed by a mass of 53.9 kilos moving with a velocity of 683 cm. per sec. ? Ans. 3.68(10) 7 gm. cm. /sec. 6. The velocity of a body is observed to increase uniformly from 428 cm. per sec. to 1635 cm. per sec. in 13.0 sec. What is the acceleration? /= 92.8 cm. /sec. 2 . 7. What would an acceleration 986 ft. per minute, per minute, be in the C. G. S. system ? /= 8.34 cm. / sec. 2 . 8. If the mile be taken as the unit of length and the day as the unit of time, what will be the value of g ? g = 4.55 (10) 7 miles / day 2 . 9. A constant force acting on a mass of 6283 gms. for 2.365 sec. changes its velocity from 5197 cm. per sec. to 9858 cm. per sec. Required the acceleration and the force. /=1971 cm. /sec. 2 ; F= 1.238(10) 7 dynes. 10. A rectangular block of stone containing 53.5 kilos has the follow- ing dimensions : 12.1 cm. by 31.7 cm. by 45.9 cm. What is its density? p = 3.04 gms. /cc. 11. A sphere has a diameter of 56.4 cm. and a mass of 65.2 kilos. What is its density ? p = 0.694 gm. /cc. 12. A cylinder has a hight of 74.3 cm., a diameter of 23.4 cm., and a density of 1.42 gms. per cc. What is its mass ? m = 45.4 kilos. 13. The mean density of the earth is 5.53 gms. per cc. What is its mass? m = 5.97(10) 27 gms. 24 MECHANICS. 14. A right circular cone whose hight is 218 cm. has a mass of 98.7 kilos and a density of 5.64 gms. per cc. What is the radius of the base ? r = 8.75 cm. 15. A tube 19.3 cm. long contains 2.54 gms. of mercury. What is its mean cross section? Ans. 0.00968 sq. cm. 16. A flask containing 51.8 gms. of salt when filled with alcohol weighs 214.2 gms. When filled with alcohol alone it weighs 182 gms. What is the density of the salt ? p = 2.14 gms./cc. 17. A flask weighs 39.74 gms. when filled with water. When 40.37 gms. of shot are inserted and the flask filled with water the weight is 76.54 gms. What is the density of the shot? p = 11.3 gms./cc. 18. A vector drawn east has a length of 16 cm., and one drawn northeast a length of 25 cm. What is their sum? Ans. 38 cm. 19. A ball is thrown south with a velocity of 36.7 ft. per sec. from a train running at a velocity of 78.4 ft. per sec. in a direction south 30 east. What is the velocity of the ball with respect to the ground ? Ans. 112 ft. /sec. 20. A particle is acted on by four forces whose magnitudes are 695 pounds, 872 pounds, 384 pounds, and 243 pounds, respectively. The angle between each force and the adjacent one is 90. What is the resultant force ? Ans. 7^=702 pounds weight. CHAPTER II. SIMPLE TYPES OF MOTION. 19. Unchanging Motion. The simplest kind of motion is unchanging motion, i.e. motion in a straight line with uni- form velocity. The equations may be written down at once from the definition. Thus, the force is zero because the motion does not change, or, F=0. The acceleration is zero because the velocity does not change, or, /=o. Also, by definition, the velocity may be written s - = const. = u, Tf whence (1) 8 = Ut. These equations contain the complete solution of this type of motion. Two of the conclusions are in apparent contradiction to experience, namely, that a body set in motion goes on forever ; and, secondly, that no force is re- quired to keep a body moving uniformly. But this is only because we have no experience of an absolutely unhampered motion, that is, of a body acted on by no force whatever. In the case of a stone sliding on smooth ice, the conditions are obviously more nearly approached than in most instances of sliding, and the approximation to the ideal case of per- fect freedom from extraneous forces is nearer. So, also, a heavy ship moving very slowly through still water loses its 26 SIMPLE TYPES OF MOTION. velocity with extreme slowness. It is considerations such as these which have led to the recognition of the simple law given above. 20. Uniformly Changing Rectilinear Motion. An im- portant type of motion is that in which a body moves with uniformly changing velocity in a straight line. Since by definition the acceleration is constant, (2) The force is also constant, and equal to the mass multiplied by the acceleration, (3) F=mf. Since the velocity increases / units per second, the gain in t seconds would be ft units of velocity, which, added to the initial velocity, gives (4) V t = V +ft. This is of course the same equation as (2). The average value of the velocity of any motion is the total distance divided by the time. Denoting it by v, (5) s = vt. In this type of motion the average velocity may be easily calculated, since the velocity increases uniformly from v to v t ; thus, This result may be represented by a diagram. To do this, let the time be plotted as abscissa and the velocity as ordinate (Fig. 15); then evidently the average hight of the line AB is given by the middle ordinate, or that of the point (7, which is one-half the sum of those at A and B\ i.e. APPLICATION TO A FALLING BODY. 27 Multiplying this value of the average velocity by the time gives for the distance traversed By the method of derivation it is evident that the value of s is represented by the v area between AB and the axis in Fig. 15. If t be eliminated be- tween (4) and (7), Summing up the result of this investigation, it is seen that whenever a con- stant force acts on a free body for a time, t, it increases its velocity by ft units, and increases its change of place by i/ 2 units of length, both of which effects are independent of the initial velocity which the body has. 21. Application to a Falling Body. The most inter- esting instance of the type of motion just discussed is that of a falling body. If the body be supposed to start from rest, v = 0. Calling the acceleration g, FIG. 15. (10) s (11) v* = 2gs, (12) F= mg = weight of m grams. It will be noted that the distances traversed are as the squares of the times; that is, for 1 second, 2 seconds, 3 seconds, 4 seconds, the corresponding distances gone are as the numbers 1, 4, 9, 16. Since the distance traveled in n seconds is proportional to ft 2 , and that in n 1 seconds to (n I) 2 , it follows that the 28 SIMPLE TYPES OF MOTION. distance traversed in the nth second is proportional to 2n 1, or the distances fallen through in the 1st, 2d, 3d, 4th seconds are, respectively, proportional to 1, 3, 5, 7, etc. If a body be thrown vertically upward with an initial velocity, v , the upward direction being called positive, the equations of motion become (13) v t = v gt, (14) s = v t %g$, (15) v? = v?-Zgs. The time and hight of rise are found by setting v t = 0, which is evidently its value, at the highest point. Thus, t- V i -, , 9 v 2 S ^7T^ % which, by comparison with (9) and (10), show that the time and hight of rise are the same as would be required for the body falling from rest to acquire the initial velocity v . 22. Atwood's Machine. A freely falling body moves too rapidly to permit observations to be conveniently made upon it in any ordinary room. This difficulty is obviated in the apparatus shown in Fig. 16, known as Atwood's Machine. Two equal brass cylinders, (7(7', are suspended by a flexible cord which passes over the light pulley P, sup- ported so as to turn with as little friction as possible. T is a swinging shelf on which Q may be placed and released at a given moment, by a clock beating seconds. R is a ring- shaped stage with a hole large enough to permit O to pass freely through it. If the slip A be placed upon O when the latter is resting upon the shelf T, then when the clock , the system CPO' will be released and begin to move AT WOOD'S MACHINE. 29 under the influence of the weight of J., with an increasing velocity until O reaches the ring stage where A will be removed. From this point the motion will be uniform until is arrested by the shelf 8. Neglect- ing friction and other resistances, the acceleration of the first motion may be calculated as follows : Let m denote the mass of each of the cylinders CO 1 . Let m f denote the mass of the slip A, M " " " " " pulley P, g " " acceleration of weight, //'..'" " " produced in the system, then, by equation 3, force acceleration = mass In applying this equation it will be neces- sary to add %M to the other masses in order to take account of the acceleration of the pulley, supposed to be a flat disc, whose motion is of a different type from that under consideration. Thus, the total mass accelerated is 2m + m' + %M, and the acting force is m!g ; hence the acceleration will be m (16) ^ By a proper choice of masses, / may be given any value between FIG. ie. 30 SIMPLE TYPES OF MOTION. g and zero, and the system be made to move slowly enough for convenient observation. If /be determined by observation from a particular set of masses, equation 16 may be used to calculate #, but the value so obtained will not be very accurate, on account of several sources of error which are present. The greatest of these is the difficulty of determining the arrival of A at the stage R, within of a second. 23. Projectile. Suppose that a heavy particle is pro- jected into empty space with a velocity, u, and at an angle, a, with the horizontal plane. Let the coordinates of the moving particle P (Fig. 17), at any instant, re- ferred to axes pass- ing through the starting point, be x and y. Then, since the only force act- ing on P is the downward force of weight, the motion parallel to the Faxis will be uniformly accelerated, but in the horizontal direction the motion will be a uniform one. If the velocities in these directions be denoted by v y and v x , respectively, M Q x FIG. 17. (17) (18) v == u cos by equation 13. Again, from equation 1, (19) x = u cos a , and from equation 14, (20) y = u sin a t \gt \ PROJECTILE. 31 Eliminating t between (19) and (20), (21) y = x tan a cos 2 a which, being an equation of the second degree, shows that the path of the body, or the trajectory as it is called, is a conic section, and, since it has but one infinite branch, it must be a parabola. The time occupied by the projectile in passing from to Q is called the time of flight. Let it be denoted by T\ then, since y = at $, equation 20 gives (22) t = ~ sina=T. y The distance OQ is called the range, which will be denoted by R. Letting t = T in equation 19, or # = in equation 21, 2u? . u* . ,, (23) X = - Sin a COS a = Sin 2a = M. 9 9 It is evident that the range is a maximum when sin 2 = 1, i.e. for a = 7r or 45. Suppose the projectile when thrown at the elevation a' has a range, R', then if a 1 = %TT a, o o Iff = sin (TT 2a) = - sin 2a = R, 9 9 or the range is the same for a and its complement. Since v y = at the highest point, equation 18 gives the time of rise, u . t = - sin , 9 which is one-half the time of flight, as it should be. Sub- stituting this value of t in equation 20, the hight of rise is found to be 32 SIMPLE TYPES OF MOTION. (24) = ^ sin 2 a = H, say. These equations are, of course, for a projectile moving in a vacuum ; but the path of a slow-moving projectile in the air will not vary much from the parabola here described. For relatively high velocities, however, the change due to the resistance of the air may be very great, but the path will always fall within the parabola. 24. Uniform Circular Motion. The characteristic prop- erties of the force necessary to the motion of a particle in a circle with uniform speed may be determined by the fol- lowing synthetic method of reasoning : 1. There is a force because there is a change of motion. If there were no force acting, the particle would move in a straight line, with constant velocity. 2. The force is constant in magnitude because the change of direction is uniform. 3. The force is directed toward the center ; for suppose it had some other direction, F, as in Fig. 18, then this force could be separated into a central component, F c , and a tangential component, F T . Now the effect of the latter would be to increase the speed, which is contrary to the definition ; therefore, there is no tan- gential component, and the force is all toward the center. 4. The force varies as the mass, since by definition F=mf. 5. The force varies inversely as the radius of the circle ; for suppose there are two circles, one twice as large FIG. 18. UNIFORM CIRCULAR MOTION. 33 FIG. 19. as the other, then, the speed being the same on both circles, in the same time the particle would describe equal arcs, AB and AB' (Fig. 19), but the change of direction, , in going from A to B' would be twice that in going from A to B, or 2a. But the force is measured by its effect, that is, by the rate of change of the direction of the motion ; therefore, the force is twice as great in the case of motion on the small circle, or, i r 6, The force varies as the square of the speed ; for sup- pose in one case the speed to be a- and in another case 2cr, then, if in the first case the particle moved from A to B (Fig. 20), in the second case it would go twice as far, AB', in the same time, or the change of direction in the second case is twice as great. But the quantity of motion, 2mcr, in the second case is twice that in the first, mo-; there- fore, the force in the PIG. 20. second case changed twice the momentum through twice the angle, or the force must have been four times as great. Therefore, 0-2. 34 SIMPLE TYPES OF MOTION. Collecting these results and denoting the central force by F c oc - , or where the value of k still remains to be determined. The exact value of the acceleration in this type may o -Vo be derived from a con- sideration of the rate at which the direction of the velocity is changed, as follows : Let the vec- tor V (Fig. 21) repre- sent the velocity when the particle is at A, and V T the velocity at the end of a small interval of time, T. Also, let o- denote the speed, and a the angle AGE. Then, by the definition of acceleration, V V /= limit r=o T Constructing the triangle abd according to the rule for subtracting vectors (Art. 16), it is seen that V r V is a vector ab, having a direction which will be ultimately par- allel to AO, i.e. toAvard the center of the circle. The mag- nitude of V T V is from the same triangle, FIG. 21. (26) ab = 2ad sin ~ = sn whence, by substitution, (27) /= limit T=0 2 which in (29) gives (31) These results may be obtained in a different manner by a consideration of the distance traversed under the action of a constant force in a straight line. Thus, suppose that the particle m moves with a discontinu- ous motion, being first drawn from A. to B (Fig. 22) by a constant force, then moving from B to C with a constant velo'city, to be again drawn a distance, (7Z>, toward the center, and so on. It is obvious that by taking the steps of this path smaller and smaller, the motion E FIG. 22. 36 SIMPLE TYPES OF MOTION. described may be made to approach uniform motion in a circle, without limit. The distance AB is called the sagitta of the arc CrAO. Let it be denoted by A, and let A = r, A O = A 0- = a, and AGr = a. Then we may read at once from the figure, aha (32) Sm 2 = a = 27 whence n (33) h = 2r sin 2 I == ~ Since by hypothesis the particle moved from A to B under the action of a constant force, the acceleration under these circumstances will be by equation 7, p. 27: 2s (34) f=~2 Substituting the value of * from equation 33 and passing to the limit, the acceleration for uniform motion in a circle becomes , 4r sin 2 J- 9 .A fl and from equation 31, 4 equating and solving for P, (36) FIG. 23. which shows that the time of revolution is independent both of the length of the string and the mass of the bob. Hence, if various masses were fastened to a point of an axis by strings of different lengths, and the system were revolved in a definite period, the masses would all arrange themselves in the same plane, as seen in Fig. 24. The principle of the conical pendulum was utilized by Watt in his "governor," which has been a familiar feature on stationary engines till within a few years. In this apparatus two balls were used, one on each side of the axis, for the sake of bal- ance, which by their change of hight with increase of speed of rotation were made to shut off the steam or admit more when the speed fell below a certain amount. Its action is not, however, sufficiently prompt to insure even moderate steadiness of running when applied to the throttle. FIG. 24. 38 SIMPLE TYPES OF MOTION. 26. Parabolic Governor. A more sensitive form of gov- ernor is that devised by Huyghens, in which the revolving masses were constrained to move on the surface of a parab- oloid of revolution. The characteristic feature of this sur- face may be derived from the consideration of the form assumed by the surface of a liquid contained in a vessel and rotated about a vertical axis. Let Fig. 25 represent a vessel containing a liquid which is rotating about the axis A A, and suppose that m is the mass of a particle of the liquid at the surface. In order to sup- ply the necessary central force suppose the weight mg is resolved into a component, F c , which is sufficient to secure the motion of the particle m in a circle of radius, ", and a component, N, which is counterbalanced by the reaction of the vessel transmitted through FIG. 25. the liquid. When the particle m is in equilibrium the direction of N must be normal to the surface, for otherwise the particle would move either up or down. Thus, from the figure, (37) mg = F c tan < = p 2 tan <, whence (38) r tan < = -r? = const. = S. Now r tan < is the subnormal AC of the point on the curve, whose value is thus shown to be independent of the position of the point. As this is a well-known property of the parab- ola, it follows that the surface assumed by the liquid is a paraboloid of revolution. PARABOLIC GOVERNOR. 39 Suppose now a particle, w, such, for instance, as a shot, placed on the surface of a paraboloid of revolution (Fig. 26), which is being whirled rapidly about the vertical axis. If the weight mg be resolved into a central force, F c , and one normal to the surface, JVJ the par- ticle will remain at rest with respect to the sur- face, for any definite period determined by (39) P = PIG. 26. where S is the constant subnormal of the surface. If, now, the period of revolution be diminished, a greater value of F c will be -required, which may be obtained by resolving mg into F' C ) having a length, jk, and N', having a length, mj. But N 1 may be replaced by a component parallel to N, and one perpendicular, i.e. parallel to the surface, and the effect of this latter will be to slide the particle up the surface, without limit. By similar reasoning, it may be shown that, if the period was increased, the particle would slide down to the bottom. Hence, if the time of revolution differs from the value of the period required by the particular surface used, the particle can have no position of equilibrium on it. In order to realize in a simple manner the condition that the balls of the governor should swing in a parabolic arc, Huyghens found it sufficient to attach each of them by a suit- able arm to a point C or C f (Fig. 26), which is the center of curvature of the paraboloid at the place where it is desired that the ball shall be in equilibrium under the rotation. 40 SIMPLE TYPES OF MOTION. 27. Drying Machines. Another important application of the principle of rotary motion is that made in the so-called centrifugal drying machines, which have come into extensive use in recent years. In order to compare the efficiency of draining and whirling, let ra (Fig. 27) be a drop of liquid adhering to the solid C. The separating force will here be the weight of the drop, namely, W= ma. Next, suppose that C with the attached drop is whirled about the axis AA r (Fig. 28) in a period, P. Then, in order that m shall move with (7 in a circle of radius, r, the attachment of the drop to the solid must sustain a force _ m mg FIG 27 which is the measure of the separating force in this case. The efficiency of whirling relative to draining is accordingly of a second, that Qr If, for instance, r were 1 foot and P ^ is to say, if the body were revolved 2400 times per minute, the comparative efficiency of whirling would be 2000. Some of the cases in which a cen- trifugal dryer is very useful are worth noting. The old method of drying clothes in the laundry was to wring or squeeze from the cleansed fabric as much water as possible, and let the rest evaporate by suspending so that the air would circulate freely about it. The modern way is to place the wet cloth in a cylinder with A' FIG. 28. CREAM SEPARATOR. 41 perforated sides and whirl it rapidly, with the result that the cloth is dried far more thoroughly than is possible by wringing. Another application is to the separation of sugar crystals from the syrup in the process of refining. Thus certain of the products of the refineries are called " centrifugal sugars." This method is also applied to the separation of oil from metal chips or waste. The modern automatic machinery used for working iron and steel, e.g. screw machines, or those used for drilling gun barrels, requires that the cutting tool be deluged with oil. The recovery of this oil from the refuse is an important item in the economical running of a large factory. 28. Cream Separator. The same principle may be used to separate two liquids of different densities, provided neither acts as a solvent on the other. New milk, for example, may be regarded as an emulsion of fat and a liquid having nearly the density of water. If it be allowed to stand, the little globules of fat slowly rise to the top on account of their lesser density. If, however, the fresh milk be whirled in a suitable cylinder, the cream and the milk will rapidly separate into two co-axial layers, with the milk on the outside. To determine the relative efficiency of the two methods, let m denote the mass of a small particle of cream, V " " volume of a small particle of cream, D " " density of the cream, m' " " mass of the same bulk of milk, D' " " density of the milk. Then, if the particle of the milk and the cream be revolved about A A' (Fig. 29) in a circle of radius, r, the force neces- 42 SIMPLE TYPES OF MOTION. sary to hold the cream in this circle will be F c= - 7 ^r-=-^r VD, which must be transmitted through the liquid as a reaction from the side of the cylinder CD. Likewise the central force required for the milk will be m r T. i7"7y *c p2 p2 U ' which must be greater than F# since D f is greater than D. Accordingly, if the pressure at the distance r is just sufficient A c mg D A: FIG. 29. + mg FIG. 30. to hold the cream in that circle, it will not be great enough to retain the particle of milk, which accordingly takes up a position further from the axis. The separating force in this case is (41) F<-F c The separating force in the case of pan setting may be obtained by the aid of the principle first annunciated by Archimedes, which asserts that, under the conditions assumed, the cream will lose a portion of its weight equal to the weight of its own bulk of milk. Hence the separating force (Fig. 30) may be written (42) mg - m'g = Vg (D - D'). LAW OF UNIVERSAL GRAVITATION. 43 Dividing (41) by (42), Fig. 31 shows one of the forms which the cream separator takes in practice. The milk is fed into the revolving bowl through the central tube F. The skim -milk is ejected into a chamber at M, and the cream ex- pelled into a separate chamber at (7. 29. Law of Universal Gravitation. The Law of Universal Gravitation, discovered by Sir Isaac Newton toward the end of the seventeenth century, FIG 3J may be stated as follows : Every particle of matter in the universe attracts every other particle with a force whose direction is that of the line joining the two particles considered, and whose magni- tude is directly as the product of the masses, and inversely as the square of the distance between them. Expressed in symbols the law is, mm' (44) jF=# , where m and m r are, respectively, the masses of the particles, r the distance between them, and O- a constant to be deter- mined by experiment. As a groundwork for this great generalization, Newton employed the results of two of the greatest astronomers who 44 SIMPLE TYPES OF MOTION. preceded him, Copernicus and Kepler. About 1500 A.D. Copernicus perceived and announced that the apparent rota- tion of the heavens about the earth could be explained by supposing the earth to rotate on an axis once in twenty- four hours. Previous to this time the earth had been re- garded as the center of the universe. He also showed that nearly all the motions of the planets, including the earth, could be explained on the assumption that these bodies revolved in circular orbits about the sun, whose position in the circle, however, was slightly excentric. The reason assigned for a circular orbit is a curious commentary on the attitude of the philosophers of that day toward nature. They said, it is manifestly improper that the heavenly bodies should move in any but perfect curves ; the circle is the only perfect curve ; therefore, the heavenly bodies move in circles. Near the close of the sixteenth century Kepler, a German astronomer, began a study of the observations which had been made by Tycho Brahe, for the most part on Mars, for the purpose of ascertaining the laws of planetary motions. After twenty years of patient labor, with no method but that of trial and error, he succeeded in deducing the following laws which bear his name : I. Each planet describes an ellipse in which the sun occupies one focus. II. The radius vector (i.e. the line which joins the center of the sun and that of the planet) describes equal areas in equal times. III. The square of the period of any planet is propor- tional to the cube of its mean distance from the sun. Various surmises were made as to the explanation of these laws, but Newton was the first to prove the law of the force which would account for the motions of all the bodies in the solar system. Having first proved that the mass of LAW OF UNIVERSAL GRAVITATION. 45 any approximately spherical and homogeneous body might be treated as if condensed at its center, he was able from Kepler's second law to show that each of the planets is kept in its orbit by a force constantly directed toward the sun. From the first law he showed that this force must vary inversely as the square of the distance from the sun. From the third law he concluded that every planet behaved exactly as any other would do if substituted for it, that is to say, the force did not depend on the kind of matter com- posing it. In other words, the constant Gr was the same for all bodies. Being now satisfied that he had found the true law of gravitation as far as the planetary motions were concerned, he endeavored to test its generality by experiment. Assum- ing that the moon is held in its orbit by the same force as that which causes a body to fall at the earth, he saw it was possible to verify the law of distances by the space fallen through in a second. Thus, in Fig. 32 let AB = a be the distance trav- ersed by the moon in one second; also, let A C= h be the distance which it has virtually fallen toward the earth ; then r by equation 33. Taking the period of the moon as 27 d. 7 h. 43 m., and r as 3.844(10) 10 cm., or sixty times the radius of the earth, h = 0.1361 cm. The distance a body falls on the earth in a second is 490 cm., or practically 3600 times as far, which proves the law of the inverse square. 46 SIMPLE TYPES OF MOTION. Newton's proof that weight depends only on quantity of matter, and not upon its kind, is explained in Art. 41. If the law of gravitation be assumed, for the case of circu- lar motion, which is very nearly that of planetary motion, Kepler's third law may be proved from it as follows : Let M = mass of the sun, m = " " a planet, P = period of the planet, r = distance of the planet from the sun ; then, by equations 31 and 44, T, C .^ whence (45) P^ Equation 45 may be used to find the mass of the sun. Thus, * The distance of the sun as determined by the aberration of light (Art. 543) is about 93,000,000 miles. The value of the gravitation constant 6r has been found with considerable precision by measurements with a torsion balance of the attraction between small metal balls. The experiments of Boys show its value to be cm. gm. sec/ The mass of any planet which has a satellite may be found by an equation of the same form as equation 46. If the distance of the satellite from the planet be denoted by r' DIMINUTION OF WEIGHT. 47 and the period of the satellite by P', the mass of the planet will be (47) Dividing equations 46 and 47, the ratio of the mass of the sun to that of any planet with satellite will be m~PV 3 30, Diminution of Weight by the Rotation of the Earth. To find the effect of the rotation of the earth upon weight, let R (Fig. 33) denote the radius of the earth, \ denote the latitude of any B point, A, r denote the distance of A from NS. Then the central force neces- sary to secure the revolution of a mass, w, at this point in the period P of the earth, will be (49) JP2 cos X. At the equator the acceleration of this force will be /-^r- Substituting the numerical values of R and P, */ * o from which it appears that if the period of the earth's rota- tion were reduced to Jy of a day, since accelerations are 48 SIMPLE TYPES OF MOTION. inversely as the squares of the periods, all bodies at the equator would lose their apparent weight. Suppose that the portion of \veight which is resolved so as to furnish the central force is denoted by F g ^ and that its other component, perpendicular to F g ^ is denoted by F then -n -r, . . (50) F t = F c sin X - sin X cos X, and 47r a wJ2 (51) F g = F c cos X = cos 2 X. The former of these forces, considered as applied to the matter composing the earth, has produced a deformation in the shape of the earth, which is changed approximately from a sphere to an oblate spheroid having an equatorial diameter twenty-six miles greater than its polar diameter. The value of F g is that portion of the total attraction of the earth which is requisite to secure the revolution of a body in a circle coincident with the surface of the earth. It follows from (51) that a body will appear to weigh more as it is car- ried toward the poles. The increment of apparent weight of a body, on account of the flattening of the earth in the polar regions, also varies very nearly as the square of the cosine of the latitude. Both of these influences on weight may be included in a formula thus : Let g denote the acceleration of weight at the pole, and &, V certain con- stants ; then the value of the acceleration at any point may be written, cos 2 X; or, by a simple transformation in the form, (52) g = a l cos 2X. The value of these constants as determined by observation EARTH'S ROTATION ON PROJECTILES. 49 shows the value of g to be, in centimeters per second, per second, (53) g = 980.606 2.5028 cos 2X. 31. Influence of the Earth's Rotation on Projectiles. - The effect of the earth's rotation on a body projected hori- zontally in any direction at the surface of the earth is to deflect it toward the right in the northern hemisphere, and toward the left in the southern. Suppose, first, that a body is projected from B to A (Fig. 33). When it leaves B it will possess an eastward velocity equal to that of the earth, and greater than that of any of the points north of it, since B moves in a larger circle. Hence, as it passes these points, it will appear to be moving toward the east, that is, to turn to the right of an observer looking in the direction of projection. On the other hand, if a body be projected from a higher into a lower latitude, it will appear to lag behind the earth's surface in the latter region, since these points have a greater eastward velocity than the velocity of the point from which it started. As has been shown in the previous article, a body moving with the earth is acted on by a component of weight, F t , which urges it toward the equator. For any body resting on the earth this component may be regarded as inoperative, since the body would have, virtually, to ascend a slope thir- teen miles in passing from a pole to the equator. If, now, such a body be projected toward the east, the central force to retain it in the circle must be greater than F c . In order to provide for this there must be a different resolution of weight, and, in consequence, F t will have a greater value. If, on the contrary, the body were thrown to the west, the values of both F c and F t would be reduced. In the first case the body, relative to op.e moving with the earth, would 50 SIMPLE TYPES OF MOTION. be urged toward the equator with greater force, in the second case with less ; or, in each case, the body would appear to turn to the right, looking in the direction of projection. Since, as has been shown, a body turns to the right when thrown in any one of four directions at right angles to each other, it follows that the body would be deflected to the right if thrown in any direction ; for this latter direction may be resolved into components along two of the former taken as axes. 32. Cyclones. The principle just deduced may be em- ployed to explain the origin of cyclones and other circu- lar storms. In the southern hemisphere their direction of N rotation is always the same as the hands of a clock; in the northern hemisphere it is in the opposite direction. Let Fig. 34 represent a map of a portion of the earth's surface north of the equator, and suppose that for some reason the region A becomes more heated by the sun's rays than the surrounding portions. The air above this region will rise, since its density diminishes as its temperature increases. The cooler portions of air rushing in from the sides are deflected to the right, producing a whirl which travels across the country in a manner determined by the other, and much less clearly understood, meteorological conditions. Violent atmospheric rotations on a small scale are called tornadoes. The dust whirls frequently seen on a windy day are little eddies in the air formed at the edge of some building or other obstacle. w TRADE WINDS. TIDES. 51 33. Trade Winds. Another illustration of the deflection of air currents, resulting from the revolution of the earth, is afforded by the trade winds which in low latitudes blow, in the northern hemisphere, with remarkable constancy from the northeast, and in the southern hemisphere from the southeast. The air over the equatorial regions becomes heated and rises. The cooler body of air flowing in to take its place has a less velocity than the earth in this region, and hence is deflected toward the west with respect to the earth's surface. The great ocean currents are a direct consequence of the prevailing direction of the wind in tropical regions, which produces a continuous drift of the surface of the water toward the west, giving rise to those great rivers in the ocean, such as the Gulf Stream in the Atlantic, or the Japan Current in the Pacific. These, too, illustrate the eastward drift of motion in the northern hemisphere, and strongly modify the climatic conditions of the northwestern portions of both continents. 34. Tides. The alternate rising and falling of the waters of the ocean, known as the tides, is a complex phe- nomenon, depending upon the attraction of the sun and moon, and the revolution of the moon about the earth. The average time between alternate high tides is 24 h. 51 m., which is the same as the average interval between the successive passages of the moon across the meridian. The interval between the passage of the moon and the next high tide at any place is called the establishment of the port. To explain why there are two high tides in a day, it is necessary to observe that the common statement that the moon revolves about the earth is not quite true, or is true only in the sense in which the earth may be said to revolve 52 SIMPLE TYPES OF MOTION. about the moon. Both the earth and the moon revolve about an axis which passes through the center of mass of the system, i.e. a point which divides the distance between the centers of the moon and the earth inversely as their masses. Since these masses are about as 1 to 80, and their distance apart is 60 radii of the earth, the position of this axis must be about 3000 miles from the center of the earth. Let M (Fig. 35) represent the moon, E the earth, and AA the common axis of revolution, and consider what would happen if the earth were covered with water to a uniform depth. On the side of the earth next the moon the attrac- tion of the latter operates to reduce the force of weight. M -o FIG. 35. But the distance of the water on this side from the axis of revolution is also smaller than elsewhere, so that the water may be gathered together about B and still remain in equi- librium while revolving about AA under the influence of diminished weight. On the side opposite the moon a greater central force than elsewhere will be necessary to keep each particle of water revolving in a circle, since the distance from the axis is greater. But the attraction of the moon conspires with that of the earth on this side, so that it is possible to have the water, heaped up about D, revolve uni- formly about the axis ALA. Hence, as the earth rotates under the moon in but little over twenty-four hours, there will be two high tides in a day. The action of the sun upon the water of the ocean is pre- cisely similar to that of the moon, but on account of its THE GYROSCOPE. 53 greater distance its tide-raising force is only two-fifths as great. At new and at full moon the solar and lunar tides unite, forming what is called a spring tide. When the moon is in quadrature the resultant is the minimum, or neap tide. When the tidal wave strikes the shores of the continents, its energy is in large measure dissipated in the form of heat. As this energy is derived from that of the earth's rotation, it is evident that the ebb and flow of the ocean must result in a lengthening of the period of the earth's rotation. It appears probable that the moon's day has been length- ened to the period of its synodic revolution by the operation of tides at a time when the moon was not a rigid body. If the solid matter of the earth were as mobile as water, it would be subject to the same tidal waves as the ocean, and there would be no rise and fall of the tides with respect to the land. If the solid portion were only a crust, the rise of the tides would be comparatively small with respect to the land, but their apparent hight would increase with the rigidity of the crust. From reasoning founded upon the observed hight of tides, Lord Kelvin has drawn the conclusion that the interior of the earth must be for the most part solid, with a rigidity greater than that of glass. 35. The Gyroscope. The circumstances of motion of an extended rigid body cannot be derived without the aid of more powerful mathematical methods than are furnished by elementary analysis ; but there is one case which, on account of its interest and importance, may be noticed here. This case, namely, the one in which a rotating rigid body is con- strained to move about a fixed point, is illustrated by the 54 SIMPLE TYPES OF MOTION. gyroscope shown in Fig. 36. R is a massive ring revolving about an axis, E, which is carried by the frame CFD. The latter is pivoted at the fixed point 0, and counter- balanced by a sliding weight, B. When the wheel R is at rest, the pivoted system behaves as any heavy bar would in the same circum- stances. For in- stance, if the counterpoise be ar- ranged so that the bar is horizontal and the standard K be rotated very slowly, the whole system may be turned with it as a result of the fric- tion on the pivot. If, on the other hand, the wheel be set spinning rapidly, the system CF.D will show a persistence of direction which is quite unaffected by the friction on the pivot when the latter is turned. If a force be applied to the bar to change its azimuth, the resulting change will be one of inclination, i.e. a change at right angles to the direction of the impressed moment. If the counterpoise be shifted so that the system, when the wheel is at rest, is overbalanced, then when the wheel is set rotating the system will on the whole appear to revolve uniformly about a vertical axis through the pivot, though the actual motion is somewhat more complicated than this. The relation of these several rotations may be most con- veniently stated by representing them as vectors. Thus, let the rotation of the wheel R about the axis E be repre- FIG. 36. PRECESSION OF THE EQUINOXES. 55 sen ted by the vector o> (Fig. 37), drawn parallel to EC. Also, suppose that the position of the counterpoise is such as to elevate the end C rotating the system about a hori- zontal axis perpendicular to EC, and let this rotation be represented by the J~^ vector 6 (Fig. 3T). Then the position ( F m 37^ of the vector o>', found by the addition of co and 9, indicates the side toward which the axis of the gyroscope rotates about the pivot under the influence of weight. 36. Precession of the Equinoxes. It was discovered as early as 120 B.C. that the point where the sun crosses the celestial equator in the spring is moving continuously west- ward on the ecliptic. At the same time and for the same reason the celestial pole describes a circle 47 in diameter in the heavens. This phenomenon, known as the pre- cession of the equinoxes, may be illus- trated by the form of gyroscope shown in Fig. 38. CD is a massive ring attached by arms on one side to the spindle G-B, whose end rests in a cup, B. If, when the ring is at rest, the axis be displaced from the vertical, it will oscillate through its position of equilibrium under the influence of weight. If, however, the ring be set spinning rapidly and then dis- placed, the axis will describe a vertical cone, BHGr. The conditions necessary for gyroscopic motion in the model, namely, a rapid rotation about an axis and the mo- ment of a force tending to change the direction of the axis, are also fulfilled in the case of the earth rotating in space. On account of its departure from a spherical form, the earth 56 SIMPLE TYPES OF MOTION. may be regarded as possessing a protuberant ring of matter in the region of the equator, which is inclined to the plane of the ecliptic (Fig. 39). Consequently, since the force of gravitation varies inversely as the square of the distance, it is evident that the attraction of the sun for the portion D must exceed that for the portion 6 7 , and will operate to restore the equator to the plane of the ecliptic. Since, how- ever, this differential attraction of the sun is small, and the FIG. 39. momentum of the earth due to its rotation is great, the pre- cessional motion is very slow, a complete gyration of the axis occupying about 26,000 years. The present position of the celestial pole is about 1J from Polaris; in 13,000 years from now it will be near Vega. The vernal equinox is at present about 30 from its first recorded position, con- sequently these observations must have been made something like 2000 years ago. 37. Nutation. The uniform precessional motion of the earth's axis is subject to two disturbances : first, an annual variation in the moment of the force exerted by the sun, due to the change of position of the earth in its orbit ; and, secondly, a variable moment exerted by the moon, with a period of about nineteen years. The effect of each of these is to impress a nodding motion on the earth's axis, so that the cone actually described is a fluted, instead of a smooth one. The influence of the moon in this phenomenon, called nutation, is several times greater than that due to the change in declination of the sun. ROTATING PROJECTILES. 57 38. Rotating Projectiles. In the case of a rapidly mov- ing body the resistance of the air becomes a very important factor in the determination of the trajectory. A familiar illustration of this fact is seen in the erratic path of a light, irregular body, such as a shell when thrown from the hand. The influence of the air upon a projectile is usually preju- dicial, though not universally, for the baseball pitcher and the boomerang thrower utilize it as the great essential of their art. It must have been early discovered that a rotating mis- sile could be thrown with surer aim, for arrows have been feathered for time out of mind. The rotation so produced practically eliminates the tendency to sidewise motion, which would be produced by any lack of symmetry in the shaft or bending which exists at the moment of release. With the introduction of gunpowder the character of mis- siles underwent a complete change. The maximum destruc- tive effect was now to be sought by making the mass and the velocity of the projectile as great as possible. As a bullet material, lead was found satisfactory, because both dense and comparatively cheap ; but the problem of attain- ing a high velocity was not solved until the invention of the modern rifled arm. By the introduction of shallow, helical groovings in the barrel, a considerable rotation is imparted to the bullet, which prevents it from tumbling, and greatly lessens the liability to sidewise deflections. This construction permits the use of a longer and hence more massive bullet, without increasing the bore. The close fit required in the barrel also serves to prevent the escape of the imprisoned gases past the projectile. The angular velocity impressed on the bullet in a modern rifle is comparatively large. For instance, if the rifling makes one turn per foot and the muzzle velocity of the 58 SIMPLE TYPES OF MOTION. bullet is 1500 feet per second, the latter, after leaving the gun, will be rotating nearly 1500 times a second. 39. Harmonic Motion. A particle moving with uniform speed in a circle, when considered with respect to its motion across a diameter, affords an example of an important type of motion, known as harmonic motion. Let p (Fig. 40) be a particle of mass m, moving with con- stant speed, o-, in a circle. Draw any diameter, AB, and suppose that time is reckoned from the instant when the particle passes A. Let the distance dp of the particle from the diameter after a time, , be denoted by #, and the corresponding angle at the center by 6. Then the com- ponent of the force F c toward this diameter will be (54) -F x = F c sm0, where the minus sign denotes that the force is in the nega- tive x direction. This force, since it is always opposite to the displacement of the particle, may be called the force of restitution. From the figure, calling the radius a, FIG. 40. (55) sin 6 = - ; whence, by substitution of the value of F c from equation 31, (56) or, (57) x = const. k, HARMONIC MOTION. 59 which shows that in harmonic motion the force varies directly as the displacement. The velocity at right angles to AB is (58) v = a- cos 6. Since the angular velocity is constant, 9 2-7T (59) -=p-; also, by definition, Substituting these values, (60) v = p- cos and (61) x = asm -p- It will be observed that the equations are now free from all distinctive reference to a circle, since P may be regarded as the period of vibration and a as the amplitude, i.e. the maximum value of the displacement. Solving equation 57 for P, I'm (62) J^ftryy which, it may be noted, does ,not involve the excursion of the particle. Hence, harmonic motion may be defined as an oscillatory motion with respect to a line, in which the period is independent of the amplitude of vibration. The name is derived from the fact that the vibrations of musical sounds are of this type. That this is so appears from the fact that when a musical string is struck the pitch of the note does not change as the amplitude of the vibration decreases. 60 SIMPLE TYPES OF MOTION. 40. Simple Pendulum. A heavy particle attached to a point by a string without sensible mass, and free to vibrate under the influence of weight, is called a simple pendulum. Let m (Fig. 41) denote the mass of the particle, let I " " length of the string, " 6 " " angular displacement, " x " " linear displacement. Then, regarding the force of restitution as supplied by weight, (63) F x = mg tan 6. If 6 does not exceed about 3, the angle x = * may be written in place of the tangent as FIG. 41. J a sufficient approximation. Substituting these values in equations 57 and 62, (64) * = X and (65> P= or the period of the simple pendulum, within the limits mentioned, depends only on its length and the acceleration of weight, and not upon the mass of the bob. The law of the pendulum, so far as it regards the relation of period and length, was first derived by Galileo, whose attention was directed to the problem by observing that a lamp suspended by a long chain in the cathedral of Pisa swung in a time which did not alter with the width of the excursion. From the independence of the period and the mass he drew the conclusion that, apart from the resistance of the air, a large PROPORTIONALITY OF WEIGHT AND MASS. 61 body would fall with the same velocity as a small one. This result he verified by dropping various bodies from the leaning tower of Pisa. 41. Proportionality of Weight and Mass. The propor- tionality of weight and mass, though ^virtually assumed by every one who had used the balance, was first demonstrated by Newton. In order to investigate the relation between these quantities, he constructed a pendulum with a hollow spherical ball, in which he successively placed various differ- ent substances, and determined the period for each kind of matter. As no alteration in the period could be observed, he concluded that the attraction of one body for another depended only on the quantity of matter in each body, and not upon its nature. 42. Determination of g by the Pendulum. The pendu- lum furnishes a very accurate method for the determination of the acceleration of weight by means of the equation (66) <7^47r 2 -^. In practice the length of the pendulum is conveniently chosen so that it beats seconds, since the period may then be easily determined from a standard clock by the method of coincidences. An observation having first been made as to whether the pendulum is gaining or losing on the clock, the time is noted when the beat of the pendulum and the clock first coincide, and again at the next coincidence, which may be several minutes later. If n be the number of seconds which elapse between two successive coincidences, the period 7? will be 2 according as the pendulum is gaining or losing. For the most accurate observations the form used is not the purely ideal simple pendulum, but a compound or 62 SIMPLE TYPES OF MOTION. physical pendulum, consisting of a loaded bar which may be suspended from either of two adjustable knife edges, L, M, (Fig. 42). One of the cylindrical masses, A, is solid, while the other, B, is hollow, so that the distribution of matter is not alike in the two ends. As will be shown in Art. 63, it is possible to find positions for these knife edges, unsymmetrical with re- spect to the center of gravity, such that the pendulum supported at either L or M will swing in the same period, which is also the period of a simple pendulum whose length is the distance between the points of suspension. This distance, defined by sharp metal edges, may evidently be determined with great precision. FIG. 42. 43. Foucault's Pendulum Experi- ment. In 1851 Foucault, a French physicist noted for his experimental ingenuity, devised a re- markable proof of the earth's rotation by means of a pendulum. It is evident from theoretical considerations, or may be simply shown by experiment, that rotation of the support about the point of suspension of a pendulum will not affect the plane in which it is swinging, provided, of course, the mass of the bob is sufficiently great not to be appreciably affected by the resistance of the air. If, for instance, a pen- dulum be set swinging at the north pole in the prime meridian, this direction would be maintained unchanged, so that if a line should be drawn on the earth coincident in direction at the start, in twenty-four hours this line would have per- FOUCAULTS PENDULUM EXPERIMENT. 63 formed a complete counter-clockwise revolution with respect to the plane of the pendulum. If the same experiment were to be tried at the equator, it is evident that there would be no motion of the earth with respect to the plane of vibration. The rate of rotation o> at any point whose latitude is \ may be found most simply by regarding the angular velocity of the earth as a vector, and taking its component in the direc- tion of the vertical through the place, thus, 7T sin X. 12 hr. Or it may be found as follows : Let PP f (Fig. 43) be the arc through which a point, P, rotates N in a time, , and let H denote the radius of the earth, r " " distance PH, b P^V, c/> " angle PHP', ^ " PJVP'. Then, since the arc PP' is common, (67) ^ = - = sinX; 9 o whence the rate at which o/r is described is (68) . s= gn t t But the angular velocity of P is = - , and therefore t 12 hr. At a latitude of 45 this rotation is at the rate of 10.5 degrees per hour. 64 SIMPLE TYPES OF MOTION. In starting the pendulum, considerable care must be exer- cised to assure its vibration exactly in a plane. This condi- tion is best attained by drawing the pendulum to one side by a thread and burning it off when the system has come to rest. The experiment was first performed by Foucault in the Pantheon at Paris, by suspending an iron ball a foot in diameter by a wire 200 feet long. The result was in entire accord with his predictions, and was received with marked interest by the scientific world as furnishing a proof of the earth's rotation quite independent of astronomical observa- tion. Belief in this rotation, it is true, had won universal acceptance before this time, but it was founded chiefly upon the improbability that the stars, which were known to be very distant, should move with a speed exactly proportional to their respective distances from the earth. 44. Examples of Harmonic Motion. Other examples of harmonic motion are furnished by a strip of wood clamped at one end and loaded at the other with a mass, m (Fig. 44), m n FIG. 44. FIG. 45. and by a helical spring attached to a bracket and weighted at the lower end, as in Fig. 45. If either of these systems be set vibrating, it will be found that the period is sensibly EXAMPLES. 65 the same whether the amplitude is a fraction of a millimeter or several centimeters. If the displacement or elongation produced by m be denoted by e, the value of k will be 7 F v mq 4?r 2 m <> *=~^=v=-pr ; whence The acceleration of weight, as found by observations on either of these pieces of apparatus, will in general be much less accurate than that found from a pendulum. EXAMPLES. 1. A body falls freely from rest for 12.6 sec. Required the final velocity and the distance traversed. v = 1.235(10) 4 cm. / sec. s = 0.77S(10) 5 cm. 2. How long will it take a body to fall 650 ft., and what velocity will it acquire? Z = 6.35 sec. fc = 204 ft. /sec. y 3. A body is thrown downwards with a velocity of 874 cm. per. sec. Required its velocity and position at the end of 16.3 sec. y = 1.68(10) 4 cm. /sec. * s = 1.44(10) 5 cm. 4. A body is thrown vertically upwards with a velocity of 827 cm. per sec. How high will it rise? s = 349 cm. 5. A body is thrown vertically upwards with a velocity of 697 cm. per sec. How long will it take to rise 195 cm., and what velocity will it then possess ? t = 0.383 sec. v = 322 cm. / sec. 6. A body is thrown downward with a velocity of 328 cm. per sec. What distance will it traverse during the 12th sec. ? Ans. 1.16(10) 4 cm. 66 SIMPLE TYPES OF MOTION. 7. A ball is thrown to a Light of 150 ft. With what velocity does it leave the hand ? v = 98.3 ft. / sec. 8. If a 5-ounce baseball be thrown with a velocity of 110 ft. per sec., and the hand traverse a distance of 3 ft. in imparting to it this velocity, required the force and the time of action, supposing this to be constant. _F=19 pounds weight. t= 0.054 sec. 9. Two bodies are dropped successively from the same point at an interval of 0.25 sec. When will they be 6.5 ft. apart? * = 0.68 sec, 10. A body is projected upwards with a velocity of 965 ft. per sec. , 6.45 sec. later a second body is thrown up. When and where do they meet? = 26.8 sec. after the departure of the second body. s = 14300 ft. 11. A body is thrown vertically downwards with a velocity of 38.5 cm. per sec.; 2.57 sec. later a body is thrown after it with a velocity of 4750 cm. per sec. When and where will the second body overtake the first? t = 1.52 sec. after departure of second body. s = 8350 cm. 12. A stone is thrown horizontally from a cliif, 364 ft. high, with a velocity of 105 ft. per sec. When and where will it strike the ground ? Time = 4.75 sec. Distance = 49 9 ft. 13. A mass of 876 gms. is attached to a spring balance which is carried upward at such a rate that the balance indicates 932 gms. What is the acceleration of the motion? /=62.7 cm. /sec. 2 . 14. A mass of 162 kilos hanging by a perfectly flexible cord drags a mass of 973 kilos along the top of a smooth table. What is the acceleration of the system, and what is the tension of the cord ? /= 140 cm. / sec. 2 . T = 1.36(10) 8 dynes. 15. Masses of 938 gms. and 762 gms., respectively, are hung by a flexible cord passing over a frictionless pulley. How far must the weights move in order to acquire a velocity of 325 cm. per sec. ? 5 = 521 cm. EXAMPLES. 67 16. What will be the tension on the string in the preceding example? r = 8.42(10) 5 dynes. 17. A particle slides down a smooth plane 326 cm. long, inclined at an angle of 45 to the horizon. Required the time of descent and the velocity with which it will reach the bottom. t = 0.970 sec. v = 672 cm. / sec. 18. If a body faU 178.3 ft. in the 6th sec., what is the value of g'l # = 32.42 ft. /sec. 2 . 19. Show that the time of descent on any chord passing through the highest point of a vertical circle is the same. t= 20. A mass descending vertically draws an equal mass 25.3 ft. up a smooth plane, inclined at an angle of 30 with the horizon, in 2.5 sec. What is the value of g ? g = 32.4 ft. / sec. 2 . 21. The record for baseball throwing is about 400 ft. What was the velocity with which the ball left the hand ? v = 113 ft. / sec. 22. If a body be projected at an angle of 30 above the horizontal, from a cliff 80 ft. high, with a velocity of 96 ft. per sec., when and where will it strike the ground ? Time = 4.2 sec. Distance = 349 ft. 23. The ratio of the masses of the moon and the earth is 0.0125, and the ratio of their diameters is 0.273. With what acceleration would a body fall at the moon's surface? /=164 cm. / sec. 2 . 24. A mass of 53.8 gms. is constrained to move in a circle of 597 cm. radius with a speed of 235 cm. per sec. What is the force and what is the period of revolution? J P=4.98(10) 3 dynes. P = 16.0 sec. 25. The distance of the moon is 3.84 (10) 10 cm., and the lunar month is approximately 27 d. 8 h. What is the acceleration of the earth's attraction at the moon? /= 0.272 cm. / sec. 2 . 26. If a skater describe a circle of 98 ft. radius with a speed of 18 ft. per sec., what should be the inclination of his body from the vertical for equilibrium? Ans. 5.9. 68 SIMPLE TYPES OF MOTION. 27. What must be the vertical distance between the mass and the point of suspension of a conical pendulum, in order that the period shall be 0.875 sec. ? h = 19.0 cm. 28. The length of a conical pendulum is 45 cm., and the radius of the circle in which the mass moves is 12.6 cm. What is the period of the pendulum ? P = 1.32 sec. 29. W T hat would be the value of if the period of a pendulum 97.31 cm. long were 1.975 sec.? g = 984.9 cm. / sec. 2 . 30. If the period of a pendulum 99.7 cm. long is 2.12 sec., what will be the period of a pendulum 822 cm. long? P = 6.09 sec. 31. What is the length of a pendulum which loses 20 sec. per day, where g = 980.3 cm. / sec. 2 ? / = 99.37 cm. 32. What is the rate of a pendulum 39.09 inches long at the same place as the preceding example? Gaining 13 sec. per day. CHAPTER III. WORK AND ENERGY. 45. Work When a body moves in the direction of a force acting upon it, the force is said to do work, and the measure of the work is denned as the product of the force by the distance moved. If the direction of the displacement is inclined to the direction of the force, the work is found by multiplying the displacement by the component of the force parallel to the displacement. Thus, if the body A (Fig. 46) is displaced to A' a distance, s, under the action of the force F, making an angle a with this direction, suppose F to be resolved into a component, F ! , parallel, and a component, F", at right angles to A A 1 . The work done by the component F" will be zero, since the body suffers no displacement in its direction. Therefore, the total work will be that done by F'. Calling this work W, (i) TT= F 1 s = F cos a s. The dimensions of work are ML 2 T~ 2 . The unit is the work done by the force of one dyne acting through the dis- 4.' T^ I V J ^ ^ g 111 ' 1 Cm ' 2 tance of one centimeter. It is symbolized by r : - , JL sec. and is called the erg. Units of work derived from the gravitational unit of force are sometimes used. Thus the kilogram-meter is the work done by a force equal to the weight of one kilogram acting 70 WORK AND ENERGY. through one meter, and the foot-pound is the work done by a force equal to the weight of one pound acting through one foot. 46. Energy. Energy is denned as the capacity of a body or system of bodies for doing work. It is convenient to distinguish two kinds of energy: 1, potential energy, or that due to the configuration of a system of bodies ; and, 2, kinetic energy, or the energy which a body possesses in virtue of its motion. An example of each is furnished by a bow and arrow. When the bow is drawn, it possesses an amount of energy due to its altered shape, which is equal to the work done in bending it. When the string in contact with the arrow is released, the latter is given a rapid motion, in virtue of which it is able to do work when it is brought to rest. The spring of a clock when wound up is another example of a system possessing potential energy, as is also a weight elevated a certain distance above the earth. The increase of potential energy of the system, which in this case con- sists of the weight and the earth, is equal to the work done in raising a mass, w, through a hight, h ; that is, mgh. When the weight is allowed to fall to earth again, it will acquire an amount of kinetic energy which may be used to do use- ful work, such, for instance, as driving a pile into the ground. 47. Kinetic Energy of a Moving Particle. It has been shown in Art. 20 that when a particle, m, is acted on by a constant force, F, for a time, , the particle will suffer a dis- placement, s = J/ 2 , due to the force, and acquire a velocity vft. Hence, the work done by the force during the process is (2) W= F-s = m TRANSFORMATIONS OF ENERGY. 71 This expression, or one-half the product of the mass by the square of its velocity, is taken as the measure of the kinetic energy of the moving particle. The dimensions of energy are ML 2 T~ 2 , the same as for work. The unit kinetic energy is the erg, or twice the kinetic energy of one gram moving with the velocity of one centimeter per second. 48. Transformations of Energy. A vibrating pendulum presents an example in which the energy is continuously changing from the potential to the kinetic form, and vice versa. Thus, at the middle of the swing the energy is all kinetic. As the pendulum rises, the potential energy in- creases and the kinetic energy diminishes, until at the extent of the swing the former becomes a maximum and the latter zero. On the return to the lowest point the energy resumes its kinetic form. The examples of energy mentioned so far have been of a purely mechanical sort. When a moving body is brought suddenly to rest by striking against an obstacle, the kinetic energy is mostly changed into heat ; as, for instance, when a piece of lead is made hot by hammering it. When carbon and oxygen are allowed to unite, as in the burning of coal, the energy of chemical separation is likewise transformed into heat. When the diaphragm of a telephone is set vibrat- ing, there is a transformation of the energy of mechanical vibration into the energy of electric currents in the wire, which are again transformed partly into the energy of vibra- tions at the receiver and partly into heat in the conductors. When a stick of resin is rubbed with fur, a part of the work expended is transformed into the energy of electrical separa- tion. Many other examples could be given, but these are sufficient to illustrate the fact that energy not only appears 72 WORK AND ENERGY. under a variety of forms, but also can be transmuted from one form into another, 49. Conservation of Energy. The doctrine of the con- servation of energy asserts that in any closed system, that is to say, one which is isolated so that it neither parts with energy from within nor receives energy from without, the amount of energy is invariable. This principle, which may be regarded as not only the greatest but the most fruitful conception which has ever been introduced into physical science, represents rather the accumulated experience of all thinking men than the dis- covery of any individual or any school of philosophy. The first comprehensive statement of the law appears to have been published by Mohr in 1837, but many philosophers before his time had assumed its truth in special cases. As early as the end of the sixteenth century, Galileo, in his dis- cussion of the laws of a falling body, recognized and used the principle that a body, in virtue of the velocity acquired in its descent, rose to the hight from which it had fallen, or would do so if the resistance of the air were removed. Huyghens went a step further, declaring that if a number of weights be set in motion the common center of gravity cannot possibly rise higher than the place it occupied when the motion began. Newton, near the middle of the seventeenth century, per- ceived that the principle of conservation, which had hitherto been considered only in connection with bodies acted on by weight, was applicable to all mechanical problems. In the scholium to his Third Law he states that " if the action of an agent be measured by the product of the force into its velocity, and if similarly the reaction of the resistance be measured by the velocities of its several parts multiplied CONSERVATION OF ENERGY. 73 into their several forces, whether they arise from friction, cohesion, weight, or acceleration, action and reaction in all combination of machines will be equal and opposite." The quantity here defined is what would now be called the rate at which work is done by the forces, but for the field covered in the statement it is essentially equivalent to the assertion that energy is conserved. By the beginning of the nineteenth century all instructed mechanicians had come to recognize that it was impossible to get any more work out of a machine than was expended upon it; and it is interesting to note that the French 11 Academy, as early as JJL75, declined to consider any further devices for securing u the perpetual motion," that is, machines which should not only keep running for an indefinite time, but also perform useful work. At that time, before the nature of heat had been deter- mined, it was not possible for any one to state whether the difference between the work supplied to a machine and the work derived from it was destroyed, or whether it had dis- appeared by passage into some other form of energy. The determination of the true nature of heat by the experiments of Rumford, of Joule, and of Davy, which will be explained more fully in Chapter XIV, gave the needed confirmation to the idea which had been more or less clearly apprehended by Carnot, Mayer, Joule, and others, that energy can neither be created nor destroyed. For having proved that a definite amount of work was always equivalent to a certain amount of heat, it was now possible to show that whatever form energy may assume it can always be made to yield an amount of work and heat, which together are equivalent to the work originally expended. It is by experiments of this kind, tried in all sorts of ways, that the law has now been established, and it is recognized 74 WOEK AND ENERGY. to be universally true that whether the energy subsist in the potential form as the energy of the visible arrangement of the parts of a system, or as the energy of molecular separation, or as the energy of electrical separation, or whether it consist in the kinetic energy of moving visible masses, or of molecular vibrations, or of wave motions in the ether, or of electrical currents, in each an'd every case the total quantity of energy in the system, when isolated from every other system, is invariable. 50. Minimum Potential Energy. It has been found that when any of the stresses on a system are removed a redis- tribution of the energy occurs, that portion of the energy in the potential form assuming a minimum value and that in the kinetic form a maximum value. This principle, some- times comprised in the statement that "potential energy tends to a minimum," is capable of application to a wide variety of problems. Thus, for instance, it furnishes a simple test for the three conditions of a system, known respectively as stable, unstable, and indifferent equilibrium. A system is in stable equilibrium if, in order to give it a small displacement, it is necessary to do work upon it, i.e. increase its potential energy. A right cone resting on its base presents an example of stable equilibrium. If, on the other hand, a slight displacement diminishes the potential energy, the system is unstable. A cone resting on its apex would be an example of unstable equilibrium. When a slight displacement produces no change in the potential MACHINES. 75 energy, the system is in indifferent equilibrium. This is illustrated by a right circular cone lying on its side on a horizontal surface. 51. Machines. A machine is a piece of apparatus by the aid of which work may be more conveniently performed. In every machine there is some point to which energy is sup- plied, and another point from which work is derived. Usually the force exerted by the latter, or working point, is greater than that applied to the former. The ratio of the force derived to the force applied is called the me- chanical advantage of the machine. This ratio may be expressed in terms of certain distances, which are more easily measured than the forces. Thus, let A (Fig. 47) be the point of a machine at which a force, -F, is applied, and B the working point. Let D be the distance moved by the point of application of the force JP, and D f that moved by the working point which exerts a force, F f . Then, if the machine is a perfect one, FIG. 47. F' D (3) or, the mechanical advantage = = . Ji JJ The mechanical advantage may thus, without knowledge of the intervening mechanism, be calculated from an observa- tion on the movements of the working point and the point at which the force is applied. 76 WORK AND ENERGY. FIG. 48. The following examples illustrate how the mechanical advantage of a perfect machine may be calculated when the arrangement of its parts is given. Screw. Let t be the pitch of a screw, that is, the distance between the threads, and I the length of the arm by which the force is applied. Then, by equation 3, the mechanical 2-rrl advantage = -- Pulleys. If in such an arrange- ment as that of Fig. 49 the weight were raised one foot, the point of application of F would move six feet; therefore, the mechan- ical advantage is 6 ; or, in general, with pulleys arranged as in the figure, the mechanical advantage is equal to the number of cords by which the weight is supported. Hydraulic Press. The hydraulic press consists of a lever, B (Fig. 50), a pump, H, and a ram, J, working in the cylinder M. Call the distance from the hand to the fulcrum Z, and the distance from the piston-rod to the fulcrum I. Then, by the geo- metrical relations of the figure, the mechanical advan- tage of the lever will be y, or, calling F the force applied at B, and F' that exerted on the bottom of the plunger in H, F' L Suppose that the plunger in H moves down a distance, c?, and that P moves up a distance, D. Because water is incompressible, the volume displaced in each case FRICTION. 77 will be the same ; or, calling a and A the respective areas of the pistons, whence the mechanical advantage of the pumps is F" d A Multiplying equations 4 and 5, the mechanical advantage of the system is found to be F al In every actual machine some of the energy is expended in work done against friction or other prejudicial resistances. FIG. 50. The ratio of the work done by the machine to the work done upon it is called the efficiency. 52. Friction. The resistance which bodies oppose to the movement of one surface on another is termed friction. It depends both on the nature and the roughness of the 78 WOEK AND ENERGY. surfaces in contact. At the commencement of the sliding it is greater than when the motion is continued, but does not change much with the relative velocity unless this becomes very great. The ratio of the tangential to the normal stress, when the sliding is about to begin, is called the coefficient of statical friction. The analogous ratio when the motion has become steady is called the coefficient of kinetic fric- tion. Its value varies greatly with the lubrication of the surfaces, and can be determined by experiment only. EXAMPLES. 1. The lower end of a ladder 15.6 m. long stands on the ground at a distance of 2.73 m. from a wall against which the upper end rests. How much work will be done in carrying 26.4 kilos up the ladder? Ans. 3.98(10) 10 ergs. 2. The diameter of the cylinder of a steam engine is 18 in. and .its length 24 in. What work will be done in each stroke of the piston if the average pressure of the steam is 110 Ibs. per sq. in.? Ans. 0.56 (10) 5 foot-pounds. 3. A mass of 2.64 Ibs. attached to a string 39.7 in. long is released after the string has been raised to the horizontal. When the pendulum has fallen to a position making an angle of 30 with the vertical, what will be the kinetic energy of the bob ? Ans. 7.56 foot-pounds. 4. If a mass of 84.5 gms. sliding down a rough inclined plane 78.2 cm. high acquire a velocity of 288 cm. per sec., how much work has been done against friction ? Ans. 2.97(10) 6 ergs. 5. 5.92 sec. after a mass of 57.8 gms. has been thrown vertically up- ward, it is observed to possess a downward velocity of 387 cm. per sec. With what energy will it reach the ground? Ans. 0.846(10) 9 ergs. 6. Given the length of a pendulum 99.3 cm., the maximum dis- placement 16.3 cm., and the mass of the bob 100 gms., show that the calculated value of the kinetic energy at the middle of the swing is equal to the work done in producing the displacement. Ans. Kinetic energy = 1.35(10) 5 ergs. 7. If a bullet having a velocity of 187 m. per sec. will pierce 8.92 cm. of a target, what velocity must the same bullet have to enter a distance of 14.8 cm. ? w = 241 m. per sec. CHAPTER IV. MECHANICS OF A RIGID BODY. 53. Motion of a Rigid Body. The treatment of the motion of an extended body may be greatly simplified by assuming that the body is rigid^ that is to say, suffers no deformation under the action of applied forces* When a body moves so that the line joining any two points of the body remains parallel to its previous position, the motion is said to be one of translation. When a body moves so that each point describes the arc of a circle having its center on a fixed straight line to which its plane is perpendicular, the motion is said to be one of rotation, and the fixed straight line is called the axis of rotation. The application of one or more forces to' a rigid body pro- duces, in general, a change both in the motion of translation and of rotation, which may be treated separately. The first of these changes has already been explained in Chapter II ; the second requires further consideration. 54. Moment of a Force. Suppose that a force, F (Fig. 51), applied to a rigid body at A, produces a rotation of the line OA ?* about the fixed point 0, through a small angle 6. Call the distance OA = p, and the arc AB = s. Taking the work done as the measure of the effect of the force in producing this rotation. (l) F-s 80 MECHANICS OF A RIGID BODY. FIG. 52. If a different force, F 1 , had been applied at A\ its effect would have been (2) F'-s' = F'p'e. Whence it appears that, in order that two different forces shall have the same effect in producing a given rotation, it is necessary that Fp = F'p*. This product of the force by the perpendicular on its direction from the center of rota- tion is called the moment of the ^F force, that is to say, the impor- tance of a force in producing a rotation. If the direction of the force is inclined to OA (Fig. 52) at an angle, a, calling the length of this line , the moment is F - 1 sin a = F sin a ?, which shows that in discussing moments it is only necessary to consider the component of F, which is at right angles to I. 55. Resultant of Two Parallel Forces. Let F l and F z (Fig. 53) be two parallel forces applied to a rigid body in the direc- tion of the lines aa' and bP. In order to find the line of application / of the resultant, such that the latter shall produce the same effect in rotation about a point, 0, as the joint action of F and F^ draw OB through the FIG. 53. RESULTANT OF TWO PARALLEL FORCES. 81 given point perpendicular to the direction of the forces. Let OA=p v OB=p v OC=p, and call the resultant R. Then in order that the rotating effect of R shall be the same as the combined effect of the forces, (3) Rp But, by definition, (4) ' whence or, p 9 p F, ( 6 ) - - ~ = -^T' P-fi F 2 By the figure, p 2 p = CB and p p l = AC; therefore, the point divides the distance AB into segments which are inversely proportional to the adjacent forces. As the point may be taken anywhere in the line AB, provided the moments are given the proper sign, say, plus when the rota- tion would be clockwise and negative in the opposite case, the motion of C at any instant will not be altered if F and J? 2 at A and B be replaced by R at 0. Taking at (7, equation 6 becomes Pl = ll Pi ^ or, that is, the moments of F l and F^ about are equal and opposite. When F l is equal and opposite to ^ 2 , R = and the solu- tion fails. This combination of forces illustrated in Fig. 54 is called a couple. Taking the moments about 0, 82 MECHANICS OF A KIGID BODY. or, the moment of a couple is equal to the product of the force into the perpendicular distance between the lines of & action of the forces. The value of a couple is, ac- cordingly, independent of O I its position in the plane. 56. Center of Parallel Forces. If any number 2 of parallel forces acting FIG -U- -5 j- x- -L in the same direction be applied to a rigid body, the resultant will pass constantly through a fixed point without reference to the particular direction which the forces may have. To prove this propo- sition, suppose that forces F v F^--- F n are applied to the body at points x^y-^z-^ FIG. 55. and let the angle which the common direction of the forces makes with the X axis be called a, that with the Y axis & and that with the Z axis 7. Denote the resultant of the forces by Rl then If R is to be the resultant of F v F%, F n , i.e. completely replace these forces, its moment about any axis must be equivalent to the combined moments of all the individual forces. CENTER OF PARALLEL FORCES. 83 Let x, ?/, 2, be a point in the line of application of the resultant. The sum of all the components parallel to the X axis is (8) R COS a = F l COS a -\- F% COS a -J- F n COS a ; similarly, for the Y axis, (9) EGOS $ = F! cos + JP a cos + j^ cos ; and for the Z axis, (10) R cos y = F 1 cos 7 + -F 2 cos 7 + " ' ' -^n cos 7- Now as the moment of R, with respect to any axis, is equal to the sum of the moments of the forces with respect to the same axis, it follows that the moment of the com- ponent of R will be equal to the sum of the moments of the components of F v F^ etc., about the same axis. Hence, taking moments about Jf, Y, and Z, (11) R cos a-y = F l cos a y l + F^ cos a # 2 + F n cos a-y n , (12) R cos ft z = F l cos @ z l + F^ cos 13 z 2 + ^cos/3-z n , (13) .ft.cos 7 x = F l cos 7 x 1 + F z cos 7 ^ n cos 7-^, whence and, similarly, Since the values of a;, ?/, 2, are independent of direction of the forces, it follows that if the system of forces, JF V F^ F n , be applied in any direction, at the same points, the 84 MECHANICS OF A RIGID BODY. resultant would always pass through a single point. This point has received the name of center of parallel forces. 57. Center of Gravity. If m v m z , - (Fig. 56) be a sys- tem of particles rigidly connected, the weights of these particles may be regarded as forces having sensibly the mi same direction. The point . G m& 6r, through which the resul- tant of this system of parallel w 4 . forces constantly passes when FlG 56 the system is turned into dif- ferent positions, is called the center of gravity. Substituting the weights of the elementary masses for the forces in equations 14, 15, and 16, the coor- dinates of the center of gravity are found to be _ Em *Lmgz (17) Em The point denned by the final form of these equations, though identical with the center of gravity, might be more properly called the center of mass, since weight does not enter into its definition. The solution of these equations in general requires the aid of more powerful mathematical methods than may profitably be introduced here, but in the simplest cases, which are also the most useful, the position of the center of gravity may be found by the application of elementary considerations alone. The Center of Gravity of a Straight Line. Since the resultant of two equal parallel forces bisects the line joining CENTER OF GRAVITY. 85 them (Art. 55), the center of gravity of two equal particles is half-way between the particles. Now, as any uniform physical line may be regarded as made up of pairs of such particles, the center of gravity of a line will be at its middle point. Center of Crravity of a Lamina having an Axis of Sym- metry. Any indefinitely thin lamina may be regarded as made up of a number of parallel lines. If it has an axis of symmetry, these lines may be chosen so that they will be bisected by this axis. Now, as the center of gravity of each of the lines is on the axis, the center of gravity of the lamina will also lie on the axis. It follows that, if the lamina have two axes of symmetry. the center of gravity will be at the inter- section. Therefore, the center of gravity of any regular plane figure is at the geo- metrical center. Center of Gravity of a Triangle. The center of gravity of a triangle, Gr (Fig. 57), is at the intersection of two median lines AD and CE. Because AB and BC are bisected at E and D, ED is parallel to A C, and __ _ AC~ ~*' Also, since the triangles AGr C and D Gr E are similar, or, Da is 1 AD. The center of gravity of any polygon may be found by dividing it into triangles, and regarding the weight of each of these applied at their respective centers of gravity. 86 MECHANICS OF A RIGID BODY. Center of Gravity of a Solid having a Plane of Symmetry. If a solid have a plane of symmetry, it may be regarded as made up of laminae which are arranged in equal pairs with respect to this plane. As the center of gravity of each pair of equal laminae is in the plane, the center of gravity of the whole solid will lie there. If there are two planes of sym- metry, it will lie on the line of intersection, and if three, at the point common to all. Hence the center of gravity of a regular solid is at its geometrical center. Center of Grravity of a Tetrahedron. Consider the tetra- hedron ABCD (Fig. 58) as made up of laminae which are similar to the base B CD. Then the centers of gravity of each of these will lie on the line AE, drawn from the vertex to the center of gravity of the base BCD, and hence the center of gravity of the tetrahedron will be found somewhere in this line. By similar reasoning it may be shown to lie in the line BF, drawn from B to F, the cen- ter of gravity of ADC. Now, as BF and AE are both in the plane AHB, they must intersect in some point, 6r, whose position is required. By the con- struction of the figure, FH=\ AH and EH=\ BH\ there- fore, EF is parallel to AB, and the triangles AGrB and GEF are similar. Whence PIG. 58. Ea_EF AB' CENTER OF GRAVITY. 87 but therefore, Since every pyramid may be divided into a number of tetrahedra having a common vertex, and their bases in the base of the pyramid, the center of gravity of a pyramid will lie in a plane drawn parallel to the base at one-fourth the altitude above it. Also, as the pyramid may be considered as made up of laminae which are polygons parallel to the base, the center of gravity of the pyramid must lie on the line joining the vertex to the center of gravity of the base, and one-fourth the distance, as was just shown. The center of gravity of a cone is likewise seen to lie on the axis, at one-fourth the distance to the vertex, since any cone may be regarded as a pyramid of an infinite number of sides. Center of Gravity of a Compound Figure. When a body is made up of several parts, of A which the centers of gravity are known, the center of gravity of the whole may readily be found by application of equations 17. For instance, let it be required to find the center of gravity of a trape- D HI ~~G zoid. This quadrilateral (Fig. 59) &<* - may be regarded as composed of a triangle and a parallelo- gram. Call the mass of the triangle BDH, m v the mass of the parallelogram BEGf-H, m^ and the mass of the trapezoid 88 MECHANICS OF A RIGID BODY. DBEGr, m m^^m^. Let u be the distance of c from the base DGr\ then Calling BE=a, D6r = 5, and the altitude^/*, and assum- ing that the density and the thickness of the body are con- stant, the value of u reduces to _ '* * (b + a) It is also evident that the trapezoid might be regarded as a portion of the triangle ADGr, in which the center of gravity C of the whole, and c 1 of the part ABE are known. Calling distances measured on the median from /, v, the mass of ADGr M and ABE= m f -, Mv m'v' b 4- 2a v = = 1* - 1 , m b-\- a where s is the portion of the median included between BE and Da. 58. Conditions of Equilibrium. If the sum of all the moments of the forces applied to a rigid body is zero, the forces will produce no change of rotation. If, in addition, the sum of the forces is zero, they will produce no change of translation. Therefore, the conditions for equilibrium may be written (18) J ^Fp = 0, (19) 1 2.F = 0, provided the moments and the forces in each case are taken with the proper algebraic sign. The origin of moments is immaterial, since every point is at rest. As a simple illus- tration of the preceding principles, suppose a bar 5 ft. long, weighing 30 Ibs., has its center of gravity 3 ft. from one MOTION OF ROTATION OF A RIGID BODY. 89 R 10] p bs. A bs. > 301 FIG. 60. ^ 251 bs. end ; if 10 Ibs. be hung on this end and 25 Ibs. on the other, required the point at which it will balance, and the weight sus- tained by the support. Let any point in the diagram (Fig. 60) be taken as the axis of moments, as, for exam- ple, the center. Call R the reaction of the support placed at a distance, jt?, from the center. Calling the downward rotation of the right-hand end positive, equations 18 and 19 here become (25 Ibs. X 2-i ft. + 30 Ibs. X i ft. - 10 Ibs. x 2 ft.) g-Bp = Q, (25 Ibs. + 30 Ibs. + 10 Ibs.) g - R =*0, whence R = 65 Ibs. i; that is, the balls exchange velocities. The laws of impact of elastic bodies may be experimen- tally demonstrated for the simple cases by the apparatus shown in Fig. 71, in which two ivory spheres are suspended so as to swing over a graduated arc. The velocities acquired by a ball in falling are proportional to the square root of the vertical hight, or, in other words, to the sagitta of the arc traversed. But the sagittas are proportional to the squares of the distances of the ex- tremities of the arcs from the vertical through the center of the circle. Hence, within the limits of the apparatus, the velocity of a ball in passing the vertical may be taken as proportional to the arc traversed. For the study of inelastic impact the balls may be ren- dered practically inelastic by sticking a piece of wax to the face of one of them. FIG. 71. BALLETIC PENDULUM. 113 82. Ballistic Pendulum. - - The velocity of a projectile may be determined by means of a sus- pended block of wood, into which the shot is fired. Let AB (Fig. 72) represent a block suspended from (7, and call its mass M. Let the mass of the shot be m, then as this system fulfills the condition of ine- lastic impact, equation 6 gives (12) m M+m* An observation of the vertical rise h of FIG. 72. the point A will give u 2 by the formula u 2 = \ltyh, whence v may be found at once from equation 12. EXAMPLES. 1. A piece of brass wire 0.1066 cm. in diameter and 27.1 cm. long is found to be stretched 0.133 cm. by the addition of a load of .454 kilo. What is Young's Modulus for the wire ? AT = 1.016(10) 10 gms. / cm. sec. 2 . 2. If a wire 76 cm. long have a period of torsional vibration of 7.52 sec., and a wire 38 cm. long of the same diameter have a period of 5.32 sec., how does the moment of restitution depend on the length of the wire, assuming that the vibrations follow the harmonic law ? Am. M & I 1 . 3. If the period of torsional vibration of a wire 0.092 cm. in diameter have a period of 7.55 sec., and that of a wire of the same length but 0.069 cm. in diameter have a period of 13.15 sec., what is the relation of the moment of restitution to the diameter of the wire? Ans. M T on the ends and the pressures on the sides, to which may be added some external force, F. Since cd is a line of equal pressure, the forces due to the pressures on the ends are equal and opposite. The pressures on the sides are perpendicular to the sur- face, since they are exerted by a fluid. The resultant force is therefore perpendicular to the axis, and in consequence the force F, which is equal and opposite to this resultant, must also be at right angles to cd. But cd is any line in the surface of equal pressure, therefore F is perpendicular to this surface. In the case of a liquid acted on by weight only, any small free surface must accordingly be a horizontal plane, and by means of the preceding article it may be shown that any horizontal plane in the liquid is a surface of equal pressure. When several fluids which do not mix are superposed, they will arrange themselves in order of increasing density from top to bottom, to fulfill the condition of minimum potential energy, and the surfaces of separation will be hori- zontal, since they are surfaces of equal pressure. 85. Levels. A spirit-level consists of a tube (Fig. 75) bent to the arc of a circle, nearly filled with alcohol, and LEVELS. 117 hermetically sealed. The small air bubble will always stand at the highest point on account of its lesser density. When mounted in a suitable base, the level may be used to set a line horizontal or" vertical. Call the upper inner radius of the tube R ; then if, when the inclination of the level is changed an angle, <, the bubble moves a distance, c?, R If, for instance, R were 60 inches, a variation of 6 min. in the inclination of the level would produce a deflection of -j^th of an inch in the bubble. A spirit- level 3 inches long would in this \ case indicate a deviation as defi- x \ / nitely as a plumb line 5 feet \ / long. Such levels may be made \ x / of any degree of sensitiveness, \ / though the precision of their reading is considerably limited by the accuracy with which it is possible to form the inner surface of the tube to a true arc of a circle. When the upper surface of the vessel is made spherical, the instrument is known as a box-level, and may be used for leveling a horizontal surface. As usually made, the box- level is not serviceable, on account of the readiness with which the alcohol evaporates. 86. Principle of Archimedes. The principle of Archi- medes may be stated as follows : a body immersed in a fluid at rest loses a portion of its weight equal to the weight of its own volume of the fluid. V FIG. 75. 118 MECHANICS OF FLUIDS. To prove it, suppose any portion, A, of a fluid at rest (Fig. 76) to be separated from the remainder by a closed sur- _ face wholly immersed. Since this portion by supposition is at rest, the pressure of the fluid on the outside of the surface must be such as to sustain the weight of the fluid within. Now, if the fluid within this imaginary surface be replaced by any other substance, it is manifest that the pressure of the fluid outside will not be modified. But this pressure caused the volume of the fluid enclosed to lose its weight. There- fore, any other body of the same shape would lose the same amount, that is, the weight of its own volume of the fluid. Call m the mass of the body, m^ the apparent mass as determined by weighing in the fluid, V the volume of the body, p f the density of the fluid ; then Archimedes' Principle may be written, (2) mff In all but the most refined observations the loss of weight, on account of immersion in air, may be neglected. If the body be immersed in a fluid whose density is greater than its own, the loss of weight is greater than the weight of the body. In such a case the body is acted on by an upward force, and if not constrained will rise till the weight of the volume of the fluid displaced is equal to the weight of the body. DETERMINATION OF DENSITY. 119 87. Determination of Density. When the volume of a solid is known, its density may be calculated directly from the definition m P = If the body is of such shape that its volume may not be readily calculated, the density may be found by weighing the body in a liquid of known density. Thus, writing F*= in equation 2, and solving for /a, (3) P = m m m If the liquid used is water, its density may be taken as unity, and the numerical value of the density of the body will be found by dividing the weight in air by the loss of weight in water. When the body is less dense than the fluid, it is necessary to use a sinker. Let w 2 be the apparent mass of the^sinker when weighed in the fluid, and M the apparent mass of both sinker and body in the fluid ; then, as the loss of weight is proportional to m -f m z M, the density is given by (4) m M p'- The density of a liquid may be obtained by the use of a flask (Fig. 77) so constructed that when its stopper is in place its content is ex- actly 100 cc. This condition is most readily effected by first making the glass stopper too long or the volume of the flask too small. Then, by grinding off the end of the stopper, the required volume may be secured as closely as may be FIG. 77. 120 MECHANICS OF FLUIDS. desired. In order to render the complete filling easy, the stopper is provided with a small hole through which the excess of liquid in the flask escapes when the stopper is pressed in. The difference between the weight of the flask when full and when empty gives the mass of the liquid. The density is obtained by dividing this mass by the volume of the flask. The density of a gas may be found in an entirely analogous manner. How- ever, on account of the relatively small density of gases, certain precautions are necessary. For example, the flask should be as large as is convenient, in order that its mass may not be too great compared to that of its contents ; and again, when the flask is weighed empty it should be exhausted, as nearly as is possible, of every kind of matter. 88. Areometer. -- The areometer is an instrument designed to indicate the density of a liquid by the depth at which it will float. It consists essentially of a 8 ' weighted glass bulb with a graduated stem (Fig. 78). Instruments intended for use in liquids lighter than water read upward from the bulb, while those designed for denser liquids read from the top down. The scale is not infre- quently arranged so as to give the percentage of one of the components in a solution. Thus, for instance, the areometer is often used to determine the concentration of commercial alcohol, or sulphuric acid, and the comparative richness of va- rious samples of new milk. In the arbitrary scale of Baume the point to which the instrument sinks in water is marked 0, and the immersion in a solution of 15 parts common salt to 85 parts water, 15. For liquids lighter than water the point of immersion in a solution of 10 parts water to 90 parts salt is marked 0, and the immersion in pure water 10. TABLE OF DENSITIES. 121 TABLE OF DENSITIES. UNIT 1 gm. per cc. . Solids. Aluminium 2.6 Ice 0.917 Antimony 6.7 Iron (cast) 7.0 to 7.6 Bismuth 9.8 (wrought) 7.3 to 7.8 Brass 8.4 Lead 11.3 Brick 2.1 Nickel 8.9 Clay 1.9 Oak 0.7 to 1.0 Copper 8.9 Platinum 21.5 Cork 0.24 Quartz 2.65 Diamond 3.5 Sand (dry) 1.4 Gas Carbon 1.9 Silver 10.5 Glass (crown) 2.5 to 2.7 Sodium 0.98 (flint) 3.0 to 6.3 Sulphur (native) 2.0 Gold 19.3 Tin 7.3 Graphite 2.3 Zinc 7.1 Liquids at C. Alcohol 0.806 Nitric Acid 1.56 Bisulphide of Carbon 1.29 Oil (linseed) 0.94 Chloroform 1.53 " (mineral) ' 0.76 to 0.83 Ether 0.736 (olive) 0.91 Glycerine 1.27 Sea Water 1.03 Hydrochloric Acid 1.27 Turpentine 0.87 Mercury 13.6 Gases at C., and 760 mm. at latitude 45. Air 0.001293 Marsh Gas 0.000715 Carbon Dioxide 0.001965 Nitrogen 0.001254 Chlorine 0.003167 Oxygen 0.001429 Hydrogen 0.0000895 Gases at C., and a pressure of 10 6 dynes per square centimeter. Air 0.001276 Marsh Gas 0.0007173 Carbon Dioxide 0.001951 Nitrogen 0.001239 Chlorine 0.003091 Oxygen 0.001411 Hydrogen 0.00008837 122 MECHANICS OF FLUIDS. Density of Saturated Water Vapor. 10 20 30 40 50 60 70 80 90 100 Air at 100 0.00000475 gm./cc. 0.00000922 0.0000170 0.0000301 0.0000509 0.0000829 0.0001306 0.0001994 0.0002959 0.0004284 0.0006062 0.0009459. 89. Barometer. The pressure of a gas is conveniently determined by comparison with the pres- sure exerted by a known column of liquid. Thus, if a tube of glass, closed at one end and more than 76 cm. long, be filled with mercury and inverted in a vessel of the same liquid, the pressure of the air will support a column of the mercury about 76 cm. high. This pressure, 76 x 13.6 x 980 = 1.013(10) 6 dynes per square centi- meter, is sometimes taken as a unit, and is called an atmosphere. Instruments used to measure the pres- sure of the air are known as barometers. A common form, called the siphon barom- eter, consists of a glass tube closed at the upper end, and bent into a /"-form at the lower end, as shown in Fig. 79. The dis- tance between the bend and the closed end must be considerably more than 76 cm. Special precautions also, are necessary to assure that the FIG. 79. FORTIES BAROMETER. 123 mercury and the tube are perfectly clean and that no air is included. The difference of level between the free surfaces of the mercury may be read from the millimeter graduations on both legs of the tube. 90. Fortin's Barometer. A portable form of barometer devised by Fortin is shown in Fig. 80. The cistern O is closed at the bottom by a leather bag, B, which may be raised and lowered by a screw, & A glass cylinder, 6r, near the top permits the upper surface of the mercury to be seen and brought in contact with the point P, which marks the zero of the barometer scale. The tube is attached to the cistern by means of a piece of chamois skin, F, which prevents the escape of the mercury but does not support any dif- ference of pressure between the outside and inside air. The glass tube is protected by an outer brass case grad- uated to millimeters and furnished with a vernier at the top, the whole being ar- ranged so that the barometer may be mounted in gimbals on a tripod. When the barometer is to be transported, the screw S is raised till the cistern and tube T are entirely full of mer- 124 MECHANICS OF FLUIDS. cury. Entrance of air into the tube and breaking of the glass by bumping of the mercury against the end are thus prevented. To obtain the true hight of the barometer the observed reading must be corrected for temperature and capillarity. 91. Aneroid Barometer. The pressure of the air may also be indicated by the deformation it produces in a metal spring. An instrument designed for this purpose is called FIG. 81. an aneroid barometer. In the usual type (Fig. 81) it consists of a metal box, 5, exhausted of air, with a top corrugated in circular ridges to give it greater flexibility, but prevented from collapsing by a stiff spring, R, attached at M. The motion of the cover as it rises and falls, with varying pres- sure, is transmitted by means of a system of levers, Z, m, t, and a chain, the kinetic energy %mv 2 of the escaping mass must equal the diminution of the potential energy of the liquid in the vessel, or, (5) v* = 2gh. This equation, known as Torricelli's theorem, was an- nounced long before the recognition of the law of conserva- tion of energy which makes its demonstration so simple. This result must not, how- ever, be used to calculate the amount of discharge from the area of the orifice, because the size of the issuing jet is considerably modified by the sliape of the opening. As shown in Fig. 89, the converging of the lines of flow produces a contraction in the area of the issuing stream at J., the value of which can, in general, be determined only by experiment. 100. Reduction of Pressure at the Side of a Moving Stream. When a column of a fluid is in motion, the pressure at any point in its side is less than if the fluid were at rest. If p 132 MECHANICS OF FLUIDS. is the density of the fluid, and p the amount of this diminu- tion, the value of the latter may be calculated by the aid of equation 5 thus: (6) p = hgp = Wp. It is necessary to observe, however, that equation 6 is only a first approximation, because the influence of viscosity has been neglected. FIG. 90. This reduction of pressure may be readily demonstrated by the apparatus shown in Fig. 90. AB is a glass tube with a constriction at C bearing a short 7-tube in commu- nication with the interior, and filled with water to the level DE. When air is blown through AB, the level at E falls and that of D rises, showing a reduction of pressure at the point C. The same thing may be shown in another way by the apparatus illustrated in Fig. 91, which consists of a tube, 6r, terminated by a disc, A. Suspended . from this by wires, on which it may slide up and down, is a light vul- canite disc, F. When air is smartly blown through the pipe 6r, the pressure in the space between the discs is so reduced that F is raised by the air pressure on the under side. JET PUMP. 133 R 101. Jet Pump. The principle just presented has been utilized in the construction of various pumps, of which Fig. 92 may serve as the representation of a widely used type. If a stream of fluid be forced through the pipe M, a sufficient reduction of pres- sure may be secured at the side of the escaping jet JV to permit the inflow at R of the same or any other fluid. The forced draft on a locomotive is obtained by what is virtually a pump of this type. FIG. 92. 102. Sprengel Air Pump. A very efficient air pump, devised by Sprengel, is shown in Fig. 93. O is a reservoir of mer- cury supported on a movable shelf, A, and connected through /and S to a vertical glass tube, F, about 2.5 millimeters in diameter. Con- nection between F and the vessel to be exhausted at R is made through a bulb, E, containing sul- phuric acid designed to absorb any water vapor that may be present. B is a manometer to indicate the degree of exhaustion attained. When the mercury is allowed to flow through the tube F in a broken stream, there is a reduction of pres- sure at J", due in part to the velocity of the moving fluid, and in part to the fact that each drop of the mercury acting like a piston FIG. 93. 134 MECHANICS OF FLUIDS. carries a small portion of the air before it. The mercury escaping at m is collected in H, and the air is released at a. The slowness of operation of the Sprengel pump is more than offset by its automatic action and great efficiency. '* An exhaustion of -0.000007 mm. or ^^^-K^th of an atmosphere, has been obtained by the use of a pump of this type. 103. Geissler's Air Pump The Torricellian vacuum, i.e. the void left by the descent of the mer- cury in the barometer tube, has been util- ized by Geissler in the construction of an air pump whose essential features are illustrated in Fig. 94. C is a glass tube widening at A into a bulb containing about a liter, and stopped above by a two-way cock, F. Connection is made at the lower end with a cistern of mercury, B, by means of a flexible rubber tube, D. One of the passages, A, shown in the enlarged draw- ing of the cock, goes straight through the plug at right angles to the T- shaped handle I ; F k W the other, begin- FIG - 9*. ning at k, passes to the end &', where it communicates with a small reservoir, 6r. Connection with the vessel to be exhausted is made through R. The cock is first set so that A is in communication with 6r and the cistern B slowly raised till A is filled and a small amount of the mercury escapes into 6r. The handle of the cock is then turned through 45, closing all the passages. If TRAJECTORY OF A LIGHT REVOLVING BALL. 135 B is now gradually lowered, the mercury falls in (7, leaving a vacuum in A. By turning the cock through an additional 45, communication is opened to R and a portion of the air expands into A. If the cock be now turned to its first position and the cistern raised, the air will be expelled through 6r, completing one stroke of the pump. 104. Trajectory of a Light Revolving Ball. The modifi- cation produced by rotation, in the trajectory of a light ball, may be shown by the following experiment. Let a tennis ball, B (Fig. 95), be suspended by a fine wire from a horizontal pulley, P, so that it may be rotated about a verti- A cal axis. If the ball when not revolving be set vibrating in a plane, the direction of this plane will change, if at all, very slowly. If, while swinging, the ball be given in addition a rapid rotation, the plane of vibra- tion will also rotate in the same direction as the ball. Suppose the ball, rotating as shown, is moving from A to b C (Fig. 96). The air current past the ball is ' assisted by the rotation at a, but retarded at b. There will, in conse- quence, be a difference of pressure between a and b which will force the ball out of the path AC&nd toward the side a. The curving of a pitched baseball or a cut tennis ball is due to a similar reduction of pressure on one side of the rotating ball. 136 MECHANICS OF FLUIDS. 