Lniv.of in. Library 53 173-7 5 5 ^ yr e 33 R. HALSTED WARD /t-s?. / .The Suru'Z.The Sun is edge. 3.Sodium,. 4. Fbtnuss ium,. 5. lithium. 6. Caesium,. 7. Rubidium.. ^Thcdlium,. 9. Calcium. 10. Strontium,. HRarium. 12. Imdium,. 13. Thosp.horus. 14. Hydro gen,. ELEMENTARY TREATISE ON NATURAL PHILOSOPHY. BY A. / PRIVAT DESCHANEL, FORMERLY PROFESSOR OF PHYSICS IN THE LYClSE LOUIS-LE-GRAND, INSPECTOR OF THE ACADEMY OF PARIS. TRANSLATED AND EDITED, WITH EXTENSIVE ADDITIONS, By J. D. EVERETT, M. A., D. 0. L., F. R. S. E., PROFESSOR OF NATURAL PHILOSOPHY IN THE QUEEN’S COLLEGE, BELFAST. IN FOUR PA RTS. PART IV. SOUND AND LIGHT. ILLUSTRATED BY 187 ENGRAVINGS ON WOOD, AND ONE COLORED PLATE. REW YORK: D. APPLETON AND COMPANY, 549 AND 551 BROADWAY. 1875. In the present Part, the chapters relating to Consonance and Dissonance, Colour, the Undulatory Theory, and Polarization, are the work of the Editor; besides numerous changes and additions in other places. The numbering of the original sections has been preserved only to the end of Chapter LX.; the two last chapters of the original having been transposed for greater convenience of treatment. With this exception, the announcements made in the “Translator's Preface,” at the beginning of Part I., are applicable to the entire work. CONTENTS—PART IV, ACOUSTICS. Chapter LIII. PRODUCTION AND PROPAGATION OF SOUND. Sound results from vibratory movement.—Vibration of straight spring.—Single and double vibration.—Period.—Amplitude.—Isochronism.—Analogy of pendulum.—Vibration of bell.—Of plate.—Nodal lines.—Opposite behaviour of sand and lycopodium.—Vibration of string.—Of air in a pipe.—Chemical harmonica.—Trevelyan experiment.—Dis¬ tinctive character of musical sound.—Periodicity.—Vehicle of sound.—Not transmitted through vacuum.—Liquids and solids can convey sound.—Mode of propagation.— Transmission of condensations and rarefactions.—Graphical representation.—Relation between period, wave-length, and velocity.—Nature of undulatory movement. — Its geometrical possibility illustrated.—Longitudinal and transverse vibrations.—Veloci¬ ties greatest at centres of condensation and rarefaction.—Propagation in an open space. —Law of inverse squares.—Regnault's experiments with sewer-pipes.—Transformations of energy in undulation.—Dissipation of sonorous energy.—Conversion into heat.— Velocity of sound in air.—Mode of determining.—Results.—Theoretical computation of velocity.—Newton’s theory and Laplace’s modification.—Velocity in gases gene¬ rally.—In liquids.—Colladon’s experiment at Lake of Geneva.—Theoretical computa¬ tion.—Allowance for heat of compression.—Velocity in solids.—Biot’s experiment.— Wertheim’s experiments and results.—Theoretical computation.—Reflection of sound. —Illustrations.—Conjugate mirrors.—Echo.—Refraction of sound.—Speaking-trum¬ pet.— Ear-trumpets.— Interference of sound-waves.—Experimental illustrations.— Interference of direct and reflected waves.—Nodes and antinodes.—Acoustic pendulum. —Beats produced by interference,.pp. 785-813. Note A. Rankine’s investigation,.pp. 813, 814. Note B. Usual investigation of velocity of sound,.pp. 814, 815. Chapter LIV. NUMERICAL EVALUATION OF SOUND. Loudness, pitch, and character.—Pitch depends on frequency.—Period and frequency are reciprocals.—Wave-length is distance travelled in one period.—Velocity is wave-length multiplied by frequency.—Character or timbre.—Musical intervals.—Gamut.—Tem¬ perament.—Absolute pitch.—Limits of musical pitch.—Minor scale.—Pythagorean scale.—Methods of counting vibrations.—Syren.—Vibroscope for writing vibrations.— Phonautograph.—Tonometer.—Pitch modified by relative motion, . pp. 816-827. Chapter LV. MODES OF VIBRATION. Longitudinal and transverse vibrations.—Transverse vibrations of strings.—Their velocity of propagation and frequency.—Sonometer.—Harmonics.—Overtones not always har¬ monics.—Strings vibrating in segments.—Segmental vibration of strings.—Sympathetic vibration or resonance. — Sounding-boards. — Longitudinal vibrations of strings.— Stringed instruments.—Transversal vibrations of rigid bodies.—Plates and bells.— IV TABLE OF CONTENTS. Tuning-fork.—Affected by temperature.—Mounted fork.—Law of linear dimensions .— Organ-pipes.—Mouth-piece.—Pitch depends on column of air.—Overtones of open and stopped pipes.—Nodes and antinodes.—Stationary undulations.—Wave-length of fun¬ damental note.—Wave-lengths of overtones.—Analogous laws for certain vibrations of rods and strings.—Application to measurement of velocity of sound in various sub¬ stances.—Reed-pipes.—Opposite effect of temperature.—Wind-instruments. — Mano- metric flames,.pp. 828-846. Chapter LYI. ANALYSIS OF VIBRATIONS. CONSTITUTION OF SOUNDS. Optical examination of sonorous vibrations. Lissajous 1 experiment.—Composition of two simple vibrations in perpendicular directions.—Unison gives an ellipse which can be inscribed in a given rectangle.—Genera 1 , equations to Lissajous’ figures.—Optical tuning.—Other modes of exhibiting the composition of rectangular vibrations.—Black¬ burn’s pendulum.—Elastic rod.—Character.—Form of vibration.—Resolution of perio¬ dic motions by Fourier’s theorem.—Every periodic vibration consists of a fundamental simple vibration and its harmonics.'—Every musical note consists of a fundamental note and its harmonics.—Constitution of a vibration defined.—Corresponds to character of resulting sound.—The harmonics which are present in a note may or may not have their origin in segmental vibrations of the instrument.—Combinations of stops in organs. —Helmholtz’s resonators.—Adaptation tomanometric flames. Human voice.—Vowel sounds.—Experiments of Willis, Wheatstone, and Helmholtz, . . . pp. 847-858. Chapter LVD. CONSONANCE, DISSONANCE, AND RESULTANT TONES. Concord and Discord.—Examples.—Consonant intervals can be more accurately identified than dissonant.—Dissonance depends on the jarring effect of beats not too slow nor too rapid.—33 beats per second give a maximum of discomfort.—Proof that all beats are due to imperfect unison .—Beating notes must be near in pitch.—Helmholtz’s calcula¬ tion of amounts of dissonance.—Discordant elements in an imperfect concord.—Resultant tones.—Difference-tones discovered by Sorge and Tartini.—Erroneously attributed to coalescence of beats.— Summation-tones.— Resultant-tones occur when small quantities of the second order are sensible.—Beats of resultant-tones, .... pp. 859-864. OPTICS. Chapter LVII. PROPAGATION OF LIGHT. Light. — Hypothesis of cether capable of propagating transverse vibrations. — Excessive shortness and excessive frequency of luminous waves.—Strength of shadows.—Recti¬ linear propagation.—Diffraction an exception.—Images produced by small apertures.— Images of sun.—Shadows ; umbra and penumbra.—Velocity of light.—Seven and a half circumferences of the earth per second.—Fizeau’s experiment with toothed-wheel.— Foucault’s experiment with rotating mirror.—Ingenious method of keeping rate of rotation constant.—Mode of reducing the observations.—Resulting velocity 298 million metres per second, or 185,000 miles per second.—Velocity deduced from eclipses of Jupiter’s satellites.—From aberration of stars.—Sun’s distance deduced from Foucault’s determination of velocity.—Photometry, principles of. — Photometers of Bouguer, Rumford, Foucault, and Bunsen,.pp. 865-882. Chapter LVIII. REFLECTION OF LIGHT. Plane of incidence and reflection.— Angles of incidence and reflection equal.—Apparatus TABLE OF CONTENTS. V for verification.—Artificial horizon.—Regular and irregular reflection. — Looking- glasses.—Speculum metal.—Silvered specula.—Plane mirrors.—Position and size of image.—Images by successive reflections.—Parallel mirrors.—Mirrors at right angles. —Kaleidoscope.—Pepper’s ghost.—Deviation produced by rotating a mirror.—Hadley’s sextant.—Spherical mirror.—Centre of curvature.—Principal and secondary axes.— Principal focus of concave mirror.—Parabolic mirrors.—Spherical aberration.—Con¬ jugate foci.—Formula for conjugate focal distances.—Formation of real images.— Position of principal focus.—March of conjugate foci.—Construction for position and size of image.—Calculation of size of image.—Phantom bouquet.—Images on screen. —Image as seen directly.—Caustic surface.—Primary and secondary foci.—Primary and secondary focal lines on a screen.—Virtual image in concave mirror.—Distinction between real and virtual images. — Convex mirrors.—Cylindric mirrors.—Anamor¬ phosis.—Ophthalmoscope and laryngoscope,.pp. 883-907. Chapter LIX. REFRACTION. Sudden change of direction.—Sunbeam entering water.—Coin in basin.—Stick appears broken.—Refractive powers of different media.—The denser usually the more refrac¬ tive.—Law of sines.—Apparatus for verification.—Index of refraction.—Table of indices.—Critical angle and total reflection.—Mirage as explained by Monge.—Camera lucida.—Caustic by refraction at a plane surface, and apparent position of virtual image.—Refraction through parallel plate.— Multiple images in plate. — Candle in looking-glass.—Prism or wedge.—Refraction through it.—Displacement of objects seen through it.—Investigation of formulae.—Geometrical construction for deviation, and proof of minimum deviation.—Conjugate foci with respect to prism in position of mi¬ nimum deviation.—Double refraction.—Iceland-spar. Ordinary and extraordinary image,.pp. 908-928. Chapter LX. LENSES. Forms of lenses.—Converging and diverging, or convex and concave.—Principal axis.— Principal focus.—Optical centre.—Secondary axes.—Conjugate foci.—Comparative sizes of object and image.—Whether image will be erect or inverted.—Investigation of formulae for focal length and conjugate focal distances.—Conjugate foci on secondary axis.—March of conjugate foci.—Minimum distance between object and real image is four times focal length.—Construction for position and size of real image.—Calculation of size.—Example.—Image on cross-wires.—Cross-wires at conjugate foci.—Aberration of lenses.—Virtual images, and formulae relating to them.—Concave lenses.—Foco- meter.— : Refraction at a single spherical surface. — Camera obscura. — Photographic camera.—Example of photographic processes.—Projection of experiments on screens.— Solar microscope.—Magic-lantern.—Photo-electric microscope, . . pp. 929-945. Chapter LXL—VISION AND OPTICAL INSTRUMENTS. Description of the eye.—Adaptation to different distances.—Binocular vision.—Data for judgment of distances.—Perception of relief.—Stereoscope.—Visual angle (plane) or apparent length.—Apparent area or solid visual angle.—Magnifying power.—Spec¬ tacles.—Magnifying lens.—Visual angle in different positions of lens.—Simple micro¬ scope.—Compound microscope.—Magnifying power computed and observed.—Astrono¬ mical telescope.—Magnifying power computed and observed.—Finder.—Bright spot.— Magnifying power deduced from comparison of object-glass with bright spot.—Terres¬ trial eye-piece.—Galilean telescope.—Its peculiarities.—Opera-glass.—Reflecting tele¬ scopes.—Herschelian and Newtonian.—Magnifying power.—Gregorian and Casse- granian.—Silvered specula.—Measure of brightness.—Intrinsic and effective.—Intrin- VI TABLE OF CONTENTS. sic brightness is •—Surfaces are equally bright at all distances.—Image formed by theoretically perfect lens has same intrinsic brightness as object; but effective bright¬ ness may be less.—Same principle applies to mirrors.—Reason why high magnification often produces loss of effective brightness.—Intrinsic brightness of image in theoreti¬ cally perfect telescope is equal to brightness of object.—Effective brightness is the same, if magnifying power*does not exceed —, and is less for higher powers.—Actual e telescopes always give images less bright than the objects.—Brightness of stars inde¬ terminate.—Light received from star increases with power of eye-piece till magnifying power is -• —Brightness of image on screen is proportional to solid angle subtended by lens.—Appearance presented to eye at focus.—Cross-wires of telescopes.—Adjustment for preventing parallax.—Line of collimation and its adjustment, . . pp. 946-972. Chapter LXII. DISPERSION. STUDY OF SPECTRA. Analysis of colours by prism.—Solar spectrum.—Modes of obtaining a pure spectrum either virtual or real.—Dark lines.—Invisible portions of spectrum.—Heating and chemical power.—Phosphorescence and fluorescence.—Ultra-violet rays not altogether invisible. —Spectroscope.—Different modes of determining positions of lines.—Train of prisms.— Use of collimator.—Different classes of spectra.—Solar, continuous, bright-line.—Spec¬ trum analysis.—Reversal of bright lines.—Analysis of solar atmosphere.—Telespectro¬ scope. — Bright lines in spectrum of sun’s edge. — Observation of prominences by method of wide slit.—Spectra of nebulae.—Displacement of lines by approach or recess. —Huggins’ recent observations.—Analysis of artificial lights.—Bodies illuminated by monochromatic light.—Chromatic aberration.—Possibility of achromatism.—Conditions of achromatism. Dispersive power.—Impossibility of complete achromatism.—Huy- genian and other achromatic eye-pieces. — Rainbows, primary, secondary, and super¬ numerary, .. . . . . pp. 973-999. Chapter LXIII. COLOUR. Nature of colour in bodies.—Opaque and transparent.—Effect of superposing coloured glasses.—Colours of mixed powders.—Mixtures of colours.—Different compositions may produce the same visual impression.—Methods of mixing colours.—By sheet of glass.—By rotating disc.—By overlapping spectra. — Distinction between mean and sum of given colours.— Colour equations.—Helmholtz’s observations with crossed slits. — Maxwell’s colour-box.—Results of observation.—Substitution of similars.—Personal differences.—All colours except purple are spectral.—Any four colours are connected by one definite relation.—Any five colours yield one definite match by taking means.— Mean of colours analogous to centre of gravity.—Sum of colours analogous to resultant of forces.—Cone of colour.—Three co-ordinates answering to hue, depth, and brightness. —Complementary colours.—Three primary colour-sensations, red, green, and blue.— Three sets of nerves.—Accidental images, negative and positive.—Colour blind vision is dichroic, the red primary being wanting.—Colour and musical pitch, pp. 1000-1011. Chapter LXIY. WAVE THEORY OF LIGHT. Principle of Huygens.—Wave-front.—Explanation of rectilinear propagation.—Spherical wave-surface in isotropic medium.—Two wave-surfaces in non-isotropic medium.— Construction for wave-front in refraction.—And in reflection.—Law of sines deduced. —Newtonian explanation of law of sines.—Foucault’s crucial experiment.—Principle of least time.—Application to reflection and refraction.—More exact statement of the TABLE OF CONTENTS. Vll principle.—Application to conjugate foci and caustics.—S /x s a minimum or maximum. —Application to terrestrial refraction.—Rays in air are concave towards the denser side.—Correction for refraction opposite to that for curvature of earth.—Calculation of curvature of a nearly horizontal ray.—Influence of pressure, temperature, and vertical change of temperature.—Average curvature 3 - or £ of earth’s curvature.—Not owing to earth’s curvature.—Curvature of inclined rays.—General formula.—Astronomical refraction.—Mirage.—Sinuous rays.—Condition of inverted images.—Analogy of total reflection.—Experiment of artificial mirage.—Diffraction fringes produced by narrow slit.—Analogy of sound.—Diffraction by a grating.—Explanation of purity of spec¬ trum. Spectra of different orders.—Angstrom’s observations.—Calculation of wave¬ lengths.—Standard diffraction-spectrum.—Contrasted with prismatic spectra.—Imagi¬ nary standard based on wave-frequency.—Examples of wave-lengths.—Colours of thin films,.pp. 1012-1031. Chapter LXY. POLARIZATION AND DOUBLE REFRACTION. Experiment of two tourmalines.—Polarizer and analyzer.—Polarization tested by variation of brightness.—Polarization by reflection.—Malus’ polariscope.—Polarizing angle.— Brewster’s criterion.—Polarization of the transmitted light.—Polarization never favours reflection. — Definition of plane of polarization.—Direction of vibration.—Polariza¬ tion by double refraction. — Explanation of double refraction in uniaxal crystals.— Wave-surface for ordinary ray spherical, for extraordinary ray spheroidal.—Absorption by tourmaline.—Nicol’s prism.—Colours produced by thin plates of selenite.—Recti¬ linear vibration changed to elliptic.—Analogy of Lissajous’ figures.—Resolution of elliptic vibration by analyzer.—Circular polarization a case of elliptic.—Why the light is coloured.—Why a thick plate shows no colour.—Crossed plates.—Plate perpendicular to axis shows rings and cross.—Changes on rotating analyzer.—Explanation of these phenomena.—Crystals are isotropic, uniaxal, or biaxal.—Rotation of plane of polariza¬ tion.—Quartz and sugar.—Production of colour.—Magneto-optic rotation.—Connection between rotation and crystalline form.—Condition of converting rectilinear into circular vibration.—Quarter-wave plates.—Fresnel’s rhomb.—Effect of combining two.—Discus¬ sion as to direction of vibration in plane polarized light.—Fresnel’s view established by Stokes.—Vibrations of ordinary light.—Polarization of obscure radiation, pp. 1032-1050. ACOUSTICS. CHAPTER LIII. PRODUCTION AND PROPAGATION OF SOUND. 629. Sound is a Vibration.—Sound, as directly known to us by the sense of hearing, is an impression of a peculiar character, very broadly distinguished from the impressions received through the rest of our senses, and admitting of great variety in its modifications. The at- D ' -P" tempt to explain the physiological ac¬ tions which constitute hearing forms no part of our present design. The business of physics is rather to treat of those external actions which constitute sound, considered as an objective existence ex¬ ternal to the ear of the percipient. It can be shown, by a variet}^ of ex¬ periments, that sound is the result of vibratory movement. Suppose, for ex¬ ample, we fix one end C of a straight spring C D (Fig. 563) in a vice A, then draw the other end D aside into the position D', and let it go. In virtue of its elasticity (§ 23), the spring will return to its original position; but the kinetic energy which it acquires in returning is sufficient to carry it to a nearly equal pig, 563.—vibration of straight spring, distance on the other side ; and it thus swings alternately from one side to the other through distances veiy gradually diminishing, until at last it comes to rest. Such move¬ ment is called vibratory. The motion from D' to D", or from D" to D', is called a single vibration . The two together constitute a double 786 PRODUCTION AND PROPAGATION OF SOUND. or complete vibration ; and the time of executing a complete vibra¬ tion is the period of vibration. The amplitude of vibration for any point in the spring is the distance of its middle position from one of its extreme positions. These terms have been already employed (§ 44) in connection with the movements of pendulums, to which indeed the movements of vibrating springs bear an extremely close resemblance. The property of isochronism, which approximately characterizes the vibrations of the pendulum, also belongs to the Fig. 564.—Vibration of Bell. spring, the approximation being usually so close that the period may practically be regarded as altogether independent of the amplitude. When the spring is long, the extent of its movements may gene¬ rally be perceived by the eye. In consequence of the persistence of impressions, we see the spring in all its positions at once; and the edges of the space moved over are more conspicuous than the central parts, because the motion of the spring is slowest at its extreme positions. As the spring is lowered in the vice, so as to shorten the vibrating portion of it, its movements become more rapid, and at the same time VIBRATION OF A BELL. 787 more limited, until, when it is very short, the eye is unable to detect any sign of motion. But where sight fails us hearing comes to our aid. As the vibrating part is shortened more and more, it emits a musical note, which continually rises in pitch; and this effect con¬ tinues after the movements have become much too small to be visible. It thus appears that a vibratory movement, if sufficiently rapid, may produce a sound. The following experiments afford additional illustration of this principle, and are samples of the evidence from which it is inferred that vibratory movement is essential to the pro¬ duction of sound. Vibration of a Bell .—A point is fixed on a stand, in such a posi- Fig. 565.—Vibration of Plate. tion as to be nearly in contact with a glass bell (Fig. 564). If a rosined fiddle-bow is then drawn over the edge of the bell, until a musical note is emitted, a series of taps are heard due to the striking of the bell against the point. A pith-ball, hung by a thread, is driven out by the bell, and kept in oscillation as long as the sound continues. By lightly touching the bell, we may feel that it is vibrating ; and if we press strongly, the vibration and the sound will both be stopped. Vibration of a Plate .—Sand is strewn over the surface of a hori¬ zontal plate (Fig. 565), which is then made to vibrate by drawing a 788 PRODUCTION AND PROPAGATION OF SOUND. bow over its edge. As soon as the plate begins to sound, the sand dances, leaves certain parts bare, and collects in definite lines, which are called noclal lines. These are, in fact, the lines which separate portions of the plate whose movements are in oppo¬ site directions. Their position changes whenever the plate changes its note. The vibratory condition of the plate is also mani¬ fested by another phenomenon, opposite—so to speak —to that just described. If very fine powder, such as lycopodium, be mixed with the sand, it will not move with the sand to the nodal lines, but will form little heaps in the centre of the vibrating segments; and these heaps will be in a state of violent agitation, with more or less of gyratory movement, as long as the plate is vibrating. This phe¬ nomenon, after long baffling explanation, was shown by Faraday to be due to indraughts of air, and ascending currents, , Flg ' 5 ?L. • ’ & ’ Vibration of String. brought about by the move¬ ments of the plate. In a moderately good vacuum, the lycopodium goes with the sand to the nodal lines. Vibration of a String .—When a note is produced from a musical string or wire, its vibrations are often of sufficient amplitude to be detected by the eye. The string thus assumes the appearance of an elongated spindle (Fig. 566). Vibration of the Air .—The sonorous body may sometimes be air, as in the case of organ- pipes, which we shall describe in a later chap¬ ter. It is easy to show by experiment that when a pipe speaks , the air within it is vibra¬ ting. Let one side of the tube be of glass, and let a small membrane m, stretched over a frame, be strewed with sand, and lowered into the pipe. The sand will be thrown into violent agitation, and the rattling of the grains, as they fall back on the membrane, is loud enough to be distinctly heard. SINGING FLAMES. 789 Singing Flames .—An experiment on the production of musical sound by flame, has long been known under the name of the chemi¬ cal harmonica. An appara¬ tus for the production of hydrogen gas (Fig. 568) is furnished with a tube, which tapers off nearly to a point at its upper end, where the gas issues and is lighted. When a tube, open at both ends, is held so as to surround the flame, a musical tone is heard, which varies with the dimensions of the tube, and often attains considerable power. The sound is due to the vibration of the air and products of combustion with¬ in the tube; and on observ¬ ing the reflection df the flame in a mirror rotating about a vertical axis, it will be seen that the flame is alternately rising and falling, its successive images, as drawn out into a horizontal series by the rotation of the mirror, resembling a number of equidistant tongues of flame, with depressions between them. The experiment may also be performed with ordinary coal-gas. Trevelyan Experiment .—A fire-shovel (Fig. 569) is heated, and balanced upon the edges of two sheets of lead fixed in a vice ; it is then seen to execute a series of small oscillations—each end being alternately raised and depressed—and a sound is at the same time emitted. The oscillations are so small as to be scarcely perceptible in themselves ; but they can be rendered very obvious by attaching to the shovel a small silvered mirror, on which a beam of light is directed. The reflected light can be made to form an image upon a screen, and this image is seen to be in a state of oscillation as long as the sound is heard. The movements observed in this experiment are due to the sudden expansion of the cold lead. When the hot iron comes in contact Fig. 568.—Chemical Harmonica. 790 PRODUCTION AND PROPAGATION OF SOUND. with it, a protuberance is instantly formed by dilatation, and the iron is thrown up. It then comes in contact with another portion Fig. 569.—Trevelyan Experiment. of the lead, where the same phenomenon is repeated while the first point cools. By alternate contacts and repulsions at the two points, the shovel is kept in a continual state of oscillation, and the regular succession of taps produces the sound. The experiment is more usually performed with a special instru¬ ment invented by Trevelyan, and called a rocker , which, after being heated and laid upon a block of lead, rocks rapidly from side to side, and yields a loud note. 630. Distinctive Character of Musical Sound.—It is not easy to draw a sharp line of demarcation between musical sound and mere noise. The name of noise is usually given to any sound which seems unsuited to the requirements of music. This unfitness may arise from one or the other of two causes. Either, 1. The sound may be unpleasant from containing discordant ele¬ ments which jar with one another, as when several consecutive keys on a piano are put down together. Or, 2. It may consist of a confused succession of sounds, the changes being so rapid that the ear is unable to identify any particular note. This kind of noise may be illustrated by sliding the finger along a violin-string, while the bow is applied. All sounds may be resolved into combinations of elementary musi¬ cal tones occurring simultaneously and in succession. Hence the study of musical sounds must necessarily form the basis of acoustics. Every sound which is recognized as musical is characterized by what may be called smoothness, evenness, or regularity; and the physical cause of this regularity is to be found in the accurate VEHICLE OF SOUND. 791 periodicity of the vibratory movements which produce the sound. By periodicity we mean the recurrence of precisely similar states at equal intervals of time, so that the movements exactly repeat them¬ selves; and the time which elapses between two successive recur¬ rences of the same state is called the period of the movements. Practically, musical and unmusical sounds often shade insensibly into one another. The tones of every musical instrument are accom¬ panied by more or less of unmusical noise. The sounds of bells and drums have a sort of intermediate character; and the confused as¬ semblage of sounds which is heard in the streets of a city blends at a distance into an agreeable hum. 631. Vehicle of Sound.—The origin of sound is alwaj^s to be found in the vibratory movements of a sonorous body; but these vibratory movements cannot bring about the sensation of hearing unless there be a medium to transmit them to the auditory apparatus. This medium may be either solid, liquid, or gaseous, but it is necessary that it be elastic. A body vibrating in an absolute vacuum, or in a medium utterly destitute of elasticity, would fail to excite our sensations of hearing. This assertion is justified by the following experiments : — 1. Under the receiver of an air-pump is placed a clock-work arrangement for producing a number of strokes on a bell. It is placed on a thick cushion of felt, or other inelastic material, and the air in the receiver is exhausted as com¬ pletely as possible. If the clock-work is then started by means of the handle g, the hammer will be seen to strike the bell, but the sound will be scarcely audi¬ ble. If hydrogen be introduced into the vacuum, and the receiver be again exhausted, the sound will be much more completely extinguished, being heard with difficulty even when the ear is placed in contact with the receiver. Hence it may fairly be con¬ cluded that if the receiver could be perfectly exhausted, and a per¬ fectly inelastic support could be found for the bell, no sound at all would be emitted. Fig. 570.—Sound in Exhausted Receiver. 792 PRODUCTION AND PROPAGATION OF SOUND. 2. The experiment may be varied by using a glass globe, furnished with a stop-cock, and having a little bell suspended within it by a thread. If the globe is exhausted of air, the sound of the bell will be scarcely audible. The globe may be filled with any kind of gas, or with vapour either saturated or non-saturated, and it will thus be found that all these bodies transmit sound. Sound is also transmitted through liquids, as may easily be proved by direct experiment. Experiment, however, is scarcely necessary for the establishment of the fact, seeing that fishes are provided with audi- Giobe with stop cock ^ory apparatus, and have often an acute sense of hearing. As to solids, some well-known facts prove that they transmit sound very perfectly. For example, light taps with the head of a pin on one end of a wooden beam, are distinctly heard by a person with his ear applied to the other end, though they cannot be heard at the same distance through air. This property is sometimes em¬ ployed as a test of the soundness of a beam, for the experiment will not succeed if the intervening wood is rotten, rotten wood being very inelastic. The stethoscope is an example of the transmission of sound through solids. It is a cylinder of wood, with an enlargement at each end, and a perforation in its axis. One end is pressed against the chest of the patient, while the observer applies his ear to the other. He is thus enabled to hear the sounds produced by various internal actions, such as the beating of the heart and the passage of the air through the tubes of the lungs. Even simple auscultation , in which the ear is applied directly to the surface of the body, implies the transmission of sound through the walls of the chest. By applying the ear to the ground, remote sounds can often be much more distinctly heard; and it is stated that savages can in this way obtain much information respecting approaching bodies of enemies. We are entitled then to assert that sound , as it affects our organs of hearing , is an effect which is propagated , from a vibrating body, through an elastic and ponderable medium. 632. 1 Mode of Propagation of Sound.—We will now endeavour to explain the action by which sound is propagated. 1 The numbering of §§ 632-638 does not correspond with the original, some transpose tions having been made. MODE OF PROPAGATION OF SOUND. 793 Let there be a plate a vibrating opposite the end of a long tube, and let us consider what happens during the passage of the plate from its most backward position a", to its most advanced position a. This movement of the plate may be divided in imagination into a number of successive parts, each of which is communicated to the layer of air close in front of it, which is thus compressed, and, in its Fig. 573.—Propagation of Sound. endeavour to recover from this compression, reacts upon the next layer, which is thus in its turn compressed. The compression is thus passed on from layer to layer through the whole tube, much in the same way as, when a number of ivory balls are laid in a row, if the first receives an impulse which drives it against the second, each ball will strike against its successor and be brought to rest. o o The compression is thus passed on from layer to layer through the tube, and is succeeded by a rarefaction corresponding to the back¬ ward movement of the plate from a' to a". As the plate goes on vibrating, these compressions and rarefactions continue to be propa¬ gated through the tube in alternate succession. The greatest com¬ pression in the layer immediately in front of the plate, occurs when the plate is at its middle position in its forward movement, and the greatest rarefaction occurs when it is in the same position in its backward movement. These are also the instants at which the plate is moving most rapidly. 1 When the plate is in its most advanced position, the layer of air next to it, A (Fig. 574), will be in its natu¬ ral state, and another layer at A h half a wave-length further on, will also be in its natural state, the pulse having travelled from A to A h while the plate was moving from a" to a'. At intervening points between A and A x , the layers will have various amounts of compression corres¬ ponding to the different positions of the plate in its forward move- ( __D* 1 ^ c r> A, Fig. 574.—Graphical Representation. 1 See § 632 a, also Note A at the end of this chapter. 794 PRODUCTION AND PROPAGATION OF SOUND. ment. The greatest compression is at C, a quarter of a wave-length in advance of A, having travelled over this distance while the plate was advancing from a to a'. The compressions at D and D x repre¬ sent those which existed immediately in front of the plate when it had advanced respectively one-fourth and three-fourths of the dis¬ tance from a" to a, and the curve AC' A x is the graphical represen¬ tation both of condensation and velocity for all points in the air between A and A x . If the plate ceased vibrating, the condition of things now existing in the portion of air A A 2 would be transferred to successive portions of air in the tube, and the curve A C' A x would, as it were, slide onward through the tube with the velocity of sound, which is about 1100 feet per second. But the plate, instead of remaining perman¬ ently at a', executes a backward movement, and produces rarefactions and retrograde velocities, which are propagated onwards in the same manner as the condensations and forward velocities. A complete wave of the undulation is accordingly represented by the curve A E' A x C' A 2 (Fig. 575), the portions of the curve below the line of Fig. 575.—Graphical Representation of Complete Wave. abscissas being intended to represent rarefactions and retrograde velo¬ cities. If we suppose the vibrating plate to be rigidly connected with a piston which works air-tight in the tube, the velocities of the particles of air in the different points of a wave-length will be iden¬ tical with the velocities of the piston at the different parts of its motion. The wave-length A A 2 is the distance that the pulse has travelled while the vibrating plate was moving from its most backward to its most advanced position, and back again. During this time, which is called the period of the vibrations, each particle of air goes through its complete cycle of changes, both as regards motion and density, The period of vibration of any particle is thus identical with that of the vibrating plate, and is the same as the time occupied by the waves in travelling a wave-length. Thus, if the plate be one leg of a common A tuning-fork, making 435 complete vibrations per second, the period will be T -^-g-th of a second, and the undulation will travel in NATURE OF UNDULATIONS. 795 this time a distance of VA° feet, or 2 feet 6 inches, which is there¬ fore the wave-length in air for this note. If the plate continues to vibrate in a uniform manner, there will be a continual series of equal and similar waves running along the tube with the velocity of sound. Such a succession of waves constitutes an undulation. Each wave consists of a condensed portion, and a rarefied portion, which are distinguished from each other in Fig. 573 by different tints, the dark shading being intended to represent condensation. 632a. Nature of Undulations.—The possibility of condensations and rarefactions being propagated continually in one direction, while each particle of air simply moves backwards and forwards about its original position, is illustrated by Fig. 575 a, which represents, in an ABC D E F AB C D E F A B C D E F A BC D E F A BCD E F A B CD E F A B CDE F , A B C DE F & A B C DE F a, A B C D EF ^ ABC D EF A B C D E Fa ABC D E Fa, Fig. 575 a.— Longitudinal Vibration. exaggerated form, the successive phases of an undulation propagated through 7 particles ABCDEFa originally equidistant, the dis¬ tance from the first to the last being one wave-length of the undula¬ tion. The diagram is composed of thirteen horizontal rows, the first and last being precisely alike. The successive rows represent the positions of the particles at successive times, the interval of time from each row to the next being xgth of the period of the undulation. In the first row A and a are centres of condensation, and D is a centre of rarefaction. In the third row B is a centre of condensa¬ tion, and E a centre of rarefaction. In the fifth row the con- a cu a'/ Oy 796 PRODUCTION AND PROPAGATION OF SOUND. densation and rarefaction have advanced by one more letter, and so on through the whole series, the initial state of things being reproduced when each of these centres has advanced through a wave¬ length, so that the thirteenth row is merely a repetition of the first. The velocities of the particles can be estimated by the comparison of successive rows. It is thus seen that the greatest forward velocity is at the centres of condensation, and the greatest backward velocity at the centres of rarefaction. Each particle has its greatest veloci¬ ties, and greatest condensation and rarefaction, in passing through its mean position, and comes for an instant to’rest in its positions ol greatest displacement, which are also positions of mean density. The distance between A and a remains invariable, being always a wave-length, and these two particles are always in the same phase. Any other two particles represented in the diagram are always in different phases, and the phases of A and D, or B and E, or C and F, are always opposite; for example, when A is moving forwards with the maximum velocity, D is moving backwards with the same velocity. The vibrations of the particles, in an undulation of this kind, are called longitudinal; and it is by such vibrations that sound is pro¬ pagated through air. Fig. 575 b illustrates the manner in which an undulation may be propagated by means of transverse vibrations, that is to say, by vibrations executed in a direction perpendicular to that in which the undulation advances. Thirteen particles A B C I) EFGHI JKLa are represented in the positions which they occupy at successive times, whose interval is one-sixth of a period. At the instant first considered, D and J are the particles which are furthest displaced. At the end of the first interval, the wave has advanced two letters, so that F and L are now the furthest displaced. At the end of the next interval, the wave has advanced two letters further, and so on, the state of things at the end of the six intervals, or of one complete period, being the same as at the beginning, so that the seventh line is merely a repetition of the first. Some examples of this kind of wave-motion will be mentioned in later chapters. 633. Propagation in an Open Space.—When a sonorous disturb¬ ance occurs in the midst of an open body of air, the undulations to which it gives rise run out in all directions from the source. If the disturbance is symmetrical about a centre, the waves will be spheri¬ cal ; but this case is exceptional. A disturbance usually produces condensation on one side, at the same instant that it produces rare- PROPAGATION IN AN OPEN SPACE. 797 faction on another. This is the case, for example, with a vibrating plate, since, when it is moving towards one side, it is moving away from the other. These inequalities which exist in the neighbour¬ hood of the sonorous body, have, how T ever, a tendency to become less marked, and ultimately to disappear, as the distance is increased. Fig. 576 represents a diametral section of a series of spherical waves. Their mode of propagation has some analogy to that of the circular A A B B c D E A B C A C D E C _ A B c E F G E _ C D £ E F G H c G H H H H I J K I J K G H I G K I J K CL K l cl K ~ a ~ CL K L cl CL a/ Fig. 575 b. —Transverse Vibration. waves produced on water by dropping a stone into it; but the par¬ ticles which form the waves of water are elevated and depressed; whereas those which form sonorous waves merely advance and retreat, their lines of motion being always coincident with the di¬ rections along which the sound travels. In both cases it is im¬ portant to remark that the undulation does not involve a movement of transference. Thus, when the surface of a liquid is traversed by waves, bodies floating on it rise and fall, but are not carried onward. This property is characteristic of undulations generally. An undu - 798 PRODUCTION AND PROPAGATION OF SOUND. lation may be defined as a system of movements in which the several particles move to and fro, or round and round , about definite points , in such a manner as to produce the continued onward transmission of a condition, or series of conditions. There is one important difference between the propagation of sound in a uniform tube and in an open space. In the former case, the layers of air corresponding to successive wave-lengths are of equal mass, and their movements are precisely alike, except in so far as they are interfered with by friction. Hence sound is transmitted through tubes to great distances with but little loss of intensity, especially if the tubes are large. 1 1 Regnault, in his experiments on the velocity of sound, found that in a conduit *108 of a metre in diameter, the report of a pistol charged with a gramme of powder ceased to be heard at the distance of 1150 metres. In a conduit of *3 m , the distance was 3810 m . In the great conduit of the St. Michel sewer, of l m> 10, the sound was made by successive reflections to traverse a distance of 10,000 metres without becoming inaudible.— D. DISSIPATION OF SONOROUS ENERGY. 799 In an open space, each successive layer has to impart its own condition to a larger layer; hence there is a continual diminution of amplitude in the vibrations as the distance from the source increases. This involves a continual decrease of loudness. An undulation involves the onward transference of energy; and the amount of energy which traverses, in unit time, any closed surface described about the source, must be equal to the energy which the source emits in unit time. Hence, by the reasoning which we employed in the case of radiant heat (§ 308), it follows that the intensity of sonorous energy diminishes according to the law of inverse squares. The energy of a particle executing simple vibrations in obedience to forces of elasticity, varies as the square of the amplitude of its excursions; for, if the amplitude be doubled, the distance worked through, and the mean working force, are both doubled, and thus the work which the elastic forces do during the movement from either extreme position to the centre is quadrupled. This work is equal to the energy of the particle in any part of its course. At the ex¬ treme positions it is all in the shape of potential energy; in the middle position it is all in the shape of kinetic energy; and at intermediate points it is partly in one of these forms, and partly in the other. If we sum the potential energies of all the particles which consti¬ tute one wave, and also sum their kinetic energies, we shall find the two sums to be equal. 1 633 a. Dissipation of Sonorous Energy.—The reasoning by which we have endeavoured to establish the law of inverse squares, assumes that onward propagation involves no loss of sonorous energy. This assumption is not rigorously true, inasmuch as vibration implies friction, and friction implies the generation of heat, at the expense of the energy which produces the vibrations. Sonorous energy must therefore diminish with distance somewhat more rapidly than ac¬ cording to the law of inverse squares. All sound, in becoming extinct, becomes converted into heat. This conversion is greatly promoted by defect of homogeneity in In the case of one of the particles, the potential energy at distance y from the position of equilibrium is half the product of force by distance, and may be denoted by ~ y 2 ; the kinetic energy is ~ (a 2 — y 1 ), a being the amplitude. The former of these quantities may be written it a 2 cos 2 d, and the latter will be ~ a 2 sin 2 6. In dealing with the series of particles which form one wave, 6 is equicrescent from particle to particle, and its limiting values differ by an entire circumference. Under these conditions, it is obvious that the mean values of cos 2 d and sin 2 0 are equal, and that each of them is equal to 800 PRODUCTION AND PROPAGATION OF SOUND. the medium of propagation. In a fog, or a snow-storm, the liquid o* solid particles present in the air produce innumerable reflections, in each of which a little sonorous energy is converted into heat. 634. Velocity of Sound in Air.—The propagation of sound through an elastic medium is not instantaneous, but occupies a very sensible time in traversing a moderate distance. For example, the flash of a gun at the distance of a few hundred yards is seen some time before the report is heard. The interval between the two impressions may be regarded as representing the time required for the propagation of the sound across the intervening distance, for the time occupied by the propagation of light across so small a distance is inappreciable. It is by experiments of this kind that the velocity of sound in air has been most accurately determined. Among the best determina¬ tions may be mentioned that of Lacaille, and other members of a commission appointed by the French Academy in 1738; that ol Arago, Bouvard, and other members of the Bureau de Longitudes, in 1822; and that of Moll, Vanbeek, and Kuytenbrouwer in Holland, in the same year. All these determinations were obtained by firing cannon at two stations, several miles distant from each other, and noting, at each station, the interval between seeing the flash and hearing the sound of the guns fired at the other. If guns were fired only at one station, the determination would be vitiated by the effect of wind blowing either with or against the sound. The error from this cause is nearly eliminated by firing the guns alternately at the two stations, and still more completely by firing them simultaneously. This last plan was adopted by the Hutch observers, the distance of the two stations in their case being about nine miles. Begnault has quite recently repeated the investigation, taking advantage of the important aid afforded by modern electrical methods for registering the times of observed phenomena. All the most careful determina¬ tions agree very closety among themselves, and show that the velo¬ city of sound through air at 0°C. is about 332 metres, or 1090 feet per second. 1 The velocity increases with the temperature, being 1 A recent determination by Mr. Stone at the Cape of Good Hope is worthy of note as being based on the comparison of observations made through the sense of hearing alone. It had previously been suggested that the two senses of sight and hearing, which are con¬ cerned in observing the flash and report of a cannon, might not be equally prompt in receiving impressions (Airy on Sound, p. 131). Mr. Stone accordingly placed two ob¬ servers—one near a cannon, and the other at about three miles distance; each of whom, on hearing the report, gave a signal through an electric telegraph. The result obtained was in precise agreement with that stated in the text. VELOCITY OF SOUND IN AIR. 801 proportional to the square root of what we have called in § 219 A the absolute temperature. If t denote the ordinary Centigrade tem¬ perature, and a the coefficient of expansion -00366, the velocity of sound through air at any temperature is given by the formula 332 y/\+at\n metres per second, or 1090 y/1 + a t in feet per second. The actual velocity of sound from place to place on the earth’s sur¬ face is found by compounding this velocity with the velocity of the wind. There is some reason, both from theory and experiment, for be¬ lieving that very loud sounds travel rather faster than sounds of moderate intensity. 635. Theoretical Computation of Velocity.—By applying the prin¬ ciples of dynamics to the propagation of undulations, 1 it is computed that the velocity of sound through air must be given by the formula v = V® (l) D denoting the density of the air, and E its elasticity, as measured by the quotient of pressure applied by compression produced. Let P denote the pressure of the air in units of force per unit of area; then, if the temperature be kept constant during compression, a small additional pressure p will, by Boyle’s law, produce a com¬ pression equal to-^-, and the value of E, being the quotient of p b}?- this quantity, will be simply P. On the other hand, if no heat is allowed either to enter or escape, the temperature of the air will be raised by compression, and addi¬ tional resistance will thus be encountered. In this case the compres¬ sion (^in §347 A) will be p ^ , l+P denoting the ratio of the two specific heats, which for air and simple gases is about 1 *41; and the value of E will be P (1 +/3). It thus appears that the velocity of sound in air cannot be less than nor greater than 1*41 Its actual velocity, as determined by observation, is nearly identical with the latter of these limiting values. It is probable that the compressions and extensions which the particles of air undergo in transmitting sound are of too brief duration to allow of any sensible transference of heat from particle to particle. 1 See Note B at the end of this chapter. 52 802 PRODUCTION AND PROPAGATION OF SOUND. The following is the actual process of calculation for perfectly dry air at 0°C., the centimetre, gramme, and second being taken as the units of length, mass, and time. The density of dry air at 0°, under the pressure of 1033 grammes per square centimetre, at Paris, is 001293 of a gramme per cubic centimetre. But the gravitating force of a gramme at Paris is 981 (§38). The density '001293 therefore corresponds to a pressure of 1033x981 units of force per unit of area; and the expression for the velocity in centimetres per second is v= / 1*41— = /l-41 1033 x 9 81 =33210 nearly; V V -001293 that is, 332 4 metres per second, or 1093 feet per second. 635 a. Effects of Pressure, Temperature, and Moisture.—The velo¬ city of sound is independent of the height of the barometer, since changes of this element (at constant temperature) affect P and D in the same direction, and to the same extent. For a given density, if P 0 denote the pressure at 0°, and a the coefficient of expansion of air, the pressure at t 0 Centigrade is P 0 (l+a£), the value of a being about Hence, if the velocity at 0° be 1090 feet per second, the velocity at t° will be 1090,^/At the temperature 50° F. or 10° C.> which is approximately the mean annual temperature of this country, the value of this expression is about 1110, and at 86° F. or 30° C. it is about 1148. The increase of velocity is thus about a foot per second for each degree Fahrenheit. The humidity of air has some influence on the velocity of sound, inasmuch as aqueous vapour is lighter than air, but the effect is comparatively trifling, at least in temperate climates. At the tem¬ perature 50° F., air saturated with moisture is less dense than dry air by about 1 part in 220, and the consequent increase of velocity cannot be greater than about 1 part in 440, which will be between 2 and 3 feet per second. The increase should, in fact, be somewhat less than this, inasmuch as the value of 1+/3 (the ratio of the two specific heats) appears to be only 1*31 for aqueous vapour. 1 635b. Newton’s Theory, and Laplace’s Modification.—The earliest theoretical investigation of the velocity of sound was that given by Newton in the Principia (book 2, section 8). It proceeds on the 1 Rankine on the Steam Engine , p. 320. VELOCITY IN GASES GENERALLY. 803 tacit assumption that no changes of temperature are produced by the compressions and extensions which enter into the constitution of a sonorous undulation; and the result obtained by Newton is equivalent to the formula ,P. p or since (§ 111 A) ^=g H, where H denotes the height of a homo¬ geneous atmosphere , and the velocity acquired in falling through any height s is V 2 g s, the velocity of sound in air is, according to Newton, the same as the velocity which would be acquired by falling in vacuo through half the height of a homogeneous atmosphere. This in fact, is the form in which Newton states his result. 1 Newton himself was quite aware that the value thus computed theoretically was too small, and he throws out a conjecture as to the cause of the discrepancy; but the true cause was first pointed out by Laplace, as depending upon increase of temperature produced by compression, and decrease of temperature produced by expansion. 635c. Velocity in Gases generally.—The same principles which apply to air apply to gases generally; and since for all simple gases the ratio of the two specific heats is 1*41, the velocity of sound in any simple gas is^/141 D denoting its absolute density at the pressure P. Comparing two gases at the same pressure, we see that the velocities of sound in them will be inversely as the square roots of their absolute densities; and this will be true whether the tem¬ peratures of the two gases are the same or different. 636. Velocity of Sound in Liquids. — The velocity of sound in water was measured by Colladon, in 1826, at the Lake of Geneva. Two boats were moored at a distance of 13,500 metres (between 8 and 9 miles). One of them carried a bell, weighing about 110 lbs., immersed in the lake. Its hammer was moved by an external lever, so arranged as to ignite a small quantity of gunpowder at the instant of striking the bell. An observer in the other boat was enabled to hear the sound by applying his ear to the extremity of a trumpet¬ shaped tube (Fig. 572), having its lower end covered with a mem¬ brane and facing towards the direction from which the sound pro- 1 Newton’s investigation relates only to simple waves; but if these have all the same velocity (as Newton shows), this must also be the velocity of the complex wave which they compose. Hence the restriction is only apparent. 804 PRODUCTION AND PROPAGATION OF SOUND. ceeded. T$y noting the interval between seeing the flash and hearing the sound, the velocity with which the sound travelled through the water was determined. The velocity thus com¬ puted was 1435 metres" per second, and the temperature of the water was 8°T C. Formula (1) of § 633 holds for liquids as well as for gases, and is easily applied to the case of water if we neglect the changes of temperature produced by compression and extension. We have stated in § 22 (Part I.) that the compressi¬ bility of water, as determined by the most re¬ cent experiments, is 0000457 per atmosphere, at the temperature of 15° C. The value of E in teTms of the units employed in § 635 is therefore anc ^ ^ the mass of a cubic centimetre of water expressed in grammes, is unity. We have therefore /E /1033 x 981 , AQMn i *-A/B = V ~r = 148920 near,y - which, reduced to metres per second, is 1489‘2. This computation applies to water at 15°, which is 7° warmer than the water of the lake. As the elasticity of water is known to increase with its temperature, the difference between the two results is in the right direction. The agreement is sufficiently close to show that the increase of elasticity from the instantaneous changes of temperature produced by sonorous undulations is insignificant in the case of water. 1 1 Sir W. Thomson has investigated, on thermo-dynamic principles, the additional pres¬ sure required to produce a given diminution of volume, when the heat of compression is not allowed to escape. He computes that the elasticity of a fluid under these circum¬ stances is to its elasticity at constant temperature as 1 + E a 2 T JC to 1, E denoting the elasticity at constant temperature, a the coefficient of expansion of the fluid per degree Centigrade, T the absolute temperature of the fluid, or the common Centigrade temperature increased by 273°, C the thermal capacity of unit volume of the liquid, and J Joule’s equivalent for a unit of heat. If E be expressed in absolute units of force per unit of area, J must be expressed in absolute units of work, and will be 42400 x 981 if the centimetre, the gramme, and the second be the units of length, mass, and time. For water at 15° C. the coefficient of expansion a is about ’00015, T is 288, and C is unity. The value of ^ a ^ J (j will be found to be about ’003, so that the heat of compres¬ sion and cold of expansion increase the effective elasticity by 3 parts in a thousand, and therefore increase the velocity of sound by 1§ part in a thousand. The same formula applies to solids if we make E denote Young’s modulus, and a the coefficient of linear expansion. For iron it gives, according to Sir W. Thomson ( Proc. R. S. E. 1865-6), an increase of about g per cent, in Young’s modulus, and therefore of | per cent, in the velocity of sound. VELOCITY OF SOUND IN SOLIDS. 805 Wertheim has measured the velocity of sound in some liquids by an indirect method, which will be explained in a later chapter. He finds it to be 1160 metres per second in ether and alcohol, and 1900 in a solution of chloride of calcium. 637. Velocity of Sound in Solids.—The velocity of sound in cast- iron was determined by Biot and Martin by means of a connected series of water-pipes, forming a conduit of a total length of 951 metres. One end of the conduit was struck with a hammer, and an observer at the other end heard two sounds, the first transmitted by the metal, and the second by the air, the interval between them being 2'5 seconds. Now the time required for travelling this dis¬ tance through air, at the temperature of the experiment (11° C.), is 2*8 seconds. The time of transmission through the metal was there¬ fore *3 of a second, which is at the rate of 3170 metres per second. It is, however, to be remarked, that the transmitting body was not a continuous mass of iron, but a series of 376 pipes, connected to¬ gether by collars of lead and tarred cloth, which must have consid¬ erably delayed the transmission of the sound. But in spite of this, the velocity is about nine times greater than in air. Wertheim, by the indirect methods above alluded to, measured the velocity of sound in a number of solids, with the following results, the velocity in air being taken as the unit of velocity:— Lead,. . 3-974 to 4-120 Steel, . . . . . 14-361 to 15-108 Tin,. . 7 338 to 7*480 Iron. . . 15-108 Gold,. . 5 "603 to 6-424 Brass, . . . . . 10-224 Silver,. . 7-903 to 8-057 Glass, . . 14-956 to 16-759 Zinc,. . 9-863 to 11-009 Flint Glass, . . 11-890 to 12-220 Copper, .... . 11*167 Oak, .... . . 9-902 to 12-02 Platinum. . . . . 7*823 to 8-467 Fir, .... . . 12-49 to 17*26 638. Theoretical Computation.—The formula serves for solids as well as for liquids and gases; but as solids can be subjected to many different kinds of strain, whereas liquids and gases can be subjected to only one, we may have different values of E, and dif¬ ferent velocities of transmission of pulses for the same solid. This is true even in the case of a solid whose properties are alike in all directions (called an isotrojpic solid); but the great majority of solids are very far from fulfilling this condition, and transmit sound more rapidly in some directions than in others. When the sound is propagated by alternate compressions and extensions running along a substance which is not prevented from 806 PRODUCTION AND PROPAGATION OF SOUND. extending and contracting laterally, the elasticity E becomes iden¬ tical 1 with Young’s modulus (§ 23). On the other hand, if uniform spherical waves of alternate compression and extension spread out¬ wards, symmetrically, from a point in the centre of an infinite solid, lateral extension and contraction will be prevented by the symmetry of the action. The effective elasticity is, in this case, greater than Young’s modulus, and the velocity of sound will be increased accord- i'igty- By the table on p. 29 the value of Young’s modulus for copper is 12,558 kilogrammes per square millimetre, or 1,255,800,000 grammes per square centimetre, and by the table on p. 89 the density of copper in grammes per cubic centimetre is 8*8. Hence, for the velo¬ city of sound through a copper rod, in centimetres per second, we have /E /1255800000 x 981 o 741 m »= VD V -F8-' = 374150 nearly, or 3741 5 metres per second. This is about 11 2 times the velocity in air. 639. Reflection of Sound.—When sonorous waves meet a fixed obstacle they are reflected, and the two sets of waves—one direct, Fig. 577.—Reflection of Sound. and the other reflected—are propagated just as if they came from two separate sources. If the reflecting surface is plane, waves di¬ verging from any centre O in front of it are reflected so as to diverge 1 Subject to the small correction mentioned in the foot-note to § 636. REFLECTION OF SOUND. 807 from a centre O' symmetrically situated behind it, and an ear at any point M in front hears the reflected sound as if it came from O'. The direction from which a sound appears to the hearer to proceed is determined by the direction along wdiich the sonorous pulses are propagated, and is always normal to the waves. A normal to a set of sound-waves may therefore conveniently be called a ray of sound. 0 I is a direct ray, and I M the corresponding reflected ray; and it is obvious, from the symmetrical posi¬ tion of the points 0 O', that these two rays are equally inclined to the surface, or the angles of inci¬ dence and reflection are equal. 640. Illustrations of Reflection of Sound.—The reflection of sonorous waves explains some well-known phenomena. If aba be an elliptic dome or arch, a sound emitted from either of the foci//will be reflected from the elliptic surface a( \ , tv r / v- Fig. 578.—Reflection from Elliptic Roof. Fig. 579.—Reflection of Sound from Conjugate Mirrors. in such a direction as to pass through the other focus. A sound emitted from either focus may thus be distinctly heard at the other, even when quite inaudible at nearer points. This is a consequence 808 PRODUCTION AND PROPAGATION OF SOUND. of the property, that lines drawn to any point on an ellipse from the two foci are equally inclined to the curve. The experiment of the conjugate mirrors (§ 311) is also applicable to sound. Let a watch be hung in the focus of one of them (Fig. 579), and let a person hold his ear at the focus of the other; or still better, to avoid intercepting the sound before it falls on the second mirror, let him employ an ear-trumpet, holding its open end at the focus. He will distinctly hear the ticking, even when the mirrors are many yards apart. 1 641. Echo.—Echo is the most familiar instance of the reflection of sound. In order to hear the echo of one’s own voice, there must be a distant body capable of reflecting sound directly back, and the number of syllables that an echo will repeat is proportional to the distance of this obstacle. Reckoning j of a second as the time of pronouncing a syllable, the space traversed by sound in this time is Fig. 580.- Echo. about 200 feet, and an obstacle must be at half this distance in order that it may be able to send back a single syllable. The sounds reflected to the speaker have travelled first over the distance 0 A (Fig. 580) from him to the reflecting body, and then back from A to O. Supposing five syllables to be pronounced in a second, and taking the velocity of sound as 1100 feet per second, a distance of 1 Sondhaus has shown that sound, like light, is capable of being refracted. A spherical balloon of collodion, filled with carbonic acid gas, acts as a sound-lens. If a watch be hung at some distance from it on one side, an ear held at the conjugate focus on the other side will hear the ticking. HEARING AND SPEAKING TRUMPETS. 809 550 feet from the speaker to the reflecting body would enable the speaker to complete the fifth syllable before the return of the first; this is at the rate of 110 feet per syllable. At distances less than about 100 feet there is not time for the distinct reflection of a single syllable; but the reflected sound mingles with the voice of the speaker. This is particularly observable under vaulted roofs. Multiple echoes are not uncommon. They are due, in some cases, to independent reflections from obstacles at different distances; in others, to reflections of re¬ flections. A position exactly midway between two parallel walls, at a sufficient distance apart, is favour¬ able for the observance of this latter phenomenon. One of the most frequently cited instances of multiple echoes is that of the old palace of Simonetta, near Milan, which forms three sides of a quadrangle. According to Kircher, it repeats forty times. 642. Speaking and Hearing Trumpets.—The complete explanation of the action of these instruments presents considerable difficulty. The speaking-trumpet (Fig. 581) consists of a long tube (sometimes 6 feet long), slightly tapering towards the speaker, furnished at this end with a hollow mouth-piece, which nearly fits the lips, and at the other with a funnel-shaped enlargement, called the bell, opening out to a width of about a foot. It is much used at sea, and is found very effectual in making the voice heard at a distance. The expla¬ nation usually given of its action is, that the slightly conical form of the long tube Fig. 5S1. Speaking-trumpet. produces a series of reflec¬ tions in directions more and more nearly parallel to the axis; but this explanation fails to account for the utility of the bell , which experience has shown to be considerable. Ear-trumpets have various forms, as represented in Fig. 582; 810 PRODUCTION AND PROPAGATION OF SOUND. having little in common, except that the external opening or bell is much larger than the end which is introduced into the ear. Mem¬ branes of gold-beaters’ skin are sometimes stretched across their interior, in the positions indicated by the dotted lines in Nos. 4 and 5. No. G consists simply of a bell with such a membrane stretched across its outer end, while its inner end communicates with the ear by an indian-rubber tube with an ivory end-piece. These light membranes are peculiarly susceptible of impression from aerial vibrations. In Regnault’s experiments above cited, it was found that membranes were affected at distances greater than those at which sound was heard. 643. Interference of Sonorous Undulations.