105. Support of a Ball on a Jet of Fluid. Let B (Fig. 97) be a ball supported at the side of a j_ fluid. It may be regarded as acted on by three p forces, namely: IF, the weight of the ball; P, an excess of air pressure on one side due to the velocity of the moving stream ; and /, the force of impact of the fluid against the ball. If these forces form a closed triangle, the ball will be in equilibrium. A position of the ball may, in general, be found for which the equilibrium will be stable. EXAMPLES. 1. What is the pressure at a depth of 76.3 cm. in a FIG 97 pool of mercury? Ans. 1.017(10) 6 dynes / cm. 2 . 2. A J7-tube is partly filled with water. How many inches of oil having a density of 0.79 gm. per cc. must be added in order to raise the water 4.5 in. in one leg above its first level ? Ans. 11.4 inches of oil. 3. What is the density of a body whose mass is 678 gms., if it weighs 235 gms. when immersed in a fluid whose density is 1.94 gms. per cc. ? p = 2.97 gms. / cc. 4. A wire 12.6 cm. long and containing 435 gms. weighs 299 gms. when immersed in water. What is the mean diameter of the wire ? d = 3.71 cm. 5. A body having a density of 2.35 gms. per cc. weighs 624 gms. when immersed in a liquid whose density is 0.827 gm. per cc. What is the mass of the body ? m = 963 gms. 6. A piece of wood containing 46.7 gms. is immersed in water by the aid of a sinker which weighs in water 75.8 gms. The combined weight of the wood and the sinker is 32.9 gms. What is the density of the wood ? P = 0.521 gm. / cc. EXAMPLES. 137 7. If the density of ice is 0.918 gm. per cc., and that of sea water is 1.03 gms. per cc., what is the volume of an iceberg exposing 697 cu. yds.? Ans. 6400 cu. yds. 8. An iron body weighing 275 gms. floats in mercury with 0.556 of its volume immersed. Required the volume and density of the body. p = 7.56 gms. / cc. ; v = 36.4 cc. 9. If the apparent mass of a body when weighed in a fluid of density p' is m', and when weighed in a fluid p" is ra", what is the mass of the body? 10. A mass of 28.1 gms. having a density of 5.59 gms. per cc. and a mass of 35.8 gms. weigh the same when immersed in water. What is the density of the second body? p =2.81 gms. / cc. 11. A sphere having a density 0.957 gm. per cc. and a volume of 168 cc. floats in a vessel of water. If a layer of oil having a density of 0.892 gm. per cc. is poured on the water so as to cover the sphere, how much of the latter will be immersed in the water ? Ans. 101 cc. 12. A cylinder of cork h cm. high, having a density />, floats on a liquid of density //. If the air above the liquid be removed, show that the cylinder will sink a distance "^ - ^ h where p" is the density of />'(/>' P") the air. 13. An ornament made of an alloy of gold and silver weighs 76.8 gms., and has a density of 18.0 gms. per cc. Assuming that the volume of the alloy is equal to the combined volumes of the components, find the amount of gold and of silver in the body. Ans. 70.2 gms. gold ; 6.6 gms. silver. 14. If a mass, m v having a density, p v weighs the same as a mass, m 2 , when both are immersed in a fluid of density p', prove that the density of the second body is Pz = - ^ -- m,o' . * -- \-m a m, Pi CHAPTER VII. SURFACE TENSION. 106. Phenomena at the Surface of a Liquid. The forces which are considered as effective in holding together the particles of a liquid or solid are termed molecular forces, since they are sensible only at insensible distances. When two fractured pieces of porcelain are placed together, even though the contact be so close that the break is imperceptible to the eye, there is no tendency for them to unite, because the parts are still separated by a distance greater than the molecular range. Two sur- FlG. 98. faces of glass, accurately plane and very clean, when pressed together will sometimes give evidence of a partial union. If, however, two freshly scraped surfaces of a more yielding substance, such as lead, be pressed together, a marked tendency to weld may be observed. In the case of liquids, these molecular forces give rise to a phenomenon at the surface which resembles in many respects the tension in a stretched membrane. Thus, suppose a (Fig. 98) represents a molecule within a body of fluid, and 5, 0, c?, e,f,g a series of molecules at equal dis- tances from a, but within the range of molecular forces. It is obvious that a would, on the whole, be attracted no more in one direction than another, and hence would be in equilibrium under these forces. If Z, on the other hand, be a particle which is nearer to a free surface than the distance of PHENOMENA AT THE SURFACE OF A LIQUID. 139 molecular range, then it is evident that the resultant of the attractions of the neighboring particles v w, w, o, p will urge it toward the interior of the body. Accordingly, since liquids have no shearing elasticity, the shape assumed by the body will be that for which the area is the least con- sistent with the given volume and the boundary conditions. The phenomena of surface tension in extended masses of liquid are usually greatly masked by the distorting effect of weight. This influence of weight on the surface of a liquid body may be eliminated by immersing the body in another liquid of its own density. Experiments in this mode are conveniently performed upon olive oil immersed in a mix- ture of alcohol and water. The density of this oil is about rm. 0.917 - -. Hence, as alcohol, whose density is 0.806 cc. cc. unites with water in all proportions, it is easy to prepare a solution in which the oil will float at any depth. If a small quantity of oil be introduced into such a solution by means of a pipette, it will assume a spherical form in accordance with the familiar fact that the sphere has a smaller area than any other surface enclosing an equal volume. Another method of avoiding the disturbance of weight is to use very thin films, such as are readily produced from a solution of soap and water. As the mass of liquid involved is quite small, its weight will exercise but little influence on the form of the film. Though the surface of a liquid behaves, in general, like a stretched elastic membrane, it nevertheless differs from such a membrane in two important particulars, namely, that liquid films, when unrestrained, contract indefinitely ; and, secondly, the tensile force is the same in all directions and independ- ent of the thickness, at least when the latter exceeds a certain very small value. 140 SURFACE TENSION. The former of these statements may be verified by the observation of a soap bubble which has been blown on a funnel, as in Fig. 99. If the small end of the funnel be left open, the sphere will slowly contract, expelling the air until the film has become a plane across the mouth of the funnel. The proc- ess does not stop here, however, but the contraction still goes on, this result being accomplished by the rise of the film in the FIG. 99. cone until it finally reaches the smallest cross section, provided the film is sufficiently stable. The second peculiarity may be exhibited by the following experiment. Let a thread be attached to one side of a wire loop (Fig. 100), and held while the wire is dipped into a soap solution so as to form a film upon it. If the lower portion of the film be now broken, the upper portion of the film, where bounded by the thread, will take an accurately circular form, AB, however it may be changed in position by pulling on the free end of the thread BC. Since the curvature of this arc is constant, it follows that the force is the same in all directions in the surface, and also, since the thickness is not constant, as is shown by the varying colors of the film, it follows that the surface tension is independent of the thickness. 107. Change in Pressure in Passing through a Liquid Sur- face. Let ABCD (Fig. 101) represent a portion of a plane liquid surface, and suppose that the force F must be applied FIG. 100. CHANGE IN PRESSURE. 141 across the line ab in order to keep the film stretched. The measure of the surface tension is then defined as the force per unit width of the film, i.e. = ab' Suppose the film now to be bent so as to form a portion of a cylin- drical surface of radius r, as in Fig. 102, and that the forces FF are resolved into compo- nents parallel and perpendicular to KO. Since each of the former components will be F sin 6, as is seen from the figure, the total normal force at the center K of the film will be (l) = 2F sin d = 2 After dividing both sides of this equation by AD, the breadth of the film, equation 1 may be written N 1 F T FIG. 102. If, now, the dimensions of the film be made smaller and smaller, it is evident that 2 GrJf- Z7) approaches the area of the film, and the left-hand member becomes the pressure just below K, due to the tension of the surface. Calling this pressure p, equation 2 becomes at the limit T (3) P = -> that is to say, in passing from the convex to the concave 142 SURFACE TENSION. side of a liquid cylindrical surface, the pressure must increase T by dynes per square centimeter, on account of the tension r in the film. Equation 3 may be extended to a surface of double curva- ture thus : Since in the case of a cylindrical surface the pres- sure due to the film depends on the tension, T, and the curvature, -, of the line AB (Fig. 102), and nothing else, it follows that if the surface be bent to a curvature, -7, in the r' line ab, which is at right angles to CD, the pressure which was due to T and - will not be altered except by an increase due entirely to the tension at right angles to AB, and the T curvature in the direction ab, i.e. by an amount, . Hence, in passing through a liquid surface of double curvature, the pressure changes by an amount, (4) p=. where r and / are the radii of curvature of two normal sec- tions at right angles to each other. It should be noted that in applying this equation to a soap film, since there are two liquid surfaces very near together, the value of T is to be taken as twice that due to a single surface. 108. Angle of Contact. If the molecular attractions of the particles of a solid for those of a liquid are greater than the attractions of the liquid molecules for each other, the liquid when brought into contact with the solid will adhere ANGLE OF CONTACT. 143 to it, in which case it is said to wet the solid. If, on the other hand, the mutual attractions of the molecules for each other exceed those exerted on them by the solid, the liquid does not wet the solid. In general, the surface of the liquid will meet that of the solid at a definite angle of contact depending on the nature of the substances actually concerned in the phenomenon. In the first-mentioned case this angle will be acute if it have any finite value, and in the second case, obtuse. The angle of contact may be measured in a variety of ways. In the method of Guy-Lussac the liquid is gradually introduced into a sphere from be- low until it reaches a level where the liquid surface is entirely plane at the edges. The required angle may then be calculated from the depth of the liquid and the dimen- sions of the sphere. . ,, . ,. , , FIG. 103. Another way is to dip a plate into the liquid, inclining it until the surface of the latter is horizontal at the point of contact, as in Fig. 103. B AC will then be the angle required. If a small amount of liquid be placed on a carefully cleaned surface (Fig. 104), the form of the drop will depend only on the surface tension, the accelera- 1( >*- tion of weight, the density of the liquid, and the angle of contact. This angle, <*, may accord- ingly be calculated from certain observed dimensions of the drop, such as the thickness, jfiT, and the vertical distance, &, between the vertex of the drop and the point on the meridian curve at which the tangent is vertical. In case the liquid wets the solid, a bubble of air may be used, as in Fig. 105. 144 SURFACE TENSION. The jangle of contact, as well as the surface tension, varies greatly with the cleanness of the surfaces supposed to be in contact. With water against FIG clean glass the angle is very small, but if the surfaces are in the least contaminated, the angle may reach, or even exceed, 90. The contact angle of mercury against glass varies from 129 to 143. 109. Capillary Phenomena. The first phenomenon of surface tension to be observed and studied was the rise of liquids in capillary tubes, so named from their fine, hair- like bore. The term capillar- ity is sometimes used in an extended sense to include all the phenomena of surface tension. When a tube is inserted in a liquid which wets it, the surface is observed to rise on FIG. 106. the outside slightly, and on the inside to a considerable hight above the hydrostatic level, as is shown in Fig. 106, provided the bore is small. The explanation is as follows: Since the liquid wets the tube, the angle of contact is acute, and the curved surface, or meniscus as it is called, must be convex downwards with an excess of pressure on the con- cave side equal to T ( -4--,}. Where the surface is hori- \r r'J zontal, as at Q and J>, the curvature is the same on each side, and the surface tension has no effect on the level. Where the surface, however, has a negative curvature, as at A 9 the CAPILLARY PHENOMENA. 145 F II FIG. 107. liquid rises to such a level that the pressure just below A is that at CD diminished by the pressure due to the hight of the column of fluid AE. When the liquid does not wet the tube, i.e. when the curvature of the surface is posi- tive, the opposite effect takes place, the surface B (Fig. 107) sinking to such a depth below the hydrostatic level that the pressure due to the liquid column is equal to that required by the curvature of the sur- face and its surface tension. The rise of liquid in a cylindrical tube may be calculated as follows : Let 3T denote the surface tension (Fig. 108), r " " radius of the tube, A " " mean hight of the liquid, p " " density of the liquid, a " " angle of contact; then the total upward force in the direc- tion of the axis is ZTTT T cos a, and the weight of the column of the liquid is FIG. 108. Since these forces are Jn equilibrium, (5) prg 7T If a > , h becomes negative, or the surface is depressed. The fact expressed by this equation, namely, that the eleva- tion is inversely as the diameter of the tube, was one of the first observed laws of capillarity. The calculation of the rise of a liquid between parallel plates is quite similar to that for tubes. If I be the length of the plates measured parallel to the 146 SURFACE TENSION. liquid surface, and d the distance between them, the total upward force will be 21T cos a, and the weight of the ele- vated column hldpy, whence (6) dpg When two glass plates are moistened with water and then brought together along a vertical edge, as in Fig. 109, the liquid appears bounded by a curve whose equation may be thus found : Let y denote the ordinate of any point in the curve, let x be the distance of the point from the common edge, and d the distance between the plates at the point xy. Then by geometry d r + Nodoid (4) 00 + + Cylinder (5) 00 00 Plane (6) - <>1 + Unduloid (7) r Catenoid (8) >M Nodoid Case 1. The condition that r shall be equal to r' deter- mines a sphere, the pressure in this case being greater within than without. This surface is readily formed by blowing a bubble and detaching it from the pipe by a sud- den motion of the hand. Case 2. The surface in this case is one form of the. figure known as the unduloid. Both curvatures are positive EXPERIMENTS ON SOAP FILMS. 149 FIG. 111. and the pressure greater within than without. It may be formed by securing a soap bubble upon two circular rings, and then separating them a certain amount, as in Fig. 111. The ends will be observed to be closed by spherical caps. Case 3. When both curvatures are positive, and the curvature of the meridian section the greater, the sur- face is a form of the nodoid for which the pressure is greater within than without. It may be formed by squeezing a soap bubble between two discs, Fig. 112. Case 4. When the curvature of the meridian section is reduced to zero, the surface is a cylinder. This surface may be formed by separating the rings an amount somewhat greater than for the ex- periment of Fig. Ill, provided the bubble is not too large. When the length of the cylinder approaches the diameter, the figure becomes un- stable. The pressure within is greater than that without, as is shown by the spherical caps. The radius of the caps is twice that of the cylinder. Case 6. The conditions here determine another form, or rather portion of the unduloid for which the meridian sec- tion has a negative curvature, and the perpendicular normal section a positive but greater curvature. It may be formed by separating the rings in Fig. Ill further than was re- quired for the cylinder. If, however, this separation (Fig. 113) be carried too far, the surface will divide into two bubbles by spontaneous constriction. The forms of the unduloid are of interest as showing how a vein of water FIG. 112. FIG. 113. 150 SURFACE TENSION. breaks up into drops, as is shown in Fig. 114. After the drops break away from the vein, they vibrate through a spherical form, becoming successively prolate and oblate spheroids. Case 7. When the curvature of the meridian sec- tion is negative, and the perpendicular normal section has a positive and numerically equal value, the sur- face is known as the catenoid. It may be formed by joining two rings wet with soap solution, breaking the films across the circles and separating the rings, o FIG. 115. FIG. lie. FIG. line as in Fig. 115. The pressure is then clearly the same on both sides of the surface. Case 8. The surface determined by these con- ditions has a form similar to that of Fig. 115, but J differs from it in having the negative portion of the i mean curvature numerically greater. As the pressure I within is less than without, this form is not con- | veniently produced in a soap film. The meridian curves of these six surfaces of revo- lution have the singular property of being in each case the roulette of the focus of a conic section, pro- duced by rolling the conic on a straight line. If, for example, an ellipse, EE 1 (Fig. 116), be rolled on the AS, the trace of the focus would be the undulating line which is the generatrix of the unduloid. When the EXPERIMENTS ON SOAP FILMS. 151 foci of the ellipse coincide, the conic becomes a circle, and the roulette a straight line, or the generatrix of the cylinder. When the eccentricity of the ellipse becomes unity by the coincidence of the foci with the extremity of the d major axis, the roulette becomes a semicircle, which is the generatrix of the sphere. A plane may be re- garded as a portion of a sphere of infinite radius. In the case of the pa- rabola the roulette is the catenary, or meridian Fm m curve of the catenoid. By rolling a hyperbola (Fig. 117) first on the branch HI and then on the branch KJ, the focus F will describe the FIG. 118. FIG. 119. curve abc - cde, which is the meridian section of the nodoid, the portion cde corresponding to case 3 and abc to case 8. 152 SURFACE TENSION. A great variety of figures may be formed by dipping wire frames into a soap solution. Figs. 118 and 119 show two such, in which the bounding surfaces are all planes. Whatever may be the form of the surface, if the pressure be the same on both sides, the curvature in one normal sec- tion will always be equal and opposite to that on the other. One familiar surface which has this property is the warped helicoid, which may be readily produced as a film on a wire bent into the form of a helix about a second wire as an axis. Whenever three liquid surfaces meet upon a line, the angle between them must be 120, since three equal forces in equilibrium when plotted as vectors form an equilateral triangle. 112. Limit of Molecular Range. In order to obtain an estimate of the distance at which the forces concerned in the phenomenon of surface tension become inoperative, Quincke made use of a glass plate covered with a wedge- shaped layer of silver. A glass rod was first laid on the plate, and the latter covered with a silver solution, from which the silver was deposited in a uniform layer except under the rod, where the thickness gradu- ally fell away. The metal was then re- moved from one-half of the plate, the surfaces carefully cleaned, and the ap- paratus partly immersed in water, as shown in Fig. 120. Since the angle of contact for glass is small, and that for silver is about 90, the water will rise on the glass surface above the level on the silver plate. But since the film is continuous, there will be near the edge of the silver a region, BC, where the line of contact curves upward to meet that on the glass. The thick- LIMIT OF MOLECULAR RANGE. 153 ness of the metal at the point B, where the molecules of the glass begin to make their influence felt through the layer of silver, will be the limit of molecular range. On converting the silver into a transparent salt, Quincke was able to esti- mate by optical methods that this distance was ^l^th of a millimeter, 5(10)~ 6 cm., or -g-ooVorth f an inch; that is to say, of a tenth of the magnitude of the wave-length of light. The colors exhibited by soap films furnish another method for estimating the range of these molecular forces. As the film gradually grows thinner, the color passes through the series of hues red, yellow, green, blue, and black, after which the film soon bursts. As this phenomenon is due to a direct relation between the thickness of the film and the wave-length of light, it furnishes a basis for the estimate that the minimum thickness which a soap film can attain is 1.2(10) 6 cm. Down to this degree of thinness there is no observed falling off in the contractile force. The principles of thermodynamics show that it is not pos- sible that this thickness can be reduced to one-hundredth of this amount without a considerable diminution of the surface tension ; for if this did not occur the energy which it would be necessary to supply to the film to reduce it to 10~ 8 cm. would be more than enough to convert the liquid into vapor at atmospheric pressure. Hence, since this diminution could occur only when the film was reduced to a layer of a few molecules and a thickness well within the molecular range, the conclusion may be drawn that the thickness of the black spot is of the same order of magni- tude as the range of the molecular forces. 113. Formation and Growth of Raindrops. The influ- ence of surface tension on the formation and growth of rain- drops was first pointed out by Lord Kelvin, who based his 154 SURFACE TENSION. -B A J FIG. 121. reasoning on the following experiment. Let C (Fig. 121) represent a vessel exhausted of air and partly filled with a liquid in which is placed a capillary tube, B. The space above the liquid will thus contain only satu- rated vapor. Now, when equilibrium has been estab- lished within <7, that is to say, when the evaporation and condensation at either of the surfaces A or B is the same, it is evident that the vapor pressure at the level A must exceed that at B by the weight of a column of the vapor of unit cross section and hight BA. But the curvature at B, being nega- tive, is less than at A, where it is zero; whence the conclusion may be drawn that when a vapor is in equilibrium with its liquid, the curvature of the surface must be greater where the pressure is greater. Since this pressure would have to be very great if the radius of curvature of a drop were very small, it is difficult to see how raindrops could begin to form, unless upon some nucleus, such as a speck of dust or a group of air particles. It is thus easy to see that a rainfall must clarify the atmosphere, by removing from it the small particles of foreign matter. Ac- cording t6 Kelvin's prin- FIG 122 ciple, the increase in size of raindrops will occur, not by the union of two or more drops which are jostled together, but by the growth of the large ones at the expense of the small ones. Thus, suppose that D and d (Fig. 122) are two drops of unequal size in a cloud. Then if the vapor pressure in the cloud is just right SURFACE TENSION AND SMALL FLOATING BODIES. 155 to preserve equilibrium at the surface of .Z), which has the smaller curvature, it will be too small for c?, which will evap- orate, increasing the pressure in the cloud and producing condensation at the surface of D. Similarly, if the pres- sure were initially right for d, it would be too large for D, so that condensation would occur, resulting in a diminution of the vapor pressure in the cloud and evaporation from d. Thus, it appears in either case that the large drops grow at the expense of the small ones. 114. Surface Tension and Small Floating Bodies. If a needle be carefully placed on the surface of still water, it may often be observed to float, although the density of steel is more than seven times that of water. The relation of the forces by which this result is effected may be seen in Fig. 123. Since the angle of contact between steel and water does not vary much from 90, as may be judged by the hemispheri- cal form which a small drop as- sumes when placed on a knife blade, the surface tension furnishes two forces whose vertical compo- nents, in the case under considera- tion, are equal to the weight of the needle diminished by the weight of the water displaced. The insects, often seen on the surface of pools, derive their support in a similar manner. The tendency often exhibited by small floating bodies on the surface of a liquid to cling to the sides of the vessel, or to collect in groups, is also a phenomenon of surface tension and may be thus explained. Suppose A and B (Fig. 124) are two plates suspended near together in a liquid which wets them, then it is obvious 156 SURFACE TENSION. that the curvature of the surface at m will be numerically greater than at I or n. But this condition, according to the law demonstrated in Art. Ill, requires a smaller pressure just below m than that above, whidh is sensibly the same as that '"" ' l FIG. 124. FIG. 125, at I or n. The difference of pressure between q and I or n thus gives rise to a resultant force which urges the bodies into contact. If the liquid does not w^et the plates, the curvature at s (Fig. 125) requires that the pressure at u shall be greater than at s, which is the same as at r or t. This condition being met by a depression of the surface at s, the bodies are urged together by the excess of the pres- sure at this level over the pressure at s. In the case where one of two plates is wet and the other not, it may be shown that they would be forced apart. Thus, if A (Fig. 126) is the plate which is wet, the liquid will rise higher at g than at A, and be depressed lower at Jc than at i. From the curvature of the surfaces it follows that the pressure to the left of gh FIG. 126. DIFFERENCE OF SURFACE TENSION. 157 is less than that to the right, i.e. atmospheric pressure, while for similar reasons the pressure to the left of ik is greater than that to the right of this region. Hence, the result of these unbalanced forces will be a separation of the plates. 115. Phenomena Depending on Difference of Surface Ten- sion. If a film of water lying on a plate be touched with a glass rod moistened with alcohol or ether, or if the rod be only approached very close to it, the water will be seen to draw back on all sides, leaving a spot entirely bare. This is due to the reduction of the surface tension where the water is mixed with the alcohol or ether. The phenomenon known as tears of strong wine, to which Proverbs xxiii: 31 is possibly a reference, has a similar explanation. When the sides of a glass containing some strong wine are moistened by shaking the liquid, the film which adheres to the glass will be seen to draw together into little ridges which slide down in fairly large drops. Just as they reach the surface, however, they may be observed to stop or even shrink back. The ridges were formed initially in those portions of the film from which the alcohol had evaporated more rapidly and the surface tension was consequently greater. On reaching the surface of the liquor the alcohol vapor reduced the surface tension suffi- ciently to permit the stronger tension at the upper side to draw the drop back. A fragment of camphor gum dropped on the surface of water is usually set gyrating in a singular manner under the action of the forces in the surface, which vary from point to point as the camphor dissolves. 116. Behavior of Oil on Water. When a drop of a mobile oil is placed on the surface of water, it will spread indefi- 158 SURFACE TENSION. nitely, or until the layer is so thin that the force of surface tension is reduced, provided the temperature of the water is sufficiently low. If, however, the water is heated, the oil may be observed to gather together in drops. The explana- tion is as follows. Let Fig. 127 represent a drop of oil on a T a water, and denote the tensions of the three surfaces by a T w , a T , and T W . Now, when the water is cold, the tension in the water-air surface exceeds the sum of the horizontal components of the ten- FIG - 127 ' 4.V 'I J -0. 'I sions in the oil-air and the oil- water surfaces ; but when the temperature is raised, although the tension in all the surfaces is diminished, the value of a T w falls off most rapidly, so that equilibrium is ultimately established between the forces at the edge, and the oil assumes a lenticular form. A curious effect is obtained by blowing on the surface of hot soup or chocolate on which there are a number of float- ing drops of oil. This is due to sudden changes in the relative surface tensions, consequent on the alteration of temperature. 117. Surface Tension as Related to Cleanness of Surfaces. The high value of the surface tension of mercury renders it exceedingly difficult to preserve a clean surface of this liquid, and the same thing is true for water, for which the surface tension, although smaller than for mercury, still considerably exceeds that of any other common liquid. In the case of water, the high value of the surface tension must be regarded rather as an advantage than a disadvantage, since this property renders it just so much more valuable as a cleanser. EXAMPLES. 159 In using benzine to remove grease spots the considera- tions already presented show that the application of the benzine to the center of the spot would naturally result in driving the grease into the clean parts of the cloth. If, however, the benzine be applied in a ring about the spot, the oil might be driven in toward the center, whence it could be absorbed by some porous substance. The diminution of the surface tension with the temperature may be utilized in the removal of grease spots by covering one side of the cloth with blotting paper or fuller's earth, and applying a hot flat- iron to the opposite side. EXAMPLES. 1. How high will water stand above hydrostatic level in a tube 0.057 cm. in diameter, assuming the angle of contact to be very small? Am. 5.8 cm. 2. How much will the level of mercury be depressed in a glass tube 0.067 cm. in diameter, the surface tension being taken as 540 dynes per cm. and the angle of contact as 135? Ans. 1.71 cm. 3. What is the pressure within a soap bubble 12.5 cm. in diameter if the tension of the liquid be taken as 80 dynes per cm., and what is the total pressure exerted by the film on the gas within ? p = 51 dynes / cm 2 ; total pressure = 25,100 dynes. 4. A horizontal disc of radius 14 cm. is held up by means of a film of water 0.21 cm. thick between it and a similar disc. Assuming that the surface of the water at the edge has a radius of curvature in a meridian plane equal to half the distance between the plates, find the weight of the lower disc and the water. Ans. 4.7 (10) 5 dynes. PART II. HEAT. CHAPTER VIII. THERMOMETRY. 118. Temperature. Two bodies are said to be at the same temperature if, on being brought in contact, neither body grows warmer. If when two bodies are brought in contact one grows warmer and the other colder, the body which becomes colder is said to be at a higher temperature than the other. This difference of temperature will be very nearly proportional to the rate at which this change goes on. To effect the change there must be a transfer from the hotter to the colder body of a definite and measurable phys- ical magnitude, called heat. No hypothesis as to its ulti- mate nature need be made for the present. 119. Scale of Temperature by Mixtures. Experiment shows that, for a given pressure, the change of a substance from one state to another takes place at a constant tempera- ture. Accordingly, in order to form a scale of temperature, the temperature of melting ice under a pressure of one atmosphere may be taken arbitrarily as zero, and the tem- perature of boiling water at the same pressure as 100. The temperature of the mixture formed by taking equal parts of boiling and freezing water might then be called 50, or, in general, the temperature of any mixture of different amounts THE TEMPERATURE SENSE. 161 of water in the standard states might be taken as numeri- cally equal to the percentage of hot water used to form it. The temperature of any body on this arbitrary scale could then be found with no other thermoscope than the tempera- ture sense. Thus, the temperature of a body which neither grew warmer nor cooler when immersed in a mixture of three parts boiling water and one part freezing water would have a temperature of 75. The scale formed by dividing the difference of tempera- ture between freezing and boiling water under the standard conditions into 100 equal steps or degrees is called the Cen- tigrade scale. The arbitrary scale of mixtures just described would coin- cide essentially with the Centigrade scale, but in practice the change of volume which accompanies the change in tem- perature of a body affords a much more convenient measure of difference of temperature. 120. The Temperature Sense. - - The temperature sense supplies quite a delicate test of equality of temperatures, provided that they are not widely different from that of the body, in portions of the same kind of matter, the range of error being about -f^ C. However, when comparison is made between different substances, the impressions received through it cannot be relied upon. Thus, if on a frosty day in winter one touch in succession several different objects, known to be of the same temperature, a piece of metal will feel very cold, a piece of wood much less so, while a woolen cloth may even seem warm. The sensation depends on the rate at which heat is lost, and this rate is a function not only of the difference of temperature between the substance and the hand, but also of the nature of the material. Since a metal conveys the heat away more rapidly than wood or wool, 162 THERMOMETRY. it feels colder. So, again, if wool conducts the heat of the body away less rapidly than the air of the room, the cloth will feel warm by contrast. If the right hand be held a few minutes in a cold bath and the left in a hot one, and both be then plunged into tepid water, the water will feel warm to the right hand but cold to the left, for the reason that the surface of the former is receiving heat, while that of the latter is losing it. 121. Expansion. It may be stated as a general law that the volume of bodies increases continuously with the FIG. 128. temperature. The few exceptions are confined to small ranges of temperature. The expansion of solids may be illustrated by the appa- ratus shown in Fig. 128. AB is a metal bar fixed at A and resting at B on a minute cylinder, which is free to roll on the plate Z>(7, and carries a pointer, P. If the temperature of the bar AB be raised by a flame applied at different points, the motion of the end B toward the right will be indicated by the revolution of the pointer. To show the expansion of liquids, let a tube of small bore with a bulb blown at the bottom, as in Fig. 129, be filled with any liquid to a point, a. If the temperature be gradu- ally raised by immersing the bulb in a vessel of hot water, the top of the column will be observed to rise from a to EXPANSION. 163 some point, b. The observed expansion in this case is really a differential effect, or the difference between the dilatation of the liquid and that of the containing vessel. The expansion of gases may be shown by invert- ing the tube filled with air in a beaker of inky water, as in Fig. 130. On warming the bulb the expanding gas will escape in bubbles from the bottom of the tube. If the source of heat be withdrawn, the air still remaining in the apparatus will contract and the inky water rise a considerable distance in the tube. Under proper precautions any one of these experi- ments might be made to yield numerical values for the change of temperature experienced by the body. The apparatus when so arranged would be called a thermometer. FIG. 129. 122. Choice of the Thermometric Substance. The choice of the thermometric substance should depend on its fulfillment of the following conditions: 1. The substance must return accurately to the same volume on being brought to its initial tempera- ture. 2. It must admit of a considerable change of temperature without change of state. A gas answers these requirements perfectly, and furnishes the most accurate measure of differences of temperature attainable, but its use is subject to the inconvenience of change of volume under change of pressure, as well as of temperature. Among the liquids, mercury fulfills the condi- tions most nearly. Its law of expansion over a wide range, say from 40 C. to 330 C-, about the temperature of melting lead, is sensibly the same as that FIG. 130. 164 THERM ONE TR Y. of air. The discrepancy between a mercurial and an air thermometer does not, from to 100, exceed 0.03 C. More- over, mercury is easily obtained, is very opaque, and hence easily seen, and does not cling to the glass. Liquids which wet glass, such as alcohol, ether, etc., are occasionally employed for moderate and for low temperatures where mercury would freeze. They offer the advantage of a much greater expansion for a given difference of temperature, and a smaller density permits the use of larger bulbs without increasing the liability to breakage, or disturbance of the reading through distortion of the bulb under the weight of the liquid. The use of volatile liquids necessitates keeping the bulb and stem at the same temperature, for otherwise vaporiza- tion into the upper portion might occur, and a false reading be v given by the difference in temperature between the bulb and the stem. The top of such a liquid thread also is not as easily read as a mercury column, on account of its transparency. 123. Air Thermometer. A convenient form of air ther- mometer devised by Jolly is represented in Fig. 131. A is a glass bulb filled with air and communicating with the tube BD through a small bore. EC and BD are two glass tubes connected by the flexible pipe FGr and filled with mercury. By raising or lowering EC the level in BD may always be brought to a fixed point scratched on the glass. The difference between the levels J^and B may be read from a graduated mirror placed behind the tubes. This difference FiO. 131. MERCURIAL THERMOMETER. 165 added to the barometric hight gives the pressure of the air in A. Call h the pressure in the bulb at the temperature C., and h the pressure at any other temperature, ; then, by the laws of gases explained below (Art. 138), or (2) The greatest source of uncertainty in the use of the air thermometer is introduced by the expansion of the glass, but this may be corrected by careful investiga- tion of the volume of the bulb throughout the range of temperature for which it is to be employed. 124. Mercurial Thermometer. The thermometer best suited to a large variety of practical purposes consists of a glass tube with a capillary bore and 100 | ending in a bulb filled with mercury. A form much used in the laboratory is shown in Fig. 132. Three processes are necessary to prepare it for use. 1. The filling: In order to introduce the mer- 50 cury, the bulb is heated so as to expel a portion of the air, and the open end of the stem is immersed in a vessel of the liquid. On cooling the bulb some of the mercury is forced in by the pressure of the outside air. The liquid thus introduced is boiled till the air is all expelled and the instrument filled only with mercury and its vapor. If the end of the stem be again dipped into the vessel of mercury, the FIG vapor will be condensed and the tube completely filled with the liquid. The whole instrument is then raised 166 THERMOMETEY. to the highest temperature it is intended to register, and the tube hermetically sealed. As the volume of glass is likely for a time to undergo sensible changes dependent on its previous history, the tube is laid away for a period of six months or more before it is marked. 2. Determination of the fixed points : To determine the freezing point, the bulb and stem, as far as the top of the column of mercury, are immersed in moist pounded ice until the level of the mercury becomes stationary. The position of this point is then carefully marked by a scratch on the glass. The boiling point is found by immersing the bulb and stem in the steam issuing from water boiling under a pres- sure of 760 mm. The stationary position of the top of the column is recorded as before by a scratch on the stem. 3. The graduation : The interval between the points of reference is now divided into 100 parts of equal volume, if the instrument is to be a Centigrade thermometer, and the graduation continued as far beyond the and 100 points as may be desired. If great accuracy is required, the errors of the graduations should be obtained by comparison with a standard instrument, or by some method of calibration. 125. Scales of Fahrenheit and Reaumur. The Centigrade scale, first constructed by Celsius in 1742, is most used for the record of scientific observations, but for the operations of daily life a different scale, devised by Fahrenheit of Dan- zig, is more familiar to the English-speaking people. On this scale the freezing point is marked 32 F. and the boil- ing point 212 F. The reason for Fahrenheit's division is not certainly known. It has been conjectured that the interval between the boiling and the freezing point was SCALES OF FAHRENHEIT AND REAUMUR. 167 divided into 180 parts from the analogous division of a semicircle into 180 degrees. But as Fahrenheit is known to have constructed many thermometers previous to the dis- covery that water had a constant boiling point for a given pressure, it seems far more probable that he chose as the upper fixed point the temperature of the human body taken in the mouth or under the armpit. The lower or zero point of his scale was the coldest then known temperature, that of a mixture of snow and salt. The interval between these points was first divided into 24 parts, and later into four times as many, i.e. 96 parts, a duodecimal division. - . The advantages of Fahrenheit's scale are the convenient size of the degrees and the rare need of nega- tive readings. Another scale common in Germany and bearing the name of Reaumur is marked zero at the freezing point and eighty at the boiling point. It has nothing to commend it. The relation between the readings of the three scales may be found thus : Let F, C, and R be the respective readings for any tem- perature. Then, since the distance between the boiling and freezing points on each scale is divided in the same ratio by the given temperature, I 212 I 100 32 80 Boiling Freezing (3) 180 _R 100~80' or, (4) 168 THERM OMETR T. 126. Maximum and Minimum Thermometer. A form of self-registering thermometer, devised by Six, is shown in Fig. 134. The bulb A is filled with glycerine. A portion of the stem BC, bent into the form of a /, contains mer- cury, the remainder being filled with alco- hol, except a space, D, designed to allow for the rise of the column B. Above the mercury in each arm is a small steel index which is held at any point where it may be left by the friction of a spring against the side of the tube, as shown in an enlarged view at K. When the liquid in the bulb A expands, the mercury rises in the left arm, pushing the index before it. But when the tem- perature falls, this index is left behind at the highest point attained. At the same time the index on the right is raised, and will in a similar manner record the lowest temperature reached. The indices may be replaced by means of a magnet. A self-registering minimum thermometer often used has a small index of glass (Fig. 134 A) placed in the column of a fluid, such as alcohol, which wets the glass. The stem of this thermometer being placed in an inclined position, so that the bulb is a little lower than the opposite end, the liquid will flow past the index with rising temperature, without disturbing its position; but when contraction occurs, so as to bring the meniscus into contact with the index, the tension of this surface carries the index back with it. A common form of maximum thermometer, which is also readily portable, is made like the ordinary mercurial ther- Jd 90" 85 15 80 -10 75 - 5 70- - C /7 - 5 60- -10 55- } > C 50- | 20 45- ;f 25 40- 30 35 35 30- -40 25- 45 20- 50 15- 55 10- GO o- >| / G5 P -70 c- 75 s I f 4 FIG. 134. FIG. 134 A. WEIGHT THERMOMETER. 169 mometer, except that the bore is very much reduced at a point near the bulb. When the temperature rises, the expansion of the mercury in the bulb forces a certain amount of the liquid past the constriction; but when the temperature falls again, the thread breaks off at this point, leaving all the column which has passed it in the tube. The instrument is set again by shaking the mercury down past the obstruction. 127. Weight Thermometer. The apparent expansion of a liquid may be obtained by weighing the overflow occa- sioned by a rise of temperature in a vessel just full at zero. Such an instrument, called a weight ther- mometer, is represented in Fig. 135. It consists of a bulb with a bent capillary stem open at the end. Let M be the mass of a liquid, say mercury, necessary to fill it at zero, and m the overflow when the temperature is raised from to t. Then, assuming the increase in volume, v, of a liquid to be proportional to the rise of tern- FlG - 135 - perature, and calling the initial volume V , (5) where a is some constant. But the volumes occupied by the masses at the temperature t are proportional to these masses; whence (6) which gives (7) V V M.-m = a (M m) , 170 THERMOMETRY. or, solving for , (8) t-~J!L where a may be determined once for all by an observation at 100 C., thus, (9) a = 128. Other Thermometers. The pressure-gauge of Bour- don, described in Art. 95, may be made to serve as a ther- mometer by filling the tube with glycerine or alcohol. Expansion or contraction of the liquid will then actuate the pointer in the same manner as would the variations in the pressure of a gas. A thermometer devised by Breguet consists of a ribbon made of three thin strips of platinum, gold, and silver, fas- tened together in the order named, and coiled into a helix so that the silver shall be on the inner face. The unequal expansion of these metals causes the helix to unwind, carry- ing a pointer over a properly graduated scale. Two methods, founded on the electrical properties of bodies, have been extensively used for estimating differences of temperature. One of these depends on the fact that if one junction in a conducting circuit, formed of dissimilar bodies, is heated or cooled, a current of electricity will be produced. The other method, of great value in the esti- mation of both high and low temperatures, is based on the variation of the electrical resistance of metals with change of temperature. Detailed description of these instruments may be more properly presented in the chapters on electricity. EXAMPLES. 171 EXAMPLES. 1. What is the temperature 92 Fahrenheit on the Centigrade scale ? Ans. 33.3 C. 2. What temperature is expressed by the same number on the Centi- grade and Fahrenheit scales? Ans. 40 C. 3. If the hight of the column of mercury which measures the pres- sure of the air in an air-thermometer at is 78.4 cm., what is the tem- perature when this hight is increased to 96.3 cm.? Ans. 62.3 C. 4. A weight thermometer weighs 104.5 gms. empty, and contains 623.5 gms. of mercury at C. At what temperature, assuming coefficient of expansion as 0.000154, will it weigh 717.6 gms.? Ans. 110 C. 5. If a weight thermometer which contains 324 gms. at zero loses 4.97 gms. by an increase of temperature of 11.9, what is the coefficient of relative expansion ? Ans. 0.00131. CHAPTER IX. EXPANSION. 129. Coefficient of Expansion. Let the measure of any dimension of a body at be called S , and its value at the temperature t under the same pressure be $, then the results of experiment may be expressed with sufficient accuracy by the linear equation (!) $ = $0 (1 + $*) where is a constant depending only on the nature of the material, and is known as the coefficient of expansion. Solving for , s-s. t increase per degree (2) 9 o ' 1 ' S size at zero or the coefficient of expansion may be defined as the ratio of the increase per degree to the size at zero. 130. Expansion of Solids. In treating of the expansion of solid bodies, three coefficients are commonly considered, according as S is regarded as a length, an area, or a volume. The equation 1 becomes in each case (3) L = L (l+\t), (4) A = A (l+', adjustable by means of a micrometer screw, at the other end. The bar A is fur- nished with a ring and cross wire at each extremity, as is shown at a, a'. The lenses on B and C are so designed that they form images of the cross wires at a and a'. In order to make an observation, the image of the cross wire at a is brought into coincidence with the wire at c and the end b fixed; then after &', b f , c' haVe been brought to a similar coincidence, with the temperature at zero, the bath about B is raised to some convenient temperature, t. A new coincidence of the cross wires is then found by shifting the lens b f a determinate distance, which is equal to the expansion of the bar L L . The coefficient of linear expression may then be calculated fr m L L COEFFICIENTS OF LINEAR EXPANSION FOR SOLIDS. SUBSTANCE. MEAN EXPANSION PER DEGKEE C. Brass 0.0000178 0.0000112 0.0000168 0.0000083 0.0000140 0.0000116 0.0000280 0.0000068 0.0000191 0.0000103 0.0000223 0.0000220 to to it a a a ti (I it it 0.0000193 0.0000188 - 0.0000120 0.0000147 0.0000119 0.0000292 0.0000092 0.0000212 0.0000132 0.0000229 0.0000298 Cast Iron . . . Copper .... Glass Gold Iron Lead . . . . Platinum . . . Silver Steel Tin Zinc . EXPANSION OF LIQUIDS. 175 132. Expansion of Liquids. In the case of fluids, the cubical coefficient is alone important ; but it is customary to distinguish two values of this coefficient, the absolute, defined by equation 5, and the apparent, or that which is observed when the fluid is contained in a vessel which serves as a measuring flask. The absolute expansion of a liquid may be obtained by the method of equilibrating columns adopted by Dulong and Petit in a very careful determination * An of this constant for mercury. The appa- ratus consisted of two tubes, A, (7, con- nected by a portion of much smaller bore, and bent as EBDF, in Fig. 137. The arm AB was surrounded with broken ice, and QD with a bath of any desired temperature. Let p , p be the densities of the mercury in the respect- 3 pjo ive branches, and h , h the corresponding hights of the surfaces a, b above the axis of the tube BD. Then, since the pressure is the same at B and D, (10) h oPo g = hpff, also for one and the same mass of the liquid, (11) V p =V P . Combining (10) and (11) and substituting in (5), (12) h = h (l + at), from which a may be found, since t is known. The results of Dulong and Petit showed that the coeffi- cient of expansion of mercury was sensibly constant between and 100, with a value i-fav or 1-8018 (10)- 4 per degree. 176 EXPANSION. However, Regnault, using a similar method, found that this coefficient increased slightly with the temperature, but that the mean value from to 100 was 1.8153 (10)~ 4 per degree. To calculate the volume at any temperature, he gave the formula y = y o (1 + 0.000181792 t + 0.000000000175 2 ). The expansion of other liquids cannot be satisfactorily expressed as a linear function of the temperature. In gen- eral, the coefficient increases with the temperature, becom- ing very large at high temperatures. The following table exhibits the deportment of some of the more familiar liquids. EXPANSION OF CERTAIN LIQUIDS. VOLUMES. TEMP. ALCOHOL. ETHER. BISULPHIDE OF CARBON. OIL OF TURPENTINE. 1. 1. 1. 1. 10 1.01050 1.01518 1.01156 1.00919 20 1.02128 1.03122 1.02350 1.01875 30 1.03242 1.04829 1.03594 1.02865 40 1.04404 1.06654 1.04901 1.03886 133. Apparent Expansion. Suppose a vessel graduated so that the reading of the scale divisions is true at zero. Let V be the volume of any fluid at zero, and V a the apparent volume at the temperature , that is, the volume measured by this vessel. These volumes are related by the equation (13) F a =F (l + o), where 1 -f at is the factor of apparent expansion. If Fis the real volume, and a the true coefficient of expansion, (14) F=F. (! + *) EXPANSION OF WATER. 177 Considering only that portion of the vessel occupied by the liquid at the temperature , it appears that this portion would have a capacity of V a cubic centimeters if the glass were cooled to zero, for by supposition the graduation is correct for this temperature. Therefore, the volume of this portion of the vessel at the temperature t will be (15) V= V a (I +fft), where g is the coefficient of expansion of the glass. Eliminating the volumes between the three equations, (16) l + at = or, approximately, (17) a = since the coefficients are small. 134. Expansion of Water. - - Water for a short range presents a remarkable exception to the law of increase of volume with rise of temperature. From to about 4 it contracts, and afterwards expands in a manner exhibited by the table on the following page. 135. Maximum Density of Water. - - The temperature of the maximum density may be roughly shown by the following experiment due to Hope. A tall beaker (Fig. 138), furnished at -the top and the bottom with a thermometer, is filled with water and surrounded at the middle with a freezing mixture of ice and salt. At first the temperature of the lower thermometer will be observed to fall grad- ually, while the upper one is little affected, thus showing that the water descends as ragjdly as it is cooled. 178 EXPANSION. TEMP. CENT. VOLUME. TEMP. CENT. VOLUME. TEMP. CENT. VOLUME. 1.000000 15 1.000735 30 1.00418 1 0.999948 16 890 35 1.0057 2 911 17 1.001057 40 76 3 889 18 235 45 97 4 883 19 424 50 1.0118 5 891 20 624 55 142 6 914 21 835 60 168 7 952 22 1.002057 65 195 8 1.000003 23 289 70 1.0225 9 068 24 530 75 256 10 147 25 78 80 288 11 239 26 1.00304 85 1.0321 12 344 27 31 90 356 13 462 28 59 95 392 14 593 29 88 100 1.0430 When the temperature of about 4 has been reached, the upper thermometer will begin to fall and continue to do so until it reaches zero. The lower one, however, remains con- stant at 4, for the reason that the densest layer stays at the bottom, while the colder but lighter layers rise to the top. A far more accurate method for determining the temperature of maximum density of water is that devised by Joule, in which he made use of two vertical cylinders about 15 cm. in diam- eter and 140 cm. high (Fig. 139), connected at the top by a channel, a, and at the bottom by a wide tube, 5, with a stopcock. If the system be filled with water and the stopcock opened, the smallest difference in density between the two columns will produce a current in the channel, the pres- ence of which may be indicated by a floating glass bead. EXPANSION OF GASES. 179 Joule proceeded to find a pair of temperatures, one above and one below the point of maximum density for which the density of water was exactly the same. By obtaining a series of such pairs of temperatures, for which the difference was smaller and smaller, the temperature of maximum density was found to be 3.95 C. within a small fraction of a degree. One of the most recent determinations of the density of water is shown in the following table. TABLE OF DENSITY OF WATER. TEMP. C. DENSITY. TEMP. C. DENSITY. TEMP. C. DENSITY. 0.999884 13 0.999443 35 0.99419 1 0.999941 14 0.999312 40 0.99236 2 0.999982 15 0.999173 45 0.99038 3 1.000004 16 0.999015 50 0.98821 4 1.000013 17 0.998854 55 0.98583 5 1.000003 18 0.998667 60 0.98339 6 0.999983 19 0.998473 65 0.98075 7 0.999946 20 0.998272 70 0.97795 8 0.999899 22 0.997839 75 0.97499 9 0.999837 24 0.997380 80 0.97195 10 0.999760 26 0.996879 85 0.96880 11 0.999668 28 0.996344 90 0.96557 12 0.999562 30 0.995778 100 0.95866 136. Expansion of Gases. Experiment has shown that in the case of a gas under constant pressure, not only is the expansion strictly proportional to the increase of temper- ature, but that all gases have sensibly the same coefficient, namely, = = 0-00367. 180 EXPANSION. This law, which may be expressed, v = v ( 1 -f - t\ for p constant, was discovered by Charles, and usually bears his name, though also termed by some writers the law of Gay-Lussac. 137. Boyle's Law. The law by which a gas was denned in Art. 69, namely, that the product of the volume by the pressure is constant as long as the temperature remains unchanged, was first established by Robert Boyle, and published by him in 1662 in his "Defence of the Doctrine touching the Spring and Weight of Air." In his investigations, Boyle made use of a ZZ-shaped tube of uniform bore, closed at the short end, as shown in Fig. 140. Mercury was first poured in till it stood at the same level in both legs, when the amount of air enclosed in the short branch was carefully noted. Small por- tions of mercury were then successively added, and the level of the columns recorded each time. The difference of these levels gave the excess of pressure of the enclosed air above that of the outside atmosphere. The volume could be read at once from the short leg. If the initial pressure be called p t and the cor- responding volume v t , and any other pressure and volume at FIG. 140. the same temperature then the relation (18) p t v t = p'v' = &, a constant, is found to be true within the limit of errors of observation. GENERAL LAW FOR A GAS. 181 Equation 18 is known as Boyle's Law among all English- speaking peoples, and most others, except the French, who style it Mariotte's Law. f 138. General Law for a Gas. The laws of Boyle and Charles may be combined so as to yield an equation which is not subject to the condi- p tion of constant tempera- ture or constant pressure. Thus, call the volume and pressure of a gas at zero respectively v and p . To find what the volume v will be if the temperature is raised to t and the pres- t=o sure changed to p : Sup- pose, first, that the vol- ume v is changed at the temperature zero until its pressure becomes p. Call this new volume v*. Then, since the temperature is constant, (19) p o v o =pv r . Now let the temperature be raised to t, while the pressure remains constant. The final volume v is given by FIG. 141. (80) V = V Substituting the value of vj from (19) in (20), - 1 which is the equation required. Clearing of fractions, (22) ^ = ^ v ( 1 + 273*)' 182 EXPANSION. which may be used to find any unknown quantity when the other four are given. Thus, it is frequently desired to know the volume at zero under a pressure of 760 mm. of mercury when the volume of a gas at a temperature, , and a pressure, A, is v. Observing that the pressures are pro- portional to the hights of the column of fluids which meas- ure them, (23) . ^-TG^rrr. 273 ' Equation 22 may be simplified by writing (24) T when (25) pv where is some constant. It is, however, obvious that if the volume and temper- ature of the gas remain constant the addition of more of the substance must increase the pressure, or must contain, implicitly, the mass w, say, C=mR, so that, finally, (26) pv = mRr, R being a constant which may be found from one observa- tion. Thus, substituting the following values determined by Regnault for air, v = ~L cc., = 1.293(10)- 3 gms., T = 273, p = hpg, where A 76 cm., ABSOLUTE ZERO. 183 = 13.60 ^^, cc. sec/ E is found to be 139. Absolute Zero. Assuming that the law, y 273 is true for all temperatures, (27) pv = 0, for = 273 C.; that is to say, either the volume or the pressure would vanish at this temperature. The con- dition of no pressure may be explained on the kinetic theory of gases by supposing the molecules reduced to rest, that is, entirely deprived of heat. The alterna- tive condition, namely, that the matter would occupy no volume, is hardly admissible. The substitution of r = + 273 in equation 22 is obviously equivalent to choosing a new zero, 273 C., below the freezing point of water. This point is called absolute zero, and the readings T on this scale, absolute temperatures. 140. Compensated Pendulums. The error in the running of a clock due to expansion or contraction of the pendulum rod may be satisfactorily compensated by the use of a mercurial bob proposed by Graham, and illustrated in Fig. 142. FIG 142 To the extremity of the rod is attached one or more jars of glass or steel containing a considerable mass of mer- 184 EXPANSION. cury. If a rise of temperature lengthen the rod, lowering the 1 center of gravity, the expansion of the mer- cury, necessarily upwards, will produce the contrary effect. By the choice of a proper amount of mercury the effective length of the pendulum may be made to remain practi- cally constant. An equivalent design proposed by Harri- son is shown in Fig. 143. From its form it is commonly known as the gridiron pen- dulum. The bars marked 8 are of steel. Their expansion will clearly lower the bob. Those marked B are of brass, and so arranged that increase of their length will raise the bob. Let L be 'the combined length of the steel rods, and L' that of the brass ones, and call X, V the corresponding coeffi- cients of expansion. Then, in order that the length of the pendulum shall be invariable, L\t L'\'t = Q. Whence FIG. 143, that is, the length of each material must be inversely as its coefficient of expansion. 141. Compensated Balance Wheel. - Fig. 144 shows the method adopted to compensate temperature changes in a chronometer. The rim of the balance wheel is constructed of two metals, the most expansible being placed on the out- side and divided by saw cuts at a and a'. FIG. 144. The effect of EXAMPLES. 185 expansion on the rim is to throw the free end of each seg- ment nearer the center, and thus correct the temperature changes in the elasticity of the hair-spring by the change in the moment of inertia of the balance wheel. EXAMPLES. 1. A steel chain is 66 feet long at 25 C. What will be its length at 0? Ans. 65 ft. 11.78 in. 2. A meter scale at 10 measures 99.981 cm. At 40 its length is 100.015 cm. What is the coefficient of expansion, and at what temper- ature is the scale correct ? X = 0.0000113 ; t = 26.8. 3. What is the length of a brass wire which increases 1.231 cm. when heated through 200 C. ? Ans. 342 cm. 4. One brass yard-measure is correct at and another at 20. What is the difference of length at the same temperature ? Ans. 0.013 in. 5. If the iron bars in a compensated pendulum are 87.5 cm. long, what should be the length of the zinc bars? Ans. 39.9 cm. 6. A platinum and a zinc wire measure respectively 251.01 cm. and 249.97 cm. at zero. At what temperature will they have the same length, and what will this length be ? t = 243; L = 251.55 cm. 7. A cast-iron ball, 5.01 cm. in diameter at zero, rests upon a copper ring 5 cm. in diameter. To what temperature must both be raised in order that the ball shall just pass through the ring? Ans. 317 C. 8. What is the area of a brass plate at 80 which measures 20.32 cm. by 15.14 cm. at 0? Ans. 308.5 sq. cm. 9. A piece of copper has a volume of 259.3 cc. at 0. What will be its volume at 101? Ans. 260.67 cc. 10. A glass flask contains 687 gms. of mercury at 70 C. What is its volume? Ans. 51.2 cc. 11. If the pressure of a gas is 8726 dynes per sq. cm. when its volume is 7375 cc., what will be its pressure at the same temperature if the volume is diminished to 1586 cc. ? Ans. 40580 dynes /cm. 2 . 186 EXPANSION. 12. What will be the volume of a gas at 382, if the volume at and the same pressure is 6580 cc. ? Ans. 15780 cc. 13. If the pressure be unaltered, what will be the volume of a gas at 572 C. which occupies 2579 cc. at 198? Ans. 4627 cc. 14. If the pressure of a gas in a vessel, whose expansion may be neglected, is 58.3 cm. of mercury at 98, what will be its pressure at a temperature of 373? Ans. 101.5 cm. Hg. 15. If the volume of a gas at zero is 2560 cc. under a pressure of 2.14 million dynes per sq. cm., what will be its volume at 95 under a pressure of 1.013 million dynes per sq. cm.? Ans. 7291 cc. 16. If 4490 cc. of a gas at 101, under a pressure of 75.4 cm. of mer- cury, have its pressure increased to 82.1 cm. and its temperature raised to 225, what will be the new volume? Ans. 5490 cc. 17. What will be the mass of a cubic meter of air at 50 under a pressure of 50 cm. of mercury ? Ans. 719 gms. 18. If the volume of a gas is 22.1 cc. under a pressure of 70.8 cm. Hg. at a temperature of 16.5, at what temperature will the volume be 19.4 cc. under a pressure of 76.0 cm. Hg.? Ans. C. 19. Find the coefficient of expansion of a gas for the Fahrenheit scale and zero. Ans. T |^ per deg. F. 20. A tube 6.0 feet long, closed at one end and containing air, is half filled with mercury. If the open end is immersed in a vessel of mer- cury, how high will the mercury stand when the barometer reads 30 inches? Ans. 12 in. 21. A cylindrical diving bell 7.1 ft. high is lowered until the top is 20 ft. below the surface of the water. How far will the water rise within? Ans. 2.9 ft. 22. A glass tube used for sounding is 15.0 inches long and open at the lower end. The inside is covered with a soluble pigment, and the tube lowered to the bottom in sea water whose density is 1.03 gms. per cc. On raising to the surface it is found that the water had entered the tube a distance of 9.3 in. What was the depth of the water? Ans. 53.9ft. CHAPTER X. CALORIMETRY. 142. Effects of Heat. The effect produced by the appli- cation of a definite quantity of heat to a body is found to depend, in general, upon the nature and quantity of matter in the body, and its condition at the time the experiment is made. For example, let equal masses of alcohol, mercury, and ice be successively exposed for the same period of time to a Bunsen flame, which may for the present purpose be regarded as a constant source of heat. The resulting changes in temperature, volume, pressure, and state will be found to be markedly different in each body. 143. Unit of Heat. Any of the effects resulting from the addition of heat to a given body, in a definite condition, might be used as the measure of a quantity of heat. The heat unit generally adopted is that quantity of heat which, added to 1 gm. of water at C., will raise its temperature to 1 C. This unit is known as the calorie. A unit one thousand times as great, and known as the large calorie, is often used in the investigations of Engi- neering. 144. Thermal Capacity. When equal masses of unlike substances, at different _ temperatures, are kept in contact until they have assumed a common temperature, it is found, in general, that this resultant temperature is different from the average of the two, from which the inference may be drawn that different bodies require different amounts of heat to raise like masses through the same difference of tern- 188 CALORIMETRY. perature. This conclusion really rests on four assumptions, which may be stated as follows: 1. All the heat gained by one body is lost by the other, and vice versa. 2. No action takes place between the bodies other than giving and receiv- ing heat. 3. If a body is made to pass through a series of states by the addition of heat, and is then allowed to cool, so as to pass through these same states in the reverse order, the quantity of heat which entered during the heating process is equal to that which left it during the cooling process. 4. The effect of a given quantity of heat does not depend on the temperature of the source. Assumptions 1 and 2, it will be observed, relate to certain conditions of the experi- ment which may be fulfilled with sufficient approximation. Numbers 3 and 4 are principles, by no means self-evident, whose truth has been established by experience. The thermal capacity of a body is defined as the quotient of the heat received by the rise of temperature produced; that is, if Q units of heat raise the temperature of a body from t to ', thermal capacity = 7-^ 145. Specific Heat. The specific heat of a substance may be defined as the thermal capacity per unit mass, that is, the heat per unit mass per unit rise of temperature. If Q units of heat raise m units of mass from t to t r degrees, then the specific heat is given by Q ' (I) 8 = m (t' t) The specific heat of substances varies a little with the temperature, but for most purposes it may be assumed as constant. SPECIFIC HEAT. 189 146. Specific Heat by Method of Mixtures. Suppose that a mass, m v of a substance having a specific heat, s v and at a temperature, t v is mixed with a mass, m^ whose specific heat is $ 2 , and whose temperature, 2 , is below t v If after an interchange of heat they assume a common temperature, , then the heat lost by the first body will be (2) Q 1 = m 1 8 1 ft *), and that gained by the second, (3) , fastened to the rods r, r, passing through the cover. A thermometer, t, with the bulb inserted in the basket, serves to indicate the temper- ature as the heating progresses. , The calorimeter proper, (7, is a cylindrical vessel made of very thin brass, contained in a larger and stronger vessel, E, but thermally insulated from it by non-conducting supports. The inner vessel contains a measured quantity of water at a known temper- 190 CALOEIMETEY. ature. To diminish loss by radiation the adjacent surfaces of both vessels are carefully polished. The calorimeter is mounted on rails so that it may be approached to J., have the basket dropped in, and then be quickly withdrawn. A wooden or cork partition, PQ, screens the calorimeter from the influence of the heating apparatus. After immersion of the heated substance in (7, the water is continually stirred till the thermometer T indicates a stationary value of the temperature, which is then noted. The specific heat of the substance may now be calculated by the method of the pre- ceding article. FIG. 145. If, however, considerable precision is desired, correction must be made for the thermal capacity of the calorimeter, and for the gain or loss of heat by radiation. The former may be determined by a separate experiment. The latter may be compensated by the following procedure suggested by Count Rumford. Let the highest temperature t reached by the calorimeter be found approximately from a preliminary experiment, and the temperature 2 f the water in the calorimeter be chosen LATENT HEAT. 191 so that it is as much below the temperature of the room as t is above it. Then while the temperature of the water is rising to that of the room the calorimeter will receive heat, but during the remainder of the process it will lose heat, since it is hotter than the temperature of the room. 148. Latent Heat. When heat is continuously applied to a solid body, its temperature rises continuously, but only to a certain point, where it begins to melt. If the mixture of solid and liquid be well stirred, further application of heat will melt more of the solid without affecting the temperature, until all of the substance has passed into the liquid state. The constant temperature at which fusion takes place is characteristic of each substance, and is called the melting point. The amount of heat required to convert a gram of a solid at the melting point into a liquid at the same temperature is called the latent heat of fusion. Quite analogous phenomena occur when a liquid under a definite pressure passes into the aeriform condition. The quantity of heat required to convert a gram of a liquid into a vapor without rise of temperature or change of pressure is called the latent heat of vaporization. 149. Ice Calorimeter. The ice calorimeter is an instru- ment for measuring quantities of heat by means of the amount of ice melted, devised by Black, the discoverer of the phenomenon of latent heat. It consists of a block of pure ice, free from bubbles, in which a cavity is hollowed out and closed by a slab of ice. To determine the specific heat of a substance a known mass of the body is raised to a convenient temperature and introduced into the cavity, which has previously been wiped 192 CALORIMETRY. dry. The temperature of the body will quickly fall to zero, and some of the ice will be melted. This water is carefully collected by a cold sponge or a piece of blotting paper, and weighed. Suppose t was the temperature of the substance when introduced, and m its mass. Let s be the specific heat, L the latent heat, i.e. the amount of heat necessary to melt a gram of ice at zero, and m' the mass of ice melted, then, mst = Lm' t or, 8 = - mt (5) The value of L could be found by introducing a mass of boiling water into the chamber, or by the method of mix- T tures thus : It has been found that if 150 gms. of water at 100 be mixed with 100 gms. of ice at the ice will be melted and that the temperature assumed by the mixture will be about 28.3 C. As the hot water lost 10,755 cal., and the water of the melted ice gained 2830 cal., the difference, 7925 cal., must have been required to melt 100 gms. of ice. Hence, FIG. 146. L = 79.25 cal. per gram. 150. Bunsen's Ice Calorimeter. A valuable form of ice calorimeter has been devised by Bunsen, in which the amount of ice melted is determined from the accompanying change of volume. The 'apparatus consists of a bent glass tube, CD (Fig. 146), furnished with a large bulb, B, into which is BUNSEN'S ICE CALORIMETER. 193 sealed a test-tube, A. The other end of the stem is termi- nated by an iron collar, E, fitted with a stopper, $, in which is inserted a graduated tube, jP, of fine bore. The collar E, the tube CD, and a portion of the bulb B are filled with mercury. The remainder of the bulb contains water care- fully freed from air. To prepare the apparatus for experiment a current of alcohol at a temperature below zero is passed through the test-tube A until the greater part of the water in B is frozen. Since water in freezing expands nearly a tenth, some of the mercury will be driven out into the stem 8 during the process. By adjusting the stopper in the mouth of the tube .Z), the extremity of the mercury column may be brought to any convenient point in the graduated stem. Some pure water is now poured into the test-tube and the apparatus packed in freshly fallen snow, till the temperature becomes zero throughout. In order to determine the thermal equivalent of a scale division, a gram of water at 100 C. is dropped into the tube. As it gives up its heat, some of the ice will be melted and the end of the mercury thread will retract. Suppose that the reading on the scale before the hot water was put in is a, and that after the temperature has again fallen to zero it is b ; a scale division will, then, correspond to 100 b-a calories. If, now, a gram of the substance to be tested, at a temper- ature of 100 C., be dropped into the tube, more ice will be melted, and the mercury thread further retracted to some point, c. The total heat abstracted from the body will be c-b 100 calories, o a 194 CALORIMETRY. and the specific heat c-b \ This instrument is particularly valuable for measuring the specific heats of substances which are obtainable only in small quantities. 151. Steam Calorimeter. A form of calorimeter depend- ing on the condensation of steam has been recently intro- duced by Dr. Joly, and is found to give more accurate results than any method yet employed. The apparatus consists, essentially, of a pan, P (Fig. 147), suspended ^ from the arm of a balance, and sur- rounded with a chamber, (7, to which steam may be admitted and allowed to escape at pleasure. In making an experiment a known mass of any substance is placed in the pan P, balanced, and allowed to attain the temperature of the chamber, which is read by means of a thermometer passing through its side. Steam is then suddenly admitted through the pipe E, so as to fill the chamber at once. As it condenses on the sub- stance the water is collected in the pan P, and weights are added to the opposite pan of the balance to restore the equi- librium. A slow circulation of steam, not enough to dis- turb the equilibrium of the balance, is kept up till the substance has attained the temperature of the steam, when the condensation, of course, ceases. Call T the temperature of the steam, L the latent heat of vaporization of water, and t the temperature of the chamber FIG. 147. STEAM CALORIMETER. 195 when the steam was first admitted. Also let m be the mass of the substance, s its specific heat, M the amount of steam condensed, and k the thermal capacity of the pan P. Then the quantity of heat abstracted from the steam during con- densation was ML, also the heat added to the substance was ms (Tt), and the heat gained by the pan k (Tt], whence (6) m8(Tt) + k(Tt) = ML. The value of k is readily found from a separate obser- vation. To secure the greatest possible accuracy a small correction must be applied for the buoyancy of steam and of air. The great merit of the steam calorimeter is that it may be used for bodies of any size and in any state, provided they are enclosed by a proper envelope of known thermal capacity. 152. Conditions which Affect the Specific Heat. The specific heat of all bodies is more or less dependent on the temperature at which the body is examined. The law of variation is not known, but the results of experiment, in the vast majority of cases, may be expressed by a formula of the form s = a -f- bt -\- c& -f- etc., where , #, c, etc., are constants. The increase of specific heat with rising temperature, in most solids, is small until the melting point is approached. Carbon is, however, a notable exception. Changes in density of solids produced by hammering, and the passage from one state to another, also largely influence the value of the spe- cific heat. The allotropic varieties of calcium carbonate and carbon differ considerably in their specific heats. 196 CALORIMETRY. SPECIFIC HEATS OF SOLIDS. Aluminum 0.2122 Antimony Bismuth 0.0507 0.0305 Brass 0.0939 Copper Glass 0.0950 0.198 Gold 0.0324 Graphite Ice 0.310 0.504 Iron 0.1124 Lead 0.0315 Magnesium 0.245 Marble 0.216 Mercury Nickel 0.03192 0.1092 Phosphorus Platinum 0.1699 0.0324 Quartz 0.19 Silver 0.0599 Steel 0.118 Sulphur Tin 0.1844 0.0559 Zinc 0.0935 SPECIFIC HEATS OF LIQUIDS. SUBSTANCE. SPECIFIC HEAT. TEMP. C. 0.5475 o Alcohol 0.6479 40 Ether 0.5290 o Ether 0.5468 30 Mercury 00333 30 Turpentine ..... 4537 40 153. Specific Heat of Water. An exact knowledge of the specific heat of water is of fundamental importance in the study of calorimetry, because all investigators have used as their unit of heat the thermal capacity of some mass of water, though at various temperatures. The fact that the specific heat of water varied with the temperature was ascer- tained in the early part of the century, but the exact nature of the variations was first established by the work of Row- land in 1879. The experiments of this investigator were of great accu- racy, and showed that the specific heat diminished by about SPECIFIC HEAT OF GASES. 197 one per cent from to 29, where its value was a mini- mum, and then increased again to the boiling point. 154. Specific Heats of Gases. The changes of volume which a solid or a liquid undergoes in a moderate rise of tem- perature are so small that no account need be taken of the pressure to which the body is subjected. In the case of a gas under constant pressure the expansion with rise of tem- perature is considerable, and the thermal capacity is inti- mately connected with the work done by this expansion. There will, accordingly, be two distinct values of the specific heat for a gas according as the pressure or as the volume is kept constant. 155. Specific Heat under Constant Pressure. The spe- cific heat of gases under constant pressure has been carefully determined by Regnault. His method was to pass a definite quantity of the gas, at a known temperature, through a series of spiral tubes surrounded by cold water. The elevation of the temperature of the water was taken as a measure of the heat given up by the gas in falling through an observed range of temperature. The researches of Regnault showed that for a body rigorously obeying Boyle's Law the specific heat at constant pressure is independent of the pressure and of the temperature. SPECIFIC HEATS AT CONSTANT PRESSURE. Air 0.2374 Bromine 0.0555 Oxygen 0.2175 Chlorine 0.1241 Hydrogen 3.4090 Nitrogen 0.2438 156. Specific Heat under Constant Volume. No satisfac- tory direct determination of the specific heat at constant volume was made until the invention of the steam calorim- 198 CALORIMETRT. eter of Joly. The experiments of this investigator indicate that the specific heat, when the volume is kept constant, increases with the density in the case of air and of carbon dioxide, but diminishes in the case of hydrogen. The chief results as far as published are : PRESSURE IN ATMOS. DENSITY. GRAMS PER CU. CM. SPECIFIC HEAT AT CON- STANT VOLUME. Air 19.51 0.0205 0.1721 Hydrogen .... Carbon Dioxide . . 7.20 0.011530 2.402 0.16841 Carbon Dioxide . . 12.20 0.019950 0.17504 Carbon Dioxide . 16.87 0.028498 0.17141 Carbon Dioxide . 20.90 0.036529 0.17305 Carbon Dioxide . . 21.66 0.037802 0.17386 157. Atomic Thermal Capacities. Law of Dulong and Petit. As early as 1819 attention was called by Dulong and Petit to a remarkable relation between the specific heat of a simple substance and its atomic weight. Consideration of such a table as that on page 196 led these investigators to announce the law which now bears their name, viz., that the product of the specific heat by the atomic weight is the same for all elementary substances. The truth of this law was later investigated by Regnault, who found that it held approximately for substances which occur in the solid state. For 32 of these substances the mean product was 6.38, the extreme values being 6.76 and 5.7. Some variation in the value of this product might be expected from the dependence of specific heat on tem- perature and molecular conditions, explained in Art. 152. EXAMPLES. 199 EXAMPLES. 1. 45.1 gms. of copper at a temperature of 99.6 was immersed in 52.5 gms. of water at a temperature of 10. The temperature of the water after immersion rose to 16.8. Required the specific heat of the copper. Am. 0.0956. 2. 72.3 gms. of a substance having a specific heat 0.874, at a tem- perature 95.6, was mixed with 129 gms. of a liquid at a temperature 23.7. The temperature of the mixture was found to be 56.3. What was the specific heat of the liquid ? Ans. 0.590. 3. 937 gms. of a substance having a specific heat 0.787, at a temper- ature 15.3, was mixed with 596 gms. of a second substance having a specific heat 0.568, and at a temperature 135. What should be the resulting temperature ? Ans. 52.9. 4. 16.1 gins, of sand at a temperature of 75 and 20 gms. of iron at 44.9 are thrown into 50 gms. of water at 3.9. What should be the temperature of the mixture? Ans. 9.9. 5. 96.8 gms. of ice at were thrown into 156 gms. of water at 98.3. What should be the temperature of the mixture? Ans. 30.3. 6. 1.08 kilos of iron at 100 were placed in an ice calorimeter and melted 155 gms. What was the specific heat of the iron ? Ans. 0.114. 7. What should be the result of mixing 51 grns. of snow at with 230 gms. of water at 20? Ans. Water at 2.0. 8. What should be the result of mixing 6.2 gms. of snow at with 7.1 gms. of water at 50? Ans. 1.7 gms. of snow, and 11.6 gms. of water. 9. What should be the result of mixing 30.2 gms. of snow at 10 with 79.8 gms. of water at 40, the specific heat of snow being taken as 0.504? Ans. Water at 5.9. 10. What change of volume would be produced in a Bunsen ice calorimeter by the insertion of 2.5 gms. of a substance at 100, having a specific heat 0.076? Ans. 0.0217 cc. 11. 15.1 gms. of mercury at 100 produce a contraction of 0.0567 cc. in a Bunsen ice calorimeter. What is the specific heat of mercury ? Ans. 0.0329. CHAPTER XL CHANGE OF STATE. 158. Laws of Fusion. When heat is continuously applied to an amorphous body, e.g. glass, resins, etc., the change from the apparently solid to the liquid condition takes place in a continuous manner ; that is to say, there is no temperature at which the substance may be said to begin to melt. In the case of most crystalline substances, however, when under a definite pressure, the change from the solid to the liquid state, or vice versa, appears to occur at a fixed temperature and to be discontinuous. At the temperature of fusion the body is capable of co- existent phases; i.e. each state in the presence of the other is stable, but at other temperatures only one of the states is so. If a definite amount of heat be applied to a body at its temperature of fusion, a definite quantity of the solid will be melted; and, conversely, if the same amount of heat be abstracted from the liquid at that temperature, the same mass will be solidified. 159. Fusion of Alloys. Alloys of two or more metals frequently have a melting point lower than that of any of the components. Thus, an alloy made 5 parts tin and 1 part lead fuses at 194 C.; another, called Rose's Fusible Metal, composed of 4 parts bismuth, 1 part tin, and 1 part lead, melts at 94 C. Other physical and chemical properties of such alloys indicate that the atoms have grouped themselves in new ways which are hardly distinguishable from the chemical molecule. TABLE OF MELTING POINTS. C. Aluminum 850 Mercury Antimony 432 Nickel Bismuth 268 Platinum Brass 1015 Silver Copper 1054 Steel, Cast Gold 1045 Sulphur Iridium 1950 Tin Iron, Cast 1200 Zinc Iron, Wrought 2000 Glass Lead 326 Paraffine FUSION OF ALLOYS. 201 C. -39 1450 1775 954 1375 115 233 433 1100 54 TABLE OF LATENT HEATS OF FUSION. UNIT, CALORIE PER GRAM. Bismuth 12.64 Platinum 27.2 Ice 79.25 Silver 21.07 Iron, Cast, Gray 23.0 Sulphur 9.37 Iron, Cast, White 33.0 Tin 14.25 Lead 5.37 Zinc 28.13 Mercury 2.83 160. Unstable Condition at Melting Point. It is fre- quently possible, by slow and undisturbed cooling, to reduce a liquid below the temperature of freezing before solidifi- cation will begin. Thus, water freed from air and covered with a layer of oil may be cooled to 12 C. without freez- ing. Similarly, drops of water suspended in a fluid of their own density have been reduced to 20 C. without change of state. If, however, a fragment of ice be dropped into water thus over-cooled, or if the vessel be jarred, solidifica- tion will begin at once. 161. Change of Volume at the Melting Point. Bodies which have a definite melting point exhibit a more or less abrupt change of volume on liquefaction. A majority of 202 CHANGE OF STATE. substances expand on melting, but there are many excep- tions, the most notable among these being water, which increases in volume from 1 to 1.0907; that is, more than one-twelfth, at the moment of congelation. This property plays an important part in the economy of nature. Thus, ice, being less dense than water, floats on the surface of ponds and rivers, where it serves as a protection to aquatic animal and vegetable life. If the density of ice were greater than that of water, it is evident that the ice would sink as fast as formed, and the entire body of water would soon be frozen solid. The stress exerted by water in solidification is very great, and usually bursts any vessel in which it is allowed to freeze. The disintegration of rocks thus effected by the frost is an important step in the preparation of the soil. Among the metals, iron, bismuth, and antimony either expand on solidifying or change very little,^and are conse- quently well suited for castings, as they retain an exact impression of the mold. TABLE SHOWING THE EXPANSION ON MELTING. METAL. DENSITY OF SOLID. DENSITY OF LIQUID. PERCENTAGE CHANGE IN VOLUME FROM SOLID TO LIQUID. Bismuth .... 9.82 8.8 10.055 8.217 Decrease 2.3 % Increase 7.1 Lead 11.4 10.37 9.93 Silver 10.57 9.51 " 11 2 Tin . . 7.5 7025 " 6 76 Zinc 72 6 48 " 11 1 Cast Iron .... 6.95 6.88 1.02 162. Volume-Temperature Diagram of Water. A valu- able method of representing the various states that a body may pass through is to plot certain of the physical coordi- nates of the state of the body, such as volume and pressure, VOLUME-TEMPERATURE DIAGRAM OF WATER. 203 or temperature, as geometrical coordinates on a diagram. Thus, let the abscissas in Fig. 148 represent temperatures, and the ordinates volumes of the unit mass of water under a constant pressure of one atmosphere. Choose the ordinate of the point C proportional to the volume of a gram of freez- ing water which is approximately 1.000116 cc. and let its abscissa be 0. Then the point J?, directly above (7, and having an ordinate 1.0907 times as great, will represent the ice at zero. The contraction of the ice, as the temperature falls, is shown by the line BA, whose slope is given by tan = 0.000457, the coefficient of expansion for ice. V o r 100 FIG. 148. From O to .Z> the water contracts, and from D to E it expands, according to the law given on p. 178. At E vapori- zation commences, and continues to 6r, the volume increasing 1696 times, but the temperature remains constant. The further expansion is along GrH, having a slope tan <= nearly. 204 CHANGE OF STATE. The condition of water which has been over-cooled would B be represented by the curve >v Co, Fig. 149. )b It is conceivable that the change from water (7 to ice B might take place along the path CabB, but since the states represented by the curve ab are in the region of complete instability there seems little probability that they can be observed. 163. Influence of Pressure on the Melting Point. It was suggested in 1849 by James Thomson, from purely theoreti- cal considerations, that water which contracts on liquefaction should have its melting point lowered by increase of pres- sure, by about 0.0075 C. per atmosphere. These predictions were shortly afterward verified by his brother, Lord Kelvin. Without presenting the mathematical theory at this point, it is not difficult to see that if a body contract on liquefac- tion an increase of pressure would assist the change, and probably allow it to take place at a lower temperature. Likewise, if a body expand on melting, a large pressure should retard the change or raise the melting point. Bunsen found that paraffine which melted at 46.3 C. under atmos- pheric pressure must be raised to 49.9 C. before it began to fuse, if the pressure was 100 atmospheres. Experiments on spermaceti and stearin showed analogous effects. 164. Regelation. When two pieces of melting ice are pressed together, they freeze along the surface of contact. This phenomenon, called regelation, may be explained by the principles of the preceding section. The pressure at the points of contact causes some of the ice to melt; the EEGELA TION. 205 water so formed escapes to a region where the pressure is less, and, since it is below zero, freezes again, cementing the blocks together. The following experiment, due to Bottom- ley, welt illustrates the melting of ice under pressure and regelation. A block of ice, A (Fig. 150), is supported on two wooden bars, B, C, and a copper wire, with two weights, Z>, E, suspended from the ends, passed over the top. The wire will be found to work its way slowly through the block, and finally fall to the floor, but the section through which it has passed will be even more firmly frozen than at first. The pressure of the wire melts the ice immediately below it, and the water thus formed escapes to the upper side, where it at once freezes, as it is free from the pressure and below zero. Incidentally, the experiment illus- trates the principle demonstrated in Art. 107, that the pressure beneath the wire varies directly as the curva- ture. For whatever the form' of the upper surface of the block may be when the experiment is started, the wire ultimately assumes a circular arc within the block, thus showing that, if the curvature is at first greater at one place than another, the more rapid descent of the wire at that point has the effect of diminishing the curvature until all parts progress uniformly. The formation of a snowball by squeezing moist snow between the hands is another example of the melting of ice under pressure and regelation. If the snow is too cold, the ball refuses to " make," for the reason that the pressure of the hand is not sufficient to bring it to the melting point. FIG. 150. 206 CHANGE OF STATE. In like manner small pieces of ice placed in a mold may be squeezed into a nearly homogeneous mass of ice, having exactly the shape of the mold, if they are subjected to suffi- cient pressure. An example of the same phenomenon, and of great inter- est on account of its intimate connection with the climate of every part of the world, is afforded by the formation and flow of glaciers. Large quantities of water evaporating in the warm regions are carried toward the poles by the pre- vailing winds of the upper air and deposited as snow within the arctic circles. As the temperature rarely rises to the melting point, the snow collects in immense amounts. The effect of the weight of the upper layers is to compact the lower portion into a solid mass of ice, from which small streams of water flow out. If, now, the ground slope toward the sea, as is generally the case in the valleys where the greatest quantities of snow collect, the relief of the pressure by melting at one point may subject another part to stress sufficient to break it. Thus, as a result of the breaking, melting, and freezing, the glacier creeps slowly toward the sea. If it were not for this method of return of the snow to the ocean, it seems probable that the accumulation of vast quantities in the polar regions would affect the stability of rotation of the earth. The glaciers which occur in high altitudes of the temperate zone have been studied by many investigators. Those of Switzer- land are found to have a velocity of one or two feet per day. 165. Evaporation. According to the molecular theory of the constitution of bodies presented in Art. 72, the mole- cules of a liquid, while not restricted to a definite position, being free to wander throughout the body, are, nevertheless, within the sphere of action of other molecules, for bodies in SUBLIMATION. 207 this state resist forces acting so as to increase their volume. If, however, a molecule happen to be moving with sufficient velocity near the free surface of a liquid, it may escape entirely from the region of attraction of the other molecules, and move in a straight path till it meets some obstacle. It is now said to be in the aeriform condition, and this change of state is called evaporation. 166. Sublimation. When a body passes by evaporation from the solid to the aeriform condition, the mode of change is termed sublimation. At atmospheric pressure, camphor, arsenic, and many less familiar substances volatilize without passing into the liquid condition. If, however, the pressure be sufficiently increased, they may be fused. A light fall of snow is often seen to dis- appear by sublimation, when the temperature is considerably below freezing. Experiment shows that for a number of substances there is a definite pressure peculiar to each, below which it is impossible to melt the body. It has been pro- posed to designate this pressure as the critical pressure. This pressure must not, however, be confused with the pres- sure of the critical state (Art. 180). 167. Ebullition. If heat be applied to a liquid under constant pressure in an open vessel, the temperature will rise, the average velocity of the molecules will be increased, and in consequence evaporation will go on at a more rapid rate. When a certain temperature, depending on the sub- stance and the pressure, has been reached, it ceases to change, bubbles begin to form at the sides of the containing vessel and rise through the liquid, growing rapidly in volume. The substance is then said to be in ebullition, and the con- stant temperature at which the vaporization goes on is called the boiling point. 208 CHANGE OF STATE. TABLE OF BOILING POINTS. UNDER A PRESSURE OF 760 MM. OF MERCURY. SUBSTANCE. BOILING POINT C. SUBSTANCE. BOILING POINT C. Alcohol . . . 77.9 Nitrogen . -194 38 5 Oxygen 184 Carbon Dioxide . - 78.2 Sulphur . . . 448.4 Chlorine . . . - 33.6 Sulphuric Acid . 338 Hydrogen . . . -243 Zinc .... 950 Ether .... 34.9 Sulphur Dioxide 10.5 Mercury . . . 350 Oil of Turpentine 159.3 168. Latent Heat of Vaporization. By a specially con- trived apparatus, too complicated for complete description here, Regnault found that the quantity of heat necessary to change a gram of water into steam at the same temperature could be expressed with considerable accuracy by the formula, = 606.5 -0.695*, where t is the temperature at which the change takes place. When the variation of the specific heat of water is taken into account, a small correction must be applied. Thus, the latent heat at 100 C. is more exactly 536.5 calories per gram, and at 200 C. 464.3 calories per gram. TABLE OF LATENT HEATS OF VAPORIZATION. SUBSTANCE. CALORIES PER GRAM. TEMPERATURE C. Alcohol QQ9 4 77 9 Bisulphide of Carbon . . . Ether 86.7 90 4 46.2 34.9 JMercurv 6 350 Oil of Turpentine .... Water 74.0 j 536.5 Regnault 159.3 100 1 535.9 Andrews UNSTABLE CONDITION AT BOILING POINT. 209 TABLE SHOWING CHANGE OF VOLUME ON VAPORIZATION. AT A PRESSURE OF ONE ATMOSPHERE. Alcohol 528 Ether 298 Oil of Turpentine . . . . . . 193 Water . . 1696 169. Unstable Condition at the Boiling Point. As early as 1777 attention was called to a variation of the boiling point by the discrepancies in the reading in thermometers on which the upper fixed point had been determined by im- mersion in water boiling under the pressure of one atmos- phere. The explanation of these variations is to be found in the existence of an unstable condition in the region of the boiling point, quite analogous to that presented by sub- stances near the temperature of liquefaction. The presence of nuclei about which vapor may begin to accumulate, and more especially the presence of dissolved air, which itself collects in small bubbles when the temperature rises, greatly favors the beginning of ebullition, so that in proportion as the liquid is freed from these, the boiling point is raised. When retarded ebullition does commence it goes on in an almost explosive manner, and the liquid bumps violently against the bottom of the vessel. Dufbur, by suspending drops of water in a mixture of linseed and clove oils, found that a drop 10 mm. in diameter could be raised to a tem- perature of 120 C., while drops 1 mm. in diameter remained liquid up to 178 C. When touched with a glass rod, or by the side of the vessel, they exploded at once. 170. Spheroidal Condition. If drops of water be placed on a metal plate which has been heated to a proper temper- 210 CHANGE OF STATE. ature, they do not vaporize at once, but roll about in little globules after the manner of mercury, or gather together in one mass which vibrates through an elliptical or stellate form. The phenomenon puzzled the first observers, who thought that the liquid had assumed a new form which they named the spheroidal state. Careful examination showed that there was no contact between the plate and the drop, thus suggesting the true explanation, namely, that the drop is supported on a cushion of its own vapor, which, being a poor conductor of heat, permits only a slow evaporation. The form assumed by the drop is fully accounted for by the laws of surface tension. 171. Cooling by Evaporation. If it be assumed that heat is the kinetic energy due to the irregular motion of the mole- cules, it is easy to see that, whenever a molecule near the surface of a liquid escapes into a free space above, the kinetic energy of the liquid must be diminished by the amount of work done in separating the molecule from the liquid and by the energy which the molecule takes with it ; that is to say, the liquid will be cooled by evaporation. When the process is sufficiently rapid and long continued, the liquid may be frozen. This was first performed by Leslie, who supported a small copper dish containing water over a vessel of sulphuric acid, and placed the whole under the receiver of an air pump. On exhausting the air the water evaporates rapidly, but the vapor is continuously absorbed by the sulphuric acid. The temperature conse- quently falls, and, if the process be conducted with sufficient celerity, it is quite possible to have the water boiling and freezing at the same moment. Very low temperatures may be produced by the vaporiza- tion of the more volatile substances. Thus, by opening a TRIPLE POINT. 211 small orifice in a vessel containing liquid <70 2 , the escaping spray cools by evaporation so rapidly that it freezes in the form of fine snow. 172. Triple Point. Let a curve be drawn on the volume temperature diagram representing the changes which a body undergoes while the pressure remains constant at p v for such a substance as water. Thus, in Fig. 151, a 1 b l represents the expansion of the solid, b^ " " contraction during melting, c^ " " changes in volume in the liquid, d l e 1 " " expansion during vaporization, e ifi in the aeriform condition. For a pressure p% less than p l the analogous changes are represented by the line ajb^d^f^ in which 6 2 e 2 and e z d 2 have approached each other. If the pressure is lowered sufficiently, these lines will ultimately meet in some line, such as J5<7; that is to say, by lowering fche pres- sure the freezing point is raised and the boiling point lowered, until a condition is reached in which water in the solid, liquid, and aeriform states may exist together in equi- librium. This state, realized in the experiment of Leslie (Art. 171), is known as the triple point. 173. Cryophorus. An instrument for showing the freez- ing of water by evaporation, invented by Wollaston and called the cryophorus, is figured in the accompanying sketch, Fig. 152. It consists of two glass bulbs, A, B, connected by a bent FIG. 151. 212 CHANGE OF STATE. tube. A small quantity of water is introduced, and, after boiling to expel the air, the whole is hermetically sealed. The experiment is commenced by running all the water into one bulb and immersing the other in a mixture of snow and salt, which has a temperature of 18 C. The condensation of the vapor in A lowers the pressure in B, permitting a rapid evaporation from the water, which in time freezes solid. The bulb is not fractured by the expansion of the ice, as might be expected, because the ice itself volatilizes so rapidly that complete contact at the sides of the bulb never occurs. 174. Freezing Machines. The principles just explained have been applied to the manufacture of ice and to refriger- n FIG. 152. FIG. 153. ation on a large scale. Some of the best-known systems make use of ammonia, according to a method introduced by Carre in 1860. The apparatus consists essentially of a boiler, A (Fig. 153), nearly filled with a strong solution of aqua ammonia, and connected by a pipe with the chamber B, which surrounds the vessel C containing the water to be frozen. When heat is applied to the boiler, the ammonia gas is driven over into B, where it is condensed in the pres- ence of a small quantity of water. Connection with A is SATURATED VAPOR. 213 now closed, and communication opened to a condenser, per- mitting a rapid evaporation of the ammonia gas from B, with consequent fall of .temperature and freezing of the water in C. 175. Saturated Vapor. When a liquid is placed in a closed, empty space, evaporation at first will go on rapidly ; but as the space above becomes filled with molecules rebounding from one another and from the sides of the vessel, some of them are reflected into the liquid and remain there. In time the vapor and its liquid come into a state of equilibrium, i.e. as many molecules are returned through the free surface as escape from it in a given time. .The vapor is then said to be saturated. 176. Pressure of a Saturated Vapor. The pressure which any vapor is able to support cannot exceed a certain maxi- mum amount, depending on the temper- ature and the nature of the substance. Any attempt to compress the vapor beyond this point (saturation) will result in the condensation of a portion into the liquid state. This behavior of a vapor may be easily shown by means of a barometer tube, A (Fig. 154), immersed in a cistern of mer- cury, B. If a few drops of ether be introduced at the bottom of the tube, it will rise to the top and quickly evapo- rate, producing a fall of the mercury below the true baro- metric hight c?, proportional to the pressure exerted by the FIG. 154. 