—When two systems of waves are traversing the same matter, the actual motion of each particle of the matter is the resultant of the motions due to each system separately. When these component motions are in the same direction the resultant is their sum; when they are in opposite directions it is their difference; and if they are equal, as well as opposite, it is zero. Very remarkable phenomena are thus produced when the two undulations have the same, or nearly the same wave¬ length; and the action which occurs in this case is called interference. When two sonorous undulations of exactly equal wave length and amplitude are traversing the same matter in the same direction, their phases must either be the same, or must everywhere differ by the same amount. If they are the same, the amplitude of vibration for each particle will be double of that due to either undulation separately. If they are opposite—in other words, if one undulation be half a wave-length in advance of the other—the motions which they would separately produce in any particle are equal and oppo¬ site, and the particle will accordingly remain at rest. Two sounds will thus, by their conjoint action, produce silence. In order that the extinction of sound may be complete, the rare¬ fied portions of each set of waves must be the exact counterparts of the condensed portions of the other set, a condition which can only be approximately attained in practice. The following experiment, due to M. Desains, affords a very direct illustration of the principle of interference. The bottom of a wooden box is pierced with an opening, in which a powerful whistle fits. The top of the box has two larger openings symmetrically placed with respect to the lower one. The inside of the box is lined with felt, to prevent the vibrations from being communicated to the box, INTERFERENCE OF SONOROUS UNDULATIONS. 811 and to weaken internal reflection. When the whistle is sounded, if a membrane, with sand strewn on it, is held in various positions in the vertical plane which bisects, at right angles, the line joining the two openings, the sand will be agitated, and will arrange itself in nodal lines. But if it is carried out of this plane, positions will be found, at equal distances on both sides of it, at which the agitation is scarcely perceptible. If, when the membrane is in one of these positions, we close one of the two openings, the sand is again agitated, clearly showing that the previous absence of agitation was due to the interference of the undulations proceeding from the two orifices. In this experiment the proof is presented to the eye. In the fol¬ lowing experiment, which is due to M. Lissajous, it is presented to the ear. A circular plate, supported like the plate in Fig. 565, is made to vibrate in sectors separated by radial nodes. The num¬ ber of sectors will always be even, and adjacent sectors will vi¬ brate in opposite directions. Let a disk of card-board of the same size be divided into the same number of sectors, and let alternate sectors be cut away, leaving only enough near the centre to hold the remaining sectors together. If the card be now held just over the vibrating disk, in such a manner that the sectors of the one are exactly over sectors of the other, a great increase of loudness will be observed, consequent on the suppression of the sound from alternate sectors; but if the card-board disk be turned through the width of half a sector, the effect no longer occurs. If the card is made to rotate rapidly in a continuous manner, the alternations of loudness will form a series of beats. It is for a similar reason that, when a large bell is vibrating, a person in its centre hears the sound as only moderately loud, while within a short distance of some portions of the edge the loudness is intolerable. 644. Interference of Direct and Reflected Waves. Nodes and Anti¬ nodes.—Interference may also occur between undulations travelling in opposite directions; for example, between a direct and a reflected sys¬ tem. When waves proceeding along a tube meet a rigid obstacle, form¬ ing a cross section of the tube, they are reflected directly back again, the motion of any particle close to the obstacle being compounded of that due to the direct wave, and an equal and opposite motion due to the reflected wave. The reflected waves are in fact the images (with reference to the obstacle regarded as a plane mirror) of the waves which would exist in the prolongation of the tube if the 812 PRODUCTION AND PROPAGATION OF SOUND. obstacle were withdrawn. At the distance of half a wave-length from the obstacle the motions due to the direct and reflected waves will accordingly be equal and opposite, so that the particles situated at this distance will be permanently at rest; and the same is true at the distance of any number of half wave-lengths from the obstacle. The air in the tube will thus be divided into a number of vibrating segments separated by nodal planes or cross sections of no vibra¬ tion arranged at distances of half a wave-length apart. One of these nodes is at the obstacle itself. At the centres of the vibrating seg¬ ments—that is to say, at the distance of a quarter wave-length plus any number of half wave-lengths from the obstacle or from any node —the velocities due to the direct and reflected waves will be equal and in the same direction, and the amplitude of vibration will ac¬ cordingly be double of that due to the direct wave alone. These are the sections of greatest disturbance as regards change of place. We shall call them antinodes. On the other hand, it is to be remem¬ bered that motion with the direct wave is motion against the re¬ flected waves, and vice versa , so that (§ 632) at points where the velocities due to both have the same absolute direction they corre¬ spond to condensation in the case of one of these undulations, and to rarefaction in the case of the other. Accordingly, these sections of maximum movement are the places of no change of density; and on the other hand, the nodes are the places where the changes of density are greatest. If the reflected undulation is feebler than the direct one, as will be the case, for example, if the obstacle is only imper¬ fectly rigid, the destruction of motion at the nodes and of change of density at the antinodes will not be complete; the former will merely be places of minimum motion, and the latter of minimum change of density. Direct experiments in verification of these principles, a wall being the reflecting body, were conducted by Savart, and also by Seebeck, the latter of whom employed a testing apparatus called the acoustic pendulum. It consists essentially of a small membrane stretched in a frame, from the top of which hangs a very light pendulum, with its bob resting against the centre of the membrane. In the middle por¬ tions of the vibrating segments the membrane, moving with the air on its two faces, throws back the pendulum, while it remains nearfy free from vibration at the nodes. Regnault made extensive use of the acoustic pendulum in his ex¬ periments on the velocity of sound. The pendulum, when thrown BEATS PRODUCED BY INTERFERENCE. 813 back by the membrane, completed an electric circuit, and thus effected a record of the instant when the sound arrived. 644a. Beats Produced by Interference.—When two notes, which are not quite in unison, are sounded together, a peculiar palpitating effect is produced;—we hear a series of bursts of sound, with inter¬ vals of comparative silence between them. The bursts of sound are called beats, and the notes are said to beat together. If we have the power of tuning one of the notes, we shall find that as they are brought more nearly into unison, the beats become slower, and that, as the departure from unison is increased, the beats become more rapid, till they degenerate first into a rattle, and then into a discord. The effect is most striking with deep notes. These beats are completely explained by the principle of interfer¬ ence. As the wave-lengths of the two notes are slightly different, while the velocity of propagation is the same, the two systems of waves will, in some portions of their course, agree in phase, and thus strengthen each other; while in other parts they will be opposite in phase, and will thus destroy each other. Let one of the notes, for example, have 100 vibrations per second, and the other 101. Then, if we start from an instant when the maxima of condensation from the two sources reach the ear together, the next such conjunction will occur exactly a second later. During the interval one of the systems of waves has been gradually falling behind the other, till, at the end of the second, the loss has amounted to one wave-length. At the middle of the second it will have amounted to half a wave¬ length, and the two sounds will destroy each other. We shall thus have one beat and one extinction in each second, as a consequence of the fact that the higher note has made one vibration more than the lower. In general, the frequency of beats is the difference of the frequencies of vibration of the beating notes. Note A. § 632. That the particles which are moving forward are in a state of compression, may be shown in the following way:—Consider an imaginary cross section travelling forward through the tube with the same velocity as the undulation. Call this velocity v, and the velocity of any particle of air u. Also let the density of any particle be denoted by p. Then u and p remain constant for the imaginary moving section, and the mass of air which it traverses in its motion per unit time is (v - u) p. As there is no permanent transfer of air in either direction through the tube, the mass thus traversed must be the same as if the air were at rest at its natural density. Hence the value of (v - u) p is the same for 814 PRODUCTION AND PROPAGATION OF SOUND. all cross sections ; whence it follows, that where u is greatest p must be greatest, and where u is negative p is less than the natural density. If p 0 denote the natural density, we have (v-u) p = v p 0 , whence— = £—; that is to say, v p the ratio of the velocity of a particle to the velocity of the undulation is equal to the conden¬ sation existing at the particle. If u is negative—that is to say, if the velocity be retrograde —its ratio to v is a measure of the rarefaction. From this principle we may easily derive a formula for the velocity of sound, bearing in mind that u is always very small in comparison with v, and that consequently the ratio of p to p 0 is very nearly unity. For, consider a thin lamina of air, whose natural thickness is bx, and let 5 p and bp be the differences between the densities and pressures respectively on its two faces. Then the equation above investigated leads to the condition-^- = —• But the time which the mov- 8p p ing section occupies in traversing the lamina is approximately —> and in this time the velocity of the lamina changes by the amount 5 u. The force producing this change of velocity is bp, or 1'41^-S p, and must be equal to the quotient of change of momentum by P time, that is to p bx. 5 u^— or to pv bu. Hence -^ = 1-41-^-. Equating this to the v b p p‘V other expression for — we have bp * = 1-41-?-, ^“1*41 P. P P‘ v P This investigation is due to Professor Rankine, Phil. Trans. 1869. Note B. § 635. The following is the usual investigation of the velocity of transmission of sound through a uniform tube filled with air, friction being neglected: Let x denote the original distance of a particle of air from the section of the tube at which the sound originates, and x + y its distance at time t, so that y is the displacement of the particle from the position of equili¬ brium. Then a particle which was originally at distance x + bx will at time t be at the distance x + bx + y + by, and the thickness of the intervening lamina, which was originally 5 x, is now bx + by. Its compression is therefore - ^ or ultimately - d -—, and if P denote S x dx the original pressure, the increase of pressure is 1-41 pAfi. The excess of pressure d x behind a lamina bx above the pressure in front is -^(1*41 P-^"\ bx, or 1*41 P bx; dx V dxl dx 1 and if D denote the original density of the air, the acceleration of the lamina will be the quotient of this expression by D.Sa;. But this acceleration is Hence we have the d t equation ^= 1 - 41 - d — V -' dt 2 D dx*' the integral of which is y = F [x-vt) +/ (x + v t ); ^/l*41 and F, f denote any functions whatever. where v denotes TRANSMISSION OF SOUND. 815 The term F (x - v t) represents a wave, of the form y = F (x), travelling forwards with velocity v; for it has the same value for t x + 5 1 and x x + v . 5 t as for t x and x x . The term f(x + vt) represents a wave, of the form y =/ (x), travelling backwards with the same velocity. In order to adapt this investigation, as well as that given in Note A, to the propagation of longitudinal vibrations through any elastic material, whether solid, liquid, or gaseous, we have merely to introduce E in the place of 1’41 P, E denoting the coefficient of elasticity of the substance, as defined by the condition that a compression-^-^ is produced by a force (per unit area) of E d y t d x CHAPTER LIV. NUMERICAL EVALUATION OF SOUND. 645. Qualities of Musical Sound.—Musical tones differ one from another in respect of three qualities;—loudness, pitch, and character. Loudness .—The loudness of a sound considered subjectively is the intensity of the sensation with which it affects the organs of hearing. Regarded objectively, it depends, in the case of sounds of the same pitch and character, upon the energy of the aerial vibrations in the neighbourhood of the ear, and is proportional to the square of the amplitude. Our auditory apparatus is, however, so constructed as to be more susceptible of impression by sounds of high than of low pitch. A bass note must have much greater energy of vibration than a treble note, in order to strike the ear as equally loud. The intensity of sonorous vibration at a point in the air is therefore not an absolute measure of the intensity of the sensation which will be received by an ear placed at the point. The word loud is also frequently applied to a source of sound, as when we say a loud voice, the reference being to the loudness as heard at a given distance from the source. The diminution of loud- ness with increase of distance according to the law of inverse squares is essentially connected with the proportionality of loudness to square of amplitude. Pitch .—Pitch is the quality in respect of which an acute sound differs from a grave one; for example, a treble note from a bass note. All persons are capable of appreciating differences of pitch to some extent, and the power of forming accurate judgments of pitch con¬ stitutes what is called a musical ear. Physically, pitch depends solely on frequency of vibration , that is to say, on the number of vibrations executed per unit time. In QUALITIES OF MUSICAL SOUND. 817 ordinary circumstances this frequency is the same for the source of sound, the medium of transmission, and the drum of the ear of the person hearing; and in general the transmission of vibrations from one body or medium to another produces no change in their fre¬ quency. The second is universally employed as the unit of time in treating of sonorous vibrations; so that frequency means number of vibrations per second. Increase of frequency corresponds to eleva¬ tion of pitch. Period and frequency are reciprocals. For example, if the period of each vibration is of a second, the number of vibrations per second is 100. Period therefore is an absolute measure of pitch, and the longer the period the lower is the pitch. The wave-length of a note in any medium is the distance which sound travels in that medium during the period corresponding to the note. Hence wave-length may be taken as a measure of pitch, pro¬ vided the medium be given; but, in passing from one medium to another, wave-length varies directly as the velocity of sound. The wave-length of a given note in air depends upon the temperature of the air, and is shortened in transmission from the heated air of a concert-room to the colder air outside, while the pitch undergoes no change. If we compare a series of notes rising one above another by what musicians regard as equal differences of pitch, their frequencies will not be equidifferent, but will form an increasing geometrical pro¬ gression, and their periods (and wave-lengths in a given medium) will form a decreasing geometrical progression. Character. —Musical sounds may, however, be alike as regards pitch and loudness, and may yet be easily distinguishable. We speak of the quality of a singer’s voice, and the tone of a musical instrument; and we characterize the one or the other as rich, sweet, or mellow, on the one hand; or as poor, harsh, nasal, &c., on the other. These epithets are descriptive of what musicians call timbre —a French word literally signifying stamp. German writers on acoustics denote the same quality by a term signifying sound-tint. It might equally well be called sound-flavour. We adopt character as the best English designation. Physically considered, as wave-length and wave-amplitude fall under the two previous heads, character must depend upon the only remaining point in which aerial waves can differ—namely their form , meaning by this term the law according to which the velo- 53 818 NUMERICAL EVALUATION OF SOUND. cities and densities change from point to point of a wave. This subject will be more fully treated in Chapter lvi. Every musical sound is more or less mingled with non-musical noises, such as puffing, scraping, twanging, hissing, rattling, &c. These are not compre¬ hended under timbre or character in the usage of the best writers on acoustics. The gradations of loudness which characterize the commencement, progress, and cessation of a note, and upon which musical effect often greatly depends, are likewise excluded from this designation. In distinguishing the sounds of different musical in¬ struments, we are often guided as much by these gradations and extraneous accompaniments as by the character of the musical tones themselves. 646. Musical Intervals.—When two notes are heard, either simul¬ taneously or in succession, the ear experiences an impression of a special kind, involving a perception of the relation existing between them as regards difference of pitch. This impression is often recog¬ nized as identical where absolute pitch is very different, and we express this identity of impression by saying that the musical inter¬ val is the same. Each musical interval, thus recognized by the ear as constituting a particular relation between two notes, is found to correspond to a particular ratio between their frequencies of vibration. Thus the octave , which of all intervals is that which is most easily recognized by the ear, is the relation between two notes whose frequencies are as 1 to 2, the upper note making twice as many vibrations as the lower in any given time. It is the musician’s business so to combine sounds as to awaken emotions of the peculiar kind which are associated with works of art. In attaining this end he employs various resources, but musical intervals occupy the foremost place. It is upon the judicious employ¬ ment of these that successful composition mainly depends. 647. Gamut. —The gamut or diatonic scale is a series of eight notes having certain definite relations to one another as regards fre¬ quency of vibration. The first and last of the eight are at an inter¬ val of an octave from each other, and are called by the same name; and by taking in like manner the octaves of the other notes of the series, we obtain a repetition of the gamut both upwards and downwards, which may be continued over as many octaves as we please. The notes of the gamut are usually called by the names THE GAMUT. 819 Do Re Mi Fa Sol La Si Do 2 and their vibration-frequencies are proportional to the numbers 1 9. JR. 1 8 4 or, clearing fractions, to A 3 JL 1 5 2 3 52 3 8 24 27 30 32 36 40 45 48 The intervals from Do to each of the others in order are called a second , a major third, a fourth , a fifth, a sixth, a seventh, and an octave respectively. The interval from La to Do 2 is called a minor third, and is evidently represented by the ratio f. The interval from Do to Re, from Fa to Sol, or from La to Si, is represented by the ratio -§-, and is called a major tone. The interval from Re to Mi, or from Sol to La, is represented by the ratio 1 -£, and is called a minor tone. The interval from Mi to Fa, or from Si to Do 2 , is represented by the ratio Fi, and is called a limma. As the square of Ft is a little greater than y, a limma is rather more than half a major tone. The intervals between the successive notes of the gamut are ac¬ cordingly represented by the following ratios 1 :— Do Re Mi Fa Sol La Si Do 2 9_ 10 ii 9_ 1_0 9. 16 8 9 158 9 8 5 Do (with all its octaves) is called the key-note, or simply the key, of the piece of music, and may have any pitch whatever. In order to obtain perfect harmony, the above ratios should be accurately main¬ tained whatever the key-note may be. 648. Tempered Gamut.—A great variety of keys are employed in music, and it is a practical impossibility, at all events in the case of instruments like the piano and organ, which have only a definite set of notes, to maintain these ratios strictly for the whole range of pos¬ sible key-notes. Compromise of some kind becomes necessary, and different systems of compromise are called different temperaments or different modes of temperament. The temperament which is most in favour in the present day is the simplest possible, and is called equal temperament, because it favours no key above another, but makes the tempered gamut exactly the same for all. It ignores the 1 The logarithmic differences, which are accurately proportional to the intervals, are approximately as under, omitting superfluous zeros. Do Re Mi Fa Sol La Si Do 51 46 28 51 46 51 28 820 NUMERICAL EVALUATION OF SOUND. difference between major and minor tones, and makes the limma exactly half of either. The interval from Do to Do 2 is thus divided into 5 tones and 2 semitones, a tone being ± of an octave, and a semitone T V of an octave. The ratio of frequencies corresponding to a tone will therefore be the sixth root of 2 , and for a semitone it will be the 12 th root of 2 . The difference between the natural and the tempered gamut for the key of C is shown by the following table, which gives the number of complete vibrations per second for each note of the middle octave of an ordinary piano:— Tempered Gamut. Natural Gamut. Tempered Gamut. Natural Ga: c . . 2587 258-7 Gr . . 387-6 388-0 D . . 290-3 291*0 A . . 435-0 431-1 E . . 325-9 323-4 B .. . 488-2 485-0 F . . 345-3 344-9 C . . 517-3 517-3 The absolute pitch here adopted is that of the Paris Conservatoire, and is fixed by the rule that A (the middle A of a piano, or the A string of a violin) is to have 435 complete vibrations per second in the tempered gamut. This is rather lower than the concert-pitch which has prevailed in this country in recent years, but is probably not so low as that which prevailed in the time of Handel. It will be noted that the number of vibrations corresponding to C is ap¬ proximately equal to a power of 2 (256 or 512). Any power of 2 accordingly expresses (to the same degree of approximation) the number of vibrations corresponding to one of the octaves of C. The Stuttgard congress (1834) recommended 528 vibrations per second for C, and the C tuning-forks sold under the sanction of the Society of Arts are guaranteed to have this pitch. By multiplying the numbers 24, 27 . . . 48, in § 647, by 11 , we shall obtain the frequencies of vibration for the natural gamut in C corresponding to this standard. What is generally called concert-pitch gives C about 538. The C of the Italian Opera is 546. Handel’s C is said to have been 499^. 649. Limits of Pitch employed in Music.—The deepest note re¬ gularly employed in music is the C of 32 vibrations per second which is emitted by the longest pipe (the 16-foot pipe) of large organs. Its wave-length in air at a temperature at which the velo¬ city of sound is 1120 feet per second, is 35 feet. The highest note employed seldom exceeds A, the third octave of the A above defined. Its number of vibrations per second is 435 x 2 8 = 3480, and MINOR AND PYTHAGOREAN SCALES. 821 its wave-length in air is about 4 inches. Above this limit it is difficult to appreciate pitch, but notes of at least ten times this num¬ ber of vibrations are audible. The average compass of the human voice is about two octaves. The deep F of a bass-singer has 87, and the upper G of the treble 775 vibrations per second. Yoices which exceed either of these limits are regarded as deep or high, though a few voices exceed them to the extent of nearly an octave. 650. Minor Scale and Pythagorean Scale.—The interval expressed by the ratio J-f- is called a diesis, and the interval f^- is called a comma. The former is the difference between a limma and a minor tone, the latter is the difference between a minor and a major tone. The signs # and b (sharp and flat) appended to a note indicate, when strictly understood, that it is to be raised or lowered by the amount of a diesis. The major scale or gamut, as above given, is modified in the following way to obtain the minor scale Do 9. 8 Re Mib Fa Sol Lab Sib Do 2 1 6 1 5 l_p _9 9 8 1 6 15 1 0 the numbers in the second line being the ratios which represent the intervals between the successive notes. It is worthy of note that Pythagoras, who was the first to attempt the numerical evaluation of musical intervals, laid down a scheme of values slightly different from that which is now generally adopted. According to him, the intervals between the successive notes of the major scale are as follows:— Do Re Mi Fa Sol La Si Do 9. 9 2 5 6 9 9_ 9. 2 5 6 8 8 243 8 8 8 243 This scheme agrees exactly with the common system as regards the values of the fourth, fifth, and octave, and makes the values of the major third, the sixth, and the seventh each greater by a comma, while the small interval from mi to fa, or from si to do, is diminished by a comma. In the ordinary system, the prime numbers which enter the ratios are 2, 3, and 5; in the Pythagorean system they are only 2 and 3; hence the interval between any two notes of the Pythagorean scale can be expressed as the sum or difference of a certain number of octaves and fifths. In tuning a violin by making the intervals between the strings true fifths, the Pythagorean scheme is virtually employed. 822 NUMERICAL EVALUATION OF SOUND. 651. Methods of Counting Vibrations. Siren.—The instrument which is chiefly employed for counting the number of vibrations corresponding to a given note, is called the siren, and was devised by Cagniard de Latour. It is represented in Figs. 583, 584, the former being a front, and the latter a back view. There is a small wind-chest, nearly cylindrical, having its top pierced with fifteen holes, disposed at equal distances round the circumference of a circle. Just over this, and nearly touching it, is a movable circular plate, pierced with the same number of holes I Fig. 583. Siren. Fig. 584. similarly arranged, and so mounted that it can rotate very freely about its centre, carrying with it the vertical axis to which it is attached. This rotation is effected by the action of the wind, which enters the wind-chest from below, and escapes through the holes. The form of the holes is shown by the section in Fig. 584. They do not pass perpendicularly through the plates, but slope contrary ways, so that the air when forced through the holes in the lower plate impinges upon one side of the holes in the upper plate, and thus blows it round in a definite direction. The instrument is driven by means of the bellows shown in Fig. 595 (§ 664). As the rotation of one plate upon the other causes the holes to be alternately opened and closed, the wind escapes in successive puffs, whose frequency THE SIREN. 823 depends upon the rate of rotation. Hence a note is emitted which rises in pitch as the rotation becomes more rapid. The siren will sound under water, if water is forced through it instead of air; and it was from this circumstance that it derived its name. In each revolution, the fifteen holes in the upper plate come opposite to those in the lower plate 15 times, and allow the com¬ pressed air in the wind-chest to escape; while in the intervening positions its escape is almost entirely prevented. Each revolution thus gives rise to 15 vibrations; and in order .to know the number of vibrations corresponding to the note emitted, it is only necessary to have a means of counting the revolutions. This is furnished by a counter, which is represented in Fig. 584 The revolving axis carries an endless screw, driving a wheel of 100 teeth, whose axis carries a hand traversing a dial marked with 100 divisions. Each revolution of the perforated plate causes this hand to advance one division. A second toothed-wheel is driven inter¬ mittently by the first, advancing suddenly one tooth whenever the hand belonging to the first wheel passes the zero of its scale. This second wheel also carries a hand traversing a second dial; and at each of the sudden movements just described this hand advances one division. Each division accordingly indicates 100 revolutions of the perforated plate, or 1500 vibrations. By pushing in one of the two buttons which are shown, one on each side of the box containing the toothed-wheels, we can instantaneously connect or disconnect the endless screw and the first toothed-wheel. In order to determine the number of vibrations corresponding to any given sound which we have the power of maintaining steadily, we fix the siren on the bellows, the screw and wheel being dis¬ connected, and drive the siren until the note which it emits is judged to be in unison with the given note. We then, either by regulating the pressure of the wind, or by employing the finger to press with more or less friction against the revolving axis, contrive to keep the note of the siren constant for a measured interval of time, which we observe by a watch. At the commencement of the interval we sud¬ denly connect the screw and toothed-wheel, and at its termination we suddenly disconnect them, having taken care to keep the siren in unison with the given sound during the interval. As the hands do not advance on the dials when the screw is out of connection with the wheels, the readings before and after the measured interval of 824 NUMERICAL EVALUATION OF SOUND. time can be taken at leisure. Each reading consists of four figures, indicating the number of revolutions from the zero position, units and tens being read off on the first dial, and hundreds and thousands on the second. The difference of the two readings is the number of revolutions made in the measured interval, and when multiplied by 15 gives the number of vibrations in the interval, whence the num¬ ber of vibrations per second is computed by division. 652. Graphic Method.—In the hands of a skilful operator, with a good musical ear, the siren is capable of yielding very accurate deter¬ minations, especially if, by adding or subtracting the number of beats, correction be made for any slight difference of pitch between the siren and the note under investigation. The vibrations of a tuning-fork can be counted, without the aid of the siren, by a graphical method, which does not call for any exer¬ cise of musical judgment, but simply involves the performance of a mechanical operation. The tuning-fork is fixed in a horizontal position, as shown in Fig. 585, and has a light style, which may be of brass wire, quill, or bristle, attached to one of its prongs by wax. To receive the trace, a piece of smoked paper is gummed round a cylinder, which can be turned by a handle, a screw cut on the axis causing it at the same THE PHONAUTOGRAPH. 825 time to travel endwise. The cylinder is placed so that the style barely touches the blackened surface. The fork is then made to vibrate by bowing it, and the cylinder is turned. The result is a wavy line traced on the blackened surface, and the number of wave¬ forms (each including a pair of bends in opposite directions) corres¬ ponds to the number of vibrations. If the experiment lasts for a measured interval of time, we have only to count these wave-forms, and divide by the number of seconds, in order to obtain the number of vibrations per second for the note of the tuning-fork. By plunging the paper in ether, the trace will be fixed, so that the paper may be laid aside and the vibrations counted at leisure. The apparatus is called the vibroscope, and was invented by Duhamel. M. Leon Scott has invented an instrument called the phonauto- graph, which is adapted to the graphical representation of sounds in Fig. 586.—Traces by Phonautograph. general. The style, which is very light, is attached to a membrane stretched across the smaller end of what may be called a large ear- trumpet. The membrane is agitated by the aerial waves proceeding from any source of sound, and the style leaves a record of these agitations on a blackened cylinder, as in DuhameFs apparatus. Fig. 586 represents the traces thus obtained from the sound of a tuning-fork in three different modes of vibration. 653. Tonometer.—When we have determined the frequency of vibration for a particular tuning-fork, that of another fork, nearly in unison with it, can be deduced by making the two forks vibrate simultaneous^, and counting the beats which they produce. Scheibler’s tonometer , which is constructed by Koenig of Paris, consists of a set of 65 tuning-forks, such that any two consecutive forks make 4 beats per second, and consequently differ in pitch by 826 NUMERICAL EVALUATION OF SOUND. 4 vibrations per second. The lowest of the series makes 256 vibra¬ tions, and the highest 512, thus completing an octave. Any note within this range can have its vibration-frequency at once deter¬ mined, with great accuracy, by making it sound simultaneously with the fork next above or below it, and counting beats. With the aid of this instrument, a piano can be tuned with cer¬ tainty to any desired system of temperament, by first tuning the notes which come within the compass of the tonometer, and then proceeding by octaves. In the ordinary methods of tuning pianos and organs, tempera¬ ment is to a great extent a matter of chance; and a tuner cannot attain the same temperament in two successive attempts. 653 a. Pitch modified by Relative Motion.—We have stated in § 645 that, in ordinary circumstances, the frequency of vibration in the source of sound, is the same as in the ear of the listener, and in the intervening medium. This identity, however, does not hold if the source of sound and the ear of the listener are approaching or receding from each other. Approach of either to the other produces increased frequency of the pulses on the ear, and consequent elevation of pitch in the sound as heard; while recession has an opposite effect. Let n be the number of vibrations performed in a second by the source of the sound, v the velocity of sound in the medium, and a the relative velocity of approach. Then the number of waves which reach the ear of the listener in a second, will be n plus the number of waves which cover a length a, that is (since n waves cover a length v). will be n + — n or At? n If the motion considered be simply that of the listener, the medium and the source being at rest relatively to each other, the case is ex¬ tremely clear, being analogous to that of a ship sailing in such a direction as to meet the waves of the sea. If, on the other hand, the listener is at rest relatively to the medium, while the source is ad¬ vancing through it, this advance shortens the wave-lengths in the medium, in the ratio , without altering the velocity of propaga¬ tion. The frequency of vibration of the particles of the medium, being equal to velocity divided by wave-length, is accordingly in¬ creased in the ratio It may be remarked that, in this case, there is a corresponding diminution of frequency of vibration in that part of the medium which lies behind the source. Different MODIFICATION OF PITCH. 827 portions of the medium will in fact vibrate with all frequencies intermediate between n and n. If the listener and the source are at rest relatively to each other, but the medium is flowing from him to the source with velocity a , the particles of the medium which flow past him will vibrate with the frequency n, but the waves will pass him with only the natural frequency n. Careful observation of the sound of a railway whistle, as an express train dashes past a station, has confirmed the fact that the sound as heard by a person standing at the station is higher while the train is approaching than when it is receding. A speed of about 40 miles an hour will sharpen the note by a semitone in approaching, and flatten it by the same amount in receding, the natural pitch being heard at the instant of passing. 1 1 The best observations of this kind were those of Buys Ballot, in which trumpeters, with their instruments previously tuned to unison, were stationed, one on the locomotive, and others at three stations beside the line of railway. Each trumpeter was accompanied by musicians, charged with the duty of estimating the difference of pitch between the note of his trumpet and those of the others, as heard before and after passing. CHAPTER LV. MODES OF VIBRATION. 654. Longitudinal and Transverse Vibrations of Solids.—Sonorous vibrations are manifestations of elasticity. When the particles of a solid body are displaced from their natural positions relative to one another by the application of external force, they tend to return, in virtue of the elasticity of the body. When the external force is removed, they spring back to their natural position, pass it in virtue of the velocity acquired in the return, and execute isochronous vibra¬ tions about it until they gradually come to rest. The isochronism of the vibrations is proved by the constancy of pitch of the sound emitted; and from the isochronism we can infer, by the aid of mathe¬ matical reasoning, that the restoring force increases directly as the displacement of the parts of the body from their natural relative position (§§ 53 a, b, c). The same body is, in general, susceptible of many different modes of vibration, which may be excited by applying forces to it in dif¬ ferent ways. The most important of these are comprehended under the two heads of longitudinal and transverse vibrations. In the former the particles of the body move to and fro in the direction along which the pulses travel, which is always regarded as the longitudinal direction, and the deformations produced consist in alternate compressions and extensions. In the latter the particles move to and fro in directions transverse to that in which the pulses travel, and the deformation consists in bending. To produce longi¬ tudinal vibrations, we must apply force in the longitudinal direction. To produce transverse vibration, we must apply force transversely. 655. Transverse Vibrations of Strings.—To the transverse vibra¬ tions of strings, instrumental music is indebted for some of its most precious resources. In the violin, violoncello, l denoting the length of the string between its points of attachment, and n the number of vibrations per second. This formula involves the following laws:— 1. When the length of the vibrating portion of the string is altered, without change of tension, the frequency of vibration varies inversely as the length. 2. If the tension be altered, without change of length in the vibrating portion, the frequency of vibration varies as the square root of the tension. 3. Strings of the same length and tension have frequencies of vibration which are inversely as the square roots of their masses (or weights). 4. Strings of the same length and density, but of different thick¬ nesses, will vibrate in the same time, if they are stretched with forces proportional to their sectional areas. All these laws are illustrated (qualitatively, if not quantitatively) by the strings of a violin. The first is illustrated by the fingering, the pitch being raised as the portion of string between the finger and the bridge is shortened. The second is illustrated by the mode of tuning, which consists in tightening the string if its pitch is to be raised, or slackening the string if it is to be lowered. The third law is illustrated by the construction of the bass string, which is wrapped round with metal wire, for the purpose of adding to its mass, and thus attaining slow vibration without undue slackness. The tension of this string is in fact greater than that of the string next it, though the latter vibrates more rapidly in the ratio of 3 to 2. The fourth law is indirectly illustrated by the sizes of the first three strings. The treble string is the smallest, and is nevertheless stretched with much greater force than any of the others. The third string is the thickest, and is stretched with less force than any of the others. The increased thickness is necessary in order to give sufficient power in spite of the slackness of the string. 657. Experimental Illustration: Sonometer.—For the quantitative illustration of these laws, the instrument called the sonometer, re¬ presented in Fig. 587, is commonly employed. It consists essen- THE SONOMETER. 831 tially of a string or wire stretched over a sounding-box by means of a weight. One end of the string is secured to a fixed point at one end of the sounding-box. The other end passes over a pulley, and carries weights which can be altered at pleasure. Near the two ends of the box are two fixed bridges, over which the cord passes. There is also a movable bridge, which can be employed for altering the length of the vibrating portion. To verify the law of lengths, the whole length between the fixed bridges is made to vibrate, either by plucking or bowing; the mov- Fig. 5S7.—Sonometer. able bridge is then introduced exactly in the middle, and one of the halves is made to vibrate; the note thus obtained will be found to be the upper octave of the first. The frequency of vibration is there¬ fore doubled. By making two-thirds of the whole length vibrate, a note will be obtained which will be recognized as the fifth of the fundamental note, its vibration-frequency being therefore greater in the ratio J-. To obtain the notes of the gamut, we commence with the string as a whole, and then employ portions of its length repre¬ sented by the fractions f, f, -J, f, ^ i To verify the law independently of all knowledge of musical inter¬ vals, a light style may be attached to the cord, and caused to trace its vibrations on the vibroscope. This mode of proof is also more general, inasmuch as it can be applied to ratios which do not corres¬ pond to any recognized musical interval. To verify the law of tensions, we must change the weight. It will be found that, to produce a rise of an octave in pitch, the weight must be increased fourfold. To verify the third and fourth laws, two strings must be employed, their masses having first been determined by weighing them. 832 MODES OF VIBRATION. If the strings are thick, and especially if they are thick steel wires, their flexural rigidity has a sensible effect in making the vibrations quicker than they would be if the tension acted alone. 658. Harmonics.—Any person of ordinary musical ear may easily, by a little exercise of attention, detect in any note of a piano the presence of its upper octave, and of another note a fifth higher than this; these being the notes which correspond to frequencies of vibra¬ tion double and triple that of the fundamental note. A highly trained ear can detect the presence of other notes, corresponding to still higher multiples of the fundamental frequency of vibration. Such- notes are called harmonics. When the vibration-frequency of one note is an exact multiple of that of another note , the former note is called a harmonic of the latter. The notes of all stringed instruments contain numerous harmonics blended with the fundamental tones. Bells and vibrating plates have higher tones mingled with the fundamental tone; but these higher tones are not harmonics in the sense in which we use the word. A violin string sometimes fails to yield its fundamental note, and gives the octave or some other harmonic instead. This result can be brought about at pleasure, by lightly touching the string at a pro¬ perly- selected point in its length, while the bow is applied in the usual way. If touched at the middle point of its length, it gives the octave. If touched at one-third of its length from either end, it gives the fifth above the octave. The law is, that if touched at ~ of its length 1 from either end, it yields the harmonic whose vibration- frequenc}^ is n times that of the fundamental tone. The string in these cases divides itself into a number of equal vibrating-segments, as shown in Fig. 589. The division into segments is often distinctly visible when the string of a sonometer is strongly bowed, and its existence can be verified, when less evident, by putting paper riders on different parts of the string. These (as shown in the figure) will be thrown off by the vibrations of the string, unless they are placed accurately at the nodal points, in which case they will retain their seats. If two strings tuned to unison are stretched on the same sonometer, the vibration of the one induces similar vibrations in the other; and the experiment of the riders may be varied, in a very instructive way, 1 Or at — of its length, if m be prime to n. RESONANCE. 833 by bowing one string, and placing the riders on the other. This is an instance of a general principle of great importance—that a vibrat¬ ing body communicates its vibrations to other bodies which are capable of vibrating in unison with it. The propagation of a sound may indeed be regarded as one grand vibration in unison; but, besides the general waves of propagation, there are waves of re- Fig. 589.—Production of a Harmonic. inf or cement, due to the synchronous vibrations of limited portions of the transmitting medium. This is the principle of resonance. 658 a. Resonance.—By applying to a pendulum originally at rest a series of very feeble impulses, at intervals precisely equal to its natural time of vibration, we shall cause it to swing through an arc of considerable magnitude. The same principle applies to a body capable of executing vibra¬ tions'under the influence of its own elasticity. A series of impulses keeping time with its own natural period may set it in powerful vibration, though any one of them singly would have no appreciable effect. Some bodies, such as strings and confined portions of air, have definite periods in which they can vibrate freely when once started; 54 834 MODES OF VIBRATION. and when a note corresponding to one of these periods is sounded in their neighbourhood, they readily take it up and emit a note of the same pitch themselves. Other bodies, especially thin pieces of dry straight-grained deal, such as are employed for the faces of violins and the sounding- boards of pianos, are capable of vibrating, more or less freely, in any period lying between certain wide limits. They are accordingly set in vibration by all the notes of their respective instruments; and by the large surface with which they act upon the air, they contribute in a very high degree to increase the sonorous effect. All stringed instruments are constructed on this principle; and their quality mainly depends on the greater or less readiness with which they respond to the vibrations of the strings. All such methods of reinforcing a sound must be included under resonance; but the word is often more particularly applied to the reinforcement produced by masses of air. 659. Longitudinal Vibrations of Strings.—Strings or wires may also be made to vibrate longitudinally , by rubbing them, in the direction of their length, with a bow or a piece of chamois leather covered with rosin. The sounds thus obtained are of much higher pitch than those produced by transverse vibration. In the case of the fundamental note, each of the two halves A C, C B (Fig. 590), is alternately extended and compressed, one being extended while the other is compressed. At the middle point C a _c_R Fig 590.—Longitudinal Vibration. First Tone. there is no extension or compression, but there is greater amplitude of movement than at any other point. The amplitudes diminish in passing from C towards either end, and vanish at the ends, which are therefore nodes. The extensions and compressions, on the other hand, increase as we travel from the middle towards either end, and obtain their greatest values at the ends. But the string may also divide itself into any number of separately- vibrating segments, just as in the case of transverse vibrations. Fig. 591 represents the motions which occur when there are three such segments, separated by two nodes D, E. The upper portion of the figure is true for one-half of the period of vibration, and the lower portion for the remaining half. STRINGED INSTRUMENTS. 835 The frequency of vibration, for longitudinal as well as for trans¬ verse vibrations, varies inversely as the length of the vibrating string, or segment of string. We shall return to this subject in § 670. A - " D «- E -*» B A < - L) -E ^- B -- - - -o - ■ --■- Fig. 591.—Longitudinal Vibration. Third Tone. 660. Stringed Instruments. — Only the transversal vibrations of strings are employed in music. In the violin and violoncello there are four strings, each being tuned a fifth above the next below it; and intermediate notes are obtained by fingering, the portion of string between the finger and the bridge being the only part that is free to vibrate. The bridge and sounding-post serve to transmit the vibrations of the strings to the body of the instrument. In the piano there is also a bridge, which is attached to the sounding-board, and communicates to it the vibrations of the wires. 661. Transversal Vibrations of Rigid Bodies: Rods, Plates, Bells.— We shall not enter into detail respecting the laws of the transverse vibrations of rigid bodies. The relations of their overtones to their fundamental tones are usually of an extremely complex character, and this fact is closely connected with the unmusical or only semi¬ musical character of the sounds emitted. When one face of the body is horizontal, the division into separate vibrating segments can be rendered visible by a method devised by Chladni, namely, by strewing sand on this face. During the vibra¬ tion, the sand, as it is tossed about, works its way to certain definite lines, where it comes nearly to rest. These nodal lines must be regarded as the intersections of internal nodal surfaces with the surface on which the sand is strewed, each nodal surface being the boundary between parts of the body which have opposite motions. The figures composed by these nodal lines are often very beautiful, and quite startling in the suddenness of their production. Chladni and Savart published the forms of a great number. A complete theoretical explanation of them would probably transcend the powers of the greatest mathematicians. Bells and bell-glasses vibrate in segments, which are never less than four in number, and are separated by nodal lines meeting in the middle of the crown. They are well shown by putting water in a 836 MODES OF VIBRATION. bell-glass, and bowing its edge. The surface of the water will im- • mediately be covered with ripples, one set of ripples proceeding from each of the vibrating segments. The division into any possible number of segments may be effected by pressing the glass with the fingers in the places where a pair of consecutive nodes ought to be formed, while the bow is applied to the middle of one of the seg¬ ments. The greater the number of segments the higher will be the note emitted. 662. Tuning-fork. — Steel rods, on account of their comparative freedom from change, are well suited for standards of pitch. The tuning-fork, which is especially used for this purpose, consists essen¬ tially of a steel rod bent double, and attached to a handle of the same material at its centre. Besides the fundamental tone, it is capable of yielding two or three overtones, which are very much higher in pitch; but these are never used for musical purposes. If the fork is held by the handle while vibrating, its motion continues for a long time, but the sound emitted is too faint to be heard except by holding the ear near it. When the handle is pressed against a table, the latter acts as a sounding-board, and communicates the vibrations to the air, but it also causes the fork to come much more speedily to rest. For the purposes of the lecture-room the fork is often mounted on a sounding- box (Fig. 592), which should be se¬ parated from the table by two pieces of india-rubber tubing. The box can then vibrate freely in unison with the fork, and the sound is both loud and lasting. The vibrations are usually excited either by bow¬ ing the fork or by drawing a piece of wood between its prongs. The pitch of a tuning-fork varies slightly with temperature, be¬ coming lower as the temperature rises. This effect is due in some trifling degree to expansion, but much more to the diminution of elastic force. 663. Law of Linear Dimensions.—The following law is of very wide application, being applicable alike to solid, liquid, and gaseous bodies: — When two bodies differing in size, but in other respects similar and similarly circumstanced, vibrate in the same mode, their vibra¬ tion-periods are directly as their linear dimensions. Their vibra- Fig. 592.—Fork on Sounding box. ORGAN-PIPES. 837 tion-frequencies are consequently in the inverse ratio of their linear dimensions. In applying the law to the transverse vibrations of strings, it is to be understood that the stretching force per unit of sectional area is constant. In this case the velocity of a pulse (§ 656) is constant, and the period of vibration, being the time required for a pulse to travel over twice the length of the string, is therefore directly as the length. 664. Organ-pipes.—In organs, and wind-instruments generally, the sonorous body is a column of air confined in a tube. To set this air Fig. 593.—Block Pipe. Fig. 594.—Flue Pipe. in vibration some kind of mouth-piece must be employed. That which is most extensively used in organs is called the flute mouth¬ piece, 1 and is represented, in conjunction with the pipe to which it is attached, in Figs. 593, 594. It closely resembles the mouth-piece of This is not the trade name. English organ-builders have no generic name for this mouth-piece. 838 MODES OF VIBRATION. an ordinary whistle. The air from the bellows arrives through the conical tube at the lower end, and before entering the main body of the pipe has to pass through a narrow slit, in issuing from which it impinges on the thin end of a wedge placed directly opposite, called the lip. This lip is itself capable of vibrating in unison with any note lying within a wide range, and the note which is actually emit¬ ted is determined by the resonance of the column of air in the pipe. Fig. 593 repre¬ sents a wooden, and Fig. 594 a metal organ-pipe, both of them being fur¬ nished with flute mouth¬ pieces. The two arrows in the sections are in¬ tended to suggest the two courses which the wind may take as it issues from the slit. The arrangements for admitting the wind to the pipes by putting down the keys are shown in Fig. 595. The bellows V are worked by the treadle P. The force of the blast can be increased by weight¬ ing the top of the bellows, or by pressing on the rod T. The air passes up from the bellows, through a large tube shown at one end, into a reservoir C, called the wind-chest. In the top of the wind- chest there are numerous openings c, d, &c., in which the tubes are to be fixed. The sectional drawing in the upper part of the figure shows the internal communications. A plate K, pressed up by a spring It, cuts off the tube c from the wind-chest, until the pin a 595.—Experimental Organ. LAW OF LINEAR DIMENSIONS. 839 is depressed. The putting down of this pin lowers the plate, and admits the wind. This description only applies to the experimental organs which are constructed for lecture illustration. In real organs the pressure of the wind in the bellows is constant; and as this pressure would be too great for most of the pipes, the several aper¬ tures of admission are partially plugged, to diminish the force of the blast. 665. The Air is the Sonorous Body.—It is easily shown that the sound emitted by an organ-pipe depends, mainly at least, on the dimensions of the inclosed column of air, and not on the thickness or material of the pipe itself. For let three pipes, one of wood, one of copper, and the other of thick card, all of the same internal dimensions, be fixed on the wind-chest. On making them speak, it will be found that the three sounds have exactly the same pitch, and but slight difference in character. If, however, the sides of the tube are excessively thin, their yielding has a sensible influence, and the pitch of the sound is modified. 666. Law of Linear Dimensions.—The law of linear dimensions, stated in a previous sec¬ tion (§ 663) as applying to the vibrations of simi¬ lar solid bodies, applies to gases also. Let two box-shaped pipes (Fig. 596) of precisely similar form, and having their linear dimensions in the ratio of 2: 1, be fixed on the wind-chest; it will be found, on making them ° Fig. 596.—Law of Linear Dimensions. speak, that the note of the small one is an octave higher than the other;—showing double frequency of vibration. 667. Bernoulli’s Laws.—The law just stated applies to the com¬ parison of similar tubes of any shape whatever. When the length of a tube is a large multiple of its diameter, the note emitted is sensibly independent of the diameter, and depends on the length alone. The relations between the fundamental note of such a tube and its overtones were discovered by Daniel Bernoulli, and are as follows:— 840 MODES OF VIBRATION. I. Overtones of Open Pipes .—Let the pipe B. (Fig. 597), which is open at the upper end, be fixed on the wind-chest; let the correspond¬ ing key be put down, and the wind gradually turned on, by means of the cock below the mouth-piece. The first note heard will be feeble and deep; it is the fundamental note of the pipe. As the wind is gradually turned full on, and increasing pressure afterwards applied to the bellows, a series of notes will be heard, each higher than its pre¬ decessor. These are the overtones of the pipe. They are the harmonics of the fundamental note; that is to say, if 1 denote the frequency of vibration for the fundamental tone, the frequencies of vibration for the overtones will be approximately 2, 3, 4, 5 . . . respectively. II. Overtones of Stopped Pipes .—If the same experiment be tried with the pipe A, which is closed at its upper end; the overtones will form the series of odd harmonics of the fundamental note, all the even harmonics being absent; in other words, the frequencies of vibration of the funda¬ mental tone and overtones will be approximately repre¬ sented by the series of odd numbers 1, 3, 5, 7 . . . Fig. 597. It will also be found, that if both pipes are of the same for Overtones ^ en gth, the fundamental note of the stopped pipe is an octave lower than that of the open pipe. 668. Mode of Production of Overtones.—In the production of the overtones, the column of air in a pipe divides itself into vibrating segments, separated by nodal cross-sections. At equal distances on opposite sides of a node, the particles of air have always equal and opposite velocities, so that the air at the node is always subjected to equal forces in opposite directions, and thus remains unmoved by their action. The portion of air constituting a vibrating segment, sways alternately in opposite directions, and as the movements in two consecutive segments are opposite, two consecutive nodes are always in opposite conditions as regards compression and extension. The middle of a vibrating segment is the place where the ampli¬ tude of vibration is greatest, and the variation of density least. It may be called an antinode. The distance from one node to the next is half a wave-length, and the distance from a node to an anti¬ node is a quarter of a wave-length. Both ends of an open pipe, and the end next the mouth-piece of a stopped pipe, are antinodes, being preserved from changes of density by their free communication with PRODUCTION OF OVERTONES. 841 the external air. At the closed end of a stopped pipe there must always be a node. The swaying to and fro of the internodal portions of air between fixed nodal planes, is an example of stationary undulation; and the vibration of a musical string is another example. A stationary undulation may always be analyzed into two component undulations equal and similar to one another, and travelling in opposite direc¬ tions, their common wave-length being double of the distance from node to node. These undulations are constantly undergoing reflec¬ tion from the ends of the pipe or string, and, in the case of pipes, the reflection is opposite in kind according as it takes place from a closed or an open end. In the former case a condensation propagated towards the end is reflected as a condensation, the forward-moving particles being compelled to recoil by the resistance which they there encounter; and a rarefaction is, in like manner, reflected as a rare¬ faction. On the other hand, when a condensation arrives at an open end, the sudden opportunity for expansion which is afforded causes an outward movement in excess of that which would suffice for equilibrium of pressure, and a rarefaction is thus produced which is propagated back through the tube. A condensation is thus reflected as a rarefaction; and a rarefaction is, in like manner, reflected as a condensation. The period of vibration of the fundamental note of a stopped pipe is the time required for propagating a pulse through four times the length of the pipe. For let a condensation be suddenly produced at the lower end by the action of the vibrating lip. It will be pro¬ pagated to the closed end and reflected back, thus travelling over twice the length of the pipe. On arriving at the aperture where the lip is situated, it is reflected as a rarefaction. This rarefaction travels to the closed end and back, as the condensation did before it, and is then reflected from the aperture as a condensation. Things are now in their initial condition, and one complete vibration has been per¬ formed. The period of the movements of the lip is determined by the arrival of these alternate condensations and rarefactions; and the lip, in its turn, serves to divert a portion of the energy of the blast, and employ it in maintaining the energy of the vibrating column. The wave-length of the fundamental note of a stopped pipe is thus four times the length of the pipe. In an open pipe, a condensation, starting from the mouth-piece, is reflected from the other end as a rarefaction. This rarefaction, on i 842 MODES OF VIBRATION. reaching the mouth-piece, is reflected as a condensation; and things are thus in their initial state after the length of the pipe has been traversed twice. The period of vibration of the fundamental note is accordingly the time of travelling over twice the length of the pipe; and its wave-length is twice the length of the pipe. In every case of longitudinal vibration, if the reflection is alike at both ends, the wave-length of the fundamental tone is twice the distance between the ends. 669. Explanation of Bernoulli’s Laws.—In investigating the theoretical relations between the fundamental tone and overtones for a pipe of either kind, it is convenient to bear in mind that the dis¬ tance from an open end to the nearest node is a quarter of a wave¬ length of the note emitted. In the case of the open pipe the first or fundamental tone requires one node, which is at the middle of the length. The second tone requires two nodes, with half a wave-length between them, while each of them is a quarter of a wave-length from the nearest end. A quarter wave-length has thus only half the length which it had for the fundamental tone, and the frequency of vibration is therefore doubled. The third tone requires three nodes, and the distance from either end to the nearest node is ^ of the length of the pipe, instead of 4 the length as in the case of the first tone. The wave-length is thus divided by 3, and the frequency of vibration is increased threefold. We can evidently account in this way for the production of the complete series of harmonics of the fundamental note. In the case of the stopped pipe, the mouth-piece is always distant a quarter wave-length from the nearest node, and this must be dis¬ tant an even number of quarter wave-lengths from the stopped end, which is itself a node. For the fundamental tone, a quarter wave-length is the whole length of the pipe. For the second tone, there is one node besides that at the closed end, and its distance from the open end is of the length of the pipe. For the third tone, there are two nodes besides that at the closed end. The distance from the open end to the nearest node is there¬ fore i of the length of the pipe. The wave-lengths of the successive tones, beginning with the fundamental, are therefore as 1, j, } . . . , and their vibration- frequencies are as 1, 3, 5, 7 . . . BERNOULLI’S LAWS. 848 Also, since the wave-length of the fundamental tone is four times the length of the pipe if stopped, and only twice its length if open, it is obvious that the wave-length is halved, and the frequency of vibration doubled, by unstopping the pipe. No change of pitch, or only very slight change, will be produced by inserting a solid partition at a node, or by put¬ ting an antinode in free communication with the external air. These principles can be illustrated by means of the jointed pipe represented in Fig. 598. 670. Application to Rods and Strings.—The same laws which apply to open organ-pipes, also apply to the longi¬ tudinal vibrations of rods free at both ends, and to both the longitudinal and transverse vibrations of strings. In all these cases the overtones form the complete series of harmonics of the first or fundamental tone, and the period of vibration for this first tone is the time occupied by a pulse in travel¬ ling over twice the length of the given rod or string In the case of longitudinal vibrations the velocity of a pulse is a/ j), M denoting the value of Young’s modulus for the rod or string, and D its density. This is identical with the velocity of sound through the rod or string, and is Fig - 598 independent of its tension. In the case of transverse pulses Pipe, in a string (regarded as perfectly flexible), the formula for the velocity of transmission, (1) § 656, may be written .y/jp F denot¬ ing the stretching force per unit of sectional area. The ratio of the latter velocity to the former is which is always a small frac- tion, since express the fraction of itself by which the string is lengthened by the force F. If a rod, free at both ends, is made to vibrate longitudinally, its nodes and antinodes will be distributed exactly in the same way as those of an open organ-pipe. The experiment can be performed by holding the rod at a node, and rubbing it with rosined chamois leather. 671. Application to Measurement of Velocity in G-ases.—Let v denote the velocity of sound in a particular gas, in feet per second, A the wave-length of a particular note in this gas in feet, and n the fre¬ quency of vibration for this note, that is the number of vibrations per second which produce it. Then \ is the distance travelled in 844 MODES OF VIBRATION. of a second, and the distance travelled in a second is v = n A, For the same note, n is constant for all media whatever, and v varies directly as The velocities of sound in two gases may thus be compared by observing the lengths of .vibrating columns of the two gases which give the same note; or if columns of equal length be employed, the velocities will be directly as the frequencies of vibra¬ tion, which are determined by observing the pitch of the notes emitted. I By these methods, Dulong, and more recently Wer- theim, have determined the velocity of sound in several f different gases. The following are Wertheim’s results, in metres per second, the gases being supposed to be at 0°C. Air, Oxygen, Hydrogen, Carbonic oxide. 331 317 1269 337 Carbonic acid, Nitrous oxide, Olefiant gas. 262 262 314 velocities given were ascer- The same principle is applicable to liquids and solids; and it was by means of the longitudinal vibra¬ tions of rods that the in § 637 tained. 672. Reed-pipes.— Instead of the flute mouth-piece above described, organ-pipes are often furnished with what is called a reed. A reed con¬ tains an elastic plate l (Figs. 599, 600) call¬ ed the tongue , which, by its vibrations, al¬ ternately opens and closes or nearly closes an aperture through which the wind passes. In Fig. 599, the air from the bellows enters first l Fig. 599.—Reed Pipe. Fig. 600.—Free Reed. WIND-INSTRUMENTS. 845 the lower part t of the pipe, and thence (when permitted by the tongue) passes through the channel 1 r into the upper part t'. The stiff wire 0 , movable with considerable friction through the hole b, limits the vibrating portion of the tongue, and is employed for tuning. Reed-pipes are often terminated above by a trumpet-shaped expansion. Fig. 599 represents a striking reed, so called because the tongue closes the orifice by striking its edges. The sound thus produced is somewhat harsh. In the free reed (Fig. 600), the tongue passes through the orifice without striking its edges, and the tone produced is much smoother. The striking reed is generally preferred in organs, its peculiar character rendering it very effective by way of contrast. It is always used for the trumpet stop. Reed-pipes can be very strongly blown without breaking into overtones. Elevation of temperature sharpens pipes with flute mouth-pieces, and flattens reed-pipes. The sharpening is due to the increased velo¬ city of sound in hot air. The flattening is due to the diminished elasticity of the metal tongue. It is thus proved that the pitch of a reed-pipe is not always that due to the free vibration of the inclosed air, but may be modified by the action of the tongue. 673. Wind-instruments.—In all wind-instruments, the sound is originated by one of the two methods just described. With the flute- pipe must be classed the flute, the flageolet, and the Pandean-pipes. The clarionet, hautboy, and bassoon have reed mouth-pieces, the vibrating tongue being a piece of reed or cane. In the bugle, trum¬ pet, and French-horn, which are mere tubes without keys, the lips of the performer act as the reed-tongue, and the notes produced are approximately the natural overtones. These, when of high order, are so near together, that a gamut can be formed by properly selecting from among them. The fingering of the flute and clarionet, has the effect sometimes of altering the effective length of the vibrating column of air, and sometimes of determining the production of overtones. In the trombone and cornet-a-piston, the length of the vibrating column of air is altered. The harmonium, accordion, and concertina are reed instruments, the reeds employed being always of the free kind. 1 The piece r, which is approximately a half-cylinder, is called the reed by organ- builders. 846 MODES OF VIBRATION. 674. Manometric Flames.—Koenig, of Paris, constructs several forms of apparatus, in which the varia¬ tions of pressure produced by vibrations of air in a pipe are rendered evident to the eye by their effect upon flames. One of these is represented in Fig. 601. Three small gas-burners are fixed at definite points in the side of a pipe, as represented in the figure. When the pipe gives its second tone, the central flame is at an antinode and remains un¬ affected, while the other two, being at nodes, are agitated or blown out. When it gives its first tone, the central flame, which is now at a node, is more power¬ fully affected than the others. The gas which supplies these burners is separated from the air in the pipe only by a thin membrane. When the pipe is made to speak, the flame at the node is violently agitated, in consequence of the changes of pressure on the back of the membrane, while those, at the ventral points are scarcely affected. The agitation of the flame is a true vibration; and, when ex¬ amined by the aid of a revolving mirror, Fig. 601.—Manometric Flames. presents the appearance of tongues of flame alternating with nearly dark spaces. If two pipes, one an octave higher than the other, are connected with the same gas flame, or with two gas flames which can be viewed in the same mirror, the tongues of flame corresponding to the upper octave are seen to be twice as numerous as the others. CHAPTER LYI. ANALYSIS OF VIBRATIONS. CONSTITUTION OF SOUNDS. 675. Optical Examination of Sonorous Vibrations. — Sound is a special sensation belonging to the sense of hearing; but the vibra¬ tions which are its physical cause often manifest themselves to other senses. For instance, we can often feel the tremors of a sono¬ rous body by touching it; we see the movements of the sand on a vibrating plate, the curve traced by the style of a vibroscope, &c. The aid which one sense can thus furnish in what seems the peculiar province of another is extremely interesting. M. Lissajous has devised a very beautiful optical method of examining sonorous vibra¬ tions, which we will briefly describe. 676. Lissajous’ Experiment.—Suppose we introduce into a dark room (Fig. 602) a beam of solar rays, which, after passing through a lens L, is reflected, first, from a small mirror fixed on one of the branches of a tuning-fork D, and then from a second mirror M, which throws it on a screen E; we can thus, by proper adjustments, form upon the screen a sharp and bright image of the sun, which will appear as a small spot of light. As long as the apparatus remains at rest, we shall not observe any movement of the image; but if the tuning-fork vibrates, the image will move rapidly up and down along the line I, I', producing, in consequence of the persistence of impressions, the appearance of a vertical line of light. If the tuning- fork remains at rest, but the mirror M is rotated through a small angle about a vertical axis, the image will move horizontally. Con¬ sequently, if both these motions take place simultaneously, the spot of light will trace out on the screen a sinuous line, as represented in the figure, each S-shaped portion corresponding to one vibration of the tuning-fork. Now, let the mirror M be replaced by a small mirror attached to 848 ANALYSIS OF VIBRATIONS. a second tuning-fork, which vibrates in a horizontal plane, as in Fig 602. - Principle of Lissajous’ Experiment. Fig. 603. If this fork vibrates alone, the image will move to and Fig. 603.—Lissajous’ Experiment. fro horizontally, presenting the appearance of a horizontal line of LISSAJOUS’ EXPERIMENT. 849 light, which gradually shortens as the vibrations die away. If both forks vibrate simultaneously, the spot of light will rise and fall ac¬ cording to the movements of the first fork, and will travel left and right according to the movements of the second fork. The curve actually described, as the resultant of those two component motions, is often extremely beautiful. Some varieties of it are represented in Fig. 604. Instead of throwing the curves on a screen, we may see them by looking into the second mirror, either with a telescope, as in Fig. 603, Fig. 604.—Lissajous’ Figures, Unison, Octave, and Fifth. or with the naked eye. In this form of the experiment, a lamp surrounded by an opaque cylinder, pierced with a small hole just opposite the flame, as represented in the figure, is a very convenient source of light. The movement of the image depends almost entirely on the an¬ gular movements of the mirrors, not on their movements of trans¬ lation ; but the distinction is of no importance, for, in the case of such small movements, the linear and angular changes may be re¬ garded as strictly proportional. Either fork vibrating alone, would cause the image to execute the particular kind of movement which we have described in § 53 a, 55 850 ANALYSIS OF VIBRATIONS. under the designation of simple vibration or simple harmonic motion; so that the movement actually executed will be the re¬ sultant of two simple harmonic motions in directions perpendicular to each other. Suppose the two forks to be in unison. Then the two simple har¬ monic motions will have the same period, and the path described will always be some kind of ellipse, 1 the circle and straight line being included as particular cases. It will be a straight line if both forks pass through their positions of equilibrium at the same instant. In order that it may be a circle, the amplitudes of the two simple har¬ monic motions must be equal, and one fork must be in a position of maximum displacement when the other is in the position of equi¬ librium. If the unison were rigorous, the curve once obtained would remain unchanged, except in so far as its breadth and height became reduced by the dying away of the vibrations. But this perfect unison is never attained in practice, and the eye detects changes depending on differences of pitch too minute to be perceived by the ear. These changes are illustrated by the upper row of forms in Fig. 604, com¬ mencing, say, with the sloping straight line at the left hand, which gradually opens out into an ellipse, and afterwards contracts into a straight line, sloping the opposite way. It then retraces its steps, moving in opposition to the arrows in the figure, and goes through the same changes again. If the interval between the two forks is an octave, we shall obtain the curves represented in the second row; 2 if the interval is a fifth, we shall obtain the curves in the lowest row. In each case the order of the changes will be understood by proceeding from left to right, 1 Employing horizontal and vertical co-ordinates, and denoting the amplitudes by a and b, we have, in the case of unison, — =sin 6, = sin (0 + B ), where /3 denotes the difference a b of phase, and 6 is an angle varying directly as the time. Eliminating 0, we obtain the equation to an ellipse, whose form and dimensions depend upon the given quanti¬ ties, a, &, j3. 2 The middle curve in this row is a parabola, and corresponds to the elimination of 0 between the equations — = cos 2 8 ,— = cos 9. The coefficient 2 indicates the double fre- 1 a a quency of horizontal as compared with vertical vibrations. The general equations to Lissajous’ figures are ~ = sin md,~ = sin (n 6 + /3), where m and n are proportional to the frequencies of horizontal and vertical vibrations. The gradual changes from one figure to another depend on the gradual change of /3, and all the figures can be inscribed in a rectangle, whose length and breadth are 2a and 2b. OPTICAL TUNING. 851 and then back again; but the curves obtained in returning will be inverted. 677. Optical Tuning.—By the aid of these principles, tuning-forks can be compared with a standard fork with much greater precision than would be attainable by ear. Fig. 605 represents a convenient arrangement for this purpose. A lens f is attached to one of the Fig. 605.—Optical Comparison of Tuning-forks. prongs of a standard fork, which vibrates in a horizontal plane; and above it is fixed an eye-piece g , the combination of the two being equivalent to a microscope. The fork to be compared is placed up¬ right beneath, and vibrates in a vertical plane, the end of one prong being in the focus of the microscope. A bright point m, produced by making a little scratch on the end of the prong with a diamond, is observed through the microscope, and is illuminated, if necessary, by converging a beam of light upon it through the lens c. When the forks are set vibrating, the bright point is seen as a luminous ellipse, whose permanence of form is a test of the closeness of the unison. The ellipse will go through a complete cycle of changes in the time required for one fork to gain a complete vibration on the other. library nN'V“RS!TY ot iitmois 852 ANALYSIS OF VIBRATIONS. 677 a. Other Modes of producing Lissajous’ Figures.—An arrange¬ ment devised in 1844 by Professor Blackburn, of Glasgow, then a student at Cambridge, affords a very easy mode of obtaining, by a slow motion, the same series of curves which, in the above arrange¬ ments, are obtained by a motion too quick for the eye to follow. A cord ABC (Fig. 605a) is fastened at A and C, leaving more or less slack, according to the curves which it is desired to' obtain; B 4 D Fig. 605 a.—B lackburn’s Pendulum. and to any intermediate point B of the cord another string is tied, carrying at its lower end a heavy body D to serve as pendulum- bob. If, when the system is in equilibrium, the bob is drawn aside in the plane of A B C and let go, it will execute vibrations in that plane, the point B remaining stationary, so that the length of the pendulum is B D. If, on the other hand, it be drawn aside in a plane perpendicular to the plane ABC, it will vibrate in this per¬ pendicular plane, carrying the whole of the string with it in its motion, so that the length of the pendulum is the distance of the bob from the point E, in which the straight line A C is cut by D B produced. The frequencies of vibration in the two cases will be inversely as the square roots of the pendulum-lengths B D, ED. If the bob is drawn aside in any other direction, it will not vibrate in one plane, but will perform movements compounded of the two independent modes of vibration just described, and will thus describe curves identical with Lissajous’. If the ratio of E D to B D is nearly equal to unity, as in the left-hand figure, we shall have curves cor¬ responding to approximate unison. If it be approximately 4, as in the right-hand figure, we shall obtain the curves of the octave. Traces of the curves can be obtained by employing for the bob a CHARACTER OR TIMBRE. 853 vessel containing sand, which runs out through a funnel-shaped opening at the bottom. The curves can also be exhibited by fixing a straight elastic rod at one end, and causing the other end to vibrate transversely. This was the earliest known method of obtaining them. If the flexural rigidity of the rod is precisely the same for all transverse directions, the vibrations will be executed in one plane; but if there be any inequality in this respect, there will be two mutually perpendicular directions possessing the same properties as the two principal direc¬ tions of vibration in Blackburn’s pendulum. A small bright metal knob is usually fixed on the vibrating extremity to render its path visible. 678. Character. — Character or timbre, which we have already defined in § 645, must of necessity depend on the form of the vibra¬ tion of the aerial particles by which sound is transmitted, the word form being used in the metaphorical sense there explained, for in the literal sense the form is always a straight line. When the changes of density are represented by ordinates of a curve, as in Fig. 575, the form of this curve is what is meant by the form of vibration. The subject of timbre has been very thoroughly investigated in recent years by Helmholtz; and the results at which he has arrived are now generally accepted as correct. The first essential of a musical note is, that the aerial movements which constitute it shall be strictly periodic; that is to say, that each vibration shall be exactly like its successor, or at all events, that, if there be any deviation from strict periodicity, it shall be so gradual as not to produce sensible dissimilarity between several con¬ secutive vibrations of the same particle. There is scarcely any proposition more important in its application to modern physical investigations than the following mathematical theorem, which was discovered by Fourier:— Any periodic vibra¬ tion executed in one line can be definitely resolved into simple vibrations, of which one has the same frequency as the given vibra¬ tion, and the others have frequencies 2, 3, 4, 5 . . . times as great, no fractional multiples being admissible. The theorem may be briefly expressed by saying that every periodic vibration consists of a fundamental simple vibration and its harmonics. 1 Mr. Hubert Airy has obtained very beautiful traces by attaching a glass pen to the bob. See Nature, Aug. 17 and Sept. 7, 1871. 854 ANALYSIS OF VIBRATIONS. We cannot but associate this mathematical law with the experi¬ mental fact, that a trained ear can detect the presence of harmonics in all but the very simplest musical notes. The analysis which Fourier’s theorem indicates, appears to be actually performed by the auditory apparatus. The constitution of a periodic vibration may be said to be known if we know the ratios of the amplitudes of the simple vibrations which compose it; and in like manner the constitution of a sound may be said to be known if we know the relative intensities of the different elementary tones which compose it. Helmholtz has shown that the character of a musical note depends upon its constitution as thus defined; and that, while change of intensity in any of the components produces a modification of char¬ acter, change of phase has no influence upon it whatever. Change of phase does however affect the form of the resultant vibration. Thus certain changes of form are admissible without change of character. The harmonics which are present in a note, usually find their origin in the vibrations of the musical instrument itself. In the case of stringed instruments, for example, along with the vibration of the string as a whole, a number of segmental vibrations are sim¬ ultaneously going on. Fig. 588 represents curves obtained by the composition of the fundamental mode of vibration with another an octave higher. The broken lines indicate the forms which the string would assume if yielding only its fundamental note. 1 The continu¬ ous lines in the first and third figures are forms which a string may assume in its two positions of greatest displacement, when yielding the octave along with the fundamental, the time required for the string to pass from one of these positions to the other being the same as the time in which each of its two segments moves across and back again. The second and fourth figures must in like manner be taken together, as representing a pair of extreme positions. The number of harmonics thus yielded by a pianoforte wire is usually some four or five; and a still larger number are yielded by the strings of a violin. The notes emitted from wide organ-pipes with flute mouth-pieces are very deficient in harmonics. This defect is remedied by combining 1 The form of a string vibrating so as to give only one tone (whether fundamental or harmonic) is a curve of sines, all its ordinates increasing or diminishing in the same pro¬ portion, as the string moves. ORGAN-PIPES. 855 with each of the larger pipes a series of smaller pipes, 1 each yielding one of its harmonics. An ordinary listener hears only one note, of Fig. 588.—String giving first Two Tones. the same pitch as the fundamental, but much richer in character than that which the fundamental pipe yields alone. A trained ear can recognize the individual harmonics in this case as in any other. It is important to remark, that though the presence of harmonic subdivisions in a vibrating body necessarily produces harmonics in the sound emitted, the converse cannot be asserted. Simple vibra¬ tions, executed by a vibrating body, produce vibrations of the same frequency as their own, in any medium to which thej' are transmit¬ ted, but not necessarily simple vibrations. If they produce com¬ pound vibrations, these, as we have seen (§ 678), must consist of a fundamental simple vibration and its harmonics. 1 The stops called open diapason and stop diapason (consisting respectively of open and stopped pipes), give the fundamental tone, almost free from harmonics. The stop absurdly called principal gives the second tone, that is the octave above the fundamental. The stops called twelfth and fifteenth give the third and fourth tones, which are a twelfth (octave + fifth), and a fifteenth (double octave) above the fundamental. The fifth and sixth tones are included in the stop called mixture. As many of our readers will be unacquainted with the structure of organs, it may be desirable to state that an organ contains a number of complete instruments, each consisting of several octaves of pipes. Each of these complete instruments is called a stop , and is brought into use at the pleasure of the organist by pulling out a slide, by means of a knob- handle, on which the name of the stop is marked. To throw it out of use, he pushes in the slide. A large number of stops are often in use at once. 856 ANALYSIS OF VIBRATIONS. 679. Helmholtz’s Resonators.-—Helmholtz derived material aid in his researches from an instrument devised by himself, and called a resonator or resonance globe (Fig. 606). It is a hollow globe of thin brass, with an opening at each end, the larger one serving for the admission of sound, while the smaller one is introduced into the ear. The inclosed mass of air has, like the column of air in an organ-pipe, a particular fundamental note of its own, depending upon its size; and whenever a note of this particular pitch is sounded in its neigh¬ bourhood, the inclosed air takes it up and intensifies it by resonance. Fig. 606.—Resonator. In order to test the presence or absence of a particular harmonic in a given musical tone, a resonator, in unison with this harmonic, is applied to the ear, and if the resonator speaks it is known that the harmonic is present. These instruments are commonly constructed so as to form a series, whose notes correspond to the bass C of a man’s voice, and its successive harmonics as far as the 10th or 12th. Koenig has applied the principle of manometric flames to enable a large number of persons to witness the analysis of sounds by resona¬ tors. A series of 6 resonators, whose notes have frequencies propor¬ tional to 1, 2, 3, 4, 5, 6, are fixed on a stand (Fig. 607), and their smaller ends, .instead of being applied to the ear, are connected each with a separate manometric capsule, which acts on a gas jet. When the mirrors are turned, it is easy to see which of the flames vibrate while a sonorous body is passed in front of the resonators. A simple tone, unaccompanied by harmonics, is dull and unin¬ teresting, and, if of low pitch, is very destitute of penetrating quality. Sounds composed of the first six elementary tones in fair propor¬ tion, are rich and sweet. VOWEL SOUNDS. 857 The higher harmonics, if sufficiently subdued, may also be present without sensible detriment to sweetness, and are useful as contribut¬ ing to expression. When too loud, they render a sound harsh and Fig. 607.—Analysis by Manometric Flames. grating; an effect which is easily explained by the discordant com¬ binations which they form one with another; the 8th and 9th tones, for example, are at the same interval as the notes Do and Re. 680. Vowel Sounds.—The human voice is extremely rich in har¬ monics, as may be proved by applying the series of resonators to the ear while the fundamental note is sung. The origin of the tones of the voice is in the vocal chords, which, when in use, form a dia¬ phragm with a slit along its middle. The edges of this slit vibrate when air is forced through, and, by alternately opening and closing the passage, perform the part of a reed. The cavity of the mouth serves as a resonance chamber, and reinforces particular notes de¬ pending on the position of the organs of speech. It is by this reson- 858 ANALYSIS OF VIBRATIONS. ance that the various vowel sounds are produced. The deepest pitch belongs to the vowel sound which is expressed in English by oo (as in moon), and the highest to ee (as in screech). Willis in 1828 1 succeeded in producing the principal vowel sounds by a single reed fitted to various lengths of tube. Wheatstone, a few years later, made some advances in theory, 2 and constructed a machine by which nearly all articulate sounds could be imitated. The best determinations of the particular notes which are rein¬ forced in the case of the several vowel sounds, have been made by Helmholtz, who employed several methods, but chiefly the two fol¬ lowing:— 1. Holding resonators to the ear, while a particular vowel sound was loudly sung. 2. Holding vibrating tuning-forks in front of the mouth when in the proper position for pronouncing a given vowel; and observing which of them had their sounds reinforced by resonance. 3 Helmholtz has verified his determinations synthetically. He em¬ ploys a set of tuning-forks which are kept in vibration by the alter¬ nate making and unmaking of electro-magnets, the circuit being made and broken by the vibrations of one large fork of 64 vibrations per second. The notes of the other forks are the successive har¬ monics of this fundamental note. Each fork is accompanied by a resonance-tube, which, when open, renders the note of the fork audible at a distance; and by means of a set of keys, like those of a piano, any of these tubes can be opened at pleasure. The different vowel-sounds can thus be produced by employing the proper com¬ binations. The same apparatus served for establishing the principle (§ 678), that the character of a musical sound depends only on constitution , irrespective of change of phase. 1 Cambridge Transactions, vol. iii. 2 London and Westminster Review, October, 1837. 3 According to Koenig (Comptes Rendus, 1870) the notes of strongest resonance for the vowels u, o, a, e, i, as pronounced in North Germany, are the five successive octaves of B flat, commencing with that which corresponds to the space above the top line of the base clef. Willis, Helmholtz, and Koenig all agree as regards the note of the vowel o, which is very nearly that of a common A tuning-fork. They are also agreed respecting the note of a (as in father), which is an octave higher. CHAPTER LVI A . CONSONANCE, DISSONANCE, AND RESULTANT TONES. 680a. Concord and Discord.—Every one not utterly destitute of musical ear is familiar with the fact that certain notes, when sounded together, produce a pleasing effect by their combination, while certain others produce an unpleasing effect. The combination of two or more notes, when agreeable, is called concord or consonance; when disagreeable, discord or dissonance. The distinction is found to depend almost entirely on difference of pitch, that is, on relative frequency of vibration ; so that the epithets consonant and dissonant can with propriety be applied to intervals. The following intervals are consonant: unison (1 : 1), octave (1 : 2), octave + fifth (1 : 8), double octave (1 : 4), fifth (2 : 8), fourth (3 :4). The major third (4 : 5) and major sixth (3 : 5), together with the minor third (5 : 6) and minor sixth (5 : 8), are less perfect in their consonance. The second and the seventh, whether major or minor' are dis¬ sonant intervals, whatever system of temperament be employed, as are also an indefinite number of other intervals not recognized in music. Besides the difference as regards pleasing or unpleasing effect, it is to be remarked that consonant intervals can be identified by ear with much greater accuracy than those which are dissonant. Musi¬ cal instruments are generally tuned by octaves and fifths, because very slight errors of excess or defect in these intervals are easily detected by the ear. To tune a piano by the mere comparison of successive notes would be be}^ond the power of the most skil¬ ful musician. A sharply marked interval is always a consonant interval. 860 CONSONANCE, DISSONANCE, AND RESULTANT TONES. 680 b. Jarring Effect of Dissonance.—According to the theory pro¬ pounded by Helmholtz, the unpleasant effect of a dissonant interval consists essentially in the production of beats. These have a jarring effect upon the auditory apparatus, which becomes increasingly dis¬ agreeable as the beats increase in frequency up to about 33 per second, and becomes gradually less disagreeable as the frequency is still further increased. The sensation produced by beats is compar¬ able to that which the eye experiences from the bobbing of a gas flame in a room lighted by it; and the frequency which entails the maximum of annoyance is in this case much smaller, on account of the greater persistence of visible impressions. The annoyance must evidently cease when the succession becomes so rapid as to produce the effect of a continuous impression. We have already (§ 644 A) described a mode of producing beats with any degree of frequency at pleasure; and this experiment is one of the main foundations on which Helmholtz bases his view. 680 c. Beats of Harmonics.—The beats in the experiment above alluded to, are produced by the imperfect unison of two notes, and indicate the number of vibrations gained by one note upon the other. Their existence is easily and completely explained by the considera¬ tions adduced in § 644 a. But it is well known to musicians, and easily established by experiment, that beats are also produced be¬ tween notes whose interval is approximately an octave, a fifth, or some other consonance; and that, in these cases also, the beats become more rapid as the interval becomes more faulty. These beats are ascribed by Helmholtz to the common harmonic of the two fundamental notes. For example, in the case of the fifth (2: 3), the third tone of the lower note would be identical with the second tone of the upper, if the interval were exact; and the beats which occur are due to the imperfect unison consequent on the devia¬ tion from exact truth. All beats are thus explained as due to im¬ perfect unison. This explanation is not merely conjectural, but is established by the following proofs:— 1. When an arrangement is employed by which the fifth is made false by a known amount, the number of beats is found to agree with the above explanation. Thus, if the interval is made to cor¬ respond to the ratio 200 : 301, it is observed that there are 2 beats to every 200 vibrations of the lower note. Now the harmonics which BEATING NOTES. 861 are in approximate unison are represented by 600 and 602, and the difference of these is 2. 2. When the resonator corresponding to this common harmonic is held to the ear, it responds to the beats, showing that this harmonic is undergoing variations of strength; but when a resonator corre¬ sponding to either of the fundamental notes is employed, it does not respond to the beats, but indicates steady continuance of its appro¬ priate note. 3. By a careful exercise of attention, a person with a good ear can hear, without any artificial aids, that it is the common harmonic which undergoes variations of intensity, and that the fundamental notes continue steady. 680 D. Beating Notes must be Near Together.—In order that two simple tones may yield audible beats, it is necessary that the musical interval between them should be small; in other words, that the ratio of their frequencies of vibration should be nearly equal to unity. Two simple notes of 300 and 320 vibrations per second will yield 20 beats in a second, and will be eminently discordant, the interval between them being only a semitone (15 : 16), but simple notes of 40 and 60 vibrations per second will not give beats, the interval between them being a fifth (2 : 3). The wider the interval between two simple notes, the feebler will be their beats; and accordingly, for a given frequency of beats, the harshness of the effect increases with the nearness of the notes to each other on the musical scale. 1 By taking joint account of the number of beats and the nearness of the beating tones, Helmholtz has endeavoured to express numeri¬ cally the severity of the discords resulting from the combination of the note C of 256 vibrations per second with any possible note lying within an octave on the upper side of it, a particular constitution (approximately that of the violin) being assumed for both notes. He finds a complete absence of discord for the intervals of uni¬ son, the octave, and the fifth, and very small amounts of discord for the fourth, the sixth, and the third. By far the worst discords are found for the intervals of the semitone and major seventh, 1 The explanation adopted by Helmholtz is, that a certain part of the ear, called Corti's organ , contains a number of elastic fibres, each of which is attuned to a particular simple tone, and is thrown into vibration when this tone, or one nearly in unison with it, is sounded. Two tones in approximate unison, when sounded together, affect several fibres in common, and cause them to beat. Tones not in approximate unison affect entirely distinct sets of fibres, and thus cannot produce interference. 862 CONSONANCE, DISSONANCE, AND KESULTANT TONES. and the next worst are for intervals a little greater or less than the fifth. 680 e. Imperfect Concord.—When there is a complete absence of discord between two notes, they are said to form a perfect concord. The intervals unison, fifth, octave, octave + fifth, and the interval from any note to any of its harmonics, are of this class. The third, fourth, and sixth are instances of imperfect concord. Suppose, for ex¬ ample, that the two notes sounded together are C of 256 and E of 320 vibrations per second, the interval between these notes being a true major third (4:5); and suppose each of these notes to consist of the first six simple tones. The first six multiples of 4 are 4, 8, 12. 16, 20. 24. The first six multiples of 5 are 5. 10, 15, 20, 25, 30. In searching for elements of discord, we select (one from each line) two multiples differing by unity. Those which satisfy this condition are 4 and 5; 16 and 15; 24 and 25. But the first pair (4 and 5) may be neglected, because their ratio differs too much from unity. Discordance will result from each of the two remaining pairs; that is to say, the 4th element of the lower of our two given notes is in discordance with the 3d element of the upper : and the 6th element of the lower is in discordance with the 5th element of the higher. To find the frequencies of the beats, we must multiply all these numbers by 64, since 256 is 4 times 64, and 320 is 5 times 64. Instead of a difference of 1, we shall then find a difference of 64, that is to say, the number of beats per second is 64 in the case of each of the two discordant combinations which we have been considering. 680f. Resultant Tones.—Under certain conditions it is found that two notes, when sounded together, produce by their combination other notes, which are not found as constituents of either. They are called resultant tones , and are of two kinds, difference-tones and summation-tones. A difference-tone has a frequency of vibration which is the difference of the frequencies of its components. A sum¬ mation-tone has a frequency of vibration which is the sum of the RESULTANT TONES. 863 frequencies of its components. As the components may either be fundamental tones or overtones, two notes which are rich in har¬ monics may yield, by their combination, a large number of resultant tones. The difference-tones were observed in the last century by Sorge and Tartini, and were, until recently, attributed to beats. The fre¬ quency of beats is always the difference of the frequencies of vibra¬ tion of the two elementary tones which produce them; and it was supposed that a rapid succession of beats produced a note of pitch corresponding to this frequency. This explanation, if admitted, would furnish an exception to what otherwise appears to be the universal law, that every elementary tone arises from a corresponding simple vibration. 1 Such an excep¬ tion should not be admitted without necessity; and in the present instance it is not only unnecessary, but also insufficient, inasmuch as it fails to render any account of the summation-tones. Helmholtz has shown, by a mathematical investigation, that when two systems of simple waves agitate the same mass of air, their mutual influence must, according to the recognized laws of dynamics, give rise to two derived systems, having frequencies which are respectively the sum and the difference of the frequencies of the two primary systems. Both classes of resultant tones are thus com¬ pletely accounted for. The resultant tones—especially the summation-tones, which are fainter than the others—are only audible when the primary tones are loud; for their existence depends upon small quantities of the second order, the amplitudes of the primaries being regarded (in comparison with the wave-lengths) as small quantities of the first order. If any further proof be required that the difference tones are not due to the coalescence of beats, it is furnished by the fact that, in certain circumstances, the beats and the difference-tones can both be heard together. 680 G. Beats due to Resultant Tones.—The existence of resultant tones serves to explain, in certain cases, the production of beats between notes which are wanting in harmonics. For example, if two simple sounds, of 100 and 201 vibrations per second respectively, are sounded together, one beat per second will be produced between 1 The discovery of this law is due to Ohm. 864 CONSONANCE, DISSONANCE, AND RESULTANT TONES. the difference-tone of 101 vibrations and the primary tone of 100 vibrations. By the heats to which they thus give rise, resultant tones exercise an influence on consonance and dissonance. Resultant tones, when sufficiently loud, are themselves capable of performing the part of primaries, and yielding what are called result - ant tones of the second order, by their combination with other pri¬ maries. Several higher orders of resultant tones can, under pecu¬ liarly favourable circumstances, be sometimes detected. OPTICS CHAPTER LVII. PROPAGATION OF LIGHT. 681. Light.—Light is the immediate externa] cause of our visual impressions. Objects, except such as are styled self-luminous, become invisible when brought into a dark room. The presence of something additional is necessary to render them visible, and that mysterious agent, whatever its real nature may be, we call light Light, like sound, is believed to consist in vibration; but it does not, like sound, require the presence of air or other gross matter to enable its vibrations to be propagated from the source to the per¬ cipient. When we exhaust a receiver, objects in its interior do not become less visible; and the light of the heavenly bodies is not pre¬ vented from reaching us by the highly vacuous spaces which lie between. It seems necessary to assume the existence of a medium far more subtle than ordinary matter; a medium which pervades alike the most vacuous spaces and the interior of all bodies, whether solid, liquid, or gaseous; and which is so highly elastic, in proportion to its density, that it is capable of transmitting vibrations with a velocity enormously transcending that of sound. This hypothetical medium is called cether. From the extreme facility with which bodies move about in it, we might be disposed to call it a subtle fluid; but the undulations which it serves to propagate are not such as can be propagated by fluids. Its elastic properties are rather those of a solid; and its waves are analogous to the pulses which travel along the wires of a piano rather than to the waves of extension and compression by which sound is propa¬ gated through air. Luminous vibrations are transverse , while those of sound are longitudinal. A self-luminous body, such as a red-hot poker or the flame of a 56 PROPAGATION OF LIGHT. 866 candle, is in a peculiar state of vibration. This vibration is com¬ municated to the surrounding sether, and is thus propagated to the eye, enabling us to see the body. In the majority of cases, however, we see bodies not by their own but by reflected light; and we are enabled to recognize the various kinds of bodies by the different modifications which light undergoes in reflection from their surfaces. As all bodies can become sonorous, so also all bodies can become self-luminous. To render them so, it is only necessary to raise them to a sufficiently high temperature, whether by the communication of heat from a furnace, or by the passage of an electric current, or by causing them to enter into chemical combination. It is to chemi¬ cal combination, in the active form of combustion, that we are in¬ debted for all the sources of light in ordinary use. The vibrations of the sether are capable of producing other effects besides illumination. They constitute what is called radiant heat, and they are also capable of producing chemical effects, as in photo¬ graphy. Vibrations of high frequency, or short period, are the most active chemically. Those of low frequency or long period have usually the most powerful heating effects; while those which affect the eye with the sense of light are of moderate frequency. 682. Rectilinear Propagation of Light.—All the remarks which have been made respecting the relations between period, frequency, and wave-length, in the case of sound, are equally applicable to light, inasmuch as all kinds of luminous waves (like all kinds of sonorous waves) have the same velocity in the same medium; but this velo¬ city is many hundreds of thousands of times greater for light than for sound, and the wave-lengths of light are at the same time very much shorter than those of sound. Frequency, being the quotient of velocity by wave-length, is accordingly about a million of millions of times greater for light than for sound. The colour of lowest pitch is deep red, its frequency being about 400 million million vibrations per second, and its wave-length in air 760 millionths of a millimetre. The colour of highest pitch is deep violet; its frequency is about 760 million million vibrations per second, and its wave-length in air 400 millionths of a millimetre. It thus appears that the range of seeing is much smaller than that of hearing, being only about one octave. The excessive shortness of luminous as compared with sonorous waves is closely connected with the strength of the shadows cast by a light, as compared with the very moderate loss of intensity pro¬ duced by interposing an obstacle in the case of sound. Sound may, RECTILINEAR PROPAGATION. 867 for ordinary purposes, be said to be capable of turning a corner, and light to be only capable of travelling in straight lines. The latter fact may be established by such an arrangement as is represented in Fig. 608. Two screens, each pierced with a hole, are arranged so that these holes are in a line with the flame of a candle. An eye placed in this line, be¬ hind the screens, is then able to see the flame; but a slight lateral dis¬ placement, either of the eye, the candle, or either of the screens, puts the flame out of sight. Fig. 608.—Rectilinear Propagation. It is to be noted that, in this experiment, the same medium (air) extends from the eye to the candle. We shall hereafter find that, when light has to pass from one medium to another, it is often bent out of a straight line. We have said that the strength of light-shadows as compared with sound-shadows is connected with the shortness of luminous waves. Theory shows that, if light is transmitted through a hole or slit, whose diameter is a very large multiple of the length of a light¬ wave, a strong shadow should be cast in all oblique directions; but that, if the hole or slit is so narrow that its diameter is comparable to the length of a wave, a large area not in the direct path of the beam will be illuminated. The experiment is easily performed in a dark room, by admitting sunlight through an exceedingly fine slit, and receiving it on a screen of white paper. The illuminated area will be marked with coloured bands, called diffraction-fringes; and if the slit is made narrower, these bands become wider. On the other hand, Colladon, in his experiments on the transmis¬ sion of sound through the water of the Lake of Geneva, established the presence of a very sharply defined sound-shadow in the water, behind the end of a projecting wall. For the present we shall ignore diffraction, 1 and confine our atten- 1 See Chap. lxiv. 868 PROPAGATION OF LIGHT. tion to the numerous phenomena which result from the rectilinear propagation of light. 683. Images produced by Small Apertures.—If a white screen is placed opposite a hole in the shutter of a room otherwise quite dark, Fig. 609.—Image formed by Small Aperture. an inverted picture of the external landscape will be formed upon it, in the natural colours. The outlines will be sharper in proportion as the hole is smaller, and distant objects will be more distinctly represented than those which are very near. These results are easily explained. Consider, a in fact, an external object AB (Fig. 610), and let 0 be the hole in the shutter. The point A sends rays in all directions into space, and among B them a small pencil, which, after passing through the opening O, falls upon the screen at A'. A' Fig. 6io.— Explanation, receives light from no other point but A, and A sends light to no part of the screen except A'. The colour and brightness of the spot A' will accordingly depend upon the colour and brightness of A; in other words, A' will be the IMAGES PRODUCED BY SMALL APERTURES. 869 image of A. In like manner B' will be the image of B, and points of the object between A and B will have their images between A' and B. A.n inverted image A' B' will thus be formed of the object A B. As the image thus formed of an external point is not a point, but a spot, whose size increases with that of the opening, there must always be a little blurring of the outlines from the overlapping of the spots which represent neighbouring points; but this will be com¬ paratively slight if the opening is very small. An experiment, substantially the same as the above, may be per¬ formed by piercing a card with a large pin-hole, and holding it between Fig. 611.—Image formed by Hole in a Card. a candle and a screen, as in Fig. 611. An inverted image of the candle will thus be formed upon the screen. When the sun shines through a small hole into a room with the blinds down, the cone of rays thus admitted is easily traced by the lighting up of the particles of dust which lie in its course. The image of the sun which is formed at its lower extremity may be either circular or elliptical, according to the aspect of the surface on which it is received. Fine images of the sun are sometimes thus formed by the chinks of a venetian-blind, especially when the sun is low, and there is a white wall opposite to receive the image. In 870 PROPAGATION OF LIGHT. these circumstances it is sometimes possible to detect the presence of spots on the sun by examining the image. When the sun’s rays shine through the foliage of a tree, the spots of light which they form upon the ground are always round or oval, whatever may be the shape of the interstices through which they have passed, provided always that these interstices are small. When the sun is undergoing eclipse, the progress of the eclipse can be traced Fig. 612.—Conical Sunbeam. by watching the shape of these images, which resembles that of the uneclipsed portion of the sun’s disc. 684. Theory of Shadows.—The rectilinear propagation of light is the foundation of the geometry of shadows. Let the source of light be a luminous point, and let an opaque body be placed so as to inter¬ cept a portion of its rays. If we construct a conical surface touching the body all round, and having its vertex at the luminous point, it is evident that all the space within this surface on the further side of the opaque body is completely screened from the rays. The cone thus constructed is called the shadow-cone, and its intersection with any surface behind the opaque body defines the shadow cast upon that surface. In the case which we have been supposing—that of a luminous point—the shadow-cone and the shadow itself will be sharply defined. THEORY OF SHADOWS. 871 Fig. 613.—Images of Sun formed by Foliage. Fig. 614.—Shadow. 872 PROPAGATION OF LIGHT. Actual sources of light, however, are not mere luminous points, but have finite dimensions. Hence some complication arises. Con¬ sider, in fact (Fig. 615), a luminous body situated between two opaque bodies, one of them larger, and the other smaller than itself. Con¬ ceive a cone touching the luminous body and either of the opaque bodies externally. This will be the cone of total shadow , or the cone of the umbra. All points lying within it are completely ex¬ cluded from view of the luminous body. This cone narrows or en¬ larges as it recedes, according as the opaque body is smaller or larger than the luminous body. In the former case it terminates at a finite distance. In the latter case it extends to infinite distance. Now conceive a double cone touching the luminous body and either of the opaque bodies internally. This cone will be wider than the cone of total shadow, and will include it. It is called the Fig. 615.—Umbra and Penumbra. cone of partial shadow , or the cone of the penumbra , All points lying within it are excluded from the view of some portion of the luminous body, and are thus partially shaded by the opaque body. If they are near its outer boundary, they are very slightly shaded. If they are so far within it as to be near the total shadow, they are almost completely shaded. Accordingly, if the shadow of the opaque body is received upon a screen, it will not have sharply defined edges, but will show a gradual transition from the total shadow which covers a finite central area to a complete absence of shadow at the outer boundary of the penumbra. Thus neither the edges of the umbra nor those of the penumbra are sharply defined. The umbra and penumbra show themselves on the surface of the VELOCITY OF LIGHT. 873 opaque body itself, the line of contact of the umbral cone being further back from the source of light than the line of contact of the penumbral cone. The zone between these two lines is in partial shadow, and separates the portion of the surface which is in total shadow from the part which is not shaded at all. 685. Velocity of Light.—Luminous undulations, unlike those of sound, advance with a velocity which may fairly be styled incon¬ ceivable, being about 298 million metres per second, or 185,000 miles per second. As the circumference of the earth is only 40 mil¬ lion metres, light would travel seven and a half times round the earth in a second. Hopeless as it might appear to attempt the measurement of such an enormous velocity by mere terrestrial experiments, the feat has actually been performed, and that by two distinct methods. In Fizeau’s experiments the distance between the two experimental stations was about 5% miles. In Foucaults experiments the whole apparatus was contained in one room, and the movement of light within this room served to determine the velocity. We will first describe Fizeau’s experiment. 686. Fizeau’s Experiment.—Imagine a source of light placed di¬ rectly in front of a plane mirror, at a great distance. The mirror will send back a reflected beam along the line of the incident beam, and an observer stationed behind the source will see its image in the mirror as a luminous point. Now imagine a toothed-wheel, with its plane perpendicular to the path of the beam, revolving uniformly in front of the source, in such a position that its teeth pass directly between the source of light and the mirror. The incident beam will be stopped by the teeth, as they successively come up, but will pass through the spaces between them. Now the velocity of the wheel may be such that the light which has thus passed through a space shall be reflected back from the mirror just in time to meet a tooth and be stopped. In this case it will not reach the observer’s eye, and the image may thus become per¬ manently invisible to him. From the velocity of the wheel, and the number of its teeth, it will be possible to compute the time occupied by the light in travelling from the wheel to the mirror, and back again. If the velocity of the wheel is such that the light is some¬ times intercepted on its return, and sometimes allowed to pass, the image will appear steadily visible, in consequence of the persistence of impressions on the retina, but with a loss of brightness propor- 874 PROPAGATION OF LIGHT. tioned to the time that the light is intercepted. The wheel employed by Fizeau had 720 teeth, the distance between the two stations was 8663 metres, and 12’6 revolutions per second produced disappearance of the image. The width of the teeth being equal to the width of the spaces, the time required to turn through the width of a tooth was i X 7 -j-Q X Yxg- of a second, that is l8 1 4 - 4 - of a second. In this time the light travelled a distance of 2x8663=17326 metres. The distance traversed by light in a second would therefore be 17,326x18,144 = 314,262,944 metres. This determination of M. Fizeau’s is believed to be somewhat in excess of the truth. A double velocity of the wheel would allow the reflected beam to pass through the space succeeding that through which the incident beam had passed; a triple velocity would again produce total eclipse, and so on. Several independent determinations of the velocity of light may thus be obtained. Fig. 616.—Fizeau’s Experiment. Thus far, we have merely indicated the principle of calculation. It will easily be understood that special means were necessary to prevent scattering of the light, and render the image visible at so great a distance. Fig. 616 will serve to give an idea of the apparatus actually employed. A beam of light from a lamp, after passing through a lens, falls on a plate of unsilvered glass M, placed at an angle of 45°, by which it is reflected along the tube of a telescope; the object-glass of the telescope is so adjusted as to render the rays parallel on emergence, and in this condition they traverse the interval between the two stations. At the second station they are collected by a lens, which foucault's experiment. 875 brings them to a focus on the surface of a plane mirror, and this mirror sends them back along the same course by which they came. A portion of the light thus sent back to the glass plate M passes through it, and is viewed by the observer through an eye-piece. The wheel It is driven by clock-work. Figs. 617, 618, 619 respect- Fig. 617.—Wheel at Rest. Fig. 618.—Total Eclipse. Fig. 619.—Partial Eclipse. ively represent the appearance of the luminous point as seen between the teeth of the wheel when not revolving, the total eclipse produced by an appropriate speed of rotation, and the partial eclipse produced by a different speed. It was found that there was some uncertainty in the phenomenon of total eclipse, probably because the light, when not entirely inter¬ cepted, might be so much reduced in amount as to be incapable of affecting the retina. The experiment of M. Fizeau was very remarkable as the first instance in which light was proved to occupy a finite time in tra¬ versing terrestrial distances; but the method is scarcely capable of giving accurate results. Foucault’s method, which we now proceed to explain, is much better fitted for rigorous determinations. 687. Foucault’s Experiment.—Foucault employed the principle of the rotating mirror, first adopted by Wheatstone in his experiments on the duration of the electric spark and the velocity of electricity (§ 437, 466). The following was the construction of his original apparatus:— A beam of light enters a room by a square hole, which has a fine platinum wire stretched across it, to serve as a mark; it is then concentrated by an achromatic lens, and, before coming to a focus, falls upon a plane mirror, revolving about an axis in its own plane. In one part of the revolution the reflected beam is directed upon a concave mirror, whose centre of curvature is in the axis of rotation, so that the beam is reflected back to the revolving mirror, and 876 PROPAGATION OF LIGHT. thence back to the hole a" which it first entered. Before reaching the hole, it has to traverse a sheet of glass, placed at an angle of 45°, which reflects a portion of it towards the observer’s eye; and the image which it forms (an image of the platinum wire) is viewed through a powerful eye-piece. The image is only formed during a small part of each revolution; but when 30 turns are made per second, the appearance presented, in consequence of the persistence of im¬ pressions, is that of a permanent image occupying a fixed position. When the speed is considerably greater, the mirror turns through a sensible angle while the light is travelling from it to the concave mirror and back again, and a sensible displacement of the image is accordingly observed. The actual speed of rotation was from 700 to 800 revolutions per second. 1 On interposing a tube filled with water between the two mirrors, it was found that the displacement was increased, showing that a longer time was occupied in traversing the water than in traversing the same length of air. This result, as we shall have occasion to point out later, is very important as confirming the undulatory theory and disproving the emission theory of light. In Fig. 619 a, a is the position of the platinum wire, L is the achromatic lens, m the revolving mirror, c the axis of revolution, M 1 It was found that, at this high speed, the amalgam at the back of ordinary looking- glasses was driven off by centrifugal force. The mirror actually employed was silvered in front with real silver. Foucault’s experiment. 877 the concave mirror, a' the image of the platinum wire, displaced from a in virtue of the rotation of the mirror; a, a images of a, a', formed by the glass plate g , and viewed through the eye-piece O. M' is a second concave mirror, at the same distance as M from the revolving mirror; T is a tube filled with water, and having plane glass ends, and L' a lens necessary for completing the focal adjust¬ ment; a " and a" are the images formed by the light which has tra¬ versed the water. 1 Foucault’s experiment, as thus described, was performed in 1850, very shortly after that of Fizeau. Some important improvements have since been introduced in the method, especially as regards the measurement of the speed of rotation of the mirror, which is evi¬ dently a principal element in the calculation. The mirror is driven by means of a bellows, furnished with a special arrangement for keeping up a constant pressure of air, and driving a kind of syren, on which the mirror is mounted. Instead of making a separate determination of the speed of rotation in each experiment, M. Fou¬ cault has adopted a very rigorous method of keeping it always at one constant value, namely, 400 revolutions per second. This is less than the speed attained in the earlier experiments; but, on the other hand, the length of the path traversed by the light between its two reflections from the revolving mirror is increased, by means of successive reflections, so as to be about 20 metres, instead of 4 as in the original experiments. 1 The distances are such that La and Lc + cM are conjugate focal distances with respect to the lens L. An image of the wire a is thus formed at M, and an image of this image is formed at a, the mirror being supposed stationary; and this relation holds not only for the central point of the concave mirror, but‘for any part of it on which the light may happen to fall at the instant considered. Let l denote the distance c M between the revolving and the fixed mirror, l' the distance c L of the revolving mirror from the centre of the lens, r the distance a L of the platinum wire from the centre of the lens, n the number of revolutions per second, V the space tra¬ versed by light in a second, t the time occupied by light in travelling from one mirror to the other and back, 0 the angle turned by the mirror in this time, and 5 the angle sub¬ tended at the centre of the lens by the distance a a between the wire and its displaced image. Then obviously t = but also t = —?— • hence V = — V 2t rro’ 0 Now the distance between the two images (corresponding to a, a' respectively) at the back of the revolving mirror is (Z-f V) 5, and is also 2 61 (§ 705 a). Hence 0 = ^- + ^j 8 tt n V s m and V = g* The observed distance a a' between the two images is equal to the dis¬ tance between a, a that is to r 5. Calling this distance cl, we have finally, 8 7r nPr v= j+n~d‘ 878 PROPAGATION OF LIGHT. The constant rate of revolution is maintained by comparison with a clock. A wheel with 400 teeth, driven by the clock, makes exactly one revolution per second. A tooth and a space alternately cover the part of the field where the image of the wire-grating (which has been substituted for the single wire) is formed. The same instan¬ taneous flashes of light from the revolving mirror which form the image, also illuminate the rim of the wheel. If the wheel advances exactly one tooth and space between consecutive flashes, its illumi¬ nated positions are undistinguishable one from another, and the wheel accordingly appears stationary. When this is the case, it is known that the mirror is making exactly 400 turns per second. A slight departure from this rate either w~ay, makes the wheel appear to be slowly revolving either forwards or backwards, and the bellows must be regulated until the stationary appearance is presented. By means of this admirable combination, Foucault has made what must be regarded as the best determination yet obtained of the velo¬ city of light. The value thus found, namely, 298 million metres per second, is smaller than that which, until a few years ago, was gene¬ rally received; but recent astronomical discussions have shown that the sun’s distance is somewhat less than was previously supposed; and when this correction is made, the astronomical determinations of the velocity of light agree well with that of Foucault. 688. Velocity of Light deduced from Observations of the Eclipses of Jupiter’s Satellites.—The fact that light occupies a sensible time in travelling over celestial distances, was first established about 1675, by Roemer, a Danish astronomer, who also made the first computa¬ tion of its velocity. He was led to this discovery by comparing the observed times of the eclipses of Jupiter’s first satellite, as contained in records extending over many successive years. The four satellites of Jupiter revolve nearly in the plane of the planet’s orbit, and undergo very frequent eclipse by entering the cone of total shadow cast by Jupiter. The satellites and their eclipses are easily seen, even with telescopes of very moderate power; and being visible at the same absolute time at all parts of the earth’s surface at which they are visible at all, they serve as signals for comparing local time at different places, and thus for determining longitudes. The first satellite (that is, the one nearest to Jupiter), from its more rapid motion and shorter time of revolution, affords both the best and the most frequent signals. The interval of time between two successive eclipses of this satellite is about 42J hours, VELOCITY OF LIGHT. 879 Fig. 620.—Earth and Jupiter. but was found by Roemer to vary by a regular law according to the position of the earth with respect to Jupiter. It is longest when the earth is increasing its distance from Jupiter most rapidly, and is shortest when the earth is diminishing its distance most rapidly. Starting from the time when the earth is nearest to Jupiter, as at T, J (Fig. 620), the intervals between successive eclipses are always longer than the mean value, until the greatest distance has been attained, as at T, J', and the sum of the excesses amounts to 16 min. 26 6 sec. From this time until the nearest distance is again attained, as at T", J", the inter¬ vals are always shorter than the mean, and the sum of the defects amounts to 16 min. 26'6 sec. It is evident, then, that the eclipses are visible 16 m. 26’6 s. earlier at the nearest than at the remotest point of the earth’s orbit; in other words, that this is the time required for the propagation of light across the diameter of the orbit. Taking this diameter as 183 millions of miles, 1 we -have a resulting velocity of about 185,500 miles per second. 688a. Velocity of Light deduced from Aberration.—About fifty years after Roemer’s discovery, Bradley, the English astronomer, employed the velocity of light to explain the astronomical pheno¬ menon called aberration. This consists in a regular periodic displace¬ ment of the stars as seen from the earth, the period of the displace¬ ment being a year. If the direction in which the earth is moving in its orbit at any instant be regarded as the forward direction, every star constantly appears on the forward side of its true place, so that, as the earth moves once round its orbit in a year, each star describes in this time a small apparent orbit about its true place. The phenomenon is explained in the same way as the familiar fact, that a shower of rain falling vertically, seems, to a person run¬ ning forwards, to be coming in his face. The relative motion of the rain-drops with respect to his body, is found by compounding the actual velocity of the drops (whether vertical or oblique) with a 1 The sun’s mean distance from the earth was, until recently, estimated at 95 millions of miles. It is now estimated at 92 or 91^ millions. 880 PROPAGATION OF LIGHT. velocity equal and opposite to that with which he runs. Thus if A B (Fig. 620 A) represents the velocity with which he runs, and C A the true velocity of the drops, the apparent velo¬ city of the drops will be represented by DA. If a tube pointed along A D moves forward parallel to itself with the velocity A B, a drop entering at its upper end will pass through its whole length without wetting its sides; for while the drop is falling along D B (we suppose with uniform velocity) the tube moves along A B, so that the lower end of the tube reaches B at the same time as the rain-drop. In like manner, if A B is the velocity of the earth, and C A the velocity of light, a telescope must be pointed along A D to see a star which really lies in the direction of AC or B D produced. When the angle BAC is a right angle (in other words, when the star lies in a direction perpendicular to that in -which the earth is moving), the angle CAD, which is called the aberration of the star, is 20"*5, and the tangent of this angle is the ratio of the velocity of the earth to the velocity of light. Hence it is found by computation that the velocity of light is about ten thousand times greater than that with which the earth moves in its orbit. The latter is easily computed, if the sun’s distance is known, and is about 18J miles per second. Hence the velocity of light is about 185,000 miles per second. It will be noted that both these astronomical methods of computing the velocity of light, depend upon the knowledge of the sun’s distance from the earth, and that, if this distance is overestimated, the com¬ puted velocity of light will be too great in the same ratio. Conversely, the velocity of light, as determined by Foucault’s method, can be employed, in connection either with aberration or the eclipses of the satellites, for computing the sun’s distance; and the first correct determination of the sun’s distance was, in fact, that deduced by Foucault from his own results. 689. Photometry.—Photometry is the measurement of the relative ^amounts of light emitted by different sources. The methods em¬ ployed for this purpose all consist in determinations of the relative distances at which two sources produce equal intensities of illumina¬ tion. The eye would be quite incompetent to measure the ratio of two unequal illuminations; but a pretty accurate judgment can be formed as to equality or inequality of illumination, at least when the bouguer’s photometer. 