214 CHANGE OF STATE. vapor in the space be. If the tube be now pushed further down into the cistern, the volume of the vapor will decrease, and its pressure, which is measured by the fall of the level of the mercury c below cZ, will increase until liquid ether begins to collect at the top of the mercury. On lowering the tube further, or raising it again, the level of the column will remain invariable as long as any liquid is present, the only observable effect being an increase or diminution of the amount of liquid ether. The condition of a saturated vapor is thus seen to depend on but two coordinates, the tempera- ture and the pressure, instead of on three which are usually required to determine the state of a body. TABLE OF MAXIMUM PRESSURES OF VAPORS. IN DYNES PER SQ. CM. TEMP. C. ALCOHOL. ETHER. CARBON BISULPHIDE. CHLOROFORM. -20 4455 9.19 XlO 4 6.31 XlO 4 -10 8630 1.53 XlO 5 1.058 x 10 5 16940 2.46 X " 1.706 x " 10 32320 3.826 X " 2.648 x " 20 59310 5.772 x " 3.975 x " 2.141 X 10 5 30 1.048 X 10 5 8.468 x " 5.799 x " 3.301 X " 40 1.783 X " 1.210 x 10 6 8.240 X " 4.927 X " 50 2.932 X " 1.687 X " 1.144 x 10 6 7.14 x " 60 4.671 X " 2.301 x " 1.554 x " 1.007 X 10 6 80 1.084 x 10 6 4.031 x " 2.711 X " 1.878 x " 100 2.265 X " 6.608 x " 4.435 x " 3.24 x " 120 4.31 x " 1.029 X 10 7 6.87 X " 5.24 x " 177. Vapor Pressure of Water. The pressures of satu- rated water vapor have been studied by Regnault, who made use of two barometer tubes surrounded by a bath whose temperature could be varied. One of the tubes (Fig. 155) ISOTHERMALS OF A GAS. 215 contained only mercury ; the other had a small quantity of water at the top of the mercury column. The difference of these columns, when corrected for capillarity and the expansion of the mer- cury, gave the value of the pressure sought. The results obtained are exhibited by the table on the following page. It will be observed that the vapor pressure at the boiling point, 100 C., is just one atmosphere, as it should be ; for the boiling point of any liquid is obviously that tempera- ture at which the vapor in a bubble is able to maintain itself against the external pres- sure on the liquid. 178. Isothermals of a Gas. If the pres- sure and volume be plotted as coordinates on a diagram, the line which represents the suc- cession of states through which a body may pass while its tem- perature remains constant is called an isothermal. In the case of a gas the isothermals will be a series of P j i \ \ \ \ equilateral hyperbolas, as appears from the ion = mRr = constant for r constant. Fig. 156 shows a series of such isothermals drawn for intervals of five de- grees. It is evident that the isothermal for any tempera- ture must lie entirely above that for a lower tempera- ture, since both the pres- sure and the volume increase with increasing temperature. FIG. 155. FIG. 156. 216 CHANGE OF STATE. TABLE OF PRESSURES OF SATURATED WATER VAPOR. IN CENTIMETERS OF MERCURY. fl cm. mere. = 1.334(10) 4 TEMP. PRESSURE. TEMP. PRESSURE. -30C. 0.0386 100C. 76.00 -25 0.0605 102 81.60 -20 0.0927 104 87.54 -15 0.1400 106 93.83 -10 0.2093 108 100.5 - 5 0.3113 110 107.5 0.4600 112 115.0 5 0.6534 114 122.8 10 0.9165 116 131.1 15 1.269 118 139.9 20 1.739 120 149.1 25 2.355 122 158.8 30 3.154 124 169.1 35 4.183 126 179.8 40 5.491 128 191.1 45 7.139 130 203.0 50 9.198 135 235.4 55 11.15 140 271.8 60 14.88 145 312.6 65 18.69 150 358.1 70 23.31 155 408.9 75 28.85 160 465.2 80 35,40 165 527.5 85 43.30 170 596.2 90 52.54 175 671.7 92 56.67 180 754.6 94 61.07 185 845.3 96 65.75 190 944.3 98 70.72 ; 195 1052 99 73.33 200 1169 ISOTHERMAL OF A VAPOE. 217 179. Isothermal of a Vapor. Suppose that a given mass of aqueous vapor be placed in a cylinder furnished with a piston, and be kept at the constant temperature of 100. Let the pressure be represented by Oa n , and the volume by Oa', Fig. 157. If the vapor be now compressed, the tem- perature remaining constant at 100, the pressure will in- crease, but not so rapidly as for a gas, since pv is something less than the constant value mRr. This series of changes will be represented by the curve ab, having an equation (7) pv where B and C are con- stants. At the point b the vapor has reached satura- tion, and further compres- sion is followed by conden- sation, the pressure all the while remaining the same until the point c is reached, where all the mass is lique- fied. At any point between c and 6, as at e, the pro- portion of liquid to vapor is given by . As the ce liquid is very incompres- sible, from the point c an immense increase of pressure cor- responds to but a very small diminution of the volume. If this process of compression were to be repeated at a higher temperature, say 200 C., a new isothermal, fgkk, would be obtained. The point of saturation g now has a greater pressure and a smaller volume. The point of com- plete liquefaction A, on the contrary, has a larger volume FIG. 157. 218 CHANGE OF STATE. than > tt^ ,, o i ^^ >>- ^^ ^^~ - >:-> ^ ' -'"" _ ^ ^ ** ( ^ -- bCbX- ^^^ ^-^ ^>' ^^, >-* #>-* . ^ ^ _- _40^. ^--- ^- < ^^^ ^ ^~-~- -^ -"" _^ VLJ- "^^ _ - -- ^^ : itm isph eres 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 32 FIG. 161. 32 30 Atmosph 20 40 60 80 ^00 ^20 140 160 180 200 220 240 260 280 300 320 FIG. 162. atures are parallel and nearly straight, but inclined to the axis of pressure in such a way as to show that pv increases 224 CHANGE OF STATE. uniformly with the pressure. If Boyle's Law were fulfilled, the lines should be horizontal and straight. The departure from a right line, which is just perceptible in the case of hydrogen, is more marked in the diagram of nitrogen (Fig. 162), where pv reaches a minimum in the vicinity of 40 atmospheres and afterwards rises. The variations, which are but little more than suggested in the diagrams of hydrogen and nitrogen, are exhibited in their entirety by the diagram of carbon dioxide (Fig. 163). 50 100 125 150 FIG. 163. 175 200 225 250 The points of minimum pv move toward the right, with rise of temperature, until about 180, where they begin to move back. Since condensation takes place below 30, the values of pv are discontinuous in this region. Little is known of the product of the volume and pres- sure of gases at low pressures, since the column of mercury, EXAMPLES. 225 by which the latter is measured, is too short to be read accurately, and the presence of the mercury vapor itself introduces considerable uncertainty. It appears probable, however, that the departure of this product from a constant value is very slight. EXAMPLES. 1. Steam at 100 is passed into 120 gms. of water at 10 until the temperature rises to 35 and. the mass of water has increased to 125 gms. What is the latent heat of steam? Ans. 535 cal. per gm. 2. 150 gms. of lead at 401 were dropped into 844 gms. of mercury at 0. What was the resulting temperature, assuming that the specific heat of melted lead is 0.0402 ? Ans. 88.4. 3. What is the least quantity of water at 10 needed to condense 100 gms. of alcohol at 78 into liquid at 15, assuming the mean specific heat of alcohol to be 0.65 ? Ans. 4.87 kilos. CHAPTER XII. SOLUTIONS. 183. Definition of a Solution. In the most general sense a solution may be defined as a homogeneous mixture which cannot be separated by mechanical means. In the aeriform condition bodies mix in all proportions. In the liquid state the miscibility is limited in amount and dependent on the nature of the substances. The solution of one solid by another is not unknown, but the cases are of comparatively little importance. The substance which is present in a solution in the largest quantity is usually spoken of as dissolving the other and may be called the solvent. The substance which is dissolved will be termed the solvend. % 184. Solution of Gases in Gases. When a homogeneous mixture of two or more gases is formed, the properties of the mixture are found to be the sum of the properties of the components. The law connecting the pressure of a mixture of gases with the pressures of the components was first established by Dalton, who found that the total pressure of a mixture was equal to the sum of the pressures which each of the gases would separately exert in the given space. Thus, for instance, suppose a number of gases having initially the volumes v v v^ v z , and pressures p v p^ p% , be placed in a vessel whose volume is FJ then the pressure of the first gas expanded to this volume would be SOLUTIONS OF LIQUIDS IN AERIFORM BODIES. 227 by the law of Boyle.. Similarly, the pressure of the second alone occupying this volume would be and so on. If, then, P denote the pressure of the mixture, by Dalton's Law, or, (2) P V= p 1 v 1 + p 2 v 2 + etc. The explanation of this law is, apparently, to be sought in the fact that the molecules are so distributed that they are able to influence each other only by their kinetic ener- gies, and not at all by those properties which are character- istic of each substance. For a similar reason it might be expected that other properties of such mixtures would be the sum of the corresponding properties of the components. This prediction has been verified for refraction and absorp- tion of light, and there is no reason to doubt that it holds true for all cases. 185. Solutions of Liquids in Aeriform Bodies. The evap- oration which takes place at the free surface of a liquid per- mits the formation of a solution of a liquid in all aeriform bodies. Experiments by Dalton on the evaporation of a liquid in contact with a gas showed that the vapor pres- sure of the liquid in a space occupied by a gas was the same as in a vacuum. Further investigations have shown that although the law is substantially accurate it cannot be re- garded as more than a first approximation, or the limiting case. The deviation from the exact law is to be assigned, 228 SOLUTIONS. 1, to the mutual actions of the molecules of the vapor and the gas, and 2, to the lowering of the vapor pressure con- sequent upon the solution of some of the gas in the liquid. (See Art. 195.) Of the solution of solids in gases nothing is definitely known, but it is probable that Dalton's Law is followed to the same extent as in the solution of liquids in gases. 186. Solution of Gases in Liquids. Liquids dissolve all gases without exception. The solutions are conveniently divided into two classes, according as the gas may or may not be completely expelled by heating under diminished pressure. The first class is subject to a law discovered by Henry, namely, that the mass of the gas dissolved by a given quan- tity of liquid is proportional to the pressure if the temper- ature remains Constant. Since the density of a gas varies directly as the pressure, Henry's Law may also be stated in this way : The volume of the gas which can be absorbed by a given quantity of liquid at a definite temperature is the same for all pressures. The ratio of the volume of the gas absorbed to the volume of the absorbing liquid is called the solubility; thus, if V= volume of liquid, v'= " " the gas, X = solubility, then, (3) X = f The value of the solubility is found to diminish with increasing temperature. SOLUTIONS OF LIQUIDS IN LIQUIDS. 229 In solutions from which the gas cannot be driven off by heating, or lowering the pressure, certain chemical changes appear to have taken place. The discussion of such solu- tions belongs rather to Chemistry than to Physics. 187. Solutions of Liquids in Liquids. The solution of one liquid by another is, in general, accompanied by a change of volume and of temperature. Thus, on mixing equal volumes of alcohol and carbon disulphide, the temperature 70 80 90 100 110 120 130 140 150 160 170 T FIG. 164. falls through 5.6 C. A mixture of ether and chloroform is accompanied by a rise of temperature of 14.4 C. and a dimi- nution of volume. The amount and direction of these changes are frequently dependent on the proportion of the components in the mixture. The proportion in which two liquids dissolve depends in a most important manner on the temperature. Solutions of liquids in liquids are sometimes divided into classes, according as they are or are not miscible in all propor- tions. Alcohol and water, chloroform and carbon disulphide, 230 SOLUTIONS. are pairs of liquids which are soluble in all proportions. On the other hand, certain pairs of liquids are soluble only within definite limits. For instance, if equal quantities of water and ether are shaken together, they will separate into two layers, the lower, an aqueous solution, containing 10% of ether, and the upper, or ethereal solution, containing 3% of water. If any liquid, A, is partially soluble in a liquid, B, then the reciprocal process will also obtain, and B will be partially dissolved by A. A valuable method of recording experiments or solutions is to plot the temperatures as abscissas and the ordinates as the percentage of one substance in the solution. Thus, in Fig. 164 the curve aaa refers to water and phenol, bbb to water and salicylic acid, cce to water and benzoic acid, ddd to water and aniline phenolate, eee to water and aniline. It will be observed that at the low temperatures there are two definite proportions in which a stable solution is formed. For instance, LH is the proportion of phenol present when it is dissolved by the water, but LK is the amount of phenol in 100 parts of the solution when phenol is the solvent, the temperature in each case being 34 C. As the temperature rises, these proportions approach equality, and above this point the liquids are completely miscible. Since there is no reason to doubt that all liquids which dissolve one another would show similar behavior at suffi- ciently high temperatures, the distinction made between two classes of such solutions is an entirely arbitrary one. 188. Solution of Solids in Liquids. Many solids when brought in contact with a liquid are dissolved up to a cer- tain limit, depending on the temperature, after which the process ceases. The liquid at this point is said to be satu- rated. DIFFUSION OF LIQUIDS. 231 The concentration of any solution may be increased to saturation by lowering the temperature, after which some of the solvend will separate if a particle of the solid be present. In the absence of such a nucleus it is possible to cool the solution below the point of saturation without a separation of the dissolved solid. The solution is then said to be super- saturated. The solubility of a solid in general increases with the temperature, but there are several important exceptions. The volume of the solution formed by dissolving a solid in a liquid is usually less than the sum of the volumes of the components, by an amount depending upon the con- centration. 189. Diffusion of Liquids. When two different solu- tions are brought in contact, a slow change of concentration is observed to go on until the composition of the whole is homogeneous. This phenomenon, called diffusion, was first investigated by Graham in the following method. A glass jar, A) was* filled with a salt solution and placed within a larger vessel, B. Water was then carefully poured into the latter until A was covered to a depth of about 3 cm. After standing for a considerable length of time, A was removed and the amount of salt present in the outer vessel deter- mined. The chief conclusion deduced from such experi- ments was that the rate of diffusion varied greatly with the nature of the dissolved substance. Arranging the substances according to the rate of diffusion, Graham found that free acid bases and neutral salts were characterized by a much greater rate of diffusion than that shown by gums, tannin, albumen, and the like. As the substances of the first men- tioned class are generally known in the crystalline form, he proposed to call them crystalloids. The second class are 232 SOLUTIONS. amorphous and were named colloids (/co\\r) = glue). Col- loids are also distinguished by the fact that they are more or less impermeable by other colloids, but do not present any marked hindrance to the diffusion of crystalloids. Graham further utilized this principle to separate a mixture of crystalloidal and colloidal substances by a method which he called dialysis. The mixture is placed in a vessel and separated from pure water by a colloidal membrane, such as parchment or animal bladder. In course of time the crys- talloids will diffuse into the water and leave the colloids behind. 190. Coefficient of Diffusion. Let the concentration of a solution be denned as the mass of the solvend divided by the space through which it is uniformly distributed, and sup- pose that the concentration at one surface of a layer of liquid of thickness d is s v and the concentration at the other sur- face is s 2 . Also suppose that a mass, m, of a substance dif- fuses through a cross section, A, of this layer in the time , then the coefficient of diffusion K may be defined by the equation -^ . (*, ,) . m = KA v 2 1 ' t. d The dimensions of K are accordingly 191. Diffusion of Gases. Gases exhibit phenomena of diffusion identical with those just described in the case of liquids, but more marked in degree. If, for instance, a small quantity of a gas possessing a characteristic odor, e.g. chlo- rine, acetylene, etc., be liberated in one corner of a room, its OSMOSE. 233 presence may be detected by the sense of smell in every part after a very short time. When two gases are separated by a porous septum, the more rapid rate of diffusion of one of the gases through the partition may establish a temporary difference of pressure between the sides. This may be shown by means of a fun- nel, A (Fig. 165), closed at the top with a plate of plaster of Paris and connected at the bottom with a glass tube, D, dipping into a vessel of colored water, O. If a beaker, B, be inverted over the funnel and filled with hydrogen, or even ordinary illuminating gas, bubbles of air will begin at once to rise through the water in C. On removing the beaker, the liquid will rise in the tube to a certain hight, after which it will slowly fall to the original level. The diffusion of the rare gas into the funnel goes on so much more rapidly than that of the air outwards that the pressure rises and air is expelled from the tube. When the beaker is re- moved, the more rapid escape of the hydrogen causes the reverse effect in the tube D. A mixture of gases may thus be separated by a method analogous to dialysis. 192. Osmose. When two liquids are allowed to diffuse through a membrane which hinders the passage of one of the substances, a considerable difference of pressure may be established between the two sides. This mode of diffusion has received the name osmose, and the difference of pressure is called the osmotic pressure. It was first described by the Abbe Nollet, who found that if a glass vessel be covered with a bladder filled with spirits of wine and immersed in FIG. 165. 234 SOLUTIONS. water, the contents of the vessel would increase so as to expand the bladder, and sometimes even burst it. Farther investigations have shown that this phenomenon has an important bearing upon the explanation" of many processes in organic life which had been loosely attributed to a hypothetical vital force. By the arrangement shown in Fig. 166, Dutrochet was able to obtain a rough measure of the osmotic pressure. A is a vessel closed at the bottom by a membrane, (7, and fitted at the top with a long open tube, B. When this vessel A is filled with a solution of gum and water and immersed in a larger vessel, Z>, containing water only, it is found that after a time the level of the fluid in B rises to a hight considerably above the level of that in D, and at the same time traces of the gum are perceptible in the water of D. Quantitative measurements of the phenom- enon by means of Dutrochet's apparatus are unsatisfactory, because an animal membrane only hinders but does not stop the passage of the dissolved substance. A great advance in the direction of measurement was made by Traube, who discovered that a pellicle formed by precipitation when two solutions are brought together is permeable by water but not by cer- D FIG. ice. a i n other substances, including the reagents by whose action the pellicle was formed. By the use of such membranes deposited on the inner surface of a clay cylinder which was afterward filled with different solutions and im- mersed in water, Pfeffer deduced the following conclusions : 1. That the osmotic pressure depends on the nature of the substance. With one per cent solutions the following pressures were observed. INVESTIGATION OF DE VRIES. 235 CM. OF MERCURY. Cane Sugar 47.1 Dextrin 16.6 Niter 178 Potassium Sulphate 193 Gum 7.2 2. That the pressure is proportional to the concentration. 3. That the pressure for a given concentration increases regularly with the temperature. 4. That the pressure is dependent on the nature of the membrane. This last conclusion is untrustworthy, since it may be shown to involve a contradiction of the Law of conservation of energy. The experimental result may be accounted for by supposing either that the membrane yielded under the considerable pressures to which it was subjected, or that the membrane was not quite impervious to the solvend. 193. Investigation of de Vries. Pringsheim discovered, in 1854, that the protoplasmic contents of an organic cell contract when the cell is brought in contact with a salt solu- tion, but that the cell wall retains its form. The explanation of the phenomenon was later found to be in the fact that the protoplasm is contained in a membrane which is perme- able by water, but is imperyious to most dissolved substances. When, on the other hand, such a cell is placed in a solution having an osmotic pressure equal to or less than that of its own contents, the protoplasm will remain in contact with the cell wall. Thus, by observing the behavior of the con- tents of the cell, it will be possible to test whether any given solution has a greater or less osmotic pressure than the cell, and to compare different solutions. By means of such a method de Vries verified the result previously obtained by Pfeffer, namely, that the pressure is 236 SOLUTIONS. proportional to the concentration of the solution, always supposed dilute, and determined solutions which contain quantities of substances dissolved in the proportion of their molecular masses exhibit equal osmotic pressures. By a similar method Bonders and Hamburger showed that solutions which exerted equal pressures at a lower tem- perature would also exert equal pressures at a higher tem- perature, i.e. that the increase of pressure was independent of the nature of the dissolved substance. Van 't Hoff has since shown, by the principles of thermodynamics, that the pressure is proportional to the absolute temperature. 194. Analogy of Osmotic to Gaseous Pressure. The laws of osmotic pressure, found by different investigators, may be summarized as follows : 1. The pressure is proportional to the concentration, i.e. inversely proportional to the volume in which a given quan- tity of the substance is dissolved. 2. The pressure is proportional to the absolute temper- ature. 3. Quantities of the dissolved substances which are pro- portional to their molecular masses exert the same pressure at the same temperature. The analogy of these laws to those of gaseous pressure is seen to be complete. The first corresponds to the law of Boyle, the second to the law of Charles, the third to the law of Avogadro. (See Art. 255.) This analogy was first pointed out and insisted on by Van 't Hoff, who showed that the formula for a gas, pv = was applicable without change in the value of R. (See Art. 138.) VAPOR PRESSURE OF SOLUTIONS. 237 The laws of osmotic pressure in dilute solutions may accordingly be reduced to the following simple statement : Dissolved substances exert the same pressure in the form of osmotic pressure that they would if gasified at the same temperature without change of volume. 195. Vapor Pressure of Solutions. The effect of a dis- solved substance on the vapor pressure of a liquid was first observed by the rise it produced in the boiling point. Since the vapor pressure of the pure liquid at the higher temper- ature would be greater than that of the atmosphere, it fol- lows that the presence of a dissolved substance lowers the vapor pressure. The law of this change may be concisely stated in a formula first proposed by Raoul, n where p is the vapor pressure of the pure liquid, p' is the vapor pressure after the addition of the substance, c is a constant nearly unity for dilute solutions, n is the num- ber of molecules of the dissolved substance, and N the number of molecules of the solvent. The formula 4 stated in words is, that the lowering of the vapor pressure bears the same ratio to the whole pres- sure that the number of molecules dissolved does to the total number. 196. Application to the Determination of Molecular Masses. By a simple transformation, equation 4 may be put in a form useful for the calculation of molecular masses. 238 SOLUTIONS. Let N= number of molecules in solvent, n= " " " " dissolved substance, M = mass of a molecule in solvent, /*= " " " " " dissolved substance, M = mass of solvent, m = " dissolved substance, then (5) \ M m n I A* Now, accepting c as unity, equation 4 yields (6) _=_!_ w pp' Substituting the values of (5), MIL__ p' Mm p p 1 whence, solving, _M p'm by which the molecular mass of the dissolved substance may be found when the lowering of the vapor pressure is known. By this means the knowledge of molecular masses has been much extended, as all methods previously used were appli- cable only to such substances as could be gasified without change. The law of Raoul does not hold for aqueous solutions of salts, strong acids, and bases, as it is found that the lower- ing of the vapor pressure in these solutions is considerably greater than is required by equation 4, as if the sol vend FREEZING POINT OF SOLUTIONS. 239 had been split up into molecules smaller than usual, i.e. dis- sociated. This hypothesis is corroborated by a character- istic peculiarity of such solutions, namely, that of being readily decomposed by the passage of an electric current, and also by certain peculiarities in their chemical reactions. 197. Freezing Point of Solutions. For reasons that are easily shown the diminution of the vapor pressure of a liquid by the presence of a substance in solution has the effect of lowering the freezing point. The freezing point is, by definition, that temperature at which the solid and its liquid are both in equilibrium with its vapor. Suppose, now, a small quantity of some salt be dissolved in the liquid, lowering its vapor pressure. The equilibrium between the solid and the vapor will be destroyed. The solid will begin to distil over into the solution at the expense of the usual amount of the latent heat of fusion, and the temperature falls to some point where the solid and the solution are again in equilibrium with the vapor. A good example of this process is seen in the familiar freezing mixture consisting of snow and salt, in which the fall of temperature is about 22 C., before equilibrium is again established. A theoretical calculation of the depression of the freezing point has been worked out by Van 't Hoff. Let T= the freezing point on the absolute scale, AT = the lowering of the freezing point, n = number of molecules of the dissolved substance, N= " " " " " solvent, p = molecular mass of dissolved substance, L = latent heat of fusion per unit mass, R = the constant in pv = 240 SOLUTIONS. then Van 't Hoff's formula may be written n -^r N The results obtained by this formula are in substantial agreement with observation, at least for dilute solutions, but it is not applicable to electrolytes in which the depres- sion is greater than would be given by the molecular mass of the substances. 198. Humidity. Humidity, or the state of the air as regards moisture, is defined as the ratio of the mass of aque- ous vapor actually present to the quantity required for satu- ration. The ratio of the masses is obviously the same as that of the densities, and also that of the pressures, if it be assumed as a sufficient approximation, that aqueous vapor obeys Boyle's Law. A great humidity is a source of considerable bodily dis- comfort either in summer or winter, but a very exact knowl- edge of this state of the air is of little importance. The humidity may be determined in three different ways, known respectively as the Chemical, the Dew Point, and the Wet and Dry Bulb methods. 199. The Chemical Hygrometer. In the chemical method a known volume of air is passed through a series of tubes containing pumice moistened with sulphuric acid, which absorbs the water vapor. The mass of the water thus col- lected, divided by the amount necessary for saturation of the same volume at the observed temperature, gives the humidity. 200. Dew Point Hygrometers. The temperature at which the aqueous vapor of the air begins to condense is called the dew point. It depends only on the amount of vapor actually present. DEW POINT HYGROMETERS. 241 A dew point hygrometer is essentially a device for cooling a body till dew or hoar frost begins to collect, at the same time indicating the temperature. A simple form of dew point hygrometer invented by Dines is shown in Fig. 167. R is a reservoir filled with ice-water, which may be allowed to flow through the pipe D into a chamber, E, and escape at A. This chamber is covered with a piece of black glass, B, and contains a delicate thermometer, T. On opening the cock (7, the passage of the cold water through E lowers the FIG. 167. temperature of B until moisture begins to collect on its sur- face. The thermometer is then read and the flow of water stopped. The temperature of the glass soon begins to rise and the moisture to evaporate. Just as it disappears from the surface the thermometer is again read and the mean of both readings taken as the dew point. It remains to show how the humidity may be calculated from the dew point. By the law of Dalton, given in Art. 184, the pressure of the aqueous vapor in the air may be regarded as the same as if the air were absent. Hence, the 242 SOLUTIONS. only pressure which keeps the vapor next to the cold body B at the point of saturation is the pressure of the aqueous vapor in the uncooled air. And, vice versa, the pressure of saturated water vapor at the temperature of the cold body is that of the uncooled air. So that finally the humidity may be found by dividing the pressure of water vapor at the dew point by the pressure at the observed temperature of the air, their values being taken from the steam table on p. 216. 201. Wet and Dry Bulb Hygrometer. The instrument most commonly used at meteorological stations for determin- ing the humidity consists of two similar delicate thermome- ters fastened side by side on a stand (Fig. 168). The bulb of one of the thermometers, W, is covered with a piece of muslin which is kept constantly moistened by means of a cotton wick attached to the muslin and dip- ping into a small vessel filled with water. If the surrounding air is not saturated, evapo- ration will occur from the wet bulb, and the consequent abstraction of heat will lower the temperature below that of the surroundings. Thus, after a time, the wet thermometer will indicate a temperature constantly below that of the dry one by an amount depending on the humidity, since the rate of evaporation is determined by the amount of aqueous vapor present in the atmosphere. In order to deduce the dew point from the readings of the thermometers, recourse is had to a set of tables which have been constructed from a long series of simultaneous observa- tions on the wet and dry bulb thermometers and a dew point hygrometer. W FIG. 168. EXAMPLES. 243 In dry weather, or when the air is quite still, the dew point, as deduced from observation of the wet and dry bulb thermometer, is usually a little high, probably because these conditions favor the elevation of the temperature of the wet thermometer by radiation from surrounding objects. EXAMPLES. 1. 2.3 liters of hydrogen under a pressure of 78 cm. of mercury, and 5.4 liters of nitrogen at a pressure of 46 cm. were introduced into a vessel containing 3.8 liters of carbon dioxide under a pressure of 27 cm. What was the pressure of the mixture ? Ans. 140 cm. of mere. 2. The temperature of the dew point is observed to be 9.2 when the temperature of the air is 14.5. What is the humidity? Ans. 0.71. 3. When the humidity is 58 % at the temperature 23, what will be the dew point? Ans. 14.3. 4. Calculate the weight of 15 liters of air at 20 saturated with water vapor, when the hight of the barometer is 75 cm. Ans. 17.66 gms. CHAPTER XIII. TRANSFERENCE OF HEAT. 202. Transference of Heat. There are three ways by which heat may be transferred from one point to another. 1. When heat is conveyed by the motion of portions of matter with which it is associated, the process is termed con- vection. It is illustrated in the familiar method of heating buildings by currents of hot air. 2. When two parts of a body differ in temperature, and heat is transmitted from the warmer to the colder portions without perceptible motion of matter, the process is called conduction. If one end of a bar of iron be placed in a furnace and the other in a vessel containing ice properly screened from the furnace, the ice will be slowly melted by the heat which is conducted along the bar. 3. When a hot body gives rise to a system of waves which traverse with a definite velocity space not filled with ponderable matter, and afterward warm a second body, the process of transfer is termed radiation. At no intermediate point, however, is the heat perceptible as such, but as a system of waves transmitted with the velocity of light, and afterward absorbed by the second body, giving rise to the observed thermal effects. The heat which reaches us from the sun is transmitted in this way. 203. Condition of Steady Flow in Conduction. When a constant source of heat is applied to one part of a body, a general rise of temperature at other points will at first be noticed, but the temperature will finally become stationary DEFINITION OF THERMAL CONDUCTIVITY. 245 at values depending on the distance of the point considered from the source of heat. Under these circumstances there is said to be a steady flow of heat from regions of higher to those of lower temperature. 204. Definition of Thermal Conductivity. Suppose two surfaces drawn in a body, on each of which the temperature has a constant but different value, say 6 l on the first and 2 on the second. Then the quantity of heat Q which will pass in a time, T, through a small area, &, on each of these iso- FIG. 169. thermal surfaces, where the distance between them is c?, is found by experiment to be proportional to d and dependent on the nature of the substance. If Q be written (D the constant k is called the conductivity. Its value, as given by equation 1, is \ / //i 7i~\ Tri If heat is measured in calories, the dimensions of k are r , 1 = [J*fAZ] [M] . PA ram] [LT] * or, in C. G. S. units, 1 gm. 1 cm. 1 sec. 246 TRANSFERENCE OF HEAT. The conductivity of a substance is found in general to diminish with increasing temperature. In some experiments of Principal Forbes upon a square -iron bar measuring 1J inches on a side, the conductivity was found to decrease from 0.207 at C. to 0.124 at 275. The thermal proper- ties of metals in this and several other respects are remark- ably like their electrical properties, as will be more fully shown hereafter. 205. Good and Bad Conductors. A certain knowledge of the conductivity of a body at moderate temperatures may be obtained by means of the temperature sense alone. It is well known that a piece of metal feels much colder than a piece of cloth at the same temperature. The explanation is that the metal conducts the heat away from the hand more rapidly than the cloth does. The low conductivity of wool makes it a suit- able fabric for the protection of the body in winter, since its heat will be carried off slowly, even when the outer surface is kept at a low temperature. Cotton and linen, which are better conductors, are for this reason more suitable for hot weather. A rough comparison of the conducting powers of metals may be made by twisting together two or more wires, and attaching to them a series of balls by means of wax, as in Fig. 170. If a lamp be ap- plied at one end, the wax will be melted as the heat flows along the wires, and the balls fall off one by one. The better conductor will ultimately melt the wax at the greater distance from the source of heat. The conductivities, as defined by equation 2, will be very nearly as the squares of these distances. It is, however, to be GOOD AND BAD CONDUCTORS. 247 noticed that the first ball does not necessarily drop soonest on the bar from which the most balls are finally ^melted. The velocity of the temperature wave will evidently be greater in proportion as the specific heat is smaller. The property upon which the rate of change of temperature during the variable stage depends has been called the diffusivity, and is defined by k (3) * = -, where s is the specific heat and Jc the conductivity. TABLE OF THERMAL CONDUCTIVITIES OF SOLIDS. SUBSTANCE. TEMPERATURE C. CONDUCTIVITY. Unit 1 gm ' "" 1 cm. 1 sec. Aluminum .... Antimony .... Bismuth to 100 to 100 to 100 0.3435 to 0.3619 0.0442 to 0.0396 0177 to 0164 Brass Copper to 100 15 0.2041 to 0.2540 0.713 to 0.9996 Flannel 10 0.0000355 Glass 0.0018 to 0021 Iron . . . 15 149 to 201 Lead. to 100 0836 to 0764 IVTarble 0012 to 00018 IVtercurv to 50 0148 to 0189 Phosphor Bronze . . Silver Snow 15 10 0.4152 1.0960 00072 Tin to 100 0.1528 to 0.1423 Wood 0.00026 to 00059 Writing Paper . Zinc 0.00012 3056 Zinc 15 2545 248 TRANSFERENCE OF HEAT. 206. Trevelyan Experiment. The thermal properties of metals are exhibited in a somewhat unusual way in an occur- rence first observed by Trevelyan, when a hot soldering iron was laid against a piece of lead. The experiment is best exhibited by means of the apparatus shown in Fig. 171. The piece .A, called the rocker, is made of brass, and grooved on one face so as to leave two projecting edges, shown at A!. If it is raised to a temperature of about 200 C. and laid on a bright cylinder of lead, B f , in the position shown, it will continue to oscillate back and forth from one FIG. 171. edge to the other for a long time, emitting a definite musical note. The explanation is that when one of the hot edges meets the surface of the lead, which has a large coefficient of expansion and a relatively small conductivity, a lump is raised with sufficient rapidity to tilt the rocker over upon the other edge, whence at once, in a similar manner, it is thrown back. The continual repetition of this process gives rise to a series of waves in the air, which produce the sensation of sound. 207. Davy's Safety Lamp. If a piece of wire gauze be lowered over a gas flame, the flame will not pass through the meshes. Similarly, if the gas be first allowed to pass through the gauze and is then lighted from above, the flame CONDUCTION IN CRYSTALLINE MEDIA. 249 will remain entirely on the upper side. The explanation is that the heat is conducted away by the metal wires so that the temperature of the gas on the side opposite to the flame does not rise high enough for ignition. This principle was utilized by Davy in the construction of a safety lamp designed to prevent the explosion of fire-damp, a dangerous mixture of gases which often collects in coal mines. By enclosing the miner's lamp in a cage of wire gauze, only so much of the explosive gas as passes through the gauze will burn, while the large body cannot ignite. 208. Conduction in Crystalline Media. In anisotropic bodies the conductivity is different in those directions in which its other properties vary. If, for instance, a slice be cut from a piece of calcite parallel to the crystallographic axis, the varying conductivity may be exhibited by covering the surface with paraffine and inserting a heated rod in a hole previously bored through the FIG center. The paraffine will be melted for a distance about the hole (Fig. 172) to a line which marks the temperature of fusion of paraffine. The form of the isothermal, thus indicated, will be an ellipse. 209. Conductivity of Liquids. The difficulties in the way of the exact determination of the conductivity of liquids are considerable. If the lower strata are heated, the change in density sets up convection currents which entirely mask the property sought. But even if these are eliminated, it still is impossible to tell how much of the heat may have been transmitted by molecular diffusion. The following table shows the observed values of the conductivity of cer- tain liquids in C.G.S. units. 250 TRANSFERENCE OF HEAT. SUBSTANCE. TEMP. CONDUCTIVITY. Alcohol 13 0.000545 Bisulphide of Carbon . . " 0.000266 Ether 0.000378 Glycerine " 0.000637 Oil of Turpentine ..." 0.000325 Water 18 0.001245 to 0.001358 210. Conductivity of Gases. In the case of gases, direct radiation increases the uncertainty which attaches to the determination of the thermal conductivity. Experiment indicates that its value is small. Poor conductors, such as felt, fur, or down, owe the property to the presence of in- terstices filled with air; for when these materials are com- pressed, to reduce the air cavities, they become much better conductors. The observed values for a few gases are as fol- lows: SUBSTANCE 756 o CONDUCTIVITY. Air O.OQ00516 Carbon Dioxide 0.0000273 Hydrogen 0.00033 Marsh Gas 0.000065 The high value found for hydrogen is consistent with a num- ber of other properties common to the elements called metals. 211. Cooling by Radiation. When a hot body is sus- pended by a poor conductor, it is found to lose heat rapidly by radiation, at a rate depending on the nature of the body and the condition of its surface, being notably greater for a blackened than for a polished one. The first experiments on the rate of cooling of a body seem to have been made by Newton, who expressed his con- clusion in the statement that the rate of cooling was propor- tional to the difference of temperature between the body and the enclosing chamber. COOLING BY RADIATION. 251 If R be the rate of loss of heat, and t v t z the respective temperatures of the radiating chamber, Newton's Law may be expressed (4) E = A(t 1 -t z ). This formula may be taken to represent the facts fairly well when t^ 2 does not exceed a few degrees, but the departure from the truth is very noticeable when the differ- ence of temperature is as much as 40 C. The matter was later investigated by Dulong and Petit in an elaborate series of experiments on the cooling of ther- mometers in an exhausted chamber. As the result of their study they proposed the formula (5) E=k ('*>), where Jc is a constant depending only on the nature of the material and the surface, and a is a number having the value 1.0077 if t is measured on the Centigrade scale. This law of Dulong and Petit has been accepted as expressing with approximate accuracy the rate of cooling within the range of their experiments, which was from 20 to 240 Centigrade. These experiments, which had been conducted with the utmost care, were reviewed in 1879 by Stefan, who found that the formula (6) R=c(ti-tj) fitted the observations more closely than the one given by the original investigators. Stefan's formula has since been deduced from theoretical considerations by both Boltzman and Galitzine. 212. Definition of Emissivity. The emissivity of a sur- face is defined as the quantity of heat lost per second, per square centimeter, per degree difference of temperature be- 252 TRANSFERENCE OF HEAT. tween the body and the walls of its enclosure under standard conditions, e.g. immersion in air at atmospheric pressure. Determinations of emissivity, undertaken by M'Farland and by Bottomley, do not appear to support Stefan's Law. 213. Calorimetry by the Method of Cooling. A method of calorimetry depending on the rate of cooling may be con- veniently used in the case of liquids, especially with those which are obtainable only in small quantities. The sub- stance to be examined is heated and placed in a thin copper vessel having a blackened surface, and supported without direct contact within a larger vessel, which is kept at a con- stant temperature by means of a large bath of water. The time occupied by the smaller vessel in cooling, through a definite range, is observed and noted. The liquid is then removed and water at the same initial temperature substi- tuted. As the external surface of the vessel is the same in both cases, the rate of cooling will be the same as before. Hence if the time of cooling through a given range is observed, the quantity of heat lost in each case may be taken as proportional to the time. Thus, let T be the time it takes the substance to cool through a range 9 l 6^ and T the time occupied by the water in cooling through the same difference of temperature. Also, Let m = mass of the liquid, m' " " " water, s specific heat of the liquid, Then ms __ T ^V~"^' or Tm 1 , ABSORPTION. 253 214. Absorption. It has already been remarked in Art. 202 that a body losing heat by radiation is a source of dis- turbance, giving rise to a train of waves which are propa- gated with the velocity of light through space not filled with gross matter. Since, in fact, these waves differ in no respect from those which produce the sensation of light, except that their wave-length may be greater, the phenomena and laws of their propagation may best be studied under the general subject of Light. These long waves are sometimes spoken of as radiant heat, but they are not heat in any strict sense of the term while they are in process of transmission ; but if they fall upon some substance which is not able to pass them on unimpaired, they are transformed into heat, and the process of the change is called absorption. In order to make clear what occurs in this case, it will be necessary to borrow an illustration from an analogous phe- nomenon in sound, known as resonance. When an elastic body, such as a column of air or a tuning fork, is capable of vibrating in one particular period, it will not be set into sen- sible vibration unless it receives a disturbance timed to that period ; so that if several systems of waves of various periods fall upon a tuning fork, those waves whose periods are dif- ferent from that of the fork pass on unaltered, but that sys- tem which is identical in period with the fork is absorbed, and the fork may be heard to emit the note peculiar to itself. If the various systems of waves radiated by a heated sub- stance fall on another body, those waves which it would emit when heated are absorbed, but most of the others are transmitted. The body is said to be athermous to the waves absorbed, but diathermous to those transmitted. The corresponding terms applied to light waves are opaque and transparent. 254 TRANSFERENCE OF' HEAT. It may be mentioned that rock salt is one of the most diathermous bodies yet examined, but further discussion of this subject will not be presented at this point, since many of the instruments used to measure radiation make use of electrical principles not yet explained. 215. Provost's Theory of Exchanges. When an isolated system of bodies is left to itself, it is known that the mem- bers of this system will ultimately, as a result of conduction or radiation, reach a common temperature. The question, whether under these circumstances the radiation has ceased or not, was first answered by PreVost in the negative. If, he reasoned, a cold body were to be placed in the midst of the system, it would immediately begin to grow warm by the radiations received ; or if one of the bodies of the system were removed and placed among colder ones, this body would begin to lose heat. Now, a cold body has not the power of acting on another at a distance so as to make it begin to emit radiations. Therefore, a body is always emitting heat at a rate depending on its temperature, regardless of the presence of other bodies, and thermal equilibrium is maintained in the system considered only by each body receiving heat from the others at the same rate that it loses it by radiation. This theory of exchanges, as it is called, was implicitly assumed in the statement of Dulong and Petit's Law in Art. 211. 216. Effect of Radiation on Thermometers. It follows from the considerations just explained that the reading of a thermometer may be greatly affected by the radiation to or from surrounding bodies, and that in order to obtain the true temperature of the air out of doors, special precautions are necessary. If the bulb be coated with polished silver, the effects of radiation may be greatly lessened, for the absorp- CROeKES 1 RADIOMETER. 255 tion of such a surface is only -fa of that of lampblack. A better arrangement, however, is that suggested by Joule, namely, to surround the thermometer by a long vertical copper tube, open at the top but closed by a cap at the lower end. If the tube and the thermometer are at the same tem- perature, the radiation between them will have no effect on the reading. But there may still be some discrepancy be- tween the temperature of the air in the tube and that out- side. In that case convection currents either up or down the tube would arise on removing the cap. To detect the presence of such currents a wire helix is suspended within the tube in such a way as to indicate their direction. The copper tube is further supplied with a jacket through which warm or cold water may be passed until the column of air is in equilibrium. If the thermometer then shows a stationary reading, it may be ac- cepted as the temperature of the air. 217. Crookes' Radiometer. An inter- esting phenomenon attending radiation and absorption in an attenuated gas was dis- covered by Crookes in 1873. It may be exhibited by means of the instrument shown in Fig. 173, called a radiometer. B is a bulb containing gas at a low pressure, say something like a centimeter of mercury. Attached to a spindle, **', and revolving with it are four vanes, FJ of mica, blackened on one side and silvered on the other. When this apparatus is placed in the sunshine, or where the radiation from a hot body falls upon it, the vanes begin to revolve in such a direction that the blackened faces move away from the source of heat and the polished ones FIG. 173. 256 TRANSFERENCE OF MEAT. toward it. The explanation is to be found in the fact that the blackened surface absorbs more heat than the bright one and acquires a higher temperature. In consequence the molecules which strike the vane bound off from this side with a greater velocity than from the other, at the same time giving the vane an impulse in the opposite direction. If the gas is too dense or too rare, the vane will receive no motion. When the instrument is placed before a very cold body, the radiation from the blackened face is more rapid, its temperature falls, and the vane will rotate in the opposite direction. 218. Dew. When small bodies on the surface of the earth are exposed to a clear sky after sunset, the loss of heat by radiation soon cools them below the temperature of saturation of the surrounding air. The moisture which is then deposited is called dew. Since, however, the tempera- ture cannot fall lower than the dew point until all the mois- ture has been abstracted from the atmosphere, it is evident that the presence of aqueous vapor in the air really serves to prevent the freezing and destruction of plant life during the night, provided the dew point is above C. Any inter- posed body, such as the clouds, by the theory of exchanges, returns a part of the heat lost by radiation and greatly diminishes the deposition of dew. The continual stirring of the air on windy nights is also unfavorable to the forma- tion of dew, since a fresh supply of heat is thus brought to the radiating body. EXAMPLES. 257 EXAMPLES. 1. A tank 6 meters long and 4 meters wide is filled with water which is covered with a layer of ice 6.6 cm. thick. If the temperature of the air is 11 , and the coefficient of conduction of ice be taken as 0.0023 gin. / cm. sec., how much heat will be transmitted through the ice per hour? Ans. 3.3(10) 6 cal. 2. The top of a steam chest containing steam at atmospheric pressure consists of a slab of stone 61 cm. long, 49 cm. broad, and 9.9 cm. thick. The top being covered with ice, it was found that 4.8 kilos of ice were melted in 29 min. What is the conductivity of the stone? Ans. 0.0072 gm. / cm. sec. 3. Water is boiled at atmospheric pressure in an iron vessel having a heating surface of 2.34 square meters and a thickness of 1.16 cm. How much water will be evaporated per hour if the surface exposed to the fire is kept at 280 and the conductivity of iron be assumed as 0.164 gm. / cm. sec. Ans. 4(10) 6 gms. 4. How much heat would be lost per square decimeter per minute by a man clothed in a fabric 0.3 cm. thick, having a conductivity 1.22(10)~ 4 gins. /cm. sec., if the temperature of the air is 5 and the temperature of the body is 30. Ans. 60 cal. CHAPTER XIV. THERMODYNAMICS . 219. Mechanical Production of Heat. Of the discovery that heat could be produced by mechanical means nothing is known, but the knowledge of the fact must have been in the possession of the race from the earliest times. Examples of such production are afforded by many familiar operations. For instance, the rubbing of a piece of steel against a rapidly revolving grindstone generates sufficient heat to raise the abraded particles of the metal to the temperature of ignition. The friction of a match against a rough surface likewise develops heat enough to ignite the phosphorus. A piece of lead may be rendered quite hot by simply striking it with a hammer, and the temperature of the water below a waterfall is raised an appreciable amount by the impact at the end of the fall. In the case of machinery which is run at a high speed, it is, in fact, very difficult to prevent the generation of a prejudicial quantity of heat at the journals. In all the examples mentioned the production of heat involves the expenditure of a certain amount of work ob- tained either from the action of some external force or from the diminution of the kinetic energy by impact. 220. Fire Syringe. Another interesting illustration of the generation of heat by the expenditure of work is afforded by the fire syringe. This instrument consists of a stout glass tube, A (Fig. 174), furnished with a close-fitting pis- ton, B. A small quantity of carbon disulphide is intro- duced into the tube, filling the chamber with a mixture of THEORIES CONCERNING THE NATURE OF HEAT. 259 air and vapor. If the piston be suddenly pushed down, suffi- cient heat may be developed by the compression to ignite the vapor, producing a bright flash. Likewise, when a gas is allowed to expand, doing work on outside bodies, its temperature falls on account of the disappearance of heat. 221. Theories Concerning the Nature of Heat. Two rival theories concerning the nature of heat existed side by side from the time of the ancient Greeks until the middle of the present century. According to one, heat was a highly elastic and self-repellent fluid termed caloric, which pervaded the interstices of all bodies. In the other doctrine, heat was ascribed to the rapid vibration of the molecules of a body. The first of these hypotheses was the generally accepted doctrine until shortly after the beginning of the present century. The history of its final overthrow marked a great advance in science, for so long as heat was accorded a material exist- ence, the establishment of the doctrine of conservation of energy was impossible. 222. The Caloric Theory. The fundamental property of heat demanded by the fluid theory was that it be indestruc- tible and uncreatable. When it was added to bodies they became warmer, and when it was abstracted they grew cooler. To account for the different thermal capacities of substances it was supposed that they had different attractions for caloric, and that some would absorb a greater quantity than others in rising through the same difference of temperature. The expansion of bodies was regarded as the natural consequence of the absorption of this self-repellent fluid. To account for 260 THERMODYNAMICS. the disappearance of heat at the change of state, Black sup- posed that caloric could exist not only in the free state, but that in such a case as the melting of ice, it became inactive or so hidden that it could not be detected by the thermome- ter, whence he termed it latent, a name which is still retained. Conduction was explained by supposing that the fluid, caloric, in consequence of the mutual repulsion of its parts, flowed from high temperatures to low, just as water runs from higher levels to lower. To explain the heating of bodies by percussion the calor- ists assumed that some of the fluid was squeezed out. In the case of friction it was supposed that the portion of the body which was reduced to powder by the abrasion of the surface had a less capacity for heat, and in this treatment gave up some of its caloric. This important postulate might easily have been submitted to experiment, but the calorists allowed their theory to rest upon bare speculation. 223. Experiments of Rumford. The first experiments for the purpose of determinating the nature of heat were undertaken by Count Rumford at the close of the last cen- tury. While engaged in boring a brass cannon he was much impressed by the amount of heat developed in the process, and, seeking to obtain exact knowledge of the matter, he prepared a special experiment, in which a blunt steel borer was pressed against the bottom of a brass cylinder provided with a thermometer to indicate the temperature. After 960 revolutions it was observed that the temperature had risen 70 F,, though the mass of metal abraded was only ^1^ of the whole. It seemed to Rumford improbable that so large a quantity of heat should be given up by the diminution of the thermal capacity of so small an amount of metal. Further experiments on massive and pulverized brass show- DAVY'S EXPERIMENT. 261 ing that the specific heat in each case was sensibly the same, Rumford rejected the hypothesis that heat was liberated by the change in the thermal capacity of bodies incident to their change of form. But if heat is a fluid, it is necessary to admit that an unlimited quantity can be generated by an insulated system of bodies, for in the experiment there had appeared no sign of diminution or exhaustion. Regarding such a conclusion as inadmissible, Rumford was led to the view " that the only thing capable of being excited and communicated in these experiments is motion." 224. Davy's Experiment. The crucial test of the caloric theory was devised by Davy in the following year, 1799. By simply rubbing two blocks of ice together he showed that they could be entirely converted into water. Now, since this change of state admittedly involves the absorption of considerable heat, the hypothesis that caloric is released by the operation is clearly disproved. 225. Mechanical Equivalent of Heat. The now univer- sally adopted theory that heat is the. kinetic energy due to the irregular motion of the molecules of a body was received with small favor during the first quarter of the century ; from that time on, however, it gained increasing support, and by 1840 its acceptance had become general. The chief contributor to this result was Dr. Joule of Manchester, who, assured that heat was a form of energy, undertook in the year mentioned to find out the numerical relation between the units of heat and work. His measurements were made upon the eleva- tion of temperature of a known mass of water when churned by the fall of a weight. The apparatus consisted essen- tially of a copper vessel, AB (Fig. 175), containing a paddle attached to a vertical roller, /, which was made to revolve 262 THERMODYNAMICS. by the descent of the weights U, U. The paddle itself (Fig. 176) was made of brass and consisted of eight sets of arms, a, which worked between stationary vanes, >, in the vessel, FIG. 175. so that the water was prevented from revolving bodily with the paddle. The roller/ could be detached from the paddle at pleasure by removing a pin, ^?, and the weights wound ! up without agitating the water. Joule's experiments extended over a number of years, and showed, when corrected as nearly as pos- sible for such sources of error as could not be eliminated, that 772 foot-pounds of work at Manchester FIG. 176. jwould raise the sec/ temperature of 1 pound of water at 60 F. through 1 F. This equivalent is denoted by J. Observing that its dimensions are those of work per unit mass, per degree, T 770 foot-pounds /i-ico/-iA\7 er s t/ = I 1 22 ^= Tr.-LOcM-LU) - ~* n s*t ' pound 1 P . gm. 1 C. ROWLAND'S DETERMINATION OF J. 263 226. Rowland's Determination of J. The determination of Joule remained the adopted value of the mechanical equivalent until 1879, when Rowland undertook the most careful and complete study of this constant which has ever been made. The method was similar to that employed by Joule, but the details were improved in several particulars. The paddles, more elaborate in design, were turned by a steam engine, thus securing a more rapid change of tem- perature, 35 C. per hour, instead of 0.62 C., obtained by Joule. The mercurial thermometer, which might introduce errors of one or two per cent, was replaced by the air thermometer. After all corrections had been applied, Rowland's experi- ments showed J= 4.187(10)7 at 15.8 C., gm. 1 C. as compared with 4.159(10) 7 , as found by Joule at the same temperature. The principal part of this difference is to be ascribed to want of agreement between the mercurial and the air thermometers. It was by this series of experiments that Rowland established the fact that the specific heat of water is not constant. (See Art. 153.) 227. Other Determinations of J. The value of the me- chanical equivalent of heat has been determined by other experimenters in a variety of ways both direct and indirect. The most reliable of the latter methods is to measure the heating of a wire by- the passage of a known electric current. There is a slight discrepancy, as may be seen from the fol- lowing table, between Joule's and the electrical method, which has not yet been accounted for. 264 THERMODYNAMICS. MECHANICAL EQUIVALENT OF HEAT. METHOD. EQUIVALENT AT 15 C. OBSERVER. FT. PDS. ERGS LB. 1 F. GM. 1 C. ( Corrected for mer- ) Fluid Friction 775 4.172(10) 7 Joule \ curial thermometer, j u a 778.3 4.189(10) 7 Rowland 11 n 776.6 4.1SO(10) 7 Miculescu Electrical 780.2 4.199(10) 7 Griffiths u 779.7 4.197(10) 7 Shuster 228. The Two Laws of Thermodynamics. The processes of reasoning employed in the discussion of problems in heat are founded on two fundamental principles relating to the conversion of heat into work, and known as the first and second laws of thermodynamics. The first law asserts that when heat is transformed into work, or work into heat, the quantity of work is equivalent to the quantity of heat. This relation, as has been seen, was first established by Joule, and may be written (i) W=Jff=Q, where W is the work measured in ergs, J the mechanical equivalent of heat, If the quantity of heat measured in calo- ries, and Q the same quantity measured in ergs. In other words, the first law declares the , identity of heat with energy and brings it within the general law of conservation of energy. The second law of thermodynamics asserts that heat can- not of itself pass from a cold to a hot body, or more formally, it is impossible, by means of inanimate material agency, to obtain work from any portion of matter by cooling it below the temperature of the coldest of surrounding objects. ADIABATIC EXPANSION. 265 There is no a priori reason for denying that work can be derived by using up the heat of a single body. For, if the ultimate material particles, to the motion of which the phe-. nomenon of heat is ascribed, be supposed large enough to be guided and controlled, it would not be impossible to pre- pare a machine which should by a suitable train of mechan- ism transfer the energy of these particles to another body and convert it into work. The second law thus reduces to the assertion that the molecules of a body are so small that it is beyond our power to manipulate them. 229. Adiabatic Expansion. A body is said to undergo an adiabatic change when its condition is altered without its gaining or losing heat to other bodies. In the case of the adiabatic expansion of a gas the relation between the pres- sure and the volume of a gas may be expressed by (2) pv* = constant, where 7 is the ratio of the specific heats. (See Art. 240.) If this equation be plotted on the P V diagram, the adiabatic line AA (Fig. 177) will be more inclined to the horizontal than the isothermal J/, or, what amounts to the same thing, if a gas be compressed from v 1 to v 2 , without permitting gain or loss of heat, the pressure rises faster than when the tem- perature remains constant. Since for a given change in volume the greater change in pressure corresponds to a FIG. 177. 266 THERMODYNAMICS. greater change in temperature, it follows that the gas is heated by compression and cooled by expansion. 230. Work Done in Compressing a Fluid. Suppose a cylinder to be filled with a fluid and fitted with a freely moving piston. In order to calculate the work done in moving the piston a distance, #, call the area of the piston A, and the average pressure during the motion p, then the work done will be (3) W=pA'X- ) but Ax equals the change in volume, say u. Substituting, (4) W=pu, a result which is independent of the area of the piston and of the volume and form of the vessel. The work done in changing the volume of a fluid may be conveniently represented on the pressure-volume diagram as fol- lows. Suppose the initial state of the body is represented by the _ point A (Fig. 178), and that its condition is altered in any man- ner until it reaches the state de- noted by the point O. If B represent an intermediate stage very near to A, the average pressure for the small change is given by o ba FIG. 178. and the diminution of volume by ba. Hence, the work done will be given by (5) W= %(Aa + Eb) ab = area b BAa. CAENOTS CYCLE. 267 Now, since the whole change from A to G may be regarded as made up of the sum of such small changes as that from A to B, the work done in changing the body from the state A to the state G, along the path BG, is proportional to the area IBCc. It will be convenient to observe a convention of signs, so that this area shall be described by a negative rotation when work is done on the body, and by a positive rotation when the body does work. Thus, CcaA will represent work done on the body, and GAac work done by the body. 231. Carnot's Cycle. One of the earliest investigators of the manner in which work is produced by heat was Sadi D T' Cold i Hot FIG. 179. Carnot. His most noteworthy contribution to this branch of science was a new method of reasoning, in which a body was made to pass through a cycle of operations. For simplicity suppose that the working substance is a gas, -A, enclosed in a cylinder (Fig. 179) furnished with a piston, 268 THERMODYNAMICS. which, with the side walls of the cylinder, is absolutely impermeable to heat, but that the bottom of the cylinder is a perfect conductor. Suppose also that B and D are two bodies maintained constantly at the respective temperatures T and T', and that C is a non-conducting stand on which the cylinder may be placed. The four distinct operations of Carnot's cycle may be thus described: First Operation. Let the cylinder and contents at the temperature T' be placed on the stand (7, and the gas com- pressed without escape of heat till its temperature rises to T. If the initial state of the gas be represented by E (Fig. 180), the change during the first opera- tion will be represented by the adiabatic line EF, and the work done by the area EFfe. Second Operation. Let the cylinder be transferred to B, and the piston allowed to rise gradually. During this process a quantity of heat, $, will flow in through the perfectly conducting bottom, and maintain the temperature constantly at r. This change is represented by the isothermal FCr (Fig. 180), and the work done by the gas is represented by the area FG-gf. Third Operation. Let the cylinder be placed on the insu- lating stand, and the gas allowed to expand until the tem- perature falls to that of the cold body B, namely, T'. This change is represented by the adiabatic GrH, and the work done by GrHhg. Fourth Operation. Let the cylinder be placed on D, and the piston pushed down till it reaches the starting point. The heat Q', which is developed by this process, is absorbed FIG. 180. REVERSIBLE CYCLE. 269 by .Z), and the temperature of the gas remains constantly T'. This change is represented by the isothermal HE, and the work done by HEeh. The body has now passed through a complete cycle of operations, and reached its initial condi- tion. The total work done during the cycle will be found by subtracting the area HEFfh, which represents the negative work, from the area F&Hhf, which represents the positive work. This difference, EFGrH, represents the work done by the body. As the working substance is left in precisely the same condition as at first, the physical results of the process are: 1, the removal of a quantity of heat, Q, at a temperature, r, from B ; 2, the performance of a quantity of work represented by the area EFGrH "; and 3, the communi- cation of a quantity of heat, Q',.&t a temperature, T', to the body D. By the -first law of the thermodynamics the relation between these quantities, measured in ergs, is (6) QQ' = w. 232. Reversible Cycle. If the working substance may be made to pass through the successive stages of the cycle in the reverse order, the process is said to be reversible. It is evident, on examination of the cycle just described, that if the engine were run backwards, the physical results of the operation would be a quantity of heat, Q', taken from D; a quantity of work, represented by the area EHCrF, done on the substance, and a quantity of heat, Q, delivered to B. In this case heat has been transferred from a cold body to a hot one, but only at the expense of mechanical work. It thus appears that the transfer of heat from one body to another, by the highly artificial process described, is a com- pletely reversible process. 270 THERMODYNAMICS. An example of an irreversible process is seen in the trans- fer of heat by conduction, for when a hot body is brought in contact with a cold one, heat passes of itself from the hot to the cold body, but never from the cold to the hot body. 233. Efficiency of an Engine. The efficiency of an engine is denned as the ratio of the work done by the engine to the quantity of heat drawn from the source. If w is the work done, and q the quantity of heat received, both being meas- ured in units of energy, then the efficiency E may be written *=f 234. Carnot's Theorem. The following important prin- ciple was discovered by Carnot. If a reversible engine, working between the temperatures t and ', receive a quan- tity of heat, , and yields a quantity, Q, to #, at the expenditure of the work W r W W The efficiency of A 2 is A and that of A 1 is ^. But by Q V hypothesis THE STEAM ENGINE. 271 that is to say, the compound engine does an available amount of work, TF 2 W l , by the expenditure of a quantity of heat, Q\ Q'%. For the source B is unaffected by the stroke, since A 2 takes Q units from it, and A 1 returns the same amount. Hence, if A 2 does more work than A v it must yield less heat to D than A l would under the same circumstances, that is to say, Q'% < Q'v so that (8) Wz-W^Q't-Q'v An amount of work is thus done by using up the heat of the colder body, and the process might be continued indefi- nitely. But this is in violation of the second law of ther- modynamics; therefore, no engine can be constructed having a greater efficiency than the reversible engine. It follows as a corollary that no reversible engine can have a greater efficiency than any other reversible engine, i.e. all reversible engines have the same efficiency. Hence the effi- ciency of a reversible engine is independent of the nature of the working substance, and consequently must depend alone on the temperatures between which the engine is worked. 235. The Steam Engine. The most important method of transforming heat into work for commercial purposes is s FIG. 181. by means of the steam engine. A sketch of a simple type is shown in Fig. 181, with the parts lettered according to the annexed scheme : 272 THERMODYNAMICS. A. Cylinder. B. Piston. C. Piston Rod. m. n. Steam-ways. E. Exhaust Port. D. Slide Valve. /. Valve Seat. F. Steam Chest, a. Steam Port. G. Valve Stem. H. Rocker. /. Rocker Shaft. L. Eccentric. K. Eccentric Strap. M. Eccentric Rod. N. Connecting Rod. P. Crank. Q. Cross Head. If steam be admitted to the chest F by the pipe $, it will pass through the open port a, force the piston B to the right, and impart a rotation to the shaft T. The slide valve D is, in consequence, moved to the left, closing the port a and allowing the steam in A to expand. .When the piston has reached the end of its stroke, the valve has moved far enough to allow steam from the chest to enter the port e, and force the piston to the left, while the steam in A escapes through , R, and E. 236. Indicator Diagram. A steam engine may be made, by means of a proper mechanism, to trace automatically the relation between the volume and the pressure of the steam in the cylinder during a complete stroke. FIG. 182. Such a pressure-volume diagram, technically known as an indicator card, is shown in Fig. 182, in which ordinates represent pressures, and the abscissas, volumes. EXPRESSION FOR THE EFFICIENCY, 273 The record of the succession of changes which went on within the cylinder is to be interpreted as follows : From a to b steam was admitted at boiler pressure ; from 5, the point of cut-off, to c, the point of release, the steam ex- panded adiabatically ; at d the pressure had fallen to its minimum value and remained sensibly constant until e was reached, when the exhaust port closed and the steam began to be compressed, as is indicated by the line ef. At/ the steam port opened again and the pressure promptly rose to its original value. The work done in this stroke may be calculated at once from the area abcdef, if the length of the stroke and the scale of the pressures are known. The rate at which work is done by engines is measured in a unit called a horse-power, which is defined by One horse-power 33,000 M- '-. mm. In the metric system (10) 7 is called a watt. sec. Accordingly 1 h.p. = 746 watts. 237. Expression for the Efficiency. The fact that the efficiency of a reversible engine is a function of the t&m- peratures alone may be concisely expressed by writing W (9) E= = 4>(t,t'), or if Q r is the quantity of heat returned to the condenser, by substituting the value of W, (10) =((, t), or 274 THERMODYNAMICS. (11) |^1-4>(M and hence t') being some other function of the temperatures. Now, referring to Fig. 180, it is evident that in passing from F to 6r, the quantity of work done, represented by FG-gf, depends on the temperature , the nature of the substance a, and the path F&, i.e. on the relation of p to v, and on nothing else. Therefore the quantity of heat received may be written (13) Q =/(*, , p, v), and, similarly, (14) Q' =/(*', , ^, V). Dividing, Q = /(*, , JP, v) Q'- f(t',s,p,v)' But since by equation 12 ^ is a function of the tempera- V tures alone, /(, , jt>, v) must be of the form JP(s, p, v) so that Therefore since is greater than $', and ^ greater than ', by the conditions of the problem / (t) is an increasing func- tion of t. Let this function for the temperature t be denoted by T, then and THEEMODYNAMIC SCALE OF TEMPERATURES. 275 The efficiency will then be completely determined as soon as the values of T and T' are found. 238. Thermodynamic Scale of Temperatures. It was suggested by Kelvin, as early as 1848, that since the ratio ~- of the quantities of heat taken in and rejected between the *c temperatures t and t f depends on these temperatures alone, the numbers r, T' might be taken to represent the tempera- tures denoted by t and t f on the Centigrade scale, thus form- ing a new or thermodynamic scale. The thermodynamic scale leads at once to the notion of an absolute zero. For, suppose r f chosen so that no heat is ejected by the engine ; the efficiency would then be unity, and all the heat taken from the source would be turned into work. But as no engine could be supposed to convert more heat into work than it received, it is impossible for T' to be nega- tive. Therefore zero is the smallest value it can have. Temperature reckoned from this zero is called absolute, and is independent of the properties of any substance. One thing still remains arbitrary on this scale, that is, the size of the degrees. This may be conveniently chosen so that there shall be one hundred degrees between the temperatures of boiling and of freezing water. If, now, a reversible engine be worked between the temperatures named, which may be denoted by r and T F , and if the quantities of heat received and rejected be measured, Q _ nr r T 100 (19) Q = ^~ = ~' from which the temperature of boiling water on the absolute scale may be found. It is not necessary actually to try the experiment, for the work done by the expansion of a substance which obeys 276 THERMODYNAMICS. Boyle's Law may be calculated by the methods of the infini- tesimal calculus. Kelvin's thermodynamic scale has in this way been shown to be identical with that of a perfect gas thermometer. 239. Correction of the Zero on the Air Thermometer. Since no body obeys Boyle's Law perfectly, the value of the absolute temperature of freezing water determined by experi- ment on gases requires a correction for the deviation from this law. A special examination of the properties of certain gases for the purpose of finding the amount of this correc- tion was undertaken by Kelvin and Joule, making use of a method which is commonly known as the porous plug experiment. The theory of the experiment may be explained by the aid of Fig. 183, which represents a tube stopped by a parti- tion, A, in which is a hole filled with a plug of cotton wool, which is forced N FIG. 183. The chamber B is filled with a through the porous plug by means of a piston moved so as to maintain a constant pressure, p, on the left side of the par- tition. After passing through the plug, the gas in is kept at a constant pressure, p', by means of the movable piston N. Call the volume of unit mass of the gas on the respective sides of the partition v and v r . Then the work done by the piston M on the unit mass in moving up to A is pv, and, similarly, the work done by this amount of gas on the piston N is p'v r . If, now, the gas is of such nature that p'v r > pv, it might be expected that the temperature would fall, in con- sequence of expansion, through the plug ; but if the product THE TWO SPECIFIC HEATS OF A GAS. 277 pv is greater on the high-pressure side there would be a heating effect. Also, if there is any vestige of molecular attraction between the particles of the gas, simple rarefaction, without doing external work, should produce a cooling effect, so that any observed change of temperature on passing through the plug is to be regarded as the algebraic sum of effects aris- ing from two different causes. The results of the observations of Joule and Kelvin, made in the manner described, are shown in the following table : EXPANSION NAME OF GAS. AT ONE ATMOS- PHERE BE- TWEEN FREEZ- ING AND BOILING POINTS MEAN COOLING EFFECT PER ATMOS- UNCOR- RECTED TEM- PERATURE OF MELTING ICE, CORREC- TION CAL- CULATED FROM COOLING ABSOLUTE TEMPERA- TURE OF MELTING (REGNAULT) PHERE. 100 EFFECT. ICE. a. a Hydrogen 0.36613 - 0.039 273.13 - 0.13 273 Air < 0.36706 + 0.208 272.44 + 0.70 273.14 Carbonic Acid 0.37100 + 1.005 269.5 + 4.4 273.9 It will be noticed that air and carbon dioxide were cooled by the expansion, as was also oxygen; but hydrogen was heated, thus showing that hydrogen stands out from the other gases in this respect, as it did in the experiments of Amagat. The experiments on air are regarded by Joule and Kelvin as the most trustworthy, so that 273.14 A. is to be accepted as the nearest approximation to the temperature of melting ice. 240. The Two Specific Heats of a Gas. If it be assumed that no work is done in separating the molecules of a gas when it expands, the excess of the heat supplied at constant pressure over that at constant volume is exactly the work done in expansion. Call v 1 the initial and v 2 the final vol- ume. Let m be the mass, p the constant pressure, and C v 278 THERMODYNAMICS. and C p the two specific heats; then the work done in expansion is (20) p(v 2 -v 1 ) = mE(t 2 -t 1 ). The excess of heat necessary to change the temperature from j to 2 over that required at constant volume is (21) (3,-Q(*2-*l) and its equivalent in units of work is (22) Jm(C p - C v ) (t 2 - tj = mR (t z - ^ ; whence (23) >->=r The ratio (24) % = y ^v may be determined from experiments on the velocity of sound in the gas (see Art. 494). In the case of air 7 = 1.408, gm also ,7= 4.187(10)* gm. 1 C. Thus O = - ^-r- = 0.1681, = 0.0686, . u C p = 0.2367. The value of this constant, found by Regnault in a direct determination, was = 0.2374. TWO ELASTICITIES OF A GAS. 279 241. Two Elasticities of a Gas. The expression strain which was taken in Art. 66 as the definition of elasticity, will have two values according as the temperature remains constant, or as heat is prevented from entering or leaving the body. In the case of gases the expressions for these two elasticities, which will be denoted respectively by E r and E^ have very simple forms. Call the initial pressure of the gas P, and the correspond- ing volume FJ and let v be the change in this volume due to a very small increase of pressure, p, at constant temper- ature, then stress _ p p T/ . ^~stain~T~^ ; V but by Boyle's Law (26) (P + p)(V-v)=PV\ that is, (27) pV=Pv+pv, or (28) -^_L P. v~ V '^ V" p which has the limit as p approaches zero. Substituting in (25), (29) E r = P. To find the other elasticity, E^ let r v T 2 (Fig. 184) be isothermals drawn on the pressure-volume diagram for a dif- ference of temperature of 1, and, similarly, rj v ?; 2 , two con- secutive adiabatics. Then, in passing from the state rj l to ?; 2 , at the constant temperature T V that is in the diagram, moving from to M, the increase of pressure is represented by the 280 THERMODYNAMICS. line LM, and the increase of volume by OL. p R F N (30) Wherefore, OL -Ljjj-L. -j-j- \J-L -r-r LM OL FIG. 184. (31) 7y ET _: V OB Likewise, if the passage from the state to the state F be without gain or loss of heat, the pressure changes by an amount rep- resented by NF, and the volume by ON. There- fore, OR V ON ' ~ OA If the mass of the gas considered be taken as a gram, the addition of a quantity of heat numerically equal to the specific heat at constant volume, C v , would produce a rise of temperature of 1 and an increase of pressure represented by 01. Let OR be the increase of pressure by the addition of Q units of heat under constant volume. Then, as these changes are proportional to the quantities of heat which produced them, Similarly, if C p units of heat be added and the pressure be kept constant, the expansion per degree is represented by OB and the expansion produced by the addition of Q units is given by OA, the quantity of heat to pass from to A being the same as from to R, since the change RA re- quires no heat. Thus, LOWERING OF THE FREEZING POINT. 281 Dividing (33) by (32), C OB _ (V C~~ ~~ OR so that, finally, (35) N n 242. Lowering of the Freezing Point by Pressure. The amount of work done by the unit mass of water when it freezes under a pressure, p, is (36) W=p(v i -V w ), where v i and v w stand respectively for the volume of the ice and of the water. If the ice be now melted in a vacuum, the substance will have passed through a complete cycle, and as it does no work in melting under this condition, equation 36 gives the total work. It is also evident that the process is a reversi- ble one, for the water might be frozen in a vacuum and melted under pressure. The heat received in melting will be exactly L, the latent heat of water. Thus, applying Carnot's theorem for the efficiency, equa- tion 18 becomes (37) JL JL r or (38) Substituting the following numerical values : = 1.013(10) 6 ^s 1 atmo., v ' 2 282 THERMODYNAMICS. gm. 1 C. = 79.25 gms. 1 C., r=:273 C., v . v w = 0.091 cu. cm., the lowering of the freezing point of water per atmosphere is found to be T -T' = 0.007586C. Experiments conducted by James Thomson showed 243. Entropy. Suppose that a substance is carried through any series of changes by a reversible process so as to return to the initial state, and let the series be repre- sented by the closed curve of Fig. 185. Divide the figure FIG. 185. up into a series of little cycles by drawing adiabatics db, cd, ef, etc., and isothermals ag, bd, ch, jf, etc., corresponding to the temperatures r v r 2 , T 3 , etc. Suppose that the quantity of heat q l is received by the body in passing from a to #, and a quantity, (? 2 , given up in passing from d to 5, and, similarly, g where the symbols have their usual signification. The natural zero of entropy is that of a body entirely deprived of heat, but as it is riot possible to bring bodies into this condition, it is more convenient to reckon entropy from an arbitrary zero, denned by some standard pressure and temperature, just as hights are measured from the arbi- trary zero of the sea level. 244. Experimental Determination of Entropy. Suppose that A (Fig. 186) is the standard state in which the entropy is taken as zero, then the entropy p of any other state, B, may be determined as follows : Call T the temperature of the standard state. Let the body in the state B expand adiabatically until the temperature falls to T and a state given by the point C. Then, keeping the temperature con- stant, let the body be brought into the state A. Now, since is on the same isentropic with B, the entire change of entropy along the path BOA takes place in going from Q to A, so that if Q is the heat abstracted in this last change, the entropy at B is given by 286 THERMODYNAMICS. 245. Change of Entropy by Equalization of Temperature. - If a small quantity of heat, r 2 . \ T 2 T l/ T 1 T 2 Therefore, all transfer of heat by the processes of con- duction and radiation from one body of a system to another increases the entropy of the system. Clausius expressed this by saying that " the entropy of the universe tends to a maximum. TABLE OF HEAT OF COMBINATION WITH OXYGEN. 1 GBAM OF COMPOUND FORMED. CALORIES OF HEAT PRODUCED. EQUIVALENT ENERGY IN ERGS. Hydrogen H 2 34000 1.43(10) 12 Carbon CO 2 8000 3.36(10) n Sulphur S0 2 2300 9.66(10) 10 Phosphorus PA 5747 2.41(10)" Zinc ZnO 1301 5.46(10)i<> Iron Fe 3 4 1576 6.62 Tin BnOj 1233 5.18 Copper CuO 602 2.53 Carbonic Oxide C0 2 2420 1.02(10)" Marsh Gas CO 2 and H 2 O 13100 5.50 " Olefiant Gas u a 11900 5.00 " Alcohol u a a 6900 2.90 246. Heat of Combination. In order to separate two molecules which are chemically united, the expenditure of a definite amount of work is necessary. Accordingly, when AVAILABLE ENERGY. 287 the molecules are allowed again to unite, they do an equiva- lent amount of work, which usually appears as heat and may be measured by a proper calorimetric apparatus. The heat of combination of oxygen with various substances has been made the subject of elaborate investigation by Andrews and Faure. Their results are shown in the preceding table. 247. Available Energy. When an engine takes a quan- tity of heat, Qg, from a source at the temperature T& and delivers a quantity, Q^ at the temperature T R , (46) units are transformed into work, but the remainder Q R , ac- cording to the second law of thermodynamics, is unavailable for the purposes of work, if the refrigerator is the coldest body of the system. If equation 46 be taken a& the measure of availability, the dissipation, or diminution of availability of a quantity of heat in passing from a body at the temper- ature TJ to another at r 2 will be that is to say, the product of the lowest temperature by the increase of entropy. Therefore, since the entropy of the universe is continually increasing, the available energy is tending toward zero. 248. Sources of Energy Available to Man. The stores of energy from which we may derive work may be classified as follows : 1. Fuel. 4. Wind. 2. Food. 5. Tides. 3. Head of Water. 6. Solar Radiation. 288 THERMODYNAMICS. The source of the energy in each of these cases, except the 5, may be traced ultimately to the sun. Thus, the dif- ferent forms of fuel, e.g. coal, wood, etc., are products of vegetable growths, which represent for present purposes definite amounts of the compounds of hydrogen and carbon chemically separated from another quantity of oxygen. The exact manner by which, in the physiological processes of plant life, this chemical separation is effected is but imper- fectly understood. In some way chlorophyll, the familiar green coloring matter of plants, appears to be able in the presence of sun- light to decompose carbon dioxide, the carbon being assimi- lated by the plant and the oxygen set free. The energy which man derives from food is likewise due to chemical separation effected by plant life, and hence traceable to the sun. The energy which is obtained by allowing water under pressure to flow through a water wheel is derived from the sun by a simple transformation governed by physical laws alone. The water is evaporated and raised by the heat of the sun, and afterward deposited in the form of rain at ele- vations above the reservoirs in which it is collected. The kinetic energy of the winds is also derived, almost exclusively, from the heat of the sun. The energy of the tides, sometimes used to turn mill wheels, is due to the kinetic energy of the relative motions of the earth, moon, and sun, combined with the potential energy of mutual gravitation. EXAMPLES. 289 EXAMPLES. 1. A mass of mercury falls from a hight of 5 meters. How much would its temperature be raised if all the heat developed were applied to this purpose ? Ans. 0.35. 2. A piece of ice weighing 8.25 kil. is dropped into a pool of water at 0. From what hight must it fall in order to melt 175 gms. ? Ans. 717 meters. 3. With what velocity must a lead bullet strike a target in order to raise its temperature 50, on the assumption that one-half the heat generated is lost? Ans. 163 meters per sec. 4. With what velocity must a lead bullet at 25 strike a target in order that the former may be melted, assuming that one-half the heat developed is dissipated ? Ans. 500 meters per sec. 5. 775 cc. of air are heated from to 100 under a constant pressure of one atmosphere. How much work will be done by the expansion ? Ans. 2.88(10) 8 ergs. 6. The thermal capacity of a system, consisting of a cotxper vessel filled with compressed air immersed in a body of water, was 10,680 cal." per degree. When 44,600 cc. of air were allowed to escape at a pressure of 76.5 cm. of mercury, the temperature of the system fell through 0.097. What value does this give for the mechanical equivalent of heat? Ans. 4.39(10) 7 cm. 2 / sec. 2 deg. C. 7. What would be the maximum efficiency of an engine working between 138 C. and 44 C. ? Ans. 20.4%. 8. The melting point of sulphur is 115, and its latent heat 9.3 cal. per gm. The density in the solid state being 2.05 gms. per cc. and 1.95 gms. per cc. in the liquid state, at what temperature will sulphur melt under a pressure of 25 atmospheres? Ans. 115.63. 9. If a specimen of coal contain 88 % carbon and 4.5% hydrogen, how, many calories will be generated by the combustion of 1 gm. of the coal? Ans. 8570 cal. CHAPTER XV. KINETIC THEORY OF GASES. 249. Explanation of the Pressure of a Gas. An early suggested explanation of the pressure of a gas was that the molecules exerted a mutual repulsion. It was found, how- ever, that it was impossible to state any law of repulsion which should make the pressure fulfill Boyle's Law, and at the same time be independent of the shape of the containing vessel. Another theory, suggested by Daniel Bernoulli, as early as 1738, was that the pressure of a gas arose from the impact of the molecules against the sides of the containing vessel. This theory was not worked out in definite form till 1848, when Joule showed, from the principles of dynamics, what the velocity of the molecules should be to produce the observed pressure. 250. Expression for the Pressure of a Gas. The funda- mental postulates of the kinetic theory are : 1, that every gaseous body which can be experimented on contains an indefinitely great number of molecules, all of the same size ; 2, that each of these molecules moves with great velocity in a straight line until it strikes the sides of the vessel, or encounters some other molecule, after which it moves off in a new direction ; 3, that the time of free motion of the molecule is very much greater than that occupied by an encounter ; 4, that the molecules influence each other by impact only. No definite assumption can be made concern- EXPRESSION FOR THE PRESSURE OF A GAS. 291 ing the behavior of the individual molecules, for even if they all had the same velocity initially, this velocity must be altered both in direction and amount at each encounter, so that they must soon appear to move in an entirely irregu- lar manner. However, the application of the laws of chance to such a system of molecules, moving in an entirely fortui- tous manner, shows that probably some molecules are moving very slowly, a very few are moving with enormous velocities, and that the others have velocities intermediate between these extremes. The best method of treating these veloci- ties, for the purpose of calculating the pressure, is to take the average of the squares of all the velocities. This quan- tity is called the mean square of the velocity, and will be denoted by V z . The square root of this quantity is called the velocity of mean square. In order to find the pressure of a gas, suppose that the latter is contained in a rectangular vessel, and let a = length of the vessel, b = breadth of the vessel, c = depth of the vessel, p = pressure of the gas, p = density of the gas, n = number of molecules, M =mass of each molecule, t = small time occupied by the impact, V 2 = mean of the squares of the velocities of all the molecules, U= v V*= velocity of mean square. On the whole, as many molecules may be regarded as moving in one direction as another, i.e. one-third may be supposed to be moving parallel to each side of the vessel. Now, if a molecule moving parallel to a, with a velocity U, 292 KINETIC THEORY OF GASES. be supposed to strike the end of the vessel, and bound back with the same velocity, its momentum will have been changed by an amount or, since J molecules are moving in this direction, the change of momentum consequent upon one impact by all the mole- cules against the end will be InMU. If each molecule be supposed to travel the length of the box without interruption, it will strike the end times in the time , so that the total change of momentum in the time t is , Ut ^nMVH InMU - ^-=i -- , 2a a and the force exerted against the end, which is measured by the time rate of change of this quantity, is F = . nM V* a Therefore, the pressure on the end whose area is be will be F _ 1 nMl , be ' abc But nMis the total mass of the gas and abc is the volume ; therefore, since Mn or, solving for F 2 , the mean square of the velocity is (2) F* = ^. BOYLE'S LAW. 293 251. Boyle's Law. If in equation 1 the volume abc be denoted by v, (3) pv = ra V*. Now, since F 2 depends only on the temperature in the same body of gas, pv = constant, for a constant temperature. Boyle's Law is thus raised from the rank of an experimental fact to a deduction from the simple assumptions made at the beginning of the preceding article. 252. Clausius' Equation of the Virial. In the case where the influence of one molecule on another is not negligible, Clausius has proved an equation which greatly extends the generality of the investigation concerning the pressure of a gas in the kinetic theory. Let w, F 2 , p, and v have the same meaning as before, but suppose now that each particle exerts an attraction, or repulsion, R, upon another particle at the distance r. The product Mr is called the virial of * the stress, and the virial of the system 3 will be the sum of the virial of the stresses which exist in it. . Let Fig. 187 represent a number of 5 particles, and let the stress between 1 Flo 187 and 2 be denoted by ^ 12 , and the dis- tance by r 12 , then the virial for all the particles taken with respect to 1 will be where the index k is to be given all values from 1 to w, and likewise for the particle 2 294 KINETIC THEORY OF GASES. and so on. Adding these expressions, the virial for the whole system becomes %?^\R kk r kk , the i being taken because R 12 R 2V etc. Clausius' equation for the virial may be written (4) 12 M F 2 = fjw + }2^Rr, in which the left-hand member denotes the kinetic energy of the particles, and the right-hand side the value found for this energy in equation 3 increased by the virial of the sys- tem. In gases like hydrogen, air, etc., the virial is very small under ordinary conditions, but if the gas be compressed it may naturally be expected to change. Solving (4), (5) pv = ^ and, comparing with the results of Art. 182, it appears that the virial is at first positive, but changes sign as p increases ; that is to say, the force between the particles is first an attraction, but afterward becomes a repulsion. 253. Van der Waals' Equation. Van der Waals has shown how to take account of the size of the molecules, and also of those forces which give rise to the phenomena of surface tension in a liquid. Boyle's Law, pv = mRr, would indicate that if the pressure were sufficiently increased the volume would approach zero, a conclusion which is not con- sistent with the known properties of matter. If the minimum value of the volume be called 5, the expression for the pressure, mRr 7? V will be free from the objection just mentioned. CALCULATION OF MOLECULAR VELOCITIES. 295 Also, if it be assumed that different portions of the fluid attract one another, the simplest supposition that can be made will be that this force is proportional to the mass of the elementary portions into which the body may be sup- posed to be divided. But the quantity of matter in an ele- ment will vary as the density p = , where m is the mass of v the body and v its volume, so that the attraction F of the force which gives rise to the surface tension may be written joe- - v v Subtracting this term from the previous value for the pressure, since it diminishes it, Van der Waals wrote mRr w 2 ~ a or V V ( where a is some constant. This equation very well repre- sents the experimental results of Andrews, presented in Art. 180. 254. Calculation of Molecular Velocities. The equation yi^Zp P permits the velocity of mean square to be calculated at once. Thus, for air at C., p = 1.013(10)^ te, cm. 2 1 gm. ~ whence, VF* = 4.85(10) 4 . sec. 296 KINETIC THEORY OF GASES. Observing that for different substances this velocity varies inversely as the density, its value for any other gas may evidently be obtained by multiplying the value just found by the square root of ratio of the density of air to that of the gas required. Thus, for hydrogen, which is 14.44 times as light as air, the velocity of mean square will be 4.85(10) 4 x Vl454 = 1.84(10) 5 , sec. or nearly a mile per second. The molecular velocity of a gas bears an important rela- tion to the elements which constitute the matter of the earth. If it were possible to give a projectile directed away from the earth a sufficiently great velocity, it would soon pass into a region where the earth's attraction would be too small to draw the body back to its surface. Now, as this velocity may be shown to be not much greater than that of the hydrogen molecule, it seems probable that if the earth ever possessed elements less dense than hydrogen they have now escaped into space. At the surface of the moon weight has but one-fifth the value of terrestrial weight, since the diameter of the moon is only one-fourth and its mass one-eightieth of that of the earth. Consequently, the lightest substance which the moon can retain must have a density twenty-five times that of hydrogen. Now, as most substances which are likely to exist in the aeriform condition are lighter than this, it might be predicted that the moon would have little or no atmos- phere. This conclusion is verified by observation, which shows that, if the moon possesses an atmosphere, it cannot exert a pressure exceeding 1 mm. of mercury. On the other hand, it is possible that a body as large as the sun may possess elements much lighter than hydrogen. LAW OF AVOGADRO. 297 255. Law of Avogadro. If two sets of molecules of dif- ferent masses are moving in a vessel, there will result from their successive encounters an exchange of energy up to a certain point. It has been shown by Maxwell that equilib- rium between two such systems will occur when the mean kinetic energy of a single molecule of each set is the same. Thus, if the molecular masses are M v M v and the mean squares of their velocities are, respectively, Fj 2 and F^, 2 , this condition of equilibrium may be stated (7) \M^ = \M^. If the pressure of the gases be denoted by p v p v the num- ber of molecules by n v n%, and the volumes by v v v v and (8) If the volumes and pressures -are the same for both gases, (9) n l M l V? If, in addition, the gases are in thermal equilibrium, i.e.. have the same temperature, by equation 7 whence, (10) n^ = n 2 . That is to say, in equal volumes of any two gases at the same temperature and pressure there is an equal number of molecules. This law was announced by Avogadro in 1811. 298 KINETIC THEORY OF GASES. If the density of the gases at the same temperature and pressure be denoted by p v p 2 , (11) by definition. Then, since by equation 10 the volumes are proportional to the number of molecules, it follows that Pi ^i (12) - = Tir Pz M* or the masses of the molecules are proportional to the abso- lute densities of the gases measured at the same temperature and pressure. Since, for thermal equilibrium, by definition, two bodies have the same temperature, equation 7 will be satisfied if the absolute temperature is taken proportional to The same result is obtained by comparing with pv = mRr. 256. Gas Density and Molecular Masses. The ratio of the density of a gas to the density of hydrogen is called its gas density. Let D = gas density of the gas, p = absolute density of the gas, p H = " " " hydrogen, M= mass of a molecule of the gas, M H = " " " " " hydrogen. Then, by the definition and equation 12, ^ = ^ = ^, or , (13) ~\ Pit MJJ M=DM H . DETERMINATION OF GAS DENSITY. 299 That is, the molecular mass of a gas can be found in terms of the hydrogen molecule when the gas density is known. Molecular masses are, however, usually expressed in terms of the hydrogen atom, i.e. half of the hydrogen molecule, which is shown to be divisible into two parts by exposing a mixture of equal volumes of hydrogen and chlorine to full sunshine. A new substance, known as hydrochloric acid, is then formed without change in pressure ; whence it follows, by the law of Avogadro, that the number of molecules in the new gas is the same as in the mixture. But since after the reaction the molecule of the acid is certainly compound, for two volumes of hydrochloric acid may be decomposed into one volume of hydrogen and one volume of chlorine, it fol- lows that the molecule in the simple gases must have con- sisted of two parts, and the reaction was due to the union of each of the parts of the hydrogen molecule uniting with one of the parts of the chlorine molecule. Accordingly, if the mass of the hydrogen atom be denoted by [A], M H =2(h], and M=2D[h], which is the familiar rule of the chemist that the molecular mass of any substance is twice its gas density. 257. Determination of Gas Density. The gas density of substances which exist ordinarily in the solid or liquid state, and which may be made to pass into the aeriform condition at a temperature below that at which glass softens, may be conveniently determined by a method devised by Victor Meyer. The apparatus consists of a cylindrical glass vessel, B (Fig. 188), drawn out into a long stem, E, which ends in an enlarged mouthpiece, .F, closed with a rubber stopper. A little below this end a branch tube, A, with upturned end, 300 KINETIC THEORY OF GASES. is arranged so as to dip below the surface of a vessel of water. Surrounding the bulb B is a larger vessel, (7, of glass, or sometimes of iron, which is intended to contain a hot liquid bath. To determine the gas density of a substance, an amount which when vaporized will not more than half fill the vessel B is weighed out and placed in a flask small enough to pass through the stem E. The bulb is now heated by immersion in a bath at a suitable temperature. During this opera- tion some of the air is driven out and escapes through the branch tube. When the whole has reached a constant temperature, and there is no further escape of air from A 9 the stopper is removed from F and the flask containing the substance is dropped into B, at the bottom of which a cushion of asbestos has been placed to prevent fracture of the glass. The stopper is at once replaced, and after one or two bubbles of air, forced out by the introduction of the stopper, have escaped from A, a graduated tube, .Z), filled with water, is carefully inverted over the orifice. The substance in the flask will be FIG IBS vaporized in a few seconds and drive out its own bulk of air which is collected in D. As soon as equilibrium is attained, the graduated tube is lowered until the level of the water is the same within and without, i.e. the pressure of the air in the tube is the same as that of the external atmosphere. The hight of the barometer is then noted, together with the temperature of the air collected and its volume, which may be read from the graduations of the tube. Let v = volume of the air collected, m f = mass of the air collected, DIFFUSION. 