881 surfaces compared are similar, and the lights by which they are illu¬ minated are of the same colour. The law of inverse squares is always made the basis of the resulting calculations; and this law may itself be verified by showing that the illumination produced by one candle at a given distance is equal to that produced by four candles at a distance twice as great. 690. Bouguer’s Photometer.—Bouguer’s photometer consists of a semi-transparent screen, of white tissue paper, ground glass, or thin white porcelain, divided into two parts by an opaque partition at right angles to it. The two lamps which are to be compared are Fig. 621.—Bouguer’s Photometer. placed one on each side of this partition, so that each of them illu¬ minates one-half of the transparent screen. The distances of the two lamps are adjusted until the two portions of the screen, as seen from the back, appear equally bright. The distances are then mea¬ sured, and their squares are assumed to be directly proportional to the illuminating powers of the lamps. 691. Rumford’s Photometer.—Rumford’s photometer is based on the comparison of shadows. A cjdindric rod is so placed that each of the two lamps casts a shadow of it on a screen; and the distances are adjusted until the two shadows are equally dark. As the shadow thrown by one lamp is illuminated by the other lamp, the compari¬ son of shadows is really a comparison of illuminations. 692. Foucault’s Photometer.—The two photometers just described 57 882 PROPAGATION OF LIGHT. are alike in principle. In each of them the two surfaces compared are illuminated each by one only of the sources of light. In Rum- ford’s the remainder of the screen is illuminated by both. In Bouguer’s it consists merely of an intervening strip which is illumi¬ nated by neither. If the partition is movable, the effect of moving it further from the screen will be to make this dark strip narrower until it disappears altogether; and if it be advanced still further, the two illuminated portions will overlap. In Foucault’s photometer there is an adjusting screw, for the purpose of advancing the parti¬ tion so far that the dark strip shall just vanish. The two illuminated portions, being then exactly contiguous, can be more easily and certainly compared. Fig. 622.—RumfcmTs Photometer. 693. Bunsen’s Photometer.— Bunsen’s photometer consists of a screen of white paper with a grease-spot in its centre. The lights to be compared are placed on opposite sides of this screen, and their distances are so adjusted that the grease-spot appears neither brighter nor darker than the rest of the paper, from whichever side it is viewed. When the distances have not been correctly adjusted, the grease-spot will appear darker than the rest of the paper when viewed from the side on which the illumination is most intense, and lighter than the rest of the paper when viewed from the other side. CHAPTER LVIIL REFLECTION OF LIGHT. 694. Reflection.—If a beam of the sun’s rays A B (Fig. 623) be admitted through a small hole in the shutter of a dark room, and allowed to fall on a polished plane surface, it will be seen to continue its course in a different direction B C. This is an example of reflec- Fig. 623.—Reflection of Light. tion. A B is called the incident beam, and B C the reflected beam. The angle ABD contained between an incident ray and the normal is called the angle of incidence; and the angle CBD contained between the corresponding reflected ray and the normal is called the angle of reflection. The plane ABD containing the incident ray and the normal is called the plane of incidence. 695. Laws of Reflection.—The reflection of light from polished surfaces takes place according to the following laws:— 1. The reflected ray lies in the plane of incidence. 884 REFLECTION OF LIGHT. 2. The angle of reflection is equal to the angle of incidence. These laws may be verified by means of the apparatus represented in Fig. 624. A vertical divided circle has a small polished plate fixed at its centre, at right angles to its plane, and two tubes travelling on its circumference with their axes always directed towards the centre. The zero of the divisions is the highest point of the circle, the plate being horizontal. A source of light, such as the flame of a candle, is placed so that its rays shine through one of the tubes upon the plate at the centre. As the tubes are blackened internally, no light passes through except in a direction almost Fig. 624.—verification of Laws precisely parallel to the axis of the tube. The observer then looks through the other tube, and moves it along the circumference till he finds the position in which the reflected light is visible through it. On examining the graduations, it will be found that the two tubes are at the same distance from the zero point, on opposite sides. Hence the angles of incidence and reflection are equal. Moreover the plane of the circle is the plane of incidence, and this also contains the reflected rays. Both the laws are thus verified. 696. Artificial Horizon.—These laws furnish the basis of a method of observation which is frequently employed for determining the altitude of a star, and which, by the consistency of its results, fur¬ nishes a very rigorous proof of the laws. A vertical divided circle (Fig. 625) is set in a vertical plane by proper adjustments. A telescope movable about the axis of the circle is pointed to a particular star, so that its line of collimation I' S' passes through the apparent place of the star. Another tele¬ scope, 1 similarly mounted on the other side of the circle, is directed downwards along the line I' R towards the image of the star as seen in a trough of mercury I. Assuming the truth of the laws of reflec¬ tion as above stated, the altitude of the star is half the angle between the directions of the two telescopes; for the ray S I from the star to the mercury is parallel to the line ST, by reason of the excessively great distance of the star; and since the rays S I, I R are equally inclined to the normal I N, which is a vertical line, the lines I'S', I' R are also equally inclined to the vertical, or, what is the same thing, 1 In practice, a single telescope usually serves for both observations. ARTIFICIAL HORIZON. 885 are equally inclined to a horizontal plane. A reflecting surface of mercury thus used is called a mercury horizon, or an artificial Fig. 625.—Artificial Horizon. horizon. Observations thus made give even more accurate results than those in which the natural horizon presented by the sea is made the standard of reference. 697. Irregular Reflection.—The reflection which we have thus far been discussing is called regular reflection. It is more marked as the reflecting surface is more highly polished, and (except in the case of metals) as the incidence is more oblique. But there is an¬ other kind of reflection, in virtue of which bodies, when illuminated, send out light in all directions, and thus become visible. This is called irregular reflection or diffusion. Regular reflection does not render the reflecting body visible, but exhibits images of surrounding objects. A perfectly reflecting mirror would be itself unseen, and 886 REFLECTION OF LIGHT. actual mirrors are only visible in virtue of the small quantity of diffused light which they usually emit. The transformation of in¬ cident into diffused light is usually selective; so that, though the incident beam may be white, the diffused light is usually coloured. The power which a body possesses of making such selection consti¬ tutes its colour. The word reflection is often used by itself to denote what we have here called regular reflection , and we shall generally so employ it. 698. Mirrors.—The mirrors of the ancients were of metal, usually of the compound now known as speculum-metal. Looking-classes date from the twelfth century. They are plates of glass, coated at the back with an amalgam of quicksilver and tin, which forms the reflecting surface. This arrangement has the great advantage of excluding the air, and thus preventing oxidation. It is attended, however, with the disadvantage that the surface of the glass and the surface of the amalgam form two mirrors; and the superposition of the two sets of images produces a confusion which would be in¬ tolerable in delicate optical arrangements. The mirrors, or specula as they are called, of reflecting telescopes are usually made of specu¬ lum-metal , which is a bronze composed of about 32 parts of copper to 15 of tin. Lead, antimony, and arsenic are sometimes added. Of late years specula of glass coated in front with real silver have been extensively used; they are known as silvered specula. A coating of platinum has also been tried, but not with much success. The mirrors employed in optics are usually either plane or spherical. 699. Plane Mirrors. — By a plane mirror we mean any plane reflect¬ ing surface. Its effect, as is well- known, is to produce, behind the mir¬ ror, images exactly similar, both in form and size, to the real objects in front of it. This phenomenon is easily explained by the laws of reflection. Let M N (Fig. 626) be a plane mir¬ ror, and S a luminous point. Bays S I, S I', S I" proceeding from this point give rise to reflected rays 10, I' O', I" 0"; and each of these, if produced backwards, will meet the normal S K in a point S', which is at the same distance behind the mirror that S is in front of s V ,IN j/ K Wj/\/ y'i / /'fe/ / S' ///' /'/ , • a " Fig. 626.—Plane Mirror. PLANE MIRRORS. 887 it. 1 The reflected rays have therefore the same directions as if they had come from S', and the eye receives the same impression as if S' were a luminous point. Fig. G27 represents a pencil of rays emitted by the highest point of a candle-flame, and re¬ flected from a plane mir¬ ror to the eye of an ob¬ server. The reflected rays are divergent (like the in¬ cident rays), and if pro¬ duced backwards would meet in a point, which is the position of the image of the top of the flame. As an object is made up of points, these principles show that the image of an object formed by, a plane mirror must be equal to the object, and symmetrically situated with respect to the plane of the mirror. For example, if A B (Fig. 628) is an object in front of the mirror, an eye placed at O will see the image of the point A at A', the image of B at B', and so on for all the other points of the ob¬ ject. The position of the image A'B' depends only on the posi¬ tions of the object and of the mirror, and remains stationary as the eye is moved about. It is possible, however, to find positions from which the eye will not see the image at all, the conditions of visibility being the same as if the image were a real object, and the mirror were an opening through which it could be seen. The images formed by a plane mirror are erect. They are not however exact duplicates of the objects from which they are formed, 1 This is evident from the comparison of the two triangles S K I, S' K I, bearing in mind that the angle N I S is equal to the alternate angle I S K, and N 10 to K S' I. Fig. 62S.—Incident and Reflected Pencils. Fig. 627.—Image of Candle. 888 REFLECTION OF LIGHT. but differ from them precisely in the same way as the left foot or hand differs from the right. The image of a printed page is like the appearance of the page as seen through the paper from the back, or like the type from which the page was printed. 700. Images of Images.— When rays from a luminous point m have been reflected from a mirror AB (Fig. 629), their subsequent course is the same as if they had come from the image m at the back of the mirror. Hence, if they fall upon a second mirror C D, an image m" of the first image will be formed at the back of the second Fig. 629.—Reflection from two Mirrors. mirror. If, after this, they undergo a third reflection, an image of m" will be formed, and so on indefinitely. The figure shows the actual paths of two rays mirs, mi' r' s'. They diverge first Fig. 630.—Parallel Mirrors. from m, then from m', and lastly from m". This is the principle of the multiple images formed by two or more mirrors, as in the following experiments. 701. Parallel Mirrors.—Let an object O be placed between two PARALLEL MIRRORS. 889 parallel mirrors which face each other, as in Fig. 630. The first reflections will form images di o x . The second reflections will form images a 2 o 2 of the first images; and the third reflections will form images a 3 o 3 of the second images. The figure represents an eye receiving the rays which form the third images , and shows the paths which these rays have taken in their whole course from the object 0 to the eye. The rays by which the same eye sees the other images are omitted, ,to avoid confusing the figure. A long row of images can thus be seen at once, becom¬ ing more and more dim as they recede in the distance, inasmuch as each reflection involves a loss of light. If the mirrors are truly parallel, all the images will be ranged in one straight line, which will be normal to the mirrors. If the the images will be ranged on th m D n % P 0 n' m r 1 i Fig. 631.—Mirrors at Right Angles. irrors are inclined at any angle, circumference of a circle, whose Fig. 632.—Mirrors at Right Angles. centre is on the line in which the reflecting surfaces would intersect if produced. This principle is sometimes employed as a means of adjusting mirrors to exact parallelism. 702. Mirrors at Right Angles.—Let two mirrors O A, 0 B (Fig. 631), 890 REFLECTION OF LIGHT. be set at right angles to each other, facing inwards, and let m be a luminous point placed between them. Images m'm" will be formed by first reflections, and two coincident images will be formed at m"' ■ by second reflections. No third reflection will occur, for the point m", being behind the planes of both the mirrors, cannot be reflected in either of them. Counting the two coincident images as one, and also counting the object as one, there will be in all four images, placed at the four corners of a rectangle. Fig. 632 will give an idea of the appearance actually presented when one of the mirrors is vertical and the other horizontal. When both the mirrors are verti¬ cal, an observer sees his own image constantly bisected by their com¬ mon section, in a way which appears at first sight very paradoxical. 703. Mirrors Inclined at 60 Degrees.—A symmetrical distribution of images may be obtained by placing a pair of mirrors at any angle which is an aliquot part of 360°. If, for example, they be inclined at 60° to each other, the number of images, counting the object itself as one, will be six. Their position is illustrated by Fig. 633. The object is placed in the sector AC B. The images formed by first reflections are situated in the two neighbouring sectors B C A', A CB'; the images formed by second reflec¬ tions are in the sectors B'CA", A' C B", and these yield, by third reflections, two coincident images in the sector B" C A", which is vertically opposite to the sector A C B in which the object lies, and is therefore behind the planes of both mirrors, so that no further reflection can occur. 704. Kaleidoscope.—The symmetrical distribution of images, ob¬ tained by two mirrors inclined at an angle which is an aliquot part of four right angles, is the principle of the kaleidoscope, an optical toy invented by Sir David Brewster. It consists of a tube containing two glass plates, extending along its whole length, and inclined at an angle of 60°. One end of the tube is closed by a metal plate, with the exception of a hole in the centre, through which the observer looks in; at the other end there are two plates, one of ground and the other of clear glass (the latter being next the eye), with a number of little pieces of coloured glass lying loosely between them. These Fig. 633.—Images in Kaleidoscope. THE KALEIDOSCOPE. 891 coloured objects, together with their images in the mirrors, form sym¬ metrical patterns of great beauty, which can be varied by turning or shaking the tube, so as to cause the pieces of glass to change their positions. A third reflecting plate is sometimes employed, the cross-section of the three forming an equilateral triangle. As each pair of plates produces a kaleidoscopic pattern, the arrangement is nearly equiva¬ lent to a combination of three kaleidoscopes. The kaleidoscope is capable of rendering important aid to designers. Fig 634.—Kaleidoscopic Pattern. Fig. 634 represents a pattern produced by the equilateral arrange¬ ment of three reflectors just described. 705. Pepper’s Ghost.—Many ingenious illusions have been con¬ trived, depending on the laws of reflection from plane surfaces. We shall mention two of the most modern. In the magic cabinet , there are two vertical mirrors hinged at the two back corners of the cabinet, and meeting each other at a right angle, so as to make angles of 45° with the sides, and also with the back. A spectator seeing the images of the two sides, mistakes them for the back, which they precisely resemble; and performers may be concealed behind the mirrors when the cabinet appears empty. If one of the persons thus concealed raises his head above the mirrors, it will appear to be suspended in mid-air without a body. 892 REFLECTION OF LIGHT. The striking spectral illusion known as Pepper's Ghost is produced by reflection from a large sheet of unsilvered glass, which is so ar¬ ranged that the actors on the stage are seen through it, while other actors, placed in strong illumination, and out of the direct view of the spectators, are seen by reflection in it, and appear as ghosts on the stage. 705 a. Deviation produced by Rotation of Mirror.—Let AB (Fig. 634? A) represent a mirror perpendicular to the plane of the paper, and capable of being rotated about an axis through C, also perpendicular to the paper; and let IC represent an incident ray. When the mirror is in the position AB, perpendicular to I C, the ray will be re¬ flected directly back upon its course; but when the mirror is turned through the acute angle A C A, the reflected ray will take the direction CR, making with the Fig. 634 A. Effect of rotating a Mirror. normal Q N an angle NCR, equal to the angle of incidence NCI. The deviation I C R of the reflected ray, produced by rotating the mirror, is therefore double of the angle I C N or A C A', through which the mirror has been turned; and if, starting from the position A B', we turn the mirror through a further angle 0, the reflected ray C R will be turned through a further angle 2 0. It thus appears, that, when a plane mirror is rotated in the plane of incidence , the direction of the reflected ray is changed by double the angle through which the mirror is turned. Con¬ versely, if we assign a constant direc¬ tion CI to the reflected ray, the direction of the incident ray R C must vary by double the angle through which the mirror is turned. 705 b. Hadley’s Sextant.—The above principle is illustrated in the nautical instrument called the sextant or quadrant, which was in¬ vented by Newton, and reinvented by Hadley. It serves for mea¬ suring the angle between any two distant objects as seen from the station occupied by the observer. Its essential parts are represented in Fig. 634 b. HADLEYS SEXTANT. 893 It has two plane mirrors A, B, one of which, A, is fixed to the frame of the instrument, and is only partially silvered, so that a dis¬ tant object in the direction A H can be seen through the unsilvered part. The other mirror B is mounted on a movable arm B I, which carries an index I, traversing a graduated arc P Q. When the two mirrors are parallel, the index is at P, the zero of the graduations, and a ray H' B incident on B parallel to H A, will be reflected first along B A, and then along A T, the continuation of H A. The ob¬ server looking through the telescope T thus sees, by two reflections, the same objects which he also sees directly through the unsilvered part of the mirror. Now let the index be advanced through an angle 0 ; then, by the principles of last section, the incident ray S B makes with H' B, or H A, an angle 2 0. The angle between S B and H A would therefore be given by reading off the angle through which the index has been advanced, and doubling; but in practice the arc P Q is always graduated on the principle of marking half degrees as whole ones, so that the reading at I is the required angle 20. In using the instrument, the two objects which are to be observed are brought into apparent coincidence, one of them being seen directly, and the other b}^ successive reflection from the two mirrors. This coincidence is not disturbed by the motion of the ship; but unpractised observers often find a difficulty in keeping both objects in the field of view. Dark glasses, not shown in the figure, are provided for protecting the eye in observations of the sun, and a vernier and reading microscope are provided instead of the pointer I. 706. Spherical Mirrors.—By a spherical mirror is meant a minor Fig. 635.—Principal Focus. whose reflecting surface is a portion (usually a very small portion) of the surface of a sphere. It is concave or convex according as the inside or outside of the spherical surface yields the reflection. The centre of the sphere (C, Fig. 635) is called the centre of curvature of 894 REFLECTION OF LIGHT. the mirror. If the mirror has a circular boundary, as is usually the case, the central point A of the reflecting surface may conveniently be called the pole of the mirror. Centre of the mirror is an ambigu¬ ous phrase, being employed sometimes to denote the pole, and some¬ times the centre of curvature. The line A C is called the principal axis of the mirror, and any other straight line through C which meets the mirror is called a secondary axis. When the incident rays are parallel to the principal axis, the re¬ flected rays converge to a point F, which is called the principal focus. This law is rigorously true for parabolic mirrors (generated by the revolution of a parabola about its principal axis). For sphe¬ rical mirrors it is only approximately true, but the approximation is very close if the mirror is only a very small portion of an entire sphere. In grinding and polishing the specula of large reflecting telescopes, the attempt is made to give them, as nearly as possible, the parabolic form. Parabolic mirrors are also frequently employed Fig. 63t>.—Theory of Conjugate Foci. to reflect, in a definite direction, the rays of a lamp placed at the focus. Rays reflected from the circumferential portion of a spherical mir¬ ror are always too convergent to concur exactly with those reflected from the central portion. This deviation from exact concurrence is called spherical aberration. 707. Conjugate Foci.—Let P (Fig. .636) be a luminous point situ¬ ated on the principal axis of a spherical mirror, and let PI be one of the rays which it sends to the mirror. Draw the normal O I, which is simply a radius of the sphere. Then 0 I P is the angle of incid- CONJUGATE FOCI. 895 ence, and the angle of reflection O I P' must be equal to it; hence 0 I bisects an angle of the triangle PI P', and therefore we have IP _ OP IP' OP" Let p,p' denote AP, A P' respectively, and let r denote the radius of the sphere. Then, if the angular aperture of the mirror is small, IP is sensibly equal to p , and I P' to p'. Substituting these approxi¬ mate values, the preceding equation becomes — = . whence pr + p'r = 2 pp'\ p r — p } or, dividing by p p' r, This formula determines the position of the point P', in which the reflected ray cuts the principal axis, and shows that it is, to the accuracy of our approximation, independent of the position of the point I; that is to say, all the rays which P sends to the mirror are reflected to the same point P'. We have assumed P to be on the principal axis. If we had taken it on a secondary axis, as at p (Fig. 636), we should have found, by the same process of reasoning, that the reflected rays would all meet in a point p' on that secondary axis. The distinction between primary and secondary axes, in the case of a spherical mirror, is in fact merely a matter of convenience, not representing any essential difference of property. Hence we can lay down the following general proposition as true within limits of error corresponding to the approximate equalities which we have above assumed as exact: — Rays proceeding from any given point in front of a concave spherical mirror , are reflected so as to meet in another point; and the line joining the two points passes through the centre of the sphere. It is evident, that rays proceeding from the second point to the mirror would be reflected to the first. The relation between them is therefore mutual, and they are hence called conjugate foci. By a focus in general is meant a point in which a number of rays meet (or would meet if produced); and the rays which thus meet, taken collectively, are called a pencil. Fig. 637 represents two pencils of rays whose foci Ss are conjugate, so that, if either of them be re¬ garded as an incident pencil, the other will be the corresponding reflected pencil. 896 REFLECTION OF LIGHT. We can now explain the formation of images by concave mirrors. Each point of the object sends a pencil of rays to the mirror, which converge, after reflection, to the conjugate focus. If the eye of the observer be placed beyond this point of concourse, and in the path of the rays, they will present to him the same appearance as if they Fig 637.—Conjugate Foci. had come from this point as origin. The image is thus composed of points which are the conjugate foci of the several points of the object. 708. Principal Focus.—If, in formula (a) of last section, we make p increase continually, the term ^ will continually decrease, and will vanish as p becomes infinite. This is the case of rays parallel to the principal axis, for parallel rays may be regarded as coming from a point at infinite distance. The formula then becomes 12 r — = — • whence »' = V r ’ r 2 that is to say, the principal focal distance is half the radius of cur¬ vature. This distance is often called the focal length of the mirror. If we denote it by /, the general formula becomes 709. Discussion of the Formula.—By the aid of this formula we can easily trace the corresponding movements of conjugate foci. If p is positive and very large, p' is a very little greater than /; that is to say, the conjugate focus is a very little beyond the princi¬ pal focus. As p diminishes, p' increases, until they become equal, in which case each of them is equal to r or 2/; that is to say, the conjugate foci move towards each other till they coincide at the centre of cur¬ vature. This last result is obvious in itself; for rays from the centre DISCUSSION OF THE FORMULA. 897 of curvature are normal to the mirror, and are therefore reflected directly back. As p continues to diminish, the two foci, as it were, change places; the luminous point advancing from the centre of curvature to the principal focus, while the conjugate focus moves away from the centre of curvature to infinity. As the luminous point continues to approach the mirror, ~ is greater than y and hence y and therefore also p\ must be nega¬ tive. The physical interpretation of this result is that the conjugate focus is behind the mirror, as at s (Fig. G38), and that the reflected Fig. 638.—Virtual Focus. rays diverge as if they had come from this point. Such a focus is called virtual, while a focus in which rays actually meet is called real. As the luminous point moves up from F to the mirror, the conjugate focus moves up from an infinite distance at the back, and meets it at the surface of the mirror. If S is a real luminous point sending rays to the mirror, it must of necessity lie in front of the mirror, and p therefore cannot be nega¬ tive ; but when we are considering images of images this restriction no longer holds. If an incident beam, for example, converges to¬ wards a point s at the back of the mirror, it will be reflected to a point S in front. In this case p is negative, and p' positive. The conjugate foci S s have in fact changed places. It appears from the above investigation that there are two prin¬ cipal cases, as regards the positions of conjugate foci of a concave mirror. 1. One focus between F and C; and the other beyond C. 2. One focus between F and the mirror; and the other behind the mirror. In the former case, the foci move to meet each other at C; in the latter, they move to meet each other at the surface of the mirror. 58 898 REFLECTION OF LIGHT. 710. Formation of Images.—We are now in a position to discuss the formation of images by concave mirrors. Let A B (Fig. 639) be an object placed in front of a concave mirror, at a distance greater than its radius of curvature. All the rays which diverge from A will be reflected to the conjugate focus a. Hence this point can be found by the following construction. Draw through A the ray A A' parallel to the principal axis, and draw its path after reflection, which must of necessity pass through the principal focus. The intersection of this reflected ray with the secondary axis through A will be the point required. A similar construction will give the conjugate focus Fig. 639.—Formation of Image. corresponding to any other point of the object; b, for example, 1 is the focus conjugate to B. Points of the object lying between A and B will have their conjugate foci between a and b. An eye placed behind the object A B will accordingly receive the same impression from the reflected rays as if the image a b were a real object. 711. Size of Image.—As regards the comparative sizes of object and image, it is obvious, from similar triangles, that their linear di¬ mensions are directly as their distances from C the centre of curvature. Again, since C F and A A' are parallel, we have aF _ a C _ ab FA' CA “ AB’ length of image _ distance of ima ge from principal focus . ^ length of object focal length and by a similar construction we can prove that 1 It is only by accident that b happens to lie on A A' in the figure. EXPERIMENT OF THE PHANTOM BOUQUET. 899 length of object _ distance of object from principal focus ^ length of image focal length This last formula affords the readiest means of calculating the size of the image when the size and position of the object are given. Both the formula (c) and (d) are perfectly general, both for concave and convex mirrors. They show that the object and image will be equal when they coincide at the centre of curvature, and that as they move away from this point, in opposite directions, that which moves away from the mirror continually gains in size upon the other. Since the lines joining corresponding points of object and image cross at the point C, which lies between them when the image is real, a real image formed by a concave mirror is always inverted. 712. Experiment of the Phantom Bouquet.—Let a box open on one Fig. 640.—Experiment of Phantom Bouquet. side be placed in front of a concave mirror, at a distance about equal to its radius of curvature, and let an inverted bouquet be suspended within it, the open side of the box being next the mirror. By giving a proper inclination to the mirror, an image of the bouquet will be obtained in mid-air, just above the top of the box. As the bouquet 900 REFLECTION OF LIGHT. is inverted, its image is erect, and a real vase may be placed in such a position that the phantom bouquet shall appear to be standing in it. The spectator must be full in front of the mirror, and at a suf¬ ficient distance for all parts of the image to lie between his eyes and the mirror. When the colours of the bouquet are bright, the image is generally bright enough to render the illusion very complete. 713. Images on a Screen.—Such experiments as that just described can only be seen by a few persons at once, since they require the spectator to be in a line with the image and the mirror. When an image is projected on a screen, it can be seen by a whole audience Fig. 641.—image on Screen. at once, if the room be darkened and the image be large and bright. Let a lighted candle, for example, be placed in front of a concave mirror, at a distance exceeding the focal length, and let a screen be placed at the conjugate focus; an inverted image of the candle will be depicted on the screen. Fig. 641 represents the case in which the candle is at a distance less than the radius of curvature, and the image is accordingly magnified. By this mode of operating, the formula for conjugate focal dis¬ tances can be experimentally verified with considerable rigour, care being taken, in each experiment, to place the screen in the position which gives the most sharply defined image. IMAGES ON SCREEN AND IN MID-AIR. 901 714. Difference between Image on Screen, and Image as seen in Mid-air. Caustics.—For the sake of simplicity we have made some statements regarding visible images which are not quite accurate; and we must now indicate the necessary corrections. Images thrown on a screen have a determinate position, and are really the loci of the conjugate foci of the points of the object; but this is not rigorously true of images seen directly. They change their position to some extent, according to the position of the observer. The actual state of things is explained by Fig. 641 A. The plane of the figure 1 is a principal plane (that is, a plane containing the prin¬ cipal axis) of a concave hemispherical mirror, and the incident rays Fig. 641 a.— Position of Image in Oblique Reflection. are parallel to the principal axis. All the rays reflected in the plane of the figure touch a certain curve called a caustic curve, which has a cusp at F, the principal focus; and the direction in which the image is seen by an eye situated in the plane of the figure is deter¬ mined by drawing from the eye a tangent to this caustic. If the eye be at E, on the principal axis, the point of contact will be F; but when the rays are received obliquely, as at E', it will be at a point a not lying in the direction of F. For an eye thus situated, a is called the primary focus, and the point where the tangent at a cuts the principal axis is called the secondary focus. When the eye is moved in the plane of the diagram, the apparent position of the 1 Figs. 641 a and 657 a are borrowed, by permission, from Mr. Osmund Airy’s Geometri¬ cal Optics. 902 REFLECTION OF LIGHT. image (as determined by its remaining in coincidence with a cross of threads or other mark) is the primary focus; and when the eye is moved perpendicular to the plane of the diagram, the apparent posi¬ tion of the image is the secondary focus. 1 If we suppose the diagram to rotate about the principal axis, it will still remain true in all positions, and the surface generated by this revolution of the caustic curve is the caustic surface. Its form and position vary with the position of the point from which the incident rays proceed; and it has a cusp at the focus conjugate to this point. There is always more or less blurring, in the case of images seen obliquely (except in plane mirrors), by reason of the fact that the point of contact with the caustic surface is not the same for rays entering different parts of the pupil of the eye. A caustic curve can be exhibited experimentally by allowing the rays of the sun or of a lamp to fall on the concave surface of a strip of polished metal bent into the form of a circular arc, as in Fig. 642, the reflected light being received on a sheet of white paper on which the strip rests. The same effect may often be observed on the surface of a cup of tea, the reflector in this case being the inside of the tea-cup. The image of a luminous point received upon a screen is formed by all the rays which touch the corresponding caustic surface. The brightest and most distinct image will be formed at the cusp, which is, in fact, the conjugate focus; but there will be a border of fainter light surrounding it. This source of indistinctness in images is an example of spherical aberration (§ 707). 714 a. Image on a Screen by Oblique Reflection. — If we attempt to throw upon a screen the image of a luminous point by means of a concave mirror very oblique to the incident rays, we shall find that no image can be obtained at all resembling a point; but that there are two positions of the screen in which the image becomes a line. Fig. 642.—Caustic by Reflection. 1 Since every ray incident parallel to the principal axis, is reflected through the principal axis. If the incident rays diverged from a point on the principal axis, they would still be reflected through the principal axis. IMAGE ON SCREEN BY OBLIQUE REFLECTION. 903 Fig. 641 b.—F ormation of Focal Lines. In the annexed figure (Fig. 641 b), which represents on a larger scale a portion of Fig. 641 a, a c,b d are rays from the highest and lowest points of the portion R S of the hemispherical mirror, which portion we suppose to be small in both its dimensions in comparison with the radius of curvature; and we may suppose the rest of the hemisphere to be re¬ moved, so that R S will repre¬ sent a small concave mirror re¬ ceiving a pencil very obliquely. Then, if a screen be held perpendicular to the plane of the diagram, at m, where the section of the pencil by the plane of the diagram is nar¬ rowest, a blurred line of light will be formed upon it, the length of the line being per¬ pendicular to the plane of the diagram. This is called the primary focal line. The secondary focal line is c d, which, if produced, passes through the centre of curvature of the mirror, and also through the point from which the incident light proceeds. This line is very sharply formed upon a screen held so as to coincide with c d and to be per¬ pendicular to the plane of the diagram. Its edges are much better defined than those of the primary line; and its position in space is also more definite. If the mirror is used as a burning-glass to collect the sun’s rays, ignition will be more easily obtained at one of these lines than in any intermediate position. 1 Focal lines can also be seen directly. In this case a small element of the mirror sends all its reflected rays to the eye, the rays from opposite sides of the element crossing each other at the focal lines, before they reach the e}'e. It is possible, in certain positions of the eye, to see either focal line at pleasure, by altering the focal adjust¬ ment of the eye; or the two may be seen with imperfect definition 1 The “elongated figure of 8” which is often mentioned in connection with the secondary focal line, is obtained by turning the screen about n the middle point of c d, so as to blur both ends of the image by bad focussing. It will be observed, from an inspection of the diagram, that c d is very oblique to the reflected rays. If we neglect the blurring of the primary line, we may describe the part of the pencil lying between the two lines as a tetrahedron, of which the two lines are opposite edges. 904 REFLECTION OF LIGHT. crossing each other at right angles. on Fig. 643.—Formation of Virtual Image. the mirror from any point of it. The experiment is easily made by employing a gas flame, turned very low, as the source of light. One line is in the plane of incid¬ ence, and the other is nor¬ mal to this plane. 715. Virtual Image in Concave Mirror. — Let an object be placed, as in Fig. 643, in front of a concave mirror, at a distance less than that of the principal focus. The rays incident as A, will be reflected as a divergent pencil, the focus from which they diverge being a point b at the back of the mirror. To find this point, we may trace the course of a ray through A parallel to the principal axis. Such a ray will be reflected to the principal focus F, and by producing this reflected ray backwards till it meets the secondary axis C A, the point b, which is the conju¬ gate focus of A, is deter¬ mined. We can find in the same way the position of a, the conjugate focus of B, and it is obvious that the image of A B will be erect and magnified. _ _ T . . T . „ 716. Remarks on Virtual Fig. 644.—Virtual Image m Concave Mirror. Images. — A virtual image cannot be projected on a screen; for the rays which produce it do not actually pass through its place, but only seem to do so. A screen placed at a 6 would obviously receive none of the reflected light whatever. CONVEX MIRRORS. 905 The images seen in a plane mirror are virtual; and any spherical mirror, whether concave or convex, is nearly equivalent to a plane mirror, when the distance of the object from its surface is small in comparison with the radius of curvature. 717. Convex Mirrors.—It is easily shown, by a simple construction, that rays incident from any luminous point upon a convex mirror, diverge after reflection. The principal focus, and the foci conjugate to all points external to the sphere, are therefore virtual. To adapt formulae (a) and (6) of the preceding sections to the case of convex mirrors, we have only to alter the sign of the term -- or y ? so that for a convex mirror we shall have r and / being here regarded as essentially positive. From this formula it is obvious that one at least of the two dis¬ tances p,p must be negative; that is to say, one at least of any pair of conjugate foci must lie behind the mirror. The construction for an image (Fig. 645) is the same as in the case of concave mirrors. Through any selected point of the object Fig. 645.—Formation of Image in Convex Mirror. draw a ray parallel to the principal axis; the reflected ray, if pro¬ duced backwards, must pass through the principal focus, and its intersection with the secondary axis through the selected point deter¬ mines the corresponding point of the image. The image of an ex¬ ternal object will evidently be erect, and smaller than the object. Repeating the same construction when the object is nearer to the mirror, we see that the image will be larger than before. The linear dimensions of an object and its image, whether in the 906 REFLECTION OF LIGHT. case of a convex or a concave mirror, are directly proportional to their distances from the centre of curvature. The image is inverted or erect according as this centre does or does not lie between the object and its image. In the case of a convex mirror the centre never lies between them (if the object be real), and therefore the image is always erect. Convex mirrors are very seldom employed in optical instruments. The silvered globes which are frequently used as ornaments, are examples of convex mirrors, and present to the observer at one view an image of nearly the whole surrounding landscape. As the part of the mirror in which he sees this image is nearly an entire hemi¬ sphere, the deformation of the image is very notable, straight lines being reflected as curves. 718. Anamorphosis.—Much greater deformations are produced by cylindric mirrors. A cylindric mirror, when the axis of the cylinder is vertical, behaves like a plane mirror as regards the angular magni¬ tude under which the height of the image is seen, and like a spherical mirror as regards the breadth of the image. If it be a convex cylin¬ der, it causes bodies to appear unduly contracted horizontally in pro¬ portion to their heights. Distorted pictures are sometimes drawn upon paper, according to such a system that when they are seen MEDICAL APPLICATIONS. 907 reflected in a cylindric mirror properly placed, as in Fig. 640, the distortion is corrected, and while the picture appears a mass of con¬ fusion, the image is instantly recognized. This restoration of true proportion in a picture is called anamorphosis. 719. Medical Applications.—Concave mirrors are frequently used to concentrate light upon an object for the purpose of rendering it more distinctly visible. The ophthalmoscope is a small concave mirror, with a small hole in its centre, through which the observer looks from behind, while he directs a beam of reflected light from a lamp into the pupil of the patient’s eye. In this way (with the help sometimes of a lens) the retina can be rendered visible, and can be minutely examined. The laryngoscope consists of two mirrors. One is a small plane mirror, with a handle attached, at an angle of about 45° to its plane. This small mirror is held at the back of the patient’s mouth, so that the observer, looking into it, is able by reflection to see down the patient’s throat, the necessary illumination being supplied by a con¬ cave mirror, strapped to the observer’s forehead, by means of which the light from a lamp is reflected upon the plane mirror, which again reflects it down the throat. CHAPTER LIX. REFRACTION. 720. Refraction.—When a ray of light passes from one transparent medium to another, it undergoes a change of direction at the surface of separation, so that its course in the second medium makes an angle with its course in the first. This changing of direction is called re¬ fraction. The phenomenon can be exhibited by admitting a beam of the sun's rays into a dark room, and receiv¬ ing it on the surface of water con¬ tained in a rectangular glass vessel. The path of the beam will be easily traced by its illumination of the small solid particles which lie in its course. The following experiment is a well-known illustration of refrac¬ tion :—A coin m n (Fig. 648) is laid at the bottom of a vessel with opaque sides, and a spectator places himself so that the coin is just hidden from him by the side of the vessel; that is to say, so that the line m A in the figure passes just above his eye. Let water now be poured into the vessel, care being taken not to displace the coin. The bottom of the vessel will appear to rise, and the coin will come into sight. Hence a pencil of rays from m must have entered the spectator's eye. The pencil in fact undergoes a sudden bend at the surface of the water, and thus reaches the eye by a crooked course, Fig. 647.—Refraction. REFRACTIVE POWERS OF DIFFERENT MEDIA. 909 in which the obstacle A is evaded. If the part of the pencil in air be produced backwards, its rays will approximately meet in a point m', which is therefore the image of m. Its position is not correctly indicated in the figure, being placed too much to the left (§ 727 a). The broken appearance presented by a stick (Fig. 649) when partly immersed in water in an oblique position, is similarly ex¬ plained, the part beneath the water being lifted up by refraction. 721. Refractive Powers of Different Media.—In the experiments of the coin and stick, the rays, in leaving the water, are bent away from the normals Z I N, Z' I' N' at the points of emergence; in the experiment first described (Fig. 647), on the other hand, the rays, in passing from air into water, are bent nearer to the normal. In every case the path which the ra} 7 s pursue in going is the same as they would pur¬ sue in returning; and of the two media concerned, that in which the ray makes the smaller angle with the normal is said to have greater refractive power than the other, or to be more highly refracting. Liquids have greater refractive power than gases, and as a general rule (subject to some exceptions in the comparison of dissimilar sub¬ stances) the denser of two substances has the greater refracting power. Hence it has become customary, in enunciating some of the laws of optics, to speak of the denser medium and the rarer medium, when the more correct designations would be more refractive and less refractive. 722. Laws of Refraction.—The quantitative law of refraction was not discovered till quite modern times. It was first stated by Snell, a Dutch philosopher, and was made more generally known by Des¬ cartes, who has often been called its discoverer. Fig. 649.—Appearance of Stick in Water. Fig. 648.—Experiment of Coin in Basin. 910 REFRACTION. Let RI (Fig. 650) be a ray incident at I on the surface of separa¬ tion of two media, and let I S be the course of the ray after refrac¬ tion. Then the angles which RI and I S make with the normal are called the angle of incidence and the angle of refraction respec¬ tively ; and the first law of refraction is that these angles lie in the same plane, or the 'plane of refraction is the same as the plane of incidence. The law which connects the mag¬ nitudes of these two angles, and which was discovered by Snell, can only be stated either by reference to a geo¬ metrical construction, or by employing the language of trigonometry. De¬ scribe a circle about the point of in¬ cidence I as centre, and drop perpen¬ diculars, from the points where it cuts the rays, on the normal. The law is that these perpendiculars R' P', S P, will have a constant ratio; or the sines of the angles of incidence and refraction are in a constant ratio. It is often referred to as the law of sines. The angle by which a ray is turned out of its original course in undergoing refraction is called its deviation. It is zero if the inci¬ dent ray is normal, and always increases with the angle of incid¬ ence. 723. Verification of the Law of Sines.—These laws can be verified by means of the apparatus represented in Fig. 651, which is very similar to that employed by Descartes. It has a vertical divided circle, to the front of which is attached a cylindrical vessel, half-filled with water or some other transparent liquid. The surface of the liquid must pass exactly through the centre of the circle. I is a movable mirror for directing a reflected beam of solar light on the centre O. The beam must be directed centrally through a short tube attached to the mirror, and to facilitate this adjustment the tube is furnished with a diaphragm with a hole in its centre. The arm 0 & is movable about the centre of the circle, and carries a ver¬ nier for measuring the angle of incidence. The ray undergoes refrac¬ tion at 0; and the angle of refraction is measured by means of a second arm 0 R, which is to be moved into such a position that the diaphragm of its tube receives the beam centrally. No refraction INDICES OF REFRACTION. 911 occurs at emergence, since the emergent beam is normal to the sur¬ faces of the liquid and glass; the position of the arm accordingly indicates the direction of the refracted ray. The angles of incidence and refraction can be read off at the verniers carried by the two arms; and the ratio of their n sines will be found constant. The sines can also be directly measured by employ¬ ing sliding - scales as indicated in the fig¬ ure, the readings be¬ ing taken at the ex¬ tremity of each arm. It would be easy to make a beam of light enter at the lower side of the ap¬ paratus, in a radial direction; and it would be found that the ratio of the sines was precisely the same as when the light entered from Fig. 651.— Apparatus for Verifying the Law. above. This is merely an instance of the general law, that the course of a returning ray is the same as that of a direct ray. 724. Indices of Refraction.—The ratio of the sine of the angle of incidence to the sine of the angle of refraction, when a ray passes from one medium into another, is called the relative index of refrac¬ tion from the former medium to the latter. When a ray passes from vacuum into any medium this ratio is always greater than unity, and is called the absolute index of refraction , or simply the index of refraction , for the medium in question. The relative index of refraction from any medium A into another B is always equal to the absolute index of B divided by the absolute index of A. The abso¬ lute index of air is so small that it may usually be neglected in com¬ parison with those of solids and liquids; but strictly speaking, the relative index for a ray passing from air into a given substance must 912 REFRACTION. be multiplied by the absolute index for air, in order to obtain the absolute index of refraction for the substance. The following table gives the indices of refraction of several sub¬ stances :— Indices of Refraction. 1 Diamond,.2*44 to 2*755 Sapphire,.1*794 Flint-glass,. 1*576 to 1*642 Crown-glass,.1*531 to 1*563 Rock-salt,. Canada balsam, ..... Bisulphide of carbon, Linseed-oil (sp. gr. *932) . . Oil of turpentine (sp. gr. *885), 1*545 1*540 1*678 1*482 1*478 Alcohol,.1*372 Aqueous humour of eye, .... 1*337 Vitreous humour,.1*339 Crystalline lens, outer coat, . . . 1*337 „ ,, under coat, . . 1*379 ,, ,, central portion, . 1*400 Sea water,.1 *343 Pure water,.1*336 Air at 0°C. and 760 mm . . . 1*000,294 725. Critical Angle.—We see, from the law of sines, that when the incident ray is in the less refractive of the two media, to every pos¬ sible angle of incidence there is a corresponding angle of refraction. This, however, is not the case when the incident ray is in the more refractive of the two media. Let S O, S' O, S" O (Fig. 652), be in- Flg. 652.—Critical Angle. cident rays in the less refractive medium, and OR, O R', O R" the corresponding refracted rays. There will be a particular direction of refraction O L corresponding to the angle of incidence of 90°. Conversely, incident rays R O, R' O, R" O, in the more refractive 1 The index of refraction is always greater for violet than for red (see Chap, lxii.) The numbers in this table are to be understood as mean values. MIRAGE. 913 medium, will emerge in the directions 0 S, 0 S', O S", and the direc¬ tion of emergence for the incident ray L 0 will be 0 B, which is coin¬ cident with the bounding surface. The angle L O N is called the critical angle, and is easily computed when the relative index of refraction is given. For let y. denote this index (the incident ray being supposed to be in the less refractive medium), then we are to have sin 90° , . 1 . —.-= fx, whence sina: = -, sin x /x that is, the sine of the critical angle is the reciprocal of the index of refraction. When the media are air and water, this angle is about 48° 30'. For air and different kinds of glass its value ranges from 38° to 41°. If a ray, as I 0, is incident in the more refractive medium, at an angle greater than the critical angle, the law of sines becomes nuga¬ tory, and experiment shows that such a ray undergoes internal reflec¬ tion in the direction O I', the angle of reflection being equal to the angle of incidence. Reflection occurring in these circumstances is nearly perfect, and has received the name of total reflection. Total reflection occurs when rays are incident in the more refractive medium at an angle greater than the critical angle. The phenomenon of total reflection may be observed in several familiar instances. For example, if a glass of water, with a spoon in it (Fig. G53), is held above the level of the eye, the under side of the surface of the water is seen to shine like a brilliant mirror, and the lower part of the spoon is seen reflected in it. Beautiful effects of the same kind may be observed in aquariums. 726. Mirage.—An appearance as of water is frequently seen in sandy deserts, where the soil is highly heated by the sun. The observer sees in the distance the reflection of the sky and of terres¬ trial objects, as in the surface of a calm lake (Fig. 654). This pheno¬ menon, called mirage , was first explained by Monge, who observed it in Bonaparte’s Egyptian expedition. The air near the ground becomes so highly heated that the density within a certain distance of the ground increases upwards. A ray, as Ma (Fig. 655), proceeding obliquely downwards, will then be ren¬ dered, by refraction, more and more nearly horizontal as it advances, until its direction is such as to bring about total reflection; and the reflected ray is then, by successive refractions, gradually elevated 59 OH REFRACTION. till it meets the eye of the observer, who thus sees an inverted image at M. Fig. 653.—Total Reflection. A kind of inverted mirage is often seen across masses of calm water, and is called looming , images of distant objects, such as ships or hills, being seen in an inverted position immediately over the objects themselves. The explanation just given of mirage will apply to this phenomenon also, if we suppose the lower strata of air to be colder and more dense than those above them. 1 1 These explanations can scarcely be regarded as satisfactory. The general subject of atmospheric refraction will be treated in Chapter lxiii. Fig. 654.—Mirage. 916 REFRACTION. Mirage may sometimes be artificially produced in the following way. A sheet-iron box is employed, with its ends cut away. Its bottom is heated externally by coals, and the observer is stationed so as to look through endwise. In general, nothing particular is observed except a fluttering due to currents of air of different den¬ sities. Sometimes, however, from causes not easily controlled, a state of calm supervenes, and one of the layers of air performs the part of a mirror, so that the observer sees the reflections of objects on the further side of the box. 727. Camera Lucida.—The camera lucida is an instrument some¬ times employed to facilitate the sketching of objects from nature. Fig. 656.—Section of Prism. Fig. 657.—Camera Lucida. It acts by total reflection, and may have various forms, of which that proposed by Wollaston, and represented in Fig. 656, 657, is one of the commonest. The essential part is a totally-reflecting prism CAMERA LUCID A. 917 with four angles, one of which is 90°, the opposite one 135°, and the other two each 67° 30'. One of the two faces which contain the right angle is turned towards the objects to be sketched. Rays incident normally on this face, as x r, make an angle greatly exceed¬ ing the critical angle with the face c d, and are totally reflected from it to the next face d a , whence they are again 1 totally reflected to the fourth face, from which they emerge normally. An eye placed so as to receive the emergent rays, will see a virtual image in a direction at right angles to that in which the object lies. In prac¬ tice, the eye is held over the corner a of the prism, in such a position that one-half of the pupil receives these reflected rays, while the other half receives light in a parallel direction outside the prism. The observer thus sees the reflected image projected on a real back¬ ground, which consists of a sheet of paper for sketching. He is thus enabled to pass a pencil over the outlines of the image, pencil image and paper being simultaneously visible. It is very desirable that the image should lie in the plane of the paper, not only because the pencil point and the image will then be seen with the same focussing of the eye, but also because parallax is thus obviated, so that when the observer shifts his eye the pencil point is not displaced on the image. As the paper, for convenience of drawing, must be at a distance of about a foot, a concave lens, with a focal length of some¬ thing less than a foot is placed close in front of the prism, in drawing distant objects. By raising or lowering the prism in its stand (Fig. 657), the image of the object to be sketched may be made to coincide with the plane of the paper. The prism is mounted in such a way that it can be rotated either about a horizontal or a vertical axis; and its top is usually covered with a movable plate of blackened metal, having a semicircular notch at one edge, for the observer to look through. 727 a. Images by Refraction at a Plane Surface.—When a point Q (Fig. 657 a), in the interior of a solid or liquid bounded by a plane surface, sends rays into air, the emergent rays, if produced back¬ wards, will all touch a certain caustic surface, which is a surface of revolution, having for axis the normal from Q to the plane surface. The caustic has a cusp, situated on this normal, at a distance from the surface equal to the distance of Q divided by the index of refrac¬ tion, and this is the position of the image as seen by an eye situated 1 The use of having two reflections is to obtain an erect image. An image obtained by one reflection would be upside down. 918 EEFEACTION. on the normal. For example, to a person looking directly down into clear water, the depth appears only three-quarters of what it really Fig. 657 a. —Caustic by Refraction. is. Wherever the eye may be situated, a tangent drawn from it to the caustic will be the direction of the visible image. The image will accordingly approach nearer to the surface as the direction of vision becomes more oblique, and will ultimately coincide with the surface. 728. Parallel Plate.—Rays falling normally on a uniform trans¬ parent plate with parallel faces, keep their course unchanged; but this is not the case with rays incident obliquely. A ray S I (Fig. 658), incident at the angle SIN, is refracted in the direction I R. The angle of incidence at R is equal to the angle of refraction at I, and hence the angle of emergence S' R N' is equal to the original angle of incidence SIN. The emergent ray R S' is therefore parallel to the incident ray S I, but is not in the same straight line with it. MULTIPLE IMAGES. 919 Objects seen obliquely through a plate are therefore displaced from their true positions. Let S (Fig. 659) be a luminous point which sends light to an eye not directly opposite to it, on the other side of a parallel plate. The emergent rays which enter the eye are parallel to the incident rays; but as they have undergone lateral displacement, their point of concourse 1 is changed from S to S', which is accordingly the image of S. The displacement thus produced increases with the thickness of the plate, its index of refraction, and the obliquity of incidence. It furnishes one of the simplest means of measuring the index of refrac¬ tion of a substance, and is thus employed in Pichot’s refractometer. 729. Multiple Images produced by a Plate.—Let S (Fig. 660) be a luminous point in front of a transparent plate with parallel faces. Of the rays which it sends to the plate, some will be reflected from the front, thus giving rise to an image S'. Another por¬ tion will enter the plate, un¬ dergo reflection at the back, and emerge with refraction at the front, giving rise to a second image S°. Another portion will undergo internal reflection at the front, then again at the back, and by emerging in front will form a third image S r The same pro¬ cess may be repeated several times; and if the luminous object be a candle, or a piece of bright metal, a number of images* one behind another, will be visible to an eye properly placed in front. All the successive images, after the first two, continually diminish in brightness. If the glass be silvered at the back, the second image is much brighter than the first, when the incidence is nearly normal, but as the angle of incidence increases, the first image gains upon the second, and ultimately surpasses it. This is due to the fact that the re¬ flecting power of a surface of glass increases with the angle of incidence. Fig. 660.—Multiple Images in Plate. 1 The rays which compose the pencil that enters the eye will not exactly meet (when produced backwards) in any one point. There will be two focal lines, just as in the case of spherical mirrors (§714, 714 a). 920 REFRACTION. If the luminous body is at a distance which may be regarded as infinite,—if it is a star, for example,— all the images should coincide, and lorm only a single image, occupying a position which does not vary with the position of the observer, provided that the plate is perfectly homogeneous, and its faces perfectly plane and par¬ allel. A severe test is thus furnished of the fulfilment of these conditions. Plates are sometimes tested, for parallelism and uniformity, by sup¬ porting them in a horizontal position on three points, viewing the image of a star in them with a telescope fur¬ nished with cross wires, and observ¬ ing whether the image is displaced on the wires when the plate is shifted into a different position, still resting on the same three points. 730. Refraction through a Prism.—For optical purposes, any por- Fig. 661.—Images of Candle in Looking-glass. Fig. 662.—Equilateral Prism. Fig. 663.—Prism mounted on Stand. tion of a transparent body lying between two plane faces which are REFRACTION THROUGH PRISMS. 921 not parallel may be regarded as a prism. 1 The line in which these faces meet, or would meet if produced, is called the edge of the prism, and a section made by a plane perpendicular to them both is called a principal section. The prisms chiefly employed are really prisms in the geometrical sense of the word. Their principal sections are usually triangular, and are very frequently equilateral, as in Fig. 662. The stand usually employed for prisms when mounted separately is represented in Fig. 663. It contains several joints. The uppermost is lor rotating the prism about its own axis. The second is for turn¬ ing the prism so that its edges shall make any required angle with the vertical. The third gives motion about a vertical axis, and also furnishes the means of raising and lowering the prism through a range of several inches. Let SI (Fig. 664) be an incident ray in the plane of a principal section of the prism. If the external medium be air, or any other substance of less refractive power than the prism, the ray in entering the prism will be bent nearer to the normal, taking such a course as I E, and in leaving the prism will be bent away from the normal, taking the course E B. The effect of these two refractions is, there¬ fore, to turn the ray away from the edge (or refracting angle) of the prism. In practice, the prism is usually so placed that I E, the path of the ray through the prism, makes equal angles with the two faces at which refraction occurs (§ 731). If the prism is turned very far from this position, the course of the ray may be alto¬ gether different from that repre¬ sented in the figure; it may, for example, enter at one face, be in¬ ternally reflected at another, and come out at the third; but we at present exclude such cases from consideration. The direction of deviation is easily shown experimentally, by admitting a narrow beam of sunlight into a dark room, and intro¬ ducing a prism in its course. It will be found that the refracted beam, in the circumstances represented in Fig. 664, is turned aside some 40° or 50° from its original course. 2 1 This amounts to saying that the word prism in optics means wedr/e. 8 The phenomena here described are complicated in practice by the unequal refrangibi- 922 REFRACTION. Since the rays which traverse a prism are bent away from the edge, the object from which they proceed will appear, to an observer looking through the prism, to be more nearly in the direction of the edge than it really is. If, for example, he looks at the flame of a candle through a prism placed so that the edge which corresponds Fig. 665.—Vision through Prism. to the refracting angle is at the top (Fig. 665), the apparent place of the flame will be above its true place. 731. Formulae for Refraction through Prisms. Minimum Deviation. —Let SI (Fig. 666) be an incident ray in the plane of a principal section A B C of a prism. Let i be the angle of incidence SIN, and r the angle of refraction M11'. Then, denoting the index of refrac¬ tion by yu, we have sin i = /i sin r. In like manner, putting r' for lity of rays of different colours (Chap, lxii.) The complication may be avoided by em¬ ploying homogeneous light, of which a spirit-lamp, with common salt sprinkled on the wick, affords a nearly perfect example. REFRACTION THROUGH PRISMS. 923 the angle of internal incidence on the second face 11' M, and i' for the angle of external refraction N' I' ft, we have sin i' — [x sin v'. The deviation produced at I is i—r, and that at I' is i'—r so that Fig. 666.—Refraction through Prism. the total deviation, which is the acute angle D contained between the rays S I, R 1', when produced to meet at o, is T> = i-r+i'-r. (1) But if we drop a perpendicular from the angular point A on the ray 11', it will divide the refracting angle BAG into two parts, of which that on the left will be equal to r, and that on the right to r', since the angle contained between two lines is equal to that contained between their perpendiculars. We have therefore A=r + r', and by substitution in the above equation I) — i + i’ — A. ( 2 ) When the path of the ray through the prism 11' makes equal angles with the two faces, the whole course of the ray is symmetrical with respect to a plane bisecting the refracting angle, so that we have Equation (^2) thus becomes D = 2 i — A, whence i = A + D 2 (3) 924 REFRACTION. This last result is of great practical importance, as it enables us to calculate the index of refraction yu from measurements of the refract¬ ing angle A of the prism, and of the deviation D which occurs when the ray passes symmetrically. When a beam of sunlight in a dark room is transmitted through a prism, it will be found, on rotating the prism about its axis, that there is a certain mean position which gives smaller deviation of the transmitted light than positions on either side of it; and that, when the prism is in this position, a small rotation of it has no sensible effect on the amount of deviation. The position determined experi¬ mentally by these conditions, and known as the 'position of mini¬ mum deviation, is the position in which the ray passes symmetrically 731a. Construction for Deviation.—The following geometrical con¬ struction furnishes a very sim¬ ple method of representing the variation of deviation with the angle of incidence:— 1. When the refraction is at a single surface, describe two circular arcs about a common centre 0 (Fig. 666 a), the ratio of their radii being the index of re¬ fraction. Then, if the incidence Fig. 666a.— General Construction ior Deviation. is from rare to dense, draw a radius O A of the smaller circle to represent the direction of the incident ray, and let NAB be the direction of the normal to the surface at the point of incidence, so that O A N is the angle of incidence. Join O B. Then O B N is the angle of refraction, since 2^ 5 = index of refraction; hence 0 B is parallel to the refracted ray. If the incidence is from dense to rare, we must draw O B to represent the incident ray, make O B N equal to the angle of incidence, and join O A. In cither case the angle A 0 B is the deviation, and it evidently in¬ creases with the angle of incidence 0 A N, attaining its greatest value when this angle (OAN" in the figure) is a right angle, in which case the angle of refraction O B" N" is the critical angle. 2. To find the deviation in refraction through a prism, describe two concentric circular arcs as before (Fig. 666 b), the ratio of their radii being the index of refraction. Draw the radius O A of the CONSTRUCTION FOR DEVIATION. 925 smaller circle to represent the incident ray, N B to represent the normal at the first surface, B N' the normal at the second surface. Then 0 B represents the direction of the ray in the prism, 0 A' the direction of the emergent ray, and A O A' is accordingly the total deviation. In fact we have OAN = angle of incidence at first surface. OBN = „ refraction „ OBN'= ,, incidence at second surface. OA'N'= „ refraction „ A O B = deviation at first surface. BOA'= ,, second ,, ABA' = angle between normals = angle of prism. Again, the deviation A 0 A', being the angle at the centre of a circle, is measured by the arc A A', which subtends it. To obtain the minimum deviation, we must so arrange matters that the angle ABA' being given (=angle of prism), the arc A A' shall be a mini¬ mum. Let ABA', aBa (Fig. 666c), be two consecutive positions, Fig. 666b. —Application to Prism. B A' and B a' being greater than B A and B a. Then, since the small angles A B a, A! B a' are equal, it is obvious, for a double reason, that the small arc A' a / is greater than A a, and hence the whole arc a a' is greater than A A. The deviation is therefore in¬ creased by altering the position in such a way as to make B A and BA' depart further from equality, and is a minimum when they are equal. 731b. Conjugate Foci for Minimum Deviation.—When the angle of incidence is nearly that corresponding to minimum deviation, a small change in this angle has no sensible effect on the amount of deviation. Hence a small pencil of rays sent in this direction from a luminous point, and incident near the refracting edge, will emerge with their 926 REFRACTION. divergence sensibly unaltered, so that if produced backwards they would meet in a virtual focus at the same distance (but of course not in the same direction) as the point from which they came. In like manner, if a small pencil of rays converging towards a point, are turned aside by interposing the edge of a prism in the position of minimum deviation, they will on emergence converge to another point at the same distance. We may therefore assert that, neglecting the thickness of a prism, conjugate foci are at the same distance from it, and on the same side, when the deviation is a minimum. 732. Double Refraction.—Thus far we have been treating of what is called single refraction. We have assumed that to each given incident ray there corresponds only one refracted ray. This is true when the refraction is into a liquid, or into well-annealed glass, or into a crystal belonging to the cubic system. On the other hand, when an incident ray is refracted into a crystal of any other than the cubic system, or Fig. 667.—Iceland-spar. into glass which is unequally stretched or compressed in different directions; for example, into unannealed glass, it gives rise in general to two refracted rays which take different paths; and this pheno¬ menon is called double refraction. Attention w T as first called to it in 1670 by Bartholin, who observed it in the case of Iceland-spar, and its laws for this substance were accurately determined by Huyghens. 733. Phenomena of Double Refraction in Iceland-spar.—Iceland-spar or calc-spar is a form of crystallized carbonate of lime, and is found in large quantity in the country from which it derives its name. It is usually found in rhombohedral form, as represented in Figs. 667, 668. To observe the phenomenon of double refraction, a piece of the spar may be laid on a page of a printed book. All the letters seen through it will appear double, as in Fig. 668; and the depth of their DOUBLE REFRACTION IN ICELAND-SPAR. 927 blackness is considerably less than that of the originals, except where the two images overlap. In order to state the laws of the phenomena with precision, it is necessary to attend to the crystalline form of Iceland-spar. At the corner which is represented as next us in Fig. 667 three equal obtuse angles meet; and this is also the case at the opposite Fig. 668.—Double Refraction of Iceland-spar. corner which is out of sight. If a line be drawn through one of these corners, making equal angles with the three edges which meet there, it or any line parallel to it is called the axis of the crystal; the axis being properly speaking not a definite line but a definite direction. The angles of the crystal are the same in all specimens; but the lengths of the three edges (which may be called the oblique length, breadth, and thickness) may have any ratios whatever. If the crystal Fig. 669.—Axis of the Crystal. is of such proportions that these three edges are equal, as in the first part of Fig. 669, the axis is the direction of one of its diagonals, which is represented in the figure. 928 REFRACTION. Any plane containing (or parallel to) the axis is called a principal plane of the crystal. If the crystal is laid over a dot on a sheet of paper, and is made to rotate while remaining always in contact with the paper, it will be observed that, of the two images of the dot, one remains un¬ moved, and the other revolves round it. The former is called the ordinary , and the latter the extraordinary image. It will also be observed that the former appears nearer than the latter, being more lifted up by refraction. The raj^s which form the ordinary image follow the ordinary law of sines (§ 722). They are called the ordinary rays. Those which form the extraordinary image (called the extraordinary rays) do not follow the law of sines, except when the plane of incidence is perpen¬ dicular to the axis of the crystal, and in this case their index of refraction (called the extraordinary index) is different from that of the ordinary rays. The ordinary index is 1 66, and the extraordinary 1 52. When the plane of incidence is parallel to the axis, the extra¬ ordinary ray lies in this plane, but the ratio of the sines of the angles of incidence and refraction is variable. When the plane of incidence is oblique to the axis, the extra¬ ordinary ray generally lies in a different plane. We shall recur to the subject of double refraction in the concluding- chapter of this volume. CHAPTEE LX. LENSES. 735. Forms of Lenses.—A lens is usually a piece of glass bounded by two surfaces which are portions of spheres. There are two prin¬ cipal classes of lenses. 1. Converging lenses or convex lenses, which have one or other of the three forms represented in Fig. 670. The first of these is called double convex, the second plano-convex, and the third concavo- convex. This last is also called a converging meniscus. All three Fig. 670.—Converging Lenses. Fig. 671.—Diverging LenseB. are thicker in the middle than at the edges. They are called con¬ verging, because rays are always more convergent or less divergent after passing through them than before. 2. Diverging lenses or concave lenses (Fig. 671) produce the opposite effect, and are characterized by being thinner in the middle than at the edges. Of the three forms represented, the first is double concave, the second plano-concave, and the third convexo-concave (also called a diverging meniscus). 60 930 LENSES. From the immense importance of lenses, especially convex lenses, in practical optics, it will be necessary to explain their properties at some length. 736. Principal Focus.—A lens is usually a solid of revolution, and the axis of revolution is called the axis of the lens, or sometimes the 'principal axis. When the surfaces are spherical, it is the line join¬ ing their centres of curvature. When rays which were originally parallel to the principal axis pass through a convex lens (Fig. 672), the ef¬ fect of the two refrac¬ tions which they un¬ dergo, one on entering and the other on leav¬ ing the lens, is to make them all converge ap¬ proximately to one point F, which is called the principal focus. The distance AF of the principal focus from the lens is called the principal focal distance , or more briefly and usually, the focal length of the lens. There is another prin¬ cipal focus at the same distance on the other side of the lens, cor¬ responding to an incid¬ ent beam coming in the opposite direction. The focal length depends on the convexity of the sur¬ faces of the lens, and also on the refractive power of the material of which it is composed, being shortened either by an increase of refractive power or by a diminution of the radii of cur¬ vature of the faces. In the case of a concave lens, rays incident parallel to the prin¬ cipal axis diverge after passing through; and their directions, if produced backwards, would approximately meet in a point F, which is still called the principal focus. It is only a virtual focus, inasmuch as the emergent rays do not actually pass through it, whereas the principal focus of a converging lens is real. Fig. 673.—Principal Focus of Concave Lens. OPTICAL CENTRE OF A LENS. 931 737. Optical Centre of a Lens. Secondary Axes.—Let 0 and O' be the centres of the two spherical surfaces of a lens. Draw any two parallel radii 01, O'E to meet these surfaces, and let the joining line IE represent a ray passing through the lens. This ray makes equal angles with the normals at I and E, since these latter are parallel by construction; hence the incident and emergent rays SI, ER also make equal angles with the normals, and are there¬ fore parallel. In fact, if tangent planes (indicated by the dotted lines in the figure) are drawn at I and E, the whole course of the ray SIER will be the same as if it had passed through a plate bounded by these planes. Let C be the point in which the line IE cuts the principal axis, and let R, R' denote the radii of the two spherical surfaces. Then, from the similarity of the triangles OCI, O'CE, we have Fig. 674.—Centre of Lens. OC = R CO' R' > ( 1 ) which shows that the point C divides the line of centres 0 O' in a definite ratio depending only on the radii. Every ray whose direc¬ tion on emergence is parallel to its direction before entering the lens, must pass through the point C in traversing the lens; and conversely, every ray which, in its course through the lens, traverses the point C, has parallel directions at incidence and emergence. The point C which possesses this remarkable property is called the centre , or optical centre , of the lens. In the case of a double convex or double concave lens, the optical centre lies in the interior, its distances from the two surfaces being directly as their radii. In plano-convex and plano-concave lenses it is situated on the convex or concave surface. In a meniscus of either kind it lies outside the lens altogether, its distances from the surfaces being still in the direct ratio of their radii of curvature. 1 x These consequences follow at once from equation (1) when applied to the several cases. 932 LENSES. In elementary optics it is usual to neglect the thickness of the lens. The incident and emergent rays SI, ER may then be re¬ garded as lying in one straight line which passes through C, and we may lay down the proposition that rays which pass through the centre of a lens undergo no deviation. Any straight line through the centre of a lens is called a secondary axis. The approximate convergence of the refracted rays to a point, when the incident rays are parallel, is true for all directions of incidence; Fig. 675.—Principal Focus on Secondary Axis. and the point to which the emergent rays approximately converge (/, Fig. 675) is always situated on the secondary axis ( acf ) parallel to the incident rays. The focal distance is sensibly the same as for rays parallel to the principal axis, unless the obliquity is consider¬ able. 738. Conjugate Foci.—When a luminous point S sends rays to a Fig. 676.—Conjugate Foci, both Real. lens (Fig. 676), the emergent rays converge (approximately) to one The distances of C from the two faces are respectively the difference between R and 0 C, and the difference between R' and O'C, and we have R _ OC = R-OC R' O'C R' - 0 7 C‘ FORMULAE RELATING TO LENSES. 933 point S'; whence it follows that rays sent from S' to the lens would converge (approximately) to S. Two points thus related are called conjugate foci of the lens, and the line joining them always passes through the centre of the lens; in other words, they must either be both on the principal axis, or both on the same secondary axis. The fact that rays which come from one point go to one point is the foundation of the theory of images, as we have already explained in connection with mirrors (§ 707). The diameters of object and image are directly as their distances from the centre of the lens, and the image will be erect or inverted according as the object and image lie on the same side or on opposite sides of this centre (§ 711). There is also, in the case of lenses, the same difference between an image seen in mid-air and an image thrown on a screen which we have pointed out in § 714. It is to be remarked that the distinction between principal and secondary axes has much more significance in the case of lenses than of mirrors; and images produced by a lens are more distinct in the neighbourhood of the principal axis than at a distance from it. 739. Formulae relating to Lenses.—The deviation produced in a ray by transmission through a lens will not be altered by substituting Fig 677.—Diagram showing Path of Ray, and Normals. for the lens a prism bounded by planes which touch the lens at the points of incidence and emergence; and in the actual use of lenses, the direction of the rays with respect to the supposed prism is such as to give a deviation not differing much from the minimum. The expression for the minimum deviation (§ 731) is 2i — 2r or 2i—A; and when the angle of the prism is small, as it is in the case of ordinary lenses, we may assume = n ; so that 2 i becomes 2fxr or juA, and the expression for the deviation becomes (/*-!) A, (1) 934 LENSES. A being the angle between the tangent planes (or between the nor¬ mals) at the points of entrance and emergence. Let x x and x 2 denote the distances of these points respectively from the principal axis, and r h r 2 the radii of curvature of the faces on which they lie. Then x - i are the sines of the angles which the J r x r 2 o normals make with the axis, and the angle A is the sum or differ¬ ence of these two angles, according to the shape of the lens. In the case of a double convex lens it is their sum, and if we identify the sines of these small angles with the angles themselves, we have A *’i ( 2 ) But if p h p 2 denote the distances from the faces of the lens to the points where the incident and emergent rays cut the principal axis, —» - are the sines of the angles which these rays make with the axis, and the deviation is the sum or difference of these two angles, according as the conjugate foci are on opposite sides or on the same side of the lens. In the former case, identifying the angles with their sines, the deviation is and this, by formula (1), is to be equal to (/* — 1) A, that is, to (/* —1) If the thickness of the lens is negligible in comparison with p h p 2 , we may regard x x and x 2 as equal, and the equation will reduce to (x_i + ^ 2 \ (3) Pi Pa \r 1 r 2 / I + i = (a-1) (! + 2). (4) Pi Pi Vi r 2 ; If p 1 is infinite, the incident rays are parallel, and p 2 is the principal focal length, which we shall denote by/. We have therefore and = (A <♦£) 1 Pi + 1 Pi f (5) ( 6 ) 740. Conjugate Foci on Secondary Axis.—Let M be a luminous CONJUGATE FOCI. 935 point on the secondary axis MOM', 0 being the centre of the lens, and let M' be the point in which an emergent ray corresponding to the incident ray MI cuts this axis. Let x denote x 1 or x 2 , the distances of the points of incidence and emergence from the principal axis, and d Fig. 678.—Conjugate Foci on Secondary Axis. the obliquity of the secondary axis; then x cos 0 is the length of the perpendicular from I upon M M', and are the sines of the angles 0 MI, OM'I respectively. But the deviation is the sum of these angles; hence, proceeding as in last section, we have x cos 6 x cos 6 ~MT 4 IT / 1 \ /X , X \ X o-vk+d = j’ (7) and when 0 is small, its cosine is sensibly equal to unity; 1 in which case the equation reduces to _L +J_=I MI MI / ( 8 ) The fact that x does not appear in equations (6) and (8) shows that, for every position of a luminous point, there is a conjugate focus lying on the same axis as the luminous point itself, at least in so far as the approximate assumptions which we have made are allowable. 741. Discussion of the Formula for Convex Lenses.—It thus appears that the formula - , 1 The quantity —L + _J_. is in fact rather greater than i; for in the first place, we have taken cos 0 as equal to unity, and in the second place we have assumed that the actual deviation is equal to the minimum deviation Correcting these inexact assumptions, we see that —^ + —L is really equal to a quantity rather greater than y, multiplied by sec 6, which is also rather greater than unity. The effective focal length is therefore rather less for oblique than for direct pencils. 936 LENSES. applies both to direct and oblique pencils, / denoting the principal focal length of the lens (supposed convex), and p, p ' the distances of a pair of conjugate foci from the lens, on opposite sides of it. This formula being identical with equation ( b ) of § 708, leads to results similar to those already deduced in the case of concave mirrors. As one focus advances from infinite distance to a principal focus, its conjugate moves away from the other principal focus to infinite distance on the other side. The more distant focus is always moving more rapidly than the nearer, and the least distance between them is accordingly attained when they are equidistant from the lens; in which case the distance of each of them from the lens is 2 f } and their distance from each other 4/. If either of the distances, as p, is less than /, the formula shows that the other distance p is negative. The meaning is that the two foci are on the same side of the lens, and in this case one of them (the more distant of the two) must be virtual. For example, in Fig. 679, if S, S' are a pair of conjugate foci, one of them S being be¬ tween the principal focus F and the lens, rays sent to the lens by a luminous point at S, will, after emergence, diverge as if from S'; and rays coming from the other side of the lens, if they converge to S' before incidence, will in reality be made to meet in S. As S moves towards the lens, S' moves in the same direction more rapidly; and they become coincident at the surface of the lens. The formula in fact shows that if ~ is very great in comparison with and positive, i must be very great and negative; that is to say, if p is a very small positive quantity, p' is a very small negative quantity. 742. Formation of Real Images.—Let A B (Fig. 680) be an object in front of a lens, at a distance exceeding the principal focal length. It will have a real image on the other side of the lens. To deter¬ mine the position of the image by construction, draw through any point A of the object a line parallel to the principal axis, meeting Fig. 679.—Conjugate Foci, one Real, one Virtual. FORMATION OF REAL IMAGES. 937 the lens in A'. The ray represented by this line will, after refrac¬ tion, pass through the principal focus F; and its intersection with the secondary axis A O determines the position of a, the focus conju¬ gate to A. We can in like manner determine the position of b, the focus conjugate to B, another point of the object; and the joining line a b will then be the image of the line AB. It is evident that if a b were the object, AB would be the image. Figs. 680, 681 repiesent the cases in which the distance of the object is respectively greater and less than twice the focal length of the lens. In each case it is evident that or the linear dimensions ao O a p’ of object and image are directly as their distances from the cen¬ tre of the lens. Again, since F 0 is parallel to a side of the triangle a A A, we have 0 A = FA = j ^ a ^ a P Fig. 681.--Real and Magnified Image. Fig. 680.—Real and Diminished Image. And by making a similar construction with respect to the other principal focus, we can prove that 0 A _ p-f' O a f We have therefore AB _ P -f _ J t a * / P' ~f ( 10 ) f denoting the focal length of the lens, and p, p’ the distances of A B, a b respectively from the lens. 938 LENSES. 743. Example.—A straight line 25 mm - long is placed perpendi¬ cularly on the axis, at a distance of 35 centimetres from a lens of 15 centimetres’ focal length; what are the position and magnitude of the image? * To determine the distance p we have l 35 lli r 35 x 15 —r = l whence p = - =26i cm - p' 15’ 1 35-15 4 For the length of the image we have o o 25 -f—= 25- 15 =18fmm. p-f 35-15 4 743a. Image on Cross-wires.—The position of a real image seen in mid-air can be tested by means of a cross of threads, or other con¬ venient mark, so arranged that it can be fixed at any required point. The observer must fix this cross so that it appears approximately to coincide with a selected point of the image. He must then try whether any relative displacement of the two occurs on shifting his eye to one. side. If so, the cross must be pushed nearer to the lens, or drawn back, according to the nature of the observed displacement, which follows the general rule of parallactic displacement, that the more distant object is displaced in the same direction as the ob¬ server’s eye. The cross may thus be brought into exact coincidence with the selected point of the image, so as to remain in apparent coincidence with it from all possible points of view. When this coincidence has been attained, the cross is at the focus conjugate to that which is occupied by the selected point of the object. By employing two crosses of threads, one to serve as object, and the other to mark the position of the image, it is easy to verify the fact that when the second cross coincides with the image of the first the first cross also coincides with the image of the second. 744. Aberration of Lenses.—In the investigations of §§ 739, 740, we made several assumptions which were only approximately true. The rays which proceed from a luminous point to a lens are in fact not accurately refracted to one point, but touch a curved surface called a caustic. The cusp of this caustic is the conjugate focus, and is the point at which the greatest concentration of light occurs. It is accordingly the place where a screen must be set to obtain the brightest and most distinct image. Bays from the central parts of VIRTUAL IMAGES. 939 the lens pass very nearly through it; but rays from the circumferen¬ tial portions fall short of it. This departure from exact concurrence is called spherical aberration. The distinctness of an image on a screen is improved by employing an annular diaphragm to cut off all except the central rays; but the brightness is of course diminished. By holding a convex lens in a position very oblique to the incident light, a primary and secondary focal line can be exhibited on a screen, just as in the case of concave mirrors (§ 714 a). The experiment, however, is rather more difficult of performance. 745. Virtual Images. —Let an object AB be placed between a con¬ vex lens and its prin¬ cipal focus. Then the foci conjugate to the points A, B are virtual, and their positions can be found by construc¬ tion from the consid¬ eration that rays through A, B, parallel to the principal axis, will be refracted to F, the principal focus on the other side. These refracted rays, if pro¬ duced backward, must meet the secondary axes OA, OB in the required points. An eye placed on the other side of the lens will accordingly see a virtual image erect, magnified, and at a greater distance from the lens than the object. This is the principle of the simple microscope. The formula for the distances D, d of object and image from the lens, when both are on the same side, is f denoting the principal focal length. 746. Concave Lens.—For a concave lens, if the focal length be still regarded as positive, and denoted by /, and if the distances D, d be on the same side of the lens, the formula becomes Fig. 682.—Virtual Image formed by Convex Lens. 1 _ \ = l_ d D / ( 12 ) which shows that d is always less than D ; that is, the image is nearer to the lens than the object. 940 LENSES. In Fig. 683, A B is the object, and a b the image. Bays incident from A and B parallel to the principal axis will emerge as if they came from the principal focus F. Hence the points a , b are deter¬ mined by the intersections of the dotted lines in the figure with the secondary axes OA, OB. An eye on the other side of the lens sees the image ab, which is always vir¬ tual, erect and dimin¬ ished. 747. Focometer. — Silbermann’s focomet¬ er (Fig. 684) is an in¬ strument for measur¬ ing the focal lengths of convex lenses, and is based on the principle (§ 741), that when the object and its image are equidistant from the lens, their distance from each other is four times the focal length. It consists of a graduated rule carrying three runners M, L, M'. The middle one L is the support for the lens which is to be examined; the other two, M, M', contain two thin plates of horn or other trans¬ lucent material, ruled with lines, which are at the same distance apart in both. The sliders must be adjusted until the image of one of these plates is thrown upon the other plate, without enlarge- Fig. 6S3.—Virtual Image formed by Concave Lens. Fig. 684.—Silbermann’s Focomoter. ment or diminution, as tested by the coincidence of the ruled lines of the image with those of the plate on which it is cast. The dis¬ tance between M and M' is then read off, and divided by 4. 747 a. Refraction at a Single Spherical Surface.—When the bound- SINGLE SPHERICAL SURFACE. 941 ing surface between two media is a portion of a sphere, the positions of conjugate foci relative to this surface are given by the equation % - 1 = +-^ 1 ; (13) TP p — r where p and p' denote the distances of object and image from the surface, both on the same side , r is the radius of curvature, and /i the index pf refraction, which is greater or less than unity according as the incidence is from rare to dense, or from dense to rare. The upper or lower sign is to be taken according as the surface is concave or convex towards the incident light. By making p infinite we see that the principal focus is at a dis¬ tance from the surface. When, as in the cornea of the human eye, Fig. 685.—Camera Obscura for Sketching. the surface is convex towards the incident light, and the incidence is from rare to dense, the refracted rays are more convergent than the incident rays, but are not so convergent as if they had been 942 LENSES. refracted through a plano-convex lens of the same convexity. The focal length of such a lens would, in fact, be only 748. Camera Obscura.—The images obtained by means of a hole in the shutter of a dark room (§ 683) become sharper as the size of the hole is diminished; but this diminution in¬ volves loss of light, so that it is impossible by this method to obtain an image at once bright and sharp. This difficulty can be overcome by employing a lens. If the objects in the external landscape depicted are all at distances many times greater than the focal length of the lens, their images will all be formed at sensibly the same distance from the lens, and may be received upon a screen placed at this distance. The images thus ob¬ tained are inverted, and are of the same size as if a simple aperture were employed instead of a lens. This is the principle of the camera obscura , one form of which is represented in Fig. 685. It is a kind of tent surrounded by opaque curtains, and having at its top a revolving lantern, containing a lens with its axis horizontal, and a mirror placed behind it at a slope of 45°, to re¬ flect the transmitted light down¬ wards on to a sheet of white paper lying on the top of a table. Im¬ ages of external objects are thus depicted on the paper, and their outlines can be traced with a pencil if desired. It is still better to combine lens and mirror in one, by the arrangement represented in section in Fig. 686. Kays Fig. 686.—Objective of Camera. Fig. 687.—Photographic Camera. PHOTOGRAPHIC CAMERA. 943 from external objects are first .refracted at a convex surface, then totally reflected at the hack of the lens, which is plane, and finally emerge through the bottom of the lens, which is concave, but with a larger radius of curvature than the first surface. The two refrac¬ tions produce the effect of a converging meniscus. The instrument is now only employed for purposes of amusement. 749. Photographic Camera.—The camera obscura employed by photographers (Fig. 687) is a box MN with a tube AB in front, containing an object-glass at its extremity. The object-glass is usually compound, consisting of two single lenses E, L, an arrange¬ ment which is very commonly adopted in optical instruments, and which has the advantage of giving the same effective focal length as a single lens of smaller radius of curvature, while it permits the employment of a larger aperture, and consequently gives more light. At G is a slide of ground glass, on which the image of the scene to be depicted is thrown, in setting the instrument. The focussing is performed in the first place by sliding the part M of the box in the part N, and finally by the pinion V which moves the lens. When the image has thus been rendered as sharp as possible, the sensitized plate is substituted for the ground glass. 1 1 The photographic processes at present in use are very venous, both optically and chemically; but are all the same in principle with the method originally employed by Talbot. This method, which was almost forgotten during the great success of Daguerre, consists in first obtaining, on a transparent plate, a picture with lights and shades reversed, called a negative; then placing this upon a piece of paper sensitized with chloride of silver, and exposing it to the sun’s rays. The light parts of the negative allow the light to pass and blacken the paper, thus producing a positive picture. The same negative serves for producing a great number of positives. The negative plate is usually a glass plate covered with a film of collodion (sometimes of albumen), sensitized by a salt of silver. The following is one of the numerous formulae for this preparation. Take Sulphuric ether,. 300 grammes Alcohol at 40°. 200 „ Gun cotton, . 5 „ Incorporate these ingredients thoroughly in a porcelain mortar; then add Iodide of potassium,.13 grammes Iodide of ammonium,. 1'75,, Iodide of cadmium,.1‘75 ,, Bromide of cadmium,.1’25 „ This mixture is poured over the plate, which is then immersed in a solution (10 per cent. 944 LENSES. 750. Use of Lenses for Purposes of Projection.—Lenses are exten¬ sively employed in the lecture-room, for rendering experiments visible to a whole audience at once, by projecting them on a screen. The arrangements vary according to the circumstances of each case, and cannot be included in a general description. 751. Solar Microscope. Magic Lantern.—In the solar microscope a convex lens of short focal length is employed to throw upon a screen a highly-magnified image of a small object placed a little beyond the principal focus. As the image is always much less bright than the object, and the more so as the magnification is greater, it is necessary that the object should be very highly illuminated. For this purpose the rays of the sun are directed upon it by means of a mirror and large lens; the latter serving to increase the solid angle of the cone of rays which fall upon the object, and thus to enable a larger por¬ tion of the magnifying lens to be utilized. The objects magnified are always transparent; and the images are formed by rays which have been transmitted through them. strong) of nitrate of silver. The film of collodion is thus brought to an opal tint, and the plate, after being allowed to drain, is ready for exposure in the camera. After being exposed, the picture is developed, by the application of a liquid for which the following is a formula; Distilled water,.250 grammes. Pyrogallic acid,. 1 „ Crystallizable acetic acid,.20 ,, When the picture is sufficiently developed, it is fixed, by the application of a solution, either of hyposulphite of soda from 25 to 30 per cent, strong, or of cyanide of potassium 3 per cent, strong, and the negative is completed. To obtain a positive, the negative plate is laid upon a sheet of paper in a glass dish, the paper having been sensitized by immersing it first in a solution of common sea-salt 3 or 4 per cent, strong, and then in a solution of nitrate of silver 18 per cent, strong. The exposure is continued till the tone is sufficiently deep, the tint is then improved by means of a salt of gold, and the picture is fixed by hyposulphite of soda. It has then only to be washed and dried.— D. PHOTO-ELECTRIC MICROSCOPE. 945 The lens employed for producing the image is usually compound, consisting of a convex and a concave lens combined. The electric light can be employed instead of the sun. The apparatus for regulating this light is usually placed within a lantern (Fig. 689), in such a position that the light is at the centre of curva- Fig. 689.—Photo-electric Microscope. ture of a spherical mirror, so that the inverted image of the light coincides with the light itself. The light is concentrated on the object by a system of lenses, and, after passing through the object, traverses another system of lenses, placed at such a distance from the object as to throw a highly-magnified image of it on a screen. The whole arrangement is called the electric or photo-electric microscope. The magic lantern is a rougher instrument of the same kind, employed for projecting magnified images of transparent paintings, executed on glass slides. It has one lens for converging a beam of light on the slide, and another for throwing an image of the slide on the screen. In all these cases the ima^e is inverted. O 61 CHAPTER LXL VISION AND OPTICAL INSTRUMENTS. 752. Description of the Eye.—The human eye (Fig. 690) is a nearly spherical ball, capable of turning in any direction in its socket. Its outermost coat is thick and horny, and is opaque except in its LEVEIUE DEL. Fig. 690.—Human Eye. anterior portion. Its opaque portion H is called the sclerotica , or in common language the white of the eye. Its transparent portion A is Galled the cornea , and has the shape of a very convex watch-glass. Behind the cornea is a diaphragm D, of annular form, called the iris. It is coloured and opaque, and the circular aperture C in its centre THE EYE. 947 is called the pupil. By the action of the involuntary muscles of the iris, this aperture is enlarged or contracted on exposure to darkness or light. The colour of the iris is what is referred to when we speak of the colour of a person’s eyes. Behind the pupil is the crystalline lens E, which has greater convexity at back than in front. It is built up of layers or shells, increasing in density inwards. This latter circumstance tends to diminish spherical aberration, the effect of which, in an ordinary lens, is to make rays which pass near the outside too convergent, as compared with those which pass near the axis. The cavity B between the cornea and the crystalline is called the anterior chamber, and is filled with a watery liquid called the aqueous humour. The much larger cavity L, behind the crystalline, is called the posterior chamber, and is filled with a transparent jelly called the vitreous humour , inclosed in a very thin transparent membrane (the hyaloid membrane). The posterior chamber is in¬ closed, except in front, by the choroid coat or uvea I, which is saturated with an intensely black and opaque mucus, called the pigmentum nigrum. The choroid is lined, except in its anterior portion, with another membrane K, called the retina, which is tra¬ versed by a ramified system of nerve filaments diverging from the optic nerve M. Light incident on the retina gives rise to the sensa¬ tion of vision; and there is no other part of the eye which possesses this property. 753. The Eye as an Optical Instrument.—It is clear, from the above description, that a pencil of rays entering the eye from an external point will undergo a series of refractions, first at the anterior surface of the cornea, and afterwards in the successive layers of the crystal¬ line lens, all tending to render them convergent (see table of indices, § 724). A real and inverted image is thus formed of any external object to which the eye is directed. If this image falls on the retina, the object is seen; and if the image thus formed on the retina is sharp and sufficiently luminous, the object is seen distinctly. 754. Adaptation to Different Distances.—As the distance of an image from a lens varies with the distance of the object, it would only be possible to see objects distinctly at one particular distance, were there not special means of adaptation in the eye. Persons whose sight is not defective can see objects in good definition at all distances exceeding a certain limit. When we wish to examine the minute details of an object to the greatest advantage, we hold it at a particular distance, which varies in different individuals, and 948 VISION AND OPTICAL INSTRUMENTS. averages about eight inches. As we move it further away, we experience rather more ease in looking at it, though the diminution of its apparent size, as measured by the visual angle, renders its minuter features less visible. On the other hand, when we bring it nearer to the eye than the distance which gives the best view, we cannot see it distinctly without more or less effort and sense of strain; and when we have brought it nearer than a certain lower limit (averaging about six inches), we find distinct vision no longer pos¬ sible. In looking at very distant objects, if our vision is not defec¬ tive, we have very little sense of effort. These phenomena are in accordance with the theory of lenses, which shows that when the distance of an object is a large multiple of the focal length of the lens, any further increase, even up to infinity, scarcely alters the distance of the image; but that, when the object is comparatively near, the effect of any change of its distance is considerable. There has been much discussion among physiologists as to the precise nature of the changes by which we adapt our eyes to distinct vision at dif¬ ferent distances. Such adaptation might consist either in a change of focal length, or in a change of distance of the retina. Observa¬ tions in which the eye of the patient is made to serve as a mirror, giving images by reflection at the front of the cornea, and at the front and back of the crystalline, have shown that the convexity of the front of the crystalline is materially changed as the patient adapts his eye to near or remote vision, the convexity being greatest for near vision. This increase of convexity corresponds to a shortening of focal length, and is thus consistent with theory. 755. Binocular Vision.—The difficulty which some persons have felt in reconciling the fact of an inverted image on the retina with the perception of an object in its true position, is altogether fanciful, and arises from confused notions as to the nature of perception. The question as to how it is that we see objects single with two eyes, rests upon a different footing, and is not to be altogether ex¬ plained by habit and association. 1 To each point in the retina of one eye there is a corresponding point , similarly situated, in the other. An impression produced on one of these points is, in ordinary cir¬ cumstances, undistinguishable from a similar impression produced on the other, and when both at once are similarly impressed, the effect is simply more intense than if one were impressed alone; or, to 1 Binocular vision is a subject which has been much debated. For the account here given of it, the Editor is alone responsible. THE STEREOSCOPE. 949 describe the same phenomena subjectively, we have only one field of view for our two eyes, and in any part of this field of view we see either one image, brighter than we should see it by one alone, or else we see two overlapping images. This latter phenomenon can be readily illustrated by holding up a finger between one’s eyes and a wall, and looking at the wall. We shall see, as it were, two trans- parent fingers projected on the wall. One of these transparent fingers is in fact seen by the right eye, and the other by the left, but our visual sensations do not directly inform us which of them is seen by the right eye, and which by the left. The principal advantage of having two eyes is in the estimation of distance, and the perception of relief. In order to see a point as single by two eyes, we must make its two images fall on correspond¬ ing points of the retinae; and this implies a greater or less converg¬ ence of the optic axes according as the object is nearer or more remote. We are thus furnished with a direct indication of the distance of the object from our eyes; and this indication is much more precise than that derived from the adjustment of their focal length. In judging of the comparative distances of two points which lie nearly in the same direction, we are greatly aided by the parallactic displacement which occurs when we change our own position. We can also form an estimate of the nearness of an object, from the amount of change in its apparent size, contour, and bearing, pro¬ duced by shifting our position. This would seem to be the readiest means by which very young animals can distinguish near from remote objects. 756. Stereoscope.—The perception of relief is closely connected with the doubleness of vision which occurs when the images on corresponding portions of the two retinae are not similar. In survey¬ ing an object we run our eyes rapidly over its surface, in such a way as always to attain single vision of the particular point to which our attention is for the instant directed. We at the same time receive a somewhat indistinct impression of all the points within our field of view; an impression which, when carefully analyzed, is found to involve a large amount of doubleness. These various impressions combine to give us the perception of relief; that is to say, of form in three dimensions. The perception of relief in binocular vision is admirably illustrated by the stereoscope , an instrument which was invented by Wheatstone, and reduced to its present more convenient form by Brewster. Two 950 VISION AND OPTICAL INSTRUMENTS. figures are drawn, as in Fig. 691, being perspective representations of the same object from two neighbouring points of view, such as might be occupied by the two eyes in looking at the object. Thus if the object be a cube, the right eye will have a fuller view of the right Fig. 692.—Stereoscope. Fig. 693.—Path of Rays in Stereoscope. face, and the left eye of the left face. The two pictures are placed in the right and left compartments of a box, which has a partition down the centre serving to insure that each eye shall see only the picture intended for it; and over each of the compartments a half¬ lens is fixed, serving, as in Fig. 693, not only to magnify the picture, but at the same time to displace it, so that the two virtual images are brought into approximate coincidence. Stereoscopic pictures are usually photographs obtained by means of a double camera, having two objectives, one beside the other, which play the part of two eyes. When matters are properly arranged, the observer seems to see the object in relief. He finds himself able to obtain single view of any one point of the solid image which is before him; and the adjust¬ ments of the optic axes which he finds it necessary to make, in shift¬ ing his view from one point of it to another, are exactly such as would be required in looking at a solid object. When one compartment of the stereoscope is empty, and the other contains an object, an observer, of normal vision, looking in in the MAGNIFYING POWER. 951 ordinary way, is unable to say which eye sees the object. If two pictures are combined, consisting of two equal circles, one of them having a cross in its centre, and the other not, he is unable to decide whether he sees the cross with one eye or both. When two entirely dissimilar pictures are placed in the two com¬ partments, they compete for mastery, each of them in turn becoming more conspicuous than the other, in spite of any efforts which the observer may make to the contrary. A similar fluctuation will be observed on looking steadily at a real object which is partially hidden from one eye by an intervening object. This tendency to alternate preponderance renders it well nigh impossible to combine two colours by placing one under each eye in the stereoscope. The immediate visual impression, when we look either at a real solid object, or at the apparently solid object formed by properly combining a pair of stereoscopic views, is a single picture formed of two slightly different pictures superimposed upon each other. The coincidence becomes exact at any point to which attention is directed, and to which the optic axes are accordingly made to converge, but in the greater part of the combined picture there is a want of coin¬ cidence, which can easily be detected by a collateral exercise of attention. The fluctuation above described to some extent tends to conceal this doubleness; and in looking at a real solid object, the concealment is further assisted by the blurring of parts which are out of focus. 757. Visual Angle. Magnifying Power.—The angle which a given straight line subtends at the eye is called its visual angle , or the angle under which it is seen. This angle is the measure of the length of the image of the straight line on the retina. Two discs at different distances from the eye, are said to have the same apparent size, if their diameters are seen under equal angles. This is the con¬ dition that the nearer disc, if interposed between the eye and the remoter disc, should be just large enough to conceal it from view. The angle under which a given line is seen, evidently depends not only on its real length, and the direction in which it points, but also on its distance from the eye; and varies, in the case of small visual angles, in the inverse ratio of this distance. The apparent length of a straight line may be regarded as measured by the visual angle which it subtends. By the magnifying power of an optical instrument, is usually meant the ratio in which it increases apparent lengths in this sense. 952 VISION AND OPTICAL INSTRUMENTS. In the case of telescopes, the comparison is between an object as seen in the telescope, and the same object as seen with the naked eye at its actual distance. In the case of microscopes, the compari¬ son is between the object as seen in the instrument, and the same object as seen by the naked eye at the least distance of distinct vision,, which is usually assumed as 10 inches. But two discs, whose diameters subtend the same angle at the eye, may be said to have the same apparent area; and since the areas of similar figures are as the squares of their linear dimensions, it is evident that the apparent area of an object varies as the square of the visual angle subtended by its diameter. The number expressing magnification of apparent area is therefore the square of. the mag¬ nifying power as above defined. Frequently, in order to show that the comparison is not between apparent areas, but between apparent lengths, an instrument is said to magnify so many diameters. If the diameter of a sphere subtends 1° as seen by the naked eye, and 10° as seen in a telescope, the telescope is said to have a magnifying power of 10 diameters. The superficial magnification in this case is evidently 100. The apparent length and apparent area of an object are respect¬ ively proportional to the length and area of its image on the retina. Apparent length is measured by the plane angle, and apparent area by the solid angle, which an object subtends at the eye. 758. Spectacles.—Spectacles are of two kinds, intended to remedy two opposite defects of vision. Short-sighted persons can see objects distinctly at a smaller distance than persons whose vision is normal; but always see distant objects confused. On the other hand, persons whose vision is normal in their youth, usually become over-sighted with advancing years, so that, while they can still adjust their eyes correctly for distant vision, objects as near as 10 or 12 inches always appear blurred. Spectacles for over-sighted persons are convex, and should be of such focal length, that, when an object is held at about 10 inches distance, its virtual image is formed at the nearest distance of distinct vision for the person who is to use them. This latter distance must be ascertained by trial. Call it p inches; then, by § 715, the formula for computing the required focal length x (in inches) is 1 _ 1 = I. io p x For example, if 15 inches is the nearest distance at which the person SPECTACLES. 953 can conveniently read without spectacles, the focal length required is 30 inches. In Fig. 691, A represents the position of a small object, and A' that of its virtual image as seen with spectacles of this kind. Over-sight is not the only defect which the eye is liable to acquire Fig. 694.—Spectacle-glass for Over-sighted Eye. by age ; but it is the defect which ordinary spectacles are designed to remedy. Spectacles for short-sighted persons are concave, and the focal Fig. 695.—Spectacle-glass for Short-sighted Eye. length which they ought to have, if designed for reading, may be computed by the formula i_ _ l l p 10 x p denoting the nearest distance at which the person can read, and x the focal length, both in inches. If his greatest distance of distinct vision exceeds the focal length, he will be able, by means of the spectacles, to obtain distinct vision of objects at all distances, from 10 inches upwards. 759. Simple Magnifier.—A magnifying-glass is a convex lens, ol 954 VISION AND OPTICAL INSTRUMENTS. shorter focal length than the human eye, and is placed at a distance somewhat less than its focal length from the object to be viewed. In Fig. 696, a b is the object, and AB the virtual image which is seen by the eye K. The construction which we have em¬ ployed for drawing the image is one which we have seve¬ ral times used before. Through the point a, the line a M is drawn parallel to the principal axis. F M is then drawn from the principal focus F; 0 a is drawn from the optical centre 0 ; and these two lines are produced till they meet in A. Distance of lens from object In order that the image may be properly seen, its distance from the eye must fall between the limits of distinct vision; and in order that it may be seen under the largest possible visual angle, the eye must be close to the lens, and the object must be as near as is compatible with distinct vision. This and other interesting properties are established by the following investigation:— Let 0 denote the visual angle under which the observer sees the image of the portion a c of the object. Also let x denote the distance cO of the object from the lens, and y the distance 0 K of the lens from the eye. Then we have , a AC AC . tan 6 = --» CK C 0 + 2/ But, by formulae (10) and (11) of last chapter, we have AC=acX, CO=x.-J—- f-x f-x Substituting these values for A C and C 0, and reducing, we have tan 6 = ac. f {x+y)f-xy (A) This equation shows that, for a given lens and a given object, the visual angle varies inversely as the quantity (x-\-y) f—xy. THE MICROSCOPE. 955 The following practical consequences are easily drawn: — (1) If the distance x + y of the eye from the object is given, the visual angle increases as the two distances x, y approach equality, and is not altered by interchanging them. (2) If one of the two distances x, y be given, and be less than /, the other must be made as small as possible, if we wish to obtain the largest possible visual angle. To obtain the absolute maximum of visual angle, we must select, from the various positions which make C K equal to the nearest distance of distinct vision, that which gives the largest value of AC, since the quotient of A C by C K is the tangent of the visual angle. Now A C increases as the image moves further from the lens, and hence the absolute maximum is obtained by making its distance from the lens equal to the nearest distance of distinct vision, and making the eye cpme up close to the lens. In this case the distance p of the object from the lens is given by the equation y--^=-y-’ where D denotes the nearest distance of distinct vision, and tan 0 is a j-OYac (y + d)- the greatest angle under which the body could be seen by the naked eye is the angle whose tangent is hence the visual angle (or its tangent) is increased by the lens in the ratio l + y, which is called the magnifying power. If the object were in the principal focus, and the eye close to the lens, the magnifying power would be y- In either case, the thickness of the lens being neglected, the visual angle is the angle which the object subtends at the centre of the lens, and therefore varies inversely as the distance of this centre from the object. For lenses of small focal length, the recipro¬ cal of the focal length may be regarded as proportional to the magnifying power. Simple Microscope .—By a simple micro¬ scope is usually understood a lens mounted in a manner convenient for the examination of small objects. Fig. 697 represents an instrument of this kind. The lens l is mounted in brass, and carried at the end of an arm. It is raised and lowered by turning the milled head V, which acts on the 95(5 VISION AND OPTICAL INSTRUMENTS. rack a. C is the platform on which the object is laid, and M is a concave mirror, which can be employed for increasing the illumina¬ tion of the object. 760. Compound Microscope. — In the compound microscope, there is one lens which forms a real and greatty enlarged image of the object; and this image is itself magnified by viewing it through another lens. In Fig. 698, a 6 is the object, O is the first lens, called the objective , and is placed at a distance only slightly ex¬ ceeding its focal length from the object; an inverted image a x b x is thus formed, at a much greater distance on the other side of the lens, and proportionally larger. O' is the second lens, called the ocular or eye-piece , which is placed at a dis¬ tance a little less than its focal length from the first image a x b x , and thus forms an enlarged virtual image of it A B, at a convenient distance for distinct vision. If we suppose the final image A B to be at the least distance of distinct vision from the eye placed at O' (this being the arrangement which gives the largest visual angle), the magnifying power will be simply the ratio of the length of this image to that of the object a b, and will be the product of the two factors and yy. The former is the magnification produced by the eye-piece, and is, as we have just shown (§ 759), 1 + j. The other factor y^- 1 is the magnification produced by the objective, and is equal to the ratio of the distances 0 If the objective is taken out, and replaced by another of differ¬ ent focal length, the readjustment will consist in altering the dis¬ tance O a, leaving the distance 0 a x unchanged. The total magnifi¬ cation therefore varies inversely as O a, that is, nearly in the inverse ratio of the focal length of the objective. Compound microscopes are usually provided with several objectives, of various focal lengths, from which the observer makes a selection according to the magni¬ fying power which he requires for the object to be examined. The powers most used range from 50 to 350 diameters. Fig. 698.—Compound Microscope. THE MICROSCOPE. 957 The magnifying power of a microscope can be determined by direct observation, in the following way. A plane reflector pierced with a hole in its centre, is placed directly over the eye-piece (Fig. 699), at an inclination of 45°, and an¬ other plane reflector, or still better, a totally reflecting prism, as in the figure, is placed parallel to it at the distance of an inch or two, so that the eye, looking down upon the first mirror, sees, by means of two successive reflections, the image of a divided scale placed at a distance of 8 or 10 inches below the second reflector. In taking an observation, a micrometer scale engraved on glass, its divisions being at a known distance apart (say of a millimetre), is placed in the microscope as the object to be magnified; and the ob¬ server holds his eye in such a position that, by means of different parts of his pupil, he sees at once the magnified image of the micrometer scale in the microscope, and the reflected and unmagnified image of the other scale. The two images will be super¬ imposed in the same field of view; and it is easy to observe how many divisions a given number of divisions of the other, large scale be millimetres, and those on the micrometer scale hun¬ dredths of a millimetre. Then the magnifying power is 100, if one of the magnified covers one of the unmagnified divisions; and is - QQ if n of the former cover N of the latter. This is on the assumption that the large scale is placed at the nearest distance of convenient vision. In stating the magnifying power of a microscope, this dis¬ tance is usually reckoned as 10 inches. A short-sighted person sees an image in a microscope (whether simple or compound) under a larger visual angle than a person Qf normal sight; but the inequality is not so great as in the case of objects seen by the naked eye. In fact, if /be the focal length of the eye-piece in a compound microscope, or of the microscope itself if simple, and D the nearest distance of distinct vision for the ob- Fig. 699. Measurement of Magnifying Power. f the one coincide with Let the divisions on the 958 VISION AND OPTICAL INSTRUMENTS. server, the visual angle under which the image is seen in the micro¬ scope is proportional to j + the greatest visual angle for the naked eye being represented by Both these angles increase as D dim¬ inishes, but the latter increases in a greater ratio than the former. When / is as small as T V of an inch, the visual angle in the micro¬ scope is sensibly the same for short as for normal sight. Before reading off the divisions in the observation above described, care should be taken to focus the microscope in such a way, that the image of the micrometer scale is at the same distance from the eye as the image of the large scale with which it is compared. When this is done, a slight motion of the eye does not displace one image with respect to the other. Instead of a single eye-lens, it is usual to employ two lenses sepa¬ rated by an interval, that which is next the eye being called the eye-glass , and the other the field-glass. This combination is equiva¬ lent to the Huyghenian or negative eye-piece employed in telescopes (§ 800). 761. Astronomical Telescope.—The astronomical refracting tele¬ scope consists essenti¬ ally (like the com¬ pound microscope) of two lenses, one of which forms areal and inverted image of the object, which is looked at through the other. In Fig. 700, 0 is the object-glass, which is sometimes a foot or more in diameter, and is always of much greater focal length than the eye-piece O'. The inverted image of a distant object is formed at the principal focus F. This image is represented at a b. The parallel rays marked 1, 2 come from the upper extremity of the object, and meet at a; and the parallel rays 3, 4, from the other extremity, meet at b. A' B' is the virtual image of a b formed by the eye-piece. Its distance from the eye can be changed by pulling out or pushing in the eye-tube; and may in practice have any value intermediate between the least distance of distinct vision and infinity, the visual angle under which it is seen being but slightly affected by THE TELESCOPE. 959 this adjustment. The rays from the highest point of the object emerge from the eye-piece as a pencil diverging from A'; and the rays from the lowest point of the object form a pencil diverging from B'. Magnification .—The angle under which the object would be seen by the naked eye is a O 6; for the rays a O, b O, if produced, would pass through its extremities. The angle under which it is seen in the telescope, if the eye be close to the eye-lens, is A' O' B' or a O' b. The magnification is therefore which is approximately the same as the ratio of the distances of the image a b from the two lenses If the eye-tube is so adjusted as to throw the image A'B' to infinite distance, F will be the principal focus of both lenses, and the magnification is the ratio of the focal length of the object-glass to that of the eye-piece. If the eye - tube be pushed in as far as is compatible with dis¬ tinct vision (the eye being close to the lens), the magnification is greater than this in the ratio D denoting the nearest distance of distinct vision, and / the focal length of the eye-piece. The magnification can be directly ob¬ served by looking with one eye through the telescope at a brick wall, while the other eye is kept open. The image will thus be superimposed on the actual wall, and we have only to observe how many courses of the latter coincide with a single course of the magnified image. If the telescope is large, its tube may prevent the second eye from seeing the wall, and it may be necessary to employ a reflecting arrangement, as in Fig. 701, analogous to that described in connec¬ tion with the microscope. Telescopes without stands seldom magnify more than about 10 diameters. Powers of from 20 to 60 are common in telescopes with 9(30 VISION AND OPTICAL INSTRUMENTS. stands, intended for terrestrial purposes. The powers chiefly em¬ ployed in astronomical observation are from 100 to 500. Mechanical Arrangements .—The achromatic object-glass O is set in a mounting which is screwed into one end of a strong brass tube A A (Fig. 702). In the other end slides a smaller tube F containing the eye¬ piece O'; and by turning the milled head V in one direction or the other, the eye-piece is moved for¬ wards or backwards. Finder. — The small telescope l , which is at¬ tached to the principal telescope, is called a find¬ er. This appendage is in¬ dispensable when the principal telescope has a high magnifying power; for a high magnifying power involves a small field of view, and consequent difficulty in directing the telescope so as to include a selected object within its range. The finder is a telescope of large field; and as it is set parallel to the principal telescope, objects will be visible in the latter if they are seen in the centre of the field of view of the former. 762. Best Position for the Eye.—The eye-piece forms a real and inverted image of the object-glass 1 at EE' (Fig. 700) ; through which all rays transmitted by the telescope must of necessity pass. If the telescope be directed to a bright sky, and a piece of white paper held behind the eye-piece to serve as a screen, a circular spot of light will be formed upon it, which will become sharply defined (and at the same time attain its smallest size) when the screen is held in the correct position. This image (which we shall call the bright spot) may be regarded as marking the proper place for the pupil of the observer’s eye. Every ray which traverses the centre of the object- glass traverses the centre of this spot; every ray which traverses the upper edge of the object-glass traverses the lower edge of the Fig. 702.—Astronomical Telescope. 1 Or it may be called an image of the aperture which the object-glass fills. It remains sen¬ sibly unchanged on removing the object-glass so as to leave the end of the telescope open. TERRESTRIAL TELESCOPE. 961 spot; and any selected point of the spot receives all the rays which have been transmitted by one particular point of the object-glass. An eye with its pupil anywhere within the limits of the bright spot, will therefore see the whole field of view of the telescope. If the spot and pupil are of exactly the same size, they must be made to coin¬ cide with one another, as the necessary condition of seeing the whole field of view with the brightest possible illumination. Usually in practice the spot is much smaller than the pupil, so that these advan¬ tages can be obtained without any nicety of adjustment; but to obtain the most distinct vision, the centre of the pupil should coin¬ cide as closely as possible with the centre of the spot. To facilitate this adjustment, a brass diaphragm, with a hole in its centre, is screwed into the eye-end of the telescope, the proper place for the eye being close to this hole. One method of determining the magnifying power of a telescope consists in measuring the diameter of the bright spot, and comparing it with the effective aperture of the object-glass. In fact, let F and / denote the focal lengths of object-glass and eye-piece, and a the dis¬ tance of the spot from the centre of the eye-piece; then F -f/ is ap¬ proximately the distance of the object-glass from the same centre, and, by the formula for conjugate focal distances, we have i = y Multiplying both sides of this equation by F and then subtract- Y -Lf Y ing unity, we have = j. But the ratio of the diameter of the object-glass to that of its image is —and j. is the usual formula for the magnifying power. Hence, the linear magnifying 'power of a telescope is the ratio of the diameter of the object-glass to that of the bright spot 763. Terrestrial Telescope.