301 t temperature of the air collected, h = pressure of the air collected in cm. of mercury, w = pressure of water vapor at t C. in cm. of mercury, m = mass of substance vaporized, D = gas density of substance vaporized, D f = gas density of air = 14.44. The volume v may be reduced to zero and 76 cm. pressure, by the equation 23, Chapter IX ; thus, h w v (14) Vn = 76 cm. . t 273 Z<3 h being corrected for the pressure of the water vapor within the tube. Then, since the density of air is , the mass of the air will be and, finally, since the masses of equal volumes of gases are as their densities, C D m p-y " U-=i 14.44. I m' 258. Diffusion. If a small hole of area s be made in the end of the box discussed in Art. 250, the rate of escape may be easily calculated as follows: One molecule strikes the y end times a second, so that if \ of the molecules be sup- al posed to move parallel to #, the end whose area is be will 302 KINETIC THEORY OF GASES. y sustain n -7 blows per second ; that is to say, the rate of 6a escape from the area s will be nVs Since n is by Avogadro's Law the same for all gases under the same conditions, the rate of escape will be proportional to the velocity of mean square, or inversely as the square root of the density of the gas. This conclusion is found to be verified by experiment, and thus offers the strongest pos- sible proof of the kinetic theory. The plaster plate used in the experiment on diffusion, described in Art. 191, is to be regarded simply as a wall with innumerable holes. The excess of pressure on one side of the plate is obviously a differential effect depending on the fact that more of the lighter molecules pass in one direc- tion than of the denser ones going the opposite way, for the mixtures in time become alike on both sides and the differ- ence of pressure vanishes. If, however, a gas, #, could be placed in a vessel, A (Fig. 189), made of such material that its walls should be absolutely impervious to FlG 189 it, and then were surrounded by another vessel, B, containing a gas, 6, to which the walls of A were completely permeable, it is evident that the gas b would diffuse into A until it exerted the same pres- sure on both sides of the wall. The total pressure in A would, however, exceed that in B by exactly the pressure of the gas a. Although such an experiment is not realizable in the case of gases, it is in the case of liquids (see Art. 192), the excess of pressure being then known as osmotic pressure, the ENEEGY OF THE GASEOUS MOLECULE. 303 explanation and the numerical value of which, as has been experimentally shown, are precisely the same as for gaseous pressure. 259. Energy of the Gaseous Molecule. The energy of a molecule will in general consist of three parts : 1, that due to the velocity of translation, already designated by ilf F 2 ; 2, that due to rotation among the component parts of the molecule; and 3, that due to the vibration of these parts. If it be assumed that each of these portions is proportional to the energy of translation, the total energy may be written where /3 is a constant greater than unity and probably equal to 1.634 for air, and the more perfect gases. The total kinetic energy of the substance will be found by multiplying by the number of molecules ; thus, (17) T = comparing with p = (18) T = ipvfr or, the kinetic energy per unit volume is (19) T v = 260. Law of Dulong and Petit. The specific heat at con- stant volume O v is numerically the quantity of heat required to raise 1 gm. through 1 C., when the volume does not change. Now, since pv is proportional to the absolute temperature, the total kinetic energy of a body of gas will be proportional to the absolute temperature. Hence, the 304 KINETIC THEORY OF GASES. increase of energy of the unit mass per degree rise of temper- ature will be found by dividing T by m and T ; thus, ** 2 mr which is reduced to heat units by dividing by J. Therefore, (20) <7.= i= } ., or, Jmr (21) MC V = I ^ = constant, nJ if ft is constant. This is the law of Dulong and Petit. 261. Determination of ft. By equation 23, Art. 240, or c Substituting the values of R and (7 V , and calling f = 7, (22) 7- 1-f & or, ft = The value of 7 is usually found from the velocity of sound in the gas by the equation (23) % 2 = 7- (see Art. 494), where u is the velocity of sound and p is the density of the gas. Combining equation 23 and F 2 = _ , the velocity of mean square is found to be VISCOSITY OF GASES. 305 (24) or ^\/F^ = l .458 win gases, for which 7 = 1.408. 262. Viscosity of Gases. The viscosity of a gas is a physical property closely connected with diffusion. Sup- pose that a stratum of a fluid is G D moving above the fixed horizontal ~~J plane AB, in the direction of the I arrow (Fig. 190), with a velocity ^ proportional to the distance from the plane, and let the velocity at the distance d be called V. The substance between the planes is undergoing a shear which increases at the rate a If the shearing stress is denoted by $, the coefficient of viscosity may be defined by S Sd < 25 ) M=:=-^ Its dimensions are d M LT A coefficient of viscosity per unit density is also some- times used. Let it be denoted by v; then (26) L 2 and its dimensions are 306 KINETIC THEORY OF GASES. This last coefficient is connected with K, the coefficient of diffusion of a gas into itself, by the equation (27) v = 0.6479 K. The viscosity of a gas determines the rate at which fine particles will settle through it. Calculations by Stokes indi- cate that a drop of water T - ^ + ^ = 0, whence ( 4) Wi gft = 0, or UNIT QUANTITY OF ELECTRICITY. 317 Also by definition, f q. = &A** (6) ifcr=*-4* therefore, since areas of similar figures are squares of their homologous lines. Comparing (5) and (7), that is to say, the function of r is the square. This proof was worked out by Cavendish, before 1775, though not published. By a careful investigation of the magnitude of the error committed in his experiments, Caven- dish further showed that the assumption that the force van- ishes within a charged conductor could not be so far wrong as to make the exponent of r differ from 2 by 0.02. The law of the inverse square was first published in 1787 by Coulomb, who had verified it very roughly by measuring the electrical forces at different distances. Modern investi- gation has shown that the error in the exponent 2 cannot exceed ^-i^. Collecting the preceding results, the law of the mutual action of one charge on another may be written, ' where F is the force reckoned positive for repulsion, q and q' are the charges, r is the distance between them, and K the dielectric constant. 272. Unit Quantity of Electricity. If two indefinitely small bodies having equal charges repel each other with the 318 ELECTBIF1CA TION. force of one dyne when placed at the distance of one centi- meter in air, the charge on each body is denned as the unit quantity of electricity. The discussion of the dimensions of this and other electrical magnitudes will be found in Chapter XXV. EXAMPLES. 1. Two small spheres separated by a distance qf 14.9 cm. in air are given charges of 39.8 and 45.1 C. G. S. units, respectively. What is the force of repulsion between them? .4ns. 8.09 dynes. 2. Two Ihiiall electrified bodies separated a distance of 12.3 cm. in a medium whose dielectric constant is 2.7 are found to attract each other with a force of 7.6 dynes. The charge on one of the bodies is 64 units. What is the charge on the other ? Ans. 48.5 gm. 5 cm.* / sec. 3. What is the distance between two small bodies charged respec- tively with 75 and 68 units, which repel one another with a force of 8.4 dynes in air ? Ans. 24.6cm. 4. Two small pith balls, each weighing 0.025 gm., are suspended from the same point by silk fibers 80 cm. long. When the balls are equally electrified, they separate under the mutual repulsion to a distance of 10.2 cm. What is the charge on each ball ? Ans. 12.7 gm.^ cm.^ / sec. 5. Two small electrified spheres, A, C, having charges respectively of 64 and 144 C. G. S. units, are placed 24 cm. apart. Where in the line AC can a third charge, B, be placed, so that it will be in equi- librium ? Ang ( AB = Q.6 cm., or 48 cm. ( CB = 14.4 cm., or 72 cm. V CHAPTER XVII. THE ELECTRIC FIELD. 273. Electric Field. Any region through which electri- cal forces are regarded as acting is called an electric field. The space may be occupied by air or other bodies, or it may be entirely devoid of ordinary matter. 274. Line of Force. A line drawn through an electric field, so as to have everywhere the direction in which a small positive charge of electricity would be urged, is called a line of electrical force. It follows from the definition that the lines of force will be directed away from a positively charged body and toward a negatively charged one ; and, further, since a positive charge can never be developed with- out the simultaneous appearance* of a negative charge some- where else, the lines of force must terminate either on the bodies in the neighborhood, or on the walls of the room, or on more remote bodies ; and where they terminate there is a quantity of electricity exactly equal and opposite to that on the part of the body from which they proceeded. 275. Intensity at a Point. The intensity at a point, or the strength of the field, is the force exerted on a small charged body divided by the number of units of electricity which it contains. It is numerically the force which would be experienced by a unit charge placed at the point in ques- tion. The intensity at a distance, r, from a point charged with a quantity, q, would be 320 THE ELECTRIC FIELD. 276. Potential at a Point. The potential at a point is defined as the work done per unit quantity of electricity in bringing a positive charge from infinity to the point con- sidered. If W be the work done in bringing a body con- taining the charge q from infinity, or other place where the force vanishes, and J^be the potential, then r-f Potential plays much the same i61e in electricity that tem- perature does in the theory of heat, or that pressure does in treating of fluids. For as heat flows from places of high to low temperature, or a fluid from high to low pressure, so electricity will flow along a conductor from places of high to low potential. An equipotential surface is a surface every point of which has the same potential. The surface of a conductor is always an equipotential surface if the electricity is at rest, for, by definition, a conductor will not maintain a difference of potential between two points. 277. Representation of an Electrical Field. By drawing lines of force and equipotential surfaces throughout a field, according to an accepted convention, it is possible to repre- sent with mathematical exactness all the electrical quantities concerned. It is usual to draw the equipotential surfaces for a constant difference of potential, and the lines of force so that if the charged surface be divided into elements hav- ing unit area, the number of lines of force which start from each element will be proportional 4o the number of units of charge upon it. According to the views of Faraday, the explanation of the mechanical action of one body upon another is to be sought, REPRESENTATION OF AN ELECTRICAL FIELD. 321 FIG. 194. not in forces of attraction and repulsion acting at a distance, but in stresses in the medium, of the nature of tension -along the lines of force and compression at right angles to them. Let A (Fig. 194) be a positively charged point, and suppose that the nega- tive charge corresponding to it has been removed to an infinite distance. It will be observed in this case that the lines of force radiate from the point. The scheme of drawing them in the figure is such that, if the surface of the sphere of unit radius about the point be divided first into zones by a system of planes perpendicular to the horizontal axis, and then into elements of unit area by a system of planes passing through this axis, the number of lines standing on each element is proportional to -^-, where q is the charge at the point. Also, since the work done in bringing up a unit charge from infinity to a point at a distance, r, from A will be the same from whichever side the body is approached, the equi- potential surfaces will be a system of concentric spheres, necessarily nearer together as r is diminished, as will be definitely proved in Art. 282. The intensity at any point of an electric field is inversely proportional to the distance between successive equipoten- tial surfaces. Tnis may be shown as follows : Let d be the distance be'tween surfaces having, respectively, potentials 322 THE ELECTRIC FIELD. V l and F^, and c^ the mean value of the intensity along the line d ; then, by definition, whence, ir (3) ^ that is to say, the intensity of an electric field at a point is equal to the (space) rate of change of the potential at that point. Equipotential surfaces and lines of force always intersect at right angles ; for, suppose that AB (Fig. 195) is a portion of any equipotential surface, and that the intensity & makes an angle, 0, with this surface. The work done in moving a unit of electricity from a to b is % (4) * V a ,F 6 = c^cos 6-ab; but, by definition, V a ^*V b \ therefore, cos 0=0, FIG. 195. or the force^is- per.pen4icula.rlo the surface. 278. Field Due to a Charged Sphere. If one of the equi- potential surfaces, B (Fig. 194), were to be transformed into a conducting shell, it is evident that the field would be un- altered, for B is still an equipotential surface and of the same numerical value, since its potential is produced by A alone. Likewise the potential of every other part of the field is the same as before. -Also, since the equipotential surfaces are at the same distance, the intensity at every point is unchanged. If, now, A be connected by a wire to -B, the lines of force within the conductor will be wiped out, but FIELD DUE TO TWO UNLIKE CHARGES. 323 the external field will remain the same, as was shown by Faraday's ice-pail experiment. Therefore, a charged spheri- cal conductor exerts the same force without its surface as it would were the electricity collected at its center. 279. Field Due to Two Unlike Charges. Fig. 196 shows the field produced by a positive charge of 10 units at A and FIG. 196. an equal negative charge at B. Near the points the equi- potential surfaces are practically spherical, but further out they become flattened on the side toward the opposite charge. The two systems of surfaces are separated by a plane, which is a surface of zero potential. Any one of the equipotential surfaces about A might be regarded as the surface of a conductor charged with 10 units of electricity, under the influence of an equal opposite charge 324 THE ELECTRIC FIELD. upon a similar conducting surface surrounding B, by simply expunging all the lines within these surfaces. It will be noticed that the intersections of the lines and the equipotential surfaces are orthogonal. If the lines of force be regarded as so many stretched elastic bands con- necting the bodies, the mechanical force exerted would be such as to draw the bodies nearer together ; that is, they show an apparent attraction. 280. Field Due to Two Like Charges. The electric field produced by two like but unequal charges at the points A and B is shown in Fig. 197. The lines of force form two FIG. 197. independent systems starting from each point and running to oppositely charged surfaces supposed to be at a great distance. FIELD ABOUT AN INSULATED CONDUCTOR. 325 The two systems are separated by a surface resembling one of the sheets of a hyperboloid of two sheets. A system of e quip o ten tial surfaces encloses each charged point, and these are in turn surrounded by a third system, of which the limiting case is a surface of two lobes meeting at the point P, where the force vanishes. If, as before, the lines of force be regarded as elastic bands, the two charged points would be drawn apart; that is, they show an apparent repulsion. 281. Field about an Insulated Conductor. Fig. 198 illus- trates roughly the in- ductive effect of a charged body, B, on a spherical conductor, A. As the conductor is brought into the field, it assumes the poten- tial at that point, and the equipotential sur- face includes A as a cavity in itself. Since as many lines of force leave the ball on one side as enter it upon the other, A has not a definite charge in the usual sense of the term, but only an apparent negative charge on the left side and an apparent positive charge on the right. This inductive action is also clearly shown by the arrangement of the equipotential sur- faces. On the left of A they have been displaced, so that there is a more rapid fall of potential than in the undis- turbed field ; that is, the force measured at any point between B and A will be greater on account of the presence of A 9 as if the latter carried a negative charge on the left side. Simi- larly, the potential of points just to the right of A is greater FIG. 198. 326 THE ELECTRIC FIELD. than in its absence, as if it had a positive charge on that side. Regarding the lines of force as elastic bands, it appears that the horizontal component of forces on the left will slightly exceed those on the right, or A will experience a resultant force, as though B attracted it. 282. Potential Due to a Charged Point. Let q be the charge at the point A (Fig. 199). The intensity at the point p 1 at a distance, r v from A will be ^ -jp-a and, similarly, Kr^ that at jt? 2 will be <^ 2 *Jr* By definition the change in Ar 2 potential between these points is the work which would be A done in carrying a unit of elec- "5"" " "p"" ' p tricity from p 2 to p v If the intensity were constant between FIG. 199. * these points, this work could be calculated at once by multiplying the force by the distance. In the case where the force varies inversely as the square of the distance, it may be shown that the work is obtained by multiplying the distance by the geometric mean value of the force between the points. Thus, (5) l > v *=XrV (r *- r ^K(7-, r)- 12 \ 1 2 If the point jt? 2 is at infinity, or the potential due to a single charge is equal to the quo- tient of the charge by the specific inductive capacity and the distance of the point considered. 283. Behavior of a Dielectric Body in an Electric Field. If a dielectric sphere is brought into a field produced by a BEHAVIOR OF A DIELECTRIC BODY. 327 charged point, A, in a medium whose specific inductive capacity is less than its own, there will be a distortion of the field like that represented in Fig. 200. The equipoten- tial surfaces are crowded away from the sphere, as in Fig. 198, though now some of the surfaces pass through it. It may be observed that the distances between surfaces within "the sphere are greater than they would be in the outside medium, showing that the intensity is less. The distortion of the equipotential surfaces has lowered the potential of points near the left side of the sphere, but raised those at the right side, as would be the case if there were a negative charge on the left side of the sphere and a positive charge on the right. This inductive action of the point A, producing an apparent charge on the sphere, is also indicated by the crowding of the lines of force into the material of the sphere, and by the shortened distance between the equipotential surfaces to the right and to the left of it. By observing the change in the angle at which the lines of force enter and leave the surface, it will appear that the sphere ought to be drawn toward the point J., since the lines on the left would have a greater horizontal resultant force than those on the right. This important conclusion may be differently stated thus : A dielectric body, placed in an electric field in a medium whose dielectric constant is less than its own, experiences a force urging it toward that part of the field where the intensity is a maximum. In a uniform field it would experience no force of trans- lation, but an elongated body would set itself so that its length would be parallel to the lines of force, as is obviously 328 THE ELECTRIC FIELD. FIG. 201. seen to be the result of regarding the lines of force in Fig. 201 as under tension. If, however, such an elongated body were placed in a non-uni- form field where the inten- sity varied greatly within a small space, the expul- sion effect, being different for different parts of the body, might be great enough to overbalance the forces which would set its length parallel to the lines of force, and cause it to set transversely with respect to these lines. In the case where the value of K is greater in the sur- rounding medium, the distortion of the field is similar to Fig. 202. The equipotential surfaces are crowded closer together within the sphere, indicating that the intensity is greater at a point inside the sphere than it would be if the medium were all such as that outside. The poten- tial of points just to the left of the sphere has been raised, and that of points on the right lowered, as if there was a positive charge placed on the side toward A and a negative charge on the side away from A. Regarding the lines of force as elastic bands, the distortion of the field is seen to have increased the angle with the horizontal where the lines meet the sphere on the left, but decreased it on the right. The horizontal component of the forces represented by the lines on the right will be in excess, and the sphere will be drawn away from A. This result may be otherwise stated thus : A dielectric body, placed in an electric field in a medium whose dielectric constant is greater than its own, FIG. 202. BEHAVIOR OF A DIELECTRIC BODY. 329 experiences a force urging it toward that part of the -field where the intensity is a minimum. In a uniform field the body would experience no force of translation, but an elongated body having a value of .ZTless than the outside medium would set itself parallel to the lines of force for reasons similar to those given in connec- tion with Fig. 201. If, however, such an elongated body be placed in a non-uniform field in which the change of in- tensity is very great, the expulsion effect, different at differ- ent parts of the body, might be great enough to overbalance what has been referred to as tension along the lines of force, and set the length of the body transversely with respect to these lines. EXAMPLES. 1. A spherical conductor 8.6 cm. in diameter is charged with 65 units of electricity. What will be the potential at a distance of 75 cm. from the surface of the sphere? Ans. 0.82 cm.' gm.*/ sec. 2. If a charge of 27 units is placed at one of the corners of an equi- lateral triangle, the length of each of whose sides is 32 cm., and a charge of 68 units at the second, what will be the intensity at the third corner of the triangle? Ans. 0.083 gm.*/ cm.* sec. 3. If charges q v q 2 are placed at points separated a distance, d, show that the surface of zero potential is a sphere of radius, 2 with its center at a distance, 2l j from q, . CHAPTER XVIII. ELECTROSTATIC INSTRUMENTS. 284. Electrical Machines. Machines designed to gener- ate charges of electricity at high differences of potential are of two types : 1, Frictional Machines, or those in which the electrification is produced and augmented by the rubbing of two dissimilar bodies ; 2, Influence Machines, or those in which a given electrification is used to generate other charges by induction. In machines of the first type a cylinder or a circular plate of glass is revolved so as to rub against a surface of leather coated with an amalgam of zinc, by which process the glass becomes positively and the rubber negatively elec- trified. The charge on the glass is collected on suitable conductors and may be used for any desired purpose. One of the best forms of the friction machine is shown in Fig. 203. P is the glass plate, which is elec- trified by revolving it be- tween the rubbers at R. The charge is collected by a pair of rings at (7, which are set on the sides toward the plate with a series of sharp points. On account of the low efficiency this type of machine has now fallen into disuse. FIG. 203. ELECTROPHORUS. 331 285. Electrophorus. The first of the induction machines was invented by Volta in 1775, and is known as the electroph- orus. It consists of a metal plate, ~ C (Fig. 204), with an insulating \ ) handle, by which it may be raised or r --- ---- JU ------ ^ -l_ -1_ -l_ .1. -U -J- .11- -U -i_ 4- -4- replaced on the base of the apparatus, a resinous disc, A, backed by the metal sole B. The disc having been electrified by rubbing with cat's fur, the plate is brought very close to the base or set upon it. Contact in this case has no perceptible effect on the charge of A, since the plate really touches the resin in but a few points which are themselves non-conducting. The electri- cal state of is then that represented in the figure. The top of C is next discharged to the ground, or the walls of the room, by touching it with the finger. If the plate be now raised, the electricity, which was strongly attracted by the negative charge of A, will be distributed over the con- ductor, and, on presenting the knuckle, will be discharged with an accompanying spark. As no part of the original charge is wasted by this process, it may be repeated any number of times. The sole B is not fl essential to the successful working of the 15 apparatus, but may retard the slow leak- age of the resinous charge from the base. +7 1 L i - ^24 C! JL i I~II-13 _/[innm 1 1 1 ] i M i i i in imiiminiii .muniim i i i b FIG. 205. The process of charging by induction, exemplified in the electrophorus, may be explained in a more philosophical way, and in accordance with the modern view of electrical phe- 332 ELE C TR OS TA TIG INS TR UMEN TS. nomena, by use of the term potential. Thus, suppose that the base A (Fig. 205a) is originally at a negative potential represented, for example, by 25 units, then as the plate C is approached to A its potential will fall almost to that of J., say to 24. If C is now touched to earth, it will take the potential of the latter, which is taken arbitrarily as zero. Also, since C is very near to the base, the potential of the latter will rise, and the state of things may be that repre- sented in Fig. 2056. Finally, if C is lifted off from the base, the potential of the former will rise and the latter fall in a way not unlike that shown in Fig. 205 , which may be connected by means of a wire with the point whose potential it is desired to examine. Under these circum- stances, the gold leaf is deflected toward the electrified plate differ- ing most from its own potential a distance which is nearly propor- tional to this difference. This particular form of instrument is known as Hankel's electrometer. 291. Quadrant Electrometer. Fig. 21 2 shows an important type of instrument, devised by Lord Kelvin for the measure- ment of small differences of potential, and called the quadrant electrometer. ft consists essentially of a metallic cylindrical box divided into four segments, A, A', B, B 1 , within which is hung, by a double thread, t, t f , a light alumi- num blade or needle, CD (Fig. 213). The bifilar suspension furnishes a small moment of restitution, which restores the needle, after a torsional displacement, to a position sym- metrical with respect to the quadrants. The opposite FIG. 212. 338 ELEC TR OS TA TIC INS Til UMEN TS. FIG. 213. quadrants AA, BB' are connected by the wires. A wire, JV, attached to the lower side of the needle dips in a dish of sulphuric acid, which serves the threefold purpose of damping the vibration of the needle, making electrical connection be- tween the binding post K and the needle, and absorbing the aqueous vapor in the case with which the electrometer is always enclosed when in use. If a constant poten- tial difference is maintained between each pair of quadrants by means of a condenser or battery and the needle is charged successively to different potentials, the angular deflections of the needle, which may be read by aid of a telescope and scale, will be proportional to the changes in potential. The instrument may also be used by charging the needle and one pair of quadrants to the same potential, while the other pair is kept at a different potential. In this case the deflection of the needle is proportional to the square of the potential difference between the quadrants. 292. Capillary Electrometer. When the surface between mercury and sulphuric acid is elec- trified, or, more properly, polarized, the surface tension is considerably affected, apparently by the release of one of the ions at the surface. This phenomenon has been utilized by Lippmann in the construction of a capillary electrometer, which has a limited application to the measure- ment of potential differences less than a volt. One form of the apparatus is shown in Fig. 214. FIG. 214. CAPACITY. 339 O is a capillary tube connected on one side with a tube, M^ filled with mercury, and on the other side with the tube $, containing dilute sulphuric acid. Platinum wires sealed into M and S, and terminating in the mercury cups P and JV, serve as electrodes. On connecting P to the positive and N to the negative pole of a small battery, the thread of mercury in the tube will fall a small distance, which may be taken as proportional to the potential difference, provided this does not approach an amount sufficient to produce continuous electrolysis. The variations in the hight of the mercury are best read by the aid of a micrometer microscope. 293. Capacity. The electrostatic capacity of a conductor is defined as the quotient of its charge by its potential. If the charge be denoted by Q, the potential by FJ and the capacity by (7, then The value of C depends only on the form and dimensions of the conductor and on the nature of the dielectric. 294. Condenser. The capacity of an insulated conductor is greatly increased by the presence of another conductor connected to the earth or to the walls of the room. Thus, let A (Fig. 215) be a conducting sphere, which is charged to a positive potential, Say ' F=16. If equipotential surfaces be FlG 215 drawn about A for unit poten- tial differences, there will be fourteen such surfaces between the conductor and the walls of the room. Now, let A be 340 ELECTROSTATIC INSTRUMENTS. surrounded with a conducting sphere, B. It will take the potential of the surface with which it coincides without altering the field. If the outer sphere be now connected to the walls of the room, its potential will fall to zero, and that of A to V= 4, so that if A be again put in communication with a reservoir of electricity maintained constantly at the potential 15, it is evident that a large quantity of electricity will have to be added to A before its potential will again rise to 15, because the presence of B at zero keeps the poten- tial of A below what it would be were A alone in the field. This arrangement of conductors is called a condenser. 295. Specific Inductive Capacity. If the space between A and B in Fig. 215 were occupied by any other insulator, the potential of A would in general be lowered, or, what amounts to the same thing, the capacity of the condenser would be increased. The original discovery of this fact was made by Cavendish, but not published. It was independently discovered by Fara- day in 1837, and became the starting point of the modern theory which assigns the electrification to the state of the dielectric rather than to the condition of the conductor. The ratio of the capacity of a condenser where the dielec- tric is a given substance to the capacity where the dielectric is air was called by Faraday the specific inductive capacity of the substance. The values of this quantity, which is identical with the dielectric constant of Art. 271, were investigated for a number of substances by comparing the capacities of a spherical condenser similar to that of Fig. 215, when different dielectrics were used ; but Faraday's results are now recognized as too small. The difficulties in the way of securing precise values of the specific inductive capacities of solids have never been overcome. It is found that, when a SPECIFIC INDUCTIVE CAPACITY. 341 condenser with solid dielectric is charged, its capacity appears to increase with the time, a phenomenon sometimes described as electric absorption or soaking in. The explanation seems to be that there is a slow yielding of the insulator under the electric stresses to which it is subjected. The following tables show the approximate values of K found by various investigators. TABLE OF SPECIFIC INDUCTIVE CAPACITIES. SOLIDS. SUBSTANCE. SPECIFIC INDUCTIVE CAPACITY. INVESTIGATOR. Hard crown glass . . Very light flint glass . Light flint glass . . Dense flint glass Extra dense flint glass Plate glass .... Shellac . 6.96 6.61 6.72 7.38 9.90 8.45 6.10 5.83 to 6.34 2 95 to 3 73 Hopkinson u it Wullner Schiller Wullner Paraffine 1 96 M u 2.32 1 85 to 2 47 Boltzmann Schiller Ebonite 2 56 Wullner 3 15 Boltzmann u 2 21 to 2 76 Schiller Sulphur 2 88 to 3.21 Wullner 3.84 Boltzmann LIQUIDS. Ether 4.2 to 4.4 Benzol .... 2.36 Carbon disulphide . 2.58 Petroleum . . . 2.02 to 2.07 342 ELECTROSTATIC INSTRUMENTS. GASES. Air 1.000 Hydrogen 0.9998 Carbon dioxide . . . 1.0008 Sulphur dioxide . . . 1.0037 Coal gas 1.0004 Vacuum 0.9985 296. Leyden Jar. The Leyden jar is a convenient form of condenser, consisting of a glass jar (Fig. 216) coated inside and out to a certain hight, 6r, with metal foil. The inner coat of the jar is connected to the knob D by means of the chain B and the rod 6 Y , inserted in the wooden stopper H. To charge the jar it is only necessary to set it upon a table or other uninsulated support and connect the knob with one of the terminals of an electrical machine. The discovery of the no. 2i6. principle of the condenser appears to have been made by Kleist, a German monk, in 1745, who showed that, if a nail be thrust through the cork of a vial in which there was a small quantity of mercury, the bottle could be very strongly electrified. A similar discovery was made the following year by Mu- schenbroek, of Leyden, while testing a notion he had, that water enclosed in a glass vessel would not lose its charge so rapidly as an exposed conductor. A wire connected with one terminal of the machine was arranged so as to dip into a vessel of water held in the hand of a pupil named Cuneus. The latter, attempting to remove the charging wire after the water had been electrified, received a violent shock. This experiment attracted wide attention and was repeated in many laboratories. "The name Leyden jar came to be applied to this particular form of condenser from the city in which the experiment originated. CAPACITY OF A SPHERICAL CONDUCTOR. 343 297. Capacity of a Spherical Conductor. Since the exter- nal field due to a charged sphere is in every respect equiva- lent to that produced by the same charge placed at its center, the potential of a charge, = Z(* + a \ = K (* b a \b a J \4-7rdf where A denotes the area of the inner sphere and d the dis- tance between the shells. It thus appears that, by the addi- tion of the outer shell connected to earth, the capacity of the sphere has been increased by the large quantity 299. Capacity of a Parallel Plate Condenser. Imagine that a portion of the spherical condenser (Fig. 217), having an area, S, is cut away. It is obvious that the capacity c of this portion will be the same part of C that 8 is of A, or, ' ii 0~ A Substituting the value of C from equation 4, KS at> KS b b _ a In case the conducting plates are planes, - = 1 ; thus letting ct d denote their separation, KS EXAMPLES. 345 EXAMPLES. 1. If a sphere 6.4 cm. in diameter and charged with 98 C. G. S. units be connected by a fine wire with a second conducting sphere whose radius is 2.5 cm., what will the charge and potential of each sphere then be? ( v = 17.2 gm.* cm.*/ sec. Ans. ) q^ = 55 gm.* cm.* / sec. ( q 2 = 43 gm.* cm.* / sec. 2. Three spheres having capacities 3 cm., 5 cm., and 8 cm. are charged to potentials 30, 24, and 18 units, respectively, and then con- nected by a fine wire. What will be their common potential? Ans. 22.1 cm.* gm.* / sec. 3. A spherical conductor having a radius of 2.7 cm. is charged to a potential of 125 C. G. S. units. On being brought in contact with a second uncharged conductor, the potential fell to 22 units. What was the capacity of the second body? Ans. 12.7 cm. 4. A spherical conductor having a radius, r v and a potential, V v is introduced into a second hollow sphere having a radius, r 2 , and a poten- tial, F 2 , and touched to it. Required the charge and the potential of each sphere after contact. Ans. F=(r 1 F 1 + r 2 F 2 )/r 2 . / cm.* sec. F = 13.2 dynes. 2. If a pole of 280 units exerts a force of 0.93 gram weight on an- other pole at a distance of 3.2 cm. from it in air, what is the strength of the second pole? Ans. 2H = 33.3 gm. 5 cm.^ / sec. 3. What is the strength of the magnetic pole which experiences a force of 12.7 dynes, in a field whose intensity is 0.179 unit? Ans. 71 gin.' cm.^/ sec. 4. If a north pole of 78 units be placed at one corner of an equi- lateral triangle measuring 15 cm. on a side, and an equal south pole at another corner, what would be the intensity and direction of the field at the third corner ? . 0.346 gm.*/ cm.* sec. parallel to the line NS. 5. A magnetic needle is suspended at some distance above and parallel to a horizontal bar magnet, which lies in the magnetic merid- ian. When the north end of the bar points northward, the period of the needle is 7.43 sec., but when the magnet is reversed, the period of the needle is 4.31 sec. What would be the period of the needle in the earth's field alone? . Ans. 5.26 sec. 6. A weight of 1 gm. placed on the upper end of a dipping needle reduced the inclination from 65.2 to 39.5. What weight would be necessary to bring it to the horizontal? Ans. 1.62 gms. 7. A horizontal magnet is suspended freely by a wire in the mag- netic meridian. On twisting the upper end of the wire through 75 the magnet is deflected through 23.4. What additional rotation of the top of the wire will be necessary to turn the magnet at right angles to the meridian ? Ans. 144. 8. If a magnet vibrate in a period of 4.98 sec., where the horizontal intensity is 0.182 unit, what will be its period where the intensity is 0.251 unit? Ans. 4.24 sec. CHAPTER XXL THE ELECTRIC CURRENT. 339. Contact Difference of Potential or Volta Effect. When two dissimilar bodies are brought in contact and after- wards separated, they are, in general, found to be oppositely electrified. For instance, let (Fig. 241) be a plate of copper connected to the earth, and Z a plate of zinc attached to an electroscope, E, and so arranged that after it has been put in communica- tion with C it may be removed from the vicinity of the lat- ter. Also, suppose that the whole experiment is carried +1 \ G T on in air. If Z be momenta- p N v rily touched to and then re- moved, the gold leaf G- will FIG. 241. be observed to move toward N, the negatively charged pole of the electroscope, indicating that the potential of Z rises as it is removed ; that is to say, Z is positively charged with respect to (7, the difference of potential reaching a value not far from one volt, when the plates are separated a great distance. This potential difference represents a difference of chemi- cal attraction in the plates for the oxygen of the air, and may be calculated from the heat of combination of oxygen with zinc and copper, respectively. If the plates had been im- mersed in some other gas, as e.g. H 2 S, the potential differ-, ence on separating the plates would have been entirely different. 386 THE ELECTRIC CURRENT. The potential difference which is observed on bringing a glass rod in contact with silk (Art. 264) has the same origin as that just described, though since the bodies are non-con- ductors the electrification will occur only at the points of contact. However, by the operation of rubbing, fresh points are brought in contact, and the small charges are piled up till the whole surface is electrified. 340. Explanation of Contact Potential Difference. If the molecule of oxygen be regarded as the union of a positively charged atom with a similar negatively charged atom, then the contact difference of poten- tial, described in the preceding article, may be understood as arising in the following manner. Let Zn and Cu (Fig. 242) be plates of zinc and copper joined by a copper wire and surrounded by an atmosphere of oxygen. The plates will then be at the same potential, and if the tem- perature is the same through- out it may be assumed that there is no electromotive force at the points of contact between the wire and the plates. At the surfaces of the zinc and copper plates there will occur a union between the metals and the negatively charged oxygen atoms, and a consequent liberation of the positive atoms ; and since zinc has a greater affinity for oxygen than copper, there will be more free positive atoms near the zinc than the copper plate. In this way there is produced a dif- ference of potential between each plate and the layer of charged atoms next to it. - + +- - o o o o o d- FlG. 242. VOLTAIC CELL. 387 These amounts, as calculated from the heats of combina- tion with oxygen, show that the layer of atoms next the zinc should be about 1.8 volts above the zinc, and that the layer next the copper should be about 0.8 volt above this plate. Accordingly, since the metals are at the same poten- tial, there will be a fall of potential of one volt in passing from the layer of atoms next the zinc through the gas to the layer next the copper. The intermediate molecules will arrange themselves along the lines of force, the positive atoms experiencing a stress in the direction of the copper plate and the negative atoms one in the direction of the zinc. If the bond uniting any two atoms of the gas is strong enough to support the stress thus imposed, the whole system will re- main, electrically, in virtual equilibrium, and such is the case with air when the plates are separated by any consider- able distance. A gas thus behaves as a non-conductor as long as the intensity of the field is not sufficient to rupture the bond uniting the parts of the molecule. If, however, this bond is not sufficiently strong to sustain the stress put upon it in the field between the plates, the atoms will pair off in new ways, so that on the whole there will be a pro- cession of positively charged atoms toward the copper and negatively charged ones toward the zinc through the gas, while in the wire there will be a flow of electricity from the copper to the zinc. Something of this sort has been observed when the metal surfaces were very clean and brought very close together. 341. Voltaic Cell. If the plates of Fig. 242 be immersed in an electrolyte, that is, a compound liquid which will not sustain a potential difference exceeding a certain small amount without chemical decomposition, there will be a continuous flow of electricity through the wire. Thus, sup- 388 THE ELECTRIC CURRENT. pose that the two metals are connected by sulphuric acid, the molecule of which is regarded as consisting of the posi- tive group, or radical H 2 , and the negative radical SO 4 . Then, if a potential difference is momentarily established, as in Fig. 242, between layers of the liquid next to the plates, the bond uniting the hydrogen to the acid radical SO 4 will be ruptured, the hydrogen going toward the copper plate, where it collects in minute bubbles. At the same time the sulphion works its way toward the zinc, and uniting with it forms ZnSO 4 . The copper thus receives a positive charge from the hydrogen, and the zinc a negative one from the sulphion ; but as they are connected by a conducting wire, a continuous discharge occurs, which, according to the ordinary convention, is known as a current from the copper to the zinc. If, however, the external circuit is broken, the copper end will be found at a certain constant difference of poten- tial above the zinc end. The arrangement just described is known as the voltaic cell. Its essential elements are two dis- similar conductors, and an electrolyte capable of a reaction with one of the conductors. Any part of an apparatus at which it is assumed that the current enters is called the anode, i.e. the way in (up) ; and the part by which the current leaves is called the kathode, i.e. the way out (down). Collectively these parts are termed electrodes. In the cell just described the zinc is the anode and the copper the kathode. 342. History of the Voltaic Cell. Previous to 1786 no electrical phenomena were known other than those which were produced by the charges obtained from the electrical machine. In the year mentioned, Galvani, having observed that some freshly prepared frogs' legs, lying near an electri- cal machine, suffered a convulsive twitch at the moment of HISTORY OF THE VOLTAIC CELL. 389 iy-samea; discharge, sought to test the influence of atmospheric elec- tricity by suspending some similar preparations on a copper hook from the iron railing of a balcony. This experiment yielded only a negative result, but on attempting to discon- nect the hook, Galvani noticed that a speci- men which came in contact with the railing experienced the same convulsion which he had before observed at the instant of an electrical discharge. The experiment at- tracted wide attention. Galvani himself attributed the phenomenon to an electric charge generated in the muscles of the leg. This explanation, however, was rejected by Volta, a professor in the University of Padua, who maintained that the true cause was the contact of dissimilar metals. In support of his view he devised an experiment equivalent to that of Art. 339. Then, pursuing the same line of proof, he tried arranging a series of discs of zinc, silver, and moistened cloth, after the manner shown in Fig. 243. On connecting the top and bottom discs of such a " pile " with the wetted fingers, a perceptible shock was ob- tained. Volta further showed that a number of compound |- v,. .->y- ; .-.-. FIG. 243. FIG. 244. metal strips, half silver and half zinc (Fig. 244), bent so as to dip into a series of cups containing dilute acid, was an arrangement electrically equivalent to his " pile." As the jars were usually set in a circle, so as to bring the extremities nearer together, it received the name Crown of Cups. 390 THE ELECTRIC CURRENT. 343. Local Action. Specimens of commercial zinc usually contain impurities, such as bits of iron, carbon, or the like, which, together with the zinc and the liquid of the cell, form a miniature battery and generate currents at the surface of the plate with consequent wasting of the metal, whether the external circuit be open or not. Such chemical action within the cell is termed local action. It may in large measure be stopped by amalgamating the surface of the zinc, the fluid amalgam serving apparently to protect the foreign particles in the zinc from contact with the electrolyte. 344. Polarization. When a current is drawn from a sim- ple voltaic cell having, for example, the elements Zn, Cu, and H 2 SO 4 , the hydrogen soon forms such a layer over the cop- per as to make it virtually a hydrogen rather than a copper plate, and in consequence the current ceases. The cell is then said to be polarized. Polarization may be reduced or entirely prevented by surrounding the positive plate with a second liquid which will be decomposed by the hydrogen. A large variety of cells have been devised in recent years, the most important of which will be noted in the following articles. It will be noticed that each has its merits and defects, and that none of them admits of universal application. With the perfection of the dynamo, the primary battery has ceased to have the commercial importance it once possessed. 345. The Daniell Cell. The first and in many respects the most important of the constant current cells was devised by Daniell in 1836. A section of this typical two-fluid cell is shown in Fig. 245. A is a vessel filled with a dilute solution of sulphuric acid o* zinc sulphate, in which is placed the negative element, THE GROVE CELL. 391 a zinc cylinder, Z, and a porous earthenware jar, */, contain- ing a solution of copper sulphate, and the positive element (7, a strip of metallic copper. The porous septum serves to keep the liquids separate, but does not offer any sensible obstruction to the passage of the hydrogen toward the copper plate. Now, as it is freed, instead of col- lecting on this plate it decomposes the cop- per sulphate, freeing the metal, whicli is deposited on the copper plate. The chemi- cal reaction may be represented by H 2 + CuSO 4 = H 2 SO 4 + Cu. An equivalent form of Daniell's cell, FlG ^ known as the gravity battery, has been extensively used for telegraphic purposes. The copper plate, with some crystals of copper sulphate, is placed in the bottom of a jar, and covered to the depth of two or three inches with a solution of the same salt. The jar is then filled with a solution of zinc sulphate, and the zinc, in the form of a branched casting, is suspended in it near the top. If the jar is undisturbed, the liquids will remain separate on account of their difference of density. The advantages of the Daniell cell are the great constancy of its potential difference, or electromotive force, as it is frequently called, at 1.08 volts while supplying a current, and the cheapness of its materials. Its disabilities are a comparatively high internal resistance, a tendency of the copper to deposit on the porous cup, and a diffusion of the copper sulphate toward the zinc plate. 346. The Grove Cell. In the cell devised by Grove the positive element, a thin sheet of platinum, and the depolar- izer, strong nitric acid, are contained in a porous cup, which is surrounded by dilute sulphuric acid and the negative 392 THE ELECTRIC CURRENT. element, an amalgamated zinc plate. The reaction within the porous cup may be represented by 3 H 2 + 2 HNO 3 = 4 H 2 + N 2 O 2 . The nitric oxide, on coming to the air, takes up another molecule of oxygen, forming the familiar red peroxide. The advantages of this cell are a high E. M. F. (about 1.9 volts), a very low internal resistance, and entire absence of polarization. It has the serious drawback of emitting the noxious fumes of nitrogen tetroxide, and must be kept under a hood. Besides the fact that it is necessary to take the cell apart and wash it thoroughly every time it is used, the original cost of the cell is considerable. 347. Bunsen Cell. The platinum in Grove's cell is sometimes replaced by a stick of carbon, in which form it is known as the Bunsen cell. Since, however, the increase of size necessitates the use of a larger amount of nitric acid, the change is one of doubtful utility. Another change, due also to Bunsen, the substitution of a chromic solution for the nitric acid, is a distinct gain in the direction of economy. When potassium bichromate, K 2 Cr 2 O 7 , is treated with strong sulphuric acid, chromium trioxide, CrO 3 , one of the best oxi- dizing agents known, is formed. By its action the nascent hydrogen in the cell is reduced to water and prevented from collecting on the positive pole. In the ordinary form of the bichromate cell the porous cup is entirely dispensed with, and both the zinc and the carbon are dipped into the same solution. This cell possesses a very high electromotive force (2.1 volts), and is free from noxious lumes ; but, as the chromic acid attacks the zinc, it is neces- sary to remove the latter when not in use. In the last-men- tioned form the cell is far from a constant one. LE CLAN CHE CELL. CLARK CELL. 393 348. Leclanche Cell. The elements of an important form of cell devised by Leclanche are zinc and carbon ; the electro- lyte is sal-ammoniac, and the depolarizer oxide of manganese. In the original construction the MnO 2 was mixed with pow- dered carbon and packed in a porous cup, which surrounded the positive element. In the more recent forms the manganic oxide and the carbon are pressed together into a stick and introduced into the NH 4 C1 together with the zinc. The advantages of this cell are a considerable potential differ- ence, say 1.5 volts, and entire freedom from local action. If tightly sealed, it may be left years without attention. In the single fluid form it polarizes rapidly, but is of the high- est value for the purposes of signaling where it remains on open circuit, except for the brief period during which it is pressed into service. 349. Clark Cell. The voltaic cell now recognized as best suited to furnish a standard of potential difference, was first made and studied by Latimer Clark in 1873. The elements are zinc arid mercury, and the electrolyte mercurous sulphate mixed with a saturated solution of zinc sulphate. The form of this cell, recommended by the National Academy of Sciences, and adopted by act of Con- gress for the purpose of the legal definition of the volt, is shown in Fig. 246. The containing glass vessel consists of two tubes, A, B, not less than 2 cm. in diameter and 3 cm. long, joined to a common neck and fitted with a glass stop- per. At the bottom of each tube a wire, 0.4 mm. diameter, is sealed into the glass. This wire in one leg, B, is covered with pure mercury, and the other, in J., with an amalgam consisting of 90 parts mercury to 10 parts zinc. A layer of a paste, about 1 cm. thick, made of mercury, mercurous sul- phate, and crystals of zinc sulphate moistened with a solution 394 THE ELECTRIC CURRENT. of zinc sulphate, is placed above the mercury in the l&gB. The paste and the amalgam are then covered with a layer of crys- tals of zinc sulphate, about 1 cm. thick, and the whole vessel filled with a saturated solution of zinc sulphate. The stop- 4 Solution - Paste of Hg 2 S0 4 -Pure Hg. A B FIG. 246. per, having been brushed with a solution of shellac, is pressed firmly in place. The potential difference of such a cell between the temperatures 10 C. and 25 C. is accurately given by volts. P.D.=r 1.434 \l -0.000802 (t 15)\. 350. Definition of Current Strength. The strength of a current in a conductor is defined as the quantity of electricity which passes any cross section per unit time. If Q denote the total and constant flow across any section of a conductor, t the time, and i the strength, or intensity, of the current, (1) UNIT OF QUANTITY AND CURRENT. 395 351. Practical Unit of Quantity and Current. For the purpose of measuring electrical currents, the unit of quan- tity, defined in Art. 272, is found to be inconveniently small. Accordingly, a new unit, called the coulomb, has been selected. It may be defined as 3 (10) 9 C. G. S. electrostatic units of quantity. This leads at once to the practical unit of current, or ampere, which may be defined as that current in which one coulomb of electricity per second flows across any section of the conductor. 352. Ohm's Law. When a constant difference of poten- tial is maintained between two points of a circuit, the cur- rent which passes every intermediate point is found to be exactly proportional to the potential difference. This result, first announced by Ohm in 1827, is known as Ohm's Law. If i denote the current and E the difference of potential, this law may be written (2) i = kE, where k is a physical constant which depends only on the nature and dimensions of the conductor, if the temperature remains constant, and is independent of the value of the electromotive force. Ohm wrote p 0) i=^ where R = - is called the resistance. k It is of interest in this connection to remark that Caven- dish, in some unpublished experiments made in 1781, antici- 396 THE ELECTRIC CURRENT. pated the law of Ohm by showing that, if the resistance was a function of the current, the exponent in the relation, R = ai n , could not exceed 0.02, a being a mere number. This result is the more remarkable because it was obtained thirty years before the invention of the galvanometer, and with no other means of judging the current than receiving the shock of a Leyden jar when discharged through a poor conductor. 353. Definition of the Ohm. The practical unit of resist- ance, called the ohm, may be defined as the resistance of that conductor in which the potential difference of one volt be- tween its terminals would generate a current of one ampere. Thus, 1 volt 1 ampere = - 1 ohm The legal definition of the ohm is as follows : " The re- sistance offered to an unvarying electric current by a column of mercury at a temperature of melting ice, 14.4521 grams in mass, of a constant cross-sectional area, and of the length of 106.3 centimeters." The cross section of this thread is very nearly 1 square millimeter. A further discussion of the practical units and their dimen- sions will be found in Art. 412. 354. Electrolysis. When a fluid on being subjected to a definite potential difference is separated into two compo- nents, the process is called electrolysis. Any compound liquid which is capable of such separation is termed an electrolyte. The two components, called ions, into which it is divided ap- LAWS OF ELECTROLYSIS. 397 pear, however, only at the electrodes. That component which appears at the anode is called the anion, or electro-negative component; likewise, that component which appears at the kathode is called the kation, or electro-positive component. If, for instance, two electrodes of platinum be placed in hydrochloric acid and connected to the poles of a battery, hydrogen will be liberated at the negative electrode and chlorine at the positive. The 'phenomenon of electrolysis was discovered acciden- tally by Carlisle and Nicholson in the year 1800. On learn- ing of Volta's invention of the pile, these experimenters set to work at once to construct one from half-crown pieces, and, thinking to make a better contact at one point of the circuit, used a drop of water to connect the wire with a metal plate. When the circuit was closed, bubbles were observed to rise through the liquid, and the odor of hydrogen gas was detected. Further investigation showed that, whenever a current was passed through water, hydrogen was given off at the negative and oxygen at the positive electrode. 355. Laws of Electrolysis. The fundamental laws of electrolysis, established by Faraday, are as follows : The mass of the ions liberated at either pole is proportional 1, to the quantity of electricity which passes any section of the circuit. 2, to the chemical equivalent of the ion, by which is meant its atomic mass divided by its valence in the com- pound electrolyzed. Thus, let q stand for the quantity of electricity which passes any point of the circuit, m the total mass liberated, A the atomic mass of the ion, and V its valence. Then A (4) m = kq, 398 THE ELECTRIC CURRENT. where k is a constant depending only on the units employed. If t denote the time of flow of the steady current z, equation 4 may be written kA . (5) m=^t = ^t, kA by writing e = . e is called the electro-chemical equivalent. It is found by experiment that one ampere, flowing for one second, deposits 0.001118 gm. of silver. Taking the atomic mass of silver as 108, that is, = 16, the value of k is found to be 1.035(10)~ 5 , from which the electro-chemical equivalents of the other elements may be calculated. Thus, by equation 5, if the atomic mass of copper be taken as 63.64, and its valence as 2, the electro-chemical equiva- lent of copper is = 0.0003294- %^- 1 ampere 1 sec. 356. Theory of Electrolysis. The facts of electrolysis are best accounted for by the theory originally proposed by Grotthuss and developed by Clausius. Each molecule of an electrolyte is assumed to be composed of two parts which are oppositely charged, and united by certain bonds usually designated as chemical. These parts may consist either of single atoms as in HC1, or of a group of atoms, which in many reactions behaves like an elementary substance, in that the group may be transferred from one compound to another without loss of identity. NO 3 and SO 4 are examples of such groups or compound radicals. It further seems proba- ble that the positive ion of the molecule does not remain united to the same individual negative ion, but that there is a frequent change of partners, and that during this inter- change some of the ions may remain for a longer or shorter time disconnected from the other portion of the chemical THEORY OF ELECTROLYSIS. 399 molecule. This assumption is justified by the fact pointed out in Art. 197, namely, that the change of the freezing point and the vapor pressure in solutions of strong bases and acids was such as to indicate the presence of a number of mole- cules greater than that calculated from other data. Important evidence that continual interchange in the parts of the molecule takes place is furnished by the behavior of the solutions of strongly combined salts in forming com- pounds in which the uniting bonds are much weaker. If the weaker compound is sufficiently insoluble, it appears as a precipitate, and justifies the conclusion that a quantity of the corresponding compound is present in every case, though from its solubility it may not be so readily detected. Suppose, now, that any compound liquid, such as hydro- chloric acid, is subjected to a difference of potential, by being placed between the electrified plates P, N (Fig. 247). The first effect will be to face all the mole- cules about, so that the negative or chlorine atom will be toward the positive electrode, and the hydrogen toward the negative electrode. If, FlG - 247> now, a molecule close to the positive electrode were to be broken up or dissociated, a process which as mentioned above is to be regarded as continually going on, the chlorine would be attracted to P, and might give up its charge, while the hydrogen would start to move toward the other electrode. Before it has gone far it may be supposed to meet a free chlorine atom, with which it pairs off. On being again set free it will make another step toward its goal, the negative electrode, which it will ultimately reach. In the same way the free chlorine atoms will gradually work their way toward the positively charged plate. 400 THE ELECTRIC CURRENT. There are thus two material currents of ions constantly moving in opposite directions through the liquid,- maintain- ing corresponding convection currents of electricity. The part of this theory which is most open to objection is the rather arbitrary way in which the ions are made to give up their charges, and the assumption that every monad atom, whatever its nature, carries the same charge, every dyad atom or ion twice as much, every triad atom or ion three times this amount, and so on. Nevertheless, the theory is a most valuable aid in the statement of the facts of electrolysis. If the potential difference between the electrodes does not exceed a certain minimum amount, there is observed a feeble current through the liquid for a time, but, however, without liberation of the ions, as if the uncombined ions moved up to the electrodes and the process stopped for lack of further dissociation, or that an atom could not be forced into com- pany of its own kind except by an electromotive force exceeding a finite magnitude. The statement that the potential difference must exceed a finite amount does not, however, apply to the case where the anode is the same metal as the kation. 357. Secondary Actions. In many cases important sec- ondary chemical processes arise from the evolution of the ions, so that the substances liberated at the electrodes are the products of the action of the ions on the solution of the electrolyte. Thus, in the electrolysis of a solution of Na 2 SO 4 , hydrogen is liberated at the kathode and oxygen at the anode. The initial products of electrolysis in this case are probably Na 2 and SO 4 , but as sodium is a highly oxidizable element, it robs the water of its oxygen, forming Na 2 O and setting the hydrogen free. At the same time the sulphion at the anode, not beting able to exist in the free state, breaks up THE VOLTAMETER. 401 into O and SO 3 . The oxygen gas is liberated, while the sulphur trioxide takes up another molecule of water, forming H 3 S0 4 . When a current is passed through water containing a small quantity of H 2 SO 4 , the products of electrolysis are exactly the components of water, two volumes of hydrogen and one of oxygen. Water was, accordingly, long regarded as a typical electrolyte. The researches of Kohlrausch have, however, shown that, in proportion as water is freed from impurities, its resistance increases, and that absolutely pure water would doubtless be a non-conductor. The hydrogen is, accordingly, to be regarded as arising from the electrolysis of H 2 SO 4 and the oxygen from the decomposition of SO 4 . It is possible, in some cases, to deposit an alloy of two metals. Thus the kathode may be plated with brass from a mixture of the cyanides of zinc and of copper. The secondary actions in the electrolytic cell are often very complicated and difficult to control. The alkali metals cannot be obtained by using weak solutions of their salts. 358. The Voltameter. Faraday, after his discovery of the law expressed by equation 5, proposed the use of the electrolytic cell as a current measurer, which he named the voltameter. Experience has shown that silver is the best sub- stance to use for deposit in such a cell. The legal definition of the ampere, by act of Congress, is "the practical equiva- lent of the unvarying current, which, when passed through a solution of nitrate of silver in water, in accordance with standard specifications, deposits silver at the rate of 0.001118 gram per second." The arrangement prescribed is as follows : The kathode shall be a platinum bowl, not less than 10 cm. in diameter and 4 cm. deep. Before use it must be washed 402 THE ELECTRIC CURRENT. successively with nitric acid, distilled water, and absolute alcohol. The anode shall be a disc of pure silver, supported by a silver rod riveted through the center. To prevent any parti- cles of silver falling on the kathode, the anode must be wrapped with filter paper properly secured. The electro- lyte shall consist of a neutral solution of pure silver nitrate, containing about 15 parts, by weight, of the nitrate to 85 parts of water. The external circuit must contain a resist- ance of at least 10 ohms, and the total current through this voltameter must not be much over one ampere. After the current has been allowed to run about half an hour, the deposit is carefully washed and dried. The current may then be calculated by the formula . m , ^ > where t =0.001118 gm. amp. x 1 sec. For the purpose of measuring stronger currents the copper voltameter, consisting of an anode of pure copper in a solu- tion of CuSO 4 , or Cu2(NO 3 ), is used. The tendency of copper toward oxidation with strong currents, or toward dissolution in the liquid with feeble ones, renders determinations with the copper voltameter less certain than those made with the silver voltameter. 359. Applications of Electrolysis. The process of elec- trolysis has been extensively used in the arts: 1. In the reduction of metals from solutions of their ores. Copper so deposited is remarkable for its purity. 2. In electroplating, where it is used chiefly to deposit a layer of gold or silver on the surface of a less precious metal. SECONDARY CELLS. 403 The solutions used are generally the double cyanide of potas- sium and gold or silver, as the case may be. Surfaces of brass are frequently nickel-plated, to prevent tarnishing, and sur- faces of steel, to prevent rusting. The latter end will not, however, be attained unless the surface has been previously coated with copper. The layer of zinc on galvanized iron, so called, is not electrolytically deposited, as the name would imply, but is a coating obtained by dipping the iron in melted zinc. 3. In electrotyping, a method of copying in reversed relief the designs of woodcuts, printing type, engraved plates, etc., in a thin sheet of electrolytically deposited copper. The sur- face to be copied is first coated with a thin layer of graphite, and then immersed in the copper solution, as the kathode. After being subjected to the action of the current for a suffi- cient time, a thin sheet of copper is deposited, which repro- duces with great fidelity the original design of the mold. This sheet is strengthened by filling the back with melted type metal. Such plates may then be used to print from, and will afford about 80,000 impressions. Most books which run through large editions are printed, not from the original type, which would wear rapidly, but from copper-plate copies of the page. 360. Secondary Cells. After a current has passed through an electrolytic cell, consisting, e.g., of two platinum electrodes and acidulated water, the plates become coated respectively with a layer of the liberated gases, so that they are no longer electrically equivalent to platinum, but are virtually plates of hydrogen and oxygen. As there are then present all the essentials of a voltaic cell, namely, two dissimilar conductors and an electrolyte, the electrodes, if disconnected from the original source of current, will show a reverse electromotive 404 THE ELECTRIC CURRENT. force, and will give a very brief current in the opposite direction on being connected by a conductor. Electrolytic cells used in this manner as a source of current are called secondary or storage batteries. By using electrodes which may be chemically changed, the energy stored in such a cell may be much increased. Plante, in 1860, devised a cell consisting of two plates of lead im- mersed in sulphuric acid. By the passage of the primary current, the surface of the anode was coated by a film of lead dioxide, and that of the kathode reduced to a spongy metallic state in which it is chemically very active. On connecting the electrodes for the discharge, the hydrogen liberated at the positive plate reduces the dioxide to monoxide, while the oxygen, going to the negative plate, forms a similar layer of oxide there. The potential difference of such a cell is from 1.8 to 2 volts. Faure, in 1881, modified the Plante cell by rolling up the lead plates with a layer of red lead and flannel between them. In charging, the red lead at the anode is peroxidized, while that at the kathode is reduced to a lower oxide, and finally to the metallic state. By this arrangement a greater layer of the working substance is obtained, and less time is required to " form " the plates. In the more modern forms, the red lead, or some equivalent paste, is pressed into holes in a lead grating or grid, where it is held by a suitable attach- ment. Secondary cells have a wide field of application and their actual efficiency is high, but the present cost of con- struction and maintenance is so great as to impair their commercial efficiency when employed on a large scale. EXAMPLES. 405 EXAMPLES. 1. What is the value of the current which would deposit 0.156 gm. of silver in 3 min. 33 sec. ? Ans. 0.655 ampere. 2. How much water should be decomposed by a current of 0.95 ampere in 58 min. ? Ans. 0.308 gram. 3. The current through an incandescent lamp is 0.682 ampere, and the potential difference 87.6 volts. What is the resistance of the lamp? Ans. 128 ohms. 4. What current will a battery of 8 Daniell cells produce through an external circuit having a resistance of 5.7 ohms, the resistance of each cell being 3.4 ohms, and the E. M. F. 1.08 volts. ? Ans. 0.263 ampere. 5. A dynamo with internal resistance of 11 ohms produces an E.M.F. of 830 volts. How many 10-ampere arc lamps in series will such a machine supply if the resistance of each lamp is 4,5 ohms ? Ans. 16 lamps. CHAPTER XXII. THE ELECTROMAGNETIC FIELD. 361. Oersted's Discovery. It had been known since 1676 that there existed some connection between electricity and magnetism, for compass needles had been reversed in polarity during thunderstorms, and steel wires magnetized by the discharge of Leyden jars, but the effects observed were too erratic to permit any systematic conclusions to be drawn from them. The first definite relation was discovered by Oersted in 1819, who found that when a conductor convey- ing a current was presented to a magnetic needle, the latter turned so as to set itself at right angles to the conductor ; and, further, that if the magnet be moved around the wire the same end would always point forward. 362. Magnetic Field Due to a Current in a Straight Con- ductor. In terms of the magnetic field, Oersted's discovery is equivalent to the statement that the lines of magnetic force produced by a current in a straight conductor, so arranged that the other portions of the circuit are at a great distance, are circles in a plane at right angles to the conductor, their direction being related to that of the current as the direction of rotation is to the advance of a right-handed screw. Thus, if A (Fig. 248) represent the end of a conductor into which a current is flowing, then the lines of force will have the direction indicated in the figure. FIG. 248. CURRENT IN A CIRCULAR COIL. 407 - The existence of such a field may be shown experimen- tally by passing a stout copper wire through a piece of card- board and scattering iron filings over the latter. If the card be gently tapped while a strong current is sent through the wire, the filings may be seen to arrange themselves about the wire in concentric circles which contract as the tapping continues. This movement of the filings across the field is an example of the motion of a paramagnetic body from places of weaker to those of stronger force. If the force on each of the poles N, S of the magnet in Fig. 248 be resolved into com- ponents parallel and perpendicular to its axis, it will appear that the magnet as a whole will not be urged in either direction parallel to the line of force, but that there will be a resultant attraction per- pendicular to this direction, which will draw the magnet toward the s ( conductor. 363. Field Produced by a Cur- rent in a Circular Coil. The field produced by a current flowing through a wire bent into the form of a circle is shown in Fig. 249. If the current is supposed to enter the plane of the paper at A and leave it at 5, then a magnet, NS, placed in the axis of the coil would set itself so that the north pole would point to the right in the figure. It may be shown, from theoretical considerations alone, that the field produced by a current flowing through a cir- FlO. 249. 408 THE ELECTROMAGNETIC FIELD. cuit is identical with that which would be produced by a magnetic shell uniformly magnetized perpendicular to its surface and bounded by the circuit, except that this state- ment must not be applied to a point within the substance of the shell. The magnetic moment of the circuit may be found by multiplying the area of the circuit by the strength of the current, and the direction of the magnetization by the familiar right-handed screw rule. Thus, in the figure, if a screw were rotated in the direction the current flows, it would advance toward the north face of the shell. 364. Ampere's Theory of Magnetism. The equivalence in magnetic action of a small plane circuit, at distances which are great compared to the dimensions of the circuit, and of a magnet whose axis is perpendicular to the plane of the circuit, was first shown by Ampere, who perceived that all magnetic phenomena could be explained in terms of electric currents. He, accordingly, propounded the hypothesis, now generally accepted, that each molecule of a magnetic sub- stance has a current circulating in it. Since magnetism does not appear to vary with the time, it is necessary to suppose that the molecular currents flow without resistance. As Ampere's theory is not inconsistent with any known facts, all the laws of magnetism might be derived from the conception of a small closed circuit, though such a method would evidently lack much of the simpli- city and directness of that employed in Chapter XX. m . 365. Force Exerted by a Current on a Pole. Let ABDE (Fig. 250) be a circuit having the portion ED bent into the arc of a circle, while the portions BA and DE ex- FIG. 250. tend in radial directions to a great distance. ROWLAND'S EXPERIMENT. 409 Call r the radius of the circle, I " length of the arc BD, m " strength of the pole placed at the center; then if a current, i, be sent through the circuit in the direc- tion BD, experiment has shown that the north pole m will be acted on by a force directed up from the paper, which may be written (1) f=k, r 2 where k is a constant depending on the units employed. 366. Rowland's Experiment. The effect on a magnetic needle of a convection current, produced by moving a charged conductor, was investigated by Rowland in 1876. The ap- paratus consisted, essentially, of a gilded ebonite disc main- tained at a high potential and revolved beneath a sensitive magnetic needle which was carefully shielded from air cur- rents and electrostatic influences. When the needle was suspended parallel to the circumference of the rapidly re- volving disc, it was deflected to the right or left according to the sign of the charge, showing that a convection current produced the same magnetic effects as a current in a wire. 367. Electromagnetic Unit of Current. Since equation 1 is an independent equation, it may be used to obtain a new definition of current strength. Thus, taking & = 1, and giving the other quantities their usual unit values, i also becomes unity. This new, or electromagnetic unit of cur- rent, expressed in words, is that current which exerts the force of one dyne on the unit magnetic pole placed at the center of a circle having a radius of one centimeter, when flowing through an arc of this circle one centimeter long. The ampere or practical unit of current is one-tenth of this electromagnetic unit. 410 THE ELECTROMAGNETIC FIELD. 368. Circular Coil in a Uniform Field. By the law stated in Art. 363, that any closed circuit is equivalent to a magnetic shell having a magnetic moment equal to the strength of the current multiplied by the area of the circuit, the turning effect experienced by a plane circuit in a uniform field may be written down at once. Let H be the strength of the field, A the area of the coil, i the strength of the current, and 6 the angle made by the north ^_ > face of the coil with the lines of V\.^\Q * force (Fig. 251). Then the me- \- * chanical moment experienced by the . > coil will be FIG. 251. M ^ HAi gin ^ The fact that any small circular current will, after the analogy of a magnetic shell, turn in a magnetic field so that its positive or north side will face along the lines of force may be differently expressed, thus: Every electric circuit will turn so as to include a maximum number of lines of force. In applying this rule the lines of the field which pass through the coil in the same direction as those pro- duced by the current are to be reckoned positive, and those which pass in the opposite direction negative. 369. Force Experienced by a Conductor in a Magnetic Field. The advantage of the above method of treating the motion of a small coil in a magnetic field is that the rule just derived is applicable to all circuits, whether movable in parts or as a whole. The generalization is due to Maxwell, who stated it essentially as follows : Every conductor conveying a cur- rent, when placed in a magnetic field, experiences a force urging it in such a direction as to increase the total induc- tion through the area bounded by the circuit. ELECTROMAGNETIC ROTATIONS. 411 FIG. 252. This principle is illustrated in the apparatus shown in Fig. 252. GrP is a movable conductor supported on a pivot, 6r, resting in a mercury cup, E, and dipping into a trough of mercury at P. A wire, ZA, joins the battery B in circuit with the cup and the trough. The current starting from the battery enters the mer- cury trough at Z>, where it divides, part going through the circuit DRP and part through DQP. These cur- rents again unite at P, and return to the battery by the path PaEZ. The lines of force due to the current from P to Cr run down through the area bounded by the circuit DRP 6r, and upward through the area bounded by DQPGr. If the induction due to the external magnetic field be downward through the area PRDQ, then the mova- ble arm PGr will experience a moment turning it from P toward (), since by such motion the induction through DRPGr is increased by the addition of posi- tive lines, and that through DQPGr is increased by the removal of negative lines. 370. Electromagnetic Rotations. The problem of continuous rotation of a portion of a circuit about a magnet, or, in other words, the continuous conver- sion of the energy of an electric current into avail- FIG. 253 a ^ e mechanical work, was first solved by Faraday in the simple device shown in Fig. 253, which consists of a tube stopped at the bottom by a cork cov- ered with a layer of mercury and pierced by a magnet, 412 THE ELECTROMAGNETIC FIELD. NS. The stopper at the other end carries a hook, from which is suspended a wire, PQ, dipping into the mercury. On passing a current through it by means of the connec- tions shown, PQ will revolve about the pole of the magnet in a manner explained in the preceding article. A great variety of other forms of apparatus have been devised, show- ing the rotation of a portion of a circuit about a magnet, or about another portion of the circuit, and the rotation of a mag- net about its axis when included in the circuit. They are all variations on the theme of Art. 369 and do not call for further explanation. Their chief interest is the place they occupy in the history of the development of the electric motor. 371. Solenoids. If a number of small, like circuits, all facing one way, be placed side by side on a common axis, they will form an apparatus whose external field does not differ sensibly from that which would be produced by a magnet built up of a series of magnetized A/ A M11M^ discs - Such a device is kn wn as a solenoid. This construction FIG. 254. . . , , is most conveniently approxi- mated by coiling a wire into the form of a helix (Fig. 254). If the current be made to traverse the coils in the direc- tion indicated, the end A will behave like the north pole of a magnet, and B like a south pole. There is, however, this distinction between a magnet and a solenoid, that within the magnet the lines of force, defined as in Art. 316, run from north to south, while within the helix they have the same direction as the induction, running from south to north. 372. Electromagnet. If the space within a helix be filled with a permeable medium, such as soft iron, the lat- ELECTROMAGNET. 413 ter will become magnetized by induction when the current passes, forming what is known as an electromagnet. On breaking the current, the magnetism nearly disappears if the iron be soft. The relation between the direction of the current and the polarity of the magnet is obviously the same as that of the solenoid. The most useful form of the electromagnet is one in which the core has a U, or horseshoe, form (Fig. 255). On closing the circuit at some point, which may be at any distance from the apparatus, the magnet will attract the armature KK'. If the core of the electromagnet is very soft, and the armature is not allowed to come into close contact with the poles, the iron will be demagnetized as soon as the current is broken, and the armature will fall back. If, however, the bar be permitted to come into intimate contact with the poles of the magnet, the residual magnetism will be so great that considerable force may be found necessary to detach the armature. The explanation is that, when the iron forms a continuous magnetic circuit, the molecules, in the strong field produced by the current, are oriented so that all face the same way, as one follows along any line of force, and each little magnetic molecule is constrained by the action of its neighbors, even after the original inducing force is removed. If, however, this arrangement is once broken by forcing off the armature, the magnetism of the core falls to an indefinitely small value depending on the retentiveness of the iron. The effects just noted also indi- cate why it is desirable that every permanent magnet should have its poles connected, when not in use, by a bar of soft 414 THE ELECTROMAGNETIC FIELD. iron, or keeper, for in this way the original setting of the molecules in the hardened steel is better guarded from derangement by the action of demagnetizing forces. When a magnet is inside a helix bearing a current, it is subject to a force in the direction of the axis, unless the arrangement of the helix and magnet is a symmetrical one. The conclusion holds true, also, if the magnet owes its mag- netism to the action of the current alone ; hence, a movable soft iron core will be drawn into a helix when made a part of a closed circuit. The mechanical advantage of such an arrangement is the readiness with which greater range of motion is secured, so that it is frequently employed in auto- matic regulating apparatus. Illustrations of its use may be seen in Figs. 260 and 278. 373. Tractive Force of a Magnet. - - The weight which can be supported by an electromagnet is limited only by the dimensions of the apparatus. If _F=the force in dynes, .Z? = the induction, A = area of contact at the pole face, it may be shown that i ^ Sw EXAMPLES. 1. If a current of 5.7 amperes flows through a coil 48 cm. in diameter, consisting of 3 turns of wire, what will be the strength of the field at the center? Ans. 0.448 grn.* / cm.* sec. 2. What must be the radius of a single coil which, when traversed by a given current, would produce a field at the center equal to that pro- duced by the same current sent through two circular concentric coils of 15 cm. and 45 cm. radius, respectively, and joined in series ? Ans. 11.2 cm. CHAPTER XXIII. GALVANOMETRY. 374. Galvanometer. The term galvanometer is commonly applied to any instrument used to measure or detect the existence of a current by a deflection produced in a mag- netic needle or its equivalent. If i denote the current and 6 the deflection of the needle, the value, of i may be written (l) i = Kf(0), in which K is a constant, and/(0) has a simple form, usually a circular function of the angle, or the angle itself. When K may be determined from the dimensions of the instrument and the strength of the earth's field alone, the apparatus is called an abso- lute galvanometer. 375. Tangent Galvanometer. The tangent galvanometer, so called because the current is calculated from the tan- gent of the angle of deflection, consists of a short needle suspended at the cen- ter, or on the axis, of one or more coils FIG. 256. of wire exactly circular in form (Fig. 256). The ends of the coils to which connection is made are brought close together, and bent at right angles to the plane of the coil so that they shall exert no influence on the needle. To find an expression for current strength in this instru- ment, suppose that AB, the plane of the coil (Fig. 257), is 416 GAL VANOME TR T. set parallel to the magnetic meridian, A being toward the north. Also suppose that the current is traversing the coil in such direction that the north pole, m, of the needle is urged to the right, and the south pole, m', to the left. Let R radius of the coil, i = current in electromag- netic units, m = strength of each pole of the needle, y , ; I = length of the needle, = deflection from plane of , the meridian, F n = force experienced by eacl FlG. 257. pole in the earth's field, F. = force experienced by each pole in the field of the coil, M H = moment experienced by the needle due to the earth, M. = moment experienced by the needle due to the coil, ff= horizontal component of the earth's field. Then, by equation 2, Art. 317, (2) F H =H\n, whence (3) M H = F H I sin 6 = ffml sin 0. Assuming that m is at the center of the coil, (4) . Fi = i -'> by equation 1, Art. 365, IF whence (5) M i = F t l cos 6 = ml - ^ cos B. THE SINE GALVANOMETER. 417 If the needle is in equilibrium, (6) M i + M a =0' 9 equating and solving for z, HR sin HE ^ (7) i = - = -^r tan 0. 2-jr cos 6 ZTT It thus appears that current strength does not depend on the magnetic constants of the needle if the dimensions of the latter are so small that both poles may be considered as at the center of trie coil. When used to measure large cur- rents, the coil of the galvanometer consists of a single turn of heavy copper wire. If feeble currents are to be measured, the coil is made of many turns of fine wire. In this case, since F i is proportional to the length of the wire, or to ZimR, where n represents the number of turns, the expression for the current becomes (8) ..- This equation is also written Ms.'^tan = Kta,n Q\ (JT Gr = =- being called the true constant of the galvanom- _/L eter and K the working constant. When great accuracy is desired, corrections must be made for the width and depth of the coil and the length of the needle. It should be noted that the value of i in each equation of this article is in electromagnetic C.G. S. units, and that to reduce it to amperes it must be multiplied by ten. 376. The Sine Galvanometer. If, instead of setting the coil of Fig. 256 parallel to the magnetic meridian, it had been 418 GAL VANOME TR Y. turned until its plane coincided with the needle while the current was running, M i would have lacked the term cos 6. The value of i would then have been H i = sin 6 ; Cr being the angle between the needle and the meridian. This form of instrument is not as convenient in use as is the tangent galvanometer, and it is little used. 377. Sensitive Galvanometers. Astatic System. Great sensitiveness in a galvanometer intended to detect small cur- rents may be secured by combining two or more strongly magnetized needles in an astatic system, i.e. one so arranged that it would experience a very small directive force in a uniform field, and suspending ^C^^-^s fid *t so ^ at different parts are affected by different coils of the same wire placed very close to the needles. One type of such galva- nometer is illustrated in Fig. 258. The needle system con- sists of an alumipum strip, ad, carrying two discs, B, C, to the backs of which are fastened pieces of magnetized watch spring, ns and Xs', the upper pair having their poles opposite to those of the lower pair. Such a system, provided the magnets are all alike and in the same plane, would be astatic, that is, would remain in equilibrium in all azimuths in a uniform field. A position of stable equilibrium is secured by a control magnet, NS, placed above the coils. The needle is suspended by a silk FlG SENSITIVE GALVANOMETERS. 419 fiber, so that the discs hang in the centers of two coils, D and E, wound in opposite directions. The face of one of these discs, B, carries a silvered mirror, and the deflection of the needle may be read with a telescope and scale. The vibrations are damped by the motion of the disc in a small air cavity. Such instruments are most often used to detect feeble currents, but may be made to give numerical values by the formula i=K0, provided 6 is small and K can be found. The usefulness of astatic instruments is seriously impaired by their sensi- tiveness to changes in the external field incident to the em- ployment of strong electric currents in the vicinity of most buildings. D'Arsonval Type. Another method of constructing a sensitive galvanometer is to suspend the coil in an intense mag- netic field. An example of this type, commonly known as the d'Arsonval galvanometer, is sketched in Fig. 259. The coil O is suspended between the poles of a strong permanent magnet, NS, by a strip of phosphor-bronze, which serves also to connect the coil with the external cir- cuit. The deflections produced by the current are read by observing through a telescope the image of a scale of equal parts in the mirror M. On breaking the circuit, the coil is restored to its initial position by the torsional elasticity of the suspension. This type of instrument possesses three valuable features. 1. It is practically independent of small variations in the external field. FIG. 259. 420 GAL VAN ONE TR Y. 2. Its oscillations may be rapidly damped by the currents induced in the coil (see Art. 414). 3. It may be made very sensitive by increasing the wind- ings and the strength of the field magnets. By the use of a shunt or divided circuit (Art. 390) the range of currents for which a sensitive galvanometer can be used may be very much extended. 378. Ballistic Galvanometer. If the duration of a transi- tory current is very brief compared with the period of the needle of the galvanometer, and there is little damping, the apparatus may be used to measure the quantity of electricity which has passed through the coils. An instrument designed for this purpose is called a ballistic galvanometer. Let Q denote the quantity of electricity which is discharged through the coil, a the angle of the first swing, and Jf the constant of the instrument. It may be shown that Q can be calcu- lated from the formula (9) Q = Ksm^' If the angle is small and the deflection is read by a tele- scope and scale, the quantity of .electricity may be taken as proportional to the first swing expressed in scale divisions. 379. Direct Reading Galvanometers. - For the measurement of currents in their industrial applications, a variety of galva- nometers have been devised which may be placed in the hands of unskilled workmen. The intensity of the current is usually indicated by the position of a pointer over a scale graduated to read amperes. An example of such an instrument, often called an ampere- WES TON AMPERE-METER. 421 meter, or ammeter, is shown in Fig. 260. CAB is a curved piece of iron suspended from the point (7, and so arranged that it is drawn into the coil D when the current passes from FtoG-. A pointer, OP, moving over an empirically gradu- ated scale, shows, within certain limits, the amount of any steady current which may be passed through the coil. On breaking the circuit, the weight of the moving system returns the pointer to the zero position. 380. Weston Ampere-Meter. Another type of industrial galvanometer, known as the Weston ampere-meter, con- FlG. 261. structed on the model of the d'Arsonval galvanometer, is shown in Fig. 261. The essential parts of the instrument are a permanent magnet, AGrB, of great constancy, a fixed soft iron cylinder, (7, which concentrates the field, and a coil 422 GALVANOMETRY. of wire, 6?, wound about an aluminum frame which turns freely on pivots V, V. The damping effect of this frame is sufficient to render the motion of the coil dead-beat, i.e. aperiodic. JE is a pointer fastened to the coil and moving over an empirically divided scale, whenever a current is passed through the coil d. On breaking the circuit, the coil is restored to the zero position by spiral springs made of FIG. 262. a non-magnetic alloy, so as to be unaffected by the field, and placed above and below. Temperature changes in the strength of the magnet are sufficiently compensated by varia- tions in the resistance of the wires. 381. Kelvin's Current Balance. Lord Kelvin has intro- duced a type of instrument for the determination of currents, in which the force exerted by one portion of a circuit on another is balanced by a sliding weight. A picture of such an electro-dynamometer is shown in Fig. 262. Its principle may be most readily understood from the diagrammatic sketch (Fig. 263). POTENTIAL GALVANOMETERS. 423 A, B, C, D are four horizontal fixed coils, between which is suspended by means of a flexible ligature of fine wires a5, a'b', a light beam carrying the coils M, N. When the cur- rent passes, the balance of the arm is disturbed by the forces exerted by the fixed coils on the movable ones, the C D end ^V being raised and M depressed. Equilib- rium is again established by sliding a counterpoise along the balance arm, which is graduated to a scale of. equal parts. It may be shown that the moment of the electromagnetic forces is proportional to the strength of the current in both the movable and the fixed coils, i.e. to i\ and since the moment of restitution of the counterpoise varies as its lever arm, if s denote the number of scale divisions which the counterpoise has been moved from its zero position, FIG. 263. (10) or, where the value of &, a constant, must be determined by experiment. 382. Potential Galvanometers. If two points, A, B (Fig. A B 264), of a circuit through which a current is flowing be connected through a galvanometer having a resistance so high that the current which passes through it does not sensibly affect the disposition of potential in the main circuit, the reading of the galvanometer may be taken as a measure of the potential difference between A and B. If the resistance PIG. 264. 424 GALVANOMETRY. R of the galvanometer circuit is known, the fall of potential by Ohm's Law will be (11) V a V b = iE. Galvanometers calibrated so as to read potential differences directly are known as potentiometers. High resistance current meters of the type shown in Figs. 260 and 261 are extensively used to measure potential differ- ences for industrial purposes. Instruments adapted to this purpose are commonly known as voltmeters. 383. Watt-Meter. Electro-dynamometers are sometimes arranged with Low-resistance fixed coils, which carry virtu- ally the whole current, and a high resistance movable coil joined as a shunt to the main cir- cuit, so as to permit the passage of a small quantity of electricity depending on the difference of potential between the points of junction. An arrangement of this sort is shown in Fig. 265. B, B' are the fixed coils carrying the main current, and (7, C' the high resistance coils pivoted so as to revolve about a horizontal axis, FlG. 265. ' and joined in parallel with B and B' at m and n. Now, since the deflection is an increasing function of the currents through both coils, it will be a function of the product of the main current and the potential difference between m and n, or, what amounts to the same thing, between the terminals P, Q. Thus, by calibrating the electro-dynamometer empirically, it may be made to indicate the rate at which energy is being supplied. Such an instru- ment is called a watt-meter. RESISTANCE OF A CONDUCTOR. 425 384. Resistance of a Conductor. The numerical defini- tion of electrical conductivity is quite analogous to that of thermal conductivity (see Art. 204). Thus, suppose that two cross sections in a conductor, at a distance, Z, and having an area, A, are maintained constantly at the potentials V\ and F 2 , then the constant electrical force which urges the electricity from places of higher to places of lower potential will be proportional to the rate of change of the potential, i.e. to I Also, since electricity behaves as an incompressible fluid, the total flow Q will vary as the area and as the time ; whence (12) goc or, (13) , = = *: where k is a constant depending only on the nature of the material. Comparing with equation 11, fC ^\. JiJL where r is called the specific resistance and k the specific conductivity. From equation 14 it appears that r is the resistance of a unit cube of the substance. 385. Variation of Resistance with the Temperature. The resistance of the metallic elements, in general, increases with rise of temperature by an amount nearly proportional to the change of temperature. 426 GAL VANOME TR Y. At 20 C. for most metals, as appears in the table follow- ing, the increase per degree is about T 4 ^ of one per cent. For alloys the temperature coefficient is considerably less. They are, accordingly, better suited to make standard resist- ance coils, since for small changes the temperature variations may be neglected. TABLE OF RESISTANCES. SPECIFIC RESISTANCE. Ohms to a centimeter cube PERCENTAGE OF VARIATION for a degree at 20 C. Silv6r annealed 1.488(10)- 0.377 " hard drawn ...... 1.616 1.580 0.388 1.616 2.036 " 0.365 " hard drawn 2.072 Zinc pressed 5.566 " 0.365 l^latinum annealed . . . . 8.057 Iron " 9.611 " Nickel " . . 12.320 " Tin pressed . . . . 13.070 " 0.365 Lead " 19 420 " 0387 German silver tl ... 20.76 " 35.110 " 389 Bismuth " 12.970 " 354 Mercury liquid ...... 94.070 " 07 A method of estimation of the temperature, from the measured resistance of a platinum wire, has recently been introduced, and affords a means of determining temperatures which are below the range of other thermometers. The apparatus arranged for this purpose is known as a platinum thermometer. PHOTO-ELECTRIC PROPERTIES OF SELENIUM. 427 In carbon the change of resistance is in the opposite direction to that in metals ; that is to say, it is a decreas- ing function of the temperature. The specific resistance of electric light carbons is 3.93(10) 6 C. G. S. units, and the decrease of resistance between and 100 C. is between ^ and Jg. The resistance of the filament of a glow lamp when hot is only about one-half what it is when cold. 386. Photo-Electric Properties of Selenium. In 1873 J. E. Mayhew discovered that the resistance of selenium, which had been carefully annealed, was less when exposed to sunlight than in the dark. Tellurium and carbon are also slightly sensitive to light. 387. Resistance of Insulators. The resistance of a num- ber of the best insulators is shown in the following table. SUBSTANCE. SPEC. RESIST. C. G. S. Units. TEMP. CENT. Mica . 8.4 10 22 20 Gutta percha ..... 4 5 10 28 24 Shellac . . 9 10 24 28 2.8 10 25 46 Paraffine 3.4 10 25 46 Glass oreater than anv of the above. The resistance of insulators decreases very greatly with rising temperature. Some of them, e.g. glass, behave as electrolytes as soon as they begin to soften. 388. Resistance Boxes. For making electrical measure- ments a number of standard resistances are necessary. These are usually constructed of a copper, zinc, and nickel alloy, and are arranged in sets so that all combinations between the lowest and highest resistance of the group may be 428 GAL VANOME TR Y. obtained. For this purpose the ends of each coil, after being wound double or non-inductively to prevent sparking and magnetic effects, are connected to a series of brass blocks, A, B, C, D (Fig. 266), and the whole attached to a con- venient frame or box. Any of the spaces between the blocks may be closed by the insertion of tapered brass T^TP 9fift plugs, which have the effect of cutting out of the circuit the resistance of the coils so joined. Combinations giving all resistances from 1 to 110 are made with the greatest facility if the individual coils have the resistances 1,2,2,5,10,20,20,50; but any required resistance may be obtained with the fewest number of coils when their resistances are in the series 20 2 1 2 2 2 3 2 4 2 ra . Eight coils in this series, for instance, give all resistances between 1 and 255. E 389. Verification of Ohm's Law. The law i=j\ may be verified within the limit of the errors of observation by either of the following methods : 1st method. Fall of potential along a wire. | a Let B (Fig. 267) be some constant source of electro- motive force, such as a FIG. 267, Daniell's cell, which pro- duces a steady current, i, in a wire, AD, of uniform cross section. Let a, 5, will never exceed -, or that from r one cell on short circuit. *"* 272 ' If n similar cells (Fig. 272) be connected in parallel, the resistance will be decreased w-fold, but the potential difference will be that of one cell, 432 GALVANOMETRY. namely, e. Hence, the current due to this arrangement will be ._ e ne \t\ v ~~ 9 n from which it appears that most will be gained when the external resistance is small. The maximum current from n cells may be obtained by a combination of the two preced- ing methods of grouping. Thus, let the cells (as in Fig. 273) be arranged in q rows, in parallel with p cells in each row. By equation 21 the current furnished by one row would be -, and by equation 22 the effect of coupling q rows will be l| ! j | i ./ '^V (23) pr -f- qR r R P This may be shown to be a maximum for - = Thus, r R 9 P suppose that - and are represented by the sides of a rect- angle. The area of this rectangle will be constant for rR rR = = constant. pq n But by geometry the square has the least perimeter of all rectangles having the same area ; that is, --\ is least for r R V P - = Hence, i (equation 23) is a maximum for this value; it is important, however, to note that this does not yield a WHEATSTONE'S BRIDGE. 433 maximum efficiency, which always involves a minimum internal resistance of the battery. 