—The astronomical telescope just de¬ scribed gives inverted images. This is no drawback in astronomical observation, but would be inconvenient in viewing terrestrial objects. In order to re-invert the image, and thus make it erect, two addi¬ tional lenses 0" O'" (Fig. 703) are introduced between the real image a b and the eye-lens O'. If the first of these 0" is at the dis¬ tance of its principal focal length from a b, the pencils which fall upon the second will be parallel, and an erect image a' b' will thus be formed in the principal focus of O'". This image is viewed through the eye-lens O', and the virtual image A' B' which is perceived by the eye will therefore be erect. The two lenses O", O'", are usually 62 962 VISION AND OPTICAL INSTRUMENTS. made precisely alike, in which case the two images a b, a' b' will be equal. In the better class of terrestrial telescopes, a different ar- Fig. 703.—Terrestrial Eye-piece. rangement is adopted, requiring one more lens; but whatever system be employed, the reinversion of the image always involves some loss both of light and of distinctness. 764. Galilean Telescope.—Besides the disadvantages just men¬ tioned, the erecting eye-piece involves a considerable addition to the length of the instrument. The telescope invented by Galileo, and the earliest of all tele¬ scopes, gives erect im¬ ages with only two len¬ ses, and with shorter length than even the astronomical telescope. O (Fig. 704) is the ob¬ ject-glass, which is con¬ vex as in the astrono¬ mical telescope, and would form a real and inverted image a b at its principal focus; but the eye-glass O', which is a concave lens, is interposed at a distance equal to or slightly exceeding its own focal length from the place of this image, and forms an erect virtual image A' B', which the observer sees. Neglecting the distance of his eye from the lens, the angle under which he sees the image is A' O' B', which is equal to a O' b, whereas Fig. 704.—Galilean Telescope. GALILEAN TELESCOPE. 963 the visual angle to the naked eye would be a 0 6. The magnification is therefore which is approximately equal to c being the prin¬ cipal focus of the object-glass. If the instrument is focussed in such a way that the image A' B' is thrown to infinite distance, c is also the principal focus of the eye-lens, and the magnification is sim¬ ply the ratio of the focal lengths of the two lenses. This is the same rule which we deduced for the astronomical tele¬ scope; but the Galilean telescope, if of the same power, is shorter by twice the focal length of the eye-lens, since the distance be¬ tween the two lenses is the difference instead of the sum of their focal lengths. This telescope has the disadvantage of not admitting of the em¬ ployment of cross-wires; for these, in order to serve their purpose, must coincide with the real image; and no such this telescope. There is another peculiarity in the absence of the bright spot above described, the image of the object-glass formed by the eye-glass being virtual. In other telescopes, if half the object-glass be cov¬ ered, half the bright spot will be obliterated; but the remaining half suffices for giving the whole field of view, though with diminished brightness. In the Galilean telescope, on the contrary, if half the object-glass be covered, half the field of view will be cut off, and the remaining half will be unaffected. The opera-glass , single or binocular, is a Gali¬ lean telescope, or a pair of Galilean telescopes. In the best instruments, both object-glass and eye-glass are achromatic combinations of three pieces, as shown in section in the figure (Fig. 705); the middle piece in each case being flint, and the other two crown (§ 794). 765. Reflecting Telescopes.—In reflecting telescopes, the place of an object-glass is supplied by a concave mirror called a speculum, usually composed of solid metal. The real and inverted image which it forms of distant objects is, in the Herschelian telescope, viewed directly through an eye-piece, the back of the observer being towards the object, and his face towards the speculum. This construction is only applicable to very large specula; as in instruments of ordinary image exists in Fig. 705.—Opera-glass. 964 VISION AND OPTICAL INSTRUMENTS. Fig. 706.—Newtonian Telescope. size the interposition of the observer’s head would occasion too serious a loss of light. An arrangement more frequently adopted is that devised by Sir Isaac Newton, and employed by him in the first reflecti ng tele¬ scope ever construct¬ ed. It is represented in Fig. 706. The specu¬ lum is at the bottom of a tube whose open end is directed towards the distant object to be examined. Therays 1 and 2 from one extre¬ mity of the object are reflected towards a, and the rays 3, 4 from the other extremity are reflected towards b. A real inverted image a b would thus be formed at the principal focus of the concave speculum; but a small plane mirror M is interposed obliquely, and causes the real image to be formed at a' b' in a symmetrical position with respect to the mirror M. The eye-lens O' transforms this into the enlarged and virtual image A' B\ Magnifying Power .—The approximate formula for the magnifying power is the same as in the case of the refracting telescopes already described. In fact the first image a b subtends, at the optical centre 0 (not shown in the figure) of the large speculum, an angle a Ob equal to the visual angle for the naked eye; and the second image a b' (which is equal to the former) subtends, at the centre of the eye-piece, an angle a' O' b' equal to the angle under which the image is seen in the telescope. The magnifying power is therefore , or, what is the same thing, is the ratio of the distance of a b from 0 to the distance of a' b' from O', or the ratio of the focal length of the speculum to that of the eye-piece. In the Gregorian telescope, which was invented before that of Newton, but not manufactured till a later date, there are two con¬ cave specula. The large one, which receives the direct rays from the object, forms a real and inverted image. The smaller speculum, which is suspended in the centre of the tube, with its back to the object, receives the rays reflected from the first speculum, and forms a second real image, which is the enlarged and inverted image of the SILVERED SPECULA. 965 first, and is therefore erect as compared with the object. This real and erect image is then magnified by means of an eye-piece, as in the instruments previously described, the eye-piece being contained in a tube which slides in a hole pierced in the middle of the large specu¬ lum. As this arrangement gives an erect image, and enables the observer to look directly towards the object, it. is specially convenient for terrestrial observation. It is the construction almost universally adopted in reflecting telescopes of small size. The Cassegranian telescope resembles the Gregorian, except that the second speculum is convex, and the image which the observer sees is inverted. 766. Silvered Specula.—Achromatic refracting telescopes give much better results, both as regards light and definition, than reflectors of the same size or weight; but it has been found practicable to make specula of much larger size than object-glasses. The aperture of Lord Rosse's largest telescope is 6 feet, whereas the aperture of the largest achromatic telescopes yet constructed is less than two feet, and increase of size involves increased thickness of glass, and conse¬ quent absorption of light. The massiveness which is found necessary in the speculum in order to prevent flexure, is a serious inconvenience, as is also the necessity for frequent repolishing—an operation of great delicacy, as the slightest change in the form of the surface impairs the definition of the images. Both these defects have been to a certain extent re¬ medied by the introduction of glass specula, covered in front with a thin coating of silver. Glass is much more easily worked than specu¬ lum-metal (which is remarkable for its brittleness in casting), and has only one-third of its specific gravity. Silver is also much supe¬ rior to speculum-metal in reflecting power; and as often as it becomes tarnished it can be removed and renewed, without liability to change of form in the reflecting surface. 1 767. Measure of Brightness.—The brightness of a surface is most naturally measured by the amount of light per unit area of its image on the retina: and therefore varies directly as the amount of light which the surface sends to the 'pupil , and inversely as the apparent area of the surface. To avoid complications arising from the varying condition of the 1 The merits of silvered specula are fully set forth in a brochure published by Mr. Browning of the Minories, entitled A Plea for Reflectors. 966 VISION AND OPTICAL INSTRUMENTS. observer, we shall leave dilatation and contraction of the pupil out of account. When a body is looked at through a small pinhole in a card held close to the eye, it appears much darker than when viewed in the ordinary way; and in like manner images formed by optical instru¬ ments often furnish beams of light too narrow to fill the pupil. In all such cases it becomes necessary to distinguish between effective brightness and intrinsic brightness , the former being less than the latter in the same ratio in which the cross section of the beam which enters the pupil is less than the area of the pupil. We may correctly speak of the intrinsic brightness of a surface for a 'particular point of the pupil; and the effective brightness will in every case be the average value of the intrinsic brightness taken over the whole pupil. In the case of natural bodies viewed in the ordinary way, the distinction may be neglected, since they usually send light in sen¬ sibly equal amounts to all parts of the pupil. To obtain a numerical measure of intrinsic brightness, let us denote by e the area of the pupil, by s a small area on a surface directly facing towards the eye (or the foreshortened projection of a small area inclined to the line of vision), and by r the distance between e and s. Then the quantity of light q which s sends to e per unit time, varies jointly as the area e, the apparent area or real solid angle and the intrinsic brightness I. We may therefore write 5 = le ^ 2 = I s and if we put w for the solid angle L which the pupil subtends at s, we have q — I sw. The intrinsic brightness of a small area s is therefore measured by -T, where q denotes the quantity of light which s emits per unit time , in directions limited by the small solid angle of divergence w. 768. Applications.—One of the most obvious consequences is that surfaces appear equally bright at all distances in the same direction, provided that no light is stopped by the air or other intervening medium; for q and w both vary inversely as the square of the dis¬ tance. The area of the image formed on the retina in fact varies directly as the amount of light by which it is formed. Images formed by Lenses. —Let A B (Fig. 707) be an object, and a b its real image formed by the lens C D, whose centre is 0. Let BRIGHTNESS OF IMAGES. 967 S denote a small area at A, and Q the quantity of light which it sends to the lens; also let s denote the corresponding area of the image, and q the quantity of light which traverses it. Then q would be identical with Q if no light were stopped by the lens; the areas S, s, are directly as the squares of the conjugate focal distances 0 A, 0 a ; and the solid angles of divergence £2 and io, for Q and q , being the A solid angles subtended by the lens at A and a (for the plane angle cad in the figure is equal to the vertical angle Ca D), are inversely as the squares of the conjugate focal distances. We have accordingly ^ = | and S£2 = sw The intrinsic brightness g ~ of the image is therefore equal to the intrinsic brightness ^ of the object, except in Fig. 707.—Brightness of Image so far as light is stopped by the lens. Precisely similar reasoning applies to virtual images formed by lenses. 1 In the case of images formed by mirrors, £2 and to are the solid angles subtended by the mirror at the conjugate foci, and are in¬ versely as the squares of the distances from the mirror; while S and s are directly as the squares of the distances from the centre of cur¬ vature ; but these four distances are proportional (§ 407), so that the same reasoning is still applicable. If the mirror only reflects half the incident light, the image will have only half the intrinsic bright¬ ness of the object. If the pupil is filled with light from the image, the effective brightness will be the same as the intrinsic brightness thus computed. If this condition is not fulfilled, the former will be less than the latter. When the image is greatly magnified as compared with the object, the angle of divergence is greatly diminished in comparison with the angle which the lens or mirror subtends at the object, and often becomes so small that only a small part of the pupil is utilized This is the explanation of the great falling off of light which is ob- 1 From the fundamental formula for refraction at a spherical surface, n =—A ■- F ’ OF AQ (Parkinson's Optics, Art. 27), where Q and F are conjugate foci, 0 Q, OF their distances from the centre of curvature, and A Q, A F their distances from the refracting surface, we n r\ have at once, by squaring both sides, p?= - • The product s w is accordingly diminished s w in the ratio of p? to 1 at the first surface of a lens, and increased again to its original value at the second surface. The conclusions deduced in the text will therefore remain true when the thickness of the lens is taken into account. 968 VISION AND OPTICAL INSTRUMENTS. served in the use of high magnifying powers, both in microscopes and telescopes. 769. Brightness of Image in a Telescope.—It has been already pointed out (§ 762) that in most forms of telescope (the Galilean being an exception), there is a certain position, a little behind the eye-piece, at which a well-defined bright spot is formed upon a screen held there while the telescope is directed to any distant source of light. It has also been pointed out that this spot is the image, formed by the eye-piece, of the opening which is filled by the object- glass, and that the magnifying power of the instrument is the ratio of the size of the object-glass to the/size of this bright spot. Let s denote the diameter of the bright spot, o the diameter of the object-glass, e the diameter of the pupil of the eye; then — is the linear magnifying power. We shall first consider the case in which the spot exactly covers the pupil of the observer’s eye, so that s = e. Then the whole light which traverses the telescope from a distant object enters the eye; and if we neglect the light stopped in the telescope, this is the whole light sent by the object to the object-glass, and is (-j) 2 times that which would be received by the naked eye. The magni¬ fication of apparent area is , which, from the equality of s and e, is the same as the increase of total light. The brightness is therefore the same as to the naked eye. Next, let s be greater than 6, and let the pupil occupy the central part of the spot. Then, since the spot is the image of the object- glass, we may divide the object-glass into two parts—a central part whose image coincides with the pupil, and a circumferential part whose image surrounds the pupil. All rays from the object which traverse the central part, traverse its image, and therefore enter the pupil; whereas rays traversing the circumferential part of the object- glass, traverse the circumferential part of the image, and so are wasted. The area of the central part (whether of the object-glass or of its image) is to the whole area as e 2 : s 2 ; and the light which the object sends to the central portion, instead of being times that which would be received by the naked eye, is only times. But is the magnification of apparent area. Hence the bright¬ ness is the same as to the naked eye. In these two cases, effective and intrinsic brightness are the same. BRIGHTNESS OF IMAGE IN TELESCOPE 969 Lastly (and this is by far the most common case in practice), let s be less than e. Then no light is wasted, but the pupil is not filled. The light received is times that which the naked eye would receive; and the magnification of apparent area is (|) 2 - The e ^ ec_ tive brightness of the image, is to the brightness of the object to the naked eye, as {^-) 2 : (f) 2 ; that is, as s 2 : e 2 ; that is, as the area of the bright spot to the whole area of the pupil. To correct for the light stopped by reflection and imperfect trans¬ parency, we have simply to multiply the result in each case by a proper fraction, expressing the ratio of the transmitted to the incident light. This ratio, for the central parts of the field of view, is about 0*85 in the best achromatic telescopes. In such telescopes, therefore, the brightness of the image cannot exceed 0 85 of the brightness of the object to the naked eye. It will have this precise value, when the magnifying power is equal to or less than and from this point upwards will vary inversely as the square of the linear mag¬ nification. The same formulae apply to reflecting telescopes, o denoting now the diameter of the large speculum which serves as objective; but the constant factor is usually considerably less than 0 8 5. It may be accepted as a general principle in optics, that while it is possible, by bad focussing or instrumental imperfections, to obtain a confused image whose brightness shall be intermediate between the brightest and the darkest parts of the object, it is impossible , by any optical arrangement whatever, to obtain an image whose brightest part shall surpass the brightest part of the object. 770. Brightness of Stars.—There is one important case in which the foregoing rules regarding the brightness of images become nuga¬ tory. The fixed stars are bodies which subtend at the earth angles smaller than the minimum visibile, but which, on account of their excessive brightness, appear to have a sensible angular diameter. This is an instance of irradiation, a phenomenon manifested by all bodies of excessive brightness, and consisting in an extension of their apparent beyond their actual boundary. What is called, in popular language, a bright star, is a star which sends a large total amount of light to the eye. Denoting by a the ratio of the transmitted to the whole incident light, a ratio which, as we have seen, is about 0*85 in the most 970 VISION AND OPTICAL INSTRUMENTS. favourable cases, and calling the light which a star sends to the naked eye unity, the light perceived in its image will be a (j) 2 , or a X square of linear magnification, if the bright spot is as large as the pupil. When the eye-piece is changed, increase of power dimin¬ ishes the size of the spot, and increases the lighf received by the eye, until the spot is reduced to the size of the pupil. After this, any further magnification has no effect on the quantity of light received, its constant value being a (°) 2 . The value of this last expression, or rather the value of a o 2 , is the measure of what is called the space-penetrating power of a telescope; that is to say, the power of rendering very faint stars visible; and it is in this respect that telescopes of very large aperture, notably the great reflector of Lord Rosse, are able to display their great superiority over instruments of moderate dimensions. We have seen that the total light in the visible image of a star remains unaltered, by increase of power in the eye-piece above a certain limit. But the visibility of faint stars in a telescope is pro¬ moted by darkening the back-ground of sky on which they are seen. Now the brightness of this back-ground varies directly as s 2 , or in¬ versely as the square of the linear magnification {s being supposed less than e). Hence it is advantageous, in examining very faint stars, to employ eye-pieces of sufficient power to render the bright spot much smaller than the pupil of the eye. 771. Images on a Screen.—Thus far we have been speaking of the brightness of images as viewed directly. Images cast upon a screen are, as a matter of fact, much less brilliant. Their brightness depends greatly on the nature of the screen, and can never exceed the bright¬ ness which the surface of the screen would exhibit if held very near the source of light. When a condensing lens is used to collect the rays of a lamp, an eye placed at the conjugate focus sees the whole lens full of light of uniform brightness, which, neglecting reflection and absorption, can be shown to be the same as that of the flame itself. 1 The illumination of a screen placed in the focus, is therefore jointly proportional to the solid angle which the lens subtends at the focus, and to the brightness of the flame; and is the same as if the screen were directly illuminated by the flame, 1 This is strictly true of intrinsic brightness, which is all that our reasoning requires. It is true for effective brightness, if the image is large enough to cover the pupil, and if the lens is at a proper distance for distinct vision. CROSS-WIRES OF TELESCOPES. 971 at a distance at which the flame itself would subtend the same solid angle. 772. Cross-wires of Telescopes.—We have described in § 743 a a mode of marking the place of a real image by means of a cross of threads. When telescopes are employed to assist in the measure¬ ment of angles, a contrivance of this kind is almost always intro¬ duced. A cross of silkworm threads, in instruments of low power, or of spider threads in instruments of higher power, is stretched across a metallic frame just in front of the eye-piece. The observer must first adjust the eye-piece for distinct vision of this cross, and must then (in the case of theodolites and other surveying instru¬ ments) adjust the distance of the object-glass until the object which is to be observed is also seen distinctly in the telescope. The image of the object will then be very nearly in the plane of the cross. If it is not exactly in the plane, parallactic displacement will be observed when the eye is shifted, and this must be cured by slightly altering the distance of the object-glass. When the adjustment has been completed, the cross always marks one definite point of the object, however the eye be shifted. This coincidence will not be disturbed by pushing in or pulling out the eye-piece; for the frame which carries the cross is attached to the body of the telescope, and the coincidence of the cross with a point of the image is real, so that it could be observed by the naked eye, if the eye-piece were re¬ moved. The adjustment of the eye-piece merely serves to give dis¬ tinct vision, and this will be obtained simultaneously for both the cross and the object. 773. Line of Collimation.—The employment of cross-wires (as these crossing threads are called) enormously increases our power of making accurate observations of direction, and constitutes one of the greatest advantages of modern over ancient instruments. The line which is regarded as the line of sight, or as the direction in which the telescope is pointed, is called the line of collimation. If we neglect the curvature of rays due to atmospheric refraction, we may define it as the line joining the cross to the object whose image falls on it. More rigorously, the line of collimation is the line joining the cross to the optical centre of the object-glass. When it is desired to adjust the line of collimation,—for example, to make it truly perpendicular to the horizontal axis on which the telescope is mounted, the adjustment is performed by shifting the frame which carries the wires, slow-motion screws being provided for this purpose. 972 VISION AND OPTICAL INSTRUMENTS. Telescopes for astronomical observation are often furnished with a number of parallel wires, crossed by one or two in the transverse direction; and the line of collimation is then defined by reference to an imaginary cross, which is the centre of mean position of all the actual crosses. 774. Micrometers.—Astronomical micrometers are of various kinds, some of them serving for measuring the angular distance between two points in the same field of view, and others for measuring their apparent direction from one another. They generally consist of spider threads placed in the principal focus of the object-glass, so as to be in the same plane as the images of celestial objects, one or more of the threads being movable by means of slow-motion screws, furnished with graduated circles, on which parts of a turn can be read off. One of the commonest kinds consists of two parallel threads, which can thus be moved to any distance apart, and can also be turned round in their own plane. CHAPTEE LXII. DISPERSION. STUDY OF SPECTRA. 775. Newtonian Experiment.—In the chapter on refraction, we have postponed the discussion of one important phenomenon by which it is usually accompanied, and which we must now proceed to explain. The following experiment, which is due to Sir Isaac Newton, will furnish a fitting introduction to the subject. On an extensive background of black, let three bright strips be laid in line, as in the left-hand part of Fig. 708, and looked at through a prism with its refracting edge parallel to the strips. We Fig. 708.—Spectra of White and Coloured Strips. shall suppose the edge to be upward, so that the image is raised above the object. The images, as represented in the right-hand part of Fig. 708, will have the same horizontal dimensions as the strips, but will be greatly extended in the vertical direction; and each image, instead of having the uniform colour of the strips from which it is derived, will be tinted with a gradual succession of colours from top to bottom. Such images are called spectra. If one of the strips (the middle one in the figure) be white, its spectrum will contain the following series of colours, beginning at the top: violet, blue, green, yellow , orange, red. 974 DISPERSION. STUDY OF SPECTRA. If one of the strips be blue (the left-hand one in the figure), its image will present bright colours at the upper end; and these will be identical with the colours adjacent to them in the spectrum oi white. The colours which form the lower part of the spectrum of white will either be very dim and dark in the spectrum of blue, or will be wanting altogether, being replaced by black. If the other strip be red, its image will contain bright colours at the lower or red end, and those which belong to the upper end of the spectrum of white will be dim or absent. Every colour that occurs in the spectrum of blue or of red will also be found, and in the same horizontal line, in the spectrum of white. If we employ other colours instead of blue or red, we shall obtain analogous results; every colour will be found to give a spectrum which is identical with part of the spectrum of white, both as regards colour and position, but not generally as regards brightness. We may occasionally meet with a body whose spectrum consists only of one colour. The petals of some kinds of convolvulus give a spectrum consisting only of blue, and the petals of nasturtium give only red. 776. Composite Nature of Ordinary Colours.—This experiment shows that the colours presented by the great majority of natural bodies are composite. When a colour is looked at with the naked eye, the sensation experienced is the joint effect of the various elementary colours which compose it. The prism serves to resolve the colour into its components, and exhibit them separately. The experiment also shows that a mixture of all the elementary colours in proper pro¬ portions produces white. 777. Solar Spectrum.—The coloured strips in the foregoing experi¬ ment may be illuminated either by daylight or by any of the ordinary sources of artificial light. The former is the best, as gas-light and candle-light are very deficient in blue and violet rays. Colour, regarded as a property of a coloured (opaque) body, is the power of selecting certain rays and reflecting them either exclusively or in larger proportion than others. The spectrum presented by a body viewed by reflected light, as ordinary bodies are, can thus only consist of the rays, or a selection of the rays, by which the body is illuminated. A beam of solar light can be directly resolved into its constituents by the following experiment, which is also due to Newton, and was the first demonstration of the composite character of solar light. SOLAR SPECTRUM. 975 Let a beam of sun-light be admitted through a small opening into a dark room. If allowed to fall normally on a white screen, it pro¬ duces (§ 683) a round white spot, which is an image of the sun. Now let a prism be placed in its path edge-downwards, as in Fig. 709; the Fig. 709.—Solar Spectrum by Newton’s Method. beam will thus be deflected upwards, and at the same time resolved into its component colours. The image depicted on the screen will be a many-coloured band, resembling the spectrum of white described in § 775. It will be of uniform width, and rounded off at the ends, being in fact built up of a number of overlapping discs, one for each kind of elementary ray. It is called the solar spectrum. The rays which have undergone the greatest deviation are the violet. They occupy the upper end of the spectrum in the figure. Those which have undergone the least deviation are the red. Of all visible rays, the violet are the most, and the red the least refrangible; and the analysis of light into its components by means of the prism is due to difference of refrangibility. If a small opening is made in the screen, so as to allow rays of only one colour to pass, it will be found, 976 DISPERSION. STUDY OF SPECTRA. on transmitting these through a second prism behind the screen, as in Fig. 709, that no further analysis can be effected, and the whole of the image formed by receiving this transmitted light on a second screen will be of this one colour. 778. Mode of obtaining a Pure Spectrum.—The spectra obtained by the methods above described are built up of a number of overlapping images of different colours. To prevent this overlapping, and obtain each elementary colour pure from all admixture with the rest, we must in the first place employ as the object for yielding the images a very narrow line; and in the second place we must take care that the images which we obtain of this line are not blurred, but have the greatest possible sharpness. A spectrum possessing these character¬ istics is called pure. The simplest mode of obtaining a pure spectrum consists in looking through a prism at a fine slit in the shutter of a dark room. The edges of the prism must be parallel to the slit, and its distance from the slit should be five feet or upwards. The observer, placing his eye close to the prism, will see a spectrum; and he should rotate the prism on its axis until he has brought this spectrum to its smallest angular distance from the real slit, of which it is the image. Let E (Fig. 710) be the position of the eye, S that of the slit. Then the extreme red and violet images of the slit will be seen at R, Y, at distances from the prism sensibly equal to the real distance of S (§ 731 b); and the other images, which compose the remainder of the spectrum, will occupy posi¬ tions between R and Y The spectrum, in this mode of operat¬ ing, is virtual. To obtain a real spectrum in a state of purity, a convex lens must be employed. Let the lens L (Fig. 711) be first placed in such a position as to throw a sharp image of the slit S upon a screen at I. Next let a prism P be introduced between the lens and screen, and rotated on its axis till the position of minimum devia¬ tion is obtained, as shown by the movements of the impure spectrum which travels about the walls of the room. Then if the screen be moved into the position RY, its distance from the prism being the Fig. 710.—Arrangement for seeing a pure Spectrum. THE PURE SPECTRUM. 977 same as before, a pure spectrum will be depicted upon it. A similar result can be obtained by placing the prism between the lens and the slit, but the adjustments are rather more troublesome. Direct sun-light, or sun-light reflected from a mirror placed outside the shutter, is necessary for this experiment, as sky-light is not suffi¬ ciently powerful. It is usual, in experiments of this kind, to em¬ ploy a movable mirror called a heliostat, by means of which the light can be reflected in any required direction. Sometimes the move¬ ments of the mirror are obtained by hand; sometimes by an ingeni¬ ous clock-work arrangement, which causes the reflected beam to keep its direction unchanged notwithstanding the progress of the sun through the heavens. The advantage of placing the prism in the position of minimum deviation is two-fold. First, the adjustments are facilitated by the equality of conjugate focal distances, which subsists in this case and in this only. Secondly and chiefly, this is the only position in which the images are not blurred. In any other position it can be shown 1 that a small cone of homogeneous incident rays is no longer a cone (that is, its rays do not accurately pass through one point) after transmission through the prism. The method of observation just described was employed by Wollaston, in the earliest observations of a pure spectrum ever obtained. Fraunhofer, a few years later, independently devised the same method, and carried it to much greater perfection. Instead of looking at the virtual image with the naked eye, he viewed it through a telescope, which greatly magnified it, and revealed several features never before detected. The prism and telescope were at a distance of 24 feet from the slit. 1 Helmholtz, Physiological Optics. Second Part, § 19. 63 978 DISPERSION. STUDY OF SPECTRA. 779. Dark Lines in the Solar Spectrum.—When a pure spectrum of solar light is examined by any of these methods, it is seen to be traversed by numerous dark lines, constituting, if we may so say, dark images of the slit. Each of these is an indication that a par¬ ticular kind of elementary ray is wanting 1 in solar light. Every elementary ray that is present gives its own image of the slit in its own peculiar colour; and these images are arranged in strict con¬ tiguity, so as to form a continuous band of light passing by perfectly gradual transitions through the whole range of simple colour, except at the narrow intervals occupied by the dark lines. Fig. 1, Plate III., is a rough representation of the appearance thus presented. If the slit is illuminated by a gas flame, or by any ordinary lamp, instead of by solar light, no such lines are seen, but a perfectly continuous spectrum is obtained. The dark lines are therefore not characteristic of light in general, but only of solar light. Wollaston saw and described some of the more conspicuous of them. Fraunhofer counted abgut 600, and marked the places of 354 upon a map of the spectrum, distinguishing some of the more con¬ spicuous by the names of letters of the alphabet, as indicated in fig. 1. These lines are constantly referred to as reference marks for the accurate specification of different portions of the spectrum. They always occur in precisely the same places as regards colour, but do not retain exactly the same relative distances one from another, when prisms of different materials are employed, different parts of the spectrum being unequally expanded by different refracting sub- tances. 2 The inequality, however, is not so great as to introduce any difficulty in the identification of the lines. The dark lines in the solar spectrum are often called Fraunhofer’s lines. Fraunhofer himself called them the “fixed lines.” 780. Invisible Rays of the Spectrum.—The brightness of the solar spectrum, however obtained, is by no means equal throughout, but is greatest between the dark lines D and E; that is to say, in the yellow and the neighbouring colours orange and light green; and falls off gradually on both sides. The heating effect upon a small thermometer or thermopile in¬ creases in going from the violet to the red, and still continues to increase for a certain distance beyond the visible spectrum at the red end. Prisms and lenses of rock-salt should be employed for this 1 Probably not absolutely wanting, but so feeble as to appear black by contrast. 2 This property is called the irrationality of dispersion. PHOSPHORESCENCE. 979 investigation, as glass largely absorbs the invisible rays which lie beyond the red. When the spectrum is thrown upon the sensitized paper employed in photography, the action is very feeble in the red, strong in the blue and violet, and is sensible to a great distance beyond the violet end. When proper precautions are taken to insure a very pure spectrum, the photograph reveals the existence of dark lines, like those of Fraunhofer, in the invisible ultra-violet portion of the spec¬ trum. The strongest of these have been named L, M, N, O, P. 781. Phosphorescence and Fluorescence.—There are some sub¬ stances which, after being exposed in the sun, are found for a long time to appear self-luminous when viewed in the dark, and this Fig. 712.—Becquerel’s Phosphoroscope. without any signs of combustion or sensible elevation of temperature. Such substances are called phosphorescent. Sulphuret of calcium and sulphuret of barium have long been noted for this property, and have hence been called respectively Canton’s phosphorus, and Bologna 980 DISPERSION. STUDY OF SPECTRA. ■phosphorus. The phenomenon is chiefly due to the action of the violet and ultra-violet portion of the sun’s rays. More recent investigations have shown that the same property exists in a much lower degree in an immense number of bodies, their phosphorescence continuing, in most cases, only for a fraction of a second after their withdrawal from the sun’s rays. E. Becquerel has contrived an instrument, called the phosphoroscope, which is ex¬ tremely appropriate for the observation of this phenomenon. It is represented in Fig. 712. Its most characteristic feature is a pair of rigidly connected discs (Fig. 713), each pierced with four openings, those of the one being not opposite but midway between those of the other. This pair of discs can be set in very rapid rotation by means of a series of wheels and pinions. The body to be examined is attached to a fixed stand between the two discs, so that it is alternately exposed on opposite sides as the discs rotate. One side is turned towards the sun, and the other towards the observer, who accordingly only sees the body when it is not exposed to the sun’s rays. The cylindrical case within which the discs revolve, is fitted into a hole in the shutter of a dark room, and is pierced with an opening on each side exactly opposite the position in which the body is fixed. The body, if not phosphorescent, will never be seen by the observer, as it is always in darkness except when it is hidden by the intervening disc. If its phosphorescence lasts as long as an eighth part of the time of one rotation, it will become visible in the darkness. Nearly all bodies, when thus examined, show traces of phosphores¬ cence, lasting, however, in some cases, only for a ten-thousandth of a second. The phenomenon of fluorescence , which is illustrated in Plate II. accompanying § 618, appears to be essentially identical with phos¬ phorescence. The former name is applied to the phenomenon, if it is observed while the body is actually exposed to the source of light, the latter to the effect of the same kind, but usually less intense, which is observed after the light from the source is cut off. Both forms of the phenomenon occur in a strongly-marked degree in the same bodies. Canary-glass, which is coloured with oxide of uranium, is FLUORESCENCE. 981 a very convenient material for the exhibition of fluorescence. A thick piece of it, held in the violet or ultra-violet portion of the solar spectrum, is filled to the depth of from -§■ to \ of an inch with a faint nebulous light. A solution of sulphate of quinine is also frequently employed for exhibiting the same effect, the luminosity in this case being bluish. If the solar spectrum be thrown upon a screen freshly washed with sulphate of quinine, the ultra-violet portion will become visible by fluorescence; and if the spectrum be very pure, the pre¬ sence of dark lines in this portion will be detected. The light of the electric lamp is particularly rich in ultra-violet rays, this portion of its spectrum being much longer than in the case of solar light, and about twice as long as the spectrum of luminous rays. Prisms and lenses of quartz should be employed for this pur¬ pose, as this material is specially transparent to the highly-refrangible rays. Flint-glass prisms, however, if of good quality, answer well in operating on solar light. The luminosity produced by fluorescence has sensibly the same tint in all parts of the spectrum in which it occurs, and depends upon the fluorescent substance employed. Pris¬ matic analysis is not necessary to the exhibition of fluorescence. The phenomenon is very conspicuous when the electric discharge of a Holtz’s machine or a Ruhmkorff’s coil is passed near fluorescent substances, and it is faintly visible when these substances are examined in bright sunshine. The light emitted by a fluorescent substance is found by analysis not to be homogeneous, but to consist of rays having a wide range of refrangibility. The ultra-violet rays, though usually styled invisible, are not altogether deserving of this title. By keeping all the rest of the spectrum out of sight, and carefully excluding all extraneous light, the eye is enabled to perceive these highly refrangible rays. Their colour is described as lavender-gray or bluish white, and has been attributed, with much appearance of probability, to fluorescence of the retina. The ultra-red rays, on the other hand, are never seen; but this may be owing to the fact, which has been established by experiment, that they are largely, if not entirely, absorbed before they can reach the retina. 782. Recomposition of White Light.—The composite nature of white light can be established by actual synthesis. This can be done in several ways. 1 . If a second prism, precisely similar to the first, but with its refracting edge turned the contrary way, is interposed in the path of 982 DISPERSION. STUDY OF SPECTRA. the coloured beam, very near its place of emergence from the first prism, the deviation produced by the second prism will be equal and opposite to that produced by the first, the two prisms will produce the effect of a parallel plate, and the image on the screen will be a white spot, nearly in the same position as if the prisms were re¬ moved. 2 . Let a convex lens (Fig. 714) be interposed in the path of the coloured beam, in such a manner that it receives all the rays, and Fig. 714.—Recomposition by Lens. that the screen and the prism are at conjugate focal distances. The imaofe thus obtained on the screen will be white, at least in its cen- tral portions. 3. Let a number of plane mirrors be placed so as to receive the successive coloured rays, and to reflect them all to one point of a Fig. 715.—Recomposition by Mirrors. screen, as in Fig. 715. The bright spot thus formed will be white, or approximately white. More complete information respecting the mixture of colours will be given in the next chapter. SPECTROSCOPE. 983 783. Spectroscope.—When we have obtained a pure spectrum by any of the methods above indicated, we have in fact effected an analysis of the light with which the slit is illuminated. In recent years, many forms of apparatus have been constructed for this pur¬ pose, under the name of spectroscopes. A spectroscope usually contains, besides a slit, a prism, and a telescope (as in Fraunhofer’s method of observation), a convex lens called a collimator , which is fixed between the prism and the slit, at the distance of its principal focal length from the latter. The effect of this arrangement is, that rays from any point of the slit emerge parallel, as if they came from a much larger slit (the virtual image of the real slit) at a much greater distance. The prism (set at minimum deviation) forms a virtual image of this image at the same distance, but in a different direction, on the principle of Fig. 711. Fig. 716.—Spectroscope. To this second virtual image the telescope is directed, being focussed as if for a very distant object. Fig. 716 represents a spectroscope thus constructed. The tube of 984 DISPERSION. STUDY OF SPECTRA. the collimator is the further tube in the figure, the lens being at the end of the tube next the prism, while at the far end, close to the lamp flame, there is a slit (hot visible in the figure) consisting of an opening between two parallel knife-edges, one of which can be moved to or from the other by turning a screw. The knife-edges must be very true, both as regards straightness and parallelism, as it is often necessary to make the slit exceedingly narrow. The tube on the left hand is the telescope, furnished with a broad guard to screen the eye from extraneous light. The near tube, with a candle opposite its end, is for purposes of measurement. It contains, at the end next the candle, a scale of equal parts, engraved or photographed on glass. At the other end of the tube is a collimating lens, at the distance ot its own focal length from the scale; and the collimator is set so that its axis and the axis of the telescope make equal angles with the near face of the prism. The observer thus sees in the telescope, by reflection from the surface of the prism, a magnified image of the scale, serving as a standard of reference for assigning the positions of the lines in any spectrum which may be under examination. This arrangement affords great facilities for rapid observation. Another plan is, for the arm which carries the telescope to be movable round a graduated circle, the telescope being furnished with cross-wires, which the observer must bring into coincidence with any line whose position he desires to measure. Arrangements are frequently made for seeing the spectra of two different sources of light in the same field of view, one half of the length of the slit being illuminated by the direct rays of one of the sources, while a reflector, placed opposite the other half of the slit, supplies it with reflected light derived from the other source. This method should always be employed when there is a question as to the exact coincidence of lines in the two spectra. The re- Mg. 7i7. flector is usually an equilateral prism. The light enters Reflecting Prism. J . r n , ° normally at one of its faces, is totally reflected at another, and emerges normally at the third, as in the annexed sketch (Fig. 717, where the dotted line represents the path of a ray. A one-prism spectroscope is amply sufficient for the ordinary pur¬ poses of chemistry. For some astronomical applications a much greater dispersion is required. This is attained by making the light pass through a number of prisms in succession, each being set in the proper position for giving minimum deviation to the rays which have USE OF COLLIMATOR. 985 passed through its predecessor. Fig. 718 represents the ground plan of such a battery of prisms, and shows the gradually increasing width of a pencil as it passes round the series of nine prisms on its way from the collimator to the telescope. The prisms are usual¬ ly connected by a special ar¬ rangement, which enables the observer, by a single movement, to bring all the prisms at once into the proper position for giving minimum deviation to the parti¬ cular ray under examination, a po¬ sition which differs considerably for rays of different refr angibili ties. 784. Use of Collimator.—The introduction of a collimating lens, to be used in conjunction with a prism and observing telescope, is due to Professor Swan. 1 Fraun¬ hofer employed no collimator; but his prism was at a distance of 24 feet from the slit, whereas a distance of less than 1 foot suffices when a collimator is used. It is obvious that homogeneous light, coming from a point at the distance of a foot, and falling upon the whole of one face of a prism —say an inch in width, cannot all have the incidence proper for minimum deviation. Those rays which very nearly fulfil this con¬ dition, will concur in forming a tolerably sharp image, in the posi¬ tion which we have already indicated. The emergent rays taken as a whole, do not diverge from any one point, but are tangents to a virtual caustic (§ 714). An eye receiving any portion of these rays, will see an image in the direction of a tangent from the eye to the caustic; and this image will be the more blurred as the deviation is further from the minimum. When the naked eye is employed, and the prism is so adjusted that the centre of the pupil receives rays of minimum deviation, a distance of five or six feet between the prism and slit is sufficient to give a sharp image; but if we employ an observing telescope whose object-glass is five times larger in diameter than the pupil of the eye, we must increase the distance between the 1 Trans. Roy. Soc. Edinburgh, 1847 and 1856. Fig. 718.—Train of Prisms. 986 DISPERSION. STUDY OF SPECTRA. prism and slit five-fold to obtain equally good definition. A colli¬ mating lens, if achromatic and of good quality, gives the advantage of good definition without inconvenient length. When exact measures of deviation are required, it confers the further advantage of altogether dispensing with a very troublesome correction for parallax. 785. Different Kinds of Spectra.—The examination of a great variety of sources of light has shown that spectra may be divided into the following classes:— 1. The solar spectrum is characterized, as already observed, by a definite system of dark lines interrupting an otherwise continuous succession of colours. The same system of dark lines is found in the spectra of the moon and planets, this being merely a consequence of the fact that they shine by the reflected light of the sun. The spectra of the fixed stars also contain systems of dark lines, which are different for different stars. 2. The spectra of incandescent solids and liquids are completely continuous, containing light of all refrangibilities from the extreme red to a higher limit depending on the temperature. 3. Flames not containing solid particles in suspension, but merely emitting the light of incandescent gases, give a discontinuous spec¬ trum, consisting of a finite number of bright lines. The continuity of the spectrum of a gas or candle flame, arises from the fact that nearly all the light of the flame is emitted by incandescent particles of solid carbon,—particles which we can easily collect in the form of soot. When a gas-flame is fed with an excessive quantity of air, as in Bunsen’s burner, the separation of the solid particles of carbon from the hydrogen with which they were combined, no longer takes place; the combustion is purely gaseous, and the spectrum of the flame is found to consist of bright lines. When the electric light is produced between metallic terminals, its spectrum contains bright lines due to the incandescent vapour of these metals, together with other bright lines due to the incandescence of the oxygen and nitro¬ gen of the air. When it is taken between charcoal terminals, its spectrum is continuous; but if metallic particles be present, the bright lines due to their vapours can be seen as well. The spectrum of the electric discharge in a Geissler’s tube consists of bright lines characteristic of the gas contained in the tube. 786. Spectrum Analysis.—As the spectrum exhibited by a com¬ pound substance when subjected to the action of heat, is frequently SPECTRUM ANALYSIS. 987 found to be identical with the spectrum of one of its constituents, or to consist of the spectra of its constituents superimposed, 1 the spec¬ troscope affords an exceedingly ready method of performing qualita¬ tive analysis. If a salt of a metal which is easily volatilized is introduced into a Bunsen lamp-flame, by means of a loop of platinum wire, the bright lines which form the spectrum of the metal will at once be seen in a spectroscope directed to the flame; and the spectrum of the Bunsen flame itself is too faint to introduce any confusion. For those metals which require a higher temperature to volatilize them, electric discharge is usually employed. Geissler’s tubes are com¬ monly used for gases. Plate III. contains representations of the spectra of several of the more easily volatilized metals, as well as of phosphorus and hydro¬ gen; and the solar spectrum is given at the top for comparison. The bright lines of some of these substances are precisely coincident with some of the dark lines in the solar spectrum. The fact that certain substances when incandescent give definite bright lines, has been known for many years, from the researches of Brewster, Herschel, Talbot, and others; but it was for a long time thought that the same line might be produced by different sub¬ stances, more especially as the bright yellow line of sodium was often seen in flames in which that metal was not supposed to be present. Professor Swan, having ascertained that the presence of the 2,500,000th part of a grain of sodium in a flame was sufficient to produce it, considered himself justified in asserting, in 1850, that this line was always to be taken as an indication of the presence of sodium in larger or smaller quantity. But the greatest advance in spectral analysis was made by Bunsen and Kirchhoff, who, by means of a four-prism spectroscope, obtained accurate observations of the positions of the bright lines in the spectra of a great number of substances, as well as of the dark lines in the solar spectrum, and called attention to the identity of several of the latter with several of the former. Since the publication of their researches, the spectroscope has come into general use among chemists, and has already led to the discovery of four new metals, cesium, rubidium, thallium, and indium. 787. Reversal of Bright Lines. Analysis of the Sun’s Atmosphere. 1 These appear to be merely examples of the dissociation of the elements of a chemical compound at high temperatures. 988 DISPERSION. STUDY OF SPECTRA. —It may seem surprising that, while incandescent solids and liquids are found to give continuous spectra, containing rays of all refran- gibilities, the solar spectrum is interrupted by dark lines indicating the absence or relative feebleness of certain elementary rays. It seems natural to suppose that the deficient rays have been removed by selective absorption, and this conjecture was thrown out long since. But where and how is this absorption produced? These questions have now received an answer which appears completely satisfactory. According to the theory of exchanges, which has been explained in connection with the radiation of heat (§ 312 c, 326), every sub¬ stance which emits certain kinds of rays to the exclusion of others, absorbs the same kind which it emits; and when its temperature is the same in the two cases compared, its emissive and absorbing power are precisely equal for any one elementary ray. When an incandescent vapour, emitting only rays of certain definite refrangibilities, and therefore having a spectrum of bright lines, is interposed between the observer and a very bright source of light, giving a continuous spectrum, the vapour allows no rays of its own peculiar kinds to pass; so that the light which actually comes to the observer Consists of transmitted rays in which these particular kinds are wanting, together with the rays emitted by the vapour itself, these latter being of precisely the same kind as those which it has refused to transmit. It depends on the relative brightness of the two sources whether these particular rays shall be on the whole in excess or defect as compared with the rest. If the two sources are at all comparable in brightness, these rays will be greatly in excess, inasmuch as they constitute the whole light of the one, and only a minute fraction of the light of the other; but the light of the electric lamp, or of the lime-light, is usually found sufficiently powerful to produce the contrary effect; so that if, for example, a spirit-lamp with salted wick is interposed between the slit of a spectroscope and the electric light, the bright yellow line due to the sodium appears black by contrast with the much brighter back-ground which belongs to the continuous spectrum of the charcoal points. By employing only some 10 or 15 cells, a light may be obtained, the yellow portion of which, as seen in a one-prism spectroscope, is sensibly equal in brightness to the yellow line of the sodium flame, so that this line can no longer be separately detected, and the appearance is the same whether the sodium flame be interposed or removed. The dark lines in the solar spectrum would therefore be accounted TELESPECTROSCOPE. 989 for by supposing that the principal portion of the sun's light comes from an inner stratum which gives a continuous spectrum, and that a layer external to this contains vapours which absorb particular rays, and thus produce the dark lines. The stratum which gives the continuous spectrum might be solid, liquid, or even gaseous, for the experiments of Frankland and Lockyer have shown that, as the pressure of a gas is increased, its bright lines broaden out into bands, and that the bands at length become so wide as to join each other and form a continuous spectrum 1 Hydrogen, potassium, sodium, calcium, barium, magnesium, zinc, iron, chromium, cobalt, nickel, copper, and manganese have all been proved to exist in the sun by the accurate identity of position of their bright lines with certain dark lines in the sun’s spectrum. The strong line D, which in a good instrument is seen to consist of two lines near together, is due to sodium; and the lines C and F are due to hydrogen. No less than 450-of the solar dark lines have been identified with bright lines of iron. 788. Telespectroscope. Solar Sierra.—For astronomical investiga¬ tions, the spectroscope is usually fitted to a telescope, and takes the place of the eye-piece, the slit being placed in the principal focus of the object-glass, so that the image is thrown upon it, and the light which enters it is the light which forms one strip (so to speak) of the image, and which therefore comes from one strip of the object. A telescope thus equipped is called a telespectroscope. Extremely interesting results have been obtained by thus subjecting to exami¬ nation a strip of the sun’s edge, the strip being sometimes tangential to the sun’s disk, and sometimes radial. When the former arrange¬ ment is adopted, the appearance presented is that depicted in fig. 2, Plate III., consisting of a few bright lines scattered through a back¬ ground of the ordinary solar spectrum. The bright lines are due to an outer layer called the sierra or chromosphere , which is thus proved to be vaporous. The ordinary solar spectrum which accompanies it, is due to that part of the sun from which most of our light is derived. This part is called the photosphere, and if not solid or liquid, it must consist of vapour so highly compressed that its pro¬ perties approximate to those of a liquid. When the slit is placed radially, in such a position that only a 1 The gradual transition from a spectrum of bright lines to a continuous spectrum may be held to be an illustration of the continuous transition which can be effected from the condition of ordinary gas to that of ordinary liquid (§ 246 a). 990 DISPERSION. STUDY OF SPECTRA. small portion of its length receives light from the body of the sun, the spectra of the photosphere and chromosphere are seen in imme¬ diate contiguity, and the bright lines in the latter (notably those of hydrogen, No. 14, Plate III.) are observed to form continuations of some of the dark lines of the former. The chromosphere is so much less bright than the photosphere, that, until a few years since, its existence was never revealed except during total eclipses of the sun, when projecting portions of it (from which it derives its name of sierra) were seen extending beyond the dark body of the moon. The spectrum of these projecting portions, which have been variously called “prominences,” “red flames,” and “rose-coloured protuberances,” was first observed during the “Indian eclipse” of 1868, and was found to consist of bright lines, including those of hydrogen. From their excessive brightness, M. Janssen, who was one of the observers, expressed confidence that he should be able to see them in full sunshine; and the same idea had been already conceived and published by Mr. Lockyer. The expectation was shortly afterwards realized by both these observers, and the chromo¬ sphere has ever since been an object of daily observation. The visi¬ bility of the chromosphere lines in full sunshine, depends upon the principle that, while a continuous spectrum is extended, and there¬ fore made fainter, by increased dispersion, a bright line in a spectrum is not sensibly broadened, and therefore loses very little of its in¬ trinsic brightness (§ 791). Very high dispersion, attainable only by the use of a long train of prisms, is necessary for this purpose. Still more recently, by opening the slit to about the average width of the prominence-region, as measured on the iinage of the sun which is thrown on the slit, it has been found possible to see the whole of an average-sized prominence at one view. This will be understood by remembering that a bright line as seen in a spectrum is a mono¬ chromatic image of the illuminated portion of the slit, or when a tele¬ spectroscope is used, as in the present case, it is a monochromatic image of one strip of the image formed by the object-glass, namely, that strip which coincides with the slit. If this strip then contains a prominence in which the elementary rays Cand F (No. 2, Plate III.) are much stronger than in the rest of the strip, a red image of the prominence will be seen in the part of the spectrum corresponding to the line C, and a blue image in the place corresponding to the line F. This method of observation requires greater dispersion than is necessary for the mere detection of the chromosphere lines; the DISPLACEMENT OF LINES. 991 dispersion required for enabling a bright-line spectrum to predominate over a continuous spectrum being always nearly proportional to the width of the slit (§ 791). Of the nebulae, it is well known that some have been resolved by powerful telescopes into clusters of stars, while others have as yet proved irresolvable. Huggins has found that the former class ol nebulae give spectra of the same general character as the sun and the fixed stars, but that some of the latter class give spectra of bright lines, indicating that their constitution is gaseous. 789. Displacement of Lines consequent on Celestial Motions.—Ac¬ cording to the undulatory theory of light, which is now universally accepted, the fundamental difference between the different rays which compose the complete spectrum, is a difference of wave- frequency, and, as connected with this, a difference of wave-length in any given medium, the rays of greatest wave-frequency or shortest wave-length being the most refrangible. JDoppler first called attention to the change of refrangibility which must be expected to ensue from the mutual approach or recess of the observer and the source of light, the expectation being grounded on reasoning which we have explained in connection with acoustics (§ 653 a). Doppler adduced this principle to explain the colours of the fixed stars, a purpose to which it is quite inadequate; but it has rendered very important service in connection with spectroscopic research. Displacement of a line towards the more refrangible end of the spec¬ trum, indicates approach, displacement in the opposite direction indi¬ cates recess, and the velocity of approach or recess admits of being calculated from the observed displacement. When the slit of the spectroscope crosses a spot on the sun’s disc, the dark lines lose their straightness in this part, and are bent, some¬ times to one side, sometimes to the other. These appearances clearly indicate uprush and downrush of gases in the sun’s atmosphere in the region occupied by the spot. Huggins has observed a displacement of the F line towards the red end, in the spectrum of Sirius, as compared with the spectrum of the sun or of hydrogen. The displacement is so small as only to admit of measurement by very powerful instrumental appliances; but, small as it is, calculation shows that it indicates a motion of recess at the rate of about 30 miles per second. 1 1 The observed displacement corresponded to recess at the rate of 41 '4 miles per second; 992 DISPERSION. STUDY OF SPECTRA. 790. Spectra of Artificial Lights.—The spectra of the artificial lights in ordinary use (including gas, oil-lamps, and candles) differ from the solar spectrum in the relative brightness of the different colours, as well as in the entire absence of dark lines. They are comparatively strong in red and green, but weak in blue; hence all colours which contain much blue in their composition appear to disadvantage by gas-light. It is possible to find artificial lights whose spectra are of a com¬ pletely different character. The salts of strontium, for example, give red light, composed of the ingredients represented in spectrum No. 10, Plate III., and those of sodium yellow light (No. 3, Plate III.) If a room is illuminated by a sodium flame (for example, by a spirit- lamp with salt sprinkled on the wick), all objects in the room will appear of a uniform colour (that of the flame itself), differing only in brightness, those which contain no yellow in their spectrum as seen by day-light being changed to black. The human countenance and hands assume a ghastly hue, and the lips are no longer red. A similar phenomenon is observed when a coloured body is held in different parts of the solar spectrum in a dark room, so as to be illuminated by different kinds of monochromatic light. The object either appears of the same colour as the light which falls upon it, or else it refuses to reflect this light and appears black. Hence a screen for exhibiting the spectrum should be white. 791. Brightness and Purity.