392. Wheatstone's Bridge. The most common method of determining an unknown resistance is by means of an apparatus called Wheatstone's Bridge. The arrangement of the parts is shown diagrammatically in Fig. 274. Four conductors, forming the arms of the bridge ABCD, are joined in the order shown. The points A, C, are connected to the terminals of a battery, E, and the points B, _Z), by the bridge wire which contains a gal- vanometer. In order to find the condition for which the current through BD van- ishes, call the resistances of the arms r v r 2"> r s"> r v an( ^ the corresponding currents i v t' 2 , i z , ^' 4 . Let V a V c be the potentials of the points A and (7. If there is to be no current through BD, the potential at B and at D must be the same. Let it be denoted by V. Then, applying Ohm's Law to each arm, (24) From the 1st and 3d of these equations, (25) r x ij = r 3 8 ; and from the 2d and 4th, (26) 434 GALVANOMETRY. Now, since all the current which flows through AB, by supposition, also goes through Hence, substituting and dividing, ?*1 ^*Q (27) -!=-$. *2 *4 One of the common forms which Wheats tone's bridge takes in practice, and known as the slide-wire bridge, is shown in Fig. 275. From A to Q a German -silver wire of uniform cross sec- tion is stretched over a scale of equal parts. The portions of the circuit AB C, indi- cated by heavy lines, are stout pieces of copper, without appreciable resistance. The gap n is closed by a known re- sistance, R, and the gap m by the resistance whose value is sought. Both the galvanometer and the battery circuits are provided with keys, K, K\ for making and breaking the cur- rent. Connection between the galvanometer and the wire AC is made by means of the sliding contact D. In order to obtain a measurement of the resistance X, a position for D is found such that, on closing K and K\ there is no deflection of the galvanometer needle. Under these conditions (28) X by equation 27, since the resistance of the portions AD and DO of a uniform wire may be taken as proportional to their lengths. POST-OFFICE PATTERN OF RESISTANCE BOX. 435 D ioooo 10 C 10 loo 1000 JJB 393. Post-Office Pattern of Resist- ance Box. Fig. 276 shows a form of resistance box which may also be used as a Wheat- stone's bridge, the unknown resist- FlG< m ance being inserted between A and B. corresponds to Figs. 274 and 275. 30'0 200 _ 100 JO _ \1NF\~ | 100\ 1000 _ 2000' 3000 4000 ^r-E The lettering FIG. 277. 394. Bolometer. The change in the electrical resistance of conductors with the tem- perature has been applied by \T Langley to the measurement of the energy radiated by a hot body. This instrument, which he calls the bolometer, consists essentially of a Wheatstone's bridge, having two strips of blackened platinum foil, $, T (Fig. 277), inserted in the arms. If, after a balance has been obtained in the usual way, one of these strips be exposed to waves radiated from the source which it is desired to examine, while the other strip is screened, the change of the resistance, in consequence of the warming of the strip, will cause a deflection of the gal- vanometer needle. In this way a difference of temperature between the strips of j^^ of a degree may be measured, and one-tenth of this amount may be detected. The bolom- eter, though not as sensitive as the radio-micrometer, has an advantage in that it may be moved to different positions during an experiment. 436 GALVANOMETEY. EXAMPLES. 1. The end A of a wire ABC is maintained at a potential of 126 volts above the end C. If the resistance of A B is 32 ohms, and that of BC is 19 ohms, what will be the current through the wire, and what the potential difference between A and ? Ans. 2.48 amperes; 79. volts. 2. If the addition of 5 ohms to a circuit reduces the current to 0.7 of its former value, how many ohms should be added to reduce the cir- cuit to one-fourth of its original value? Ans. 35 ohms. 3. The resistance of 10.15 gms. of mercury, when contained in a uniform tube 92.1 cm. long, was found to be 1.059 ohms. What was the specific resistance of the mercury ? Ans. 0.932 (10)- 4 ohm X cm. 4. What would be the resistance of a copper wire 137 cm. long, and having a diameter of 0.038 mm.? Ans. 19.3 ohms. 5. If the resistance of a wire, 0.068 cm. in diameter, is 16.3 ohms, what would be the resistance of a wire of the same length and material having a diameter of 0.024 cm.? Ans. 131 ohms. 6. How much German-silver wire, 0.083 cm. in diameter, would be required to make a 6-ohm coil, if r = 20.76(10)-? Ans. 1560cm. 7. What length of iron wire, 0.86 cm. in diameter, will have the same resistance as a piece of copper wire 632 meters long and 0.125 cm. in diameter? Ans. 4960 meters. 8. A piece of copper wire 18.12 meters long has a resistance of 3.028 ohms. What would be the length of a coil of the same wire which had a resistance of 22.65 ohms? Ans. 135.5 meters. 9. It is found that there is no current through the galvanometer in a Wheatstone's bridge, when contact is made at a point 53.8 cm. from the end of the slide wire which is 100 cm. long. If the resistance in the homologous arm of the bridge is 38 ohms, what is the other resistance? Ans. 32.6 ohms. 10. A certain piece of wire has a resistance of 0.082 ohm to the meter. Required the resistance between the angles A and C of a skele- ton triangle, ABC, made of this wire, in which AB is 3.7 meters long, BC is 5.2 meters long, and CA is 2.6 meters long. Ans. 0.165 ohm. EXAMPLES. 437 11. Two wires which separately have resistances of 37.3 ohms and 45.8 ohms are joined in parallel so that a total current of 78.4 amperes flows through the system. What is the resistance of this portion of the circuit, and what current flows through each branch ? Ans. C 1 = 43.2 amperes ; C% = 35.2 amperes. 12. A piece of wire having a resistance of 36 ohms is bent into a square and its ends soldered together. The diagonally opposite corners M t N are then connected to the poles of a battery whose resistance is 12 ohms. Compare the current through the battery with that which would pass if M and N were connected by a single straight piece of the same wire. Ans. C l ==l.l7Q (7 2 . 13. A galvanometer having a resistance of 146 ohms is shunted by a coil whose resistance is 65 ohms. If the other resistance in the cir- cuit is 30 ohms, compare the current through the galvanometer with and without the shunt. Ans. C^ = 0.723 C 2 . 14. If a galvanometer having a resistance, G, is shunted through a resistance, S, and the other resistance of the circuit be denoted by R, what resistance must be added to keep the main current constant ? G* Ans. (jr-\- O 15. What shunt will be necessary to reduce the sensitiveness of a galvanometer having a resistance of 278 ohms to T ^ of its normal value? Ans. 2.81 ohms. 16. When a galvanometer having a resistance of 270 ohms is shunted with a 30-ohm coil, it is found that the current through the galvanom- eter is halved. What was the resistance of the external circuit in the first case? Ans. 33.8 ohms. 17. A certain galvanometer having a resistance of 65 ohms cannot safely carry a current of more than y 1 ^ of an ampere, (a) With what resistance should it be shunted to measure a 2.5 ampere current? (Z>) If when so arranged the galvanometer shows a current of 0.084 ampere, what is the current through the circuit ? Ans. (a) 2.7 ohms; (6) 2.11 amperes. 18. What current must be sent through a tangent galvanometer con- sisting of 5 turns of wire bent into a circle 45 cm. in diameter, in order that the needle shall be deflected 30 where H =0.18 C. G. S. units? Ans. 0.74 ampere. CHAPTER XXIV. RELATIONS BETWEEN HEAT AND ELECTRICITY. 395. Heating of Conductors. - - When a current flows through a conductor, an amount of heat is evolved equal to the diminution of the potential energy of the electricity. Let R be the resistance between two points of a homoge- neous conductor having, respectively, the potentials V 1 and V v and suppose that a quantity of electricity, $, passes any cross section in* the time , then, by the definition of poten- tial, the work done by the electricity is also, by Ohm's Law, whence To reduce this to secondary heat units, it must be divided by the mechanical equivalent of heat. If i is measured in amperes, R in ohms, t in seconds, and H in calories, (2) H= 0.24 t a 12*. Equation 1 was first experimentally verified by Joule, and is often called Joule r s Law. Also, as the heating of the conductor, always supposed homogeneous, is independent of the direction of the current, to distinguish it from certain other phenomena observable in a non-homogeneous circuit it is often called the Joule, or irreversible heat effect. ELECTRIC LIGHTING. 439 When the current passing through a wire, stretched hori- zontally between two points, is gradually increased, the wire will be seen first to sag, then to redden, and, passing through various degrees of luminosity, finally melt. Many useful applications of this heating effect are made in the mechanic arts. Instruments which are likely to be damaged by the passage of too strong a current may be effectively protected by inserting in the circuit a strip of metal which will melt as soon as the current reaches a dangerous amount and break the circuit. Generation of heat by electric currents has many marked advantages over the generation by the use of fuel, in the ease with which the production is controlled, and the directness with which the heat may be applied, as, for instance, in welding, cooking, heating of rooms, blasting, etc. 396. Electric Lighting. Any body heated to incandes- cence by the passage of the electric current furnishes an easily controlled source of light. Metallic threads are not available on account of the danger of fusion, whenever the current rises above a definite amount. Carbon has been found to be the most suitable substance for use in incandes- cent lamps, as it does not pass through the liquid state at ordinary pressures. If it is heated to incandescence in the air, it gradually wastes by oxidation. This result may be avoided by enclosing the thread in a vessel exhausted of air. The filament of the familiar glow-lamp consists of a strip of bamboo, or other vegetable fiber, carbonized by a special process, and sealed into a globe exhausted to the highest possible degree. The ends of the filament are attached to platinum wires, which pass through the glass and form the electrodes of the lamp. An ordinary 16-candle lamp for use on a IQO-volt circuit has a resistance of about 160 ohms 440 RELATIONS BETWEEN HEAT AND ELECTRICITY. when hot, and takes a current of about 0.6 ampere. It may be said to possess a probable life of 1000 hours of illumina- tion. If a greater current is forced through the filament, the carbon will disintegrate, and the life of the lamp be much shortened. The electric lamp possesses a great superiority over the older method of lighting by combustion, in the fact that there is less heat generated, and no deleterious gases are evolved. Yet even the best electric lamp must be re- garded as an extravagant mode of producing light, since 90 per cent of the energy supplied is lost as heat. 397. The Electric Arc. The flash obtained by the disrup- tive discharge of a conductor may be made continuous by maintaining two conductors at a constant difference of poten- tial. The light is shown by the spectroscope to be due to the incandescence of particles of the electrodes and of the surrounding gas. The luminous and conducting track may be increased in length by separating the terminals beyond the original striking distance. From its curved form it is called the electric arc. When the arc is formed between two carbon points, they grow very hot and together with the vapor become the source of an exceedingly brilliant light. This result was first obtained by Davy in 1800. The tem- perature of the arc, estimated by Violle as 3500 C., and by Rosetti as 4800 C., is the highest which can be artificially obtained, and is sufficient to volatilize all substances. 398. The Arc Lamp. As the carbon points in the electric arc slowly waste away by oxidation, some automatic device to feed them up is necessary, if the light is to be continu- ously maintained. The mechanism should also be capable of approaching the points within striking distance when the arc is interrupted, and afterward separate them to the proper THERMO-ELECTROMOTIVE FORCE. 441 working distance. These conditions are attained in practice in a great variety of ways by means of an electromagnetic control. A typical form of regulator is shown diagrammatically in Fig. 278. The weight of the upper carbon and holder is nearly balanced by the soft iron core (7, suspended over the wheel 6r, so that, when no current is passing, the point A rests on B. D is a shunt, and E a series coil, so arranged that they act differentially on the core O. When a current passes through the circuit HABEK, the core is drawn into E, and the points J., B are separated a little, after the arc is established. As the resistance in- creases, a greater portion of the current will pass through D until the movable system is in equilibrium with a proper separation of the carbons. By this means the resistance of the arc is maintained practically constant. FJG 27g The carbon pencils in common use are formed of powdered coke made into a paste and baked in the form of rods about a centimeter in diameter. In an arc lamp of from 1000 to 2000 candle-power a potential difference of from 40 to 50 volts is necessary. The usual current is from 5 to 10 amperes, and the consumption of the carbons, roughly, 2.5 cm. per hour. 399. Thermo-Electromotive Force. In 1822 Seebeck dis- covered that, if one of the junctions of a circuit consisting of two dissimilar metals was maintained at a different tempera- ture from the other, an electric current was produced in the circuit. Thus, if the circuit consist of copper and bismuth, and the junction .//"(Fig. 279) be heated, a continuous cur- rent will flow from the bismuth to the copper across the hot 442 RELATIONS BETWEEN HEAT AND ELECTRICITY. junction, the work done by the current being mechanically equivalent to the heat which is absorbed at the warmer junction. This electromotive force of contact, which is evidently a function of the tem- perature, may be designated as the Seebeck effect. H Bi FIG. 279. 400. Peltier Effect. The converse of the Seebeck phe- nomenon was discovered by Peltier in 1834, who found that if a current was made to pass the junction of two dissimilar metals, the joint was heated or cooled according to the direction of the current. Let Fig. 280 represent a circuit consisting of bismuth and copper, of which both junctions are initially at the same temperature. Peltier's discovery is best explained by as- suming that there is a small contact electromotive force at the junctions in the direction of the full arrows. If, now, a current be sent through the circuit in the direction of the feathered arrow, work will be done by the current at TF, and by the electromotive force at 0. The first junc- tion will accordingly be warmed, and the second cooled. The electromotive force at the junction of two metals, as determined by this method, must be carefully distinguished from the difference of potential which is found on join- ing two metals in air or other gas and then separating them. The Peltier effect, described in this article, is distinguished irom the Joule effect by its reversible character. FIG. 280. THERMOELECTRIC SERIES. 443 The equation for the heating of a portion of a circuit con- taining two metals may be written where P is a constant called the coefficient of Peltier effect. It is numerically represented by the heat which is absorbed at the junction when the unit current flows for a unit time. 401. Thermoelectric Series. If the different metals be arranged in order according to the electromotive force devel- oped per degree difference of temperature between two junc- tions at ordinary temperatures, the metals will fall in the following series. The adjoined numbers, called thermoelec- tric hights, give the electromotive force in C. G. S. units f ^ volt Y per degree Centigrade difference of tempera- ture, with respect to lead at 20 C. It has been shown by experiment that, if two metals are separated in a circuit by several intermediate metals main- tained at a common temperature, the E. M. F. is the same as if these metals were joined directly and the junction raised to the given temperature. THERMOELECTRIC HIGHTS AT 20 C. Tellurium .... 50200 Tin 10 Antimony . . 2G40 to 600 Lead Iron . 1750 JMercurv 41 8 Copper (electrolytic) 380 Zinc fpure^ 370 German silver . Nickel . -1175 2200 Silver 300 Bismuth . 4500 to 9700 The sign is chosen so that a current running from a higher to a lower metal in the table generates heat at that junction. 444 HE LA TIONS BE T WEEN HE A T AND ELE C TRIG I TY. 402. Thermoelectric Inversion. It was discovered by Gumming that the order of certain metals in the preceding series was different at high from that at low temperatures. Thus, if one junction in a copper-iron circuit be heated, and the other be kept at 20 C., the current will flow from the copper to the iron at the hot junction, and the electromotive force will increase until the temperature of 284 C. is reached. If, however, the temperature is still increased, the electro- motive force will begin to decrease, and finally the current will set in the opposite direction across this junction. This reversal may be obtained more quickly by raising the tem- perature of the cooler junction. These experiments show that, if both junctions be above a certain temperature, T, the current will flow from the iron to the copper across the hotter junction ; that is, in a direction opposite to that which it takes when all the circuit is below T. The temperature T is called the neutral point, or temperature of inversion. It follows from the preceding discussion that if one of the junctions be kept at the neutral point, and the other junction be made either hotter or colder, the current will set from the copper to the iron through the junction at the neutral tem- perature. 403. Thomson Effect. Since, in the arrangement just mentioned, the neutral junction has no effect, and the E.M.F. of the Peltier effect is in the wrong direction to produce the current, if the second junction is below T, it is evident that there must exist a certain electromotive force in an un- equally heated conductor. This conclusion was first reached by Lord Kelvin (Sir William Thomson), who predicted from the laws of thermodynamics, and afterward showed experi- mentally, that a current from a hot portion to a cold cooled iron but warmed copper. This phenomenon has received the THERMOELECTRIC DIAGRAM. 445 name of the Thomson effect. It is reversible in "the same sense as the Peltier effect. Tait has shown that if T denote the absolute temperature of the hot junction, T' that of the cold junction, and r n the temperature of the neutral point, the electromotive force of any circuit may be expressed by the equation (3) U=a(T-T>)tT n -%(T + T f )l where a is a constant, depending only on the nature of the two metals composing the circuit. 404. Thermoelectric Diagram. A valuable method of representing the thermoelectric properties of different sub- stances is to plot the absolute temperatures as abscissas, and the thermoelectric hights, i.e. the increase of elec- tromotive force relative to lead per degree rise of temperature, as ordinates. On such a diagram (Fig. 281) the thermoelectric properties of iron will be well represented, for ordi- nary temperatures, by the line IF', denned by Absolute Temperature O 1734 _ 4.87 r FIG. 281. t being measured on the Centigrade scale ; and copper, by the line (7(7, given by 136 + 0.95 *. The point _ZV, where these lines intersect, is the neutral point. To illustrate the use of such a diagram, draw the ordi- nates corresponding to the abscissas Or and Or', the absolute 446 RELATIONS BETWEEN HEAT AND ELECTRICITY. temperatures, respectively, of the cold and hot junctions of a copper-iron pair ; then, since thermoelectric hight multiplied by temperature gives electromotive force, the area bcy/3 rep- resents an electromotive force of the Peltier effect. But, as this again is proportional to a quantity of work or heat, the same area may represent the heat absorbed when a current flows from copper to iron through the hot junction. In the same way daa& may represent the heat developed, or the Peltier effect, at the cold junction. It may also be shown that the absorption of heat in the copper by the flow of current from the cold portion to the hot, i.e. from a to 6, can likewise be represented by the area abfta, while that in the iron will be given by cdfy. Accordingly, the total heat absorbed is pro- portional to aabcdb, and that evolved to daaS. The differ- ence, abed, is the amount transformed into the electrical energy of the current. In general, the E. M. F. of any thermoelectric chain is given by the area bounded by the ordinates corresponding to the temperatures of the junctions and the lines of the metals in the diagram. The direction of the current is found by traversing the boundary so as always to keep the area on the left. Heat will be absorbed wherever the current sets from a lower to a higher thermoelectric hight, and devel- oped where the current runs from a higher to a lower level. If it is found that the current in one part of the circuit is opposite to that at another, then the areas corresponding o c 50 100 150 200 250 FIG. 282. 300 350 400 THERMOPILE. 447 to these opposing electromotive forces must be subtracted, in order to get the resultant electromotive force, which will have its direction determined by the boundary of the larger area. The diagram from Tait on the preceding page shows the thermoelectric properties of a number of the more important substances. 405. Thermopile. Although the E. M. F. of a single thermoelectric couple, as shown by the table on p. 443, is extremely small, it is possible, by connecting a number of such pairs in series, as shown in Fig. 283, so that alternate junctions, p, r, , P may be heated, to obtain a resultant electromotive force comparable to that from a voltaic cell. Such an arrange- ment is known as a thermopile. One of its principal applications has been to the detection of small differences of temperature. If a series of thermoelectric couples be ar- ranged in a block so that the radiations from a hot body may fall on the odd junctions, and if the terminals of the pile be connected through a sensitive galvanometer, a very small elevation of temperature at the exposed face of the thermo- pile will be indicated by the deflection of the galvanometer mirror. By the use of such a piece of apparatus, Melloni made his experiments, now classic, on the diathermancy and emissive powers of various bodies. For the production of currents sufficient for many laboratory purposes, the pile takes the cylindrical form (Fig. 284), which is built up of a number of rings, each containing ten couples made of iron and an antimony-zinc alloy, properly supported and insulated. The interior junctions, which are heated by a central gas jet, 448 RELATIONS BETWEEN HEAT AND ELECTRICITY. are embedded in fire clay to protect them from contact with the flame. A pile of 120 such elements will give an E.M.F. FIG. 284. of about 8 volts and have a resistance of about 3 ohms. As a current generator the thermopile is but little used, for the E.M.F. is small and unsteady. 406. Radio-Micrometer. A very sensitive instrument for detecting absorbed radiations, first devised by d'Arsonval, but independently invented and greatly improved by Boys, is the radio-micrometer shown in Fig. 285. N and $ are the poles of a strong magnet, between which is suspended a coil consisting of a single loop of copper wire, the purest attainable. To one end of this wire is soldered a small piece of antimony, J., and to the other a piece of bismuth, B, these in turn being attached to a small projecting strip of copper, 0. The thermal element is itself surrounded by a tube of soft iron, H, in order to shield its diamagnetic material from the influence of the poles. The RADIO-MICROMETER. 449 radiations from the source which it is proposed to study are allowed to fall on the small strip of cop- per, where they are absorbed, pro- ducing an electromotive force in the thermoelectric pair. Although this E.M.F. is exceedingly minute, the resistance of the circuit is also very small, and a sufficient current is produced to cause a deflection of the mirror. This instrument has been found capable of indi- cating differences of temperature of the order of one-millionth of a degree Centigrade, and is .said to be able to detect the radiations FlG. 285. from a candle two miles distant. In comparison with the bolometer, it has the disadvantage that it cannot be moved during an experiment. EXAMPLES. 1. How much heat would be generated by a current of 2.32 amperes flowing through a wire whose resistance is 75 ohms for 30 minutes? Ans. 175,000 cal. 2. How much will the temperature of 125 gms. of water be raised by a current of 1.39 amperes flowing for half an hour through a coil having a resistance of 5.04 ohms immersed in it? Ans. 33.5. 3. A 16-candle lamp requires a potential difference of 65 volts, and a current of 0.82 ampere. How many watts per candle-power are absorbed by the lamp ? Ans. 3.33 watts per candle. 4. What power will be required to light 78 incandescent lamps, if the P. D. required for each lamp be 65 volts, and tjie current 0.83 ampere? Ans. 4210 watts. CHAPTER XXV. DIMENSIONS AND UNITS OF ELECTRICAL QUANTITIES. 407. Systems of Electrical Units. The dimensions of electrical quantities take on different forms according as they are defined with or without reference to magnetic phe- nomena. There have arisen, in consequence, two systems of absolute units, designated respectively as the electrostatic and the electromagnetic system. Since most of the phe- nomena of electricity, which are experimentally measured, have to do with their magnetic effects, the second system is relatively of the greater importance. As, however, the electromagnetic C. G. S. units have in practice been found of inconvenient size, they have been replaced for commercial and most scientific purposes by a third or practical system. 408. Electrostatic System. Quantity. The electrostatic system of units is founded on the definition of quantity derived from the equation Kr* setting [Q] for the dimensions of quantity, and observing that [F] = MLT-\ and [r]=Z, As nothing is at present known concerning the dimensions of K, it will be allowed to stand in the formula. The advan- tage of this retention will be more fully seen hereafter. If in each case the K is divided out, or suppressed, the units so obtained will belong to what is called the electrostatic ELECTROSTATIC SYSTEM. 451 system. Thus, from the equation above, the C. G. S. elec- trostatic unit of quantity is denned as that charge which when concentrated at a point, at a distance of one centimeter from an equal charge, exerts the force of one dyne when the intervening medium is air. This unit may be symbolized by gmJ cm.^ sec. The electrostatic units have received no special names. Intensity or Strength of an Electric Field. Electrical in- tensity is the force per unit charge. Its dimensions are those of force divided by quantity. Thus, (2) The C. G. S. unit is one dyne per unit charge, or cm.i sec. Potential. Potential is the work per unit charge. Its dimensions are those of work divided by quantity, or The C. G. S. unit is the erg per unit charge, or sec. Capacity. Capacity is the charge per unit difference of potential. Its dimensions are The C. G. S. unit is 1 cm. 452 ELECTRICAL QUANTITIES. Electric Current. A current in a conductor is the quan- tity of electricity which passes any cross section per unit of time. Its dimensions are thus The C. G. S. unit is that current in which the unit quan- tity passes in a second. It is symbolized by gm.2 cm. ^ sec. 2 Resistance. By Ohm's Law, resistance is the quotient of potential difference by current. Thus, the dimensions of resistance are 1 sec The C. G. S. unit is - - : , the same formula as for slow- 1 cm. ness, or the reciprocal of velocity. But no physical signifi- cance can be attached to this until some justification is found for the assumption that ^Tis of no dimensions. 409. Electromagnetic System. Quantity of Magnetism, or Magnetic Pole. The electromagnetic system of units is founded upon the definition of the unit quantity of magnet- ism derived from the equation Substituting the dimensions of force and distance, the dimensions of m become (7) [m] ELECTROMAGNETIC SYSTEM. 453 the symbol for magnetic permeability being retained as in the case of K, because nothing can be predicated concerning its dimensions. The electromagnetic system of units, as commonly under- stood, is that which is found by suppressing the dimensions of fji from the more general formulas. The C. G. S. unit pole in the electromagnetic system is denned as that pole which, placed at a distance of one centi- meter from an equal pole, will exert the force of one dyne, when the intervening medium is air. This unit may be symbolized by gin. 2 cmJ sec. Strength of Magnetic Field. The strength of a field, or the magnetic intensity at a point, is the force per unit pole. Its dimensions are, accordingly, Magnetic Moment. Magnetic moment is the moment ex- perienced by a magnet divided by the strength of the field. Its dimensions are thus Intensity of Magnetization. Intensity of magnetization is the magnetic moment per unit volume. Its dimensions are thus 454 ELECTRICAL QUANTITIES. Magnetic Induction. Taking the definition of magnetic induction from -g _ the dimensions of B become (11) Current Strength. The fundamental fact of electromag- netism is, that a coil conveying a current, i, and bounding an area, A, in a medium whose permeability is /t, perfectly simulates a magnet whose magnetic moment is (12) M = pi A, from which the dimensions of i are found to be The same result may be found from the expression for the magnetic field produced at the center of a circular coil of radius, jR, by a current, i, namely, whence (15) [t] The C. G. S. electromagnetic unit of current is that cur- rent which, flowing in a conductor one centimeter long (see Fig. 250), would produce a force of one dyne on a unit pole at the distance of one centimeter. It may be symbolized by * ^ sec. Either equation 12 or 14 may be used to find the dimen- sions of the magnetic quantities in the electrostatic system, in terms of the dimensions of i. These values will be found in the table on page 456. ELECTROMAGNETIC SYSTEM. 455 Quantity of Electricity. The quantity of electricity which passes any section of a conductor is the current multiplied by the time; hence, the dimensions of [Q] are (16) [] = W [2 7 ] = [M^-*]- Potential Difference, or Electromotive Force. Potential, or the work per unit charge, has the dimensions The C. G.S. electromagnetic unit of electromotive force is symbolized by sec/ Resistance. By Ohm's Law the dimensions of resistance are The C. G. S. electromagnetic unit of resistance is sym- cm. bolized by " It will be noted that the dimensions of sec. resistance become those of velocity only under the arbitrary suppression of the dimensions of //,. Capacity. The dimensions of capacity, as determined by the equation (7= ,, are (19) [0 ]= The C. G. S. electromagnetic unit of capacity is sym- bolized by '- cm. 456 ELECTRICAL QUANTITIES. For convenience of reference, the preceding results are collected in the following table. DIMENSIONS OF ELECTRICAL QUANTITIES. NAME. SYM- BOL. IN TERMS OF K. IN TEKMS OF Quantity of electricity . Electric intensity . . . Potential Electromotive force . . Capacity Specific inductive capacity Current strength . Resistance Quantity of magnetism, or magnetic pole . . . Strength of field . . . Magnetic moment . . . Intensity of magnetization Permeability Magnetic induction '-IJT-* c K i R m H M I B LK K Now, since a quantity of electricity is physically the same thing, whether measured in terms of electrostatic or electro- kinetic phenomena, it is evident that its dimensions should be the same in either case. Equating the values of q from the table, \L\ 1 (20) ELECTROMAGNETIC UNITS. 457 that is to say, in order that the two expressions for quantity should be congruent, K~^ /*"* must have the dimensions of velocity. Also, since all the other magnitudes are derived from (?, it is obvious that the first column of dimensions in the table will become congruent with the second by the same substitution. 410. Ratio of the Electrostatic to the Electromagnetic Units. Let the electrostatic unit of quantity, i.e. the value ob- tained by suppressing K, be denoted by \_Q^\ = \_M^L% 2 7 " 1 ], and, similarly, the electromagnetic unit by \_Q m ~\ = [_M*lfi]. Let q e be the number which expresses the measure of a defi- nite quantity of electricity in electrostatic units, and q m the number which expresses the same quantity in electromag- netic units. Then, taking the quotient of these two concrete quantities, = ?[ | = f = ' 2 - These equations enable one to pass from one system of units to another, and also indicate several other methods of measuring v, which, on trial, have given essentially the same value as that already stated. 411. Electromagnetic Theory of Light. That the re- markable coincidence between v and the velocity of light is not accidental clearly appears from the work of Maxwell, who showed that an electromagnetic disturbance would be propagated through vacuous space with the velocity of light. He was thus led to propound the theory that light itself is an electromagnetic phenomenon. In the case where JTand p are not both unity, theory shows that the velocity of propagation should be (24) p and, although experiments on the velocity of light in different media do not show so complete a verification of this relation as in the case of empty space, they nevertheless indicate that PRACTICAL UNITS. 459 the principal part of the velocity is contained in this formula. Equation 24 may also be derived from the dimensional for- mulas of Art. 409. Thus, let K, without brackets, be the numerical measure of the specific inductive capacity of the medium, and /*- its permeability; then, measuring a given quantity of electricity by both the electrostatic and the elec- tromagnetic definition, (25) whence (26) (27) 412. Practical Units. It is found that if the measure- ment of such electrical quantities as are met with in practi- cal work be expressed in C. G. S. units, the numbers are generally either very large or inconveniently small. For ex- ample, the electromotive forces of all the ordinary forms of voltaic cells are included between one hundred and two hun- dred millions of electromagnetic units ; so, too, the resistance of a mile of ordinary telegraph wire is not far from ten thou- sand millions of electromagnetic units. This difficulty has been met by the formation of a new, or Practical System, founded upon the electromagnetic system, in which the fun- damental units chosen are L = (10) 9 centimeters, M = 10~ n grams, T= 1 second. The electrical units so derived, with their names, are shown in the following table : 460 ELECTRICAL QUANTITIES. TABLE OF PRACTICAL UNITS. VALUE IN C. G. S. Electro- Electro- MAGNITUDE. NAME. magnetic static Units. Units. Resistance Ohm . . . 10 9 i lO" 11 Potential Difference, J or Volt . . . 10 8 i 10- 2 Electromotive Force ) Current Ampere . . lO" 1 3(10) 9 Quantity Coulomb . 1Q- 1 3(10) T I Joule, 10' ergs. .Lnergy ) ( Ampere x Volt X Second > ( Joule per second ) . . . Power = J f. . . . . Watt, 10 7 ergs per sec. I Ampere x Volt > The joule, or unit of energy, is the work which would be done in carrying a coulomb between two points differing in potential by one volt, ^tt; is also the work which would be done by an ampere in flowing for one second through a resistance of one ohm. The watt is the rate at which work is done by an ampere, flowing through a circuit having the resistance of one ohm. The prefix mega- is sometimes used in connection with a physical unit to designate a quantity one million times as great. Similarly, micro- signifies one-millionth of the unit to which it is prefixed, while kilo- and milli- stand, respec- tively, for one thousand and one-thousandth. The legal definition of the ohm has been given in Art. 353, the volt in Art. 349, and the ampere in Art. 358. CHAPTER XXVI. INDUCTION OF CURRENTS. 413. Faraday 's Discovery. From consideration of the fact that a charged body will electrify another, Faraday was led to investigate whether a conductor, in which a current was flowing, might not also excite, by influence, a state sim- ilar to its own in a neighboring conductor. A series of experiments, extending over seven years, yielded nothing till 1831, when he observed that at the instant of making and breaking a current in the vicinity of a circuit containing an unusually sensitive galvanometer, the needle was momenta- rily disturbed. Although this was not exactly what Fara- day was seeking, namely, a permanent change in the state of the galvanometer circuit, it proved to be a discovery of the highest practical importance. 414. Phenomena of Current Induction. By further ex- amination of the state of a secondary circuit, under various circumstances, Faraday showed that there were several dif- ferent, though not essentially distinct, ways in which cur- rents could be induced in any closed circuit. 1. By variation of the primary current. Let A (Fig. 286) be the primary circuit containing a vol- taic battery, and some device by which the current may be made and broken, or have its intensity varied. Let B be the secondary circuit, containing a galvanometer so placed as to be unaffected by the direct influence of the battery current. It is then found that when the primary current is started, the galvanometer indicates an inverse 462 INDUCTION OF CURRENTS. current through the parallel portion of the secondary bb f , that is, one in a direction opposite to that in the primary aa\ but if the strength of the primary current is maintained constant, the induced current immediately disappears. If the primary current be now stopped, an induced current is observed in the secondary, which has the same direction through bb' as the original current through aa'. It will also be found that any increase in the strength of the primary produces an inverse current in the secondary, while ja decrease in the primary current produces a direct current in the sec- ondary ; but as long as the primary current remains constant a b' FIG. 286. there will be no effect in the secondary circuit. The induced currents will be much increased by arranging the two wires in concentric insulated coils, and still more by introducing a soft iron core into the coils. Entirely similar phenomena of self-induction take place within the primary circuit when its current is varied. While the current is increasing there is an induced inverse electro- motive force, and while the current is decreasing a direct one. The latter may be readily shown by opening a circuit containing an electromagnet, while the current is flowing, when a bright spark will be formed at the gap. No such effect, however, is observed when the circuit is closed. It will be noted that in the case of self-induction, just as in the oscillatory discharge of a Leyden jar, electricity behaves as if it possessed momentum whose changes are pro- portional to the force applied and the time during which it acts. LAWS OF CURRENT INDUCTION. 463 2. Induction by motion of the primary with respect to the secondary circuit. Let the primary and secondary circuits be so arranged as to contain the coils L and M (Fig. 287) in close proximity. Then, if while the current is flowing in the primary the coils be approached, a brief inverse current will be induced A v FIG. 287. in M, but if the coils be separated, a direct current will be induced. Also, any variation in the form of the secondary circuit will give rise to similar effects. 3. Induction by the motion of a magnet with respect to a secondary circuit. If the coil L of Fig. 287 be replaced by a magnet, with its north pole in the same direction as the north face of the coil, the phenomena produced by approaching or withdraw- ing the magnet from M will be the same as those obtained by moving the primary circuit. 415. Laws of Current Induction. The laws of current induction may be thus stated : Any change in the magnetic field, with respect to a con- ductor, induces a current in the conductor whose direction is such as to oppose the change which produced it, and^the induced electromotive force is proportional to the rate of change of the field^) This rule for the direction of the induced currents, so far as it relates to those arising from the actual motion of the conductors or of magnets, was first stated by Lenz, and is commonly known as Lenz's Law. 464 INDUCTION OF CURRENTS. Another statement of the law of induced currents, in the language of Faraday, is, that the induced electromotive force is proportional to the number of lines of force which are cut per second by the con- ductor. In this connection Fleming has given the fol- lowing rule for determining the direction of the current. Let the thumb of the right .Motion hand ( F 'g- 288 ) to P inted in the direction of the dis- placement, and the index finger in the direction of the lines of force, then the middle finger, set at right angles to the plane of the other two, will give the direction of the current. Another statement of the law of induced currents, due to Maxwell, is, that the induced electromotive force in any circuit is equal to the rate of decrease of the magnetic induc- tion through the circuit. The direction of the induced current is here called posi- tive with respect to the field, when the relations are those considered in the right- handed screw rule of JG? Art. 362. FlG. 288. FIG. 289. 416. Application of the F T .,, , . . Rule. In illustration of the preceding rules, suppose that a portion of the circuit is a horizontal frame, ABCD (Fig. 289), upon which is placed a slider, FG-, and that the magnetic induction is downward through the circuit. If, now, the slider be moved toward APPLICATION OF THE RULE. 465 the end BC, the induction will be diminished, and by Max- well's rule a current will be induced in the positive direction with respect to the lines of force, i.e. GrBCF. Again, let PQ (Fig. 290) represent a portion of a primary circuit, and VSTU the secondary circuit. Then, if a cur- rent start to flow in the primary from P to Q, Q . \v , \ \ u the region about this portion of the conduc- tor may be regarded as surrounded by circular lines of force which expand as the current increases. Then, since the induction through the area enclosed by the second- ary is increasing, the direction of the induced, current will be negative with respect to the lines of force, i.e. in the direction STUV. Suppose, for instance, that the north end of a magnet is thrust into the coil PQRO (Fig. 291). Then, since by Lenz's Law the induced current must oppose the motion, PIG. 290. K FIG. 291. J FIG. 292. the face of the circuit nearest N will be north, or the cur- rent will flow in the direction RQPO when the circuit is closed. As another example, let HIJK (Fig. 292) be a coil which may be made to revolve about a horizontal axis, aa', and suppose that it is placed in a uniform field, so that the lines UNIVERSITY OF 466 INDUCTION OF CURRENTS. of force are perpendicular to the plane of the coil. Call the angle Ial f , made by the coil at successive positions with the vertical 0, then, as 6 increases from to ^ the induction is 2 decreasing, and the induced current will have the direction 7T HIJK. From to TT the magnetic induction through the coil is increasing, but as it has the opposite direction through the circuit the induced current will still be HIJK. Further application of the rule will show that from 6 = TT to 2?r the current will have the direction KJIH. If 6 is described uniformly with the time, it may be shown that the induced electromotive force E can be written (1) JE=Asm0, where A is a constant. 417. Verification of the Law of Induced E. M. F. The law that the induced electromotive force is proportional to the rate of change of induction may be verified by placing a ballistic galvanometer in the circuit of Fig. 291, when it will be found that the first swing of the needle is the same, whether the magnet is withdrawn from the coil in ^ of a second or a second. Therefore, since the same quantity flows through the galvanometer when the time is ^V as great, the electromotive force must have been 25 times as large. 418. Coefficient of Self -Induction. The coefficient of self- induction may be defined as the total induction through a cir- cuit per unit of the current which produces it. If N denote the induction and i the current, the coefficient of self-induc- tion L is given by (2) N= Li. INDUCTION COIL. 467 Since the dimensions of N are those of B multiplied by an area, those of self-induction will be W The C. G. S. electromagnetic unit of self-induction is, accordingly, 1 cm., and the practical unit, called the henry, is (10) 9 cm. 419. Induction Coil. Since the total magnetic induction through any circuit is altered by varying the number of turns in the secondary, it is evident that the induced electromotive force resulting from any given change in the primary current may be varied at will. Instruments designed to produce an induced current with an average electromotive force greater or less than that of the primary current are called transformers, or induction coils. 420. Ruhmkorff Coil. The earlier forms of induction coils were designed to obtain spark discharges at consider- able potential differences by use of the current from a bat- tery. Fig. 293 shows dia- grammatically the arrange- ment of parts in a common form of induction coil, often called the Ruhmkorff coil, from one of its later improvers. The primary circuit of stout wire is coiled in one or a very few layers about a core, MN, made of a bundle of soft iron wire to exclude Foucault currents (Art. 423). The secondary circuit is of fine wire often many miles in length, wound outside the primary, and insulated from it with great care. 468 INDUCTION OF CURRENTS. To guard against sparking between different layers of the secondary, it is usually wound in a number of disc-like coils, joined in series, but separated by insulating partitions. The primary current flowing from the battery is rendered inter- mittent by a vibrator, /, which consists essentially of a spring with a mass of iron at the end. When the current mag- netizes the soft iron core NM, I is attracted and opens the circuit at K. The spring flies back by its own elasticity, FIG. 294. closing the circuit, when I is again attracted and the process repeated. If the gap A is not too great, a spark will pass each time the primary circuit is broken. As the self- induction of the primary circuit is considerable, a spark is also likely to form at K, when the current is broken, thus lengthening the time of decay of the field about the sec- ondary. To prevent this, a condenser, (7, of large capacity is placed in the primary. Under these circumstances, the induced electromotive force will be either insufficient to form an arc in the primary when the circuit is opened, or RUHMKORFF COIL. 469 its duration will be greatly shortened. On account of the increased rapidity in the change of the field, the spark at A is considerably lengthened. At the moment when the primary is closed, the self-induction lengthens the time dur- ing which the current rises to its full value, and, in conse- quence, there is in general no spark in the gap A at the "make." A working form of induction coil is shown in Fig. 294, with essentially the same lettering as Fig. 293. The condenser is here enclosed in the base. The introduction of a condenser into the secondary circuit produces a notable alteration in the character of the spark. The striking distance is smaller, but the brightness of the spark is much increased, and is usually oscillatory in char- acter. When there is no condenser in the circuit, the sparks ), where (f> is a quantity determining the phase of the current. It is shown in Analysis that the mean value of sin 2 between and 2?r is J-, therefore the virtual electromotive force or and, similarly, 434. Lag and Lead. When there is self-induction in the circuit, the current lags behind the impressed E. M. F. This may be represented in a diagram in which the abscissas rep- resent time, the ordinates of the full line (Fig. 313) E. M. F., and the ordinates of the dotted line current strength. It is seen that the current does not reach E/ its maximum till a certain time, i^ t^ after the electromotive force has attained its maximum value. This time expressed as a fraction of the ... FIG. 313. period is termed the difference in phase. It appears in the equations as an angle, <, which is the same fraction of a complete revolution that 2 ^ is of the period, and is called the lag. The mean power, p m , or heating effect of an alternating current, may be shown to be (7) p m = E [/ i v Go^,^. 482 INDUCTION OF CURRENTS. This value evidently vanishes for $ = , or a difference of phase of a quarter period. Such a current, in engineering parlance, is called wattless. If the circuit have appreciable electrostatic capacity, in equation 4 may become positive, and is then called the lead. 435. Virtual Resistance. Alternating currents do not obey Ohm's Law, but a formal resemblance may be obtained by writing Wjf (8) R v = ^, l v where R v is called the virtual resistance, or impedance. Its value may be shown to be (9) R v where R is the true resistance, n the frequency, and L the self-induction. Substituting this value in equation 8, and transforming by the aid of equations 5, 6, and 4, the value of the instanta- neous current becomes (10) If the circuit also contain an appreciable electrostatic ca- pacity, (7, the value of the impedance must then be written 436. Transmission of Electrical Energy. Suppose that a wire which is used to transmit a steady current, i, has a potential, Fj, at one end, and V 2 at the other. Call the TBANSFOEMERS. 483 resistance of the circuit R ; then, if W be the total energy supplied, Art. 395, (11) = Ri 2 + iV 2 , the first term representing the rate at which energy is lost by heating, and the second the rate at which energy is deliv- ered at the further end of the wire. It thus appears that for a given current the energy deliv- ered will be greater, the larger F^ is made and the smaller R is made. The great cost of copper will not permit in an ex- tensive system any considerable reduction of the resistance. Successful transmission of power must accordingly be sought in the use of as high voltages as possible. Alternating currents are most often employed on account of the facility with which they may be transformed and applied to a variety of purposes. 437. Transformers. The cores of induction coils used to transform alternating currents are always laminated and usu- ally form a closed magnetic circuit. The details of the con- struction of a common type of transformer are shown in Fig. 314. The core is built up of pieces of sheet iron, about ^ mm. thick, stamped out in the form shown at A. The primary and secondary coils c, c 2 are first wound on frames, and the stamp- ings inserted by raising the flaps /, /', as shown at B, care being taken to alternate the joints. Each stamping is insulated from its neighbor by a sheet of paper, and the whole securely bolted together. FIG. 314. 484 INDUCTION OF CURRENTS. 438. Motors. Almost any dynamo may be used to trans- form the energy of electrical currents into mechanical work. When a current is sent through the field magnets and the arma- ture, the latter experiences a moment which will keep it turn- ing against considerable resistance as long as a current is sup- plied. Dynamo-electric machines designed to work in this way are called motors. Continuous current motors differ so little from the dynamos that a detailed description is unnecessary. An ordinary alternator, when used as a motor, is at a disadvantage, in that it is not self-starting, and that the field magnets must be separately excited. These difficulties , , may be overcome by the use of a rotary field, first suggested by Baily in 1879. One arrangement for producing such a field is indicated in Fig. 315, which represents a coil wrapped continuously about a soft-iron ring. If the terminals of an alternator were connected to this coil by the wires a and 5, the portions of the ring at A and at B would become alternately north and south poles with every reversal of the current. Likewise, if c and d were connected to another alternator, C and D would be points with similarly changing polarity. If, now, both alternators be worked at the same time, but so arranged that the current through cd is a quarter period behind that in a, the poles at A and B will travel about the ring in the direction ADBO. A magnet, NS, pivoted at the center of the ring, would be dragged around by the changes in the field, and, similarly, a conductor would be set rotating under the influence of the eddy currents produced. FIG. 315. MOTOR DYNAMOS. 485 Rotating fields can be produced by various other combinations of currents, and have been successfully applied to the con- struction of self-starting motors of high efficiency. The revolving part of such machines, technically known as a rotor, is a sort of wire cage built up on a properly lam- inated core. It is unconnected with outside circuits, and acquires its magnetic properties solely by the currents which are induced by changes of the field. 439. Motor Dynamos. By using a motor to run a dynamo, it is evident that from a given current any required current may be obtained. Instead of using two separate machines, the same result may be secured more simply by making the armature with two independent windings. Such an arrange- ment is called a motor-dynamo, or a continuous current trans- former if the original and derived currents are continuous. Other machines of this type are called continuous-alternate or alternate-continuous transformers, according to the nature of the change effected by the machine. EXAMPLES. 1. Two parallel rods, 135 cm. apart, are joined at one end by a wire and at the other by a sliding rod. When this rod is moved parallel to itself at a velocity of 720 cm. per sec., what electromotive force will be induced in the circuit where the vertical component of the earth's magnetism is 0.453 C. G. S. unit? Ans. 4.4(10)~ 4 volts. 2. A coil of wire, consisting of 25 turns wound upon a core 40 cm. in diameter, is rotated 30 times a second upon a vertical axis in a field where H = 0.18 ** i ' What is the mean induced electromotive cm. 3 sec. force? Ans. 0.00678 volt. 3. A copper disc, 18 cm. in diameter, is rotated 27 times a second in a field whose intensity is 732 C. G. S. units. What will be the induced E. M. F. between the center and the circumference ? Ans. 0.0503 volt. CHAPTER XXVII. TELEGRAPH AND TELEPHONE. 440. The Needle Telegraph. The first system of lines for the electrical transmission of signals, on an extensive scale and for commercial purposes, was laid by Wheatstone and Cooke in 1837. The circuits were five in number, each containing a galvanometer needle which might be deflected to the right or left between stops, at will. By agreeing that different combinations of position of the needles should represent the letters of the alphabet, any desired message might be transmitted. It was afterward found that a single needle and circuit was sufficient, since the alphabet could be represented equally well by successive deflections to the right or left. In 1837 also Steinheil showed that a return wire was unnecessary if the extremities of the line were con- nected to the earth. 441. Morse Instrument. In 1831 Henry had shown that, if a current was made and broken at one point of a circuit, the attraction and release of the armature of an electro- magnet at another part of the circuit would produce a suc- cession of clicks which might be interpreted as signals. Six years later Morse arranged a similar electromagnet so that it would print a succession of dots and dashes, and invented a code which, with a few modifications, has been universally adopted. It was soon found, in practice, that the message could be read with ease from the click of the armature alone, so that the marker is dispensed with, except where an auto- matic record is desired. When messages are transmitted DUPLEX TELEGRAPHY. 487 over considerable distances, and the currents are very feeble, the receiving instrument, or " sounder," is placed in a local battery circuit which is opened and closed by the lightly pivoted armature of an electromagnet called a relay and operated by the current in the line wire. 442. Duplex Telegraphy. In order to increase the work- ing capacity of a single line, various methods have been devised to effect the simultaneous transmission of several messages. The duplex system, which permits one message to be sent from each end of the line at the same time, makes use of M FIG. 316. the principle of Wheats tone's Bridge. Its working may be understood by the aid of Fig. 316. Suppose that L is the line wire, and let the analogous parts at each end be denoted by large and small letters respec- tively. K is the key, E the battery, and S the sounder. A, B, and D are three coils whose resistances with that of the line and its opposite end connections form a proportion, and may be regarded as the arms of the bridge. If, now, the key jfiTbe closed, the current flowing from the battery E and dividing at the point P will not affect S, since, by the ar- rangement of resistances, M and N are at the same potential. 488 TELEGEAPH AND TELEPHONE. The current going through the line wire will divide at w, and a portion passing through s will operate this sounder. Now, since by the arrangement of the resistances the battery e is unable to change the potential difference between m and w, it is evident that the operation of K and s is the same whether k is open or closed, that is to say, messages may be sent in opposite directions at the same time. In order to secure satisfactory working in long lines, it is necessary to introduce condensers at and c. 443. Diplex System.. A method for sending two mes- sages in the same direction at the same time, invented by ^PlTTSi \f m * e K w/ l*Fl ] 1 ] ( _i 1 1 l+ b FIG. 317. Edison and known as the diplex sys- tem, is shown in Fig. 317. At r is a polarized relay, that is, an instrument containing a permanent magnet so arranged that its armature responds to either a strong or a weak current in one direction, but not at all to a current in the opposite direction. R is a second relay sensitive to currents in either direction, provided they exceed a definite value. B and b are batr teries of high and low electromotive force respectively. K and k are metal keys furnished with projecting fingers at a, /, and g, which may be brought into contact with the springs c, e, and i. If, when the connections are made as MULTIPLEX TELEGRAPHY. 489 shown, k be depressed, the negative pole of b will be con- nected to the earth through igm, the positive pole to the line through nacp, and r will be operated by a feeble posi- tive current. If, on the other hand, K alone be depressed, the positive pole of B and 6, which are joined in series, will be grounded through dcefm, and the negative pole connected to the line through i and h. In this case R will be affected since the current is strong, but r will not respond because the current is in the wrong direction. If K and k are both depressed, the positive pole of the double battery will be con- nected to the line through deep, and the negative pole to the earth through igm. In this case a strong positive current will operate both relays. By combining the two preceding systems, it is possible to send two messages simultaneously in each direction. Such an arrangement is called the quadruplex system. 444. Multiplex Telegraphy. If the transmitter send a periodically alternating or intermittent current, and the re- ceiving instrument be so arranged as to respond only to a FIG. 318. current of a special period, then by using instruments of different periods a number of messages may be sent at the same time. A method for sending several messages at successive inter- vals of time so short that they are practically simultaneous, was devised by Meyer in 1873. Suppose that the line wire L (Fig. 318) is connected with two arms, R, r, which revolve rapidly with exact synchro- 490 TELEGRAPH AND TELEPHONE. nism, making contact with a number of sectors, , and produces a corresponding fluc- tuation in the strength of the battery current. CHAPTER XXVIII. PASSAGE OF ELECTRICITY THROUGH GASES. 449. Discharge of a Conductor Immersed in a Gas. - The passage of electricity through gases gives rise to an extensive series of complex phenomena. Only a few of the more important features are touched upon in the following articles. A gas can be regarded as an insulator only at moderate temperatures, even for bodies charged to a low potential. As it is heated, it begins to conduct electricity with a facility depending on the molecular decomposition resulting from the rise of temperature. Experiment indicates that the con- duction is convec^ive, and that free atoms are essential to the process. The gaseous molecule does not appear to be capa- ble of receiving a charge. Flames are found to be very good conductors, and exhibit certain properties not unlike electro- lytes. For instance, if two dissimilar wires are connected, and the free ends dipped into the flame, there will be an elec- tromotive force around the circuit, amounting in some cases to three or four volts. The spark discharge of a conductor at low potential in a gas seems also to depend on a fine dust resulting from the decomposition of the electrodes. The amount of disinte- gration of the kathode is very marked when the spark from an induction coil is passed through an exhausted tube. The glass about the kathode often receives a perceptible metallic film by the deposit of particles torn from the adjacent elec- trode. The amount of this disintegration depends on the na- ture of the kathode, but seems to be unaffected by that of 496 PASSAGE OF ELECTRICITY THROUGH GASES. the anode. If ultra-violet light be allowed to fall on a nega- tively charged conductor, a pronounced leakage occurs, due, apparently, to a disintegration of the surface under the action of the light. The order of sensitiveness of metals to this light effect is roughly that of Volta's Contact Series. Nc evidence of similar loss of a small positive charge has been observed. 450. Spark Discharge. The greatest electromotive in- tensity which a gas can sustain, sometimes called its electric strength, depends not alone on the gas, but also on the mate- rial of the electrodes, the state of their surface, their size, shape, and distance apart, and on variations of the electric field, either in time or space. 451. Spark Length and Potential Difference. The poten- tial difference required to produce sparks between two slightly convex surfaces in air at normal pressure may be expressed as a linear function of the length of the spark. Bailie's experiments lead to the relation (i) V= 4.997 + 99.593 I for sparks over 2 mm. long, where V is the potential differ- ence in electrostatic units, and I the length of the spark in centimeters. Experiment indicates that for very short sparks there is a minimum value of potential necessary to produce a spark, and that to produce a spark having a length greater or less than the critical value corresponding to this minimum, a greater potential difference is required. If the spark length is kept constant and the curvature of the equal electrodes, starting with a plane, is changed, Bailie finds that the potential difference increases with the curva- ture and attains a maximum. This critical value of the cur- DISCHARGE WITHOUT ELECTRODES. 497 vature decreases with diminishing spark length. The spark potential diminishes with the pressure until the latter reaches a certain critical value depending on the length of the spark, the nature of the gas, the shape and size of the electrodes and that of the vessel containing the gas ; but further dimi- nution of the pressure is accompanied by increase of potential difference. The potential difference required to produce a spark of given length in hydrogen is much less than in air. Carbon dioxide is stronger than air for short sparks, but weaker for long ones. 452. Discharge without Electrodes. The phenomena of discharge through rarefied gases are much simplified when metallic electrodes are not used. In this case the discharge may be produced by bringing an exhausted tube into a rap- idly alternating field. A con- Inductor venient method of producing such a field is shown in Fig. 322. The inner coatings of two Leyden jars, D, F, are connected to the terminals A, B, of a Holtz machine or an induction coil, and the outer coatings to a wire in which a few turns are made at C. If an exhausted bulb FIG. 322. be introduced into this coil, when sparks are made to pass between A and B the rapid oscillations which are set up in the wire produce a sufficient electromotive intensity to cause a bright discharge within the bulb. In order to prevent electrostatic effects in the glass, the coil should be connected to the earth. 498 PASSAGE OF ELECTRICITY THROUGH GASES. When the pressure of the air within the bulb is consider- able, no discharge can be obtained ; but when it is reduced to about 1 mm. of mercury, a thin thread of reddish light may be observed within the plane of the coil. As the pressure is diminished, the color changes to white, the luminosity increases and attains a maximum, while the discharge ap- pears as a very bright and well-defined ring. As the pres- sure is still further diminished, the luminosity also decreases ; and when a good vacuum has been obtained, no discharge at all passes. The critical pressure in tubes without elec- trodes is very much less than in tubes in which they are present. The discharge experiences considerable difficulty in pass- ing across the junction of a rarefied gas and a metal. If, for instance, a metal diaphragm be extended completely across the bulb, the discharge will not cross the metal plate, but breaks up into two sepa- rate circuits, as shown in Fig. 323. The high resistance of a rarefied gas in a tube provided with the usual electrodes has been found to depend chiefly on the difficulty experienced by the discharge in passing from the electrodes into the gas. Experiments on the conductivity of rarefied gases, when the discharge is confined entirely to the gas, show that the molec- ular conductivity, i.e. the specific conductivity divided by the number of molecules per unit volume, is enormously greater than that of the best electrolytes, and higher even than for silver and copper. The conductivity of rare gases does not obey Ohm's Law, but increases with the electromotive intensity, as might be expected if the discharge is due to the splitting up of the molecules. FIG. 323. ELECTRODELESS DISCHARGE. 499 453. Effect of Magnet on Electrodeless Discharge. A magnet deflects the discharge through a tube without elec- trodes in much the same way as it does a flexible conductor conveying a current. If, for instance, the horizontal bright ring discharge of the preceding article be introduced into a horizontal magnetic field, the ring in those regions where it is at right angles to the lines of force will be divided into two parts, one being raised and the other lowered, for the reason that the discharge is oscillatory. The magnetic field in this case has also the effect of increasing the difficulty of the discharge. If, however, the lines of force are in the direction of the discharge, the effect of the magnet is to facilitate the discharge. It has been suggested that the streamers in the aurora borealis are intimately connected with this effect of a magnetic field, since the rare air in which the dis- charges occur is electrically weakest along the lines of force. 454. Discharge between Electrodes in a Rare Gas. When the discharge passes between electrodes in a tube containing gas at about i mm. pressure, the following phenomena may be observed: 1. A velvety glow, often appearing in patches, runs over the surface of the kathode k (Fig. 324). An opaque body placed withki this glow casts a shadow on the negative electrode. 2. From I to b there is a comparatively dark region, called the Orookes\ or first dark space. Its length increases as the pressure diminishes, though not in the same proportion. The luminous bound- ary of this dark space is, roughly, a surface parallel to the surface of the kathode. 500 PASSAGE OF ELECTRICITY THROUGH GASES. 3. Next to the dark space comes a luminous region, bp> called the negative glow, or column, of variable length, but entirely independent of the position of the anode. The negative glow is stopped by any substance against which it strikes, and is checked by too great a restriction of the space between the walls of the tube. The effect of a magnet on the negative column may be described as that which would be observed if it consisted of a paramagnetic substance with- out weight and having perfect freedom of motion. 4. Following the negative glow is the Faraday, or second negative dark space, very variable in length and sometimes wanting. 5. The remainder of the tube, quite up to the anode, is occupied by a luminous space, called the positive column. This region often exhibits striking periodic variations in its luminosity, known as striae. The distance between the bright parts increases with the diameter of the tube, provided the striations reach its sides. This distance also increases as the density of the gas diminishes. The striae often have an irregular motion of translation which obscures their distinct- ness. When observed in a revolving mirror, many discharges appear striated which seem continuous when examined by direct vision. It is not improbable that the striations exist at pressures much greater than those at which the striae are usually observed. The positive column always takes the shortest path through the gas to the negative electrode, and is re- garded as made up of a succession of discharges by which the electricity passes between the electrodes. Its length increases with the length of the tube. The negative dark space and the negative glow are merely local effects depending on the circumstances of the passage of the electricity from the gas to the kathode. By the use of a revolving mirror and a DISCHARGE OF ELECTRODES IN RARE GAS. 501 long discharge tube, J. J. Thomson has measured the velocity with which the flash travels through the positive column, and finds that in air at i mm. pressure, in a tube 5 mm. in diameter, the velocity of discharge is rather more than half the velocity of light, the luminosity always traveling from the positive to the negative electrode. The positive column is deflected by a magnet in the same way as a perfectly flexi- ble wire would be when carrying a current in the direction of the discharge through the tube. 6. Negative rays or molecular streams. In a highly exhausted tube the luminous effects are chiefly confined to the glass, as if gaseous particles were projected FIG. 325. at right angles from the kathode, and had the power of excit- ing luminescence in any substance which will phosphoresce in ultra-violet light. If either a conducting or insulating screen be placed between the electrodes and the walls of the tube, a shadow is thrown on the glass, that is to say, the phos- phorescence is stopped at all points behind the screen (Fig. 325). The light emitted from any phosphorescing substance within the tube shows a band spectrum characteristic of the 502 PASSAGE OF ELECTRICITY THROUGH GASES. substance. Crookes has further shown that, by using a por- tion of a spherical shell for a kathode, the negative rays may be concentrated on a platinum wire which is made to glow. When the rays are allowed to fall on vanes, such as are used in the radiometer, these are made to revolve as if bom- barded by particles from the negative electrode. The kathode ray is deflected in a magnetic field in the same way as it would be if it consisted of a stream of negatively charged particles moving away from the kathode. Two pencils of these rays exert a mutual repulsion which is consistent with the same hypothesis. 455. Kathode Photography. Hertz discovered that the glass of a Crookes tube would phosphoresce when covered with thin flakes of gold leaf, and afterward showed that these metal flakes were more transparent to the kathode radiations than sheets of mica of equal thickness. In 1894 Lenard, by using an aluminum window, succeeded in bring- ing the kathode rays out into the air, where their presence was detected by the blackening of a photographic plate and the production of luminescence. He also found that dense bodies were rather more opaque than rare ones. These radiations passed through hydrogen more readily than through oxygen at the same pressure ; but if the hydro- gen was compressed to the same density, it was as opaque as the oxygen. He also obtained a shadow photograph on a sen- sitive plate enclosed within a box with an aluminum front. In 1896 Rontgen found that high-vacua tubes which had been covered with black paper produced luminescence in phosphorescent substances, and that bodies differed widely in their ability to transmit these radiations. He found, for instance, the flesh of the hand so much more transparent than the bones that the latter produced a visible shadow on THE AEG DISCHARGE. 503 a fluorescent screen. He also made photographs of objects contained in an aluminum box, and of the bones in living subjects. The Rontgen radiations seem to differ from those inves- tigated by Lenard, in the distance from the source at which they are perceptible, and by their not being deflected by a magnet. 456. The Arc Discharge. The arc discharge is charac- terized by the incandescence of both terminals, the passage of a considerable current, and a comparatively small poten- tial difference. This potential difference is virtually inde- pendent of the current, and may be expressed in terms of the length of the arc , by the equation (2) V= a + II, where a and b are constants. When Fis measured in volts, and I in centimeters, a and b are, roughly, numbers of the order of 30 and 50 for carbon, provided I is not very small. The great fall of potential occurs close to the anode. It seems probable that the term a (equation 2) is connected with the work required to disintegrate the electrodes. This disintegration is a very marked feature of the arc discharge, and is not confined to the kathode, as in the case of the passage of the electricity through an exhausted tube. In the case of carbon the loss of the anode is considerably greater than that of the kathode. When the current is con- tinued for some time, the former is hollowed into a cup-like form, while the kathode takes a pointed shape. 457. Theory of the Electric Discharge. The phenomena of electric discharge are best explained by the view that the pas- sage of electricity through air or through any other gas, 504 PASSAGE OF ELECTRICITY THROUGH GASES. as well as through an electrolyte, is effected by chemical changes. Chemical decomposition on this view, then, is not an accidental accompaniment of the discharge, but the essen- tial feature without which it could not occur. The nature of these chemical changes is most clearly represented by the idea, introduced by Grotthus, of a chain of gaseous particles alternately charged positively and negatively, oo oo 0- and oriented by the electrical forces of the field. Suppose, for instance, that P and N b c d e f g h O 00 00 00 $ a be d e f g h (Fig. 326) are two electrodes, FlG . 326. and that ab, cd, ef are a series of molecules in such a chain. It may be shown by calculation that it is unlikely that any separation will take place from the unaided agency of the external field. But if the bond between two atoms is nearly broken by a collision, the forces of the field may complete the separation. Then, if a is freed, it will go to P, b will join to * 1 1 1 n 1 i i ' i 1 . i i 1 ' ' ' ' ' ' 1 1 * i > ' 1 i < ' 1 T > 4 . i i FIG. 334. Since the forces of restitution requisite for a transverse vibration arise from the shearing elasticity of the medium, transverse waves can be transmitted only by solids. Light-waves are, however, transverse ; but the vibrations in this case must be regarded as of an electrical nature. 467. Longitudinal Waves. If the displacements of P (Fig. 334) be plotted parallel to the direction of propagation, the particles will now have the arrangement shown at Q, which represents the position of a row of particles, normally equidistant, at any instant while transmitting a longitudinal wave. 516 WAVES. At r is a point of rarefaction, or minimum pressure, and at k is a point of condensation, or maximum pressure. An in- stant later both of these points will have moved to the right, if the differences of phase are regarded the same as in P. The forces of restitution necessary for the propagation of such a wave are those arising from the compressural elasticity of the medium. Accordingly, all forms of matter are capable of transmitting waves of rarefaction and condensation, of which sound-waves are the most important example. Since a clearer representation of a longitudinal wave may be obtained by plotting either the displacement or the pressure as ordinates, that method will be adopted in the following articles. 468. Equation of a Wave. --To find the equation of a wave, let A, i,j\ k, etc. (Fig. 335), be a continuous row of Y J k particles which at any instant are displaced to the positions A', z v , j 7 , k', according to the harmonic law. Take the origin of coordinates at the point A, and suppose that time is reck- oned from the instant when h is passing its mean position. The displacement of this point at the time t may then be written = a sin == a sin ADDITION OF WAVES. 517 where a is the amplitude, P the period, and a the angle in the circle of reference. The displacement of the point i at the distance x from the origin will be (3) y = a sin ft = a sin (a 8), where /8 is an angle less than a by an amount, S, in case the wave is moving to the right. If the wave-length be denoted by X, whence, by substitution in equation 3 ft x\ (5) y = asm 2ir ( - J, which is the equation sought. The number 8 determines what is called the difference of phase between the points h and i. It is variously referred to as a fraction of the period, as a fraction of the wave-length, or as an angle in the circle of reference. 469. Addition of Waves. Since, when two displacements of a particle occur simultaneously, neither is modified by the FIG. 336. other, as was shown in Art. 15, the resultant, arising from the superposition of two systems of waves, may be obtained by the geometric addition of the individual displacements of each particle. For instance, let a l l l c l (Fig. 336) and 518 WAVES. a 2 b 2 c 2 be two waves which are simultaneously traversing the row of particles abc. Then the displacement aa', bb', cc', of the resultant wave may be found by taking the vector sum aa^ -j- aa 2 ; bb 1 -j- bb^ etc. Several special cases are of interest. 1. Two waves of the same length, but differing in phase, combine to produce a wave of the same length, for the dis- FlG. 337. placement of any particle, a (Fig. 337), in the resultant wave, due to the addition of the displacements aa^ and aa^ will be exactly repeated at every other point, #, which is distant one or more wave-lengths from a. 2. Two waves of the same period and amplitude, but FIG. 338. differing in phase by a wave-length, combine (Fig. 338) to produce a wave of double amplitude. 3. Two waves of the same period and amplitude, but FIG. 339. differing in phase by a half wave-length, mutually annul each other (Fig. 339). The waves in this case are said to interfere. ADDITION OF WAVES. 519 4. Two waves which differ slightly in length combine to produce a wave of varying amplitude as illustrated by the heavy line in Fig. 340. In the case of sound-waves this system gives rise to the phenomenon of beats. FIG. 340. The preceding results may also be obtained analytically. Thus, to add two waves having the same period, write = w in equation 5, then (6) y l = j sin (tot Sj), (7) ?/ 2 = a 2 sin (tot - 8 2 >. Expanding and adding, (8) yi + y* sin cot (a x cos 8 l + a 2 cos 2 ) cos cot (a^ sin Sj -j- a 2 sin 8 2 ). Introducing two new symbols, defined by ( A cos 7 .= j cos Sj -J- a 2 cos 8 2 ( ^1 sin 7 = #j sin ^ + ^2 s ^ n ^2' ^! + y a = ^4. sin cot cos 7 A cos < sin 7 (10) = A sin (&> 7) ; whence it is seen that two sine waves having a common period combine to produce a new sine wave of the same period, but with amplitude A and phase constant 7, the values of which from equations 9 are (12) tan = z#ja 2 cos (Oj Sa sin S ^a cos 8 520 WA VES. When one wave is a whole wave-length behind the other, or the difference of phase is any multiple of 2?r, (k being some integer), whence (13) A = a l -\- a 2 . When one wave is an odd half wave-length behind the other, or the difference of phase is any multiple of TT, B 1 S 2 = (2& + 1) TT, or (14) A = a 1 a 2 . If, in addition, a l = a^ the wave vanishes. To investigate the case where one period is slightly greater than the other, it will be convenient to write equation 5 in the form (15) y = a sin %7rn It -- V which may be done by aid of the relations n and v = n\. Suppose, now, that the amplitudes of both waves are the same, but that the frequency of one is m and that of the other is n. The equation of the resultant wave is then (16) y l + y 2 = a sin 2?rm ( t -\-\-a sin 2?rn (t- Making use of the trigonometric relation (17) sin a + sin /3 = 2 sin -~ cos equation 16 may be written (18) STATIONARY WAVES. 521 This equation may be most readily interpreted by writing it in the form / x\ A sin 2-n-N It -- I * (19) which shows that the resultant wave has a frequency which is the mean of its components. Its amplitude, how- ever, is variable, both in time and place, as is indicated in Fig. 340. The value of A, (20) n A 2a cos 2?r x\ -- ] V J shows that, as the waves pass any given point, their ampli- tude varies from to 2a, (m n) times per second. 470. Stationary Waves. When two equal waves traverse a row of particles in opposite directions, the resulting wave remains stationary. Let A (Fig. 341) be a wave moving to the right, and B a similar wave moving to the left. At the point 6, midway between the points of zero displacement x and y on each FIG. 341. wave, draw the ordinates W and bb". Since V and b lr are homologous points on equal waves, these ordinates are equal, and b will be a point of zero displacement on the resultant wave. Also, since the waves are equal and moving with 522 WAVES. the same velocity, the increase of bb' in any given time will be the same as the increase of bb", and b will remain a point of no displacement ; that is to say, the wave remains sta- tionary. The intermediate points of the row vibrate together from zero displacement to a certain maximum value, depending upon their position in the wave. The points of the row which remain at rest are termed nodes, and those which vibrate through the greatest amplitude, distant a quarter wave-length from the nodes, are called antinodes. The analytical expression for a stationary wave is obtained by adding y l = a sin (at &), a wave traveling to the right, to y^=.a sin (cot -f- S), a similar wave traveling to the left. Substituting and t J. in equations 11 and 12 A = 2a cos S, and 7 = 0, which, in equation 10, gives y 1 -{- y^ 2a cos 8 sin o> = 2a cos sin o)^, (21) _ which is the equation of a stationary wave. It is seen from the absence of the phase constant that each particle is in the same phase ; the amplitude, however, varies from point to point, and nodes occur at distances which are multiples of a half wave-length. WAVES IN A STRETCHED CORD. 523 471. Waves in a Stretched Cord. The phenomena of lon- gitudinal and transverse waves may be conveniently studied by the aid of a long helical cord, made by winding hard brass wire about a small rod. If one end be fastened to the wall and the cord be held taut, a smart blow with the finger against the cord will cause a transverse wave to run along FIG. 342. the cord (Fig. 342). At the fixed end. this wave is reflected and will return to J., but with the important modification that a crest is returned as a trough and vice versa. In this case reflection is said to occur with change of phase. When the cord is attached by means of a long thread, as in Fig. 343, the reflection occurs without change of phase, the wave being returned in the same form as it advanced. A division of the energy, however, occurs at B, depending on the relative masses of the cord and the thread. By giving to the cord a series of properly timed impulses, the reflected and the advancing waves may be made to com- bine so as to produce a stationary wave. The same cord may be made to illustrate longitudinal FIG. 343. waves by plucking the coils apart with a hard point, though in this case the pulse will, in general, move so fast as to be followed with difficulty by the eye. When one end of the cord is fixed, reflection takes place without change of phase ; a condensation is reflected as a condensation, and a rarefac- tion as a rarefaction. 524 WAVES. The laws of reflection may be established by consideration of a row of balls (Fig. 344) which may be supposed attached by a series of elastic bonds. If the ball p be struck, a com- pression will run through the row, and the ball r will fly off ; but as it is attached to q by a bond which will transmit a oococoo FIG. 344. tensile stress, it originates a rarefaction which will traverse the row from right to left. If the row be composed of balls of two sizes, as in Fig. 345, the change in the velocity of the ball b may be found by the equations of impact (Art. 81). Let the masses of the balls COOCH2OXD a be FIG. 345. be m and ra', and their velocities before impact v and 0, and the velocities after impact u and u f . By equations 8 and 9, P. in, u = m m , . m - v, and u' = - - v, m -\- m m -f- m from which it appears that u is in the same or the opposite direction to v, according as m is greater or less than m f . Ac- cordingly, when a longitudinal wave passes from one medium to another, a part of the energy is reflected and a part trans- mitted, the phase of the reflected wave being changed, i.e. a half wave-length lost, when the reflection occurs in the denser medium. In the rarer medium the phase of the wave is unaltered. VELOCITY OF A TRANSVERSE WAVE. 525 By consideration of a row of balls united by bonds pos- sessing a shearing elasticity (Fig. 346), it may be shown in an analogous manner that the direction of the velocity of a v transverse impulse im- | parted to a row of par- CyCjCjCX m/ A A A ) tides sucn as m w iH be reversed or continued at FIG. 346. m, according as m is less or greater than m f . The reversal of the velocity of m will evidently return a crest in the reflected wave in place of a trough in the advancing wave. Hence, whenever a transverse wave passes from one medium to another, a part of the energy is reflected and a part transmitted, the phase of the reflected wave being changed, i.e. a half wave-length lost, when the reflection occurs in the rarer medium. In the denser medium the phase of the wave is unaltered. 472. Velocity of a Transverse Wave. LeiDABH(Fig. 347) be a perfectly flex- ible massive cord passing about a pulley, (7, and driven with a constant velocity. Let T= tension in the cord, v = velocity of the cord, R = radius of curvature at A, I length of small arc DB, m mass of the length I of the cord, IJL = mass per unit length. H FIG. 347. The normal force at A, due to the tension, supposed to be applied at B and 2), is (22) F=2T sin a. 526 WAVES. The force per unit length, when the point B is regarded as very close to A, is F 2Ta T < 23 > T= =5- But by the law of uniform motion in a circle or, F m v 2 v 2 <24) -l=Jlt =li R- If the velocity of the cord be such that the pressure against the pulley just vanishes, v* T (25) *z=Jt or, (26) i these circumstances the pulley might be removed Without altering the form of the cord at BD, and as all motion is relative, it is a matter of indifference whether the cord be regarded as running around the bend, or the bend be regarded as running around the cord with a velocity, v. Adopting the latter view, equation 26 shows that a trans- verse wave will travel along a stretched cord with a velocity equal to the square root | : . j : : | : j of the tension divided by I j I I : : the mass per unit length. v 473. Velocity of a j ^g Compressural Wave. - Let Fig. 348 represent a portion of a medium through which a compressural wave is passing from left to right, and suppose that P v P 2 are VELOCITY OF A COMPRESSURAL WAVE. 527 two planes which, keeping a constant distance apart, move along with the velocity of the wave so as always to maintain the same relative position with respect to it. Under these circumstances a certain quantity of matter may be regarded as flowing through any area, J., of the planes. To find its amount Let MJ = the actual velocity of vibration of a particle close raj = the mass which flows through the area A in P v P 1 = the volume per unit mass near P v v = velocity of the wave, T = an interval of time so short that u-^ remains sen- sibly constant. Then, since the velocity of the matter with respect to P 1 is Mj v, the length of the portion of the substance which passes in the time r is (u v) r, and its mass is ( U v)r-A (27) Wl = i-l > ; Pi similarly, the mass which passes P 2 will be (u* v) T - A (28) m, = V 2 ' Now, since the distance between the planes is invariable, and the density of the intercepted substance does not change, for the planes always occupy the same position with respect to the wave, as much matter will pass in through P 2 as passes out through P r Therefore, m 1 = m 2 = m, say. Making these substitutions, (29) Wj = V ^!, (30) =v _??5t2. 528 WA VES. Now, although the same quantity of matter issues from the plane f > l as entered at -P 2 , it suffers a change of momentum from the operation of the force of elasticity of the medium. Calling this force F, and measuring it by the rate at which it changes the momentum, T, -i . n n , , A . ~ (31) F= - - -- 2 = __ (/3 2 - /3J, by equation 30. Again, in the definition of the coefficient of elasticity, E= - , if ft is the volume of unit mass of the undisturbed strain o _ o medium, the strain will be 2 - ; whence (32) E= - ~-~~ A bj e( l uation 31 ' But for the undisturbed medium vrA (33) m = ~ft~^ which, substituted in equation 32, gives H- Replacing - by the density /?, and solving for v, (34) V which is the velocity of a compressural wave in any medium, provided the temperature remains constant. If the trans- mission is adiabatic, this formula becomes (35) v where 7 is the ratio of the two specific heats, as explained in Art. 240. COMPOSITION OF VIBRATIONS. 529 474. Composition of Vibrations. Lissajous's Figures. - A case of composition of vibrations of considerable interest is that in which a particle is displaced by two harmonic vibrations in directions at right angles, particularly when the periods of the two motions are in the ratio of two small whole numbers. The curves described by the particle may be found by eliminating t from the equations x = a sin (27rnt S), (36) (37) y = b sin The character of the different curves may be more readily studied by the following geometric construction. Suppose first, for simplicity, that both periods of vibration are the same, and that they take place, respec- tively, in the lines RQ and TS (Fig. 349), Let a'V - - k'l' be the circle of reference for the first vibration, and a"b" - - - k"l" that for the second. Suppose, also, that the difference of phase is such that when the vertical displacement of the vibrating point P is TA, the horizontal displacement is TP. The actual position of the point will then be A, which may be determined by the abscissa through a' and the ordinate through a". In like manner b r and b lf determine B, the position of P after a short lapse of time, with similar con- structions for other points. The resulting curve for this case FlG. 349. 530 WAVES. is an ellipse inscribed in a rectangle having sides equal to twice the respective amplitudes. The effect of al- tering the difference of phase will be to shift the position and dimensions of the ellipse, leaving it always inscribed in the same rectangle, of the periods is 2 to 1, an application of the preceding method gives the curve of Fig. 350. Fig. 351 shows some of the forms assumed as the differ- ence of phase is changed from to' TT when the ratios FIG. 350. When the ratio FIG. 351. WATER WAVES. 531 are i ? ^ or |. From TT to 2-Tr the curve runs through the same series in the inverse order. As the values of n and n f are increased, the figures become very complicated; there is, however, a method of regarding them, introduced by Lissajous, after whom they are named, which affords a clear conception of the changes the figures undergo as the periods are altered. Let one of the directions of vibration of the point P, for the case where the periods are equal, be RQ (Fig. 352), and suppose that the other is replaced by uni- form motion in a circle TCSL The com- ^J^ _ bination of these two motions will give a curved path, L CFI, lying upon the surface of a cylinder. If the eye be placed at a great distance in the plane of TCS, the horizontal motion of the point P will appear rectilinear, and the curve LCFI as one of the forms of the ellipse (Fig. 349). By revolving the cylinder about its axis, the projected curve may be made to pass through all the forms which would be obtained by varying the difference of phase of two rectilinear vibrations between and 2?r. If the surface of the cylinder be unrolled, it is obvious, from the mode of generation of the curve LCFI, that the latter will be developed as a simple sine curve. From the preceding principles it appears that Lissajous's figures may be regarded as the projection of sine curves traced upon the surface of a cylinder. 475. Water Waves. When the surface of a body of water, or other liquid, is disturbed, a system of waves is formed of a type essentially distinct from those previously described. "-^ /TT p'\ "^ <. ^.-^L" \P' s ~~ X. n f FIG. 352. 532 WAVES. Although for small amplitudes water waves are hardly dis- tinguishable from those arising from transverse vibrations, it is obvious that they cannot be of this type, since the lower layers of the liquid must then be alternately rarefied and compressed. The actual motion of the particles of water in transmitting a surface-wave of considerable dimensions, say longer than a foot, is one of revolution in a vertical curve under the influence of weight, as appears on observation of a moving liquid carrying fine particles of any substance in suspension. In deep water these curves are exactly circles. 476. Lyman's Apparatus. Many interesting properties of water waves are exhibited in the apparatus shown in Fig. 353, designed by Professor C. S. Lyman. The particles of water at the surface are represented by little studs, .A, B, , attached to a series of cranks, which revolve simultaneously, and the wave itself by a flexible wire passing through holes drilled in these studs. The motion of the particles below the surface diminishes according to the exponential law. At a depth of an eighth LYMAN'S APPARATUS. 533 of a wave-length the excursions of the particles represented by #, b - - - are only half as great as at the surface, and at the depth of a half wave-length the water is virtually quiescent. The wave form belongs to a family of curves known as troehoids, and may be generated by rolling a circle on the FIG. 354. line FCr (Fig. 354), while some point, />, is allowed to trace its own path. The curve is never symmetrical with respect to the line of centers, the maximum curvature always being numerically greater than the minimum. When the describing point is chosen on the circumference of the rolling circle, the roulette takes the special form known as the cycloid, the part in this case corresponding to the crest of the wave rising to a cusp. JN (Fig. 353) is a circle, having a circumference equal to the length of the wave. It may be shown that the period of the wave is that of a pendulum whose length is equal to the radius of this circle. Calling this radius R, the wave-length X, and the period P, (38) J R = A, (39) 2*91 whence 534 WAVES. or the velocity of the wave increases as the square root of the wave-length. The wire JK in the figure, pivoted at J and constantly passing through the point E, has always the direction of the resultant force passing through the particle E\ for, letting r denote the radius OE, the central force, acting on the particle m at E, will be , which, combined with the expression for P in equation 39, gives whence it follows that, if EO is taken proportional to the central force on the particle, and JO to the weight, JE will represent in magnitude and direction the resultant force on the particle. The model also shows that the direction of this force is always at right angles to the surface of the wave, as it should be. When F c = mg, the particles at the pointed crest of a wave lose their weight and are readily blown off in the wind. Since in forming a wave more water is taken out of the trough than is put into the crest, the line of centers is some- what higher than the original surface LM. The vertical wires AaaJ, BbV of the model illustrate the peculiar swaying motion which takes place in lines ini- tially vertical, or the strain to which a vertically floating board would be subjected. Similarly, the bending of the wire ABCDEFCr shows the strain on a horizontal body at the surface. The distortion suffered by a rectangular block of water is shown by ABla, BCcb, etc., which assists one to understand why a wrecked vessel is so quickly broken up by the waves. RIPPLES. . 535 477. Ripples. In the previous discussion of water waves it has been assumed that the waves were of considerable size. In the case where their dimensions are not of such size that the weight of water displaced in each wave may be regarded as very great with respect to the force of surface tension, the expression for the wave velocity must be altered by the addi- tion of a term containing this force. The equation then takes the form where T denotes the surface tension and p the density of the liquid. It thus appears that the velocity of the wave will increase with diminishing wave-length, if this is very small, and also with increasing wave-length, if this is large. When the two terms are equal, i.e. when P9 * p the wave will have a minimum velocity. This minimum velocity for water is about 23 - . which sec. corresponds to a wave-length of 1.72 cm. Wavelets in which the surface tension is the preponder- ating force, that is, those whose wave-lengths are less than 1.7 cm., are designated as ripples. The existence of a minimum velocity for water waves offers a simple explanation for the singular fact that the sur- face of a small body of water may remain perfectly calm, although the air above it is not so ; for, if a variation in the air pressure should create a slight wave disturbance, the ripples would run away from the disturbing cause as long as the velocity of the air currents remained less than 1.7 '-. sec. It appears further, from equation 42, that there are in gen- 536 WAVES. eral both a ripple and a wave, which travel with the same velocity. These may be shown experimentally by moving a small rod dipped vertically into water, with a velocity slightly greater than 23 cm. per sec. The ripples in this case may be ob- served advancing in a group with, and in front of, the rod, while the waves follow at the same speed behind. If the motion of the rod be quickened, the wave-length of the rip- ples will be shortened, while that of the waves is increased. The dependence of the wave-length and velocity of ripples upon surface tension has been utilized by Lord Rayleigh to determine the value of surface tension in various liquids. The method in brief is to produce a series of stationary rip- ples by the vibrations of a style, attached to a tuning fork of known period, and dipping into a trough of the liquid. The wave-lengths are then carefully observed and the surface tension calculated from these data. 478. Wave Propagation in Three Dimensions. Suppose a disturbance takes place at a point, p (Fig. 355), in an iso- tropic medium. Since it will be propa- gated with the same velocity in all direc- tions, after a certain lapse of time the disturbance will have spread so as to affect similarly a series of particles lying in the surface of a sphere, abc. A disturbance propagated in this manner is termed a spherical wave, the wave front being de- fined as the locus of all particles having the same phase. 479. Huyghens's Principle. Suppose that, when the wave originating at p has reached the point b (Fig. 356), the vibra- tions of every other particle are reduced to rest. The point REFLECTION AND REFRACTION. 537 b now becomes a center of disturbance, and an instant later a small wave from this center will have reached b'. But the motion of b is just the same whether the neighboring parti- cles are at rest or vibrating. Therefore, every point of the wave front passing through bode must be regarded as a source of disturbance, and as sending out little waves which an in- stant later will have reached b'c f d f e'. But, since the only wave which can be physically p. recognized at a 1 - b f g' is the spherical one having the center p, it follows that any wave may be regarded as ema- nating either from an original source, p, or from a series of points through which the wave FIG 356 front passed an instant before. This mode of regarding the disturbance at any point of a wave front as the resultant of the separate disturbances which would have been produced by each point in the wave front acting singly at some earlier position, is knowri as Huyghens's Principle. Stokes has shown by analysis that each of the elementary waves mutually destroy each other, except at the surface which is their envelope. 480. Reflection and Refraction. Suppose that AS (Fig. 357) represents the smooth interface between two media of different density, the denser one being below. By a smooth surface is meant one in which the roughnesses are small 538 WAVES. compared to the wave-lengths. Thus, a masonry wall would be smooth in this sense to long sea waves, or ordinary sound waves, but to such short waves as produce the sensation of vision, only the most highly polished surfaces can properly be called smooth. Suppose, also, that kl and op are the fronts of a train of waves moving in the direction of the arrow. Experience shows that when op strikes AB there arise, in general, two new systems of waves with changed directions ; those such as oq, moving in the first medium, are known as reflected waves, and those FIG 35 7> such as oq f , transmitted through the medium on the other side of the bounding surface, are called refracted waves. To determine the laws of reflection and refraction it will be sufficient to observe that at the surface AS there is a por- tion of the wave surface which is common to the three sys- tems. Let the angle between the incident wave po and the interface, namely, poB, be denoted by i ; the angle between the reflected wave oq and AB, namely, qoB, be denoted by /o; and the angle between the refracted wave oq' and AB, namely, Aoq r , be denoted by r. Also, call the velocity of the waves in the first medium v, and in the second v f ; then, since the velocity in the direction of the interface of that portion of the wave common to all is the same in each system, V V V 1 (43) -; ;=- = - , sin ^ sin p sin r whence (44) sin p = sin i, REFLECTION AND REFRACTION. 539 and sin i , v (45) = .. sm r v' From equation 44, since i is not identically equal to />, for there is known to be a change of direction, (46) p = 7r i, which is the general law for reflection for all types of wave surfaces. It is frequently stated by saying that the angle of inci- dence is equal to the angle of reflection ; but this statement, in strictness, requires the additional explanation that the angle of reflection is to be measured in the opposite direction from the angle of incidence. Equation 45 is the law of refraction. Since the ratio of the velocities in the two media is constant, it may be denoted by a single letter, say w, which is called the index of refrac- tion. Writing the law in the ordinary form, (47) sin i = n sin r. Instances of reflection and refraction are common in the phenomena of sound, light, radiant heat, so called, and in elec- trical waves ; but, since the direction of propagation of light- waves only can be accurately determined by our senses, the general discussion of the consequences of these laws may be postponed to the chapters on light. CHAPTER XXXI. SOUNDS AND THEIR RELATIONS. 481. Sound. Sound may be defined as a sensation pecul- iar to the auditory nerves. A sound sensation is normally produced by a series of waves originated by a vibrating body and transmitted to the ear by an elastic medium, usually the air. A sounding body is easily recognized to be in vibration. Thus, for example, a stretched string when sounding takes the appearance of an elongated spindle with an indistinct or hazy appearance about the middle. If a light body, like a pith-ball, be arranged so as to rest against a sounding rod or plate, the ball will be visibly agitated. The sound in each case ceases if the body is touched in such a way as to stop its vibration. The sounding column of air in an organ pipe may be shown to be in vibration by lowering a hori- zontal membrane, covered with a few grains of sand, by a thread into a pipe furnished with a glass window. When the pipe is made to speak, the sand is, in general, thrown into violent agitation. To show that air is the ordinary medium of transmission of sound-waves, it is sufficient to place a bell, actuated by clock work, under the receiver of an air pump, taking care to support it so that its vibrations cannot be transmitted through the attachment. On exhausting the receiver, the sound grows fainter and fainter, and at last practically ceases. If the receiver be first filled with hydrogen, and then ex- hausted, the diminution of sound will be more marked, the CHARACTERISTICS OF A MUSICAL SOUND. 