—The laws which determine the bright¬ ness of images generally, and which have been expounded at some length in the preceding chapter, may be applied to the spectroscope. We shall, in the first instance, neglect the loss of light by reflection and imperfect transmission. Let A denote the 'prismatic dispersion , as measured by the angular separation of two specified monochromatic images when the naked eye is applied to the last prism, the observing telescope being re¬ moved. Then, putting m for the linear magnifying power of the but 12'0 of this must be deducted for the motion of the earth in its orbit at the season of the year when the observation was made. The remainder, 29'4, is therefore the rate at which the distance between the sun and Sirius is increasing. In a more recent paper, read while this volume was going through the press, Dr. Hug¬ gins gives the results of observations with more powerful instrumental appliances. The recess of Sirius is found to be only 20 miles per second. Arcturus is approaching at the rate of 50 miles per second. Community of motion has been established in certain sets of stars; and the belief previously held by astronomers, as to the direction in which the solar system is moving with respect to the stars as a whole, is fully confirmed. BRIGHTNESS AND PURITY 993 telescope, m A is the angular separation observed when the eye is applied to the telescope. We shall call mA the total dispersion. Let Q denote the angle which the breadth of the slit subtends at the centre of the collimating lens, and which is measured by focanen^i^of^e n s • Then Q is also the apparent breadth of any absolutely monochromatic image of the slit, formed by rays of minimum devia¬ tion,as seen by an eye applied either to the first prism, the last prism, or any one of the train of prisms. The change produced in a pencil of monochromatic rays by transmission through a prism at minimum deviation, is in fact simply a change of direction, without any change of mutual inclination; and thus neither brightness nor apparent size is at all affected. In ordinary cases, the bright lines of a spectrum may be regarded as monochromatic, and their apparent breadth, as seen without the telescope, is sensibly equal to 0. Strictly speaking, the effect of prismatic dispersion in actual cases, is to increase the apparent breadth by a small quantity, which, if all the prisms are alike, is proportional to the number of prisms; but the increase is usually too small to be sensible. Let I denote the intrinsic brightness of the source as regards any one of its (approximately) monochromatic constituents; in other words, the brightness which the source would have if deprived of all its light except that which goes to form a particular bright line. Then, still neglecting the light stopped by the instrument, the bright¬ ness of this line as seen without the aid of the telescope will be I; and as seen in the telescope it will either be equal to or less than this, according to the magnifying power of the telescope and the effective aperture of the object-glass (§ 769). If the breadth of the slit be halved, the breadth of the bright line will be halved, and its brightness will be unchanged. These conclusions remain true so long as the bright line can be regarded as practically monochromatic. The brightness of any part of a continuous spectrum follows a very different law. It varies directly as the width of the slit, and inversely as the prismatic dispersion. Its value without the ob- serving telescope, or its maximum value with a telescope, is A i, where i is a coefficient depending only on the source. The purity of any part of a continuous spectrum is properly mea¬ sured by the ratio of the distance between two specified mono¬ chromatic images to the breadth of either , the distance in question being measured from the centre of one to the centre of the other. 64 994 DISPERSION. STUDY OF SPECTRA. This ratio is unaffected by the employment of an observing telescope, and is The ratio of the brightness of a bright line to that of the adjacent portion of a continuous spectrum forming its back-ground, is -yJ* assuming the line to be so nearly monochromatic that the increase of its breadth produced by the dispersion of the prisms is an insigni¬ ficant fraction of its whole breadth. As we widen the slit, and so increase 0, we must increase A in the same ratio, if we wish to preserve the same ratio of brightness. As y is increased indefinitely, the predominance of the bright lines does not increase indefinitely, but tends to a definite limit, namely, to the predominance which they would have in a perfectly pure spectrum of the given source. The loss of light by reflection and imperfect transmission, increases with the number of surfaces of glass which are to be traversed; so that, with a long train of prisms and an observing telescope, the actual brightness will always be much less than the theoretical bright¬ ness as above computed. The actual purity is alwaj^s less than the theoretical purity, being greatly dependent on freedom from optical imperfections; and these can be much more completely avoided in lenses than in prisms. It is said that a single good prism, with a first-class collimator and telescope, (as originally employed by Swan,) gives a spectrum much more free from blurring than the modern multiprism spectroscopes, when the total dispersion m A is the same in both the cases com¬ pared. 792. Chromatic Aberration.—The unequal refrangibility of the different elementary rays is a source of grave inconvenience in con¬ nection with lenses. The focal length of a lens depends upon its index of refraction, which of course increases with refrangibility, the focal length being shortest for the most refrangible rays. Thus a lens of uniform material will not form a single white image of a white object, but a series of images, of all the colours of the spectrum, arranged at different distances, the violet images being nearest, and the red most remote. If we place a screen anywhere in the series of images, it can only be in the right position for one colour. Every other colour will give a blurred image, and the superposition of them all produces the image actually formed on the screen. If the object be a uniform white spot on a black ground, its image on the screen ACHROMATISM. 995 will consist of white in its central parts, gradually merging into a coloured fringe at its edge. Sharpness of outline is thus rendered impossible, and nothing better can be done than to place the screen at the focal distance corresponding to the brightest part of the spec¬ trum. Similar indistinctness will attach to images observed in mid¬ air, whether directly or by means of another lens. This source of confusion is called chromatic aberration. 793. Possibility of Achromatism.—In order to ascertain whether it was possible to remedy this evil by combining lenses of two different materials, Newton made some trials with a compound prism com¬ posed of glass and water (the latter containing a little sugar of lead), and he found that it was not possible, by any arrangement of these two substances, to produce deviation of the transmitted light without separation into its component colours. Unfortunately he did not extend his trials to other substances, but concluded at once that an achromatic prism (and hence also an achromatic lens) was an impos¬ sibility ; and this conclusion was for a long time accepted as indis¬ putable. Mr. Hall, a gentleman of Worcestershire, was the first to show that it was erroneous, and is said to have constructed some achromatic telescopes; but the important fact thus discovered did not become generally known till it was rediscovered by Dollond, an eminent London optician, in whose hands the manufacture of achro¬ matic instruments attained great perfection. 794. Conditions of Achromatism. — The conditions necessary for achromatism are easily explained. The angular separation between the brightest red and the brightest violet ray transmitted through a prism is called the dispersion of the prism, and is evidently the differ¬ ence of the deviations of these rays. These deviations, for the position of minimum deviation of a prism of small refracting angle A, are (p — 1) A and (p" — 1) A, p and p" denoting the indices of refraction for the two rays considered—§ 739, equation (1)—and their difference is {p — p) A. This difference is always small in comparison with either of the deviations whose difference it is, and its ratio to either of them, or more accurately its ratio to the value of (p— I) A for the brightest part of the spectrum, is called the dispersive power of the substance. As the common factor A may be omitted, the formula for the dis¬ persive power is evidently * If this ratio were the same for all substances, as Newton supposed, achromatism would be impossible; but in fact its value varies greatly, £tnd is greater for flint than for crown glass. If two prisms of these 996 DISPERSION. STUDY OF SPECTRA. substances, of small refracting angles, be combined into one, with their edges turned opposite ways, they will achromatize one another if (/*"-//) A, or the product of deviation by dispersive power, is the same for both. As the deviations can be made to have any ratio we please by altering the angles of the prisms, the condition is evidently possible. The deviation which a simple ray undergoes in traversing a lens, at a distance x from the axis, is y> f denoting the focal length of the lens (§ 739), and the separation of the red and violet constituents of a compound ray is the product of this deviation by the dispersive power of the material. If a convex and concave lens are combined, fitting closely together, the deviations which they produce in a ray traversing both, are in opposite directions, and so also are the dis¬ persions. If we may regard x as having the same value for both (a supposition which amounts to neglecting the thicknesses of the lenses in comparison with their focal lengths) the condition of no resultant dispersion is that 1 dispersive power x — has the same value for both lenses. Their focal lengths must there¬ fore be directly as the dispersive powers of their materials. These latter are about ‘033 for crown and ‘052 for flint glass. A converg¬ ing achromatic lens usually consists of a double convex lens of crown fitted to a diverging meniscus of flint. In every achromatic com¬ bination of two pieces, the direction of resultant deviation is that due to the piece of smaller dispersive power. The definition above given of dispersive power is rather loose. To make it accurate, we must specify, by reference to the “fixed lines,” the precise positions in the spectrum of the two rays whose separa¬ tion we consider. Since the distances between the fixed lines have different propor¬ tions for crown and flint glass, achromatism of the whole spectrum is impossible. With two pieces it is possible to unite any two selected rays, with three pieces any three selected rays, and so on. It is considered a sign of good achromatism when no colours can be brought into view by bad focussing except purple and green. 795. Achromatic Eye-pieces.—The eye-pieces of microscopes and astronomical telescopes, usually consist of two lenses of the same kind of glass, so arranged as to counteract, to some extent, the spherical and chromatic aberrations of the object-glass. The positive eye-piece, THE RAINBOW. 997 invented by Ramsden, is suited for observation with cross-wires or micrometers; the negative eye-piece, invented by Huyghens, is not adapted for purposes of measurement, but is preferred when distinct vision is the sole requisite. These eye-pieces are commonly called achromatic, but their achromatism is in a manner spurious. It con¬ sists not in bringing the red and violet images into true coincidence, but merely in causing one to cover the other as seen from the posi¬ tion occupied by the observer’s eye. In the best opera-glasses (§ 764), the eye-piece, as well as the ob¬ ject-glass, is composed of lenses of flint and crown so combined as to be achromatic in the more proper sense of the word. 796. Rainbow.—The unequal refrangibility of the different ele¬ mentary rays furnishes a complete explanation of the ordinary phe¬ nomena of rainbows. The explanation was first given by Newton, who confirmed it by actual measurement. It is well known that rainbows are seen when the sun is shining on drops of water. Sometimes one bow is seen, sometimes two, each of them presenting colours resembling those of the solar spectrum. When there is only one bow, the red arch is above and the violet below. When there is a second bow, it is at some distance outside of this, has the colours in reverse order, and is usually less bright. Rainbows are often observed in the spray of cascades and fountains, when the sun is shining. In every case, a line join¬ ing the observer to the sun is the axis of the bow or bows; that is to say, all parts of the length of the bow are at the same angu¬ lar distance from the sun. The formation of the pri¬ mary bow is illustrated by Fig. 719. A ray of solar light, falling on a spherical drop of water, in the direc¬ tion SI, is refracted at I, then reflected internally from the back of the drop, and again refracted into the air in the direction I' M. If we take different points of incidence, we shall obtain differ¬ ent directions of emergence, so that the whole light which emerges 998 DISPERSION. STUDY OF SPECTRA. from the drop after undergoing, as in the figure, two refractions and one reflection, forms a widely-divergent pencil. Some portions of this pencil, however, contain very little light. This is especially the case with those rays which, having been incident nearly normally, are returned almost directly back, and also with those which were almost tangential at incidence. The greatest condensation, as re¬ gards any particular species of elementary ray, occurs at that part of the emergent pencil which makes the smallest obtuse angle or the greatest acute angle with the direction of incidence. As the ob¬ tuse angle is the measure of the deviation, the direction of greatest condensation is the direction of minimum deviation. It is by means of rays which have undergone this minimum deviation, that the observer sees the corresponding colour in the bow; and the devia¬ tion which they have undergone is evidently equal to the angular distance of this part of the bow from the sun. The minimum deviation will be greatest for those rays which are most refrangible. If the figure, for example, be supposed to represent the circumstances of minimum deviation for violet, we shall obtain smaller deviation in the case of red, even by giving the angle IA I' the same value which it has in the case of minimum deviation for violet, and still more when we give it the value which corresponds to the minimum deviation of red. The most refrangible colours are accordingly seen furthest from the sun. The effect of the rays which undergo other than minimum deviation, is to produce a border of white light on the side remote from the sun; that is to say, on the inner edge of the bow. 1 The condensation which accompanies minimum deviation, is merely a particular case of the general mathematical law that magnitudes remain nearly constant in the neighbourhood of a maximum or minimum value. A small parallel pencil SI incident at and around the precise point which corresponds to minimum deviation, will thus 1 When the drops are very uniform in size, a series of faint supernumerary bows, alter¬ nately purple and green, is sometimes seen beneath the primary bow. These bows are produced by the mutual interference of rays which have undergone other than minimum deviation, and the interference arises in the following way. Any two parallel directions of emergence, for rays of a given refrangibilty, correspond in general to two different points of incidence on any given drop, one of the two incident rays being more nearly normal, and the other more nearly tangential to the drop than the ray of minimum deviation. These two rays have pursued dissimilar paths in the drop, and are in different phases when they reach the observer’s eye. The difference of phase may amount to one, two, three, or more, exact wave-lengths, and thus one, two, three, or more supernumerary bows may be formed. The distances between the supernumerary bows will be greater as the drops of water are smaller. This explanation is due to Dr. Thomas Young. THE RAINBOW. 999 have deviations which may be regarded as equal, and will accord ingly remain sensibly parallel at emergence. A parallel pencil in cident on any other part of the drop, will be divergent at emer¬ gence. The indices of refraction for red and violet rays from air into water are respectively l° t 8 and 1 ^ r 9 , and calculation shows that the distances from the centre of the sun to the parts of the bow in which these colours are strongest should be Fig. 720. —Production of Secondary Bow. the supplements of 42° 2' and 40° 17' respectively. These results agree with observation. The angles 42° 2' and 40° 17' are the distance from the antisolar 'point , which is always the cen¬ tre of the bow. The rays which form the secondary bow have undergone two internal reflections, as repre¬ sented in Fig. 720, and here again a special con¬ centration occurs in the direction of minimum deviation. This devia¬ tion is greater than 180°, and is greatest for the most refrangible rays. The distance of the arc thus formed from the sun’s centre, is 360° min¬ us the deviation, and is accordingly least for the most refrangible rays. Thus the violet arc is nearest the sun, and the red furthest from it, in the secondary bow. Some idea of the relative situations of the eye, the sun, and the drops of water in which the two bows are formed, may be obtained from an inspection of Fig. 721. CHAPTER LXIIL COLOUR. 797. Colour as a Property of Opaque Bodies.—A body which reflects (by irregular reflection) all the rays of the spectrum in equal propor¬ tion, will appear of the same colour as the light which falls upon it; that is to say, in ordinary cases, white or gray. But the majority of bodies reflect some rays in larger proportion than others, and are therefore coloured, their colour being that which arises from the mixture of the rays which they reflect. A body reflecting no light would be perfectly black. Practically, white, gray, and black differ only in brightness. A piece of white paper in shadow appears gray, and in stronger shadow black. 798. Colour of Transparent Bodies.—A transparent body, seen by transmitted light, is coloured, if it is more transparent to some rays than to others, its colour being that which results from mixing the transmitted rays. No new ingredient is added by transmission, but certain ingredients are more or less completely stopped out. Some transparent substances appear of very different colours according: to their thickness. A solution of chloride of chromium, for example, appears green when a thin layer of it is examined, while a greater thickness of it presents the appearance of reddish brown. In such cases, different kinds of rays successively disappear by selec¬ tive absorption, and the transmitted light, being always the sum of the rays which remain unabsorbed, is accordingly of different com¬ position according to the thickness. When two pieces of coloured glass are placed one behind the other, the light which passes through both has undergone a double process of selective absorption, and therefore consists mainly of those rays which are abundantly transmitted by both glasses; or to speak broadly, the colour which we see in looking through the combination COLOURS OF MIXED POWDERS. 1001 is not the sum of the colours of the two glasses, but their common part. Accordingly, if we combine a piece of glass coloured red with oxide of copper, and transmitting light which consists almost entirely of red rays, with a blue or violet glass of about the same depth of tint, and transmitting hardly any red, the combination will be almost black. The light transmitted through two glasses of different colour, and of the same depth of tint, is always less than would be trans¬ mitted by a double thickness of either; and the colour of the trans¬ mitted light is in most cases a colour which occupies in the spectrum an intermediate place between the two given colours. Thus, if the two glasses are yellow and blue, the transmitted light will, in most cases, be green, since most natural yellows and blues when analyzed by a prism show a large quantity of green in their composition. Similar effects are obtained by mixing coloured liquids. 799. Colours of Mixed Powders.—“In a coloured powder, each par¬ ticle is to be regarded as a small transparent body which colours light by selective absorption. It is true that powdered pigments when taken in bulk are extremely opaque. Nevertheless, whenever we have the opportunity of seeing these substances in compact and homogeneous pieces before they have been reduced to powder, we find them transparent, at least when in thin slices. Cinnabar, chromate of lead, verdigris, and cobalt glass are examples in point. “When light falls on a powder thus composed of transparent par¬ ticles, a small part is reflected at the upper surface; the rest penetrates, and undergoes partial reflection at some of the surfaces of separation between the particles. A single plate of uncoloured glass reflects ^ of normally incident light; two plates X V, and a large number nearly the whole. In the powder of such glass, we must accordingly con¬ clude that only about of normally incident light is reflected from the first surface, and that all the rest of the light which gives the powder its whiteness comes from deeper layers. It must be the same with the light reflected from blue glass; and in coloured powders generally only a very small part of the light which they reflect comes from the first surface; it nearly all comes from beneath. The light reflected from the first surface is white, except when the reflection is metallic. That which comes from below is coloured, and so much the more deeply the further it has penetrated. This is the reason why coarse powder of a given material is more deeply col¬ oured than fine, for the quantity of light returned at each successive reflection depends only on the number of reflections and not on the 1002 COLOUR. thickness of the particles. If these are large, the light must pene¬ trate so much the deeper in order to undergo a given number of reflections, and will therefore be the more deeply coloured. “The reflection at the surfaces of the particles is weakened if we interpose between them, in the place of air, a fluid whose index of refraction more nearly approaches their own. Thus powders and pigments are usually rendered darker by wetting them with water, and still more with the more highly refracting liquid, oil. “If the colours of powders depended only on light reflected from their first surfaces, the light reflected from a mixed powder would be the sum of the lights reflected from the surfaces of both. But most of the light, in fact, comes from deeper layers, and having had to traverse particles of both powders, must consist of those rays which are able to traverse both. The resultant colour therefore, as in the case of superposed glass plates, depends not on addition but rather on subtraction. Hence it is that a mixture of two pigments is usually much more sombre than the pigments themselves, if these are very unlike in the average refrangibility of the light which they reflect. Vermilion and ultramarine, for example, give a black-gray (showing scarcely a trace of purple, which would be the colour obtained by a true mixture of lights), each of these pigments being in fact nearly opaque to the light of the other.” 1 800. Mixtures of Colours.—By the colour resulting from the mix¬ ture of two lights, we mean the colour which is seen when they both fall on the same part of the retina. Propositions regarding mixtures of colours are merely subjective. The only objective differences of colour are differences of refrangibility, or if traced to their source, differences of wave-frequency. All the colours in a pure spectrum are objectively simple, each having its own definite period of vibra¬ tion by which it is distinguished from all others. But whereas, in acoustics, the quality of a sound as it affects the ear varies with every change in its composition, in colour, on the other hand, very different compositions may produce precisely the same visual im¬ pression. Every colour that we see in nature can be exactly imitated by an infinite variety of different combinations of elementary rays. To take, for example, the case of white. Ordinary white light consists of all the colours of the spectrum combined; but any one of the elementary colours, from the extreme red to a certain point in yellowish green, can be combined with another elementary colour 1 Translated from Helmholtz’s Physiological Optics, § 20. MIXTURE OF COLOURS. 1003 on the other side of green in such proportion as to yield a perfect imitation of ordinary white. The prism would instantly reveal the differences, but to the naked eye all these whites are completely undistinguishable one from another. O 801. Methods of Mixing Colours.—The following are some of the best methods of mixing colours (that is coloured lights):— 1. By combining reflected and transmitted light; for example, by looking at one colour through a piece of glass, while another colour is seen by reflection from the near side of the glass. The lower sash of a window, when opened far enough to allow an arm to be put through, answers well for this purpose. The brighter of the two coloured objects employed should be held inside the window, and seen by reflection; the second object should then be held outside in such a position as to be seen in coincidence with the image of the first. As the quantity of reflected light increases with the angle of incidence, the two colours may be mixed in various proportions by shifting the position of the eye. This method is not however adapted to quantitative comparison, and can scarcely be employed for combining more than two colours. 2. By employing a ro¬ tating disc (Fig. 722) composed of differently coloured sectors. If the disc be made to revolve rapidly, the sectors will not be separately visible, but their colours will appear blended into one on account of the per¬ sistence of visual impres¬ sions. The proportions Fig. 722.—Rotating Disc, can be varied by varying the sizes of the sectors. Coloured discs of paper, each having a radial slit, are very convenient for this purpose, as any moderate number of such discs can be combined, and the sizes of the sectors exhibited can be varied at pleasure. The mixed colour obtained by a rotating disc is to be regarded as 1004 COLOUR. a mean of the colours of the several sectors—a mean in which each of these colours is assigned a weight proportional to the size of its sector. Thus, if the 360 degrees which compose the entire disc consist of 100° of red paper, 100° of green, and 160° of blue, the intensity of the light received from the red when the disc is rotating will be only ^ of that which would be received from the red sector when seen at rest; and the total effect on the retina is represented by of the intensity of the red, plus ^ of the intensity of the green, plus of the intensity of the blue; or if we denote the colours of the sectors by their initial letters, the effect may be symbolized by the formula — - B + '^ G + 1 6 j Denoting the resultant colour by C, we have the symbolic equation 10R+10G+16B = 36C; and the resultant colour may be called the mean of 10 parts of red, 10 of green, and 16 of blue. Colour-equations, such as the above, are frequently employed, and may be combined by the same rules as ordinary equations. 3. By causing two or more spectra to overlap. We thus obtain mixtures which are the sums of the overlapping colours. If, in the experiment of § 778, we employ, instead of a single straight slit, a pair of slits meeting at an angle, so as to form either an X or a V, we shall obtain mixtures of all the simple colours two and two, since the coloured images of one of the slits will cross those of the other. The display of colours thus obtained upon a screen is exquisitely beautiful, and if the eye is placed at any point of the image (for example, by looking through a hole in the screen), the prism will be seen filled with the colour which falls on this point. 802. Experiments of Helmholtz and Maxwell.—Helmholtz, in an excellent series of observations of mixtures of simple colours, em¬ ployed a spectroscope with a V-shaped slit, the two strokes of the V being at right angles to one another; and by rotating the V he was able to diminish the breadth and increase the intensity of one of the two spectra, while producing an inverse change in the other. To isolate any part of the compound image formed by the two over¬ lapping spectra, he drew his eye back from the eye-piece, so as to limit his view to a small portion of the field. But the most effective apparatus for observing mixtures of simple colours is one devised by Professor Clerk Maxwell, by means of which any two or three colours of the spectrum can be combined in MIXTURE OF COLOURS. 1005 any required proportions. In principle, this method is nearly equi¬ valent to looking through the hole in the screen in the experiment above described. Let P (Fig. 723) be a prism, in the position of minimum deviation; L a lens; E and R conju¬ gate foci for rays of a particular refrangibility, say red; E and V conju¬ gate foci for rays of an¬ other given refrangibi¬ lity, say violet. If a slit Fig. 72s._principla of Maxwell’s Colour-box. is opened at R, an eye at E will receive only red rays, and will see the lens filled with red light. If this slit be closed, and a slit opened at Y, the eye, still placed at E, will see the lens filled with violet light. If both slits be opened, it will see the lens filled with a uniform mixture of the two lights; and if a third slit be opened, between R and Y, the lens will be seen filled with a mixture of three lights. Again, from the properties of conjugate foci, if a slit is opened at E, its spectral image will be formed at R Y, the red part of it being at R, and the violet part at Y. The apparatus was inclosed in a box painted black within. There was a slit fixed in position at E, and a frame with three movable slits at R Y. When it was desired to combine colours from three given parts of the spectrum, specified by reference to Fraunhofer’s lines, the slit E was first turned towards the light, giving a real spectrum in the plane R Y, in which Fraunhofer’s lines were visible, and the three movable slits were set at the three specified parts of the spectrum. The box was then turned end for end, so that light was admitted (reflected from a large white screen placed in sunshine) at the movable slits, and the observer, looking in at the slit E, saw the resultant colour. 803. Results of Experiment.—The following are some of the prin¬ cipal results of experiments on the mixture of coloured lights:— 1. Lights which appear precisely alike to the naked eye yield identical results in mixtures; or employing the term similar to express apparent identity as judged by the naked eye, the sums of similar lights are themselves similar. It is by reason of this phy¬ sical fact, that colour-equations yield true results when combined according to the ordinary rules of elimination. 1006 COLOUR. In the strict application of this rule, the same observer must be the judge of similarity in the different cases considered. For 2. Colours may be similar as seen by one observer, and dissimilar as seen by another; and in like manner, colours may be similar as seen through one coloured glass, and dissimilar as seen through another. The reason, in both cases, is that selective absorption depends upon real composition, which may be very different for two merely similar lights. Most eyes are found to exhibit selective absorption of a certain kind of elementary blue, which is accord¬ ingly weakened before reaching the retina. 3. Every colour, except purple, is similar to a colour of the spec¬ trum either pure or diluted, and all purples are similar to mixtures of red and blue with or without dilution. By diluting a colour we mean mixing it with white, gray, or black. Brown colours are obtained by diluting red, orange, or yellow of feeble intensity. 4 Between any four colours, given in intensity as well as in kind, one colour-equation subsists; expressing the fact that, when we have the power of varying their intensities at pleasure, there is one. defi¬ nite way of making them yield a match, that is to say, a pair of similar colours. Any colour (intensity included) can therefore be completely specified by three numbers, expressing its relation to three arbitrarily selected colours. This is analogous to the theorem in statics that a force acting at a given point can be specified by three numbers denoting its components in three arbitrarily selected directions. 5. Between any five colours (intensity included) a match can be made in one definite way by taking means; 1 for example, by mount¬ ing the colours on two rotating discs. If we had the power of illu¬ minating one disc more strongly than the other in any required ratio, four colours would be theoretically sufficient; and we can, in fact, do what is nearly equivalent to this, by employing black as one of our five colours. Taking means of colours is analogous to finding centres of gravity. In following out the analogy, a colour (given in kind merely) must be represented by a material point given in position merely, and the intensity of the colour must be represented by the mass of the material point. The means of two given colours will be represented by points in the line joining two given points. The means of three given colours will be represented by points lying 1 Propositions 4 and 5 are not really independent, but represent different aspects of one physical (or rather physiological) law. CONE OF COLOUR. 1007 within the triangle formed by joining three given points, and the means of four given colours will be represented by points within a tetrahedron whose four corners are given. When we have five colours given, we have five points given, and of these generally no four will lie in one plane. Call them A, B, C, D, E. Then if E lies within the tetrahedron A B C D, we can make the centre of gravity of A, B, C, and D coincide with E, and the colour E can be matched by a mean of the other four colours. If E lies outside the tetrahedron, let the planes which contain the tetrahedron be produced indefinitely. Then if E lies in the external solid angle* which is vertical to the solid angle A of the tetrahedron, the point A lies within the tetra¬ hedron EBCD, and the colour A is the match. Lastly, let E lie in the external space which is separated from the tetrahedron by the plane BCD. Then the point where this plane is cut by the line joining A E represents the match, for it is a mean of A, E, and is also a mean of B, C, D. With six given colours, six different matches can be made, and six colour-equations will thus be obtained, the consistency of which among themselves will be a test of the accuracy both of theory and observation, as only three of the six can be really independent. Experiments which have been conducted on this plan have given very consistent results. 804. Cone of Colour.—All combinations of colour (intensity in¬ cluded) can be represented geometrically by means of a cone or pyramid within which all possible colours will have their definite places. The vertex will represent total blackness, or the complete absence of light; and colours situated on the same line passing through the vertex will differ only in intensity of light. Any cross- section of the cone will contain all colours, except so far as intensity is concerned, and the colours residing on its perimeter will be the colours of the spectrum ranged in order, with purple to fill up the interval between violet and red. It appears from Maxwell’s experi¬ ments, that the true form of the cross-section is approximately trian¬ gular, 1 with red, green, and blue at the three corners. When all the colours (intensity included) have been assigned their proper places in the cone, a straight line joining any two of them passes through colours which are means of these two; and if two lines are drawn from the vertex to any two colours, the parallelogram constructed 1 The shape of the triangle is a mere matter of convenience, not involving any question of fact. 1008 COLOUR. on these two lines will have at its further corner the colour which is the sum of these two colours. A certain axial line of the cone will contain white or gray at all points of its length, and may be called the line of white. It is convenient to distinguish three qualities of colour which may be called hue , depth , and brightness. Brightness or intensity of light is represented by distance from the vertex of the cone. Depth depends upon angular distance from the line of white, and is the same for all points on the same line through the vertex. Paleness or lightness is the opposite of depth, and is measured by angular nearness to the line of white. Hue or tint is that which is often par excellence termed colour. If we suppose a plane, containing the line of white, to revolve about this line as axis, it will pass succes¬ sively through different tints; and in any one position it contains only two tints, which are separated from each other by the line of white, and are complementary. Red is complementary to.Bluish green. Orange ,, ,, .Sky blue. Yellow ,, ,, .Violet blue. Greenish yellow „ .Violet. Green „ ,, .Pink. Any two colours, of complementary tint, give white when mixed in proper proportions; and any three colours can be mixed in such proportions as to yield white, unless they are all on the same side of a plane drawn through the line of white. According to Maxwell, the orange and yellow of the spectrum can be exactly reproduced by mixtures of red and green, and the extreme colours of the spectrum (crimson and violet) can be reproduced (approximately at least) by mixtures of red and blue. 805. Three Primary Colour-sensations.—All authorities are now agreed in accepting the doctrine, first propounded by Dr. Thomas Young, that there are three elements of colour-sensation; or, in other words, three distinct physiological actions, which, by their various combinations, produce our various sensations of colour. Each is excitable by light of various wave-lengths lying within a wide range, but has a maximum of excitability for a particular wave-length, and is affected only to a slight degree by light of wave-length very different from this. The cone of colour is theoretically a triangular pyramid, having for its three edges the colours which correspond to these three wave-lengths; but it is probable that we cannot obtain THREE PRIMARY COLOUR-SENSATIONS. 1009 one of the three elementary colour-sensations quite from admixture of the other two, and the edges of the pyramid are thus practically rounded off. One of these sensations is excited in its greatest purity by the green near Fraunhofer’s line b, another by the extreme red, and the third by a part of the spectrum lying somewhere in deep blue or violet, its precise position being difficult to determine by reason of the feebleness of the light at this end of the spectrum. Helmholtz ascribes these three actions to three distinct sets of nerves, having their terminations in different parts of the thickness of the retina—a supposition which aids in accounting for the approxi¬ mate achromatism of the eye, for the three sets of nerve-terminations may thus be at the proper distances for receiving distinct images of red, green, and blue respectively, the focal length of a lens being shorter for blue than for red. Light of great intensity, whatever its composition, seems to pro¬ duce a considerable excitement of all three elements of colour-sensa¬ tion. If a spectroscope, for example, be directed first to the clouds and then to the sun, all parts of the spectrum appear much paler in the latter case than in the former. The popular idea that red, yellow, and blue are the three prima¬ ries, is quite wrong as regards mixtures of lights or combinations of colour-sensations. The idea has arisen from facts observed in con¬ nection with the mixture of pigments and the transmission of light through coloured glasses. We have already pointed out the true interpretation of observations of this nature, and have only now to add that in attempting to construct a theory of the colours obtained by mixtures of pigments, the law of substitution of similars cannot be employed. Two pigments of similar colour will not in general give the same result in mixtures. 806. Accidental Images.—If we look steadily at a bright stained- glass window, and then turn our eyes to a white wall, we see an image of the window with the colours changed into their com- plementaries. The explanation is that the nerves which have been strongly exercised in the perception of the bright colours have had their sensibility diminished, so that the balance of action which is necessary to the sensation of white no longer exists, but those elements of sensation which have not been weakened preponderate. The sub¬ jective appearances arising from this cause are called negative acci¬ dental images. Many well-known effects of contrast are similarly explained. White paper, when seen upon a background of any one 65 1010 COLOUR. colour, often appears tinged with the complementary colour; and stray beams of sunlight entering a room shaded with yellow holland blinds, produce blue streaks when they fall upon a white table¬ cloth. In some cases, especially when the object looked at is painfully bright, there is a positive accidental image; that is, one of the same colour as the object; and this is frequently followed by a negative image. A positive accidental image may be regarded as an extreme instance of the persistence of impressions. 807. Colour-blindness.—What is called colour-blindness has been found, in every case which has been carefully investigated, to consist in the absence of the elementary sensation corresponding to red. To persons thus affected the solar spectrum appears to consist of two decidedly distinct colours with white or gray at their place of junc¬ tion, which is a little way on the less refrangible side of the line F. One of these two colours is doubtless nearly identical with the normal sensation of blue. It attains its maximum about midway between F and G, and extends beyond G as far as the normally visible spectrum. The other colour extends a considerable distance into what to normal eyes is the red portion of the spectrum, attaining its maximum about midway between D and E, and becoming deeper and more faint till it vanishes at about the place where to normal eyes crimson begins. The scarlet of the spectrum is thus visible to the colour-blind, not as scarlet but as a deep dark colour, probably a kind of dark green, orange and yellow as brighter shades of the same colour, while bluish-green appears nearly white. It is obvious from this account that what is called “ colour-blind¬ ness " should rather be called dichroic vision, normal vision being distinctively designated as trichvoic. To the dichroic eye any colour can be matched by a mixture of yellow and blue, and a match can be made between any three (instead of four) given colours. Objects which have the same colour to the trichroic eye have also the same colour to the dichroic eye. 808. Colour and Musical Pitch.—As it is completely established that the difference between the colours of the spectrum is a difference of vibration-frequency, there is an obvious analogy between colour and musical pitch; but in almost all details the relations between colours are strikingly different from the relations between sounds. The compass of visible colour, including the lavender rays which lie beyond the violet, and are perhaps visible not in themselves, but by COLOUR AND MUSICAL PITCH. 1011 the fluorescence which they produce on the retina, is, according to Helmholtz, about an octave and a fourth; but if we exclude the lavender, it is almost exactly an octave. Attempts have been made to compare the successive colours of the spectrum with the notes of the gamut; but much forcing is necessary to bring out any trace of identity, and the gradual transitions which characterize the spectrum, and constitute a feature of its beauty, are in marked contrast to the transitions per saltum which are required in music. CHAPTER LXIY. WAVE THEORY OF LIGHT. 809. Principle of Huygens. 1 —The propagation of waves, whether of sound or light, is a propagation of energy. Each small portion of the medium experiences successive changes of state, involving changes in the forces which it exerts upon neighbouring portions. These changes of force produce changes of state in these neighbouring por¬ tions, or in such of them as lie on the forward side of the wave, and thus a disturbance existing at any one part is propagated onwards. Let us denote by the name wave-front a continuous surface drawn through particles which have the same phase; then each wave-front advances with the velocity of light, and each of its points may 7 be regarded as a secondary centre from which disturbances are continu¬ ally propagated. This mode of regarding the propagation of light is due to Huygens, who derived from it the following principle, which lies at the root of all practical applications of the undulatory theory: The disturbance at any'point of a wave-front is the resultant (given by the parallelogram of motions) of the separate disturbances which the different portions of the same wave-front in any one of its earlier positions, would have occasioned if acting singly. This principle involves the physical fact that rays of light are not affected by crossing one another; and its truth, which has been experiment¬ ally tested by a variety of consequences, must be taken as an indica¬ tion that the amplitudes of luminiferous vibrations are infinitesimal in comparison with the wave-lengths. A similar law applies to the resultant of small disturbances generally, and is called by writers on dymamics the law of “superposition of small motions/’ It is analo¬ gous to the arithmetical principle that, when a and b are very small fractions, the product of 1+a and 1+6 may be identified with 1 For the spelling of this name see remarks by Lalande, Memoires de VAcademie, 1773. RECTILINEAR PROPAGATION. 1013 1 +« + &, the term a b, which represents the mutual influence of two small changes, being negligible in comparison with the sum a + 6 of the small changes themselves. 810. Explanation of Rectilinear Propagation. — In a medium in which light travels with the same velocity in all parts and in all directions, the waves propagated from any point will be concentric spheres, having this point for centre, and the lines of propagation, in other words the rays of light, will be the radii of these spheres. It can in fact be shown that the only part of one of these waves which needs to be considered, in computing the resultant disturbance of an external point, is the part which lies directly between this external point and the centre of the sphere. The remainder of the wave-front can be divided into small parts, each of which, by the mutual interference of its own subdivisions, gives a resultant effect of zero at the given point. We express these properties by saying that in a homogeneous and isotropic medium the wave-surface is a sphere , and the rays are normal to the wave-fronts. This class of media includes gases, liquids, crystals of the cubic system, and well- annealed glass. If a medium be homogeneous but not isotropic, disturbances emanating from a point in it will be propagated in waves which will retain their form unchanged as they expand in receding from their source, but this form will not generally be spherical. The rays of light in such a medium will be straight, proceeding directly from the centre of disturbance, and any one ray will cut all the wave-fronts at the same angle; but this angle will generally be different for different rays. In this case, as in the last, the disturbance produced at any point may be computed by merely taking into account that small portion of a wave-front which lies directly between the given point and the source,—in other words, which lies on or very near to the ray which traverses the given point. A disturbance in such a medium usually gives rise to two sets of waves, having two distinct forms, and these remarks apply to each set separately. The tendency of the different parts of a wave-front to propagate disturbances in other directions besides the single one to which such propagation is usually confined, is manifested in certain phenomena which are included under the general name of diffraction. The only wave-fronts with which it is necessary to concern our¬ selves are those which belong to waves emanating from a single 1014 WAVE THEORY OF LIGHT. point,—that is to say, either from a surface really very small, or from a surface which, by reason of its distance, subtends a very small solid angle at the parts of space considered. 811. Application to Refraction.— When waves are propagated from one medium into another, the principle of Huygens leads to the following construction:— Let AE (Fig. 724) represent a portion of the surface of separation between two media, and A B a portion of a wave-front in the first medium; both portions being small enough to be regarded as plane. Fig. 724.—Huygens’ Construction for Wave-front. Then straight lines CA, DBF, normal to the wave-front, represent rays incident at A and E. From A as centre, describe a wave-surface, of such dimensions that light emanating from A would reach this surface in the same time in which light in air travels the distance B E, and draw a tangent plane (perpendicular to the plane of incid¬ ence) through E to this surface. Let F be the point of contact (which is not necessarily in the plane of incidence). Then the tan¬ gent plane E F is a wave-front in the second medium, and A F is a ray in the second medium; for it can be shown that disturbances propagated from all points in the wave-front A B will just have reached E F when the disturbance propagated from B has reached E. For example, a ray proceeding from m, the middle point of the line A B, will exhaust half the time in travelling to the middle point a of A E, and the remaining half in travelling through a f, equal and parallel to half of A F. When the wave-surfaces in both media are spherical, the planes of incidence and refraction ABE, AFE coincide, the angle BAE (Fig. 725) between the first wave-front and the surface of separation is the same as the angle between the normals to these surfaces, that APPLICATION TO REFLECTION. 1015 is to say, is the angle of incidence; and the angle A E F between the surface of separation and the second wave-front is the angle of refraction. The sine of the former is 5-?, and the sine of the latter E A is AF The ratio is therefore BE AF' But B E and A F are the Fig. 725.—Wave-front in Ordinary Refraction. E A sin r distances travelled in the same time in the two media. Hence the sines of the angles of in¬ cidence and refraction are di¬ rectly as the velocities of pro¬ pagation of the incident and refracted light. The relative index of refraction from one medium into another is there¬ fore the ratio of the velocity of light in the first medium to its velocity in the second; and the absolute index of refraction of any medium is inversely as the velocity of light in that medium. 812. Application to Reflection.—The explanation of reflection is precisely similar. Let CA,DE (Fig. 726) be parallel rays incident at A and E; A B the wave-front. As the successive points of the wave-front arrive at the reflecting surface, hemispherical waves di¬ verge from the points of inci¬ dence; and by the time that B reaches E, the wave from A will have diverged in all direc¬ tions to a distance equal to B E. If then we describe in the plane of incidence a semi¬ circle, with centre A and radius equal to B E, the tangent E F to this semicircle will be the wave - front of the reflected light, and A F will be the reflected ray corresponding to the incident ray CA. From the equality of the right-angled triangles ABE, E F A, it is evident that the angles of incidence and reflection are equal. 813. Newtonian Explanation of Refraction. — In the Newtonian theory, the change of direction which a ray experiences at the bound- 1016 WAVE THEORY OF LIGHT. ing surface of two media, is attributed to the preponderance of the attraction of the denser medium upon the particles of light. As the resultant force of this attraction is normal to the surface, the tan¬ gential component of velocity remains unchanged, and the normal component is increased or diminished according as the incidence is from rare to dense or from dense to rare. Let /a denote the relative index of refraction from rare to dense. Let v, v' be the velocities ol light in the rarer and denser medium respectively, and i, i' the angles which the rays in the two media make with the normal. Then the tangential components of velocity in the two media are v sin i, v sin i r respectively, and these by the Newtonian theory are equal; whence '= sin f=u; whereas according to the undulatory theory k = 1 In the Newtonian theory, the velocity of light in any medium is di¬ rectly as the absolute index of refraction of the medium; whereas, in the undulatory theory, the reverse rule holds. The main design of Foucault’s experiment with the rotating mir¬ ror (§ 687), in its original form, was to put these opposite conclusions to the test of direct experiment. For this purpose it was not neces¬ sary to determine the velocity of the rotating mirror, since it affected both the observed displacements alike. The two images were seen in the same field of view, and were easily distinguished by the green¬ ness of the water-image. In every trial the water-image was more displaced than the air-image, indicating longer time and slower velo¬ city ; and the measurements taken were in complete accordance with the undulatory theory, while the Newtonian theory was conclusively disproved. 814. Principle of Least Time.—The path by which light travels from one point to another is in the generality of cases that which occupies least time. For example, in ordinary cases of reflection (except from very concave 1 surfaces), if we select any two points, one on the in¬ cident and the other on the reflected ray, the sum of their distances from the point of incidence is less than the sum of their distances from any neighbouring point on the reflecting surface. In this case, since only one medium is concerned, distance is proportional to time. When a ray in air is refracted into water, if we select any two points, 1 Suppose an ellipse described, having the two selected points for foci, and passing through the point of incidence. If the curvature of the reflecting surface in the plane of incidence is greater than the curvature of this ellipse, the length of the path is a maximum, if less, a minimum. This follows at once from ‘ the constancy of the sum of the focal distances in an ellipse. PRINCIPLE OF LEAST TIME. 1017 one on the incident and the other on the refracted ray, and call their distances from any point of the refracting surface s, s' respectively, and the velocities of propagation in the two media v, v', then the sum of ^ and p is generally less when s and s' are measured to the point of incidence than when they are measured to any neighbouring point on the surface. p is evidently the time of going from the first point to the refracting surface, and p the time from the refracting surface to the second point. The proposition as above enunciated admits of certain exceptions, the time being sometimes a maximum instead of a minimum. The really essential condition (which is fulfilled in both these opposite cases) is that all points on a small area surrounding the point of incidence give sensibly the same time. The component waves sent from all parts of this small area will be in the same phase, and will propagate a ray of light by their combined action. When the two points considered are conjugate foci, and there is no aberration, this condition must be fulfilled by all the rays which pass through both; and the time of travelling from one focus to the other is the same for all the rays. Spherical waves diverging from one focus will, after incidence, become spherical waves converging to or diverging from its conjugate focus. An effect of this kind can be beautifully exhibited to the eye by means of an elliptic dish contain¬ ing mercury. If agitation is produced at one focus of the ellipse by dipping a small rod into the liquid at this point, circular waves will be seen to converge towards the other focus. A circular dish exhi¬ bits a similar result somewhat imperfectly; waves diverging from a point near the centre will be seen to converge to a point symmetri¬ cally situated on the other side of the centre. When the second point lies on a caustic surface formed by the reflection or refraction of rays emanating from the first point, all points on an area of sensible magnitude in the neighbourhood of the point of incidence would give sensibly the same time of travelling as the actual point of incidence, so that the light which traverses a point on a caustic may be regarded as coming from an area of sensible magnitude instead of (as in the case of points not on the caustic) an excessively small area. An eye placed at a point on a caustic will see this portion of the surface filled with light. As the velocity of light is inversely proportional to the index of 1018 WAVE THEORY OF LIGHT. refraction n, the time of travelling a distance s with constant velocity may be represented by ^s, and if a ray of light passes from one point to another by a crooked path, made up of straight lines s 1 ,s 2 ,s 3 , . . . . lyingin media whose absolute indices are /u 3 , . . . , the expres¬ sion /u 1 s r +ju 2 s 2 +yu 3 s 3 + . . . represents the time of passage. This expression, which may be called the sum of such terms as fis, must therefore fulfil the above condition; that is to say, the points of incidence on the surfaces of separation must be so situated that this sum either remains absolutely constant when small changes are sup¬ posed to be made in the positions of these points, or else retains that approximate constancy which is characteristic of maxima and minima. Conversely, all lines from a luminous point which fulfil this condition, will be paths of actual rays. 815. Terrestrial Refraction. 1 —The atmosphere may be regarded as homogeneous when we confine our attention to small portions of it, and hence it is sensibly true, in ordinary experiments where no great distances are concerned, that rays of light in air are straight, just as it is true in the same limited sense that the surface of a liquid at rest is a horizontal plane. The surface of an ocean is not plane, but approximately spherical, its curvature being quite sensible in ordinary nautical observations, where the distance concerned is merely that of the visible sea-horizon; and a correction for curvature is in like manner required in observing levels on land. If the observer is standing on a perfectly level plain, and observing a distant object at precisely the same height as his eye above the plain, it will appear to be below his eye, for a horizontal 'plane through his eye will pass above it, since a perfectly level plain is not plane , but shares in the general curvature of the earth. It is easily proved that the apparent depression due to this cause is half the angle between the verticals at the positions of the observer and of the object observed. But experience has shown that this apparent depression is to a consider¬ able extent modified by an opposite disturbing cause, called terres¬ trial refraction. When the atmosphere is in its normal condition, a ray of light from the object to the observer is not straight, but is slightly concave downwards. This curvature of a nearl}^ horizontal ray is not due to the curva¬ ture of the earth and of the layers of equal density in the earth’s atmosphere, as is often erroneously supposed, but would still exist, 1 For the leading idea which is developed in §§ 815-817, the Editor is indebted to suggestions from Professor James Thomson. CURVATURE OF RAY. 1019 and with no sensible change in its amount, if the earth's surface were plane, and the directions of gravity every where parallel. It is due to the fact that light travels faster in the rarer air above than in the denser air below, so that time is saved by deviating slightly to the upper side of a straight course. The actual amount of curvature (as determined by surveying) is from J to of the curvature of the earth; that is to say, the radius of curvature of the ray is from 2 to 10 times the earth’s radius. 816. Calculation of Curvature of Ray.—In order to calculate the radius of curvature from physical data, it is better to approach the subject from a somewhat different point of view. The wave-fronts of a ray in air are perpendicular to the ray; and if the ray is nearly horizontal, its wave-fronts will be nearly ver¬ tical. If two of these wave-fronts are produced downwards until they meet, the distance of their intersection from the ray will be the radius of curvature. Let us consider two points on the same wave- front, one of them a foot above the other; then the upper one being in rarer air will be advancing faster than the lower one, and it is easily shown that the difference of their velocities is to the velocity of either as 1 foot is to the radius of curvature. Put f t for the radius of curvature in feet, v and v + lv for the two velocities, p. and p — $p for the indices of refraction of the air at the tw ; o points. Then we have — = — == ^ = dfi, nearly. (1) p v fX Now it has been ascertained, by direct experiment, that the value of /* — 1 for air, within ordinary limits of density, is sensibly pro¬ portional to the density (even when the temperature varies), and is ‘0002943 or - 3l V o at the density corresponding to the pressure 760 mm (at Paris) and temperature 0°C. The difference of density at the two points considered, supposing them both to be at the same tempera¬ ture, will be to the density of either as 1 foot is to the “height of the homogeneous atmosphere'’in feet, which call H (§ 111 A). Then ~i will be g, and the value of — in (1) may be written ~p .^ -I) = g0*- 1 ). = H 3400‘ W Hence p is 3400 times the height of the homogeneous atmosphere. But this height is about 5 miles, or of the earth’s radius. The 1020 WAVE THEOKY OF LIGHT. value of p is therefore about 4^ radii of the earth. This is on the assumptions that the barometer is at 760 mm , the thermometer at 0°C., and that there is no change of temperature in ascending. If we depart from these assumptions, we have the following consequences:— I. If the barometer is at any other height, the factor ^ remains unaltered, and the other factor p — 1 varies directly as the pressure. II. If the temperature is t° Centigrade, H is changed in the direct ratio of 1 -f- a t, a denoting the coefficient of expansion. The first factor g is therefore changed in the inverse ratio of l-fa£. The second factor is changed in the same ratio. The curvature of the ray therefore varies inversely as (1 +a£) 2 . III. Suppose the temperature decreases upwards at the rate of - of a degree Centigrade per foot. The expansion due to 1 of a degree S/a difference of density Centigrade is The first factor -~ v or- will therefore become g — —which, if we put n = 540 (corresponding to 1° Fahr. in 300 feet), and reckon H as 26,000, is approximately 26^00 ~ 1470 00 or S ( 1 —The second factor of the expression for - p is unaffected. It appears, then, that decrease of temperature upwards at the rate of 1°C. in 540 feet, or 1° F. in 300 feet (which is the gene¬ rally-received average), makes the curvature of the ray five-sixths of what it would be if the temperature were uniform. 1 Combining this correction with correction II., it appears that, with a mean temperature of 10°C. or 50°F., and barometer at 760 mm , the curvature of a nearly horizontal ray (taking the earth’s curvature as unity) is 5 6 = —— nea: 5-5 ,rly. This is in perfect agreement with observation, the received average (obtained as an empirical deduction from observation) being \ or i. 817. Curvature of Inclined Rays.—Thus far we have been treating of nearly horizontal rays. To adapt our formula for ^ ( (2) § 816) to the case of an oblique ray, we have merely to multiply it by cos 0, 1 If the temperature decreases upwards at the rate of 1°C. in n feet, or 1°F. in n' feet the first factor of the expression for (which would be g at uniform temperature) becomes 1 96 \ 1 /, 53 \- approximately g C ° r H\ VI MIRAGE. 1021 0 denoting the inclination of the ray to the horizontal, or the inclina¬ tion of the wave-front to the vertical. For, if we still compare two points a foot apart, on the same wave-front, and in the same vertical plane with each other and with the ray, their difference of height 3 v • will be the product of 1 foot by cos 0, and -- will therefore be less than before in the ratio cos 0. Hence it can be shown that the earth’s curvature, so far from being the cause of terrestrial refraction, rather tends in ordinary cir¬ cumstances to diminish it, by increasing the average obliquity of a ray joining two points at the same level. The general formula for the curvature of a ray (lying in a vertical plane) at any point in its length, may be written H( 1 - 9 >- i)cos '’ 1 /, 53 x 008 e > (3) n denoting the number of feet of ascent which give a decrease of 1° C., and n' the number of feet which give a decrease of 1° F. The unit of length for H and may be anything we please. 818. Astronomical Refraction.—Astronomical refraction, in virtue of which stars appear nearer the zenith than they really are, can be reduced to these principles; but it is simpler, in the case of stars not more than 70° or 80° from the zenith, to regard the earth and the layers of equal density in the atmosphere as plane, and to assume that the final result is the same as if the rays from the star were refracted at once out of vacuum into the horizontal stratum of air in which the observer’s eye is situated. If 0 be the apparent and z the true zenith distance of the star, we shall thus have sin z —g sin 0 , whence it may be shown that the value of z'—z, in terms of p^dius ’ * s approximately Qu— 1) tan 0 . 819. Mirage.—Of the three circumstances which govern the curva¬ ture of a ray in air at any point of its course, viz. pressure, tempera¬ ture, and rate of change of temperature, the third is in general the most important. If the temperature decreases upwards at the rate of g 3 of 1°F. per foot of height, the correction for refraction will vanish altogether, and rays will be straight. A more rapid decrease than this will render them concave upwards. 1022 WAVE THEORY OF LIGHT. Fig. 727, which is intended to explain the mirage of the desert, 1 is constructed on the hypothesis that a stratum of air, extending from the ground to a little above the observer’s eye E, has so rapid a decrease of temperature upwards, that the curvature of rays which traverse it is four times greater than the opposite curvature of rays in the air above it. On tracing the course of the different rays repre¬ sented in the figure as entering the eye from the object AB, it is obvious that there will be two images; one of them A'B' erect and lifted up, the ottier A"B" inverted and depressed. It is scarcely necessary to remark that vertical distances are greatly exaggerated in the figure as compared with horizontal distances. In general, if there be a plane stratum of air of maximum density, with regular diminutions of density on both sides, rays which cross 1 See § 726 for a description of the phenomena. MIRAGE. 