541 reason being that less energy is transmitted to the glass by hydrogen than by air, on account of the difference in density (Art. 471). The ear broadly distinguishes noises and musical sounds. The former arise from vibrations which are entirely irregular ; the latter result from periodic vibrations. It is obvious that sound-waves in air must be of the com- pressural type, since no other can be transmitted by a fluid. 482. Characteristics of a Musical Sound. The ear recog- nizes a musical sound to be capable of variation in three particulars, known as pitch, loudness, and quality. 483. Pitch. The pitch of a note is determined by the frequency or number of vibrations per second, an acute note having a greater frequency than a grave one. Since velocity, wave-length, and frequency are connected by the equation v = n\, any given note may also be designated by its wave- length in air. 484. Loudness. The loudness or intensity of a sound is found to depend upon the energy of the vibrations trans- mitted to the ear, and to increase with it. As a first approxi- mation, the loudness of a note may be taken as proportional to the amplitude of its vibration. It is obviously impossible to state a sensation with great precision, but experiment indicates that a more exact expres- sion for the loudness of a sound involves the pitch of the note, and the logarithm of the energy reaching the ear. 485. Quality. Difference in quality or timbre is a term used to express the distinction between notes of the same pitch and equal loudness, emitted by different instruments, such as, e.g., the violin, the clarinet, or the human voice. 542 SOUNDS AND THEIR RELATIONS. The quality of a note was first shown by Helmholtz to be determined by the presence of a series of notes 'simply related to the fundamental or tone of lowest pitch. Plotting the pressure at any point, in a medium transmit- ting sound-waves, as a function of the time, the pitch of a given note may be represented by the length, the loudness by the amplitude, and the quality by the form of the wave. 486. Siren. A convenient instrument for producing notes of any required pitch is the siren, invented by Caignard de la Tour. It consists of a wind chest, A (Fig. 358), having a top perforated with a circular row of holes. Close to this, on the upper side, is a metallic disc, B, pierced in an anal- ogous manner, and so arranged that it will rotate freely on a vertical axis. The direction of the holes is inclined to the face of plates, those in the disc sloping in a direction opposite to those in the top. Air, on being forced into the chest by a bellows, escapes through the open- ings in the top ; but since, in its passage, the direction of the current is changed, a certain pressure will be exerted against the sides of the holes, setting the disc in rotation. As a result of this rotation, the air escapes in a succession of puffs which give rise to musical sounds of definite pitch. In order to count the number of revolu- tions, the spindle carries at the upper portion a screw which can be made to engage with a wheel of one hundred teeth, _Z), to whicli is attached a pointer moving over a graduated dial. By the side of D, and connected with it, so that it is advanced one tooth for every revolution of D, FIG. 358. VIBROSCOPE. 543 is a similar wheel, .Z7, the whole serving as an automatic counter of the number of revolutions made by the perforated disc. In order to study notes which are simply related, the siren is usually made with more than one row of holes in the disc, and provided with stops, so that the air can be admitted to one or all rows at pleasure. When it is desired to obtain the frequency of a note, wind is admitted to the chest uittil the note emitted by the siren is judged to be in unison with the given note, after which the speed is kept as nearly constant as possible, and the train of counters thrown in gear with the spindle for a certain period of time, which is carefully noted. The product of the number of holes in the row used, by the number of revolutions per sec- ond, gives the frequency of the note sounded. As it is quite possible to make an error of one revolution, such a determination may be in doubt as much as ten vibrations. 487. Vibroscope. A tuning fork may be made to record its frequency automatically by attaching a style to one of the prongs so that it will leave an undulating trace upon a piece of smoked paper wrapped about a revolving cylinder, as in Fig. 3. By arranging a pendulum so as to close an elec- tric circuit every second, a spark is made to pierce the paper at the point of the style and mark equal intervals of time. The frequency of the fork is thus readily found by counting the number of undulations between the punctures. The free period of the fork will be influenced slightly by the mass of the style, its friction against the paper, and, when the fork is electrically driven, by the forced character of the vibrations. If, however, the period of a particular fork be carefully deter- mined under definite conditions from its graphical record, the frequency of any other fork may be obtained from it with great accuracy by Lissajous's method of comparison (Art. 510). 544 SOUNDS AND THEIR RELATIONS. 488. Musical Intervals. If the outer and inner rows in the siren, having, respectively, sixteen and eight holes, be open at the same time, the resulting tones are recognized to be in a concordant relation to each other, called the octave, whatever the absolute pitch of the notes may be. In general, when two notes differ in pitch, they are said to be separated by a musical interval, which is measured by the ratio of the frequency of the higher to that of the lower note. If in the siren the rows containing twelve and eight holes are opened, the ear will recognize an interval known as the Fifth. Simi- larly, the rows of sixteen and twelve holes, respectively, will yield the Fourth. 489. Musical Scales. It has been recognized as a funda- mental principle of the musical compositions of all nations that the alteration of pitch in any melody takes place by steps, or intervals, and not by continuous transition. The particular succession of intervals by which a composition ad- vances from one note to its octave is called a musical scale. The major scale is a succession of eight notes which are related to the fundamental and variously designated as shown in the following scheme, the first note being taken as 256 IP gp -zs I] r vsLJ 256 288 320 341 384 427 480 512 c' d' e' f g' a' 6' c Do Re Mi Fa Sol La Si Do I 9 1 I 1 | V 1 First. Second. Third. Fourth. Fifth. Sixth. Seventh. Octave. vibrations per second, which is practically the middle C of the piano, or the fundamental tone of an organ pipe two feet long. MUSICAL SCALES. 545 The first three methods of representation determine the absolute pitch of the notes, but the last three refer only to their relative positions on the scale. The intervals between each note and the next higher are : Major Minor Half- Major Minor Major Half- Second Second Tone Second Second Second Tone I V if I V- I if The smallest interval recognized in music is J-^, the dif- ference between the major and minor seconds. It is called a comma. The various other octaves are usually designated by subscripts and accents as follows: m C t (16ft.) C(8ft.) c(4ft.) c'(2ft.) c"(lft.) c'"(Jft.) c iv (|ft.) c v 32 64 128 256 512 1024 2048 4096 The notes of the scale are often changed by an interval of a chromatic semitone. Theoretically this interval may have two values: Jf, the minor semitone; and || |^ = Y||? the major semitone ; but the difference is hardly perceptible by most ears, and the ordinary musical notation does not distin- guish between them. A note changed by a semitone is called sharp when raised, and flat when lowered, e.g. B when raised a semitone is designated by B#, read B sharp, and when lowered by B(>, read B flat. The scale of twenty tones, ob- tained by introducing these new tones, is called the chromatic scale. 546 SOUNDS AND THEIR RELATIONS. In addition to the two scales already denned, there is a group of three scales designated collectively as the minor mode. They are, in the key of C : C D Eb F G At? B C i I f * I t V ! I if J 9- I if 44 if CD Eb F G A B C i f f t i f V ? I if V I V I if C D Eb F G Ab Bl? C i I f t I t f I if I if I V- The first is usually an ascending instrumental scale, the second an ascending vocal scale, and the third a descending scale, though usage is not perfectly settled in these respects. The one interval common to all, and that which gives a minor chord its peculiar character, is the minor third, J. 490. Transposition. In order to accommodate different voices or instruments, the Do or keynote of a composition may be transferred from its first position to any other note of the scale. Suppose, for instance, it is desired to begin a CHANGES m scale on D = | n. Then, KEY OF D. KEY OF c . calculating the frequencies * and comparing them with e * ' si !!'! the key of C, it appears that Fa * ' 7 = * '^ '^ besides the keynote D, the G g l Q 7 _ 81 and B are right, and that E La. . . . Jgft = V- anc ^ ^ differ from the true Si .... J^5 si . is. . 2 scale only by . the negligi- Do. . . . | = | ble interval |l. The notes THE TEMPERED SCALE. 547 F and C, however, must be raised a chromatic semitone before they can be introduced into the new scale. Instead of writing the sharps before these notes every time they occur, they are usually placed at the beginning of the staff, forming what is called the signature of the key, thus : 491. The Tempered Scale. In order to perform a piece of music in just intonation in all the keys in present use, a scale containing at least fifty-three tones would be required. To voices, strings, or trombones this is of no moment, but in any instrument of fixed tone, such as the piano or organ, it is practically impossible to control so many notes. For these instruments a compromise scale, known as the scale of equal temperament, is used. In it the octave is divided into twelve equal intervals represented by the number v^ = 1.059, thus ignoring the difference between the major and minor tones, and identifying the neighboring chromatics. The following table exhibits the difference between the true and the tem- pered scale. Do Re Me Fa Sol La Si Do Just * 256 288 320 341.3 384 426.7 480 512 Even 256 287.3 322.5 341.7 383.6 430.5 483.2 512 The only accurately tuned interval in such a scale is the octave, the others being more or less false. In a melody the difference is hardly noticeable, but when several notes are sounded in a chord, especially on an instrument giving sus- tained tones, the contrast between the tempered and the true scale is more marked. Music rendered in the scale of even temperament is decidedly inferior to that played in just intonation. 548 SOUNDS AND THEIR RELATIONS. 492. Limits of Audible Sound. The greatest and the least numbers of vibrations which are capable of producing a musi- cal sound vary somewhat with the individual ear and its age. According to Helmholtz the gravest note, having a definite pitch, is about thirty vibrations per second. Below this, the auditory nerves are no longer excited uniformly throughout the whole time of a vibration, though it may be mentioned that some other investigators have thought that they were able to perceive musical sounds as low as sixteen vibrations per second. The range of audible sounds is about ten octaves, say from (^ = 32 to c viu = 32768, though many persons of fair hearing cannot perceive notes above c vu = 16384. By means of a whistle of adjustable length, such as is shown in Fig. 359, it is easy to produce a series of short waves, gradually passing beyond the limit of hearing. The highest note employed in the orchestra is d y on the piccolo. 493. Harmonics. When the frequency of one note is an exact multiple of another, the second is said to be a har- monic of the first. Thus, the first sixteen harmonics of C = 64 are, in the ordinary musical notation, C c g c' e' tf tf> c" d" e" f"$ g" a" b"b ft" c'" 64 128 192 256 320 384 448 512 576 640 704 768 832 896 960 1024 The sixth, tenth, twelfth, and thirteenth overtones, or upper partials as they are sometimes called, are not used in music, and their positions on the staff can be only approxi- mately marked. EXAMPLES. 549 EXAMPLES. 1. Show that the interval from g to d f is a true Fifth. 2. Write the scale beginning Do = |n. 3. A siren of 15 holes is speeded up till it emits the note g". How many revolutions per minute does it make? Ans. 3072. 4. The frequency of three notes are 504, 630, and 945 per second. What are the intervals between them ? f Third. Ans. \ Fifth. / Seventh. CHAPTER XXXII. PROPAGATION OF SOUND-WAVES. 494. Velocity of Sound- Waves. The velocity of a com- pressural wave has been shown in general to be P where E is the elasticity of the medium under constant tem- perature, and p its density. In the case of sound-waves in air, the changes in volume must be regarded as adiabatic, for which case the formula becomes (Art. 241) p being the pressure and 7 the ratio of the specific heats. For air, at C. and one atmosphere pressure, dynes = 1.014 (10)' cm. whence PQ = 1.293 (10)- 3 &>. 7 = 1.405; = 331.8^: = 1089 sec. sec. The mean of ten careful determinations of the velocity of sound by as many different observers has shown it to be, at zero and a pressure of one atmosphere, met. = 331.6 sec. DOPPLER'S PRINCIPLE. 551 The greatest and least values were 332.4 and 330.6. Since at the temperature t C. / 4- \ P-~Po\ 2T3 / (equation 22, p. 181), the velocity of sound at any tempera- ture in a gas will be This velocity is not sensibly affected by the pitch or the intensity of the sound. From equation 1 it appears that the velocity of sound in different gases should vary inversely as the square root of the density, a conclusion which is entirely verified by experiment. 495. Doppler's Principle. If an observer be approaching or receding from a fixed source of sound, it is obvious that the number of waves meeting the ear will be greater or less than the actual frequency of the sound by the number of wave-lengths passed over in a second. Thus, if the actual frequency be n, the apparent frequency w', the velocity of sound v, and the velocity of the observer relative to the source w, it is evident that u (2) n = n > or n \ on substituting the value of X. The alteration in pitch of a bell or whistle, due to relative motion, may often be observed on a passing locomotive. 496. Interference of Sound. An illustration of the gen- eral principle that two waves in opposite phase, when super- posed, annul each other, is afforded by the tuning fork. 552 PROPAGATION OF SOUND-WAVES. If a fork, after being set in vibration, be slowly revolved about its axis close to the ear, or before a resonant cavity, it will be found that there are four positions in which the sound attains a maximum intensity. The general character of the phenomenon may be under- stood from Fig. 360. Let A, B be two prongs of a tuning fork, each of which is vibrating in a similar manner, but in the opposite phase, as indicated by the arrows. Without attempting to state the exact condition of every point in the field, it may be seen that at any point between the prongs their displacements are such as to increase the pressure. Similar conditions obviously ob- tain for the regions d and e, on each side of the axis AB. In the vicinity of the point f the displacement of A will produce a diminution of pressure, and that of B an increase ; but the effect of A will predominate, since this prong is nearer. As one passes from the region / to the region d, there must be a line of equilibrium where there is no change of pressure, and hence extinction of sound. The positions of no sound lie nearly in planes passing through the axis of the fork, and making angles of 45 with its face. If, while the fork is held so that the ear is 011 a line of silence, a tube be slipped over one of the prongs without touching it, the sound will be at once restored. 497. Beats. A beat is a periodic variation in the inten- sity of a sound produced by the interference of two waves having slightly different periods. BEATS. 553 If the pitch of one of two forks which are exactly in uni- son be lowered by attaching a small coin to one of its prongs, by means of wax, and both be simultaneously sounded, the observer will hear bursts of sound at intervals which decrease as the difference in pitch between the forks is increased. A similar result may be obtained by slightly warming one of the forks. The mutual reinforcement and destruction of two waves are shown diagrammatically in Fig. 340, Art. 469, where it has also been proved that the number of beats a second is equal to the difference in frequency of the notes, provided they are near together. Beats may be produced by swinging a fork of acute pitch back and forth in front of a wall. In this case those waves which reach the observer directly, when the fork is moving toward the wall, will, according to Doppler's principle, be slightly lowered in pitch, while, on the other hand, those waves which reach the ear only after reflection from the wall will be slightly raised in pitch. The direct and re- flected systems are then in a condition to inter- fere and produce beats. If two organ pipes, originally in unison, be brought slightly out of tune by means of a slider, 8 (Fig. 361), which opens a hole at the top of one of the pipes, very strong beats may be produced. When the pipes are exactly in tune, and attached to the same wind chest, the FIG. 361. note obtained from the two sounded together is relatively feeble, for the reason that the air columns natu- rally take a form of alternate vibration, such that if there is a condensation at the mouth of one, there will be a rarefac- tion at the mouth of the other. The resulting waves differ 554 PROPAGATION OF SOUND-WAVES. by a half wave-length, and hence annul each other. A feather placed at the mouth of either will show that the air makes essentially the same vibrations as if one pipe were sounded alone. If a screen be interposed between the mouths of the pipes, the sound will be restored. The existence of beats between two notes is a most valu- able assistance in the accurate tuning of musical instru- ments, as the ear possesses but a limited ability to estimate such intervals as the Fifth or Fourth without their aid. EXAMPLES. 1. What is the wave-length of rf iv in air at 0? Ans. 5.64 in. 2. At what temperature would the velocity of sound-waves in air be 349 meters per sec.? Ans. 28.7. 3. How long will it take the sound of a signal gun to reach an observer 3.2 miles away if the temperature of the air is 18 C.? Ans. 15 sec. 4. Calculate the velocity of sound in hydrogen at zero. Ans. 1.261(10) 5 cm. /sec. 5. With what velocity would sound travel in water at 11 for which the coefficient of elasticity is 2.1(10) 10 gm. / cm. sec. 2 ? Ans. 1.45(10) 5 cm./sec. 6. Calculate the velocity of sound in steel. Ans. 0.522(10) 6 cm. / sec. 7. If the velocity of a sound in a gas be 340 meters per sec. at 16, what will the velocity be at a temperature of 168 if the pressure of the gas be doubled? Ans. 420 met. per sec. 8. If a note sounded on a train, which is approaching an observer at a velocity of 48 miles an hour, has an apparent frequency of 384 vibrations per second, what is its actual frequency ? Ans. 359 per sec. 9. At what velocity must the hearer approach a source of sound in order to raise the pitch a major semitone ? Ans. 40.6 miles per hour. CHAPTER XXXIII. SONOROUS BODIES. 498. Vibrating Columns of Air. The laws of vibrating air columns may be easily deduced from a knowledge of the velocity of a compressural wave and the nature of the reflection from a free or a closed end. Thus, suppose a com- pression starts at A (Fig. 362) and travels along the tube to B. Supposing the end open, it will be reflected as A\ a rarefaction (Art. 471), which on its return to A is re- flected as a con- densation and is in the same state as FIG. 362. at the start. Since in the time of one vibration it has traveled twice the length L of the tube, the fundamental wave-length is 2Z, and its frequency n = ry Whenever a succession of pulses is transmitted along the tube, those advancing combine with those returning to form a stationary wave. The node, or place of maximum variation of pressure, will be at the center, and the antinodes, or places of unchanged pressure, will be at the open ends. For simplicity of statement it is convenient to consider the displacement of the particles of air at different points along the tube. Drawing these as ordinates above or below 556 SONOROUS BODIES. the side of the tube, as in Fig. 362, the position of the node and the length of the wave are clearly shown. Open tubes may also vibrate with two, three, four, or more nodes, which must be placed so that an antinode will come at the end of the tube. These cases are represented at CD and EF ; it is easily seen that wave-lengths correspond- ing to these overtones are, respectively, L and |Z-. From the preceding reasoning it follows that an open pipe yields the complete series of harmonics 1, 2, 3, 4, 5, 6, . In a pipe closed at one end it is evident that there must always be a node at the stopped end. Hence, the different modes of vibration will be those rep- resented in Fig. 363, having wave- lengths, respec- FIG. 363. tively, 4Jv, |X, -|-_L. The fundamental u of a stopped pipe is, accordingly, an octave lower than that of an open pipe, and its overtones are the odd notes in the harmonic series, their frequencies being the order of the numbers 1, 3, 5, 7, 9, . These relations between the fundamental and the overtones were discovered by Daniel Bernoulli, and are commonly known as Bernoulli's Laws. The assumption in this discussion that there is no vari- ation of pressure at the open end of a pipe is not strictly true. It is, accordingly, found that the expressions for the wave-length require a correction, unless the length of the pipe is very great compared to its cross section. In the case of a cylindrical tube of radius R the actual wave-length is greater than the length of the tube by 0.6J2 for each open MOUTHPIECES. 557 end. For a square pipe this correction is roughly equal to the depth of the pipe. The nature of the material of which the pipe is constructed may be neglected except when it is extremely thin. It may be stated as a general law that if two bodies of air are geometrically similar, and similarly set in vibration, the pitches will be in the inverse ratio of their homologous linear dimensions. 499. Mouthpieces. The mouthpieces by which the tones of musical pipes are generated may be grouped in three classes, according as the current of air is blown across a sharp edge, or through a reed, or between membranous tongues. The mouthpiece used on the flute pipes of an organ is shown in Fig. 364. When a current of air is blown across the edge of a solid, it gives rise to a hissing sound, which may be regarded as due to a series of a great number of wave-lengths lying within ill-defined limits. Under cer- tain circumstances, as in the rapid movement of a whip through the air, or in the whistling of the wind, the sound often approaches a definite pitch. The experiments of Strouhall indicate that a current of air flowing past a cylindrical wire gives rise to a sound whose frequency varies directly as the velocity of the current and inversely as the diameter of the wire, but depends on noth- ing else, provided the temperature remain constant. It was also found that when the period of this note approached any of the free periods of the wire the intensity of the sound was greatly increased. FIG. 364. 558 SONOROUS BODIES. The excitation of flute pipes and all whistles is to be ex- plained in the same way. When the limits of the periods of the disturbances, which are caused by the rush of the current of air across the edge of the mouthpiece, include the funda- mental period of the pipe, it will resound in its lowest tone. If the velocity of the air current is now gradually increased, the fre- quency of the exciting disturbances FlG 365 will also increase, and as they come into the vicinity of the harmonics of the pipe it will speak in the pitch of its various overtones. In the flute (Fig. 365) the embouchure is merely an oval B a FIG. 367. hole, against the edge of which a current of air from the player's mouth is directed. In the second class of mouthpieces the vibrations of the pipe are induced by a vibrating tongue or reed, which alternately opens and closes an aperture in which it is placed. The reed most often used in organ pipes, and shown in Fig. 366, is known as a striking reed. The metal tongue I is slightly curved, so that it rolls itself over the aperture r, closing it gradually. The period of the reed, which is more or less coerced by that of the air in the pipe, may be altered and the pipe tuned by pushing down the wire d. FIG. 366. MOUTHPIECES. 559 The free reed used in the accordion, harmonium, and rarely in pipes, is shown in Fig. 367. The tongue in this case swings through the aperture, interrupting the air current, whose direction is indicated by the arrows. In these instruments the free reed retains its own period unaltered. The mouthpiece of the clarinet is furnished with a striking reed made of thin cane (Fig. 368). In the oboe and the bassoon the mouthpiece is in the form of a small wedge, the sides being formed of two tapering slips of cane, leaving a narrow slit at the apex (Fig. 369). The period of these labial reed tongues is determined, for the most part, by that of the air column in the pipe, since FIG. 369. FIG. 370. the instruments which employ them are capable of yielding a great number of notes. An example of a mouthpiece formed by membranous tongues is shown in Fig. 370. It is made by cutting off a gutta-percha tube obliquely on both sides, and stretching two strips of india-rubber over the points thus formed, so as to 560 SONOROUS BODIES. leave a narrow slit between them. If air be blown through the slit in either direction, the membranes will emit a note depending on the body of air in the tube. There are but two examples of such membranous tongues which have any importance in music the lips as used in playing the brass wind instruments, and the larynx in singing. In the trumpet, and instruments of that class, the mouth- piece has the form of a cup (Fig. 371), which when applied to the lips permits them to act as stretched membranes, determining the vibration of the air in the tube. The frequency of vibration depends in part upon the tension of the lips, and in part on the pressure of the air exerted by the performer. 500. The Larynx. The apparatus of the voice in man consists essentially of two elastic folds, known as the vocal cords c, c, Fig. 372, in which A represents a diagrammatic FIG. 371. FIG. 372. vertical section and B the position of the cords in tone pro- duction, as seen from above. These folds, when stretched across the head of the windpipe from front to back, leave a KUNDT'S TUBES. 561 small aperture between them, not unlike the model of Fig. 370. The frequency of vibration is determined by the form of the slit and the tension of the membrane, both of which may be altered at pleasure. 501. Kundt's Tubes. The relative velocity of sound in gases and solids may be determined by the following simple method due to Kundt. The apparatus consists essentially of a tube of glass, AB (Fig. 373), fitted at one end with a movable stopper, $, and at the other with a tightly fitting cork, J5, in which is fastened another tube or rod, (7J9, about a meter long and a centimeter in diameter. This rod is also terminated by a disc of cork which nearly fills the larger _ AS _______ _______ TO m C B D FIG. 373. tube. If the inner surface of AC be coated with a light dust, and a moist cloth be drawn along the rod D till it is set in longitudinal vibration, yielding its fundamental note, dust figures will be formed within the tube. By moving the stopper out or in, the dust may be made to collect at the nodes whenever the length of the air column is an exact multiple of the wave-length of the sound in air, which may then be taken as twice the distance between two nodes, n, m. But the wave-length in the rod is four times the distance BD. Hence, if V a be the velocity of sound in air, and V g be the velocity in glass, By introducing any other gas into the tube it is evident that the velocity of sound in this gas might be found in terms of the velocity in air. 562 SONOROUS BODIES. By the use of such a tube Kundt found that the velocity of sound relative to air was in steel 15.2, in carbon dioxide 0.8, in illuminating gas 1.6, and in hydrogen 3.56, numbers which are in substantial agreement with those calculated from formulas already given. 502. Resonance. It is easy to show, experimentally, that when an elastic body, possessing a definite period, is subjected to a series of impulses, it will not be sensibly affected unless the period of these impulses corresponds to that of the body, when it may finally receive sufficient energy to set it in vibration with considerable amplitude. For example, let a tuning fork be struck and held over the mouth of a cylindrical vessel (Fig. 374) having a depth greater than the quarter wave-length of the note given by the fork. If water be gradually poured in, it will be found that at a certain level the sound of the fork will be strongly reinforced ; but above or below this level the column of air within does not FIG 374 greatly affect the intensity of the sound. This phenomenon, in which one body sets in vibration another having a similar period, is termed resonance. The principle is extensively used in musical instruments to reinforce the sound. Thus the cords of all stringed instruments are attached to a sounding box with an air cavity, by the aid of which the energy is given out at a more rapid rate. 503. Resonators. For the purpose of investigating the existence of partial tones in any note, Helmholtz devised the instrument which he called a resonator. It consisted of a globe of thin brass enclosing a mass of air possessing a defi- THE VIBRATIONS OF STEINGS. 563 nite period of its own. On one side there was a small open tube, b (Fig. 375), designed to fit into the meatus of the ear, and opposite it another opening, #, by which the enclosed body of air might receive vibrations from without. When one ear is stopped and such a resonator applied to the other, practically the only sound heard is that having the same pitch as the resonator. By using a J FIG. 375. series of such instruments, Helmholtz was first enabled to make a satisfactory analysis of a com- pound note into its partial tones. 504. The Vibrations of Strings. The fundamental vibra- tion of a musical string fixed at both ends may be regarded as identical with those giving rise to waves of the stationary type in a cord of unlimited length whose nodes correspond to the ends of the string. The period of a stationary wave is obviously the time it takes one of the progressive waves of which it is composed to move over a wave-length, i.e. twice the distance between the adjacent nodes. Hence, if I be the length of a string fastened at both ends and n its frequency, by equation 2, Art. 466, or, substituting the value of v from equation 26, Art. 472, 1 T The influence of the length, the tension, and the linear Density on the pitch of strings is illustrated qualitatively in instruments of the violin type. Thus, as the length of the string is shortened by running the finger down the fin- 564 SONOROUS BODIES. ger board, the pitch rises. The pitch is also raised by in- creasing the tension in the process of tuning. In the lower strings the period of the open string is lengthened by wind- ing it with wire, thus increasing its linear density without affecting its elasticity. For studying the relation between the length of a string and the pitch, the instrument shown in Fig. 376, and known as the sonometer, is a convenient one. It consists essentially of a wire stretched over two fixed bridges on a sounding-box. Between these a movable bridge is arranged to slide over a scale of equal parts so that the string may be stopped at any desired length. The string is set in vibration, either by plucking or bow- ing. If the lengths chosen FIG 376 are successively f, f , f , f , f, T 8 ^, and i, the resulting notes will be found to be those of the major scale. By stopping the string at one-third or one-fourth of its length, the twelfth and the double octaves may be obtained. In these cases the existence of nodes and loops may be demonstrated by placing little paper riders on the string. If, for instance, in the case last mentioned the riders are placed at points an eighth of the string's length apart, and the string is bowed while the finger is placed against a point one-fourth of the length from one end, all the riders, except those at points dividing the string into four equal parts, will fall off, showing the exist- ence of nodes at these points. It is not difficult to recognize the presence of several over- tones when the string is vibrating in its gravest mode. Thus, if the string be plucked at a distance of one-fourth of its length from the end, and then stopped at the center, the fundamental tone will cease, but the octave may still be TRANSVERSE VIBRATIONS OF RODS. 565 heard. Likewise, if the string be plucked at the center, and then stopped at one-third of its length, the twelfth will be recognized, and similarly with a large number of har- monics. It is, however, to be noticed that the series of these harmonics changes with the place where the string is struck, since this point can never be a node. Thus a string struck at the center will yield only the odd harmonics. If a stretched wire be plucked near the extremity, the quality of the note is noticeably different, on account of the prominence given to some of the upper harmonics. 505. Transverse Vibrations of Rods. The expression for the frequency of a vibrating rod may be shown to have the form Ad \E (4) n = where E is Young's Modulus, p the density, I the length of the rod, d its thickness in the direction of vibration, and A a constant depending on the mode of fastening and the num- ber of nodes, but which cannot be determined completely without recourse to experiment. When a bar having both ends free vibrates so as to form two nodes, these are situ- ated approximately at one- fifth of the length of the l^'^ysF-- -"'"^/^^^ bar from the end. If a bar be laid on strings or sharp-edged corks at these points, as in Fig. 377, and struck with an ivory mallet, the note corresponding to the above mode of vibration will be obtained. In this way the dependence of the pitch on the length and thickness of the bar, stated in equation 4, may be verified. Thus it will be found: 566 SONOROUS BODIES. 1. That two bars having the same length and thickness will have the same note, whatever their breadth. 2. That if one of two bars having the same thickness has a length VJ times that of the other, its tone will be an octave higher. 3. That if one of two bars having the same length is but half as thick as another, its tone will be an octave lower. The frequency of the note of a bar vibrating in any other mode than the fundamental is quite different from that oc- curring in the case of strings. Beginning with the case of a rod fastened at one end, if its fundamental be denoted by O l = (1.2)%, the overtones will be approximately, C, g $ d" -d> bj +/ VI (1.2) 2 n (3) 2 n (5) 2 n (7) 2 n (9) 2 rc (ll) 2 n 1 6 17J 34i. 56,1 84 It will be observed that, beginning with the first overtone, the frequencies increase as the squares of the odd numbers, and hence form an inharmonic series. In the case of a rod free at both ends, that is, vibrating with two nodes, as in Fig. 377, the series is the same as be- fore, except that it begins on (3 2 )w as the fundamental. The series for a bar fixed at both ends is the same as for one free at both ends. Vibrating rods are used, to a limited extent, for purposes of music in several ways. Four of their applications are enumerated below. In the xylophone the sonorous bodies are strips of wood supported at the nodal points and tuned to the notes of the scale through two or more octaves. In the Geneva music box the notes are produced by the me- chanical plucking of the teeth of a metal comb, which are loaded so as to increase their period. The triangle is a bent bar occasionally used in the orchestra. TUNING FOEK. 567 Chimes are sometimes made of long metal tubes instead of bells, as they can be more satisfactorily tuned. The vibrations of reeds of organ pipes are largely coerced by the columns of air which they set in vibration. 506. Tuning Fork. Another important example of the vibrating bars is the familiar tuning fork, commonly used as a standard of pitch on account of its freedom from all varia- tions except those dependent upon temperature. If a fork, after being set in vibration, be held in the hand, it will con- tinue in motion for a long time ; but, since it gives up its energy to the air slowly, the sound will be very feeble. If, however, the stem of the fork be pressed against the top of a table, the latter experiences a series of forced vibrations, and the intensity of the sound is greatly increased, though it is necessarily of shorter duration, since the energy is given out at a more rapid rate. The manner in which the fork is able to set the sounding board in vibration may be readily understood from Fig. 378. A. shows how the nodes approach the center as the bar is bent more and more. B shows that, while the prongs of the fork are swinging back and forth, the bow where the stem is attached is moving up and down, but in an am- plitude so small that contact with the board does not at once bring it to rest. For use in the lecture room, large tuning forks are mounted upon a wooden box, open at one or both ends, and of such length that the enclosed air has quite exactly the period of the fork, thus rendering the tone full and pure. ,B FIG. 378. 568 SONOROUS BODIES. A rise of temperature diminishes the elasticity of the steel, and has the effect of lowering the pitch of the fork slightly. When the fork is set vibrating by bowing, the discordant overtones are very feeble and evanescent. The form of the wave in this case may be very exactly represented by the sine curve, but the quality of its tone, musically considered, lacks character. 507. Longitudinal Vibrations of Rods. The existence of longitudinal vibrations in rods may be rendered visible by means of the apparatus shown in Fig. 379. An ivory ball suspended by two threads is arranged so as to rest against the end of a brass rod clamped at the center. If the outer end of the rod be stroked with a piece of rosined leather, it will emit an acute musical R note, and if the rubbing be continued vigorously, the amplitude of the vibrations may be increased so as to impart a considerable impulse to the ball, which is driven away as often as it touches the end of the rod. The frequencies of the different modes of vibration are related to each other as the natural series of numbers, as may be proved by the reasoning used for organ pipes (Art. 498). Neither the torsional nor longitudinal vibrations of rods are used in music. 508. Vibrations of Plates. The modes of vibration of plates were first investigated experimentally by Chladni, who found the position of the nodal lines by scattering fine sand uniformly over the plate held by a clamp, and then bowing the edge. VIBRATIONS OF PLATES. 569 If a square plate, clamped at the center, be touched at the middle of one side while the edge is bowed at the corner, it will vibrate in its gravest mode, and the sand will collect along lines shown in Fig. 380. The direction of motion of any segment changes on cross- ing a nodal line, as is indicated by the signs plus and minus. If the plate be touched at the corner and bowed at the FlG. FIG. 382. FIG. 383. middle of a side, the nodal lines will now coincide with the diagonals (Fig. 381), and the note be a Fifth higher than that of the previous example. Figs. 382 and 383 are two other examples, among a great variety of Chladni's figures, corresponding to the acuter modes of vibration. The theoretical discussion of vibrating plates, and of mem- branes, which are somewhat analogous, is too difficult to be presented here. Instruments like the cymbal and snare-drum have small musical value and are used solely to accent the rhythm of 570 SONOEOUS BODIES. military music. The kettledrum, used in the orchestra, permits of tuning, but simply that it shall not disturb the harmony of the other instruments. 509. Bells. The modes of vibration of a bell resemble those of a plate in that the surface is divided by nodal lines, and the overtones form an inharmonic and often discordant series. The least number of segments in which a bell may vibrate is four. Thus, if the full circle (Fig. 384) represent the rim of the bell, the distortion resulting from a blow at some point on the \Q side would be represented by the ellipse MN. By virtue of its elas- ticity it will immediately return not only to its original form, but will pass beyond, to a shape represented by PQ. If the bell is entirely sym- metrical with respect to its axis of figure, the points 0, 0, 0, will remain relatively at rest, so that it will be possible to distinguish a difference of intensity of the sound opposite the points M and 0. If, however, there be any want of symmetry in the figure, or homogeneity in the rim, these nodes may slowly revolve about the rim, producing the beats commonly heard when a large bell is struck. The form of a bell best adapted to suppress inharmonious partials can only be determined by trial. FlG. 385. The one adopted for large bells, after many centuries of experiment, is similar to that shown LISSAJOUS'S OPTICAL TUNING. 571 in Fig. 885. The tones vary with the thickness of the rim near the mouth, technically known as the sound bow. A bell may be tuned to a certain extent by chipping away a portion of the metal, but the pitch of the fundamental can be determined only by copying a pattern known to give the required note. 510. Lissajous's Optical Tuning. The relation between the frequencies of two forks may be very conveniently stud- ied by a method devised by Lissajous. Let (Fig. 386) represent a fork bearing a small mirrror at the extremity of one prong vibrating vertically, and let M be a similar fork vibrating horizontally. If a beam of light, R, be allowed to fall on the mirrors so as to form a small spot on the screen S, when the fork L is set vibrating alone, this spot, on account of the persistence FlG. 386. of the impression on the retina, will appear drawn out into a vertical line. Likewise, if M were set vibrating alone, the spot would trace a horizontal line. If both forks are set mov- 572 SONOROUS BODIES. ing at the same time, their vibrations will be compounded and will produce the characteristic figures described in Art. 474. Thus the interval between the forks may be judged from the form of the figure, and the exactness of the tuning by its constancy of form. It will be evident, from the explanation of Art. 474, that when the tuning is imperfect the change of the figure through a complete cycle means that one fork has gained or lost one vibration over the other, which corre- sponds to what the ear recognizes as a beat. In practice the ap- paratus takes the more convenient form shown in Fig. 387. A micro- scope is arranged on a stand so that the ocular and tube shall remain fixed, while the objec- tive, which is attached Flo> 387 . to a prong of a stand- ard fork, is capable of vibrating in a vertical direction. A'ny small point observed through the microscope while this fork is sounding will consequently appear drawn out into a line. In order to make the vibrations of the fork continuous, the prongs are arranged between the poles of an electromagnet so that they automatically interrupt the current about the magnet with each vibration. The fork to be examined is placed by the side of the first, so that the directions of their vibrations shall be at right angles. A bright point on the second fork, such as a globule STANDARD PITCH. 573 .of mercury, observed by the aid of the vibration microscope, wj[ll, when both forks are set vibrating, appear to describe the Lissajous figures, by which the interval between the forks may be obtained with greater precision than is attain- able with the ear. 511. Standard Pitch. The starting point, or tuning note, for an orchestra is a/, the note in the second space of the treble clef. In the practice of musicians its absolute value has varied considerably and undergone, on the whole, a marked rise. In 1751 Handel used a fork, #' = 422.5. In 1891 a prominent New York maker was tuning his pianos to a fork, a' = 451.7. The pitch at present receiving widest recognition is that adopted by the Paris Conserva- toire, namely, a r = 435 at 15 C. Several circumstances seem to have influenced the consid- erable rise of pitch noted. An error might be introduced in copying a standard fork by neglecting the influence of tem- perature. In making a new fork the prongs are left a little too long, so that the note is at first too grave. The pitch is then raised to unison by filing away a small amount of the metal. This manipulation will probably raise the tempera- ture so that, even if the tuning be perfect when it leaves the workman's hand, after it has been allowed to cool, its note will be sharp by a small amount. It is thus possible that after a succession of copies a considerable error might arise. Historically the chief variation of orchestral pitch is to be ascribed to the makers or players of wind instruments, who, by raising the pitch, secure greater brilliancy of tone. It is also probable that this is true of piano makers. 512. Determination of Absolute Pitch. The most impor- tant methods available for the determination of absolute pitch are, in order of increasing accuracy: 574 SONOROUS BODIES. 1. The Siren (Art. 486). The difficulty of maintaining a constant velocity of rotation, and of judging a unison be- tween a note of its screaming character with one of softer quality, introduces an uncertainty of at least 2 or 3 per cent, and frequently more. 2. The Vibroscope (Art. 487). It is manifest that the free period of the fork will be influenced somewhat by the mass of the style, by its friction against the paper, and, when it is electrically driven, by the forced character of the vibrations. If, however, the period of a particu- lar fork is carefully determined under certain definite condi- tions, from its graphical record the frequency of any other fork may be obtained from it with great accuracy by Lissajous's method of comparison. 3. Konig's Clock-Fork. In this method a large fork, hav- ing a frequency of about 64 vibrations per second, is made to serve as the isochronous ele- ment of a clock in place of the usual balance spring (Fig. 388). By observing the rate of this clock, the period of the fork may be determined to almost any degree of accuracy. Then, by means of the vibration microscope with which it is sup- plied, the period of any other fork may be obtained in the usual way. FIG. 388. EXAMPLES. 575 EXAMPLES. 1. Find the frequency and wave-lengths of the first three tones emitted by an open pipe 67 cm. long when blown with air at 0. Ans. n = 247 per sec., x x = 134 cm. n 2 = 4S5 per sec., X 2 = 67 cm. n 3 = 742 per sec., X 3 = 44.7 cm. 2. What must be the length of a stopped pipe whose fundamental note should have a frequency of 520 vibrations per second ? -4ns. 6.3 inches. 3. A rod 76 cm. long, when vibrating longitudinally in its funda- mental mode, yields a note of 4520 vibrations per second. What is the velocity of the wave in it? .. met. Ans. 6870 sec. 4. If an organ pipe be warmed from 16 to 127 C., how much will the note which it emits be affected ? Ans. About the interval from c to d$. . 5. What alteration in the pitch of a whistle in air will be made by ffin blowing it with a gas whose density is 0.7173(10) ~ 3 - ? Ans. A Fourth. 6. A glass rod 73.2 cm. long, clamped at the center and rubbed with a rosined cloth, emits its fundamental note of 2780 vibrations per second. Calculate the velocity of sound in glass. Ans. 4.07 (10) 5 cm. per sec. 7. 213 cm. of a uniform wire weigh 4.79 grams. What should be the note emitted by a piece of wire 47 cm. long, when stretched by a force of 24,800 grams' weight? Ans. n = 350 per sec. 8. If a vibrating string, when stretched with a weight of 16 pounds, emits the note a, what weight must be added to make it yield c' ? Ans. 7.04 pounds. 9. The length of the e string of a violin is 33 cm., and it has a mass of 0.125 gm. What is its tension when tuned to 640 vibrations per second? Ans. 0.676(10) 7 dynes. CHAPTER XXXIV. COMPOUND TONES. 513. Beats of Upper Partials. In the case of simple tones beats are produced only when the pitch of the notes is nearly the same, but beats may be produced between com- pound tones whenever any of their partial tones nearly coin- cide, or the prime of one tone approaches the upper partial of another. Thus, if two notes differing by a seventh be sounded, the first overtone of the lower note will beat with the higher note, the number of beats being, as in the case of simple tones, the numerical difference of their frequencies. If, now, the interval between the primes be increased, the number of beats will grow less and less until the octave is reached, when they entirely disappear, thus affording a very sensitive test of the accuracy of tuning. The perfectly tuned intervals of the octave, twelfth, or fifth, being undisturbed by beats, are especially satisfactory to the ear. For this reason they are called consonant inter- vals, in distinction from the less harmonious, such as the second or seventh, which are termed dissonant. The following examples in musical notation show the I DISSONANT INTERVALS. 577 coincidences and beating notes among the upper partials of the most important intervals, arranged in the order of their selection by the ear. The primes in each case are denoted by half notes and the overtones by quarter notes. 514. Dissonant Intervals. The disagreeable impression produced on the ear by a dissonant interval is due, on the theory developed by Helmholtz, in every case to the presence of beats, which give an intermittent character to the. sound. Why an intermittent stimulation of the auditory nerves should produce a more unpleasant sensation than an even stronger but continuous excitation may be better understood by comparing the analogous behavior of certain other nerves. In general, any considerable stimulation of a nerve deadens its sensibility, i.e. renders it less sensitive to succeeding ex- citation, but if left to itself the nerve quickly recovers its original sensitiveness. For instance, on passing from a dark room into a brilliantly illuminated one the light may at first appear so intense as to blind one, or to produce a painful sensation. But the continued action of the light, although partially excluded through the contraction of the pupil, soon diminishes the sensibility of the optic nerves so that the dis- comfort is relieved. If, however, the intensity of the light be intermittent, as in a flickering gas flame, where periods of illumination are followed by intervals during which the nerves of the eye recover, the irritation produced will evi- dently be much greater than that from a steady source. The same holds true of the nerves of touch, so that the scrap- ing of the finger-nail, or the light brushing of a feather over the skin, is more annoying than the same pressure contin- uously applied. The effect of an intermittent or beating note on the ear may thus be regarded as comparable with that of a flicker- 578 COMPOUND TONES. ing gas flame on the eye, or a tickling feather on the skin, producing a more irritating and unpleasant sensation than would be occasioned by a continuous tone. Helmholtz found that, as the number of beats was increased, a maximum of discomfort was reached at about 33 per sec- ond, for notes of medium pitch, after which, on account of the persistence of impression, the intermittent character is less easily distinguished, just as the eye is unable to differ- entiate- a succession of impressions exceeding a certain fre- quency. A glowing stick, for instance, when whirled with even a moderate velocity, appears as a continuous ring of fire. 515. Development of the Scale. The fact that the con- sonant intervals of the octave, fifth, and fourth, could be obtained by using similar strings, whose lengths were in the ratios J, J, J, respectively, was discovered by Pythagoras four centuries before the Christian era, and modern physi- cists have known since the middle of the eighteenth century that the intervals of the scale could be expressed by ratios between the first five numbers, or their simple multiples ; but previous to the investigations of Helmholtz no one could give any other explanation for these remarkable re- lations than that the human mind had a peculiar pleasure in simple ratios. In the light of Helmholtz's theory of consonant intervals, it is easy to see that the selection of these particular notes of the scale by all civilized nations has been determined chiefly by the natural relationships which exist between the keynote and the other tones. If after sounding one note the octave is struck, the ear hears again nothing but what was present in the lower com- pound note, though the intensities may be different. In sounding the fundamental and the twelfth, there is a repeti- COMBINATIONAL TONES. 579 tion in the upper note of some of the partials of the first note. Proceeding in this manner, selecting those notes whose natural relationship is such that they have at least one partial tone the same in each series of overtones, the following notes are obtained belonging to a single octave and arranged in the order of their relationship to the tonic c : c c' g f a e eb 1 f I I I I f Neglecting the minor third, these notes form all of the major scale except two notes, which are inserted to divide the rather long intervals from c to e and from a to c'. The value of 6 ^- may be regarded as having been selected either to form the major third above #, or, what is more prob- able, so as to form a leading note to the octave c r . The position of d = | was probably chosen to form the fifth of G- in the octave below. 516. Combinational Tones. Two notes when sounded together may produce tones whose frequencies are, respec- tively, the sum and difference of the frequencies of the com- ponents. The difference tones were discovered by Sorge, and independently by Tartini, after whom they are frequently called the tones of Tartini. The summation tones were first predicted by Helmholtz, from a theory which he had developed to account for differ- ence tones, and afterwards found by experiment. Combinational tones may be formed between the over- tones of two compound notes as well as between the primes, and even in some cases between the combinational tones themselves. The resultant tones of the first order, arising from the usual harmonic intervals, are as follows, the gen- 580 COMPOUND TONES. erating tones being denoted by half notes and the resultant tones by quarter notes: Octave Fifth Fourth Octave Fifth Sixt SUMMATION TONES Maj. Third Maj. Third Min. Sixth The summation tones are much more difficult to perceive than the difference tones, which may be readily obtained from the siren, harmonium, or two tuning forks, when any of these instruments are sounded loudly. Since combinational tones are observed only when the generating tones have considerable intensity, they are, in the theory of Helmholtz, regarded as the resultant of a secondary system of waves arising from a type of vibration which may no longer be treated as harmonic for considerable amplitudes. Assuming the force of restitution to be a quadratic func- tion of the displacement, Helmholtz showed that secondary systems of waves would be produced whose frequencies were, respectively, the sum and difference of those of the harmonic primaries. VIBRATION OF BOWED STRINGS. 581 517. Vibration of Bowed Strings. By the aid of the vibra- tion microscope, Helmholtz was able to determine the precise character of the vibration of bowed strings. The vibrations of the standard fork being performed in a horizontal line and those of a string very nearly in unison with it, in a vertical line, the series of figures shown in Fig. 389 was obtained. FIG. 389. As any of Lissajous's figures may be regarded as the projec- tion of a curve which has been plotted in terms of the time and the displacement and then wrapped about a cylinder (Art. 474), it is evident that the curve representing the dis- placement as a function of the time in the figures of Fig. 389 may be obtained by simply developing the cylinder. The result thus obtained for a point near the middle of the string, when the bow bites well, is shown at A (Fig. 390). If the point observed is nearer the end, the curve takes the form A ^ ^ , FIG. 390. shown at B (Fig. 390). These figures enable us to form a clear conception of the process of bowing. When the rosined strands of the bow are applied to the string, they adhere and the string is displaced a short distance by the motion of the bow, since for small stresses rosin behaves like a viscous, fluid. But as soon as the force of restitution in the string exceeds a certain value, the string will break away from the bow, for rosin behaves like a brittle solid to stresses 582 COMPOUND TONES. FIG. 391. exceeding a definite amount, and fly back to the opposite side of its position of rest, where it is again gripped by the bow, to be again drawn aside. The string thus receives a series of impulses which have sensibly the period of its own free vibration. Since Fig. 390 defines the form of the wave, it is possible to determine from these lines the presence and intensity of the upper partials. The fundamental form of the vibrations of a violin string is nearly independent of the place where it is bowed, differ- ing in this respect greatly from plucked or struck strings. When the point of bowing was not near the end of the string, Helmholtz found that little crumples were introduced on the lines of the vi- brational figure, as in Fig. 391. If a fundamental wave, with its five upper partials vary- ing in amplitude as the wave-length, be added geometrically, the resulting curve (Fig. 392) is seen to bear a very close re- semblance to Fig. 391. Helmholtz concludes from his figures that, in a violin string which speaks well, all the upper par- tials are, in general, present, but with an intensity which diminishes as the square of their frequency increases. FIG. 392. QUALITY OF VOCAL SOUNDS. 583 518. Quality of Vocal Sounds. The different qualities of vocal sounds are determined chiefly by the form of the glottis and the tension of the vocal cords, but the intensity of the overtones is considerably modified by the resonance which takes place in the cavity of the mouth. In order to produce a smooth, pure tone, it would seem to be necessary that the edges of the vocal cords be straight and capable of close and perfect alinement without striking. The quality of scream- ing and rasping voices would seem to be due to imperfect fulfillment of these conditions, permitting the vocal cords to overlap and strike together. At least these sounds resemble those produced by striking reeds. Helmholtz is of the opinion that in the speaking voice the vocal cords act as striking tongues. Hoarseness in colds is due in part to the relaxation of the vocal cords incident to their inflammation, and in part to flakes of mucus which rest in the glottis, disturbing the motion of the cords. By the aid of resonators it has been shown that six or eight partials are commonly present, though in different degrees of intensity, depending on the form of the cavity of the mouth. The fact that this cavity is tuned to different pitches may be easily recognized in the quality of prolonged whispered vowels. By means of the resonance excited by tuning forks, Helmholtz found that this cavity for different (German) vowels was tuned as follows : / /' 6'b b"b d" f f f f * -H TH -t i i 1 f U On O 584 COMPOUND TONES. 519. Synthesis of Vowel Sounds. By means of a series of tuning forks with resonators which could be adjusted so as to produce any required intensity of each note, Helniholtz was able to imitate a number of the vowel sounds. His results are exhibited in the following table, in which the upper row shows the harmonic series of forks used, and the common musical marks of loudness, pp, p, mf, /, ff, indicate the inten- sity of each tone used to produce the vowel sound designated at the left. i i>b 2 V\) 3 /" 4 b"b 5 d'" 6 /" 7 a'"b 8 b'"b u f o mf f P A mf mf mf f f A f f P ff ff E mf mf ff ff ff It was further found possible, with the apparatus men- tioned, to imitate the quality of the sounds produced by several sorts of organ pipes. From these and similar experiments Helmholtz drew the important conclusion that the quality of a compound tone depends solely on the number and intensity of its partial tones, and not upon their difference of phase. 520. Musical Quality of Sounds. The nearly simple tones obtained from tuning forks and wide-mouthed stopped pipes, although free from roughness, are not satisfactory musical sounds on account of their dull or colorless quality. Com- pound tones, consisting of a prime accompanied by a series of moderately loud overtones up to the sixth, such as those of open organ pipes, the French horn, or the softer tones of the human voice, are described, musically, as full and rich. ANATOMY OF THE EAR. 585 When only the harmonics, corresponding to the odd series of numbers, are present, as in stopped organ pipes, the tones are described as hollow if there are but few overtones pres- ent. When a large number of upper partials are present, such sounds are termed nasal. When overtones above the seventh are very distinct, the quality of the notes is piercing and rough on account of the dissonances which they form with one another. Such tones are often called wiry, reedy, or brassy, after the nature of the instruments which produce them. 521. Anatomy of the Ear. The apparatus of hearing in man consists of three well-marked portions, distinguished as the external, the middle, and the internal ear. The external ear consists of the auricle (pinna), the familiar appendage at FIG. 393. M, meatus ; m, malleus ; i, incus ; s, stapes ; mt, membrana tympani ; t, tympanum ; fr, fenestra rotunda ; c, cochlea ; K, vestibulum ; h, horizontal semicircular canal ; va, vertical anterior semicircular canal; vp, vertical posterior semi- circular canal ; E, Eustachian tube. 586 COMPOUND TONES. the side of the head, and the auditory canal (meatus), M(}?ig. 393, which shows a section through the right ear). In the lower animals the auricle may assist in locating the direction of a source of sound, but in man its influence on hearing is insignificant. The meatus is closed by a membrane (membrana tympani), mt, commonly called the ear-drum, which is stretched across the tube with its outer surface slightly concave. The middle ear is an air cavity (tympanum), t, completely shut off from the external ear by the drum, and from the inner ear by a bony wall. This wall is pierced by two mem- brane-covered holes, known re- spectively as the oval window (fenestra ovalis) and the round window (fenestra rotunda). The tympanic cavity contains a chain of three small bones, known as the hammer (malleus), m, the anvil (incus), i, and the stirrup (stapes), s. The articulation of Mm FIG 394 these ossicles is more plainly shown in Fig. 394. The handle Mm of the malleus is attached to the inner side of the tym- panic membrane a little below the center, and the head of the stapes to the membrane of the oval window. The middle ear communicates with the external air by means of the Eustachian tube E (Fig. 393), extending into the upper part of the throat, but generally closed except in the act of swallowing. By closing the mouth and nostrils and compressing the air in the pharyngeal cavity, the tube may be forced open and the pressure in the tympanum in- creased, an effect which is easily recognized by a charac- teristic crackling sensation in the oar. Upon swallowing, however, the pressure will be again equalized. ANATOMY OF THE EAR. 587 The internal ear, or labyrinth, is a partly membranous and partly bony organ embedded in very dense bone and filled with a watery fluid. It consists of three parts: the vesti- bule, v (Fig. 393); the semicircular canals, A, vp, va; and the cochlea, c. The outer wall of the vestibule is pierced by the two membrane-covered windows already mentioned. The lower, or round window, permits a slight expression of the fluid whenever the membrane of the oval window is pushed in by the stirrup. Opening out of the vestibule, and continuous with it, is the cochlea, consisting of a spiral tapered tube, so named from its resemblance to a snail shell. From the central col- umn of bone a partition, Sp (Fig. 395), extends outward toward the center of the cochleal canal, where it is met by two membranes, the membrana vestibu- laris, mv, and the membrana basilaris, mb. The upper gallery thus formed ends in the vestibule and is called the scala vestibuli. The lower one, extend- ing to the round window, is known as the scala tympani, and the central one the ductus cochlearis. A portion of the auditory nerves, rising through the axis of the cochlea, pass over the bony partition and terminate on a multitude of rod-like fibers at (7, called, from their discov- erer, the rods of Corti. These rods form a double series, about 3000 in number, increasing somewhat in size as they approach the vertex of the cochlea. The fibers of each pair are joined at the top and spread at the bottom, where they rest on the membrana basilaris, so as to form a sort of arch over it, as shown in Fig. 396. The epithelium, near the terminals of another portion of the auditory nerves which pass into the vestibule, is covered 588 COMPOUND TONES. with a number of stiff elastic hairs, extremely well adapted for moving sympathetically with the fluid in which they are immersed. In the vicinity of these surfaces there are also found a number of calcareous concretions, known as otoliths, designed apparently to produce a mechanical irritation in the nerves of the surface upon which they rest, whenever they are agitated. 522. Functions of the Several Parts of the Ear. When a train of sound-waves falls on the ear, the agitation of the air near the drum will set this membrane in a similar vibra- tion. By means of the ossicles of the middle ear these vibrations are reduced in amplitude and transmitted to the fluid of the labyrinth, and thence to the ciliated cells of the vestibule and the basilar membrane with the superposed organ of Corti. The function assigned to the last-named parts is one in accord with the well-known ability of the ear to distinguish a complex wave, not as the eye does, by its peculiar form, i.e. a succession of varying total displacements, but by resolv- ing it into its component harmonic vibrations. The only mechanical analogy of the analysis of a compound periodic motion into simple harmonic vibrations is that afforded by the phenomena of sympathetic vibration. Thus, for instance, if all the dampers be raised from the strings of a piano and one of the vowels be spoken (or, better, sung) close to the THE SEVERAL PARTS OF THE EAR. 589 sounding board, only those strings will be set in vibration which have the same frequency as the simple tones which were contained in the given note. In this case the echo, or resonance, of the strings is a sufficiently perfect copy of the original sound to permit the original vowel to be easily recognized. In the case of the ear it is not possible to ascertain which parts, if any, vibrate sympathetically with' each note ; but the whole construction of the basilar membrane and the arches of Corti seem so well adapted for performing independent vibrations, that it appears most probable that it is by the aid of this apparatus that we are enabled to resolve any musical sound into its simple components. The tuning is apparently determined by the breadth at different points of the membrana basilaris, which increases in width from about 0.041 mm. at the base of the spiral to 0.495 mm. at its apex. As already pointed out, there is a corresponding, though not proportionate, increase in the Corti's arches as they approach the vertex of the spiral. Under these circumstances, those portions of the membrane in unison with the higher notes must be looked for near the round window, and those corresponding with the deeper near the vertex. The ciliae of the vestibule may be regarded as assisting in the perception of squeaking, hissing, or crackling sounds. The semicircular canals A, va, vp (Fig. 393) are so ar- ranged that one lies essentially in a horizontal plane, another parallel to the median plane, and the third in a vertical right and left plane. They seem to have no connection with hear- ing, but rather to be the organ of another sense, viz. that of equilibration. At least the destruction of one or more of the semicircular canals in the lower animals is accompanied by re- markable disturbances in their ability to preserve equilibrium. 590 COMPOUND TONES. 523. Duration of Aural Impression. When the rapidity of beats between two notes exceeds a certain number, differ- ent for different pitches, the ear no longer recognizes the sound as an intermittent one. By examination of the small- est number of beats which one note makes with another, in order that the sensation shall be continuous and the interval an essentially consonant one, Mayer has found the following values for the duration of different tones : NOTE. BEATS. DURATION. C = 64 16 sec. T V = 0.0625 c = 128 26 0.0384 c' = 256 47 0.0212 g' = 384 60 0.0167 c" = 512 78 0.0128 e" = 640 90 0.0111 g" = 768 109 0.0091 e'" = 1024 135 0.0074 524. Resonance of External Canal. According to the researches of Helmholtz, the air in the external auditory canal plays the part of a resonator for sounds in the region be- tween e""=2640 and g""= 3168, strongly reinforcing these tones. To some ears these notes when loud are almost painful. It is for the same reason that the stridulant note of the cricket seems relatively so intense. The sound is much weakened by applying a small tube to the ear, so as to lower the pitch of the resonance chamber. It is a notable fact that the human voice is especially rich in those overtones to which the ear is peculiarly sensitive. CHAPTER XXXV. MUSICAL INSTRUMENTS. 525. Organ. The church organ is a great assemblage of metal and wood pipes, sounded by means of air supplied under pressure by a bellows and admitted through valves operated by keys under the control of a single performer. The pipes are usually arranged in four departments, each a more or less complete instrument with a keyboard of its own, and known respectively as the great organ, the choir organ, the swell organ, and the pedal organ. The keys of the first three are arranged in as many banks or manuals, one above the other in the order mentioned. The last is operated by a row of foot-keys or pedals, whence its name. Each department of the organ is composed of a number of sets of pipes or "stops " having the same quality through a certain range of pitch. The different octaves on the organ are usually distinguished by the length of the open pipe used to give the prime. . Thus, the octave to B is called the eight-foot octave. The peculiar quality of the various stops is determined in part by the kind of mouthpiece (Art. 499), though more especially by the form and dimensions of the pipe, which rein- force certain overtones and extinguish others. In the stops called mixtures the note is produced by sounding together a number of pipes, each tuned to the individual tones of the harmonic series. Several forms of organ pipes are shown in Fig. 397. The method of tuning organ pipes varies with the nature of the pipe. Closed wood and metal pipes are tuned by means 592 MUSICAL INSTRUMENTS. of a sliding stopper. Open metal pipes are usually flattened by expanding the upper end of the pipe by means of a cone 1. Principal (4 foot). 2. Spitz-note (8 and 4 foot). 3. Twelfth (3 foot). 4. Cornet. 5. Flute (8 and 4 foot). 6. Trumpet (8 and 4 foot). 7. Vox humana (8 foot). 8. Bombarde, or double reed (16 and 8 foot). 9. Mixture (4 ranks). driven into it, and sharpened by contracting this orifice. Delicate stops are often tuned by means of ears (Fig. 397), which may be bent over the mouth. Open pipes are some- VIOLIN. - VIOLA. 593 times tuned by means of a sheet of lead projecting over the top, and sometimes by means of a slide which opens a slit at the top (Fig. 361). All reed pipes are tuned by shortening or lengthening the tongue of the reed by means of a wire resting against it (see Fig. 366). The pitch of organ pipes is often seriously impaired by the changes of temperature, which affect the velocity of sound within the pipes (Art. 494). The disturbances from this cause are further aggravated by the fact that the reed pipes are not affected to the same extent as the flute pipes. 526. Violin. The most important of bowed instruments is the violin. It consists of four gut strings, the lowest cov- ered with silvered copper wire, stretched over a bridge on a wooden chest (Fig. 398). The strings are tuned to fifths as follows : #, d', a', e rr , by means of pegs over which they pass. The intermediate and higher notes are obtained by stopping the strings with the tips of the fingers, which are pressed against the finger board. The compass is about three octaves and a half, the high notes being obtained by touching the string lightly so as to make it yield the harmonics. The vibrational form of the string discussed (Art. 517) is not exactly that, of the wave which reaches the ear, for the string does not itself readily com- municate its vibrations to the air: This transfer of energy is effected by the sonorous body of the instrument, which natu- rally favors some of the partials at the expense of others. 527. Viola. The viola is similar to the violin, except that it is somewhat larger. It has four strings of gut, the FIG. 398. 594 MUSICAL INSTRUMENTS. lower two of which are covered with silvered copper wire. They are tuned to fifths as follows : to c", and its harmonics are those of an open pipe. 400. 533. Clarinet. The clarinet (Fig. 402) is a wood instrument of cylindrical bore, ending in a bell at one end and a mouthpiece with a single beating reed at the other (Fig. 368). The effective length of the tube is altered by opening holes in its side. The clarinet yields only the odd series of harmonics, as obviously should be the case from the variations of pressure at the mouthpiece incident to the opening and closing of the reed. This instrument has two principal registers, differing by a twelfth, the lower one extending from e to >'&, and the upper, obtained by opening a FIG. special hole, from b f to e"#. By cross fingering a still higher octave may be obtained. Clarinets are made in several keys to avoid the difficulties of execution which arise in trying to play all music on one instrument. 596 MUSICAL INSTRUMENTS. 534. Saxophone. The saxophone (Fig. 403) is an instrument resembling the clarinet in many re- spects, but differing from it in being made of brass with a bent conical tube. Its usual compass is from b' to /'", but it is made in several other keys. The saxophone, though much esteemed in military music, is but little used in the orchestra. 535. Trumpet. The trumpet is the most ancient of wind instruments, and may be regarded as the prototype of all instruments with cupped mouth- pieces. It consists of a long brass tube, tapering near the outer end into a bell (Fig. 404). Its notes are those of an open tube, about as in the diagram below. On account of the * difficulty in striking it, the lowest note is not used. The trumpet used in or- chestral music is furnished either with a slide like the trombone (Art. 537), or with valves like the cornet (Art. 538). FIG. 403. FIG. 402. FIG. 404. =t 3 u FRENCH HOEN. TROMBONE. 597 536. French Horn. The French horn (Fig. 405) consists of a tube about seventeen feet in length, terminating in a wide bell, and yielding the same series of harmonics as the trumpet. It possesses, however, a more mellow quality of tone, determined in part by the shape of the mouthpiece (Fig. 371 #), and in part by the shape of the bell. Its pitch may be altered by the use of auxiliary tubes, called crooks, by which the acoustic length of the instrument may be in- creased or diminished. The missing notes of its scale were formerly inter- polated by partially stopping the bell with the hand, but the modern orches- tral instrument has been given a fixed chromatic scale by the introduction of a slide and valves. 537. Trombone. The trombone (Fig. 406) is a modified form of trumpet, in which the length of the tube is capable FIG. 405. FIG. 406.- of continuous variation, throughout a considerable length, by means of the slide SL. It is the only one of the wind instru- ments which possesses a perfect scale ; its compass is from about E to b"b. 538. Cornet. The cornet (Fig. 407) is essentially a trumpet which has been modified by the introduction of 598 MUSICAL INSTRUMENTS. a b c FIG. 407. three slides through which the wind may be made to pass, by depressing the valves, or pistons, a, 6, c. The effective length of the tube is thus changed by amounts necessary to flatten the open tones by one, two, or three semitones. The pitch of the cornet may be altered by changing the length of the tube be- tween the mouthpiece and the instrument. Cornets are made in several sizes and keys, but their usual compass is from c to g"'. 539. Saxhorn. The saxhorn, or tuba, is a name given to a family of bass wind instruments of the general type of that shown in Fig. 408. They have the .usual harmonics of an open pipe, the chromatics and intermediate tones being obtained by the use of pistons, as in the cornet. Saxhorns have a compass of about three octaves, and being made in various sizes and keys, furnish all the necessary foundation tones for the military band. They possess the notable advantage of a uniform finger- ing. The lowest of all the contra-bombardon, on account of its weight and size, is coiled in circular form so as to be carried on the shoulders of the performer. FIG. 408. PART V. LIGHT. CHAPTER XXXVI. NATURE AND PROPAGATION OF LIGHT. 540. Meaning of the Word Light. The word light is commonly used in two distinct senses : 1, to designate the sensation which is characteristic of the organ of vision ; and, 2, as the name for the usual cause of that sensation. This double meaning of the word would cause little inconvenience if there were always a definite relation between the sensation and its cause ; but this is far from being the case, since, as will be shown hereafter, the same sensation may be produced in a variety of ways. In order to avoid confusion, the first use of the word has been confined to a single chapter the Forty-Fourth. In that chapter light is treated as a sensation alone. Elsewhere it is regarded as a phenomenon of wave motion entirely apart from the sense organ which reveals its existence to us. By white light are meant such waves as are emitted by a solid body at a very high temperature, as, for example, the incandescent lime in the calcium light. Any other collec- tion of waves, even though indistinguishable by the unas- sisted eye from these, is not white. Likewise, by yellow light, green light, etc., are meant those waves of definite length which will excite in the normal retina the sensations yellow, green, etc. coo NATURE AND PROPAGATION OF LIGHT. 541. Rectilinear Propagation. -- In a homogeneous me- dium, light originating at a luminous point may be regarded as propagated in straight lines, so that the shadow of any obstacle in the path of the light may be determined to the first approximation by the locus of tangents drawn through the luminous point and the bounding line of the opaque body, according to the principles of geometry. If the source of light is a lumi- nous body, iS (Fig. 409), the shadow of an opaque body, B, on a screen at M will not be sharply FIG. 409. defined, for the reason that a point, such as Pj for instance, though screened from the portion of the lumnKus disc ac, is still illuminated by the portion be. The region of total shadow is called the umbra, and that of partial and varying shadow the penumbra. Even when the source of light may be regarded as a point, the edge of the shadow is not perfectly distinct, for the reason that points near the edge of the obstacle them- selves become new centers of disturbance, giving rise to a phenomenon known as diffraction (see Chapter XXXIX). 542. Shadow Pictures. If light proceeding from any luminous object be allowed to pass through a minute hole in an opaque screen, and fall upon a white sheet in a dark room, an inverted picture of the object will be formed. VELOCITY OF LIGHT. 601 Thus, in Fig. 410, light proceeding from JJ9, and passing through the hole H in straight lines, will illuminate minute areas at ab on the screen in proportion to the brightness of each source, so that there will appear on the screen a figure re- producing the outlines, 1 U 1..L FlG " 410 ' color, and varying bright- ness of the original object, but with a distinctness which diminishes as the size of the hole is increased. 543. Velocity of Light. Earner's Method. The first definite knowledge concerning the velocity with which light is prop- agated was deduced by Romer, a Danish astronomer, in 1675, from consideration of certain peculiarities which had been observed in a series of eclipses of a satellite of Jupiter. The mean time between successive eclipses of this satellite, the first, was about forty-two and one-half hours ; but when the earth was approaching Jupiter this period was slightly shortened, and when it was receding the period was lengthened. Thus, suppose iS (Fig. 411) represents the sun, E, the earth, and J, Jupi- ter. If the first obser- vation was made when , Jupiter was in opposi- / tion, that is at J in the figure, it was found that when Jupiter came to conjunction at J 7 , the eclipse did not appear to occur till about 1000 seconds later than its calculated time of immersion, whence Romer concluded m FIG. 411. 602 NATURE AND PROPAGATION OF LIGHT. that this time must have been required for the light to traverse the difference of path between J'E' and JE, that is, the diameter of the earth's orbit. Assuming this distance to be 186,000,000 miles, the velocity of light is found to be about 186,000 miles per second. Bradley's Method. In 1727 Bradley, while seeking to determine the parallax of the fixed stars, discovered that they appeared to describe a small orbit about their mean position in the period of a year. The explanation of this phenomenon, given by its discov- erer, leads to an expression by which the velocity of light may be determined. To derive this, suppose first- that P (Fig. 412) is a point moving with a velocity represented by PQ above a sheet of paper, AB, which is moved in a direction at right angles to PQ with a velocity RQ. The mo- tion of the point with respect to the paper, as shown by its trace, will be PR. In a similar manner, if RQ rep- resent the velocity of the earth in its orbit, and PQ the velocity of light coming from a star in a direction per- pendicular to the plane of the ecliptic, the light will appear to come in the direction PR ; that is, the star will appear displaced from its actual position by the angle RPQ. Callings the velocity of the earth in its orbit, D the distance of the earth from the sun, a the angle of aber- ration, and T the period, one year, the velocity of light is r> FIG. 412. (1) 2-rrD V = u cot a = cot a. Taking D = 93(10) 6 miles, and a = 20.5", the value of v is found to be practically the same, as that quoted above. If the VELOCITY OF LIGHT. 603 velocity of light be assumed as determined by one of the succeeding methods, equation 1 may be used to calculate D. The distance of the sun so determined is regarded as the most accurate yet known. Fizeau's Method. The first purely experimental deter- mination of the velocity of light was made by Fizeau in 1849. Light from a bright source, 8 (Fig. 413), was allowed to fall upon a piece of unsilvered glass, 6r, whence it was reflected, passing between two teeth of a cogwheel, W, to a mirror several miles distant. From this it was again reflected to the first station, where, passing through the same gap in the wheel and the glass 6r, it entered the eye of the observer. The wheel was then set revolv- ing at a gradually increasing velocity until the light was cut off from the eye, an occurrence M FIG. 413. which happened whenever the time occupied by the light in traveling from the wheel to the mir- ror and back was equal to the time it took the tooth to move forward its own width and block the gap through which the light passed. The speed of the wheel being observed at this instant, the velocity of light can be calculated at once when the distance between the stations is known. Thus, suppose that this distance is d, and that the wheel must make n revolutions per second in order to produce an eclipse of the light. Calling m the number of teeth in the wheel, and as- suming that the width of a tooth is equal to the width of a space, the velocity of light is (2) v = Admn. 604 NATURE AND PROPAGATION OF LIGHT. The velocity found by Fizeau, 314,000,000 meters per second, is somewhat in excess of the true value. FoucauWs Method. The most accurate determinations of the velocity of light have been obtained by an arrangement of apparatus originally devised by Foucault and shown in Fig. 414. S is a slit by which sunlight may be admitted, Cr is a piece of plane glass, L a lens, R a plane mirror which may be revolved rapidly in the direction shown, and M a fixed concave mirror. When the mirror R is at rest, light entering at 8 falls on R and is reflected to Jf, by which it is sent back over its previous course to Cr. The portion re- flected from this surface is then observed by the eye placed at 8'. If, however, the mirror R be given a rapid rota- tion, then, while the FlG 414 light travels from R to M and back again, the mirror will have turned through a small angle, so that the light on its second reflection from R will depart in a direction slightly different from RS, and may be observed at some point, as S rr . Thus, from a knowledge of the distances &R, RM, S'S", and the angular velocity of the mirror, the velocity of light may be calculated. The most recent deter- minations of this velocity by Michelson and by Newcomb indicate a value in a vacuum of 2. 9986(1 0) 10 cm. /sec. 544. Theories Concerning the Nature of Light. Two theo- ries of the nature of light have been held, known respec- tively as the wave theory and the emission theory. In the last quarter of the seventeenth century Huyghens published a paper in which he explained the familiar phenomena of THEORIES CONCERNING THE NATURE OF LIGHT. 605 light by means of avaves in a medium which pervades all space and is called the luminiferous ether. The reasoning was so convincing, the explanations so simple, and the ex- periments supporting his views so apt, that it could hardly have failed to receive at an early day the universal accept- ance which it now commands, except for the influence of a single philosopher, then living, who was greater than Huy- ghens himself. A few years earlier, in 1669, Newton had commenced his labors in the field of optics, by which, largely on account of the fame and authority which he had won in the domain of mechanics and astronomy, he established a theory of light which remained almost unquestioned for nearly a century and a half. Newton supposed light to con- sist in extremely small particles of matter projected from shining bodies with enormous velocities. It has since ap- peared that this hypothesis was not only less fruitful than that of Huyghens, but even within the comparatively limited range of optical phenomena known to Newton and his con- temporaries it was less probable. It was left to Fresnel to establish the wave theory by a remarkable series of papers, extending from 1815 to his death in 1827, upon a foundation which leaves no room for doubt. There is one experiment which must be regarded as a crucial test between the two theories, though historically it came too late to be of any service in overthrowing the emis- sion theory, and that is the velocity of light in different media. According to the theory developed by Newton, light should move with a greater velocity in an optically dense medium, such as water, than in a rare one, such as air, while on the wave theory the velocity should be greater in air than in water. The apparatus of Fig. 414 was designed primarily for the determination of this question, and by it Foucault showed that the velocity of light ,was considerably greater in 606 NATURE AND PROPAGATION OF LIGHT. air than in water, though he did not make any estimate of the ratio. Its ralue has been recently determined in this manner by Michelson, who found it to be 1.330 for yellow light a number in substantial agreement with that found by refraction (Art. 555). Very much remains to be learned as to the character of the ether and its relations to the forms of ponderable matter; but it now seems certain that light-waves involve an electri- cal displacement at right angles to the direction of propaga- tion, as first suggested by Maxwell. It also appears that waves of all lengths traverse the ether with the same velocity, for otherwise the reappearance of Jupiter's moons after an eclipse would be accompanied by changing color, resulting from the successive arrival of waves of different length. 545. Light a Wave Phenomenon. To prove that any phenomenon is due to- wave motion it is sufficient to show, 1, that it is periodic, and, 2, that it is propagated with a finite velocity. As light has already been shown to have a _p j, c velocity, it only remains to prove its periodicity. This may be done by the aid of a lens of small curvature pressed against a piece of plate glass On looking at this by reflected light, the center will appear dark, but surrounded by p- a succession of bright and dark rings, if the light be monochromatic, or colored, if the incident light be white. As there is clearly nothing periodic about the appa- ratus, it follows that the periodicity must inhere in the light, which is accordingly shown to be a wave phenomenon. The explanation of the rings is as follows : a portion of the light falling upon the surface LL' at a is reflected in the LIGHT A WAVE PHENOMENON. 607 direction T, and another portion which has passed through to the surface PP' will be reflected from the point b also in the direction T. Now, as the waves in the line bT have had to traverse a path longer than that of the waves in aT by double the distance ab, it is evident that they must be some- what behind those of Ta in phase. Also, since the reflection at b takes place in the rarer medium, there will be an addi- tional loss of half a wave-length in those waves reflected from the plane surface. At the center of the lens the differ- ence of phase is just a half wave-length, and the two systems destroy each other. Proceeding outward from the center, the difference of phase increases until it becomes a whole wave-length, when the two systems conspire, forming a bright ring. A little farther out extinction will again occur, the bright and dark spaces being thus repeated for a considerable distance from the center when light of a single wave-length is used. When the inci- dent light is white, the rings are colored, but only about seven alternations can be traced. If the air film be illuminated successively by the light in the solar spectrum from red to blue, the diameter of the rings will gradually diminish, indi- cating that the red waves are longer than the blue, in some- thing like the ratio of two to one. By a measurement of the diameter of one of the rings and the curvature of the lens, the wave-length of any color of the spectrum might be determined. Thus, suppose that the radius of the surface LU (Fig. 415) is R, and the distance of the m th bright ring from the center of the lens is d. The distance ah for per- d 2 pendicular incidence may be taken as (Art. 24). Now, 2iit the condition that there shall be a bright spot at this point requires that this double distance shall be equal to an odd number of half wave-lengths, or, 608 NATURE AND PROPAGATION OF LIGHT. (3) 2ab = m\ + i\ = (2 By substitution this becomes (2m Calculation made in this way shows that yellow light has a wave-length of about 45^^ of an inch, or 0.000056 cm. EXAMPLES. 1. What is the length of the cone of the umbra in the earth's shadow cast by the sun, and the diameter of a cross section of it at the distance of the moon? The distance of the sun may be taken as 1.49(10) 13 cm., its diameter as 109 radii of the earth, and the mean distance of the moon as 3.84(10) 10 cm. -4ns. Length, 1.38(10) u cm. ; Diameter, 0.92(10) 9 cm. 2. In an experiment by Fizeau on the velocity of light the distance between stations was 8663 meters. When the wheel which had 720 teeth made 12.6 revolutions per second, the light was eclipsed. What value does this give for the velocity? Ans. 3.143(10) 10 cm. / sec. CHAPTER XXXVII. REFLECTION AND REFRACTION. 546. Modification of Waves at the Boundary of Two Media. The laws governing refraction and reflection of plane waves, at a boundary plane separating two media, have already been derived in Art. 480. The discussion will now be extended to the modification of waves at any surface. Suppose that FIG. 416. A A (Fig. 416) is the interface between two media, and that is a source of light-waves in the medium in which they are propagated with the greater velocity. When the wave front reaches the point , this point may be regarded as a center of disturbance from which, by Huyghens's Principle, a secondary wave would, after a brief time, have spread back- ward to g, and, if the medium were everywhere the same, for- 610 REFLECTION AND REFRACTION. ward to o 1 . In the case assumed it will only have reached some point, p, nearer to a. A moment after o reaches a the point o 1 in the wave front will arrive at a v which in turn becomes a new center of disturbance. The secondary wave from this point will be found passing through q 1 in the first medium, and through p 1 in the second, at the instant that the wavelet from o is at p and q, the distance a 1 q 1 being taken less than 0^' by the distance 0^, and a 1 p l in the same ratio to a,]pi that ap is to ao'. In a similar manner the secondary waves from any other point in the interface may be con- structed. The envelope of all the little spheres through is the front of the reflected wave, and, similarly, a ' ,A surface through ppip 2 is the front of the refracted wave. This construction is ob- viously perfectly general and may be applied to all cases of reflection and refraction. 547. Optical Images. When light proceeding from a point is so modified that (1) it proceeds K o, o_ FIG. 417. or (2) appears to proceed from another point, the second point is called the image of the first. In the first case the image is said to be real; in the second case, virtual. In general, when a wave . diverging from a point is modified by reflec- V-- tion or refraction at a surface, the curva- ture of the wave is so changed that it is no longer spherical and no image will be formed. If, however, the boundary between the two media, J., A (Fig. 416), is spher- ical, those portions of the new wave fronts within a few degrees of the axis qo 1 may be regarded as approximately spherical with centers P and $, which are, respectively, the REFLECTION FROM A PLANE SURFACE. 611 reflected and refracted images of 0, In the case represented they are, according to the definition, both virtual. Under the limitations mentioned, a formula may be derived which contains implicitly the whole theory of mirrors and lenses. To find this, let the curvature , at any point of a surface, H be called positive when the center of curvature is in the same direction from the surface as the propagation of the light. In Fig. 417 let 7 curvature of the interface, C= " " " incident wave front 0, 0', 0", 1 " " " modified " " o v 0/, 0/', y = the common half chord gk, f " sagitta ag, x= " " og, x i= " " iff- Then, by geometry, (1) ? = i^ 2 7; * = iy(7; 3^=1 y*C v If p = the ratio of the velocity of light in the medium to the right of AA to that in the medium to the left, (2) ?-2i=p(?-a), whence, by substitution of the values from equation 1, 7 #1 = P (7 #), or (3) C l = P C+(l-p) % which is the equation sought. 548. Reflection from a Plane Surface. When light is reflected from a plane surface, it is to be observed that its velocity is reversed in direction, and that the curvature of a plane is zero. Substituting p 1 and 7 in equation 3, (4) Ci = -<7; 612 REFLECTION AND REFRACTION. N FIG. 418. perverted image. that is to say, the light appears to proceed from a point as far behind the mirror as the object is in front. The image is, therefore, virtual. Regarding any figure, ABCD (Fig. 418), as an assemblage of luminous points, it follows that its image in a plane mirror, MN, is an equal figure, A'B'C'D', symmetrically placed with re- spect to the mirror, but differ- ing from the object in this respect : that, seen from the mirror, B is on the right of A ; while in the image, B' is on the left of A'. This change is called perversion, and the image a It follows, from the results just obtained, that, to the perception of an eye placed in front of an unbounded polished plane, like a mirror, the half of the universe behind the mirror is annihilated and replaced by a perverted copy of the half in front of it. If the mirror is not unbounded, the space behind may still be regarded as occupied by a perverted copy of what lies before, and the bounded mirror as a window through which we can look into this space. From these simple considerations it is obvious that the smallest mirror by which a person can see his whole figure is one having half the length and width of the observer, and that quite independently of his distance from the mirror. If two mirrors, DE, EF, be placed at right angles to each other, as in Fig. 419, one of the mirrors forms a perverted D FIG. 419. REFLECTION FROM A SPHERICAL SURFACE. 613 image of the region between the two, and the other a per- verted image of the first image, so that in the region F'ED' there is formed an unperverted image of the region between the mirrors. To a person familiar only with his features as they appear in a single mirror, any want of symmetry, as in the arrangement of the teeth for example, becomes very strik- ing when seen in an unperverted image. 549. Reflection from a Spherical Surface. - - To apply formula 3 to reflection from a spherical surface it is suffi- cient to set p = 1 when the curvature of the reflected wave becomes (5) Ci= which signifies that the curvature of the incident wave is increased by twice the curvature of the mirror after reversal of its direction. Calling the radii of the incident wave, the reflected wave, and the surface of the mirror, respectively, r, r v and R, equation 5 may be written (6) - --- 1 r,~ E r 7 ' For the degree of approximation assumed in proving the formula r and r^ may be taken as the distance of the lumi- nous point and its image from the mirror. Case 1. Concave Mirrors. Making 7 negative, then C l 2 v the angle of incidence on the second surface < x ', and the corresponding angle of refraction < 2 . The difference in direction between WX and UVis called the deviation. Let it be denoted by 8; then (8) It is obvious from the figure that a decrease of < x cor responds to an increase of / and by consequence of < 2 , hence, the deviation is an increasing function of the index of refraction. When <' > sin- 1 ( - ) there will be total reflection. W 618 REFLECTION AND REFRACTION. Also, since v this condition may be written (/>! < a sin" 1 ( ). The different deviation of waves of W different wave-lengths will be discussed in Art. 580. 554. Minimum Deviation. If in Fig. 425 the light be sup- posed to traverse the prism in the reverse direction XWVU, it is obvious that the deviation would be the same as before, whence it follows that for every angle of incidence within the limit of emergence there is another value, $ 2 , for which the deviation will be the same. Suppose that i and i f are such a pair of values, i being less than i v , and let the angle of incidence be gradually changed from i to i'. During this process 8 will also vary, but since at the end it resumes its initial value it must have passed through a maximum or minimum. If, now, other pairs of values of i and i' be taken, making the difference i f i smaller and smaller, it is obvious that the stationary value of S occurring in the change from i to i 1 must correspond to the case where the difference van- ishes, that is, when, in Fig. 425, = 2 . That 8 is a mini- mum and not a maximum may be shown most simply by an experiment in which the direction of the incident light is varied, though it may be also proved analytically. When = < 2 , we have also fa = a-, whence > = and sin cf> (9) n = -. s & This result is of great practical importance, since it enables the index of refraction to be calculated from the measured values of the minimum deviation and the angle of the prism. INDICES OF REFRACTION. 619 555. Indices of Refraction. The relative velocity of light in two media is always taken in such sense that the ratio shall be greater than unity. K V Q W ~ ~ ' then, obviously, n Qk = n Ql ' n l2 ' ' n k-\,k' The ratio of the velocity of light in a vacuum to the velocity in any medium is called the absolute index of re- fraction of that medium. As the velocity of light, in gen- eral, diminishes with the wave-length, a statement of the index of refraction of any substance beyond two or three sig- nificant figures requires the specification of the wave-length also. The following table gives the approximate value for sev- eral important substances. Water 1.33 Canada Balsam . . . 1.54 Sea Water 1.34 Rock Salt 1.54 Alcohol 1.37 Glass, Crown . *. 1.51 to 1.54 Turpentine 1.47 " Flint . .1.56 to 1.78 Linseed Oil .... 1.48 Diamond . . . 2.47 to 2.75 Carbon Bisulphide . . 1.68 Lead Chromate . 2.5 to 2.97 Air 1.00029 556. Refraction through Lenses. Pieces of transparent substances such as glass, quartz, etc., bounded by polished curved surfaces, are called lenses. Those which are thicker 620 REFLECTION AND REFRACTION. in the center than at the edge are called positive lenses, and those which have the edges thicker than the center are called negative lenses. The for- mer may produce a real image of an object, but the second never can. Some of the common forms of both types are FlG . 42 6. shown in Fig. 426. To investigate refraction through a lens suppose that Fig. 427 represents a portion of a medium bounded by two spherical surfaces, y v y 2 . By equation 3 the light-waves converging to the point have their curvature modified by the first surface, so that it becomes (10) C l = Cp l + (1 Pi) 7r This wave surface spreads with uniformly decreasing radius until it reaches the second surface at a distance, , from the first. Its radius has now become , and its curvature the reciprocal of this, namely, (ii > ^'=dv At the sec- ond surface the wave will undergo an- other refraction, which may be calculated as before; thus, (12) O 2 = p z C 1 ' + y z (l-p 2 ). In the case most commonly considered, the waves emerge into the air and the thickness is regarded as negligible. FIG. 427. REFRACTION THROUGH LENSES. 621 Setting Pz = = n, and = 0, equation 12 becomes (13) C7 a =a+(w-l)(7 1 -7a) whence it appears that the curvature of a wave is altered by refraction through a thin lens by an amount (n 1) (y 1 *y 2 ). This expression is called the power of the lens and may be denoted by $. If the incident waves are plane, (70 and O 2 = , and the radius of the emergent waves is , which is * called the focal length of the lens and will be denoted by/. Equation 13 is often written in the form 1 11 where r 2 is the distance of the image, and r the distance of the image from the lens. Discussion of Equation. The case of a double convex lens may be used to show the application of the formula and illus- trate the relations between the object and image. Thus, making 7 2 negative, <7 2 will be positive for O negative and numerically less than ; that is to say, a real image will be formed of all points farther than the focal distance from the lens, and these images will lie between infinity and the prin- cipal focus. For negative and numerically greater than $, <7 2 is negative, or the image of any point nearer than the focal distance is virtual, and lies between the lens and infin- ity. For converging waves, <7 2 is always positive ; that is, the waves are brought to a focus between the principal focus and the lens. For double concave lenses equation 13 becomes Ci=<7-(n-l)( 7l + 7l ). The images of all points are virtual, lying between the lens and the principal focus. Converging waves having a curva- 622 REFLECTION AND REFRACTION. ture numerically greater than will, after refraction, converge to a point. 557. Magnification by Lenses. Let If (Fig. 428) be the center of curvature of the surface 7. Let be a luminous point, O l its image, and A another point very close to 0. Draw a line through A and K. The image of AO will be at A^. Let be the incident wave surface converging to A, and C 1 the refracted wave FlG m surface. Then by the law of refraction, since the angles are small, the angle between the surface 7 and O 1 is p 1 times the angle between 7 and (7, or whence, calling the lengths A l O l = a^ and A = a, a, O ?"f-^T Similarly, after refraction at the second surface of a thin lens, whence, observing that p z - p 1 = 1, a^_C_ a~ <7 a ' or the sizes of object and image are directly as their distances from the lens. When the ratio is positive, that is, when the & object and its image are on the same side of the lens, the image is erect ; but when object and image are on opposite sides, the image is inverted. EXAMPLES. 623 EXAMPLES. 1. Two plane mirrors are placed parallel and facing each other at a distance of 20 cm. Required the distances of the three nearest images, in each mirror, of an object placed 8 cm. from one of the mirrors. ( 8, 32, and 48 cm. from first. ' { 12, 28, and 52 cm. from second. 2. A circular disc is placed parallel to, and 3 feet in front of, a wall. Required the size and shape of the mirror which, placed on the wall, shall enable an observer standing 8 feet in front of the wall to see the exact outline of the disc. Ans. A circular mirror, three-fourths of the diameter of the disc. 3. Show that when the sun is shining obliquely on a vertical plane mirror an object placed just in front of the mirror may cast two shadows besides the direct one. 4. AB and AC are two plane mirrors inclined to each other at an angle of 15. Required the angle at which light-waves should fall on A C from a point in AB, in order that they should, after three reflections, proceed in a direction parallel to AB. Ans. 45. 5. A mirror is made to revolve about a vertical axis 25 times a second. If a horizontal beam of light is allowed to fall on the mirror from a fixed source, required the velocity at which the reflected beam would traverse a circle 78 cm. in diameter having its center on the axis of the mirror. Ans. 1.23(10) 4 cm. / sec. 6. Let AB and CB be two mirrors inclined to each other at an angle, a. Also, let p' be the image in AB of any point, p, placed be- tween the mirrors, and p" the image of p' in CB. Show that the angu- lar separation of pBp /f is twice the angle between the mirrors. 7. An object 0.96 cm. long is placed at a point 35 cm. in front of a concave mirror having a focal length of 30 cm. Required the size and position of the image. Ans. 5.8 cm. long ; 210 cm. in front. 8. What is the radius of a spherical mirror which forms an image at a distance of 46.2 cm. in front of the mirror when the object is placed 153 cm. from the vertex? Ans. R= 71.0 cm. 624 REFLECTION AND REFRACTION. 9. Required the radius of curvature and position of a mirror" which would form on a w r all a three times magnified image of a gas flame at a distance of 80 cm. from the wall. Ans. R = GO cm. ; distance, 120 cm. from wall. 10. What will be the size of the image of the sun formed by a mirror having a radius of 275 cm., the diameter of the sun being taken as 32mm.? Ans. 1.28cm. 11. An object 3.2 cm. long is placed at a distance of 6 cm. in front of a convex mirror having a focal length of 12 cm. What will be the position and size of the image? Ans. 4 cm. behind ; length, 2.1 cm. M 12. If an object be placed ai a distance of 25 cm. in front of a con- cave mirror having a curvature of 0.0167 per cm., what will be the posi- tion arid size of the image ? Ans. 150 cm. behind the mirror, and magnified six times. 13. What will be the apparent depth of a lake 27.3 feet deep ? Ans. 20.5 feet. 14. If an eye immersed in a fluid whose index of refraction is 1.42, look out through a horizontal surface, what will be the greatest ap- parent zenith distance of a star? Ans. 44 46'. 15. Find the radius of the circle on the upper surface beyond which light-waves, emitted by a luminous point at the bottom of a layer of liquid 4.2 cm. deep and having an index of refraction of 1.25, will cease to emerge. Ans. 5.6cm. 16. When a layer of liquid 4.65 cm. deep is poured upon a dot on a glass plate, the position of its image, as found by the necessary change in the focus of a microscope, is 1.37 cm. above the plate. What is the index of refraction of the liquid? Ans. n = 1.42. 17. If the index of refraction from water into another liquid is 1.23, what will be the index of refraction for light passing from air into this liquid? Ans. n = 1.64. 18. The relative index of refraction for two media is 1.26. If the absolute index for the first is 1.38, what will be the velocity of light in the second medium? Ans. 1.72(10) 10 cm. / sec. EXAMPLES. 625 19. What would be the minimum deviation produced by a prism whose angle is 1.3, for which n = 1.54? Ans. 45.4 min. 20. The minimum deviation produced in monochromatic light by a prism whose angle is 45.05 was 26.67. What is the index of refrac- tion? Ans. n= 1.530. 21. Prove that the focal power of a glass lens, n = 1.5, when im- mersed in water is only of its power when immersed in air. 22. The radii of curvature of a thin double convex lens are 46.4 cm., and the index of refraction 1.53. What is its focal length? Ans. /==43.S cm. 23. Required the focal length of a thin lens which forms an image at a distance of 30.3 cm. behind the lens, when the object is placed 91.1 cm. in front. Ans. /=22.7 cm. 24. An object is placed 59 cm. in front of a positive lens whose focal length is 14.9 cm. Required the magnification. Ans. Three times. 25. If the nearest distance of distinct vision for a far-sighted person is 2 ft. 11 in., what should be the focal length of the spectacles he would require for reading ? Ans. /=14in. 26. If the greatest distance of distinct vision for a myopic eye is 3.9 in., what should be the focal length of the proper reading spectacles? Ans. /=6.4 in. 27. A positive lens placed at a distance of 5.2 cm. from a luminous object forms an image on a screen. When the lens is moved a distance of 23 cm. nearer the screen, another image is formed. What is the focal length of the lens? Ans. y=4.4 cm. 28. A positive lens placed at a distance of 12.7 cm. from a screen forms on it an image six times the size of the object. What is the focal length of the lens? Ans. /=10.9 cm. 29. A luminous object placed a distance, d, in front of a screen has an image thrown on the latter by means of a convex lens. On moving the lens toward the object another image is formed which is a times as great as the first. Required the focal length of the lens. An,./- V "_ d. 626 REFLECTION AND REFRACTION. 30. A luminous point is placed on the axis of a hemispherical lens of radius R, at a distance, d, in front of the plane surface. If the image formed by reflection at this surface coincides with that formed by refraction at the plane and reflection at the spherical surface, show TT> that the index of refraction of the glass is n = _ ~ . 31. Show that if light-waves fall at an angle of 60 on a sphere whose refractive index is -\/3, they will emerge after one internal reflection parallel to their original direction. 32. A small air bubble is embedded in a sphere of glass whose radius is 7.03 cm. and index of refraction 1.4, at a distance of 5.98 cm. from the nearest point of the surface. What will be its apparent depth when observed from this side? Ans. 5.64 cm. 33. The upper surface of a thin convex lens has a radius of curva- ture of 10 in., and an index of refraction of 1.6. The lower face with a radius of 15 in. is just immersed in water. Find the position of the two principal foci. Downward waves, / x =17.1 in. Upward waves, / 2 = 12.9 in. 34. If the eye be placed close to the surface of a sphere of glass, show that the image which the eye sees of itself would be | of its natural size. 35. Find the focal length of a glass sphere. 1 n Ans. f= -It. n 1 CHAPTER XXXVIII. ELEMENTARY THEORY OF OPTICAL INSTRUMENTS. 558. Camera Obscura. If a white screen be placed at the position A l B l (Fig. 429), where the real image of any object, AB, is formed by a lens, L, and surrounded by the blackened walls of a box so that no light-waves except those from the object AB can fall upon it, each point of the paper will appear bright or dark according to the brightness of the correspond- ing point in the object ; in short, the surface of the paper will appear as a faithful but in- verted picture of the , . A FIG. 429. object. An instrument so constructed is called a camera obscura. Used formerly only as a scientific toy, or occasionally as an aid in draw- ing the outlines of an object, it has become, since the invention of the various methods of fixing an image on a sen- sitive plate, one of the most important optical instruments. Since it is necessary to have the surface upon which the pic- ture is formed at a distance from the lens depending upon the distance to the object, the sides of the photographic cam- era are made extensible like the flexible portion of a bellows, and the adjustment of the focus is made before the sensitive plate is introduced. The magic lantern and the solar micro- scope are cameras in which the object is quite close to the lens and its image remote. The only additional feature in each of these instruments is an arrangement by which the object may be very strongly illuminated. 628 THEORY OF OPTICAL INSTRUMENTS. 559. The Eye. Considered as an optical instrument, the eye is simply a camera obscura. The lens system consists of the cornea, a (Fig. 430), a chamber, m, filled with the aque- ous humor, and a harder transparent body, o, known as the crystalline lens. The chamber of the eye is darkened by a, cornea; 6, sclerotic; c, sheath of optic nerve; (7, choroid ; e, ciliary muscle ; /, ciliary process ; gg, iris ; h, optic nerve with artery in center ; i, papilla ; k, fovea centralis; I, retina ; m, anterior chamber of aqueous humor ; n, pos- terior chamber of aqueous humor ; o, crystalline lens ; g, suspensory ligament of lens ; r, vitreous humor. a black opaque membrane (choroid), d, just within the white skin (sclerotic), b, which forms the visible outer substance of the eyeball. Inside this dark coat is the delicate white membrane of nerve ends, I, called the retina, and forming the sensitive surface upon which the images of external objects are depicted. The cavity of the eyeball, r, is filled with a transparent gelatinous mass, known as the vitreous humor. Just in front of the crystalline lens is a colored THE EYE. 629 opaque diaphragm (iris), gg, pierced with a circular aperture (pupil), by which the quantity of light admitted to the eye is regulated. The only striking difference, from an optical standpoint, between the eye and the photographic camera is the mode of adapting the apparatus to the varying distance of the object. Instead of modifying the distance between the screen and the lens, the power of the lens is altered by changing its thickness at the middle. This capacity for "accommodating" the power of the lens to the use required is very remarkable in young children, enabling them to see objects with perfect sharpness from a distance of three or four inches up to infinity. A great diminution in the accommo- dation occurs about the twentieth and another about the fortieth year. If the eye during the earlier age is employed almost exclusively for near objects, the lens is apt to assume a permanently thickened form, so that it is too powerful for waves having a very remote source. Such an eye is called short-sighted, or myopic. In order to see distant objects, its power must be reduced by means of negative lenses, i.e. spec- tacle glasses thinner in the middle than at the edge. For purposes of comparison, the distance of distinct vision for an object or its image in front of the eye is arbitrarily chosen as ten inches. Thus, suppose, for instance, that the greatest distance of distinct vision in a myopic person has been found by trial to be r inches. It is then obvious that the focal power, < = -, of the negative lens which shall reduce the original curvature, - , of the waves to will be given by the relation from which the focal length may be calculated at once. 630 THEORY OF OPTICAL INSTRUMENTS. In eyes having no disposition toward myopia, the power of thickening the lens at will is gradually lost, and the form becomes more and more continuously that proper for vision of remote objects. Such an eye is called far-sighted, or presbyopic. For the purpose of clear vision of near objects its power must be increased by the aid of positive lenses. If the nearest distance of distinct vision in a presbyopic eye has been found to be r inches, then the focal power of the lens which will be necessary to increase the curvature of the wave to will be given by 1 (2) - + * = r ' 10 in. In the normal eye the loss of range of accommodation is probably continuous through life, but it does not usually pro- gress far enough to take the nearest point of distinct vision beyond convenient reach before the fifth decade, when the necessity of aids to distinct vision becomes evident. Many persons, however, are born with eyes myopic, either from exces- sive power of the lens or cornea, or from abnormal axial length of the eyeball ; so, too, instances are not rare where deficiency of refractive power is congenital, in which cases positive lenses must be used even for remote objects. Another defect by no means rare is for the eye to have different powers in dif- ferent planes. It can be recognized by the differing distinct- ness with which horizontal and vertical lines in a brick wall may be seen, or by the elongated appearance of a star. This fault is known as astigmatism. It may be compensated by a lens with cylindrical instead of spherical surfaces. Since the eye is optically a camera obscura, it follows that the image depicted on the retina is inverted with respect to the object. Why this inversion does not appear in the sen- SIMPLE MICROSCOPE. 631 sation is a question of psychology rather than of optics. It is sufficient to note here that an inverted image on the retina corresponds to the sensation of an erect object and vice versa. 560. Simple Microscope. When an object or its image is within the range of distinct vision, i.e. in front of the eye and at a greater distance than six inches, it can be distinctly seen if large and bright. If an object is too small to be seen with distinctness, it may be made to appear larger by bring- ing it nearer the eye, so that at half the distance it appears twice as large. But if this means of increasing the apparent size is carried too far, the power of the eye becomes insufficient to change convex wave surfaces from the object to concave ones having their centers on the retina. Un- der these circumstances the power of the eye lens may be increased by means of a positive lens close to the eye. A lens so used (Fig. 431) is called a simple microscope. The magnifying power of a microscope is defined as the ratio of the apparent size of the image to the size of the object as seen at a distance of ten inches. The object AB (Fig. 431) is always a little nearer the lens than the focus; but if the eye be placed very near the lens, ten inches divided by the focal length may be taken as an approximate value of the magnifying power of a simple microscope. 561. Galilean Telescope. The principle of the telescope seems to have been first discovered by Franz Lippershey, a spectacle maker of Middelburg, in 1608. In the following year Galileo, while visiting Venice, learned for the first time that a marvelous instrument had been discovered in Holland, 632 THEORY OF OPTICAL INSTRUMENTS. which would enable an observer to see a distant object with the same distinctness as if it were at only a small fraction of its real distance. Soon after his return to Padua, where he held the position of professor of mathematics, he attacked the problem independently, and with such success that in a few months he sent to the Venetian Senate a more perfect instrument than they had been able to procure from Holland. Six months later, by means of a telescope magnifying but thirty times, he discovered four satellites of Jupiter, soon after following it by the discovery of the mountains on the moon and the variable phases of Venus, which incontestably established the Copernican theory of the solar system. On account of his discoveries and improvements, the name of Galileo has always been connected with the form of telescope which he used. The Galilean telescope consists of a negative lens, a (Fig. 432), close to the eye, and a positive lens, 6, at a certain dis- tance from the first depending on its power. For the sake of simplicity it may be assumed that the negative lens of the telescope just neutralizes the lens of the eye. This supposition entails no loss of generality in the conclu- sion drawn from it, and r: ^'~~_.^ isinsubstan- tial accord with the fact in the instru- FlG. 432. ment as it now survives. In this case the waves suffer no change of curvature in passing into the eye, and if the lens have such a position that waves from a distant source have their centers of curvature after passing it on the retina, the conditions of distinct vision are met. The effect of the u A Vf 1 " -- ' GALILEAN TELESCOPE. 633 instrument is thus merely to increase the virtual size of the eye, and with it, as appears from the theory of the camera, the size of the image on the retina in the same ratio. In one particular only does this apparatus, the Galilean tele- scope and the eye, differ from a large eye, and in that dif- ference lies the limitation of this form of instrument. In the eye the iris, which limits the portion of the lens upon which the light- waves fall, is close to the lens and remote from the retina. In this magnified eye the iris is relatively near the retina and remote from the lens. From this it follows that light corresponding to different points of the image passes through different parts of the lens b. This is evident from Fig. 432, for waves which pass through the pupil of the eye and converge to c l come from a portion of the objective near C", while those waves which produce the point c in the retinal image proceed from a different portion, C. A point in the object which gives rise to the waves which, after passing the objective, have their center at , and the focal length of the objective F, approximately F 10 1 '" Accordingly, since the magnification of the ocular is I0 in L the total magnification is ^ J * The compound microscope is in no respect superior in theory to the simple microscope, but in practice it is difficult to make simple microscopes of very great power, say of a 638 THEORY OF OPTICAL INSTRUMENTS. magnification much greater than 250, because of the extreme minuteness of the lenses required. 565. Terrestrial Telescope. If the ocular of the astro- nomical telescope be replaced by a compound microscope, an erect image of the distant object will be seen. This arrange- ment constitutes the terrestrial telescope, or spy-glass. It is necessarily longer than the astronomical telescope, and less perfect because it is subject to the unavoidable defects of a larger number of lenses. An image may be inverted by successive reflection as well as by refraction, and, with a proper sequence of reflections, as was first shown by Poro, it is possible to invert an image without changing the direction of the light from it. Such devices are coming into extensive use for terrestrial telescopes of low power largely on account of their compactness. In the discussion of the purely geometrical principles in- volved in the more important forms of optical instruments, the magnification has been shown to depend upon the powers of the lenses in a simple manner. It will be shown in Chap. XLI that the efficiency of the instruments is determined almost wholly by the effective diameters of the lenses, and that their powers play but an insignificant r61e. 566. Sextant. The sextant (Fig. 436) is an instrument, invented by Hadley, for the purpose of measuring the angu- lar displacement between two distant objects. FF is an arm pivoted at the center of a sixty degree sector, and traversing a graduated arc, AA. Fastened to this arm and turning with it is a mirror, B. A second mirror, (7, silvered only on the half next to the frame, is fixed in a position parallel to the first mirror, when the radial arm is at zero. DE is a small telescope directed toward (7, and intended to assist the observer in viewing a distant object. SEXTANT. 639 Suppose that light is falling on the instrument from two distant points in the directions HO and SB respectively. Then, by turning the arm F through a proper angle, it is possible to arrange the mirrors so that one-half of the pupil placed at E shall receive light coming from H through the unsilvered portion of (7, while the other half receives light from the second source 8 after reflection from the surfaces of both mirrors. It may now be shown that when FIG. 436. the image of one object is thus brought into coincidence with a second, the angle through which the mirror has been turned is one-half the angular separation of the objects as seen by the observer. Let 0, 0' (Fig. 437) be the two mirrors, and a the angle between them. Also, let SO be the direction of the incident light on the first mirror, and draw the normals NO, N f O'. Measuring the angles in the usual direction, let $ = ]% OR 1 , the deviation at the first mirror, 8' = E'O'E", the deviation at the second mirror, i = SON, the angle of incidence on the first mirror, i' = O'N', the angle of incidence on the second mirror. Then, by the law of reflection, 640 THEORY OF OPTICAL INSTRUMENTS. But in the triangle OO'N' the exterior angle NOG' is equal to the sum of the opposite interior angles ON' 0' and OO'N'-, that is, i = i' -|- a, or i i' a ; therefore the deflection is or the angle For convenience of observation the limb of the sector is divided so as to read half-degrees as if they were whole degrees. Several plates of colored glass are provided at L and K in order to diminish the intensity of the light when it is too bright for the eye. The sextant is especially useful at sea in determining the distance of the sun above the horizon at any time, since the apparent coincidence of these objects is not affected by the motion of the ship. If the altitude of the sun be ob- served when it is on the meridian, the latitude of the place may be found when the solar declination is known. Likewise, if the altitude of the sun be taken in the middle of the afternoon when it is near the prime vertical, the local time of the place may be found in terms of the known lati- tude and the time at Greenwich, as shown by the ship's chronometer, and hence the longitude may be calculated, EXAMPLES. 1. What will be the magnifying power of a telescope of which the objective has a focal length of 610 cm. and the ocular one of 1.27 cm.? Ans. 480. 2. What would be the magnifying power of an opera-glass if the focal length of the objective be 10.2 cm. and that of the ocular 3.5 cm.? Ans. 2.9. CHAPTER XXXIX. INTERFERENCE. 567. Phenomena of Limited Wave Surfaces. Suppose Fig. 438 to represent a wave-front limited by the parallel edges ab of a slit in a screen and moving toward the center jt?, which is thus by definition the image of the point from which the wave took its origin. According to Huyghens's Principle, each point of the wave-front ab must be regarded as the center of a disturbance which is propagated in all directions through the medium. Let p l be a point so chosen that its distance from a is a half wave-length greater ^ than its distance from 5; that is, so that aa' equals a half wave-length. Then a disturbance setting out FIG. 438. -~^z from a will reach p l at the same time as one from b starting a half period later. In short, the dis- placement at p 1 due to the wavelet from a will be exactly equal and opposite to that from &, so that the effects of this pair of points in the wave ab will perfectly neutralize each other. But as these two points are the only ones in the limited wave-front ab so related to p v the effects of all the other pairs of points which might be chosen in the wave-front, symmetrically placed with respect to its middle point, will only partially counteract each other. For points between p and p 1 the destruction of the elementary waves is 642 INTERFERENCE. even less complete, for there is no pair of points which wholly destroy each other's effect. Next consider the point p 2 such that its distance from a is a whole wave-length greater than its distance from b. In this case the disturbance from the middle point of the wave-front will be a half wave-length behind that from b and will destroy its effect ; but for every point between b and the middle of the wave-front a corre- sponding point between the middle and a can be found, which is a half wave-length farther from p 2 ; hence the effects of all the elementary waves at p 2 will be nil, and the medium there will remain undisturbed. By extension of this reason- ing it appears that in general there will be motion due to the effect of bounded wave-front except where the difference of distance from a and b is a whole number of wave-lengths. When the difference is an odd number of half wave-lengths, the disturbance is greater than at any closely lying point, because the conditions of self-destruction are most widely departed from. On the other hand, it is obvious that the absolute value of the disturbance decreases very rapidly on leaving the position of the geometrical image p, because a larger and larger number of pairs of mutually destructive centers can be found. The light which is found outside the geometrical image is called diffracted light, and a large class of analogous phenomena are embraced under the general term diffraction. From the preceding discussion the conclusion may be drawn that a limited concave wave-front forms not a simple image at its geometrical center, but a series of images of which the middle one corresponds in place with the geomet- rical image and is by far the strongest, while it is symmetri- cally flanked on both sides by a series of secondary images rapidly diminishing in brightness. It is further evident from the figure that pp^, which is one-half the width of the central DIFFRACTION THROUGH A CIRCULAR APERTURE. 643 image, bears the same ratio to the distance ap that aa' does to ab ; that is, (1) 568. Diffraction through a Circular Aperture. The case in which the wave-front is bounded by a circular aperture is the most common and interesting case in optics, for it is that of practically all optical instruments. The phenomena are rather more complex, but not different in kind. As might be expected, the diffracted image in this case becomes a cir- cular area surrounded by concentric rings, but their radii are slightly different from the values derived for the fringes in Fig. 438. Thus, calculation shows that the radius of the first dark ring is 1.2 times as great as from p to p%. 569. Conditions necessary for Observance of Diffraction Phe- nomena. The conditions implied in the preceding reason- ing and necessary for the realization of the results described are somewhat rigorous. First, in order to secure regular phe- nomena about the region ^>, the series of waves must have a uniform length, since the distance from the primary to the secondary images depends on this length. Again, if the open- ing through which the waves come is less than a wave-length, there is no point, such as p v whose distance from one edge of the aperture is a whole wave-length greater than that from the other ; consequently the most striking peculiarity of the effect, i.e. regions of quiescence, are wholly wanting. In the case of sound-waves, in which the length for a note of medium pitch is about four feet, it is obvious that the aper- ture would have to be a number of feet across in order to produce the required .effect. But in cases where the apertures are as great as this, being of the same order .as the dimensions of the enclosures in which the observations would be likely 644 INTERFERENCE. to be made, the reflection from the surrounding wall would entirely mask the effects sought ; at least to one not guided by theory in his research. Thus, it is not surprising that this particular class of phenomena in sound-waves failed of recog- nition until after their discovery in the case of light- waves. Finally, if the aperture is many times larger than the length of the wave, the brighter secondary images will lie very close to the primary, so close in fact as to escape our powers of perception. This, indeed, is generally the case with light- waves. Since to the eye a star appears as a round point of light and not surrounded by a series of concentric circles, although the waves from the source are bounded by the circular edge of the iris, the conclusion may be drawn that light-waves are very many times shorter than the diameter of the pupil, or, in other words, an eighth of an inch is very many times greater than the wave-length of light. By looking through a needle hole in a card at a very bright point, such as a distant electric light, or more conveniently a bright bead or a thermometer bulb in the sunshine, the cen- tral disc and the concentric rings become evident at once. Moreover, in accordance with theory, the smaller the hole in the card, the larger the disc and its surrounding rings, though of course the fainter, because less light is allowed to enter the eye. If the luminous point, or the artificial star, as it is some- times called, is very brilliant, two or three or even more rings may be observed ; but if a less brilliant source is employed, the outer rings, which decrease very rapidly in brightness, will be imperceptibly faint. 570. Diffraction through a Double Aperture. The modifi- cation in the image produced by a concave wave surface passing through two small holes in a screen is of special DIFFRACTION THROUGH A DOUBLE APERTURE. 645 interest because leading to a ready means of measuring the wave-length of light. Suppose ab and a'b f (Fig. 439) to represent two circular holes through which the wave-front whose center is at p passes. Then . the light which passes through ab forms an image in the region about p consisting of a central disc and concentric rings, as has been already shown. So, too, the light which passes through a'b' forms a similar image at the same place. If the waves which pass through the former opening have no definite relations to those which pass through the latter, as would be the case, for example, if the waves through ab came from one artificial star and those through a'b' from another, the effect will be to make the image twice as bright, since twice as much light passes through the screen. But if the waves at the two apertures are FIG. 439. congruent, which would be the case if they came from the same source and had been subjected to the same conditions before reaching the screen, the image will be profoundly changed. In this case the disc and the con- centric rings will still be present, but they will be crossed by a series of dark and nearly straight lines perpendicular to the line joining the centers of the holes. That this should be so follows from the consideration of the prin- ciples involved. The distance from the center to the first dark ring, which may be called the radius of the primary image, is, as has been explained, 1.2 times the wave-length multiplied by the ratio of ap to ab. But, starting from the point p long before the point p^ the dark region due to ab alone, is reached, we come to a* position, p', such that the dif- 646 INTERFERENCE. ference of its distances from a and a' is a half wave-length, and hence the effect produced at p' by the portion of the wave from a is destroyed by that produced by a r . If, how- ever, the center a is neutralized by the center a', the effect of every other point within the region ab will be neutralized by the corresponding point in a'b 1 '. Consequently there will be no light at p'. But this reasoning is equally applicable to a point, p' f , which is at three half wave-lengths' greater dis- tance from a than from a', and also when this difference of path is any odd number of half wave-lengths. Hence there will be a series of dark lines as described. It will be observed that the diameter of the image is inversely as the diameter of the aperture, but that the dis- tance apart of the dark lines is inversely as the distance between the holes. This phenomenon may be easily seen by making two needle holes in a card at a distance considerably less than the diam- eter of the pupil of the eye, and looking through them at an artificial star. 571. Measurement of Wave-Length. The preceding exper- iment affords a method of determining the wave-length of light with considerable precision by means of a very simple apparatus. For this purpose make a series of pairs of holes in a card with a needle and look through them at an artificial star. If the pair of holes are separated by an interval of a twelfth of an inch or more, no lines across the image of the star will be seen ; but if the interval is a twentieth of an inch or less, the lines become very distinct. Having selected a pair of holes which are at the limit of the resolving power of the eye, i.e. at such distance apart that the lines can just be seen, let the distance between the holes be measured with a finely divided scale. If this scale is divided to hundredths FRESNEL'S EXPERIMENTS. 647 of an inch, it will be possible to measure this distance, which may be denoted by D, much closer than a two-hundredth of an inch. Neither of the other quantities, p'p", ap, can be easily measured directly, but their ratio may be found as fol- lows : Draw a series of parallel lines on a piece of white paper in ink, making the width of the lines about equal te the spaces. Fasten this paper to the wall and find the dis- tance at which they can just be seen as separate lines when brightly illuminated. The ratio of the distance apart of the lines to this distance is the required ratio, since by supposi- tion the system of interference lines was also just visible as separate lines. It then follows, from the discussion already given in connection with Fig. 438, that the wave-length of light is D times this ratio. For instance, it was found by one observer that, looking through certain pairs of holes at a thermometer bulb in sunshine, no lines could be seen through the first pair of holes, very distinct lines were vis- ible through a second, and the finest possible lines through a third. Measurement of the intervals gave 0.08 in., 0.05 in., and 0.065 in., respectively. For that observer's eye, then, .Z) = 0.065 in. A series of parallel lines ^V mcn a P ar ^ were next drawn on a piece of paper, and it was found that they could just be seen as separate lines, in full sunlight, at a dis- tance of twelve feet. The ratio of the interval between the lines to their distance from the observer was accordingly ^-g^, whence the wave-length of light is %^-%-f- in., or -%-%^-Q-Q of an inch. 572. Fresnel's Experiments. Instead of allowing the light- waves to pass through two holes, as in Fig. 439, an experi- ment due to Young, certain other arrangements of apparatus proposed by Fresnel are frequently used, when all the meas- urements on the interference fringes may be made directly. 648 INTERFERENCE. In one of these a piece of glass, known as a Iriprism, having an angle, ABO (Fig. 440), differing very little from 180, is A placed before a brightly illuminated slit, s, of which it forms by refraction two images, s' and s", near together. The B FIG. 440. light which has passed through the prism very near B is in condition to interfere and will form fringes near p 1 ^ as if s', s" were real sources of light. In another arrangement Fresnel substituted for the prism two mirrors, BA, BC (Fig. 441), whose planes make a very small angle with each other, so as to form two images FIG. 441. G of a slit, s, at s r and s". Interference fringes will be found at p f , p", where the difference of path is an odd number of half wave-lengths. 573. Interference Phenomena in White Light. The ex- planation of interference phenomena may be quite easily extended so as to include the color effects due to varying wave-lengths. It is only necessary to observe that the images produced have dimensions in direct ratio to the wave-length of the light which forms it. Accordingly the image of a white artificial star consists of a disc of which the center is white, since all the waves are there represented, but having a red or orange margin. The rings immediately surrounding INTERFERENCE PHENOMENA IN WHITE* LIGHT, 649 the disc are blue on the inner side and red on the outer. Though this is the true description of the image, it is impos- sible to recognize it as such because of the limit of our per- ceptions ; for if the aperture be made large so as to make the colors brilliant, the disc and rings will be too small to be clearly perceived. On the other hand, if the opening be very small, so that the various features of the image are large enough to be obvious, the light will be so faint that the colors are unrecognizable, just as we are unable to name colors readily in even very bright moonlight. With two or more apertures, however, better success may be obtained. In the case of two holes the resulting image may be regarded as composed of an indefinite number, all having the same center, but of sizes increasing regularly from the violet to the red. As we go outward from the center the chromatic separation will become greater, until finally we reach a point beyond which every color, though of course not every wave- length, will be represented at all points. In all such regions the image will appear white and the immediate effects of interference vanish. This limit is found to be practically reached in the case of white light in the eighth or ninth band, so that in white light no more than seventeen or nineteen interference bands are visible even under favorable circum- stances, though with light of one color it is sometimes pos- sible to see many thousands. The chief difficulty in seeing the colors in the experiment with two apertures is the fineness of the lines and the faint- ness of the colors; but with holes a thirtieth of an inch apart or less, the lines are sufficiently coarse, though if many bands are to be seen the holes should be small. Since greatly increasing the number of apertures, pro- vided they are symmetrically arranged, produces little change except increased brightness, the color effects may be made 650 INTERFERENCE. very obvious by this means. An artificial star seen through the web of a uniform feather will thus show very beautiful effects. Though nature and art present numerous examples of such structure, this class of phenomena is not familiarly known, because these bodies are more often looked at than looked through. Perhaps the only common example where both regularity of structure of the screen and smallness of the source of light are met is when an electric light is observed through a silk umbrella. 574. Gratings. A diffraction grating is a piece of trans- parent glass or polished metal ruled with a great number of equidistant parallel lines by means of a very fine diamond point. These grooves are practically opaque, for they scatter the light in all directions. If light passing through a slit be allowed to fall on such a grating held with the rulings par- allel to the slit, a number of spectra, of great regularity and purity where they do not overlap, may be observed on either side of the direct path of the light. Gratings are of much practical interest as furnishing a per- fectly normal spectrum and the most accurate means for meas- uring the wave-lengths of light. To show how this may be done, suppose that Fig. 442 represents a highly magnified portion of a grating, and that plane- waves are falling on the grating from the left. Each point of those portions of the wave-fronts which fall in the spaces becomes a new center of disturbance and propagates light in all directions. Let p be a distant point* in the direc- tion ap such that the difference in path in the wave which reaches it from a is a whole wave-length behind that from a. Then, dividing up the spaces ab and cd into pairs of point^ related in the same way as a and c, it follows that the effect GEA TINGS. 651 of the limited wave-fronts ab and cd is to produce light at p. Similarly, the light sent by each consecutive pair of wave- fronts in the direction ap will be in the same phase as that from the first pair and simply add to the intensity at p. Call the distance between the rulings ac, , and the angle okq, 6, then from the figure ac 1 (2) sin0 ac Now, as the angle 6 may easily be found by observation, and s may be measured, X may be calculated very simply. If the incident light be white, as the angle 6 increases with FIG. 442. the wave-length the resulting spectrum will be arranged with the violet* deviated least, and the red most. If the direction ap be chosen so that ac 1 is equal to 2\, 3X, etc., similar spectra of the 2d, <5d, or higher orders are formed on each side of the central image, though the purity of those beyond the second may be greatly impaired by the overlap- ping of the lower spectra. In actual observations the flatness of the incident waves is secured by allowing them to pass through a lens system, called a collimator, and the measure- ment of the angle 6 is assisted by the use of a telescope with cross wires. 652 INTERFERENCE. 575. Reflection Gratings. The gratings just described are known as transmission gratings, but entirely similar results may be obtained by using a piece of polished specu- lum metal ruled in the same way. The only modification in the theory is the replacing of the slit by its image pro- duced by the polished surface. The mechanical difficulties of ruling upon glass are somewhat greater than upon metal. Professor Rowland has introduced the important modifi- cation of ruling the gratings on concave spherical surfaces, and is thus enabled to dispense with the collimator and the object glass of the observing telescope. Such a grating is especially adapted for photographic work. By the use of the grating at Johns Hopkins University having 160,000 lines in the space of six inches, the solar spectrum has been photographed in sections which, placed end to end, extend over a length of thirty feet, and measurements of the wave- lengths of light made which are universally recognized as the best now attainable. 576. Wave-Lengths. The wave-lengths in air of the principal lines of the solar spectrum are shown in the follow- ing table. These measures are expressed in what is known as the Angstrom unit, or tenth-meter, i.e k - meter, or 0.00000001 cm. wm .4 = 7621 5 1 = 5183.791 B = 6870.186 .F= 4861.527 O= 6563.054 a = 4340.634 D l = 5896.357 h = 4102.000 D 2 = 5890.186 H= 3968.625 J? x = 5270.495 K= 3933.825 U = 5269.723 IRIDESCENCE. 653 Taking the velocity of light as 3(10) 10 cm. per second, the frequencies of the waves in the visible spectrum are found to lie between 3.94(10) 14 and 7.63(10) 14 vibrations per sec- ond, which corresponds to an interval of about an octave in sound. According to Langley, the solar spectrum extends in the infra-red to wave-length 0.00027 cm., with a frequency of 1.1(10) 14 vibrations per second; while the radiations from terrestrial bodies below 100 C. extend to 0.0015 cm., with a frequency of 2 (10) 13 vibrations per second. The shortest measured wave-length in the ultra-violet is 1854 tenth- meters, with a frequency of 1.618 (10) 15 vibrations per second. These limits are being rapidly extended by con- temporary investigations. In general the frequency of a wave is unchanged when it passes from one medium to another, but its length changes in the inverse ratio of the index of refraction. The length of any wave in a vacuum may thus be found by multiplying the wave-length in any medium by the index of refraction for that medium. 577. Iridescence. The peculiar color effect, which de- pends upon the direction in which a grooved reflector is observed and upon the direction of the source of light, is called iridescence. Mother-of-pearl, which is deposited in smooth, reflective layers, shows this in a marked degree on account of its structure. That it is due to the structure alone may be readily proved by taking a print of a piece of bright mother-of-pearl on white wax, when it will be found that the surface of the wax also shows iridescent colors. Feathers, which have at the same time regular structure and brilliant luster, exhibit the same phenomenon. Most of the beauty of peacock feathers and all that of the iridescent 654 INTERFERENCE. feathers of the male birds of the turkey, pigeon, and hum- ming-bird families are thus to be explained. If the source of light is very large, or if the structure of the reflecting surface is somewhat irregular, the colors are less pronounced, or may be wholly wanting, the only effect being a remarkable change of luster with varying obliquity. The effects familiar in white mother-of-pearl, in the gem called cat's-eye, and in satin spar, are of this kind. The variety of feldspar called labradorite, however, has such reg- ularity of fibrous structure that a polished surface illuminated by a source of light of moderate extent will show most vivid colors. 578. Colors of Thin Plates. The colors exhibited by thin plates are a phenomenon of interference produced, not by breaking the wave surface into limited portions, but by securing a difference of path by reflection from two slightly separated surfaces in a manner precisely the same as in the case of Newton's rings (Art. 545). Thus, if a plate have a thickness of half the wave-length of red light, then the sys- tem of waves which are reflected from the second surface of the plate will differ from those reflected from the front sur- face by a half wave-length in the red, and the two systems will be mutually destructive as regards these particular waves, so that the effect upon the retina would be that of white light minus red, i.e. cyan-blue, as will be more partic- ularly shown in Art. 617. If the plate be a little thinner, say one-half the yellow wave-length, this color would be absent from the combined effect and the plate would appear blue. Again, suppose the plate to have a thickness of a wave- length of red light, then the retardation of the waves re- flected from the back would be | wave-lengths, and the COLORS OF THIN PLATES. 655 mutual destruction of this particular wave-length would ensue with a corresponding color. This is essentially the explanation of the production of colors by reflection from thin plates, such as a soap film or a thin layer of oil on the surface of water which is not hot. There is a notable peculiarity of the colors produced in this way which is worthy of consideration, as it greatly affects the character of the phenomena. It depends on the fact that the colors are not those of the various wave-lengths, as in the case of the prismatic spec- trum, or even those of the sum of several wave-lengths, as in the case of many diffraction phenomena, but they are the colors proper to white light after being deprived of one or several definite wave-lengths. Thus, suppose we have a plate of such thickness that the yellow is destroyed in the reflected light, the remainder is blue. But if the thickness is such that the retardation of the second system is f wave- lengths of red, it will be J wave-lengths of the shorter green waves ; hence the color will be white minus such red and green. Now red and green light combined make yellow; hence, white deprived of these hues will be blue, but paler than that produced in the former way, which is called blue of the first order. So a thicker plate would destroy three wave-lengths, and, if the middle one were yellow, would again yield a blue, but paler than either of the others. If this increase of thickness be so far extended as to effect a destruction of the wave-lengths of a series of colors which when combined would produce white, the difference between their sum and white light would also be white. It thus appears that only films of a very few wave-lengths in thick- ness can produce colors by reflection. All these consequences of theory can be observed and veri- fied very conveniently in soap films, either by watching a 656 INTERFERENCE. moderate-sized soap bubble protected from currents of air, or, better still, by dipping a wire ring two or three inches in diameter into a solution of soap, whence it can be removed with a film across it. .If this be held in an inclined posi- tion where the reflection of the sky from it can be seen as the water slowly flows to the lower side, a series of hori- zontal colored bands will appear, of which the upper ones correspond to a thinner plate. It will be observed that the. colors repeat themselves, but grow paler with each repetition. Sir Isaac Newton, who first studied the colors of thin plates and their causes, devised an apparatus, which has already been illustrated and described in Art. 545, for show- ing all the phenomena with great regularity. The rings exhibited by this air film are universally known as Newton's Rings. The colors thus produced may be frequently seen in par- tially fractured glass or ice. Quartz, as well as other crys- tals, not uncommonly have internal fractures which show brilliant colors of this origin, and the magnificent play of colors in the precious opal is explained in the same way. If a piece of polished steel be heated in the air, the surface is oxidized to a varying depth, depending primarily on the temperature. This thin sheet of oxide reflects colored light and affords a valuable aid to the mechanic in judging of the proper temperature, corresponding to a certain hardness in the process of tempering. 579. Diffraction Patterns. It follows, from the discussion in connection with Fig. 439, that if three holes be made in the card, there must be a set of lines across the disc for each ' pair of holes. For example, three holes at equal distances in a card held before the eye yield a series of hexagonal DIFFRACTION PATTERNS. 657 images arranged like cells in a honeycomb, though it would appear simply as three systems of equidistant lines crossing the disc under mutual angles of 60. More important is the deduction which may be drawn from the indifference as regards the position of the holes before the eye, provided only that the light from no one of them is cut off from the retina by the iris. That this position is un- essential follows obviously from the discussion of Fig. 439, where the iris is quite left out of the question. Hence, we might have two systems of holes, of exactly the same size and configuration, so far apart that the light from one system would not materially modify that from the other. In this case the effect would be only to double the quantity of light which forms the image. But the second system of holes need not be remote from the first if there are no new distances introduced except multiples of the original dis- tances. Hence, if a piece of cardboard, such as is used for worsted embroidery, be held before the eye while looking at the artificial star, the effect is the same in kind as though only four of the holes were transparent, though much brighter, and the separate images are smaller. The phenomena presented by small luminous points seen through fine and regularly woven fabrics such as silk, lawn, bolt cloths, wire cloth, etc., are of this kind. Many feathers will exhibit beautiful effects in this way. A modification of the phenomenon may be made by holding a perforated cardboard in front of the objective of a telescope directed toward a bright star. In this case we virtually increase the dimensions of the eye and can use correspond- ingly coarse structure in the screen. Many of the phenom- ena of diffraction are of surprising beauty, and in all of these, where the image is bright and large, brilliant colors are seen, the explanation of which will be discussed in Chapter XLII. 658 INTERFERENCE. EXAMPLES. 1. The following observations were made upon the nickel line in the orange of the second spectrum of a diffraction grating ruled with 14,440 lines to the inch. Readings to the right 69 44' and 69 44.5', readings to the left 345 41' and 345 40'. What was the wave-length of the light corresponding to this line? Ans. X = 5888 (10)- 10 meters. 2. The deviation of the F line in the second spectrum of a diffrac- tion grating was found to be 41 26.8'. What was the distance between the rulings? Ans. 1.468 (10)~ 4 cm. CHAPTER XL. DISPERSION. 580. Dispersion. For simplicity of treatment it has been tacitly assumed that the light- waves which were modified by reflection, refraction, or interference were all of the same kind. In general, however, the light emitted by a luminous point consists of all wave-lengths within a considerable range. In reflection from large surfaces the composite character of the light has no effect on the images formed ; but in the case of interference, which depends upon the length of the waves, the phenomenon becomes clearly more complicated, and, though not obvious from what has preceded, the phenomena of refraction are considerably modified. This may be made at once evident by allowing sunlight to pass through a prism. If the light be received upon a distant white screen, it will be found that, instead of a white spot where the deflected light falls, there will be a brilliantly colored strip, the end nearest the original position of the light being red, and that most remote violet. The intermediate colors in order from the red are orange, yellow, green, green-blue, blue, and violet. The change from one of these hues to the next is absolutely continuous, so that the number of colors is limited only by the number of names at our command to designate them. Since the change in direction of propa- gation of the waves depends only on their less velocity in glass than in air, it follows that those waves which pro- duce the sensation of red move less slowly than those which give rise to the sensation of orange, and than others which are deviated still more. This separation according to wave- 660 DISPERSION. length is called dispersion, and the resulting colored image of the source is called a spectrum. Sir Isaac Newton was the first to investigate the phe- nomenon in a scientific manner and to fix its terminology, using, however, the color names green, blue, indigo, and violet instead of green, green-blue or cyan-blue, and violet, which modern writers have found more appropriate. New- ton's discovery of greatest importance was, that after light is thus modified, any one color suffers no further change on passing through another prism. His conclusion was that ordinary white light is compound and made up of an in- definite number of hues, of which seven are recognized by familiar color names. He also showed that if different colors were united, either by allowing them to fall on a con- cave mirror and reflecting them to a point, or by passing them through a similar prism turned in an opposite direction, the result was light like that from the original source. Thus, both by analysis and synthesis he demonstrated the composite nature of white light. Newton also found that like prisms of different substances would produce quite dissimilar amounts of dispersion ; but fortunately for the development of practical optics, his con- clusion that dispersion, which should be regarded as a sec- ondary phenomenon of refraction, increases as the refractive power was an error. The explanation of dispersion, as due to differences of velocity in the prism, may be corroborated in various ways. Thus, for instance, the experiment described in Art. 545, in which the size of the rings varied continuously as the prism was rotated so as to allow all colors from red to blue to fall on the air film, shows that the length of the waves decreases continuously from the red to the blue end of the spectrum. Or, again, if a thermometer bulb be placed in the sunshine DISPERSION. 661 and viewed through a prism, the artificial star will appear as a fine linear spectrum. If, then, the screen with two needle holes be placed close to the eye between it and the prism, with the line joining the holes at right angles to the spectrum, the spectrum will appear broadened and traversed by a series of fine horizontal lines whose separation diminishes quite uni- formly from the red to the violet. The conclusions from these experiments are : 1, that the velocity of light-waves in glass decreases continuously with decreasing wave-length; 2, that waves of different length falling on the retina produce differ- ent sensations, the longest waves awakening the sensation of red and the shortest that of violet. The length of the red waves, as shown in Art. 545, is about twice that of the violet. The change in the index of refraction with the wave-length for a few important substances is shown in the following table. TABLE OF INDICES OF REFRACTION. SUBSTANCE. WAVE-LENGTH IN CM. INDEX. TEMPERA- TURE C. Carbon Bisulphide . . . u u Water . . . 0.0000589 ' 485 589 1.624 1.648 1 334 25 25 16 (i 485 1 338 16 u 434 1 341 16 Rock Salt 589 1 544 24 n 485 1.553 24 it U 434 1.561 24 Flint Glass 589 1 651 it U. 485 1 665 u u 434 1 677 Crown Glass . . . . . u it 589 485 1.517 1 524 u u 434 1 5 9 9 662 DISPERSION. 581. Character of Refracted Images as Affected by Dispersion. The images formed by the reflection of white light are not affected by its composite character. In the case of refraction, however, the simple geometric image which would be formed by monochromatic light is in general replaced by a spectrum of the object whenever it emits or is illuminated by white light. Thus, for instance, refraction at a plane surface, as in Fig. 421, forms a virtual spectrum of the source of which the violet end is nearer the refracting surface. The action of a plate is to form a virtual spectrum with the blue end nearer the plate; but this effect will hardly be detected, except in the case of a very thick plate when the image is seen obliquely. Thus, a white pebble seen vertically downward through deep water still appears white, but if observed in a direction away from the vertical, the colors of the spectrum appear very distinctly. So also a prism, instead of forming a virtual image of the source displaced toward the thin edge of the prism, and approached by a fraction of the thickness of the prism, as appeared in Fig. 424, in reality forms a spec- trum with the blue end more displaced than the red end, and also brought a little nearer the prism. 582. Dispersive Power. The value of minimum devia- tion of light through a prism was shown, in Art. 554, to be 8 = 2i - a. Observing that when the light is incident nearly perpen- dicularly on a prism of small angle, a condition fulfilled in most lenses, sin i i n = = - approximately ; sin r r whence 2z 2nr = na, which substituted above gives (1) B = (n-l) a . POSSIBILITY OF ACHROMATISM. 663 If n A and n u are the indices of refraction for the red and the violet waves respectively, the dispersion, that is, the an- gular separation between these two rays, may be written (2) (n H -n A }a. If n D be the index of the brightest part of the spectrum, *W r M the ratio of the diSpersion to the mean deviation, -^ is n D -\ called the dispersive power of the substance, and n H n A the coefficient of dispersion. 583. Possibility of Achromatism. Newton, after making experiments on the refractive indices of glass and water, drew the conclusion that the dispersion of all substances was pro- portional to the deviation, or, in other words, that their dis- persive power was constant, and hence that it was impossible to correct the chromatic aberration of a lens by the combina- tion of two or more refracting substances. Newton's conclusion was disputed, though unsue^gSfully, by Euler about a hundred years later. The criticism had the good effect, however, of leading to a more careful study of the phenomena. In this John Dolland met with brilliant success. Repeating an experiment of Newton's with a prism of water opposed to a prism of glass, he found that deviation of light could be produced without accompanying dispersion into prismatic colors. He also found that two varieties of glass, known as crown, or common window-glass, and flint glass, which is character- ized by the presence of a considerable amount of lead oxide, possessed very different dispersive powers. Thus, if G (Fig. 443) is a prism produc- ing a dispersion, (n 2 f n^) a f , and F is another prism, opposed to the first, and having a dispersion, FIG. 443. UNIVERSITY ^s^gAUfOgg^^ 664 DISPERSION. (n 2 " Wj") a", the transmitted light will suffer no dispersion when or the angles of the prisms must be inversely as the coeffi- cients of dispersion. Thus, for instance, if for the flint n e r - n D ' = 1.66028 - 1.63503 = 0.02525, and for the crown, n " - n D " = 1.54165 - 1.52958 = 0.01207, then ^ = 2.09. It also follows that a positive lens of crown combined with a negative lens of flint, as shown in Fig. 444, would yield a nearly colorless image. Since the dispersive power of a prism is slightly different for different wave- lengths, it is not possible to secure achromatism throughout the whole spectrum. It may be noted, in passing, that twenty years before Dolland's success Mr. Chester Moor Hall had invented and made achromatic telescopes ; but PIG. 444. * this fact remained unknown to the world of science until after Dolland's telescopes became famous. CHAPTER XLI. MAXIMUM EFFICIENCY OF OPTICAL INSTRUMENTS. 584. Limit of Power of Simple Lens. It has been shown in Art. 560 that the office of the simple magnifier is to form a virtual image of an object placed near its principal focus at a conveniently greater distance, or, in other words, to render convex wave surfaces flatter after passing the lens. Assum- ing that this may be done perfectly, the magnification may be increased indefinitely by increasing the power of the lens. An increase of the power, however, requires increased curva- ture of the lens surfaces, and decreased distance between the lens and its object. For example, a glass lens of spherical form which would make an object appear 100 times larger than it would at a distance of ten inches from the eye would require the object to be within ^ o f an inch from the sur- face of the glass. If the magnification were 1000 times, this distance, called the working distance of the lens, would be reduced to ^Q of an inch. It is obvious that this fact would put a practical limit to the useful power on account of the difficulties of illumination and adjustment. The working distance might be considerably increased in special cases by using material of greater refractive power than glass, such as diamond, sapphire, or garnet. In the early part of the century many experiments were tried with these substances, Taut without the advantages hoped for by their advocates. There is, however, another source of limitation in power depending on the absolute wave-length of the light-waves. For it is obvious that to increase the power of a lens, since this increase depends on the increase of curvature of the 666 EFFICIENCY OF OPTICAL INSTRUMENTS. refracting surfaces in the end, the diameter of the lens must be decreased, and hence the diameter of the wave surface, after passing the lens, must be less with greater magnifying power. Now the experiment on the resolving power of the eye, described in Art. 571, taken in connection with Fig. 438, shows that vision through a hole much less than a sixteenth of an inch in diameter becomes notably impaired because such dimensions are not very great compared to the length of a wave. From this it follows that the minute details of an image could not be recognized if the diameter of the lens is much less than a sixteenth of an inch. This may be ren- dered clear by considering that, if two points in the object are very close together, the image of these points will appear as a disc of determinate size, and if the separation of these points is no greater than the diameter of the discs, they can- not be seen as two points, but only as one. Increasing the power of the lens will not help, for though it increases the apparent separation of the images, it at the same time, on account of the necessarily diminished diameter, increases the diameter of the disc which represents the image of the point in the same ratio. Experience shows that, when the aperture of the pupil is reduced to a thirtieth of an inch, the indis- tinctness due to this cause becomes very obvious. Conse- quently a lens smaller than this in diameter can no longer add to the power of vision, since each point in the image appears as a disc, and each line as a stripe, of which the diameter and thickness increase with the magnification. It is obvious that the greatest diameter which a lens used as a simple magnifier can have is twice the focal length, since in this case the incident wave surface is hemispherical. Hence, by what precedes, nothing is gained by using a lens having a focal length less than ^ of an inch, that is, by Art. 560, a magnifying power greater than 600, which may EFFICIENCY OF THE TELESCOPE. 667 be taken as the theoretical limit. It will be observed that this conclusion is independent of the material of the lens, but that air immersion is tacitly assumed. The practical limit would probably be found considerably below this ; at least, it is tolerably certain that no discoveries have ever been made with simple microscopes magnifying more than 250 or 300 times. 585. Maximum Efficiency of the Telescope. The result of Art. 568 is of great importance, since it leads at once to the maximum efficiency of a telescope. Thus, calling the diam- eter of the aperture of the objective D, and the focal length F, by what precedes, the image of any star in the telescope will be a disc whose diameter is 2.4 F The angular diameter of the image is this quantity divided by F. As- suming the wave-length of light to be ^ g^-^o of an inch, the angular diameter of this disc in a one-inch telescope would be 10". 75, that is, two stars separated by an interval of 10". 75 would appear to touch, if the light could be traced quite out to the place of the dark ring. In fact, however, the edge of such a disc is so faint that it appears much smaller than this calculation would show, and two stars under most favorable circumstances at somewhat less than half this distance, can just be seen as two distant objects. Consequently the clos- est double stars which can be seen with a perfect telescope may be found by dividing 4".56 by the diameter of the objective in inches. It is clear that, if the images of two points are not separated, they cannot be made to appear separate by any increase of magnifying power in the ocular, just as, beyond a certain extent, nothing more can be found by magnifying a photograph. 668 EFFICIENCY OF OPTICAL INSTRUMENTS. It has thus been shown that the diameter of the objective determines the upper limit of power as well as the lower. 586. Magnification of the Compound Microscope. The dif- ficulties arising from too close an approximation of the eye and lens to an object may be obviated by the compound microscope, for in this case the eye is removed by a little more than the length of the tube from the object, while a deficiency of power in the objective can be compensated by increased power in the ocular. There is thus no necessary relation either between the working distance or the power of the objective and the total magnification. If the value of the magnification in Art. 564 be written in the form , it may be noticed that -- is essentially / -* J the expression found for the magnification of a telescope, and that - is the magnifying power of the objective con- sidered as a simple microscope. Thus, looking simply at the analytical form of this expression, it appears that instead of considering the compound microscope as composed of an objective to form an image, and an ocular to magnify it, it is possible, as far as the magnification is concerned, to regard the objective as forming, at a very great distance, an enlarged image which is viewed by the rest of the instrument used as a telescope. By the aid of this highly artificial but permissi- ble conception it is possible to apply at once results already found for the telescope.- In that discussion it appeared that the highest useful power was about thirty times the diameter of the objective aperture expressed in inches. Let this diameter, say D, be expressed in terms of the focal length of the objective by the relation (1) D = 2NF, GREATEST RESOLVING POWER. 669 where N is some number, and F the focal length of the microscope objective. The expression for the useful magnifi- cation then becomes (2) M= 30D = 30 - 2NF - = Jc Ji Since, as was shown in Art. 584, the maximum value of D for air immersion is 2F, the greatest value of JVwill be 1, and therefore the highest useful magnification for the com- pound microscope with dry objective will be 600 times, which is the same as that found for the simple microscope. An important conclusion which may be drawn from the relation M= 300 is that the ultimate useful power of a compound microscope depends on the ratio of the effective diameter of the rear surface of the objective to its focal length, and not upon the power of the objective or the length of the tube. Thus, the efficiency of a perfect microscope is determined by the value of N alone, which may be used to characterize an objective. It is called the numerical aperture. 587. Greatest Resolving Power. The greatest resolving power of a telescope was seen in Art. 585 to be 4". 56 divided by the number of inches in the aperture of the objective, or 4".56 ' Reducing the seconds to circular measure, the linear separation of two points at the distance F would be g^Vo~o of an inch when N 1. That is to say, the finest structure which can be seen in white light (\ -%\$$) ^7 means of a dry objective is 90,000 repetitions per inch. Since, as appeared in Art. 585, the resolving power of a telescope increases with diminishing wave-length, the denn- ing power of the microscope will be slightly greater for blue 670 EFFICIENCY OF OPTICAL INSTRUMENTS. light. For example, if light of the same wave-length as the F line in the solar spectrum be used, the number rises to 100,000, which may be regarded as the practical maximum for vision. For a photographic plate, however, the denning power is somewhat greater, say by 15 to 20. per cent. The magnification necessary to exhibit the structure pre- viously mentioned may be determined by the following con- siderations. A good eye will just recognize a system of lines separated by intervals of from 60 to 70 seconds. Taking the 2" 28 resolving power of a telescope as ' , the magnification requisite will be 70 - SQNF approximately. Z.Zo ~NF Thus, a total magnification by the microscope of Jb will just enable a keen eye to see the structure, while twice this value will certainly quite reach the limit imposed by the length of the light-waves, even with inferior eyes. This, then, is the meaning of the ultimate useful power being 600JV. 588. Angular Aperture. The largest angular extent of wave surface which the objective can transmit is called its an- gular aperture. The sine of half this angle, usually denoted by -ZV, is called the numerical aperture, and constitutes the true measure of the optical power of a microscope. 589. Hemispherical Front. Between the years 1850 and 1860 was introduced the practice of making the anterior lens of high-power objectives of a single piece of crown glass nearly hemispherical in shape. Although the value of this HEMISPHERICAL FRONT. 671 innovation stands almost unapproached, it is impossible to name definitely its inventor. It is usually attributed to Amici, but is also claimed by Wenham. When the wave transmitted by a lens is of constant curva- ture, i.e. either flat or spherical, the lens is free from spheri- cal aberration, and, leaving out of consideration the chromatic aberration which can be corrected by other means, the result- ing image is geometrically perfect. In general, spherical refracting surfaces, the only ones which can be made with precision, do not secure this, and the only practical method of correcting the spherical aberration is to combine convex and concave surfaces, so that the opposite errors shall annul each other as far as possible. There is, however, one special case in which the refraction at a spherical surface is geometrically perfect, and the peculiar advantage of the hemispherical front lens is that it utilizes this case. Let Fig. 445 represent a piece of glass bounded on the left by a polished spherical surface having c as its center, the line FIG. 445. cq as the axis of figure, and let p l represent a luminous point more remote from the surface than the center. Light-waves diverging from such a point will, in general, lose their spheri- 672 EFFICIENCY OF OPTICAL INSTRUMENTS. cal form on refraction ; but there is one position of this point for which the waves suffer refraction with considerable change of curvature, but still remain strictly spherical. To find its position, suppose that light proceeding from p 1 in the direc- tion of a has its direction changed so that it appears to pro- ceed from j? 2 . Call the radius of the sphere H, and its index of refraction n. Let the angle p-^ac = (f> v p 2 ac 2 , cp^a = V and cp 2 a = 2 . Then, by trigonometry, cp sin d>o (3) ^"^^0"' Ct* bill l/o and ca _ sin 6 l cp 1 sin (^ Multiplying these equations, cp sin < sin 0, sin l (5) . . ^ = n . TT- cpi sin 9j sin u 2 sin c/ 2 It is obvious that if ^^ in this equation is constant, the sin 2 position of p z will be independent of the position of a, that is, the direction of the incident light. This condition will be fulfilled if it is possible to set 6 l = 2 and 2 < p when equa- tion 5 would become (6) ^ = Now this condition may evidently be imposed, for since in the triangles cp-^a and cp z a the angle at c is common, it amounts to making these triangles equiangular. Hence, because these triangles are similar, cf)-i ca - x = , or ca cp n (7) cp, - APPLICATION TO THE MICROSCOPE. which, combined with equation 6, shows 673 (8) cp 2 = nR R Therefore, when both points are denned by equation 8, p z is a perfect optical image of p l for refraction through the spherical surface to the left of the source. The construction fails for the surface to the 'right, since the condition 1 = (f> 2 cannot be fulfilled when The magnification is seen directly from the figure to be / C p ~Wi~ 590. Application to the Microscope. The preceding theory shows a means by which an object embedded in a sphere of glass, and a short distance from the center, can be replaced by a virtual image n 2 times as great, and absolutely without fault, except so far as dependent on the small variations of n with different wave-lengths. The limit of advantage to be obtained in this manner may be derived as follows : Let p 2 (Fig. 446) be the virtual image of the point p 1 magnified n 2 times, and lorn an indefinitely thin lens system everywhere equidistant from p^, which will FIG. 446. 674 EFFICIENCY OF OPTICAL INSTRUMENTS. render wave surfaces from p 2 exactly flat. In the sense of the previous analysis, the system is just that which, combined with a telescope, will constitute a compound microscope. To adapt the expression for the highest useful magnifica- tion given on page 669 to this case, it is sufficient to notice that the power of this objective system is n 2 times that of lorn, or that the maximum aperture of the telescope is 20p 2 sin Ip 2 o. Comparing Fig. 446 with Fig. 445, this angle is seen to T) be equal to bp 2 c, or cbp v and sin cbp 1 = The aperture is, 9 -j I\MJ accordingly, ? or - of that considered on page 669. There- fore the power being n 2 times greater, and the aperture -' ilf the magnification will be n times greater, and the highest use- ful power 600wJV". This formula, derived on the supposition that the object is embedded in the lens, still needs a slight modification, for in practice it is necessary that the distance between the lens and the object shall be slightly variable in order to admit of focusing. In order to fulfill the latter condition, the object is mounted under a thin cover glass, W (Fig. 447), whose influ- ence may be neglected, and the space L between it and the FIG. 447. hemispherical lens H filled with a liquid. Under these cir- cumstances the waves proceeding from suffer an additional refraction at the flat surface of the lens, by which their curva- ture is changed in the ratio of the index of refraction of the APOCHBOMATIC OBJECTIVE. 675 fluid to that of the lens. Hence, the expression for the high est useful magnification becomes n where n r is the index of refraction of the immersion fluid. In the case of dry objectives n f = 1 and the limit of res- olution, 90,000 lines to the inch, is unaltered. For water immersion, n' = 1.3, the limit becomes 120,000, while fora homogeneous immersion, that is, a fluid having the same refractive power as glass, the limit rises to 135,000. In this case the refraction of the cover glass will introduce no error. Since the greatest known value of the index of refraction for any transparent medium is about 2.5, there is no hope of ever making visible a greater number of lines than 250,000 to the inch. Since the highest powers require the employ- ment of more than a full hemisphere, as appears from Fig. 445, the difficulties of forming and mounting such objectives make them very costly. It should be remarked that the immersion fluid was origi- nally introduced by Amici, to reduce the loss of light from the face of the objective, and that its theoretical improvement of the resolution was not anticipated before its advantage had been experimentally found. 591. Apochromatic Objective. The most important ad- vance in recent years in the correction of some of the defects which seemed inseparable from the old construction is the invention by Professor Abbe of what he calls the apochro- matic objective. Suppose the objective, which, according to our previous convention, makes a perfect virtual image at an infinite dis- tance, to be divided into two portions as in Fig. 448, the lower of which, /, forms a magnified virtual image of the ob- 676 EFFICIENCY OF OPTICAL INSTRUMENTS. ject at a finite distance, and the uppermost, M, a corrected virtual image of this at an infinite distance. Now since all transparent substances refract short waves more than long, the image of o formed by I (Fig. 448 A), as- sumed to be somewhat under-corrected, will be a complex of colored images from r to 5, the blue image being most -^ remote. The obiect of the Jw * system u is to convert all the complex system of wave sur- faces, having their centers from r to 6, into plane wave surfaces. In other words, the system u must have a higher power for red than for blue waves, and at the same time be nearly free from spherical aberration for all colors. There is no difficulty in meet- ing these conditions for wave surfaces of moderate angular extent bv the combination of FIG. 448. J negative lenses of flint glass with positive lenses of crown glass ; but when the wave sur- faces are large, as they must be in powerful objectives, it is found that the power of the upper system is always relatively too small at the margin for the short light-waves. The lack of flexibility in the means of correction arises from the fact that in all known glasses a great increase in dispersive power is invariably accompanied by an increase of refractive power. The use of fluids in lenses, though suc- cessfully applied by Abbe and Zeiss, is practically debarred on account of the difficulty of keeping them in good order. APOCHROMATIC OBJECTIVE. 677 A satisfactory correction of the defect mentioned has been secured by Abbe in an entirely different way. If the lens system u, instead of standing as close as possible to Z, be removed to a considerable distance, as in Fig. 448 B, the change of curvature which has been produced by u is dimin- ished, and the power of the objective is correspondingly de- creased; but at the same time the difference in curvature between the red and the blue is decreased in a much greater ratio. Thus, if the curvature of the red light-waves is half as great in the second case, the difference will be one-fourth as great ; or if the curvature is reduced three times by mak- ing the distance between I and u twice as great as that between I and r, the difference of curvature will be reduced nine times, and so on. The new construction consequently admits of varying the relation between the two functions of the system u within wide limits, and yields another arbitrarily variable element for the attainment of a closer correction. This, with a skill- ful selection of materials to reduce to a minimum the far less serious defect of what is called secondary chromatic aberra- tion, constitutes the important optical apparatus to which the inventor has given the name apochromatic objective. The construction of this objective entails, however, a defect in the complete instrument which is of considerable interest on account of its general nature. The expression for the focal power of a lens shows that the magnification in a virtual image by the sys- tem I is greater for short waves than for long, and conse- quently the whole objective will magnify a blue object more than a red one, unless the system u exactly reverses this 678 EFFICIENCY OF OPTICAL INSTRUMENTS. relation, i.e. unless it changes the curvature of the wave surfaces of short wave-length less than those of longer waves, and less by just the proper amount. Now whatever be the case with the system u in Fig. 448 A, it is clear that the cor- FlG. 449. responding system in Fig. 448 B cannot correct the defect in question in the anterior* portion, because the separation has been made for the sake of reducing the relative difference of curvature of the wave surfaces. Thus, it appears that in this objective and, in general, in any system in which correction for color is obtained by lenses separated by a considerable APOCHBOMATIC OBJECTIVE. 679 distance from uncorrected lenses, the images of an object, though all in the same plane, and therefore achromatic in the ordinary sense of the word, differ materially in magnitude with differing color. This defect produces a characteristic imperfection in the seeing with all high power microscopes. Abbe corrects this defect by adding a compound lens to the ocular, of such construction that it makes its power greater for red light than for blue in the same ratio as the excess of magnification of the objective lies in the opposite direction. These two elements combined, namely, the apochromatic ob- jective and compensating ocular, form the modern perfected microscope. The accompanying cut (Fig. 449) of an instrument may be regarded as one of the best existing models. The appa- ratus below the stage is for the purpose of securing at will illumination from any desired direction. It is known as Abbe's illuminator. CHAPTER XLIL OPTICAL PHENOMENA OF THE ATMOSPHERE. 592. Optical Phenomena of the Atmosphere. There are a large number of phenomena, some beautiful, some merely curious, which depend upon the modification which light undergoes in its passage through the atmosphere. In cer- tain cases these phenomena are a necessary consequence of the optical properties of the air alone, or at least de- pendent upon invariable constituents, as, for example, the blue color of the sky, mirages, and looming, and scintilla- tion of the stars. In other cases the cause is to be sought in bodies temporarily constituting a portion of the atmosphere, as in the rainbow, corona, and halo. 593. Color of the Sky. If the atmosphere were perfectly transparent, i.e. if the light-waves could be transmitted through it without loss, the sky would appear quite black, except where a bright spot marked the presence of a star or a planet. In short, the sky of day would differ from that of night only by the presence of a brighter star, the sun. On the other hand, if the air should only transmit a portion of the wave energy, converting the remainder into heat or some. other form of energy unappreciable by the eye, the sky would bear no closer resemblance to its familiar appearance, for it would still remain black, except at points in the direction of the stars where a portion of the light might penetrate. Professor Langley has demonstrated that what is ordinarily called a clear atmosphere is very slightly opaque to light- COLOR OF THE SKY. 681 waves. The colored appearance of the sky may be under- stood from the following reasoning: Suppose the atmosphere is perfectly transparent, and hence the sky black with a high sun, and that small drops of water are distributed at infrequent intervals, say one to a thousand cubic feet of space. The sky would send light to the eye from all directions, since each drop would scatter the light which fell upon it, though the brightness of the direct sun- light might not be notably diminished. Such a sky would be called hazy. If the number of these small drops should be continually increased, the density of the haze would grow, while the brightness of the direct sunlight would pro- gressively diminish until the sky became overcast and the sun invisible. Now imagine the process reversed; that is, suppose the drops are removed, leaving the distribution fairly uniform until the sky is no brighter than an ordinary clear sky. Then suppose the drops to be reduced in size, but at the same time increased in number at such a rate that the total quantity of light from the sky remains the same. This reduction of size, however, introduces an entirely new ele- ment into the consideration, for when it is carried so far that the particles of water have a diameter which is small com- pared to the wave-length of light, they would be incapable of reflecting these waves, precisely as a floating body on the ocean would be unable to reflect waves whose lengths are great compared to its own dimensions, although it would offer a perfect barrier to the passage of short waves. Before reaching this condition of extreme tenuity, however, a range of dimensions will be passed, which, though small when com- pared to the wave-length of red light, are not small compared to the wave-length of blue or violet light. Such particles would reflect violet and blue light more copiously than orange 682 OPTICAL PHENOMENA OF THE ATMOSPHERE. and red light. It is under these conditions, and for this reason, that the clear sky is blue. It is obviously immaterial of what the particles are made, provided they are small and not coagulated; hence, when light is reflected from a cloud of smoke which is not too dense, especially if the background is black, so that only such light reaches the eye, it appears pale blue. The blue of opal and opalescent bodies has its origin in a similar manner, as has also the color of blue eyes and that of the deep sea. It is well known to painters that generally a mixture of a white with a black paint gives a strongly bluish gray, al- though there may be no suggestion of this color in either of the components. This, too, may be explained by a species of selective reflection depending on the minuteness of the reflecting particles. It follows from these considerations that sunlight which has come through the atmosphere has lost more in short than in long waves, and consequently the hue of such light is somewhat yellow. If the light has passed a very long dis- tance through the air, as when the sun is near the horizon, we may have, with a great diminution in the strength of the waves, a practically complete stoppage of the short waves. This would give yellow, orange, or red, depending on the completeness of the selective action, and also, as will appear in the study of the phenomena of color sensation, to a con- siderable extent upon the absolute intensity of the light com- ing to the eye. The colors are strongest* after the sun is below the horizon and the light is received only indirectly from reflecting clouds, for the path of the light through the air is much longer than when the source is above the horizon. An analogous phenomenon is seen in a light smoke, which appears blue against a dark background, as already noted, AERIAL PERSPECTIVE. 683 but yellow when the background sends more light to the eye than the smoke itself. The greenish blues and the blue greens, which are not infrequently seen in the sunset sky, are probably always a physiological effect of contrast. One of the most notable effects of the opacity, or more properly opalescence, of the atmosphere is the change it pro- duces in the aspect of distant objects. Thus, two surfaces, the one light and the other dark, lose something of their con- trast as they recede from the eye, the one becoming darker by the absorption of its light through the intervening air, and the other brighter by the superadded light diffused by the air. If the atmosphere is free from coarse dust particles and relatively large particles of water, this added light is blue. Often only a short distance is required to give a strongly blue hue to a sunlit rock, for instance, much less than a mile, while distant hills are always of a strong violet or blue in a clear day. This effect goes under the general name of aerial perspective, and affords the readiest means of estimating the distances of remote objects on land. In a very clear, dry atmosphere it is greatly diminished, so that a moist climate adds greatly to the grandeur of mountain scenery. 594. Atmospheric Refraction . The velocity of light- waves in air of the prevailing density at the surface of the earth is about three parts in ten thousand less than in a vacuum. In consequence of this retardation taken in connection with the decreasing density of the air at higher altitudes, light enter- ing the atmosphere at its greatest obliquity in equatorial regions has its course changed by about 35 minutes, or not far from the diameter of the sun. The sun therefore appears to be just above the horizon, when, were there no air, it would appear to be just below it. 684 OPTICAL PHENOMENA OF THE ATMOSPHERE. The effective lengthening of the day is about four minutes at the equator ; at higher latitudes the effect would be greater, and in extreme cases it may amount to many hours. 595. Looming. Occasionally temporary inequalities in atmospheric density give rise to peculiar phenomena of re- fraction, known as looming and mirage.* These phenomena in their simpler forms are not so uncommon but that it is* possible to obtain a knowledge of them from observation at least a hundred days in the year. Let AB (Fig. 450) represent a distant upright object, and SB the level surface of the earth or of water. Now, suppose that the temperature of the air above SB decreases up to FIG. 450. the level EC, where it is lowest, and that above this hight it either increases or remains constant. This arrangement, though unusual on a large scale, may obtain under conditions to be noted later. Since the velocity of light-waves in air depends upon the density, and this, in turn, depends upon the temperature and the hight above the surface of the earth, it is evident that only in the direction CE will such waves be propagated without deflection. In all other directions the line of propagation will be a curved one. For instance, a wave surface below CE, and moving toward the left, would tend constantly upward because its upper edge, moving in a denser medium, would advance more slowly than the lower edge. LOOMING. MIRAGE. 685 Likewise the path of a similar wave above CE would curve downwards. . For a discussion of the consequences of these refractions it will be necessary to assume a position for the eye of the observer. Suppose it to be at E. Then the path of that portion of a wave surface having its origin in A would have some such form as AaE, and A would appear at A' in the direction of the tangent at E. Any point between A and C would send out waves whose paths to the eye would be more curved than the one marked ; hence they too would be seen above their real position, but raised in a greater ratio than A. Accordingly, AQ would be increased in apparent hight, the increase being greater in the lower part, but the width would remain unchanged. This phenomenon is known as looming. It will be noticed that these conclusions are quite applicable to ordinary atmospheric conditions, since in that case the density of the atmosphere decreases from the earth upwards at a diminishing rate. Thus, at a hight of three and a half miles the density is reduced about a half, while to reduce it the remaining half, a further ascent of more than a hundred miles is necessary. For the reasons mentioned, the Eight of a mountain can- not be determined accurately from the distance at which it is visible at sea, nor, conversely, can the dimensions of the earth be found from the measured horizon of a mountain whose hight is known. 596. Mirage. Light-waves emitted by the point A (Fig. 450) may come to the eye by a path of the form indicated in the line AbE. The apparent position of A would now be at ^4." in the direction of the tangent to the curve at E. Any other point of A C will also be a source of wave surfaces, which, becoming distorted by differing velocities of propagation in 686 OPTICAL PHENOMENA OF THE ATMOSPHERE. different directions, will reach E along two different paths, one lying between AaE and CE, and the other, for the greater part of its course, between AbE and CE. No light from points immediately below C will reach the eye, because, the paths being continually curved upwards until they pass the layer of maximum density, the reversal of curvature there experienced will not be sufficient to make any part of the waves pass through E. Hence, under the conditions sup- posed, two images will be visible from the point E, one erect and elongated, but in a ratio continuously decreasing from the foot to the top ; the other inverted, meeting A at (7, and extending downwards to that point of CB, or of the ground between S and B, which is the first to emit light-waves any part of which passes through the place of the eye. It does not appear that the inverted image is either lengthened or shortened, since no supposition has been made as to the law of change of density; but there will be no alteration of the image in width. It must be borne in mind that the diagram greatly exaggerates the effect of irregular refraction, the changes in direction being in general only a fraction of a degree, instead of many degrees, as in the figure. If the eye be placed below the level of maximum density, the most notable change would be that the point of the lower portion of the object, or of the ground whose image marks the limit of the inverted image of AC, would be lowered so that more of the region above C would appear in this simu- lated reflection. The condition most frequently met in the ordinary phase of mirage, riot only of the desert, but also of that extremely common in our own latitudes, is that in which the eye is placed above the level of maximum density. In this case the portion of the ground, or lower part of CB, which can be seen is increased, and the portion of this line which is in- MIRAGE. 687 visible and replaced by an inverted image of AC is corre- spondingly reduced. Moreover, since the curvature of the paths from A to the place of the eye will, in general, be less, the looming will be less pronounced. In a broad, sandy plain with a low horizon, which has be- come heated by the sun, the lower region of the atmosphere may become much warmer than the air a few inches higher, provided there is no wind sufficiently strong to secure a tolerably homogeneous mixing. Under these circumstances there would be a certain region of the plain, at a distance increasing with the hight of the eye, which would be visible, but beyond that only an inverted image of the sky could be seen. The appearance would be virtually that of a smooth body of water, an effect which would be greatly increased should there be any object to break the horizon, such as a tree, for the appearance would then reproduce the familiar, reflections from a lake. In our own regions the conditions most favorable for the production of a lower layer of air of higher temperature than that of the air above are found Over considerable bodies of water. For example, in the early hours of clear, quiet morn- ings late in summer or early in the fall, the atmosphere is often cooled by nocturnal radiation to a temperature consid- erably below that of the water, which retains much of the heat accumulated during the long summer days. A portion of the heat of the water is imparted to the air in contact with it, and all the conditions that have been considered are then present. The layer of optically rarer air may be only a few inches thick, in which case it is necessary to bring the eye quite close to the surface of the water in order to see very marked effects. There are, however, very few clear, windless mornings in the autumn when it is not possible, in any of the New England bays or harbors, to see distant vessels ac- 688 OPTICAL PHENOMENA OF THE ATMOSPHERE. companied by an inverted image beneath them. In search- ing for such phenomena it is often advantageous to employ a small telescope. Interesting imitations of the mirage may be found by look- ing at distant objects along a wall, or straight board fence, which has been warmed by the rays of the sun, especially if the surface is moist. The aid rendered by the moisture is due in part to the fact that the velocity of light-waves in a mixture of steam and air is somewhat greater than in air of the same temperature and pressure, but chiefly because the water vapor stops more of the heat radiated from the surface, and thus suffers a greater rise of temperature. Here, of course, the looming is absent, because the air more remote from the wall is homogeneous, but the double image with its limits is very clearly shown. In high latitudes over fields of ice much more complicated cases of mirage are not uncommon. There it sometimes happens that there is a deep layer of cold air higher up, the eye being far below the region of transition. In this case it is easy to show by a course of reasoning similar to that fol- lowed in Fig. 450 that it may be possible to see a distant object through the lower nearly homogeneous layer in its true position with an inverted image above it separated by a greater or less distance, according to the hight of the transi- tion region. Since the upper paths are more curved than the surface of the earth, the inverted image may, under ex- ceptional circumstances, be seen when the object itself is below the horizon, and therefore invisible. This phenomenon seems not to be very rare. 597. Scintillation. The phenomenon of twinkling, or scintillation, of the stars was first explained by Arago as depending upon the lack of optical uniformity in the atmos- SCINTILLATION. 689 phere. Several phases of the phenomena may be noted as follows : 1. It is only the stars that scintillate, the planets appearing quite steady to the eye, with the exception, pos- sibly, of Mercury, which is not only very small, but can only be seen near the horizon, where the conditions are most favor- able for the effect. 2. If the finger be pressed gently against the right eye, so that its image of a twinkling star is slightly displaced from that of the left eye, it may be noticed that these two images flash quite independently of each other ; or, if the star be observed through the biprism of Fig. 440, the two images seen with one eye will not change their intensity simultaneously. 3. If, when a scintillating star is observed through a telescope, the telescope be given a sudden motion by striking it with the hand, a ribbon of light with constantly changing colors may be seen. All of these observations may be thus explained : Suppose a bright point to be the source of light-waves which come to the eye through the atmosphere, then those wave surfaces which fall upon any area, say one-half of the pupil of the eye or of the objective of the telescope, will in general, on account of the irregularity of the atmosphere, have suffered quite different modifications from those which fall upon the other half. If, now, this difference is such that one portion is retarded any number of half wave-lengths, the two por- tions will be mutually destructive, and that particular wave- length will be wanting. Should this wave-length be long, the star would appear for an instant greenish ; if of medium length, the color would be purplish ; and if short, orange or yellow. In a faint star the colors would escape detection without the aid of a telescope, and only variations of inten- sity would be seen. Anything tending to make the air more homogeneous would reduce the amount of scintilla- tion. It is for this reason that the phenomenon is most 690 OPTICAL PHENOMENA OF THE ATMOSPHERE. pronounced in a dry atmosphere, and is almost wanting over tropical seas. It is evident that stars near the zenith should appear more steady than those at lower altitudes, since the light has trav- ersed a minimum extent of the air. Though a single point of a planet may scintillate strongly, since the light from dif- ferent points of its disc will have suffered very different modifications, it might be anticipated that the average bright- ness of the sum of all the points would vary but little, which is indeed the fact. 598. Coronas. Coronas are a series of colored circles seen about the sun or moon, when covered by very light clouds. They are distinguished from the larger circles, called halos, not only by their smaller and variable size, but also by the arrangement of the colors, the inner edge being blue and the outer edge red, which is the reverse order of their occurrence in halos. Fraunhofer showed that coro- nas may be perfectly imitated by scattering very small, circular, opaque bodies, such as lycopodium powder, in an entirely irregular manner over the surface of glass, and look- ing through it at the sun or moon. By the use of a telescope, so as to magnify the effect, and by looking at a much smaller source of light, such as a star or planet, he was able to secure the same result by employing a number of equal discs of tin foil irregularly placed in front of the objective. These experiments show that the phenomenon is a diffraction effect. It was seen in Art. 573 that when a bright point was observed through a very small round hole it would appear as a disc surrounded by a series of concentric rings having a blue color within and a red without. If another hole of the same size were perforated in the card, it was found that the CORONAS. 691 disc and rings, now twice as bright, remain, but crossed by a series of dark lines. If the number of holes be increased, retaining the same size, it is found that the disc and rings maintain their positions, but with constantly increasing brightness and complexity of intersecting systems of dark lines. Should the number be increased indefinitely, it may be concluded that the dark lines would become too numerous to be seen, and the final effect would be the same as that of a single aperture of the size chosen, though multiplied in brightness by a number equal to that of the holes. That such is the fact has been established mathematically by Verdet, and may be verified by looking at an artificial star through a piece of tin foil perforated with a large number of small equal holes, or by piercing a sheet of paper with a large number of irregularly distributed holes even as great as one- tenth of an inch in diameter, and looking through it with a telescope at a bright star. In order to employ these facts in the explanation of coro- nas, it will be necessary to make use of what is known as the principle of Babinet, which may be thus stated : If illu- mination occurs at any point on account of the presence of an opaque screen, however complicated, between the source and that point, then, if all the transparent portions of the screen be made opaque, and the opaque portions transparent, the quantity of light at that point will remain unchanged. The proof of this follows from the considerations of Fig. 439. The reason that light appears at jt? 2 , etc., is because the wave surface is limited ; consequently the opaque portion of the screen cuts off what would, if added to the waves at p v ex- actly destroy them ; in other words, waves of the same inten- sity, but differing by one-half wave-length in phase. This is indeed but a special case, but the reasoning is perfectly general, and the principle may be accepted as governing all cases. 692 OPTICAL PHENOMENA OF THE ATMOSPHERE. By means of this principle it is now possible to pass at once from the case of the screen irregularly perforated with uniform circular holes to the glass plate covered with irregu- larly disposed opaque discs or spheres, and from that to small spheres of water suspended in the atmosphere, since these are essentially opaque, because almost all of the light which passes through them is greatly changed in direction. The essential conditions of a corona are, therefore, suspended particles of water, uniform in size, and so small that the dif- fraction rings due to them shall be considerably larger in angular dimensions than the sun or moon. As the spheres grow larger, the corona becomes smaller, and vice versa. Coronas thus furnish a guide as to whether the droplets of moisture are increasing or diminishing, and an indication of value for predicting changes of weather. Several other phenomena may be mentioned having an analogous cause and appearance to coronas. Thus, many ob- servers may recognize a system of colored circles surround- ing an arc light seen against a dark background, particularly just after awaking from sleep. These become especially marked after a severe blow upon the eye, and remain some- times for many months with decreasing brightness and in- creasing dimensions. They are attributed to a slight opacity 3 in the epithelial cells of the cornea. Occasionally such circles are to be seen about a light when observed through a sheet of glass upon which there is a considerable deposit of moisture from the air. An effect not unlike the corona FlG - 451 - may be obtained by passing light through a small opening in a screen, 88' (Fig. 451), and reflecting it from a concave silver-on-glass mirror, MM', so EAINBOW. 693 that the light passes directly back through the hole. If the glass be now breathed upon, so as to tarnish the surface, light which has been diffused at the front surface and then regularly reflected will interfere with light which, first regu- larly reflected, has suffered diffusion at emergence, and will produce a series of colored rings upon the screen. 599. Rainbow. Rainbows are produced by refraction of direct sunlight falling on spherical drops of water. In gen- eral very little light will be received from such an illumi- nated drop, if remote, for that which leaves the drop, either after refraction or after reflection from the outer surface, or after having suffered refraction combined with interior reflec- tions, will be sent in every direction, so that little can fall upon the small area of the pupil of the eye. There are, however, certain direc- tions for which this state- ment does not hold, as may be shown by the aid of Fig. 452. Let o be the center of a drop of water, and AB a plane wave surface moving from left to right and falling upon it. The portion of the wave which meets the drop at a will have its direction of propagation changed and also have its curvature made concave, since the velocity is less in water than in air. Let ac be the new direction of the wave. At c the wave will be reflected, its concavity reduced, and its direction changed to that of the chord cb, which is of the same length FIG. 452. 694 OPTICAL PHENOMENA OF THE ATMOSPHERE. as ac, since, by the law of reflection, the angles on either side of the radius drawn to c are equal. The wave, accordingly, meets the surface at b at the same angle that it left the point #, and hence, on emergence, will suffer a change of curvature equal and opposite to that experienced on entering the drop. If, then, the reflection at c should change the concave surface into an equal convex one, the emergent surface would be plane. If the index of refraction of the sphere be less than 2, this condition may always be met by arranging the incidence so that c is the center of the refracted wave at a. This is not, how- ever, the only case in which a plane incident wave emerges as a plane wave, as may be seen in Fig. 453. Assuming here a larger angle at a, there results a greater deviation and change in curvature by refraction, so that the wave surface is concave until it reaches the center ^; it is thence convex to the first point of reflec- tion, where its convexity is dimin- FlG 453 ished. By a proper choice of the point #, this reflection may be made to render the convex waves plane ; in which case they would move on without change until they became concave by an- other reflection, with a center at c 2 ; thence they would move as convex waves to 6, where they would be again refracted into plane waves. Proceeding in a similar manner, it may be shown that there is an indefinite number of ways in which the same result may be attained, the geometrical condition being that, if there is an odd number of interior reflections, the middle one must correspond to a point where the curvature of the RAINBOW. 695 wave surface is infinite, while, if the number of reflections is even, the middle chord, which marks the path of the waves, corresponds to a region where the curvature is zero. The analytical conditions may be found as follows : Let the angle of incidence at a be called i, and the angle of refraction r. The deviation, or change in direction, of the wave between its entrance at a and emergence at b may be written down at once from the figure (1) 8 = i r + TT 2r + i r = TT 2 (2r z), or, after m reflections, (2) g = 2 (i r) + m (TT 2r) = m-rr 2 \ (m + 1 ) r i} . Suppose that when i is increased by a small amount, , r increases by the amount i/r. Substituting these values, the deviation for the slightly altered angle of incidence becomes In order that the emergent wave shall be plane, it is neces- sary that portions of the wave entering very near a shall emerge with the same deviation. Comparing equations 2 and 3, it appears that this will be the case when (4) <=(w + l)ijr. To find the value of i corre- sponding to this condition, draw two circles (Fig. 454) whose radii have the ratio of the index of refraction n. Let pov = i, and draw pu parallel to ov. The intersection of pu with the outer circle determines the angle qov = r, 696 OPTICAL PHENOMENA OF THE ATMOSPHERE. because, by construction, sin pov = n sin qov. Similarly, dra\Ving u'q' near and parallel to uq, pop' = , and qoq' = ^. Draw ps and qt perpendicular to ov. Then, because their sides are perpendicular, the triangle psp' is similar to puo, and qtq' to quo ; therefore the angle p'ps = i, and q'qt = r. Hence sp , tq -L- = cos ^ and -JL cos r. PP W Observing that sp = tq, pp' cos i = qq' cos r ; but pp' op'4>, qq' = n- op- fa and ^-(m + 1)^; whence (5) (w + 1) cos z = w cos r. Combining this with * sin i = n sin r, we have, finally, ^ (6) COS I = ' m 2 -\- 2m This condition of a stationary value of the deviation is also one that determines either a maximum or a minimum devia- FORMATION OF THE BOWS, 697 tion. It is easy to see in this case that it is not a maximum, for when the incident light passes through the center of the drop m = 1 and 8 = TT, which is greater than that of equation 6. Substituting the indices of refraction for water, the follow- ing values of the minimum deviation are found : 8 m RED. VIOLET. 1 . . 138 140 2 . . 231 235 3 . . 309 313 4 . . 403 410 600. Formation of the Bows. From the preceding table it is seen that all raindrops upon which the sun shines at an- gular distance between 42 and 40 from the antisolar point, that is, 138 to 140 from the sun, will appear bright, and con- sequently a portion of a ring will be seen against the sky, if the sun is less than 42 above the horizon. So, too, drops at a distance of from .51 to 55 from the antisolar point will form a similar bright bow. The minus sign in this radius signifies that the light received from any drop in the second- ary bow has entered the lower instead of the upper side, as in Fig. 453. The secondary bow is necessarily much fainter than the primary, because its light has ^suffered two partial reflections, and this condition must hold with stronger rea- son in bows of higher order. This fact is, without doubt, sufficient to account for the absence of the fifth bow, which is 486 from the sun, or of a radius of 54 about the anti- solar point. Were the third, or even the fourth, as favor- ably situated for observation, it would certainly be visible at times. As, however, these bows are only about 50 from the 698 OPTICAL PHENOMENA OF THE ATMOSPHERE. sun, and that portion of the sky is always strongly illumi- nated by light which has been transmitted through the drops, it is not remarkable that they have never been seen. It follows from the table of deviations that the primary bow is red on the outside and violet on the inside, while in the secondary bow the order of these colors is reversed. The prismatic colors cannot be very pure on account of the angu- lar dimensions of the sun. Since the deflection of the light which forms the bows is a minimum, it follows that no drops nearer the sun than 138 can send any light to the eye by a single interior reflection, and none nearer than 231 can send any light by two interior reflections ; therefore those drops which are situated between the bows, being nearer in one direction than the first limit, and nearer in the other direction than the second, can send no light of either modification. This accounts for the rela- tive darkness of the sky between the two bows, which forms a notable feature of a well-developed rainbow. 601. Supernumerary Bows. Since the light which pro- duces the bows comes from portions of the plane incident waves which have suffered a minimum deviation, it follows that there are parts of these waves above the most effective portions which have the same deviation, such light coming from drops just under the primary bow, or just above the secondary. But such portions will have passed through dif- ferent lengths of water, and will therefore be in a condition to interfere. It is easy to see that this will produce a series of repeti- tions of each color, just inside the primary and outside the secondary, of rapidly diminishing brightness and at an an- gular distance decreasing with increasing size of the drops. These bands, known as supernumerary bows, are most often SUPERNUMERARY BOWS. 699 seen under the highest part of a bright inner bow, and, more rarely, as an accompaniment of the outer bow. The conditions of distinctness, namely, uniformity of size and smallness of the drops, are such as are more likely to obtain at a higher altitude. Another effect of interference is to change slightly the dimensions of the bows, decreasing the apparent diameter of the inner and increasing that of the outer bow. This change, though small for the ordinary rainbow, may amount to sev- eral degrees in the white bow, of which the description follows. Occasionally a very bright primary bow is seen, with only a tinge of red on the outside and of blue on the inside. It is formed when a bright sun shines on a dense wall of mist, where the droplets are sufficiently large to give a tolerably definite reflection, but at the same time differ greatly among themselves as to size. It is always smaller in angular diam- eter than the colored bow, for the reasons just stated. The lunar rainbow appears almost colorless, on account of its faintness, just as foliage loses nearly every trace of color by moonlight. This is, however, merely a physiological effect. Sometimes arcs of rainbows are seen on a sward when cov- ered with dewdrops, or in a spray thrown up by the bow of a boat. In such cases, although the image on the retina is always a circular arc, yet, since the neighboring objects per- mit an estimation of the distance of the drops from which the light proceeds, we ascribe to the bow the form of the projection of this circle on the surface. Hence, if the sun has an altitude greater than 42, the bow will appear as an arc of an ellipse ; if less, as an arc of a hyperbola. The history of the theory of the rainbow is an interesting one. Descartes gave the first geometrical theory in 1637, 700 OPTICAL PHENOMENA OF THE ATMOSPHERE. without, however, accounting for the existence of colors. This explanation was added by Newton in 1704. The ex- planation of supernumerary bows was suggested by Young in 1804, but first completely worked out by Airy in 1836. 602. Halos. Very frequently a circle of about 22 radius, red inside and bluish without, and showing indis- tinctly some of the intermediate prismatic colors, is seen in a slightly hazy sky to surround the sun or moon. On rare occasions this is accompanied by complicated series of curves and bright areas, some of which may be vividly colored and others quite colorless. Such curves are called halos, and the limited bright areas are called parhelia, or sun-dogs. These are all attributable to minute crystals of ice floating in the atmosphere, and their theory is pretty completely worked out at least for all the common features. Thus, the ordinary circle of 22 radius is explained as follows: A minute crystal of ice is known to take, generally, the form of a right hexagonal prism, of which any two alternate pris- matic faces would form a 60 prism. From the measured index of refraction of ice it may be demonstrated that such a prism would have a minimum deviation of about 22. If, now, we imagine the air filled with a host of such little crys- tals floating in it, we may easily see that all of them which are at nearly this angular distance from the sun, and which at the same time chance to have approximately the proper orientation, will refract light towards the eye, while none of those nearer the apparent position of the smn can do so, since the angle of 22 is a minimum angle. On the other hand, crystals more remote from the sun than these can also divert light towards the eye, but in view of the fact that to do so requires a very delicate adjustment of orientation, the pro- portion of those thus favorably situated must very rapidly HAL OS. 701 decrease with increasing angular distances from the sun. These considerations lead us to expect a sudden increase in the luminosity of the sky, when such crystals are abundant, at a distance of 22 from the sun, followed by a pretty rapid reduction as we go outwards. When regard is paid to the different refractive power for different wave-lengths, the cause of the color of the circle is evident. Fro. 455. The dihedral angle of 90, formed by the sides and bases of the hexagonal prisms, also produces a circular halo in a similar manner, but of 46 radius. This is necessarily much fainter than the 22 circle, and is in fact very infrequent. The other phenomena, some of which are represented in Fig. 455, are also due to suspended crystals of ice, but to those which have a prevailing constancy of direction of their axes. That small bodies of a definite form would, in falling 702 OPTICAL PHENOMENA OF THE ATMOSPHERE. through a quiescent atmosphere, assume a general likeness of direction is at least probable. At any rate, an assumption of this kind explains all of the more common complications of halos. Thus, crystals having their crystalline axes verti- cal cause the familiar sun-dogs, which are almost as common as the smaller circle, and the tangent arc to the outer circle. To such crystals also is ascribed the horizontal colorless band passing through the sun, sometimes extending com- pletely around the heavens, called the parhelic circle ; but to explain the tangent arcs to the 22 circle, and a number of other rarer features, we must assume the presence of crystals with their axes horizontal. CHAPTER XLIII. RADIATION AND ABSORPTION OF LIGHT-WAVES. 603. Selective Absorption. Although the vibrations of light- and heat-waves must be regarded as electrical in their ultimate nature, they obey a law of emission and absorp- tion quite analogous to those already found for the purely mechanical vibrations of sound-waves, in that if the body is capable of vibrating so, as to emit waves of certain definite periods, then whenever waves of these periods fall upon the body they will be absorbed. The character of the absorp- tion which takes place in any substance may be conveniently studied by analyzing the transmitted light with a prism, in which case the spectrum appears crossed by dark bands cor- responding to the missing waves. Thus, the spectrum of light which has passed through a piece of red glass shows that it transmits the red waves copiously and some of the orange, but that the green and blue are entirely absorbed. Likewise a piece of cobalt glass will be found to cut off all the bright red, the orange, and yellow waves, but to transmit all of the violet and blue with some of the green, and, singu- larly enough, a band of dark red very near the end of the spectrum. A solution of chlorophyll, the green coloring matter of plants, exhibits dark bands in the red, yellow, green, and violet, and a dilute solution of permanganate of potash shows several characteristic bands in the green. If a piece of red glass be heated till it becomes luminous, and then placed in a dark room, it will be seen to emit a bluish green light, that is, waves of those periods which it absorbed 704 RADIATION AND ABSORPTION OF LIGHT-WAVES. when cold, thus illustrating the proportionality of its emissive and absorptive power. These phenomena are more strikingly exhibited in gases on account of the freedom of the molecules to vibrate in their own peculiar periods. Thus, if the light emitted by incandescent sodium vapor at a moderate temperature be ana- lyzed by means of a prism of high dispersive power, it will be found that the sodium emits only waves of two definite wave-lengths in the yellow. On the other hand, if the light from some incandescent solid be allowed to pass through a sodium-tinged flame at a lower temperature than that of the first source, say through the flame of an alcohol lamp with salted wick, it will now be found that the spectrum is crossed by two black lines situated exactly at the place previously occupied by the bright sodium lines. 604. Kinds of Spectra. Spectra are often classified for convenience as continuous and discontinuous. The former are emitted by incandescent solids or liquids, and are char- acterized by the presence of all waves from the red to a higher limit determined by the temperature. Discontinuous spectra are marked by the absence of par- ticular waves, and are of two sorts, the absorption spec- trum arising from the passage of white light through an absorbing medium, and the spectrum emitted by an incan- descent gas. 605. Spectroscope. The spectroscope is an instrument conveniently arranged for analyzing and observing the light emitted by various sources. The disposition of parts in a form often used in the chemical laboratory is shown in Fig. 456. The prism, P, used to form the spectrum, rests on a table, A, at the center, SPECTROSCOPE. 705 supported by leveling screws so that it may be readily ad- justed. On either side of the prism, and movable about the vertical axis of the table, are placed a telescope, T, and a tube, (7, known as the collimator, furnished at the outer end with a narrow slit, $ and at the inner end with a posi- tive achromatic lens. The light which it is desired to ex- amine is first concentrated on the slit. The waves diverging from this point have their curvature reduced to zero by the lens of the collimator. Falling on the prism at the FIG. 456. angle of minimum deviation, they suffer dispersion, and after passing through T appear to diverge from points in the field of the telescope, which differ with different wave- lengths. For purposes of comparison a bright scale of equal parts in L is often so arranged that its image by reflection is made to coincide with the spectrum ; or a portion of the slit is illuminated by light from some standard source, and the other portion with the waves to be investigated, in which 706 RADIATION AND ABSORPTION OF LIGHT-WAVES. case the spectra appear placed one above the other. When the instrument is furnished with a graduated circle to meas- FIG. 457. ure the deviation produced in passing through the prism, it is called a spectrometer, of which a simple form is shown in Fig. 45T. 606. Spectrum Analysis. It has been found that each element in the gaseous state emits a discontinuous spectrum, differing greatly in complexity, but always so char- acteristic that there is not even a remote resemblance of one to another. Since, also, it is found that at the temperature of gaseous incandescence the spec- trum of a compound consists of the sum of spectra of the constituent elements, the spectroscope furnishes a valuable aid to qualitative analysis. Since it re- quires but a very small quantity of a substance to yield its characteristic spectrum, a number of rare elements that had previously escaped detection have been discovered by the spectroscope. Among these may be mentioned caesium, rubidium, thallium, indium, gallium, scandium, ytterbium. FIG. 458. In the study of gases the electric discharge through SOLAR SPECTRUM. 707 an exhausted tube, such as that shown in Fig. 458, is usually em- ployed. 607. Solar Spectrum. The spectrum of sun- light was first observed by Wollaston, in 1802, to be crossed by a num- ber of dark lines. These lines were independ- ently discovered and carefully studied fifteen years later by Fraun- hofer, after whom they ^ are usually named the | Fraunhofer lines. The more prominent have been designated by the letters of the alpha- bet, as in Fig. 459. The corresponding wave-lengths in air are given in the table on p. 652. Fig. 460 is a copy of a photograph of a portion of the solar spectrum, nearly all of which lies beyond the accepted visual limit. The let- tering of the prominent 708 RADIATION AND ABSORPTION OF LIGHT-WAVES. lines has been extended from Fraunhofer's system. The obvious explanation of the Fraunhofer lines is that the principal light of the sun is emitted from an incandescent central mass (photosphere), and that the missing waves are absorbed either by the gaseous envelope about the sun (chromosphere) or in our own atmosphere. Nearly all the lines of the solar spectrum have been found to be identical with those of the emission spectra of known elements, from which it may be certainly concluded that these substances are present in the sun. Sodium, hydrogen, iron, calcium, H K I L \ 1 1 II llli li .lUI'HHliilii i In I i ! FIG. 460. lithium, aluminium, titanium, chromium, carbon, manganese, nickel, copper, magnesium, with perhaps thirty others, un- doubtedly exist. Antimony, arsenic, gold, boron, bismuth, mercury, have not as yet been identified, but it by no means follows that these substances are absent. On the other hand, it is of interest to note that an unknown bright yellow line in the chromosphere, which has been for many years provisionally assigned to a substance called helium, has recently been identified with the spectrum of a gas obtained from the rare mineral cleveite. A and B, in Fig. 459, are oxygen lines due to absorption in the earth's atmosphere. The double D line is produced by sodium, and the group b by magnesium. C', F, and h are due to hydrogen. The preceding principles may be applied to the stars, and their chemical nature inferred in the same manner. Many nebulae and a few stars emit bright-line spectra. DISPLACEMENT OF LINES. ABSORPTION. 709 608. Displacement of Lines. The principle which was explained in Art. 495 has several important applications in the case of light. It was shown in the article mentioned that when a source of waves was approaching an observer at 77 a velocity, u, the frequency was increased in the ratio If, now, u is quite small with respect to v, a condition always fulfilled in the case of light, the wave-length, since it varies inversely as the frequency, would be decreased in the same ratio ; that is, by the amount - X = Since the mole- v n cules of a luminous gas are in motion- with all velocities between certain limits, say -j- u and M, it follows that every line of the spectrum must possess a finite breadth, although it corresponds to a single frequency of vibration. The displacement of known lines in the spectrum has also furnished a means of estimating the proper motion of fixed stars in the line of sight, the speed of rotation of the sun about its axis, and the velocity of the gasos in a solar eruption. 609. Absorption. Body Color. The experiments de- scribed in Art. 603 show that when a body exerted selective absorption the color of the body was the result of the com- bination of those waves which were allowed to pass, so that a piece of yellow glass was yellow, not because it colored the light, but because it absorbed the blue waves. Likewise if waves penetrate a short distance into a sub- stance and are then reflected out, its color will be that of the light it transmits. The color so determined, which is that of most natural objects, is' known as body color. When, for instance, a piece of blue glass is placed in front of a piece of yellow glass, obviously that light will pass the combination which is transmitted by both, namely, green. This accounts 710 RADIATION AND ABSORPTION OF LIGHT-WAVES. for the fact that a mixture of blue and yellow pigments appears green, which is not at all the color of a mixture of yellow and blue light. Sometimes the amount of absorption for different waves varies with different thicknesses. In this case the color of a thick layer of the substance may differ much from that of a thin one. For instance, glass colored with a cobalt salt appears blue in a thin plate, but violet in a thick one. /Surface Color. Certain substances, notably such metals as gold and copper, and many of the aniline dyes, exert the power of selective reflection ; that is, they completely reflect certain waves in addition to the amount sent back in ordinary reflection, and transmit others. The light which is trans- mitted by thin films of such substances is complementary (Art. 613) to that which is reflected. 610. Anomalous Dispersion. The law found in Art. 574 showed that in a diffraction spectrum the deviation of the different wave-lengths was equal to an angle whose sine varied directly as the wave-length. By inclining the grat- ing to the direction of the incident light the deviation from the normal to the grating may be made so small that it may be taken as proportional to the wave-length. Such a spec- trum is called a normal spectrum. In the case of dispersion through a prism, however, the F G H se P arat i on f tne Fraunhofer lines in different parts of the spectrum is not alike for any two substances. B b b E F G H In. fact no substances have yet been FIG. 46i. found which will afford coinci- dences between more than two lines in each spectrum (Fig. 461). Hence complete achromatism cannot be produced by any combination of two lenses. ANOMALOUS DISPERSION. 711 In vacuous space the velocity of light appears to be inde- pendent of the wave-length. When, however, light enters ponderable media, since the waves are not indefinitely great compared to the elementary structure, it might be anticipated that the velocity of the waves and the period of free vibra- tion of the ether would be considerably modified, and prob- ably according to a complex law. In substances without selective absorption, i.e. those whose molecules do not have natural periods approaching that of the waves, the velocity is an increasing function of the wave-length or period of the light under consideration. A form of this function given by Cauchy, and found to agree moderately well with observa- tion throughout a considerable range, is where 619. The Young-Helmholtz Theory of Color Sensation. - The theory which up to the present affords the most satis- factory explanation of the phenomena of color sensation was proposed by Young in 1802, but having since been con- siderably extended by Helmholtz, it usually bears the name of its joint authors. Its fundamental assumption is that the nerve termini of the eye are of three distinct sorts, which when stimulated give rise to the sensations of red, green, and blue, respectively. The stimulation by different wave-lengths of light, of each of these sets of nerves, which may for convenience be styled the red nerves, the green nerves, and the blue nerves, is shown graphically in Fig. 467, where the ab- scissas determine the wave-lengths, and the ordinates of each curve represent the excitation of the nerve for that wave-length. The maximum for the first set is just below the orange, for the second set in the yellowish green, and for the third set just above the blue. In general, light of a single wave-length produces a simultaneous excitation of all three sets of nerves. Thus, light near the D line stimulates the blue nerves feebly, it ABNORMAL COLOR VISION. 723 the green nerves strongly, and the red nerves nearly as much, giving a resulting sensation, yellow. When all three sets are stimulated to the same degree, the impression received is white. Since no wave-length produces a sensation into which a single set of nerves enters alone, it follows that no color will appear perfectly saturated. It is further evident from the figure that the saturation of a mixture of red and violet, i.e. purple, is nearly as great as that of the spectral colors, and that green is the least saturated. When the intensity of the light is so low as to produce a very feeble excitation of the nerves of the eye, it is not pos- sible to distinguish the impression of one wave-length from that of another, and all appear alike gray. 620. Abnormal Color Vision. Strong evidence in favor of the Young-Helmholtz theory is to be found in the simple explanation which it affords of the phenomena of defective color vision. In certain persons the red perceptive nerves appear to be lacking. Consider, for instance (Fig. 468), the character of the sensation experienced by a red-blind person, i.e. one who entirely lacks the red per- ceptive fibers. The red waves produce a moderate stimulation of the green nerves, but the effect on the blue nerves is very feeble. These waves accordingly appear to such a person as a green of low luminosity and strong saturation. Passing up the spectrum, the green of maximum saturation and considerable intensity will be reached at #, about halfway between l^and G-. Beyond this point the intensity increases somewhat, but the color becomes paler till white is reached at w. 724 SENSATIONS OF COLOR. The hue of shorter waves is bluish, but the saturation increases with diminishing wave-length. Two waves, such as r and #, may appear to a red-blind person alike in hue and intensity, but a normal eye would recognize the shorter waves as having a lower intensity. The visual sensations of a green-blind person, that is, one lacking the perceptive nerves for green, may be predicted in a simple manner from Fig. 469. All waves from red to green pro- duce a sensation of red with a maximum intensity near the spectral orange, but of gradually diminishing saturation. At w the sensation is white or gray, beyond which the saturation and luminosity both increase. If a green-blind person suc- ceeds in distinguishing red and green, it is usually by the aid of the different intensities of the two sensations. The color sensations of the blue-blind are exhibited by Fig. 470. In this case there is evidently no confusion of red and green, but the spectrum will appear of low satura- tion throughout the first half, with a white band in the yellow. All waves beyond this produce a sen- sation of green of dimin- ishing intensity and increasing saturation. A feeble violet would hardly be distinguishable from black. Abnormal color vision of analogous peculiarity would obvi- ously be produced if one set of nerves, instead of being en- tirely lacking, was considerably enfeebled with respect to the others. All these defects are classed under the general head of color blindness. Its prevalence is found to be about 4 per cent in men and ^ per cent in women. DETECTION OF COLOR BLINDNESS. 725 621. Detection of Color Blindness. Holmgren's method of testing for color blindness is as follows : A large variety of colored worsteds, including red, orange, yellow, yellow-green, green, blue-green, blue, violet, purple, pink, brown, and gray, with several variations of luminosity and saturation in each hue, are placed in a pile upon a table. A sample skein of pale, pure green is laid at one side of the pile, and the sub- ject instructed to place with the sample all other skeins which at all resemble it, without reference to a match. If in this selection any reddish or bluish hues, or a gray, are classed with the green sample, the person is color-blind. In order to decide the nature of the defect, a sample magenta skein is shown, and a selection of resembling colors from the pile requested. If in this second test blues or violets are classed with the magenta, the person is red-blind ; if gray or greens are associated with the magenta, he is green-blind; if red or orange are placed alongside the magenta, he is blue- blind. 622. Insufficiency of Young-Helmholtz Theory. This theory of color sensation represents very satisfactorily the conditions of ordinary vision, and is therefore of great value and interest ; but it certainly fails in the case of feeble visual impressions, and must be in some way modified if complete- ness is to be hoped for. Probably the prevailing tendency to this end is the assumption of a fourth sensation, inde- pendent of color, which may be called the sensation of bright- ness, and which is supposed to respond to a stimulus much too feeble to awaken the sensation of color. With this addi- tion it is possible to conceive of complete color blindness without insensibility to any portion of the spectrum, which seems to be the condition of the extreme marginal portions of the retina. There are even some reasons for supposing 726 SENSATIONS OF COLOR. that the cones of the retina are alone concerned with sensa- tions which may be described in the language of the Young- Helmholtz theory, while the rods are the means for producing this sensation of brightness. EXAMPLES. 1. A gas flame placed at a distance of 96 cm. from the grease spot of a Bunsen photometer is found to give the same illumination as a standard candle at a distance of 31 cm. What is the candle-power of the flame? Ans. 9.6 C. P. 2. Two sources of light, of 2 and 32 candle-power respectively, are placed 180 centimeters apart. Where must a screen be placed on the line joining them in order that it shall receive equal illumination from each source? Ans. 36 cm., or 60 cm. from smaller light. CHAPTER XLV. POLARIZATION. 623. Polarization of Light. When light-waves exhibit different properties in different directions at right angles to the line of propagation, the light is said to be polarized. Since in a longitudinal wave it is impossible that there should be any difference between one side and another, it follows that light-waves must be of the transverse type. If the paths of all the points in a wave surface are alike, the light is perfectly polarized. When the paths are straight lines the polarization is called plane, but they may be ellipses or circles, in which cases the polarization is called elliptical or circular polarization, respectively, with the added term of right or left, if it is convenient to designate the direction in which the particle moves in the ellipse. In the case of plane polarized light the plane of polarization is denned as the one which is perpendicular to the direction of vibration. When a train of light-waves is reflected or refracted at the surface of a transparent medium, such as glass, it suffers, in general, a more or less complete polarization, as may be shown by reflecting from a second surface, when the intensity of the reflection will be found to depend upon the relative azimuths of the two planes of reflection. The explanation is that the vibrations of the light which was reflected from the first surface were reduced for the most part to a direction parallel to the surface, and those of the refracted light, similarly, were confined to a direction perpendicular to the plane of incidence, or, in other words, the reflected waves are polarized in the plane of incidence, 728 POLAEIZA TION. and the refracted waves at right angles to it. The amount of the polarization, which is the same in both the reflected and refracted trains, depends upon the angle of incidence, and reaches a maximum when the direction of the refracted ray is at right angles to the reflected ray. From this it follows that the angle of refraction is the complement of the angle of incidence, or calling the latter, the angle of polarization, or, (l) tan a = n. For glass this angle has a value of 57 or 58. Polarization, even at the polarizing angle, is not quite complete in isotropic substances except for those having a value of n = ~LA6. If a beam of polarized light be allowed to fall at the polar- izing angle upon a second surface of glass (Fig. 471 a) held PIG. 471. parallel to the first, the light will all be reflected and none refracted; but if the second glass be revolved about the direction of the incident light as an axis, through a right angle (Fig. 471 #), no light will be reflected, all the waves being refracted. The first mirror in this experiment may be termed the polarizer, and the second the analyzer. 624. Refraction in Non-Isotropic Media. When a disturb- ance enters a medium which possesses different properties in different directions, in general the original disturbance is DOUBLE REFRACTION OF LIGHT. 729 propagated as two systems which correspond, respectively, to the fastest and the slowest modes of vibration of the particles in the line along which the disturbance moves. A mechan- ical illustration of this principle may be seen in a rod of elliptical or oblong cross section. If any portion of this rod be displaced transversely in a direction not coincident with either of the principal axes of the cross section, and suddenly released, two waves will run along the rod, one in the plane containing the major axis of the section, and the other at right angles to it, and each traveling with a velocity peculiar to that mode of vibration. In a somewhat analogous manner, when a train of light- waves falls upon a crystalline substance, the disturbance within the medium is propagated as two sets of waves, polar- ized in planes at right angles to each other and giving rise to the phenomenon of double refraction, presently to be described in detail. That the transmitted light is polarized may be shown by placing two slices of tourmaline cut par- allel to the axis in a beam of light. If these slices are laid one upon the other in the same relative position, a certain amount of light will pass both sections; but if the slices are rotated till they cross at right angles (Fig. 472), the light which passed the first will be entirely FlG 472 stopped at the second. In this particular crystal one of the two refracted trains of waves mentioned above is so rapidly absorbed that it does not emerge from the first section. 625. Double Refraction of Light. The phenomenon of double refraction was discovered in 1669 by Bartholinus, 730 POL ARIZ A TION. who found that when a single incident beam fell upon a crystal of Iceland spar (calcite) it was divided into two beams, one of which obeyed the ordinary law of refraction, and the other a complex and hitherto unknown law. The two systems of waves are, accordingly, distinguished as the ordinary and extraordinary waves. Double refraction occurs in all homogeneous anisotropic media, but is most conveniently studied in crystallized minerals. In every doubly refracting crystal there is at least one, and in many two directions, called optic axes, in which bifur- cation of the ray does not occur. In accordance with this principle, all crystals, except those of the cubic system which are singly refracting, are classified in two systems as uni- axial or biaxial crystals. 626. Wave Surfaces in Uniaxial Crystals. The expla- nation of the phenomenon of double refraction in uniaxial crystals, given by Huyghens soon after its discovery, was that any disturbance in such a medium spread as two wave surfaces, which, determined by the principle now bearing his name, were for the ordinary waves a sphere and for the extra- ordinary waves a spheroid (Fig. 473). Each of these surfaces has an axis in common which is coincident in direction with the optic axis ; when -a train of plane waves falls upon such a crystal, the refracted waves;are also plane and may be found as follows. Let ^l#,(Figl!474) represent the surface of a uniaxial crystal and p C the direction of the optic axis, taken for simplicity in the plane of the paper. Also, let ppip 2 be the plane incident wave perpendicular to the plane of the paper. While the wave-front is traveling the distance p^p'z in the air, suppose the spherical disturbance originating at p has spread in the crystal to Co and that the spheroidal wave WAVE SURFACES IN UNI AXIAL CRYSTALS. 731 FIG. 473. has reached Ce. Likewise, in the same time, let the disturb- ance in the wave-front at p l have moved to p\ in the air and to o' and e' in the crystal. Then, by Huyghens's principle^ the new wave-fronts corresponding to pp 2 will be the envelopes of the elementary dis- turbances, which in this case are seen to be planes p' 2 o'o and p'^e'e, tangent, respec- tively, to the sphere at o, to the spheroid at e, and passing through a line at p\ perpendicular to the plane of incidence. Confining the FIG. 474. 732 POLARIZATION. attention to a small pencil of light incident in the direction ip, it is seen to be divided into two refracted pencils, the ordinary po, which is perpendicular to the wave-front in the plane of incidence, and obeying in every respect the ordinary law of refraction, and the extraordinary pencil pe, drawn from the point of tangency e of the plane through the per- pendicular at p' y pe is, in general, neither perpendicular to the wave-front p'ye'e, nor does it lie in the plane of inci- dence, though it happens to in the figure, because the optic axis is taken in the plane of incidence. In one particular case, however, when the plane of incidence is perpendicular to the optic axis (Fig. 475), the extraordinary ray is both perpendicular to its wave-front and in the plane of incidence. For this case the velocity of the extraor- dinary wave bears a constant ratio to the velocity in air for FIG. 475. / all angles of incidence, its value being known as the extraordinary index of refraction. Thus, denoting Pip' v the velocity in air, by , the velocity po of the ordinary wave by v , and the velocity pe of the extraordinary wave by v e , the two indices of refraction are Those crystals in which n e > n are called positive, and those in which n e < n are called negative. In the former, of which ice and quartz are examples, the ellipsoid lies within the PILE OF PLATES. 733 sphere ; in the latter, as for example in Iceland spar and tour- maline, the ellipsoid lies without the sphere (see Fig. 473). The ordinar}' ray is polarized in a plane passing through the optic axis, and perpendicular to the refracting surface, and technically known as a principal plane ; the extraordinary ray is polarized in a plane at right angles to this. 627. Pile of Plates. Since the ratio of the velocities of light at emergence is the reciprocal of that at entrance, it follows from Brewster's Law, equation 1, that the angle of polarization for incidence in the denser medium is the com- plement of that in the rarer. Hence, if light fall on a plate of glass at the polarizing angle, the refracted portion will strike the second surface also at the polarizing angle, and the portion reflected at this surface will be polarized. Like- wise, if there be a series of plates parallel to the first, the transmitted ordinary light will meet each successive surface at the polarizing angle, and the reflected portions will all be po- lauized in the same plane. This arrangement, known as a "pile of plates," forms quite an effi- cient and inexpensive polarizer. To secure satisfactory results, it is necessary to employ at least a dozen layers of thin plate glass. C r"^ c f FlG. 476. 628. Nicol's Prism. The most effective means of pro- ducing a beam of polarized light is by double refraction. In order to secure one train of waves free from the other, Nicol devised the prism shown in Fig. 476. An elongated 734 POLARIZATION. rhomb of calc-spar is cut by a plane passing through the obtuse angles A, A' and parallel to the diagonal DB. The faces of the section are then polished and cemented together again by means of Canada balsam. Now, this substance has an index of refraction less than that of the ordinary wave and greater than that of the ex- traordinary, so that the extraordinary wave will be trans- mitted without sensible change, while the ordinary wave is totally reflected and hence removed from the field. 629. Foucault's Prism. Foucault's prism (Fig. 477) dif- fers from Nicol's in the substitution of an air film in place of the layer of Canada balsam. Since the critical angle for the ordinary ray in Iceland spar is about 37 14', and for the extra- ordinary ray 42 23', it is evident that the angle of incidence on the section must be intermediate between these values in order that the extraordinary ray shall be transmitted, and the ordinary totally reflected. This construction is not only more easily made, but also requires less than two-thirds as much material as is necessary for a Nicol's prism, a matter of considerable importance in the present scarcity of Iceland' spar. There is, however, a considerable loss of light by reflection from the air film and the angular field is much smaller. 630. Hartnack's Prism. The usefulness of the ordinary Nicol's prism in connection with a lens system is consider- ably impaired by the loss of light reflected from the oblique ends, by the influence of imperfections in these surfaces upon the definition on account of their inclined position, -and by DOUBLE IMAGE PRISMS. 735 the small angular limits of the field, which are determined by the indices of refraction and the position of the section plane. Fig. 478 shows an improved form of polarizing prism devised by Hartnack and Prazmoski, in which they have remedied the first two defects by cutting the ends of the prism PQ and RS so that the light A a Q enters and leaves them normally. By giving the section plane PS a position perpendicular to the optic axis, and cementing the two halves with linseed oil, which has an index of refraction identical with the extra- ordinary index of Iceland spar, the field is increased from less than 20 to 35. The size of the rhomb necessary to cut such a prism is shown at ABCD, while the size necessary for a Nicol of the same thickness is shown at abed. d D FIG. 478. 631. Double Image Prisms. By using a prism of a double refracting substance, two images of a bright source of light may be obtained with the transmitted beams polarized in planes at right angles. .0 When a single prism is em- ployed, it is cut so that the refracting edge is parallel to the optic axis, because the FIG. 479. difference of deviation be- tween the ordinary and the extraordinary rays is then greatest. The dispersion may be corrected by a reversed prism of glass placed in front. Rochon varied this construction by making the front prism 736 POLARIZATION. FIG. of the same material as the back one, but with the face AB (Fig. 479) perpendicular to the optic axis. Light incL dent on the face AB perpendicu- larly is not modified until it reaches AC, where double refraction oc- curs; the ordinary ray continues without deviation, since the index of refraction is the same in both parts, while the extraordinary ray is bent toward the base or the edge of ADC, according as the crystal is positive or negative. A greater deviation may be secured by the construction shown in Fig. 480, due to Wollaston, in which the face of the front prism is cut parallel to the optic axis, with the edge A perpendicular to it. In this case, although there is no de- viation in the first prism, the waves are divided into two sys- tems, with vibrations respectively parallel and perpendicular to AB, which traverse the crystal with different velocities. On entering the A CD, the planes of the vibra- tions do not change, but that system which before had the greater velocity will now have the less, and vice versa. Thus, each system will be deviated by nearly equal amounts, but in opposite directions. 632. Wave Surface in Biaxial Crystals. FIG. 48i. In biaxial crystals CONICAL REFRACTION. 737 neither of the refracted rays, in general, obeys the ordinary law of refraction, but there are two directions in which both refracted rays coincide. The phenomena were first ex- plained by Fresnel on the assumption that a disturbance in such a crystal was propagated as a wave surface of two sheets whose form might be given by an equation of the fourth degree, a portion of which is shown in Fig. 481. The intersections of this surface with the co-ordinate planes are shown in Fig. 482. aa', 65', and cc' are arcs of circles, and b f c, c f a, and a'b are arcs of ellipses. OM, the perpendicu- lar to the common tangent MN, and a line symmetrically placed on the left side of OZ are the optic axes, for in this direction of propagation only one plane wave-front exists. The position of the two re- fracted plane waves resulting from an incident plane wave may be found by the same con- struction as that employed in Fig. 474, namely, by drawing through the trace of the incident plane two planes tangent to Fresnel's wave surface. The lines drawn from the center of the surface to the points of tangency give the direction of the refracted rays. 633. Conical Refraction. It was pointed out by Sir Wil- liam Hamilton that if the refracted plane wave should have such a position, MN (Fig. 482), that it was tangent to both sheets of the wave surface, this plane would touch the sur- face in a circle, and any line drawn from to this circle of contact would be a possible direction of the refracted ray. FIG. 482. 738 POLARIZATION. The truth of this prediction was tested by Lloyd in the fol- lowing experiment. A piece of aragonite, CD (Fig. 483), was cut with its faces perpendicular to the bisector of the optic axes, and a sheet of metal foil pierced with a small hole, 0, attached to D the upper face. A pencil of light was admitted through a hole, /S Y , in the screen AB, and received on FlG 483 a second screen, FGr. On changing the position of the crystal slowly, until the proper angle of incidence was secured, the two images of S suddenly spread out into a ring, showing that the path of the light through the crystal was the surface of the cone OMN. This case is distinguished as internal conical refraction. A second peculiarity, likewise suggested by Hamilton and verified by Lloyd, relates to the direction OP (Fig. 482). By the construction of Art. 626 this would be the direction of the ray corresponding to any one of the infinite number of plane waves which could be drawn tangent to the surface at P. Thus, if light were to fall upon the crystal in the direction of any one of the elements of a certain cone, it would be doubly refracted at the surface, so that one of the rays would in every case coincide with the singular FJG 4g4 direction OP. On emerging from a second surface parallel to the first, these rays should again be refracted so as sensibly to regain their original INTERFERENCE OF POLARIZED LIGHT. 739 direction, forming a conical shell. To test this, a beam of light, converging at the proper angle, was allowed to fall on the piece of aragonite at (Fig. 484), and emerge through a small aperture, P, in a piece of tin foil, which cut off all rays except those transmitted along OP. The rela- tive positions of and P were slowly changed, and when the adjustment was complete, a bright annulus of light was seen on looking into the aperture at P. This phenomenon is known as external conical refraction. 634. Interference of Polarized Light. The experiments of Fresnel and Arago show that in order that two beams of polarized light shall interfere in the same manner as ordinary light, it is necessary: 1, that the beams of light shall be polarized in the same plane ; and, 2, that they have a com- mon origin. If, when the analyzer is set so as to extinguish a beam, a thin section of a doubly refracting crystal is interposed between the analyzer and the polarizer, the light will be restored and also colored, the hue depending on the thick- ness of the crystal. By turning the plate in its own plane, two positions will be found in which the light is extinguished, and two for which it is a maximum. If, on the other hand, the analyzer be revolved, the saturation will dimmish till white is reached, when the color changes to the comple- mentary hue and the saturation increases. If the analyzer be replaced by a double image prism, the two fields will have complementary colors, except at the spot where they overlap, which will be white. The explanation is as follows: Let OP (Fig. 485) repre- sent the direction and amplitude of vibration of the incident plane polarized light ; $$, the principal plane of the thin crys- tal section ; and A A, the principal plane of a doubly refract- 740 POLAR1ZA TWN. ing crystal, used as an analyzer. On entering the crystal section the plane polarized light is divided into two trains, which traverse the plate with different velocities, the ordi- nary component having an amplitude and direction of vibra- tion, CO, and the extraordinary an amplitude and direction, CE. After passing the analyzer, both these component vibrations are subdivided into pairs, Co, Ce, and Co', Ce', which have been differently retarded and are in condition to interfere. The in- tensity of the ordinary and ex- traordinary image will, accord- ingly, be different for different wave-lengths ; and if the incident light is white, the images will appear colored, but of comple- mentary hues, since their sum should give the original set of waves, provided none are lost by absorption If the preceding experiment be varied by passing the light through two thin crystalline plates before it is received in the analyzer, the images will appear complementary as before, but the hues will change with the rotation of the analyzer, or of either plate. FIG. 485. 635. Double Refraction in Strained Trans- parent Substances. Any unequal alteration in the elasticity of a homogeneous transparent solid by a stress produces a double refraction, which is easily recognized when the body is ex- amined by polarized light. Thus, for instance, if a block of glass, held by a clamp, as in Fig. 486, be placed before the FIG. 486. EINGS AND CROSS. 741 analyzer and examined by polarized light, it will be found that a slight pressure by the screw on the block brings out an elaborate colored pattern. Tempered glass, that is, glass which has been heated and suddenly cooled, likewise ex- hibits the presence of unequal stresses when examined by the polariscope. In thick pieces of glass, such as are used for lenses of considerable size, it is difficult to anneal the blocks so that they will not show lack of uniformity by this sensitive test. Dr. Kerr has established the fact that fluid as well as solid dielectrics, when subjected to electrostatic stresses, are modified so as to become double refracting. FIG. 487. 636. Rings and Cross. If a plate cut from a uniaxial crystal be placed before the analyzer, and observed in convergent plane po- larized light, a series of brilliantly colored rings interrupted by a rectangular bright or dark cross (Fig. 487) may be observed. Let MN (Fig. 488) represent the crystal section, and sup- pose that light diverging from the point falls on the plate in any direction, OX. Also, suppose that PP is the plane of vibration of the light transmitted by the polarizer, and AA that by the analyzer. When the light meets the surface at X it will be doubly refracted, the vibrations in one of the rays taking place at right angles to OX, since this line is in a principal plane of the crystal and the other in the plane of OX. When the point considered is at X 1 or X", the crystal section evidently transmits only those vibrations which are parallel to PP, and these are completely extinguished by 742 POLARIZATION. the analyzer. Hence the regions PP and AA will appear dark. At any other points the crystal will exhibit the colors of thin plates, but arranged in rings about the axis, since the thickness of the plate traversed varies with the angle of in- M N FIG. 488. cidence. On rotating the analyzer 90 the cross appears white, and the colors change to their complementary value. The analogous but more complicated phenomena presented by biaxial crystals are shown in Fig. 489. The succession FIG. 489. of forms illustrates the changes which occur as the crystal section is rotated 45 between the crossed polarizer and analyzer. The appearance of these figures is so varied and characteristic as to furnish, in practically all cases, a satis- factory means of identifying any doubly refracting crystal. CIRCULAR POLARIZATION. 743 637. Circular Polarization. Although, as has been pointed out, two beams of light polarized in planes at right angles cannot produce destructive interference, they may neverthe- less combine so as to produce a vibration of a special charac- ter. The vibratory form resulting from such composition, when the periods are the same, has been shown in Art. 474 to be an ellipse inscribed in a rectangle whose sides are the component vibrations. Plane polarized light which has been passed through a doubly refracting plate and has acquired this special form of vibration is said to be elliptically polar- ized. Polarized light which has suffered reflection from a metal surface is also, in general, elliptically polarized. When the difference of phase between the rectangular vibrations is just a quarter period and the amplitudes are the same, the vibrational form reduces to a circle. Circu- larly polarized light may be produced by passing plane polar- ized light through a thin sheet of mica, so chosen that the retardation of one train of waves over the other is just a quarter wave-length. Such a crystal section is known as a quarter-wave plate. The light which has passed through it will appear equally bright in all positions of the analyzer. When a quarter-wave plate, used to produce circularly polarized light, is rotated 90 in its own plane, the light is still circularly polarized, but the sense of vibration in the circle will be reversed, i.e. right-hand circular polarization will be altered to left-hand, and the contrary. 638. Rotary Polarization. Quartz is a uniaxial crystal in which the extraordinary wave surface lies completely within the ordinary wave surface, and the vibrations are, in general, elliptically polarized in opposite directions. In con- sequence, the speed of .the two waves along the axis of the 744 POLARIZATION. M crystal is different with opposite circular polarizations. It is easy to see that the resultant of two opposite circular vibrations of the same period is a simple harmonic vibration. Thus, if p,p' (Fig. 490) be two points moving with the same period in a circle, and symmetrically placed with respect to the line MN, then by resolving each circular motion into simple harmonic motions parallel and perpendicular to this line, it is seen that the perpendicular components annul each other, leaving a rectilinear vibration in the line MN. In an analogous manner the oppositely polarized beams through the quartz produce plane polarized light on emergence, but because one train of waves traveled faster than the other the plane of vibration of the emer- gent light will have been rotated through a certain angle proportional to the thick- ness of the quartz plate and depending on the wave-length of the light. If the in- cident light is white, the emergent light after passing the analyzer will exhibit colors depending on the position of its plane of vibration. Certain specimens of quartz and other crystals produce a rotation of the plane of polarization to the right and others to the left. The rotation, for different wave-lengths, produced by a plate of quartz one millimeter thick, cut perpendicular to the axis, and having a temperature of 20 C., is shown in the following table. o 12.G7 15.75 17.32 21.68 21.73 27.54 32.77 42.60 51.19 EOTAEY POWER OF LIQUIDS. 745 The rotations in yellow light for several other crystals are approximately as follows. SUBSTANCE. FORMULA. a FOR l mm THICK- NESS. HgS 32.5 Sodium Chlorate .... NaCLO 3 3.5 Sodium Bromate .... NaBrO, 2.8 Hyposulphate of Potash . . K 2 S 2 6 8.4 " " Calcium CaS 2 O 6 + 4H 2 O 2.1 " Lead . . PbS 2 6 + 4H 2 5.5 " " Strontium SrS 2 6 + 4H 2 1.6 639. Rotary Power of Liquids. Many liquids and even vapors possess a power of rotating the plane of polarization similar to that of crystals, but in a far less degree. In solids this power depends on the structure, and is lost when the body is fused or dissolved. In fluids, however, the power appears to be inherent in the molecule. The rotation produced bj- any active liquid increases directly as the thick- ness of the layer through which the light passes, and also varies with the wave-length of light used and with the tem- perature. The rotary powers of liquids are usually stated in what is known as specific rotation, i.e. the rotation in degrees, per decimeter, per unit, density. If a be the rotation of the plane of polarization in degrees, I the thickness of the layer of the fluid in decimeters, and p its density, the specific rotation [<*] may be written When an active liquid is dissolved in an inactive one, the rotary power is also a function of the concentration of the 746 POLARIZATION. solution and of the nature of the solvent. If p denote the percentage composition of the solution, i.e. the number of grams of the solute in 100 grams of the solution, the spe- cific rotation of a solution may be defined by l-p-p l-c where c denotes the concentration or the number of grams of the solute in 100 cc. of the solution. A knowledge of the rotary properties of substances fur- nishes a valuable method of investigating either the character or the concentration of a solution. This method of analysis is extensively used in the determination of sugars, whence it is commonly known as saccharimetry. The specific rotations of a few sugars for sodium light are exhibited in the following table, rotation to the right being denoted by the plus, and rotation to the left by the minus sign. LIMITS OF PER- NAME. TEMP. C CENTAGE [<*]/>- COMPOSITION,^. Cane Sugar . . . 15 10 to 50 + 66.94 - 0.012 p Lactose .... 20 Oto 36 + 52.53 constant Maltose .... 20 10 + 136.75 to 136.96 Glucose .... 20 to 100 + 47.73 + 0.0155;? Levulose .... 20 2 to 30 _ 91.90- 0.111 p 640. Saccharimetry. - An analyzer, such as a Nicol's prism, in which the position of the plane of polarization is judged by a maximum or minimum intensity of the field, is not sufficiently precise for quantitative measures in rotary polarization. For this end it is found that better results SAC CHA RIME TR Y. 747 may be obtained by modifying the apparatus so that the field appears double, with a different illumination in each portion, except at the critical position. The latter may be judged N D P N FIG. 491. with considerable precision by the contrast in the two parts of the field. Laurent's Saccharimeter. The arrangement of parts in Laurent's saccharimeter is shown in Fig. 491. PNis a prism producing a beam of plane polarized light, and S a tube with glass ends, containing a solution whose rotary power is to be examined. AN is a Nicol analyzer, FIG. 492. whose angular position is accurately indicated by the gradu- ated circle and vernier, V, shown in Fig. 492, which repre- sents the complete instrument. The distinguishing feature 748 POLARIZATION. of this apparatus is a disc, D (Fig. 491), one-half of which is a plane glass plate, APE (Fig. 493), and the other half a quartz plate, A QB, cut parallel to the axis and of such thick- ness that one train of waves is retarded a half wave-length over the other. Suppose that AB is the direction of the optic axis of the quartz, and that PO is the plane of vibration of the incident light. This light will be transmitted by the glass plate without change, but it will be doubly refracted by the quartz, the component vibrations having directions re- spectively parallel and perpendicular to the axis. On emergence these components will re-combine to produce plane polarized light with its plane of vibration, OQ, rotated through an angle, 2POB, since the re- tardation of one component was just sufficient to reverse its direction. If the light be now examined by the analyzer AN (Fig. 491), the parts of the field will appear unequally illuminated, except when its principal plane bisects the angle between OP and OQ. The analyzer having been set so that both halves are equally bright, the substance to be examined is interposed at 8. The angle through which the analyzer must be turned to produce equal illumination in both portions of the field is the angle through which the plane of polarization has been rotated. As this rotation would be different for different wave-lengths, the light used is either that emitted by a sodium- tinged flame, or the yellow obtained by sifting white light throtigh a plate of bichromate of potash. SoleiVs Saccharimeter. Fig. 494 shows a type of saccha- rimeter invented by Soleil, which works upon a somewhat different principle from the preceding. The disc at B in SA C CHA RIME TR Y. 749 this case consists of two semicircular quartz plates of oppo- site sign and of same thickness. There will, in general, be some wave-length for which the rotation of the plane of polari- zation has been a right angle in each half of the field, so that its vibrations are now parallel to a certain line, say DD f . White light, transmitted first through the Nicol P and then through the biquartz B, will thus, in general, appear FIG. 494 colored, but of a different hue in each half of the field when examined through the analyzing Nicol A. If, however, the principal plane of the analyzer is perpendicular to DD f both parts of the field will have the same hue, because all the other wave-lengths are present in the same proportion in each half. By a proper choice of thickness of the quartz, this common color may be made any one desired. The hue usually selected is a violet, produced by the extinction of the 750 POLARIZATION. yellow waves in white light, and sometimes called the sensitive tint, since by a minute rotation of the analyzer in either direc- tion one-half the field becomes blue and the other half red. If the tube containing the solution to be examined be in- troduced between B and ft when the field is of uniform hue, the consequent rotation of the plane of polarization will pro- duce a marked disparity in the color of the two sections of the field. In order to determine the amount of this rotation, a com- pensator is provided at E, which consists of a right-handed quartz plate, ft and two left-handed quartz wedges, abc and ado, arranged to slide one over the other, so as to vary their combined thickness by a measured amount. By a proper adjustment of the wedges it is clearly possible to produce such a right- or left-handed rotation of the plane of polari- zation as exactly to neutralize that produced by the solution. The value of a scale division on the compensator may be reduced to angular measure by an observation made on some substance having a known rotation. The German instruments of this type are graduated so that one scale division corresponds to a rotation of the plane of polarization through 0.346. For commercial tests, 26.0 gms. of cane sugar in 100 cc. of aqueous solution is termed, a normal solution. It is assumed that if the sugar is pure a column of this solution 20 cm. long would produce a rota- tion, as read on the compensator scale, of 100 divisions. If the sugar is impure, the reading of the scale will give the percentage of saccharose present in the sugar. In order to adapt the instrument for use with a colored liquid, a regulator, consisting of a Nicol N' and a quartz plate, ft, is provided, so that the hue of the light entering the polarizer N may be altered at will. Thus, by selecting a hue complementary to that of the light transmitted by the substance, the so-called sensitive tint may always be obtained. MAGNETO-OPTIC ROTATION. 751 641. Magneto-Optic Rotation. In 1845 Faraday discov- ered that a dense boro-silicate of lead, when placed in a strong magnetic field, acquired the ability to rotate the plane of polarization of light transmitted in the direction of the field. When the light was propagated in the same direction as the lines of force, the rotation was positive, as determined by the familiar right-handed screw rule; but when passed in the opposite direction, the rotation was negative. In a direction perpendicular to the field, the plane of polarization was unaltered. Later experiments indicate that probably all substances are similarly modified by the magnetic forces. It will be observed that there is this difference between magneto-optic rotation and that of crystals ; namely, that if the light, having traversed a section of quartz in one direc- tion, be reflected and made to traverse the plate in the oppo- site direction, the rotation will be undone, while in the case of the magnetic field the rotation is doubled. 642. Nature of Ordinary Light. Common light does not exhibit any evidence of polarization. Accordingly, it must be assumed that the character of its vibrational form is con- tinually changing, so that in a brief period these vibrations are performed in all azimuths per- haps in a manner resembling A or B (Fig. 495). At least it is known that if the plane of polarization of light be revolved with a period less than the duration of visual impression, about a tenth of a second, all trace of polarization disappears. The phenomenon of interference bands, of which several thousands nfay be counted in homogeneous light, shows that the vibra- tions of light must possess a certain degree of uniformity; 752 POL A RIZA TION. but since about fifty millions of vibrations are performed in a tenth of a second, it is obvious that there is room for great diversity which might entirely escape detection in the aver- age impression made on the eye. Accordingly, if this ir- regularity be present in two streams of common light from different sources, or from different parts of the same source, there is no reason why the streams should interfere. MISCELLANEOUS EXAMPLES. 1. A battery of 20 cells in series has been improperly set up. On introducing two similar cells, so that they reinforce or oppose the bat- tery, the current in the circuit is found to be in the ratio of 4 to 3. How many cells of the battery were reversed ? Ans. 3. 2. A piece of zinc weighing 48.3 gms., at a temperature of 10.7, was immersed in a current of steam at 100 and found to condense 0.762 gm. of steam. What was the specific heat of the zinc ? Ans. 0.0947. 3. At what temperature will the pitch of a pipe, giving 124 vibra- tions per second at 10, be raised a Fifth ? Ans. 364 C. 4. A stone dropped from a cliff is heard to strike the ground 7.3 seconds later. Assuming the velocity of sound to be 1150 feet per sec., what is the hight of the cliff ? Ans. 724 feet. 5. How many beats per second would be heard if /'# were simulta- neously sounded on the natural and on the even tempered scale ? Ans. 1.6. 6. Notes of 225 and 336 vibrations per second, in which the first and second upper partials are present, are simultaneously sounded. What beats will be heard and whence do they arise? Ans. 3 per sec. 7. In an experiment with Kundt's tube the length of the rubbed glass rod was 900 cm., and the nodal points were found at the marks 0, 63, 127, 188, 252, on a scale divided to centimeters. What was the velocity of sound in glass ? _ met. Ans. 4880 sec. INDEX. The numbers refer to pages. Abbe, apochromatic objective, 676. Aberration, of light, 602 ; spherical, 671, 676 ; chromatic, 636, 662. Abnormal color vision, 723. Abridged method of calculation, 3. Absolute system of units, 13 ; tem- perature, 183 ; zero, 183, 275. zero, correction of, on air ther- mometer, 276. Absorption, of heat waves, 253; selective, 703. Acceleration, 16; of weight, 18. Accommodation, 629. Achromatism, 663. Addition of waves, 517. Adiabatic expansion, 265. Aeriform condition, 101, 102. Air, columns, vibrating, 555 ; pump, 129; thermometer, 164. Airy, rainbow, 700. Alternating currents, 480. Alternator, 475, 479. Amagat, experiments on gases, 222. Amid, hemispherical front lens, 671. Amorphous, 103. Ampere, theory of magnetism, 408. Ampere, the, 395, 409, 460; legal, 401. Ampere-meter, 421. Amplitude of vibration, 59. Analyzer, 728. Andrews, experiments, 218, 286, 295. Angle, 15 ; of contact, 142. Angstrom unit, 652. Angular velocity, 16. Anisotropic media, refraction in, 729. Anode, 388. Anomalous dispersion, 710. Antinode, 522, 555. Aperture, 670. Arago, rotation, 472. Arc, discharge, 503; electric, 440; lamp, 440. Archimedes, principle of, 117. Area, dimensions and unit, 14. Areometer, 120. Armature, 474 ; disc, 475, 479; drum, 475 ; ring, 475, 476 ; of an electro- magnet, 413. Astatic system, 418. Astigmatism, 630. Athermous, 253. Atmosphere, optical phenomena of, 680 ; unit of pressure, 122. Atmospheric electricity, 353. Atom, 105. Atwood's machine, 28. Audibility, limits of, 548. Auricle, 585. Available energy, 287. Avogadro, 297. 754 INDEX. B = magnetic induction, 375, 454. /3, the ratio of the total energy of the gaseous molecule to its kinetic energy, 303. Bdbinet, principle of, 691. Bailie, spark length, 496. Baily, rotary field, 484. Balance, 12; current, 422. Balance wheel, compensated, 184. Ball, curved pitching, 135 ; supported on jet, 136. Ballistic galvanometer, 420 ; pendu- lum, 113. Barometer, aneroid, 124 ; as weather glass, 125 ; Fortin's, 123 ; measure- ment of altitude, 125 ; siphon, 122. Barrett, recalescence, 380. Bars, vibration of, 565. Bartholinus, double refraction, 729. Basilar membrane, 587. Bassoon, 595; mouthpiece, 559. Beats, 519, 521, 552; in bells, 570; of upper partials, 576. Bell, 570. Bell, Graham, telephone, 492. Bernoulli, gaseous pressure, 290; laws, 556. Biaxial crystals, wave surface in, 736. BioVs hypothesis, 367. Black, 716. Black, latent heat, 191, 260. Body color, 709. Boiling point, table, 208. Bolometer, 435. Boltzmann, molecular constants, 307 ; on Stefan's law, 251. Bottomley, experiment, 205. Bourdon, pressure gauge, 126 ; as thermometer, 170. Bowed strings, 581. Boyle, law of, 102, 180, 222, 293, 294 ; correction of, 222. Boys, gravitation, 46 ; radio-microm- eter, 448. Bradley, velocity of light, 602. Brake, observations on Mars, 44. Brassy sounds, 585. Breguet, thermometer, 170. Brewster, 728, 733. Brightness, sensation of, 725. Brittle, 100. Brush discharge, 349. Bunsen, cell, 392; ice calorimeter, 192; melting point of paraffine, 204; photometer, 719. C = electrostatic capacity, 339. C. G. S. system, 13. Cable, submarine, 490. Cagniard de la Tour, continuity of state, 218 ; siren, 542. Caliper, micrometer, 11; vernier, 10. Caloric, theory, 259. Calorie, 187. - Calorimeter, ice, 191; Bunsen's, 192; steam, 194; water, 189. Calorimetry by the method of cool- ing, 252. Camera obscura, 627. Camphor gum on water, 157. Capacity, electrostatic, 339, 451, 455, 458, 460; of a sphere, 343; of a spherical condenser, 343 ; of a par- allel plate condenser, 344; ther- mal, 187. Capillary, electrometer, 338; phe- nomena, 144. Carbon, resistance of, 427. Carlisle, electrolysis, 397. Carnot, cycle, 267 ; theorem, 270. Carre, freezing machine, 212. Carriers, 333. Catenoid, 150. Cauchy, dispersion, 711. INDEX. 755 Cavendish, discovery of Ohm's law, 395; law of electrical force, 317; specific inductive capacity, 340. Cells, grouping of, 431. Center, of gravity, 84 ff. ; of mass, 84; of parallel forces, 82. Centigrade scale, 161. Centimeter defined, 8. Central force, 32 ff. Characteristic curves, 478. Charge, 310, 313. Charged conductor, energy of, 353. Charles, law of, 180. Chimes, 567. Chladni, figures, 568. Chlorophyll, 288, 703. Chromatic, aberration, 662 ; polar- ization, 739 ; scale, 545. Chronograph, 7. Chronometer, 6. Circuit, field due to, 408. Circular, aperture, diffraction through, 643; polarization, 743. Clarinet, 559, 595. Clark cell, 393. Clausius, electrolysis, 398 ; entropy, 286 ; virial, 293. Clock, 6 ; fork, 574. Closed pipes, 556. Coefficient, of dispersion, 663; of fric- tion, 77 ; of elasticity, 106 ; tables, 110; of expansion, 172; of self- induction, 466. Coercive force, 383. Coil, circular, in a uniform field, 410. Colloids, 233. Color, blindness, 723; detection of, 725; diagram, 719; sensation, 714; characteristics of color sensation, 716 ; of sky, 680 ; of thin plates, 655 ; non-spectral, 721. Combination tones, 579, 580. Comma, 545. Commutator, 471. Compass, mariner's, 369. Compensated, balance wheel, 184; pendulum, 183. Complementary colors, 715, 721. Composition of vibrations, 529. Compound, pendulum, 93 ; tones, 576 ; winding, 478. Compressibility of aeriform bodies, 222. Compressing a fluid, work done in, 266. Compressural wave, 515; velocity of, 526. Concave mirror, 613. Concert pitch, 573. Conduction, electric, 310; of heat, 244; in crystals, 249. Conductivity, electrical, 425; ther- mal, of gases, 250 ; of liquids, 249; tables, 250; of solids, 245; tables, 247. Conductor in a magnetic field, 410. Conductors, thermal, good and bad, 246. Conical, pendulum, 36 ; refraction, 737. Connectors, 333. Conservation, of electricity, 313 ; of energy, 72. Conservatoire, Paris, 573. Consonant intervals, 576. Contact, difference of potential, 385, 386. Continuity, of state, 218 ; below the critical temperature, 221. Contraction of films, 140. Convection, 244. Convex mirror, 614. Cooke, telegraph, 486. Copernicus, 44. Cord, waves in a, 523. Cornet, 598. 756 INDEX. Coronas, 690. Corti, rods of, 587, 589. Coulomb, law of electrical force, 317 ; law of magnetic action, 359. Coulomb, the, 395, 460. Couple, 81. Cream separator, 41. Crest of a wave, 514. Critical, angle, 615 ; state, 220, 221 ; temperature of magnetization, 379. Crookes, dark space, 499; tube, 501, 502 ; radiometer, 255. Cryophorus, 211. Crystalline, lens, 628 ; structure, 103. Crystalloids, 231. Crystals, 104. Cuneus, 342. Cupped mouthpiece, 560. Current, electric, 394, 452, 458, 460 ; electromagnetic unit, 409; field produced by, 406, 407; strength, 454. Curvature, 142. Cycloid, 533. Cyclones, 50. Cymbal, 569. Dalibard, lightning rod, 354. Dalton, law of pressures, 226, 227. Daniell cell, 390. D'Arsonval, galvanometer, 419 ; ra- dio-micrometer, 448. Davy, conservation of energy, 73 ; electric arc, 440 ; safety lamp, 248; test of caloric theory, 261. Declination, magnetic, 363 ; tables, 364. Density, 14; determination of, 119; flask, 119 ; tables, 121 ; of water, 179. Derived units, 13, 14. Descartes, rainbows, 699. Deviation by a prism, 617. De Fries, Osmotic pressure, 235. Dew, 256 ; point, 240. Dialysis, 232. Diamagnetic, 372 ; substances, 383. Diathermous, 253. Dielectric, 312; constant, 315, 340; tables, 341 ; body in a field, 326. Difference tones, 579, 580. Diffraction, 642 ff. ; patterns, 657. Diffusion, of gases, 232, 301 ; of liquids, 231 ; coefficient of, 232. Diffusivity, 247. Dimensions, 14, 450 ff. ; table for electrical and magnetic quantities, 456. Dines, hygrometer, 241. Dip, magnetic, 363, 367 ; tables, 364. Diplex telegraphy, 488. Discharge, affected by a magnet, 499 ; brush, 349 ; disruptive, 347 ; from points, 348; in a rare gas, 499 ; of a conductor in a gas, 495 ; oscillatory, 350; theory of elec- tric, 503. Discomfort due to beats, 578. Dispersion, 659; anomalous, 710; coefficient of, 663. Dispersive powers, 662. Displacement, direction of in polar- ized light, 512. Dissonant intervals, 576, 577. Divided circuit, 429. Dolland, achromatism, 663. Donders, osmotic pressure, 236. Doppler's principle, 551 ; in light, 709. Double, bass, 594; image prism, 735 ; refraction, 729 ; in strained substances, 729. Drum, 569; armature, 475, 477; of the ear, 586. Drying machines, 40. Ductile, 100. INDEX. 757 , superheated water, 209. Dulong and Petit, coefficient of ex- pansion of mercury, 175; law of, 198, 303 ; rate of cooling, 251. Duplex telegraphy, 487. Duration, of aural impression, 590 ; of spark, 348. Dutrochet, osmotic pressure, 234. Dynamo, 474, 477, 478 ; motor, 485. e electro-chemical equivalent, 398. E. M. F., electromotive force, 391, 455. Ear, anatomy of, 585; functions of parts, 588. Earth's rotation, effect on weight, 47 ; on projectile, 49. Ebullition, 207. Efficiency, of a machine, 77 ; of an engine, 270 ; expression for, 273. Elasticity, 99 ; volume, 106 ; shear- ing, 107 ; two elasticities of a gas/ 279. Electric, attractions and repulsions, 309, 310, 325, 327, 328 ; discharge, 347 ; field, 319 ; representation of, 320 ; light, 439 ; waves, 506. Electrical machines, 330. Electricity, 309 ; quantity of, 313 ; conservation of, 313 ; not energy, 352. Electrification, 309 ; nature of a separation, 314 ; of the atmos- phere, 355. Electro-chemical equivalent, 398. Eleetrodeless discharge, 497. Electro-dynamometer, 422. Electrolysis, 396-398. Electrolyte, 310, 396. Electromagnet, 412. Electromagnetic, rotations, 411; sys- tem, 452 ; ratio to electrostatic, 457 ; theory of light, 458. Electrometer, capillary, 338; quad- rant, 337 ; HankePs, 336. Electromotive force, 391, 455. Electrophorus, 331 ; reciprocal, 332. Electroplating, 402. Electroscope, 311. Electrostatic system, 450, 457. Electrotype, 403. Elements of a voltaic cell, 388. Emissivity, 251. Energy, 70; classified forms of, 74; ^xjonservation, 72 ; electricity not energy, 353 ; of a charged con- ductor, 353 ; of gaseous molecule, 303 ; sources of, 287. Entropy, 282 ; change by equaliza- tion of temperature, 286 ; experi- mental determination, 285. Equilibration, organ of, 589. Equilibrium, conditions of, 88 ; sta- ble and unstable, 74. Equipotential surface, 320 ff . Erg, 69. Establishment of a port, 51. Euler, achromatism, 663. Eustachian tube, 586. Evaporation, 207 ; cooling by, 210. Ewing, hysteresis, 382 ; three stages of magnetization, 381. Excitation, modes of, 477. Exciter, 480. Expansion, 162 ; absolute and appar- ent, 175, 176; of gases, 179; co- efficient of, 172 ; tables for solids, 174 ; for liquids, 176; of water, 178. Expulsion of a body from electric field, 327, 329; from magnetic field, 371. Extraordinary wave, 730. Eye, 628. F= force, 17. &= electric intensity at a point, 319. 758 INDEX. f= acceleration, 16. Fahrenheit scale, 166. Falling body, 27. Farad, the, 460. Faraday, dark space, 500 ; disc ma- chine, 473 ; dielectric, 312 ; elec- tromagnetic rotations, 411; hollow cube, 314; ice-pail experiment, 311 ; induction, 461 ; influence of the medium, 320; laws of elec- trolysis, 397 ; liquefaction of gases, 218 ; paramagnetic and diamag- netic bodies, 372 ; magneto-optic rotation, 751 ; specific inductive capacity, 340; voltameter, 401. Far-sightedness, 630. Faure, heat of combination, 287 ; secondary cell, 404. Fedderson, oscillatory discharge, 351. Field, about insulated conductor, 325; behavior of dielectric body in an electrostatic, 326 ; behavior of bodies in a magnetic, 371 ; electric, 320, 321, 451 ; due to a charged point, 321 ; a charged sphere, 322; to two unlike charges, 323 ; to two like charges, 324 ; earth's, 363, 368 ; magnetic, 360 ; strength of magnetic,' 360, 453 ; strength of electric, 451. Fife, 594. Fire syringe, 258. Fizeau, velocity of light, 603. Flames, conduction of, 495. Flat, 545. Fleming, rule for induced currents, 464. Floating bodies and surface tension, 155. Fluid, 100, 101. Fluorescence, 713. Flute, 558, 594. Flux, 373. Foot-pound, 70. Forbes, conductivity of iron, 246. Force, 17; due to a current, 408; law of electrical, 314 ; law of mag- netic, 358. Fortin, barometer, 123. Foucault, pendulum experiment, 62 ,- prism, 734 ; velocity of light, 604. Franklin, discharge from points, 349 ; kite experiment, 353. Fraunhofer lines, 707. Freezing, machines, 212; mixture, 239 ; point, lowered by pressure, 281 ; of solutions, 239. Frequency, 515. Fresnel, diffraction, 647 ; wave sur- face, 737 ; wave theory, 605. Friction, 77. Frictional electric machines, 330. Fundamental magnitudes, 5. Fusion, latent heat of, 191 ; laws of,, 200; of alloys, 200. G = gravitation constant, 46. g = acceleration of weight, 27, 48, 61. 7 = ratio of .specific heats, 278, 305. Galileo, pendulum, 60; telescope, 631. Galton, whistle, 548. Galvani, experiment with frogs' legs, 388. Galvanometer, 415 ; ballistic, 420 ; d' Arson val, 419, 421 ; direct read- ing, 420; potential, 423; sensi- tive, 418 ; sine, 417 ; tangent, 415 ; Weston, 421. Gas, 102; density, 298, 299; ex- pansion of, 179; general law of, 181. Gay-Lussac, law of, 180; nieasure- ment of angle of contact, 143. Geissler, air pump, 134 ; tube, 499, 706. INDEX. 759 Gilbert, magnetization of the earth, 363. Glaciers, 206. Glow lamp, 439. Governor, parabolic, 38 ; Watt's, 37. Graham, compensated pendulum, 183; diffusion, 231. Gram, 11. Gramme, ring armature, 476. Gratings, 650. Gravitation, Newton's law, 43 ; con- stant, 46. Gravitational unit, of force, 18; of pressure, 99 ; of work, 69. Gravity battery, 391. Gray, telephone transmitter, 492. Griffiths, determination of J, 264. Grotthus, chain of ions, 504; elec- trolysis, 398. Grouping of cells, 431. Grove cell, 391. Gulf stream, 51. l Gyroscope, 53. H= horizontal component of earth's magnetism, 364. H = strength of magnetic field, 36.0, 375. Hadley sextant, 638. Hall, achromatism, 664. Halo, 700. Hamilton, conical refraction, 737. Hankel, electrometer, 336. Harmonic motion, 58, 64. Harmonics, 548. Harrison, compensated pendulum, 184. Hartnack prism, 734. Hautbois, see Oboe, 595. Hazy sky, 681. Heat, 71, 160; and electricity, 439; effects of, 187; of combination, 286 ; produced by work, 258 ; spe- cific, 188; transference of, 244; theories concerning, 259 ; unit of, 187. Heating of conductors, 438. Helicoid, 152. Helmholtz, gravest note, 548 ; quality of a note, 542 ; of a compound tone, 584; resonator, 562; sum- mation tones, 580; theory of dissonance, 577 ; vibration of strings, 581 ; Young-Helmholtz theory, 722, 725. Hemispherical front, 670. Henry, oscillatory discharge, 351 ; solution of gases in liquids, 228; telegraph, 486. Henry, the, 467. Herschel, reflecting telescope, 637. Hertz, electric waves, 507; kathode rays, 502. Holmgren, detection of color blind- ness, 725. Holtz, influence machine, 333. Hope, experiment on maximum den- sity of water, 177. Hopkinson, temperature changes in magnetization, 379. Horizontal pendulum, 309. Horn, French, 597. Horrocks, transit of Venus, 634. Horse-power, 273. Hue, 716. Hughes, microphone, 492. Humidity, 240. Hunning^s telephone, 493. Huyghens, conservation of energy, 72 ; construction, 536 ; for refrac- tion in uniaxial crystals, 730; nature of light, 604 ; parabolic governor, 38 ; principle of, 536. Hydraulic press, 76. Hydrokinetic analogy to magnetic induction, 372. 760 INDEX. Hygrometer, chemical, 240; dew point, 240 ; wet and dry bulb, 242. I = intensity of magnetization, 361, 453. i = current strength, 394. Iceland spar, 628. Ice-pail experiment, 311. Image, optical, 610. Impact, 111. Impedance, 482. Inclination, magnetic, 363. Incus, 586. Index of refraction, 539, 661. Indicator diagram, 272. Induced electromotive force, law of, 463, 466. Induction, electrostatic, 312 ; coil, 467 ; of currents, 461 ; laws of, 463; magnetic, 370, 375; hydro- kinetic analogy, 372. Inductors, influence machine, 332; dynamo-electric machinery, 474. Influence machine, 330. Inharmonic series, 566. Insulator., 311; resistance of, 427. Intensity, electrical, at a point, 319, 451 ; as shown by equipotential surfaces, 321; magnetic, 375; of magnetization, 361, 453. Interference of light, 607, 641 ; in white light, 648; of polarized light, 739; of sound, 551. Intermittent stimulation, 57" Interval, musical, 544. Ion, 396. Iridescence, 653. Iron, magnetic constants of, 376. Irreversible heat effect, 438. Isoclinic,,365. Isodynamic, 366. Isogonic, 365. [217. Isothermal, of a gas, 215 ; of a vapor, J = mechanical equivalent of heat, 264. Jolly, air thermometer, 164 ; speci- fic heat at constant volume, 198 ; steam calorimeter, 194. Joule, arrangement of thermometer, 255; conservation of energy, 73; effect, 438 ; kinetic theory of gases, 290; maximum density of water, 178; mechanical equivalent of heat, 261 ; porous plug experi- ment, 276. Joule, the, 460. Just intonation, 547. K = specific inductive capacity or dielectric constant, 315, 340, 341. Kathode, 388; rays, 501, 602. Keeper, 414. Kelvin, compass, 369; current bal- ance, 422 ; definition of induction and intensity, 375 ; experiment with shoemaker's wax, 101 ; growth of raindrops, 153; molecular range, 153; porous plug experi- ment, 276 ; quadrant electrometer, 337 ; rigidity of earth, 53 ; siphon recorder, 491; thermodynamic scale, 275 ; water-dropping collec- tor, 354. Kepler, laws of, 44. Kerr, double refraction in strained dielectrics, 741. Ketteler, dispersion formula, 712. Kilo-, as prefix, 460. Kilogram, 12. Kilogram-meter, 69. .-j^* Kinetic, energy, 70 ; theory tff gases, 290. Kleist, 342. Kohlrausch, electrolysis of water, 401. Konirj, clock-fork, 574. Kundt, tubes, 661. INDEX. 761 X = wave-length, 515. Labyrinth, 587. Lag, 481. Lamps, electric, 440. Lane, unit jar, 352. Langley, bolometer, 435 ; opacity of air, 681 ; spectrum, 653. Larynx, 560. Latent heat, of fusion, 191 ; tables, 201 ; of vaporization, 208. Laurent, saccharimeter, 741. Lead, 481. Leclanche cell, 393. Lenard, electrification of waterfalls, 355 ; kathode rays, 502. Length, measurement of, 9; unit of, 8. Lens, 620 ; limit of power of, 665. Lenz, law, 463, 465. Leslie, freezing by evaporation, 210. Levels, 116. Leyden jar, 342. Light, a wave phenomenon, 606; definition, 599 ; electromagnetic theory, 458; nature of ordinary, 751 ; propagation of, 600 ; waves, 515. Lighting, electric, 439. Lightning. 353, 355. Limit of resolving power, 669. Limits of audibility, 548. Line of force, cuts level surface 'at right angles, 322, 324 ; electrical, 319; magnetic, 360; mode of drawing, 321. Lippershey, telescope, 631. Lippmann, electrometer, 338. Liquid, 101 ; continuity of liquid and aeriform states, 218 ; phenom- ena at surface, 138 Lissajous, figures, 529 ; optical tun- ing, 571. Lloyd, conical refraction, 738. Local action, 390. Lode-stone, 357. Lodge, electrical resonance, 511. Longitudinal waves, 515. Looming, 684. Loudness, 541. Luminescence, 712. Luminosity, 716. Lyman, wave apparatus, 532. M = moment of a force, 79. M = magnetic moment, 361, 453. m = mass, 12. m = quantity of magnetism, 359. n = permeability, 359. Machines, 75. Magnet, 357 ; action of one on an- other, 361. Magnetic, attraction and repulsion, 358; charts, 365; constants of iron, 376; field, 360, 375; field due to current, 406, 407 ; force experienced by conductor in mag- netic field, 410 ; force, law of, 358; induction, 370, 375; merid- ians, 367 ; moment, 361, 453 ; mo- ment of a circuit, 408, 410, 454 ; pole, 357 ; shell, 408. Magnetism, 357 ; Ampere's theory of, 408 ; quantity of. 359, 452 ; terrestrial, 363. Magnetization, a molecular phcnom enon, 380; affected by tempera- ture, 379; intensity of, 361, produced by electric discharge, 406 ; three stages of, 381. Magneto, 474. Magneto-optic rotation, 751. Magnification, by lenses, 622 ; of compound microscope, 668; of mirrors, 614. Maheiv, photo-electric properties of selenium, 427. 762 INDEX. Major scale, 544. Malleable, 100. Malleus, 586. Manometer, 126. Manuals, 591. Mariner's compass, 369. Marriotte, law of, 181. Mass, measurement of, 12 ; unit, 11. Matter, characteristics of, 6. Maximum efficiency of optical in- struments, 665. Maximum and minimum thermom- eter, 168. Maxwell, color diagram, 720 ; color discs, 715; electromagnetic theory of light, 458; force on conductor bearing a current, 410 ; molecular constants, 307 ; rule for current induction, 464 ; velocity of elec- tromagnetic disturbance, 507. Mayer, duration of sound sensation, 590. Mean free path, 307. Mean square, of current, 480; of velocity, 291. Measurement, of physical magni- tudes, 1. Meatus, 586. Mechanical, advantage, 75 ; equiva- lent of heat, 261, 263. Mega-, as prefix, 460. Melloni, 447. Melting point, 191 ; affected by pres- sure, 204 ; tables, 201. Meniscus, 144. Meter, 8. Meyer, multiplex telegraphy, 489. Meyer, Victor, gas density, 299. Michelson, velocity of light, 604, 606. Micro-, as prefix, 460. Microphone, 492. Microscope, compound, 637 ; limit of resolving power, 669 ; magni- fication, 668 ; micrometer, 9 ; sim- ple, 631. Miculescu, determination of J., 204. Minimum deviation, of prism, 018; of raindrop, 695, 696. Minimum potential energy, 74. Minor scale, 546. Mirage, 685. Mixtures, scale of temperatures, by, 160. Mohr, conservation of energy, 72. Molecular, mass, and gas density, 298 ; masses determined by lower- ing of vapor pressure, 237 ; range, 152; theory, 104; velocities, 295. Molecules, 104; size and number, 306. Moment, of a force, 79 ; of inertia, 91 ; about parallel axes, 93 ; of a magnet, 361 ; of momentum, 91. Momentum, 17. Moon, atmosphere of, 296 ; produc- tion of tides, 51 ; used by Newton to verify law of gravitation, 45. Morse code, 486. Motion, quantity of, 17 ; of rotation, 79 ; of translation, 79 ; types of, 25; unchanging, 25; uniform cir- cular, 32 ; uniformly changing rec- tilinear, 26. Motor, electric, 484 ; dynamo, 485. Mouthpieces, 557. Multiple arc, conductors in, 430. Multiplex telegraphy, 489. Muschenbroeck, 342. Musical, interval, 544 ; quality, 584 ; scale, 544. Music box, 566. Myopia, 629. ( = frequency, 515; n \ = index of refraction, 539, 618. Natter er, investigation of Boyle's law, 222. INDEX. 763 Near-sightedness, 629. Needle telejgraph, 486. Negative, Crystals, 732 ; glow, 500; rays, 50|. Neutral point, 444. Newcomb, velocity of light, 604. Newton, achromatism, 663 ; conser- vation ofj energy, 72 ; corpuscular theory, 605; law of universal grav- itation, 43; mass proportional to weight, 12 ; rainbow, 700 ; rate of cooling, 250 ; reflecting telescope, 636; rings, 606, 656; spectrum, 660. Nicholson, electrolysis, 397 ; revolv- ing doubler, 333. Nicol prism, 733. Node, 522, 555. Nodoide, 148, 151. Noise, 541. Nollet, osmotic pressure, 233. Non-conductor, 311. Non-spectral colors, 721. Normal spectrum, 710. Notation in powers of ten, 5. Numeric, 1. Numerical aperture, 669, 670. Nutation, 56. Objective, 634, 668. Oboe, 559, 595. Ocular, 635 ; circle, 635. Oersted, effect of current, 406. Ohm, law of, 395; verification of law, 428. Ohm, the, 396. Oil, behavior on water, 157. Opalescence, 682. Opaque, 253. Open pipes, 556. Optical image, 610. Ordinary wave, 730.. Organ, church, 591. Oscillator, 507. Oscillatory discharge, 350, 355 ; pe- riod of, 506. Osmose, 233. Osmotic pressure, 234, 302 ; analogy to gaseous pressure, 236. Overtones, 548. p = pressure, 98, 99. P. D. = potential difference, 391, 455. Pacinotti, dynamo, 476. Page, sounds accompanying magnet- ization, 491. Paraboloid of revolution, 38. Parallel, conductors in, 430. Paramagnetic, 372 ; bodies, 378. Partials, 548. Pedals, 591. Peltier, effect, 442, 446. Pendulum, simple, 60; compound, 62, 93. Penumbra, 600. Period, of magnetic needle, 364 ; of oscillatory discharge, 506. Permeability, 359. Perversion, 612. Pfeffer, osmotic pressure, 234. Phase, change of, in reflection, 523- 525; coexistent phases, 200; dif- ference of, 517. Phosphorescence, 501, 713. Photo-electric properties of selenium, 427. Photography, kathode, 502. Photometry, 717, 718. Piccolo, 548, 594. Pile of plates, 733 ; Volta's, 389. Pipe, organ, 555, 591. Pitch, musical, 541 ; determination of, 573 ; standard, 573. Plane mirror, 611. Plante, secondary cell, 404. 764 INDEX. Plastic, 99. Plates, vibration of, 568. Platinum thermometer, 426. Pohl, commutator, 472. Points, discharge from, 348. Polarimeter, 747. Polarization, of a cell, 390 ; of light, 727. Polarizer, 728. Pole, celestial, 56 ; magnetic, 357, 367, 452 ; of a magnet, 357, 363 ; unit, 359. Positive, column, 500 ; crystals, 732. Potential, at a point, 320 ; due to a charged point, 326; electric, 451, 455, 458 ; energy, 70. Potentiometer, 424. Pound, relation to the kilogram, 12 ; force, 18. Practical, system of units, 450, 459, 460; unit of quantity, 395; of current, 395. Precession, 55. Presbyopia, 630. Pressure, 98; change in passing through a liquid surface, 140 ; gauge, 126 ; hydrostatic, 99 ; in the kinetic theory, 290 ; in a fluid at rest, 115 ; reduction at side of moving stream, 131 ; surface of constant, 116; volume diagram, 217, 265. Prtvost, theory of exchanges, 254. Pringsheim, protoplasmic contents of a cell, 235. Projectile, 30. Propagation of light, 600. Pulley, 76. Pumps, 127 ; air, 129 ; Geissler, 134 ; jet, 133 ; Sprengel, 133. Pythagoras, discovery of consonant ratios, 578. q = quantity of electricity, 313. Quality, musical, 541, 584; of vocal sounds, 583. Quantity, of electricity, 313, 317, 450, 455, 458 ; of magnetism, 359, 452. Quarter-wave plate, 743. Quartz, 743. Quincke, molecular range, 152 ; sur- face tension, 147. R = constant for gases, 182. p= density, 14, 119. Radian, 15. Radiant heat, 253. Radiation, 244 ; cooling by, 250 ; effect on thermometers, 254. Radiometer, 255. Radio-micrometer, 448. Radius of gyration, 92. Rainbow, 693. Raindrops, formation and growth, 153. Range, of a projectile, 31. Raoul, vapor pressure of solutions, 237. Rayleigh, ripples and surface ten- sion, 536. Reaumur, scale of, 166. Recalescence, 380. Reed, 558. Reedy, 585. Reflection, 523, 537 ; of compres- sural wave, 524 ; of transverse wave, 525; of light, 609. Refraction, 537 ; at a plane surface, 615 ; atmospheric, 683 ; double, in anisotropic media, 729 ; of light, 609 ; of magnetic lines of force, 374 ; through a plate, 616 ; through a prism, 616; throu lenses, 619. Regelation, 204. r INDEX. 765 Regnault, compressibility of gases, 222; expansion of mercury, 176; pressure of water vapor, 214 ; spe- cific heat at constant pressure, 278. Reis, telephone, 491. Relationship of notes, 579. Relay, 487. Replenishes, 333. Residual magnetism, 382. Resinous charge, 310. Resistance, 425, 452, 455, 458 ; boxes, 427, 435; effect of temperature, 425 ; tables, 426, 427. Resolvipg power of telescope, 669. Resonance, 511, 562; of external canal, 590. Resonator, 508, 562. Resultant of two parallel forces, 80. Retina, 628. Return shock, 355. Reversible, cycle, 269; heat effect, 442 ; pendulum, 62, 94. Rigid body, motion of, 79 ; rotation of, 89. Rigidity, simple, 109; tables, 110. Ring armature, 476, 477. Rings and cross, 741. Ripples, 535. Rise of liquid, in a tube, 145 ; be- tween plates, 146. Rods, vibrations of, 565, 568. Roeiner, velocity of light, 601. Rontgen rays, 502. Rosetti, temperature of arc, 440. Rotary, field, 484 ; polarization, 743 ; power of liquids, 745. Rotor, 485. Roulettes of the conic sections, 150. Rowland, . concave grating, 652 ; d'Arsonval galvanometer, 419 ; experiment on electric convection currents, 409 ; mechanical equiva- lent of heat, 263 ; specific heat of water, 196. Ruhmkorff, coil, 467. Rumford, arrangement of calorime- ter, 190; conservation of energy, 73 ; controverted caloric theory, 260; experiments in mechanical production of heat, 260 ; photom- eter, 718. s= distance traversed, 15. 0- = speed, 16, 33. Saccharimetry, 746. Safety lamp, 248. Sagitta, 36. Saturated, color, 721 ; vapor, 213. Saturation, magnetic, 378, 382 ; color, 716. Saxhorn, 598. Saxophone, 596. Scalar, 18. Scale, development of, 578 ; musi- cal, 544-547; of length, 9; of temperature, 160, 166. Scintillation, 688. Screw, 76. Second, the, 6. Secondary, actions, 400 ; cell, 403. Seebeck effect, 441. Self-induction, 462; coefficient of, 466. Semicircular canals, 587, 589.- Semitone, 545. Sense, temperature, 161. Sensitive, galvanometer, 418; tint, 750. Series winding, 477. Sextant, 638. Shadow pictures, 600. Sharp, 545. Shear, 98. Shunt, 430 ; winding, 477. Shuster, determination of J., 264. 766 INDEX. Siemens, armature, 475. Signature, 547. Sine galvanometer, 417. Siphon, 1.30 ; recorder, 491. Siren, 542. Sky, color of, 680. Soap films, 147, 656. Solar spectrum, 707. . Soleil, saccharimeter, 748. Solenoid, 412. Solid, 100. Solubility, 228. Solution, 226 ; gases in gases, 226 ; gases in liquids, 228 ; liquids in aeriforms, 227 ; liquids in liquids, 229 ; solids in liquids, 230. Solvend or solute, 226. Solvent, 226. Sonometer, 564. Sonorous bodies, 555. Sorge, difference tones, 579. Sound, 540 ; medium of transmis- sion of, 540 ; waves, 515 ; velocity of, 550, 561. Sounder, telegraph.. 487. Space, 5. Spark, 347 ; discharge, 496 ; length, 496. Specific, conductivity, electrical, 425; gravity, 15 ; heat, 188 ; by mix- tures, 189 ; conditions affecting specific heat, 195 ; tables of, 196- 198; two specific heats of a gas, 277 ; inductive capacity, 340, 341 ; magnetic inductive capacity, 359 ; resistance, 425. Spectacle glasses, 629. Spectra, kinds of, 704. Spectral colors, 714. Spectrometer, 706. Spectroscope, 704. Spectrum, 660; analysis, 706; solar, 707. Speed, 16. Spherical mirrors, 613. Spheroidal condition, 209. Sprengel, air pump, 133. Stampings, 483. Standard pitch, 573. Stapes, 586. Stationary waves, 521, 522. Steady flow of heat, condition of, 244. Steam, engine, 271 ; line, 219 ; tables, 216. Stefan, law of cooling, 251. Steinheil, return wire, 486. Stokes, descent of fine particles, 306. Stops, 591. Storage battery, 403. Strain, 99. Strength, of a body, 100 ; of an elec- tric field, 319, 451 ; of magnetic field, 360 ; of current, 394 ; of water, 101. Stress, 98 ; shearing, 98. Striae, 500, 505. Strings, vibrations of, 563 ; of bowed strings, 581. Strouhall, experiments of, 557. Structure, 103. Sublimation, 207. Submarine telegraph, 490. Supernumerary bows, 698. Surface color, 710. Surface tension, 138, 141, 147 ; table, 147 ; and floating bodies, 155 ; and cleanness of surfaces, 158. Susceptibility, 375. T = absolute temperature, 183. Tait, thermo-electromotive force, 445. Tangent galvanometer, 415. Tartini, tones of, 579. Tears of strong wine, 157. Telegraph, 486. INDEX. 767 Telephone, 486. Telescope, astronomical, 634 ; Gali- lean, 632 ; maximum efficiency, 667 ; resolving power, 669 ; reflect- ing, 636 ; terrestrial, 638. Temper, 105. Temperature, difference of, 160; sense, 161. Tempered scale, 547. Tension, 98; in a liquid film, 139, 141. Terrestrial magnetism, 363; Biot's hypothesis, 367; charts, 365; ta- bles, 364. Testa coil, 470. Thermal capacity, 187. Therm odynamic scale of tempera- ture, 275. Thermodynamics, two laws of, 264. Thermoelectric, diagram, 445 ; hight, 443 ; inversion, 444. Thermo-electromotive force, 441. Thermometer, 163 ; air, 164 ; mer- curial, 165; platinum, 170, 426; weight, 169 ; scales of Fahrenheit and Reaumur, 166. Thermometric substance, choice of, 163. Thermopile, 447. Thin plates, colors of, 654. Thomson, James, lowering of melt- ing point, 204, 282. Thomson, Joseph J., discharge with- out electrodes, 497 ; flying Faraday tubes, 510 ; velocity of discharge, 501. Thomson, Sir William (see Lord Kelvin) ; effect, 444. Tides, 51. Timbre, 541. Time, 5, 6. Toepler-Holtz machine, 333. Torricelli, theorem, 131. Torsion, 108. Total reflection, 615. Tourmaline, 729. Tourniquet, 349. Tractive force of a magnet, 414. Trade winds, 51. Trajectory, 31 ; of a light ball, 135. Transformers, 483. Transmission of electrical energy, 482. Transmitter, 493. Transparent, 253. Transposition, 546. Transverse wave, 514; velocity of, 525. Traube, osmose, 234. Trevelyan, rocker, 248. Triple point, 211. Trochoid, 533. Trombone, 597. Trough, 514. Trumpet, 596. Tube of force, 3.60, 373. Tuning, 554 ; methods of, 591, 592 ; fork, 567. Two fluid cells, 390. Tympanum, 586. Ultra violet light, 496. Umbra, 600. Unchanging motion, 25. Unduloid, 148. Uniaxial crystals, wave surface in, 730. Unit jar, 351. Units, see the names of the quantities. Unstable condition, at the melting point, 201 ; at the boiling point, 209. Upper partials, beats of, 576. F = electric potential, 320. v = velocity of light, 457, 458, 602. 768 INDEX. Van der Waals, equation, 294. Van'tHoff, osmotic pressure, 236; lowering of freezing point, 239. Vapor, 103; pressure of solutions, 237, and molecular masses, 237 ; table of vapor pressures, 214. Vaporization, latent heat of, 208. Variation of earth's field, 368. Varley, cable messages, 490. Vector, 18-21. Velocity, 15; change of, in impact, 112 ; of efflux, 131 ; of electric discharge, 501 ; of light, 602 ; of mean square, 291 ; of sound, 550, 561 ; effect of temperature, 551. Verdet, 691. Vibration, of a sounding body, 540 ; of strings, 563. Vibroscope, 543, 574. Viola, 593. Violin, 593. Violle, temperature of the electric arc, 440. Violoncello, 594. Virial, 293. Virtual currents, 40 ; electromotive force, 480 ; image, 610 j. resistance, 482. Viscosity, 110 ; of gases, 305. Viscous, 100. Vitreous charge, 310. Vocal, cords, 560, 583 ; sounds, qual- ity of, 583. Volt, the, 393, 460. Volta, contact theory, 389; effect, 385 ; electrophorus, 331 ; pile, 389. Voltaic cell, 387, 388. Voltameter, 401. Voltmeter, 424. Volume, 14 ; change in melting, 201, 202 ; change in vaporization, 209. Volume-temperature diagram of water, 202. Foss, machine, 334. Vowels, analysis of, 583 ; synthesis, 584. Water, density of, 177, 179; expan- sion of, 177, 178 ; vapor, 122, 214, 216. Water-dropping collector, 354. Water waves, 531. Watt, governor, 37. Watt, the, 460. Watt-meter, 424. Wave, 514 ; addition, 517 ; equation, 516; length, 515; electric, 507, 512, 513 ; light, 515 ; sound, 515. Wave-lengths of light, 646,N*52. Weighing, 13. Weight, 6; affected by rotation of earth, 47; effect on liquid surface, 139 ; of a mass, 18 ; proportion- ality of weight and mass, 61. Weights, i.e. standard masses, 13. Weston, ampere-meter, 421 Wheatstone, bridge, 433, <34 ; du- ration of spark, 348 ; elegraph, 486. White, 715, 720. Wimshurst, machine, 335. Wollaston, cryophorus, 211; spec- trum, 707. Work, 69. Xylophone, 566. Young, rainbow, 700. Young-Helmholtz theory, 722, 725. Young's modulus, 107. Zeiss, microscope, 679. OF THE UNIVERSITY OF *UFORHih<