1023 it at a very small angle will be bent round to it again, and may thus cross it backwards and forwards any number of times. Generally speaking, an image formed after an odd number of such crossings will be inverted, and after an even number erect. Images of high orders (that is, formed by rays which have made numerous cross¬ ings) will be crowded together in the neighbourhood of. the true direction of the object, and will merely produce confusion. In every case of inverted images, the rays which form them must have crossed one another an odd number of times, and images formed by rays which have crossed one another an even number of times, or not at all, will be erect. Reversal of curvature in rays is not neces¬ sary for inversion, nor for the formation of multiple images; and a phenomenon bearing a strong resemblance to total reflection may be produced by a thin stratum of air in which the refractive index diminishes very rapidly in one direction only. Suppose the stratum to be horizontal, and to diminish in refractive index upwards. Then rays entering it nearly horizontally from below will be bent down¬ wards in traversing it; and if their original inclination to the verti¬ cal exceeds a certain limiting angle (analogous to the critical angle in total reflection), they will not be able to get through it, but will emerge again at its lower side. Two nearly parallel rays entering it ip. the same vertical plane will cross one another within it, and emerge in the same directions as if they had been reflected from a horizontal mirror in or above it. An observer receiving such rays after emergence will see images elevated and inverted; and he may at the same time see the real objects directly by rays which have not reached the stratum. A stratum below the observer’s eye, and rapidly diminishing in refractive index downwards, may in like manner produce inverted images below the real objects. Masses of air of different refractive indices, arranged in irregular forms, may produce double or multiple images; but these will generally be so blurred and distorted as to be barely recognizable. In dealing with air, it is not important to distinguish between difference of density and difference of refractive index; for /x -1 is sensibly proportional to the density even in comparing dry air with moist, or hot with cold. An extremely beautiful imitation of mirage may be obtained by arranging, in a vessel with plate-glass sides, three clear liquids, one above another, such that the middle liquid, while intermediate in density, has the highest index of refraction. Yery distinct triple 1024 WAVE THEORY OF LIGHT. images (the middle one inverted) may thus be obtained of all the objects in a landscape. 1 820. Diffraction Fringes.—When a beam of direct sunlight is admitted into a dark room through a narrow slit, a screen placed at any distance to receive it will show a line of white light, bordered with coloured fringes which become wider as the slit is narrowed. They also increase in width as the screen is removed further off. If they are viewed through a piece of red glass which allows only red rays to pass, they will appear as a succession of bands alternately bright and dark. To explain their origin, we shall suppose the sun’s rays (which may be reflected from an external mirror) to be perpendicular to the plane of the slit, 2 so that the wave-fronts are parallel to this plane, and we shall, in the first instance, confine our attention to light of a particular wave-length; for example, that of the light transmitted by the red glass. Then, if the slit be uniform through its whole length, the positions of the bright and dark bands will be governed by the fol¬ lowing laws:— 1. The darkest parts will be at points whose distances from the two edges of the slit differ by an exact number of wave-lengths. If the difference be one wave-length, the light which arrives at any instant from different parts of the width of the slit is in all possible phases, and the disturbance produced by the nearer half of the slit cancels that produced by the remoter half. If the difference be n wave-lengths, we can divide the slit into n parts, such that the effect due to each part is thus nil . 3 1 The experiment succeeds well with saturated and filtered solution of alum for the lowest liquid, pure water for the highest, and clear whisky, with white sugar dissolved ia it, for the intermediate liquid. The solution of alum should be introduced first, then the water, and lastly the sugared whisky, its specific gravity (which increases with the quan¬ tity of sugar) having been carefully adjusted by trial before introducing it. It should be introduced (by means of a pipette) in sufficient quantity to form a layer about a quarter of an inch thick. This combination of liquids, in a vessel 6 inches square, has been found to retain its pow r er of giving triple images for several days. 2 That is, to the plane of the two knife-edges by which the slit is bounded. This condition can only be strictly fulfilled for a single point on the sun’s disk. Every point on the sun’s surface sends out its own waves as an independent source ; and waves from one point cannot interfere with waves from another. In the experiment as described in the text the fringes due to different parts of the sun’s surface are all produced at once on the screen, and overlap each other. 3 The following explanation will serve to establish the legitimacy of the reasoning here employed:— Each element of the length of the slit tends to produce a system of circular rings (the screen being supposed parallel to the plane of the slit). If the width of the slit is uniform. DIFFRACTION. 1025 2. The brightest parts will be at points whose distances from the two edges of the slit differ by an exact number of wave-lengths plus a half. Let the difference be n-\- J; then we can divide the slit into n inefficient parts and one efficient part, this latter having only half the width of one of the others. Each colour of light has its own alternate bands of brightness and darkness, the distance from band to band being greatest for red and least for violet. The superposition of all the bands constitutes the coloured fringes which are seen. This experiment furnishes the simplest answer to the objection formerly raised to the undulatory theory, that light is not able, like sound, to pass round an obstacle, but can only travel in straight lines. In this experiment light does pass round an obstacle, and turns more and more away from a straight line as the slit is narrowed. When the slit is not exceedingly narrow, the light sent in oblique directions is quite insensible in comparison with the direct light, and no fringes are visible. “We have reason to think that when sound © passes through a very large aperture, or when it is reflected from a large surface (which amounts nearly to the same thing), it is hardly sensible except in front of the opening, or in the direction of reflection.’' 1 There are several other modes of producing diffraction fringes, which our limits do not permit us to notice. We proceed to describe the mode of obtaining a pure spectrum by diffraction. 821. Diffraction by a Grating.—If a piece of glass is ruled with parallel equidistant scratches (by means of a dividing engine and diamond point) at the rate of some hundreds or thousands to the inch, we shall find, on looking through it at a slit or other bright line (the glass being held so that the scratches are parallel to the slit), that a number of spectra are presented to view, ranged at nearly equal distances, on both sides of the slit. If the experiment is made under favourable circumstances, the spectra will be so pure as to show a number of Fraunhofer’s lines. Instead of viewing the spectra with the naked eye, we may with advantage employ a telescope, focussed on the plane of the slit; or we may project the spectra on a screen, by first placing a convex lens so as to form an image of the slit (which must be very strongly these systems will be precisely alike, and will have for their resultant a system of straight bands, parallel to the slit and touching the rings. These are the bands described in the text. Hence, to determine the illumination of any point of the screen, it is only necessary to attend, as in the text, to the nearest points of the two edges of the slit. 1 Airy, Undulatory Theory. Art. 28 1026 WAVE THEORY OF LIGHT. illuminated) on the screen, and then interposing the ruled glass in the path of the beam. A piece of glass thus ruled is called a grating} A grating for diffraction experiments consists essentially of a number of parallel strips alternately transparent and opaque. The distance between the “fixed lines” of the spectra, and the distance from one spectrum to the next, are found to depend on the distance of the strips measured from centre to centre, in other words, on the number of scratches to the inch, but not at all on the relative breadths of the transparent and opaque strips. This latter circumstance only affects the brightness of the spectra. Diffraction spectra are of great practi¬ cal importance— 1. As furnishing a uniform standard of reference in the comparison of spectra. 2. As affording the most accurate method of determining the wave-lengths of the different eleipentary rays of light. 822. Principle of Diffraction Spectrum. —Let GG (Fig. 728) be a grating, re¬ ceiving light from an infinitely 2 distant point lying in a direction perpendicular to the plane of the grating, so that the wave-fronts of the incident light are parallel to this plane. Let a convex lens L be placed on the other side of the grating, and let its axis make an acute angle 0 with the rays incident on the grating. Then the light collected at its principal focus F consists of all the light incident upon the lens parallel to its axis. Let s denote the distance between the rulings, measured from centre to centre, so that if, for 1 The Hon. J. W. Strutt has recently shown that very effective gratings may be made by copying engraved glass gratings photographically. The method appears likely to be of great practical service, engraved glass gratings of sufficient size being extremely expensive and difficult to procure. 2 It is not necessary that the source should be infinitely distant (or the incident rays parallel); but this is the simplest case, and the most usual case in practice. Fig. 728. Principle of Diffraction Spectrum. DIFFRACTION SPECTRUM. 1027 example, there are 1000 lines to the inch, s will be 1 - o 1 o 0 of an inch; and suppose first that s sin 0 is exactly equal to the wave-length \ of one of the elementary kinds of light. Then, of all the light which falls upon the lens parallel to its axis, the left-hand portion in the figure is most retarded (having travelled farthest), and the right-hand portion least, the retardation, in comparing each transparent interval with the next, being constant, and equal to s sin 0, as is evident from an inspection of the figure. Now, for the particular kind of light for which X=s sin0, this retardation is exactly a wave-length, and all the transparent intervals send light of the same phase to the focus F; so that, if there are'1000 such intervals, the resultant amplitude of vibration of F is 1000 times the amplitude due to one interval alone. For light of any other wave-length this coincidence of phase will not exist. For example, if the difference between A and s sin 0 is A, the difference of phase between the lights received from the 1st and 2d intervals will be x 0 X 0 0 - A, between the 1st and 3d yoVo between the 1st and 501st t %°q- A, or just half a wave-length, and so on. The 1st and 501st are thus in complete discordance, as are also the 2d and 502d, &c. Light of every wave-length except one is thus almost completely destroyed by interference, and the light collected at F consists almost entirely of the particular kind defined by the condition X = s sin 9. (1) The purity of the diffraction spectrum is thus explained. If a screen be held at F, with its plane perpendicular to the prin¬ cipal axis, any point on this screen a little to one side of F will •receive light of another definite wave-length, corresponding to an¬ other direction of incidence on the lens, and a pure spectrum will thus be depicted on the screen. 823. Practical Application.—In the arrangement actually em¬ ployed for accurate observation, the lens L L is the object-glass of a telescope with a cross of spider-lines at its principal focus F. The telescope is first pointed directly towards the source of light, and is then turned to one side through a measured angle 0. Any fixed line of the spectrum can thus be brought into apparent coincidence with the cross of spider-lines, and its wave-length can be computed by the formula (1). The spectrum to which formula (1) relates is called the spectrum of the first order. 1028 WAVE THEORY OF LIGHT. There is also a spectrum of the second order, corresponding to values of 0 nearly twice as great, and for which the equation is 2 X = sin d. (2) For the spectrum of the third order, the equation is 3 X = sin 6; (3) and so on, the explanation of their formation being almost precisely the same as that above given. There are two spectra of each order, one to the right, and the other at the same distance to the left of the direction of the source. In Angstrom s observations, 1 which are the best yet taken, all the spectra, up to the sixth inclusive, were observed, and numerous independent determinations of wave-length were thus obtained for several hundred of the dark lines of the solar spectrum. The source of light was the infinitely distant image of an illumi¬ nated slit, the slit being placed at the principal focus of a collimator, and illuminated by a beam of the sun's rays reflected from a mirror. The purity of a diffraction spectrum increases with the number of lines on the grating which come into play, provided that they are exactly equidistant; and may therefore be increased either by in¬ creasing the size of the grating, or by ruling its lines closer together. The gratings employed by Angstrom were about { of an inch square, the closest ruled having about 4500 lines, and the widest 1500. As regards brightness, diffraction spectra are far inferior to those obtained by prisms. To give a maximum of light, the opaque inter¬ vals should be perfectly opaque, and the transparent intervals per¬ fectly transparent; but even under the most favourable conditions, the whole light of any one of the spectra cannot exceed about of the light which would be received by directing the telescope to the slit. The greatest attainable intrinsic brightness in any part of a diffraction spectrum is thus not more than of the intrinsic brightness in the same part of a prismatic spectrum, obtained with the same slit, collimator, and observing telescope, and with the same angular separation of fixed lines. The brightness of the spectra partly depends upon the ratio of the breadths of the transparent and opaque intervals. In the case of the spectra of the first order, the best ratio is that of equality, and equal departures from equality in opposite directions give identical results; for example, if the breadth 1 Angstrom, Recherches sur la Spectre Solaire. Upsal, 1868. STANDARD SPECTRUM. 1029 of the transparent intervals is to the breadth of the opaque either as 1 : 5 or as 5 : 1, it can be shown that the quantity of light in the first spectrum is just a quarter of what it would be with the breadths equal. When a diffraction spectrum is seen with the naked eye, the cornea and crystalline of the eye take the place of the lens L L, and form a real image on the retina at F. 824. Standard Spectrum.—The simplicity of the law connecting ‘ wave-length with position, in the spectra obtained by diffraction, offers a remarkable contrast to the “irrationality " of the dispersion produced by prisms. Diffraction spectra may thus be fairly regarded as natural standards of comparison; and, in particular, the limit¬ ing form (if we may so call it) to which the diffraction spectra tend, as sin 6 becomes small enough to be identified with 0, so that devia¬ tion becomes simply proportional to wave-length, is generally and deservedly accepted by spectroscopists as the absolute standard of reference. This limiting form is often briefly designated as “the diffraction spectrum;” it differs in fact to a scarcely appreciable extent from the first, or even the second and third spectra furnished in ordinary cases by a grating. The diffraction spectrum differs notably from prismatic spectra in the much greater relative extension of the red end. Owing to this circumstance, the brightest part of the diffraction spectrum of solar light is nearly in its centre. The first three columns of numbers in the subjoined table indicate the approximate distances between the fixed lines B, D, E, F, G in certain prismatic spectra, and in the standard diffraction spectrum, the distance from B to G being in each case taken as 1000:— Flint-glass. Angle of 60°. Bisulphide of j Carbon. Angle ot 60°. Diffraction, or Difference of Wave-length. Difference of W ave-freq uency. B to D, . . 220 194 381 278 D to E, . . 214 206 243 23 i e to r, 192 190 160 184 F tp G, . . 374 410 216 306 1000 1000 1000 1000 In the standard diffraction spectrum, deviation is simply propor¬ tional to wave-length, and therefore the distance between two colours represents the difference of their wave-lengths. It has been sug¬ gested that a more convenient reference-spectrum would be con- 1030 WAVE THEORY OF LIGHT. structed by assigning to each colour a deviation proportional to its wave-frequency (or to the reciprocal of its wave-length), so that the distance between two colours will represent the difference between their wave-frequencies. The result of thus disposing the fixed lines is shown in the last column of the above table. It differs from pris¬ matic spectra in the same direction, but to a much less extent than the diffraction spectrum. It has been suggested by Mr. Stoney as extremely probable, that the bright lines of spectra are in many cases harmonics of some one fundamental vibration. Three of the four bright lines of hydrogen have wave-frequencies exactly proportional to the numbers 20, 27, and 32; and in the spectrum of chloro-chromic acid all the lines whose positions have been observed (31 in number) have wave-frequencies which are multiples of one common fundamental. 825. Wave-lengths.—Wave-lengths of light are commonly stated in terms of a unit of which 10 10 make a metre,—hence called the tenth-metre. The following are the wave-lengths of some of the principal “fixed lines” as determined by Angstrom: 1 — WAVE-LENGTHS IN TENTH-METRES. A 7604 E . 5269 B 6867 F . 4861 C 6562 Gr 4307 d 2 . 5895 H, . 3968 I>1 . 5889 H 2 . . 3933 The velocity of light as determined by Foucault is 298 million metres per second, or 298 x 10 16 tenth-metres per second. The num¬ ber of waves per second for any colour is therefore 298 x 10 16 divided by its wave-length as above expressed. Hence we find approxi¬ mately : — For A. 392 millions of millions per second. „ I>.506 „ „ „ >> H. 754 „ ,, ,, 826. Colours of Thin Films. Newton’s Rings.—If two pieces of glass, with their surfaces clean, are brought into close contact, coloured fringes are seen surrounding the point where the contact is closest. They are best seen when light is obliquely reflected to the eye from the surfaces of the glass, and fringes of the complementary colours may be seen by transmitted light. A drop of oil placed on the surface of 1 The wave-lengths of the spectral lines of all elementary substances will be found in Dr. W. M. Watts’ Index of Spectra. NEWTON S RINGS. 1031 clean water spreads out into a thin film, which exhibits similar fringes of colour; and in general, a very thin film of any transparent substance, separating media whose indices of refraction are different from its own, exhibits colour, especially when viewed by obliquely reflected light. In the first experiment above-mentioned, the thin film is an air-film separating the pieces of glass. In soap-bubbles or films of soapy water stretched on rings, a similar effect is produced by a small thickness of water separating two portions of air. The colours, in all these cases, when seen by reflected light, are produced by the mutual interference of the light reflected from the two surfaces of the thin film. An incident ray undergoes, as ex¬ plained in § 729, a series of reflections and refractions; and we may thus distinguish, for light of any given refrangibility, several systems of waves, all of which originally came from the same source. These systems give by their interference a series of alternately bright and dark fringes; and when ordinary white light is employed, the fringes are broadest for the colours of greatest wave-length. Their super¬ position thus produces the observed colours. The colours seen by transmitted light may be similarly explained. The first careful observations of these coloured fringes were made by Newton, and they are generally known as Newton's rings. CHAPTER LXV. POLARIZATION AND DOUBLE REFRACTION. Fig. 729.—Tourmaline Plates. 827. Polarization.—When a piece of the semi-transparent mineral called tourmaline is cut into slices by sections parallel to its axis, it is found that two of these slices, if laid one upon the other in a particular relative position, as A, B (Fig. 729), form an opaque combination. Let one of them, in fact, be turned round upon the other through various angles (Fig. 729). It will be found that the combination is most transparent in two posi¬ tions differing by 180°, one of them ab being the natural position which they originally occupied in the crystal; and that it is most opaque in the two positions at right angles to these. It is not necessary that the slices should be cut from the same crystal. Any two plates of tourmaline with their faces parallel to the axes of the crystals from which they were cut, will exhibit the same phenomenon. The experiment shows that light which has passed through one such plate is in a peculiar and so to speak unsymmetrical condition. It is said to be plane-polarized. According to the undulatory theory, a ray of common light contains vibrations in all planes passing through the ray, and a ray of plane- polarized light contains vibrations in one plane only. Polarized light cannot be distinguished from common light by the naked eye; and for all experiments in polarization two pieces of apparatus must be employed—one to produce polarization, and the other to show it. The former is called the polarizer , the latter the analyzer; and every apparatus that serves for one of these purposes will also serve for the other. In the experiment above described, the plate next the eye is POLARIZATION BY REFLECTION. 1033 the analyzer. The usual process in examining light with a view to test whether it is polarized, consists in looking at it through an analyzer, and observing whether any change of brightness occurs as the analyzer is rotated. When the light of the blue sky is thus examined, a difference of brightness can always be detected accord¬ ing to the position of the analyzer, especially at the distance of about 90° from the sun. In all such cases there are two positions, differing by 180°, which give a minimum of light, and the two posi¬ tions intermediate between these give a maximum of light. The extent of the changes thus observed is a measure of the com¬ pleteness of the polarization of the light. 828. Polarization by Reflection.—Transmission through tourmaline is only one of several ways in which light can be polarized. When a beam of light is reflected from a polished surface of glass, wood, ivory, leather, or any other non-metallic substance, at an angle of from 50° to 60° with the normal, it is more or less polarized, and in like manner a reflector composed of any of these substances may be employed as an analyzer. In so using it, it should be rotated about an axis parallel to the incident rays which are to be tested, and the observation consists in noting whether this rotation produces changes in the amount of reflected light. Mains’ Polariscope (Fig. 730) consists of two reflectors A, B, one serving as polarizer and the other as analyzer, each consisting of a pile of glass plates. Each of these reflectors can be turned about a horizontal axis; and the upper one (which is the analyzer) can also be turned about a vertical axis, the amount of rotation being mea¬ sured on the horizontal circle C C. To obtain the most powerful ef¬ fects, each of the reflectors should be set at an angle of about 33° to the vertical, and a strong beam of common light should be allowed to fall upon the lower pile in such a direction as to be reflected vertically upwards. It will thus fall upon the centre of the upper pile, and the angles of incidence and reflection on both the piles will be about 57°. The observer looking into the upper pile, in such a 10.34 POLARIZATION AND DOUBLE REFRACTION. direction as to receive the reflected beam, will find that, as the upper pile is rotated about a vertical axis, there are two positions (differing by 180°) in which he sees a black spot in the centre of the field of view, these being the positions in which the upper pile refuses to reflect the light reflected to it from the lower pile. They are 90° on either side of the position in which the two piles are parallel; this latter, and the position differing from it by 180°, being those which give a maximum of reflected light. For every reflecting substance there is a particular angle of in¬ cidence which gives a maximum of polarization in the reflected light. It is called the 'polarizing angle for the substance, and its tangent is always equal to the index of refraction of the substance; or what amounts to the same thing, it is that particular angle of incidence which is the complement of the angle of refraction, so that the refracted and reflected rays are at right angles. 1 This important law was discovered experimentally by Sir David Brewster. The reflected ray under these circumstances is in a state of almost complete polarization; and the advantage of employing a pile of plates consists merely in the greater intensity of the reflected light thus furnished. The transmitted light is also polarized; it diminishes in intensity, but becomes more completely polarized, as the number of plates is increased. The reflected and the transmitted light are in fact mutually complementary, being the two parts into which common light has been decomposed; and their polarizations are accordingly opposite, so that, if both the transmitted and reflected beams are examined by a tourmaline, the maxima of obscuration will be obtained by placing the axis of the tourmaline in the one case parallel and in the other perpendicular to the plane of incidence. It is to be noted that what is lost in reflection is gained in trans¬ mission, and that polarization never favours reflection at the expense of transmission. 829. Plane of Polarization.—That particular plane in which a ray of polarized light, incident at the polarizing angle, is most copiously reflected, is called the plane of polarization of the ray. When the polarization is produced by reflection, the plane of reflection is the 1 Adopting the indices of refraction given in the table § 724, we find the following values for the polarizing angle for the undermentioned substances:— Diamond, . . . 67° 43' to 70° 3' Flint-glass, . . 57° 36' to 58° 40' Crown-glass,. . 56° 51' to 57° 23' Pure Water,. 53° 11' Air,.45° THEORY OF DOUBLE REFRACTION. 1035 plane of polarization. According to Fresnel’s theory, which is that generally received, the vibrations of light polarized in any plane are perpendicular to that plane (§ 841). The vibrations of a ray reflected at the polarizing angle are accordingly to be regarded as perpendi¬ cular to the plane of incidence and reflection, and therefore as parallel to the reflecting surface. 830. Polarization by Double Refraction.—We have described in §732 some of the principal phenomena of double refraction in uniaxal crystals. We have now to mention the important fact that the two rays furnished by double refraction are polarized, the polarization in this case being more complete than in any of the cases thus far dis¬ cussed. On looking at the two images through a plate of tourmaline, or any other analyzer, it will be found that they undergo great varia¬ tions of brightness as the analyzer is rotated, one of them becoming fainter whenever the other becomes brighter, and the maximum brightness of either being simultaneous with the absolute extinction of the other. If a second piece of Iceland-spar be used as the analyzer, four images will be seen, of which one pair become dimmer as the other pair become brighter, and either of these pairs can be extin¬ guished by giving the analyzer a proper position. 831. Theory of Double Refraction.—The existence of double refrac¬ tion admits of a very natural explanation, on the undulatory theory. In uniaxal crystals it is assumed that the. elasticity of the luminifer¬ ous sether is the same for all vibrations executed in directions perpen¬ dicular to the axis; and that, for vibrations in other directions, the elasticity varies solely according to the inclination of the direction of vibration to the axis. There are two classes of doubly-refracting uniaxal crystals, called respectively 'positive and negative. In the former the elasticity for vibrations perpendicular to the axis is a maximum; in the latter it is a minimum. Iceland-spar belongs to the latter class; and as small elasticity implies slow propagation, a ray propagated by vibrations perpendicular to the axis will, in this crystal, travel with minimum velocity; while the most rapid pro¬ pagation will be attained by rays whose vibrations are parallel to the axis. Consider any plane oblique to the axis. Through any point in this plane we can draw one line perpendicular to the axis; and the line at right angles to this will have smaller inclination to the axis than any other line in the plane. These two lines are the directions of least and greatest resistance to vibration; the former is the direc- 1036 POLARIZATION AND DOUBLE REFRACTION. tion of vibration for an ordinary, and the latter for an extraordinary ray. The velocity of propagation is the same for the ordinary rays in all directions in the crystal, so that the wave-surface for these is spherical; but the velocity of propagation for the extraordinary rays differs according to their inclination to the axis, and their wave- surface is a spheroid whose polar diameter is equal to the diameter of the aforesaid sphere. The sphere and spheroid touch one another at the extremities of this diameter (which is parallel to the axis of the crystal), and the ordinary and extraordinary rays in this par¬ ticular direction coincide, so that the double refraction becomes single. The course of the two rays produced in the crystal by a given ray incident on a plane face, may be determined by Huygens’ construc¬ tion, which has been described in § 811. The ordinary index is the ratio of the velocity in air to the velocity of the ordinary ray. The extraordinary index (so called) is the ratio of the velocity in air to the velocity of the slowest extraordinary rays in the case of positive crystals, or to the velocity of the swiftest extraordinary rays in the case of negative crystals. In both cases the extraordinary index is that value of s i ne ~ 0 f re f rac tion W “ 1C “ differs most from the ordinary S index. The extraordinary index is applicable to refraction at a plane surface parallel to the axis, when the plane of incidence is perpendicular to the axis. Tourmaline, like Iceland-spar, is a nega¬ tive uniaxal crystal; and its use as a pol¬ arizer depends on the property which it possesses of absorbing the ordinary much more rapidly than the extraordinary ray, so that a thickness which is tolerably transparent to the latter is almost com¬ pletely opaque to the former. 832. Nicol’s Prism.—One of the most convenient and effective contrivances for polarizing light, or analyzing it when pol¬ arized, is that known, from the name of its inventor, as Nicol’s prism. It is made by slitting a rhomb of Iceland-spar along a diagonal plane acbd (Fig. 731), and cementing the two pieces together in their natural position by Canada balsam, a substance whose refractive index E Fig. 731.—Nicol’s Prism. ELLIPTIC POLARIZATION. 1037 is intermediate between the ordinary and extraordinary indices of the crystal. 1 A ray of common light SI undergoes double refraction on entering the prism. Of the two rays thus formed, the ordinary ray is totally reflected on meeting the first surface of the balsam, and passes out at one side of the crystal, as o 0; while the extraordinary iay is transmitted through the balsam as through a parallel plate, and finally emerges at the end of the prism, in the direction eE, parallel to the original direction SI. This apparatus has nearly all the convenience of a tourmaline plate, with the advantages of much greater transparency and of complete polarization. In Foucault’s prism, which is extensively used instead of Nicol’s, the Canada balsam is omitted, and there is nothing but air between the two pieces. This change has the advantage of shortening the prism (because the critical angle of total reflection depends on the relative index of refraction of the two media), but gives a smaller field of view, and rather more loss of light by reflection. 833. Colours produced by Elliptic Polarization.—Very beautiful colours may be produced by the peculiar action of polarized light. For example, if a piece of selenite (crystallized gypsum) about the thickness of paper, is introduced between the polarizer and analyzer of any polarizing arrangement, and turned about into different directions, it will in some positions appear brightly coloured, the colour being most decided when the analyzer is in either of the two critical positions which give respectively the greatest light and the greatest darkness. The colour is changed to its complementary by rotating the analyzer through a right angle; but rotation of the piece of selenite, when the analyzer is in either of the critical positions, merely alters the depth of the colour without changing its tint, and in certain critical positions of the selenite there is a complete absence of colour. Thicker plates of selenite restore the light when ex¬ tinguished by the analyzer, but do not show colour. 834. Explanation.—The following is the explanation of these appearances. Let the analyzer be turned into such a position as to produce complete extinction of the plane-polarized light which comes to it from the polarizer; and let the plane of polarization and the plane perpendicular thereto (and parallel to the polarized rays) be 1 a and b are the corners at which three equal obtuse angles meet (§ 733). The ends of the rhomb which are shaded in the figure are rhombuses. Their diagonals drawn through a and b respectively will lie in one plane, which will contain the axis of the crystal, and will cut the plane of section acbd at right angles. The length of the rhomb is about three and a half times its breadth. 1038 POLARIZATION AND DOUBLE REFRACTION. called the two 'planes of reference. Let the slice of selenite be laid so that the polarized rays pass through it normally. Then there are two directions, at right angles to each other, which are the directions of greatest and least elasticity in the plane of the slice. Unless the slice is laid so that these directions coincide with the two planes of reference, the plane-polarized light which is incident upon it will be broken up into two rays, one of which will traverse it more rapidly than the other. Referring to the diagram of Lissajous' figures (Fig. 604), let the sides of the rectangle be the directions of greatest and least elasticity, and let the diagonal line in the first figure be the direction of the vibrations of an incident ray,—this diagonal accord¬ ingly lies in one of the two planes of reference. In traversing the slice, the component vibrations in the directions of greatest and least elasticity will be propagated with unequal velocities; and if the incident ray be homogeneous, the emergent light will be elliptically polarized; that is to say, its vibrations, instead of being rectilinear, will be elliptic, precisely on the principle 1 of Blackburn's pendulum (§ 677 a). The shape of the ellipse depends, as in the case of Lis sajous’ figures, on the amount of retardation of one of the two com¬ ponent vibrations as compared with the other, and this is directly proportional to the thickness of the slice. The analyzer resolves these elliptic vibrations into two rectilinear components parallel and per¬ pendicular to the original direction of vibration, and suppresses one of these components, so that only the other remains. Thus if the ellipse in the annexed figure (Fig. 732) represent the vibrations of the light as it emerges from the selenite, and CD, EF be tan¬ gents parallel to the original direction of vibration, the perpendicular distance between these tangents, AB, is the component vibration which is not sup¬ pressed when the analyzer is so turned that all the light would be suppressed if the selenite were re¬ moved. By rotating the analyzer, we shall obtain vibrations of vari¬ ous amplitudes, corresponding to the distances between parallel tan¬ gents drawn in various directions. For a certain thickness of selenite the ellipse will become a circle, Fig. 732.—Colours of Selenite Plates. 1 The principle is that, whereas displacement of a particle parallel to either of the sides of the rectangle calls out a restoring force directly opposite to the displacement, displace¬ ment in any other direction calls out a restoring force inclined to the direction of displace¬ ment, being in fact the resultant of the two restoring forces which its two components parallel to the sides of the rectangle would call out. COLOUKS OF SELENITE PLATES. 1039 and we have thus what is called circularly-'polarized light, which is characterized by the property that rotation of the analyzer produces no change of intensity. Circularly-polarized light is not however identical with ordinary light; for the interposition of an additional thickness of selenite converts it into elliptically (or in a particular case into plane) polarized light (§ 840). The above explanation applies to homogeneous light. When the incident light is of various refrangibilities, the retardation of one component upon the other is greatest for the rays of shortest wave¬ length. The ellipses are accordingly different for the different elemen¬ tary colours, and the analyzer in any given position will produce unequal suppression of different colours. But since the component which is suppressed in any one position of the analyzer, is the com¬ ponent which is not suppressed when the analyzer, is turned through a right angle, the light yielded in the former case plus the light yielded in the latter must be equal to the whole light which was incident on the selenite. 1 Hence the colours exhibited in these two positions must be complementary. It is necessary for the exhibition of colour in these experiments that the plate of selenite should be very thin, otherwise the retarda¬ tion of one component vibration as compared with the other will be greater by several complete periods for violet than for red, so that the ellipses will be identical for several different colours, and the total non-suppressed light will be sensibly white in all positions of the analyzer. Two thick plates may however be so combined as to produce the effect of one thin plate. For example, two selenite plates, of nearly equal thickness, may be laid one upon the other, so that the direc¬ tion of greatest elasticity in the one shall be parallel to that of least elasticity in the other. The resultant effect in this case will be that due to the difference of their thicknesses. Two plates so laid are said to be crossed. 835. Colours of Plates perpendicular to Axis.—A different class of appearances are presented when a plate, cut from a uniaxal crystal by sections perpendicular to the axis, is inserted between the polar¬ izer and the analyzer. Instead of a broad sheet of uniform colour, 1 We here neglect the light absorbed and scattered; but the loss of this does not sensibly affect the colour of the whole. It is to be borne in mind that the intensity of light is mea¬ sured by the square of the amplitude, and is therefore the simple sum of the intensities of its two components when the resolution is rectangular. 1040 POLARIZATION AND DOUBLE REFRACTION. we have now a system of coloured rings, interrupted when the analyzer is in one of the two critical positions, by a black or white cross, as at A, B (Fig. 733). 836. Explanation.—The following is the explanation of these ap- Fig. 733.—Rings and Cross. pearances. Suppose, for simplicity, that the analyzer is a plate of tourmaline held close to the eye. Then the light which comes to the eye from the nearest point of the plate under examination (the foot of a perpendicular dropped upon it from the eye), has traversed the plate normally, and therefore parallel to its optic axis. It has therefore not been resolved into an ordinary and an extraordinary ray, but has emerged from the plate in the same condition in which it entered, and is therefore black, gray, or white according to the position of the analyzer, just as it would be if the plate were re¬ moved. But the light which comes obliquely to the eye from any other part of the plate, has traversed the plate obliquely, and has undergone double refraction. Let E (Fig. 734) be the position of the eye, E O a perpendicular on the plate, P a point on the circumference of a circle described about O as centre. Then, since E 0 is parallel to the axis of the plate, the direc¬ tion of vibration for the ordinary ray at P is perpendicular to the plane E O P, and is tangen¬ tial to the circle. The direction of vibration for the extraordinary ray lies in the plane E 0 P, is nearly perpendicular to E 0 (or to the axis), if the angle 0 E P is small, and deviates more from perpendicularity to the axis as the angle O E P increases. Both for this reason, and also on account of the greater thickness traversed, the retardation of one ray upon the other is greater as P is taken further from O; and from the symmetry of the circumstances, it Fig. 734. Theory of Rings and Cross. THEORY OF RINGS AND CROSS. 1041 must be the same at the same distance from O all round. In con¬ sequence of this retardation, the light which emerges at P in the di¬ rection P E is elliptically polarized; and by the agency of the analyzer it is accordingly resolved into two components, one of which is sup¬ pressed. With homogeneous light, rings alternately dark and bright would thus be formed at distances from O corresponding to retarda¬ tions of 0, 1,1J, 2, 2§, . . . complete periods; and it can be shown that the radii of these rings would be proportional to the numbers 0, VI, V2, V3, V4, Vo, V6: . . The rings are larger for light of long than of short wave-length; and the coloured rings actually exhibited when white light is employed, are produced by the superposition of all the systems of monochromatic rings. The monochromatic rings for red light are easily seen by looking at the actual rings through a piece of red glass. Let 0, P, Fig. 735, be the same points which were denoted by these letters in Fig. 734, and let A B be the direction of vibration of the light incident on the crystal at P. Draw A C, D B parallel to O P, and complete the rectangle A CBD. Then the length and breadth of this rect¬ angle are approximately the direc¬ tions of vibration of the two com¬ ponents, one of which loses upon the other in traversing the crystal. The vibration of the emergent ray is represented by an ellipse inscribed in the rectangle A C B D (§ 676, note 2); and when the loss is half a period, this ellipse shrinks into a straight line, namely, the diagonal C D. Through C and D draw lines parallel to AB; then the distance between these, parallels represents the double amplitude of the vibration which is trans¬ mitted when there has been a retardation of half a period, and is greater than the distance between the tangents in the same direc¬ tion to any of the inscribed ellipses. A retardation of another half period will again reduce the inscribed ellipse to the straight line A B, as at first. The position D C corresponds to the brightest and A B to the darkest part of any one of the series of rings for a given wave-length of light, the analyzer being in the position for sup¬ pressing all the light if the crystal were removed. When the analyzer is turned into the position at right angles to this, A B corresponds to the brightest, and D C to the darkest parts of the rings. It is to 67 1042 POLARIZATION AND DOUBLE REFRACTION. be remembered that amount of retardation depends upon distance from the centre of the rings, and is the same all round. The two diagonals of our rectangle therefore correspond to different sizes of rings. If the analyzer is in such a position with respect to the point P considered, that the suppressed vibration is parallel to one of the sides of the rectangle (in other words, if 0 P, or a line perpendi¬ cular to 0 P, is the direction of suppression) the retardation of one component upon the other has no influence, inasmuch as one of the two components is completely suppressed and the other is completely transmitted. There are, accordingly, in all positions of the analyzer, a pair of diameters, coinciding with the directions of suppression and non-suppression, which are alike along their whole length and free from colour. Again if P is situated at B or at 90° from B, the corner C of the rectangle coincides with B or with A, and the rectangle, with all its inscribed ellipses, shrinks into the straight line AB. The two diameters coincident with and perpendicular to AB are therefore alike along their whole length and uncoloured. The two colourless crosses which we have thus accounted for, one of them turning with the analyzer and the other remaining fixed with the polarizer, are easily observed when the analyzer is not near the critical positions. In the critical positions, the two crosses come into coincidence; and these are also the positions of maximum black¬ ness or maximum whiteness for the two crosses considered separ¬ ately. Hence the conspicuous character of the cross in either of these positions, as represented at A, B, Fig. 733. As the analyzer is turned away from these positions, the cross at first turns after it with half its angular velocity, but soon breaks up into rings, some¬ what in the manner represented at C, which corresponds to a posi¬ tion not differing much from A. 837. Biaxal Crystals. — Crystals may be divided optically into three classes:— 1. Those in which there is no distinction of different directions, as regards optical properties. Such crystals are said to be optically isotropic. 2. Those in which the optical properties are the same for all direc¬ tions equally inclined to one particular direction called the optic axis, but vary according to this inclination. Such crystals are called uniaxal. BIAXAL CRYSTALS. 1043 3. All remaining crystals (excluding compound and irregular for¬ mations) belong to the class called biaxal. In any homogeneous elastic solid, there are three cardinal directions called axes of elasti¬ city, possessing the same distinctive properties which belong to the two principal planes of vibration in Blackburn’s pendulum (§ 677 a) ; that is to say, if any small portion of the solid be distorted by for¬ cibly displacing one of its particles in one of these cardinal directions, the forces of elasticity thus evoked tend to urge the particle directly back; whereas displacement in any other direction calls out forces whose resultant is generally oblique to the direction of displacement, so that when the particle is released it does not fly back through the position of equilibrium, but passes on one side of it, just as the bob of Blackburn s pendulum generally passes beside and not through the lowest point which it can reach. In biaxal crystals, the resistances to displacement in the three cardinal directions are all unequal; and this is true not only for the crystalline substance itself, but also for the luminiferous aether which pervades it, and is influenced by it. 1 The construction given by Fresnel for the wave-surface in any crystal is as follows:—First take an ellipsoid, having its axes parallel to the three cardinal direc¬ tions, and of lengths depending on the particular crystalline sub¬ stance considered. Then let any plane sections (which will of course be ellipses) be made through the centre of this ellipsoid, let normals to them be drawn through the centre, and on each normal let points be taken at distances from the centre equal to the greatest and least radii of the corresponding section. The locus of these points is the complete wave-surface, which consists of two sheets cutting one another at four points. These four points of intersection are situated upon the normals to the two circular sections of the ellipsoid, and the two optic axes, from which biaxal crystals derive their name, are closely related to these two circular sections. The optic axes are the directions of single wave-velocity , and the normals to the two circular sections are the directions of single ray-velocity. The direction of advance of a wave is always regarded as normal to the front of the wave, whereas the direction of a ray (defined by the condition of traversing two apertures placed in its path) always passes through the centre of the wave-surface, and is not in general normal to the front. Both these pairs of directions of single velo- 1 The cardinal directions are however believed not to be the same for the aether as for the material of the crystal. 1044 POLARIZATION AND DOUBLE REFRACTION. city are m the plane which contains the greatest and least axes of the ellipsoid. When two axes of the ellipsoid are equal, it becomes a spheroid, and the crystal is uniaxal. When all three axes are equal, it be¬ comes a sphere, and the crystal is isotropic. Experiment has shown that biaxal crystals expand with heat unequally in three cardinal directions, so that in fact a spherical piece of such a crystal is changed into an ellipsoid 1 when its tem¬ perature is raised or lowered. A spherical piece of a uniaxal crystal in the same circumstances changes into a spheroid; and a spherical piece of an isotropic crystal remains a sphere. It is generally possible to determine to which of the three classes a crystal belongs, from a mere inspection of its shape as it occurs in nature. Isotropic crystals are sometimes said to be symmetrical about a point, uniaxal crystals about a line, biaxal crystals about neither. The following statement is rather more precise:— If there is one and only one line about which if the crystal be rotated through 90° or else through 120° the crystalline form remains in its original position, the crystal is uniaxal, having that line for the axis. If there is more than one such line, the crystal is isotropic, while, if there is no such line, it is biaxal. Even in the last case, if there exist a plane of crystalline symmetry, such that one half of the crystal is the reflected image of the other half with respect to this plane, it is also a plane of optical symmetry, and one of the three cardinal directions for the aether is perpendicular to it. 2 Glass, when in a strained condition, ceases to be isotropic, and if inserted between a polarizer and an analyzer, exhibits coloured streaks or spots, which afford an indication of the distribution of strain through its substance. The experiment is shown sometimes with unannealed glass, which is in a condition of permanent strain, sometimes with a piece of ordinary glass which can be subjected to force at pleasure by turning a screw. Any very small portion of a piece of strained glass has the optical properties of a crystal, but different portions have different properties, and hence the glass as a whole does not behave like one crystal. The production of colour by interposition between a polarizer and 1 This fact furnishes the best possible definition of an ellipsoid for persons unacquainted with solid geometry. 2 The optic axes either lie in the plane of symmetry, or lie in a perpendicular plane and are equally inclined to the plane of symmetry. For the precise statement here given, the Editor is indebted to Professor Stokes. ROTATION OF PLANE OF POLARIZATION. 1045 an analyzer, is by far the most delicate test of double refraction. Many organic bodies (for example, grains of starch) are thus found to be doubly refracting; and microscopists often avail themselves of this means of detecting diversities of structure in the objects which they examine. 838, Rotation of Plane of Polarization.—When a plate of quartz (rock-crystal), even of considerable thickness, cut perpendicular to the axis, is interposed between the polarizer and analyzer, colour is exhibited, the tints changing as the analyzer is rotated; and similar effects of colour are produced by employing, instead of quartz, a solu¬ tion of sugar, inclosed in a tube with plane glass ends. If homogeneous light is employed, it is found that if the analyzer is first adjusted to produce extinction of the polarized light, and the quartz or saccharine solution is then introduced, there is a partial restoration of light. On rotating the analyzer through a certain angle, there is again complete extinction of the light; and on com¬ paring different plates of quartz, it will be found that the angle through which the analyzer must be rotated is proportional to the thickness of the plate. In the case of solutions of sugar, the angle is proportional jointly to the length of the tube and the strength of the solution. The action thus exerted by quartz or sugar is called rotation of the plome of polarization, a name which precisely expresses the observed phenomena. In the case of ordinary quartz, and solutions of sugar-candy, it is necessary to rotate the analyzer in the direc¬ tion of watch-hands as seen by the observer, and the rotation of the plane of polarization is said to be right-handed. In the case of what is called left-handed quartz, and of solutions of non-crystal- lizable sugar, the rotation of the plane of polarization is in the opposite direction, and the observer must rotate the analyzer against watch-hands. The amount of rotation is different for the different elementary colours, and has been found to be inversely as the square of the wave-length. Hence the production of colour. 839. Magneto-optic Rotation.—Faraday made the remarkable dis¬ covery that the plane of polarization can be rotated in certain cir¬ cumstances by the action of magnetism. Let a long rectangular piece of “heavy-glass” (silico-borate of lead) be placed longitudinally between the poles of the powerful electro-magnet represented in Fig. 432 (Part III.), which is for this purpose made hollow in its axis, 1046 POLARIZATION AND DOUBLE REFRACTION. so that an observer can see through it from end to end. Let a Nicol’s prism be fitted into one end of the magnet, to serve as polarizer, and another into the other end to serve as analyzer, and let one of them be turned till the light is extinguished. Then, as long as no current is passed round the electro-magnet, the interposition of the heavy-glass will produce no effect; but the passing of a current, while the heavy-glass is in its place between the poles, produces rotation of the plane of polarization in the same direction as that in which the current circulates. The amount of rotation is directly as the strength of current, and directly as the length of heavy-glass traversed by the light. Flint-glass gives about half the effect of heavy-glass, and all transparent solids and liquids exhibit an effect of the same kind in a more or less marked degree. A steel magnet, if extremely powerful, may be used instead of an electro-magnet; and in all cases, to give the strongest effect, the lines of magnetic force should coincide with the direction of the trans¬ mitted ray. Faraday regarded these phenomena as proving the direct action of magnetism upon light; but it is now more commonly believed that the direct effect of the magnetism is to put the particles of the transparent body in a peculiar state of strain, to which the observed optical effect is due. In every case tried by Faraday, the direction of the rotation was the same as the direction in which the current circulated; but cer¬ tain substances 1 have since been found which give rotation against the current. The law for the relative amounts of rotation of differ¬ ent colours is approximately the same as in the case of quartz. The direction of rotation is with watch-hands as seen from one end of the arrangement, arid against watch-hands as seen from the other; so that the same piece of glass, in the same circumstances, behaves like right-handed quartz to light entering it at one end, and like left-handed quartz to light entering it at the other. The rotatory power of quartz and sugar appears to depend upon a certain unsymmetrical arrangement of their molecules, an arrange¬ ment somewhat analogous to the thread of a screw; right-handed and left-handed screws representing the two opposite rotatory powers. It is worthy of note that the two kinds of quartz crystallize in different forms, each of which is unsymmetrical, one being like 1 One such substance is a solution of Fe 2 Cl 3 ('old notation) in methylic (not methylated) alcohol. CIRCULAR POLARIZATION. 1047 the image of the other as seen in a looking-glass. Pasteur has con¬ ducted extremely interesting researches into the relations existing between substances which, while in other respects identical or nearly identical, differ as regards their power of producing rotation. For the results we must refer to treatises on chemistry. 840. Circular Polarization. Fresnel's Rhomb.—We have explained in § 834 the process by which elliptic polarization is brought about, when plane-polarized light is transmitted through a thin plate of selenite. To obtain circular polarization (which is merely a case of elliptic), the plate must be of such thickness as to retard one com¬ ponent more than the other by a quarter of a wave-length, and must be laid so that the directions of the two component vibrations make angles of 45° with the plane of polarization. Plates specially pre¬ pared for this purpose are in general use, and are called quarter- wave 'plates . They are usually of mica, which differs but little in its properties from selenite. It is impossible, however, in this way to obtain complete circular polarization of ordinary white light, since different thicknesses are required for light of different wave-lengths, the thickness which is appropriate for violet being too small for red. Fresnel discovered that plane-polarized light is elliptically polarized by total internal reflection in glass, whenever the plane of polarization of the inci¬ dent light is inclined to the plane of incidence. The rectilinear vibrations of the incident light are in fact resolved into two components, one of them in, and the other perpendicular to, the plane of incidence; and one of these is retarded with respect to the other in the act of reflection, by an amount depending on the angle of incidence. He determined the magnitude of this angle for which the retardation is precisely ^ of a wave-length; and constructed a rhomb, or ob¬ lique parallelopiped of glass (Fig. 736), in which a u r . r 1 Two Fresnel’s Rhombs. ray, entering normally at one end, undergoes two suc¬ cessive reflections at this angle (about 55°), the plane of reflection being the same in both. The total retardation of one component on the other is thus \ of a wave-length; and if the rhomb is in such a position that the plane in which the two reflections take place is at an angle of 45° to the plane of polarization of the incident light, the emergent light is circularly-polarized. The effect does not vary i’ig. 736. 1048 POLARIZATION AND DOUBLE REFRACTION. much with the wave-length, and sensibly white circularly polarized light can accordingly be obtained by this method. When circularly-polarized light is transmitted through a Fresnel’s rhomb, or through a quarter-wave plate, it becomes plane-polarized, and we have thus a simple mode of distinguishing circularly-polarized light from common light; for the latter does not become polarized when thus treated. Two quarter-wave plates, or two Fresnel’s rhombs, may be combined either so as to assist or to oppose one another. By the former arrangement, which is represented in Fig. 736, we can convert plane-polarized light into light polarized in a perpendi¬ cular plane, the final result being therefore the same as if the plane of polarization had been rotated through 90°. The several steps of the process are illustrated by the five diagrams of Fig. 737, which Fig. 737.—Form of Vibration in traversing the Rhombs. represent the vibrations of the five portions A C, C D, D d, do ca of the ray which traverses the two rhombs in the preceding figure. The sides of the square are parallel to the directions of resolution; the initial direction of vibration is one diagonal of the square, and the final direction is the other diagonal; a gain or loss of half a com¬ plete vibration on the part of either component being just sufficient to effect this change. 841. Direction of Vibration of Plane-polarized Light.—The plane of polarization of plane-polarized light may be defined as the plane in which it is most copiously reflected. It is perpendicular to the plane in which the light refuses to be reflected (at the polarizing angle); and is identical with the original plane of reflection, if the polariza¬ tion was produced by reflection. This definition is somewhat arbi¬ trary, but has been adopted by universal consent. When light is polarized by the double refraction of Iceland-spar, or of any other uniaxal crystal, it is found that the plane of polariza¬ tion of the ordinary ray is the plane which contains the axis of the crystal. But the distinctive properties of the ordinary ray are most naturally explained. by supposing that its vibrations are perpendi¬ cular to the axis. Hence we conclude that the direction of vibration VIBRATIONS OF ORDINARY LIGHT. 1049 in plane-polarized light is normal to the so-called plane of polariza¬ tion, and therefore that, in polarization by reflection, the vibrations of the reflected light are parallel to the reflecting surface. This is Fresnel’s doctrine. MacCullagh, however, reversed this hypothesis, and maintained that the direction of vibration is in the plane of polarization. Both theories have been ably expounded; but Stokes contrived a crucial experiment in diffraction, which con¬ firmed Fresnel’s view; 1 and in his classical paper on “Change of Re- frangibility,” he has deduced the same conclusion from a considera¬ tion of the phenomena of the polarization of light by reflection from excessively fine particles of solid matter in suspension in a liquid 2 842. Vibrations of Ordinary Light.—Ordinary light agrees with circularly-polarized light in always yielding two beams of equal intensity when subjected to double refraction; but it differs from cir¬ cularly-polarized light in not becoming plane-polarized by transmis¬ sion through a Fresnel’s rhomb or a quarter-wave plate. What, then, can be the form of vibration for common light A It is probably very irregular, consisting of ellipses of various sizes, positions, and forms (including circles and straight lines), rapidly succeeding one another. By this irregularity we can account for the fact that beams of light from different sources (even from different points of the same flame, or from different parts of the sun’s disc), cannot, by any treatment whatever, be made to exhibit the phenomena of mutual interference; and for the additional fact that the two rectangular components into which a beam of common light is resolved by double refraction, cannot be made to interfere, even if their planes of polarization are brought into coincidence by one of the methods of rotation above described. Certain phenomena of interference show that a few hundred con¬ secutive vibrations of common light may be regarded as similar; but as the number of vibrations in a second is about 500 millions of millions, there is ample room for excessive diversity during the time that one impression remains upon the retina. 843. Polarization of Radiant Heat.—The fundamental identity of radiant heat and light is confirmed by thermal experiments on polarization. Such experiments were first successfully performed by Forbes in 1834, shortly after Melloni’s invention of the thermo¬ multiplier. He first proved the polarization of heat by tourmaline; 1 Cambridge Transactions. 1850. 3 Philosophical Transactions, 1852; pp. 530, 531. 1050 POLARIZATION AND DOUBLE REFRACTION. next by transmission through a bundle of very thin mica plates, inclined to the transmitted rays; and afterwards by reflection from the multiplied surfaces of a pile of thin mica plates placed at the polarizing angle. He next succeeded in showing that polarized heat, even when quite obscure, is subject to the same modifications which doubly refracting crystallized bodies impress upon light, by suffering a beam of heat, after being polarized by transmission, to pass through an interposed plate of mica, serving the purpose of the plate of selenite in the experiment of § 833, the heat traversing a second mica bundle before it was received on the thermo-pile. As the interposed plate was turned round in its own plane, the amount of heat shown by the galvanometer was found to fluctuate just as the amount of light received by the eye under similar circumstances would have done. He also succeeded in producing circular polarization of heat by a Fresnel’s rhomb of rock-salt. These results have since been fully confirmed by the experiments of other observers. PROBLEMS. [Words inclosed within square brackets [ ] have been interpolated by the Editor.] I.—DYNAMICS AND HYDROSTATICS. 1. Two projectiles are successively discharged vertically upwards from the same point, with a velocity of 100 metres per second. What must be the interval of time between their discharges that the second may move for 8*7 sec. before meeting the first ? 2. In an Atwood’s machine the equal weights at the two ends of the thread are each 100 grammes. What must be the additional weight laid upon one of them that the space traversed in the first two seconds of the fall may be 4 decimetres ? 3. A body is thrown horizontally from the top of a tower 100 m. hig;h, with a velocity of 30 metres per sec. When and where will it strike the ground ? 4 Two bodies are successively dropped from the same point, with an interval of jr of a second. When will the distance between them be 1 metre ? 5. A stone is dropped into a well, and after 2 sec. is heard to strike the bottom. What is the depth ? 6. Explain the well-known fact that a straight stick, loaded with lead at one end, can be more easily balanced vertically on the finger when the loaded end is upwards than when it is downwards. 7. Find the centre of gravity of a sphere 1 decimetre in radius, hav¬ ing in its interior a spherical excavation whose centre is at a distance of 5 centimetres from the centre of the large sphere [and whose radius is 4 centimetres]. 8. Two small spheres, one of lead, weighing 100 grammes, the other of ivory, weighing 25 gr., are connected together by a thread 0*75 cm. long, and mounted on the rod of the centrifugal force apparatus [Fig. 37]. Find the position of [unstable] equilibrium. a 1052 PROBLEMS. 9. A round table is supported on one central leg. At what points of its circumference must weights of 4, 5 and 6 kilog. be placed, that the resultant pressure may act at the centre ? 10. A glass globe is full of air at atmospheric pressure (750 mm.) It is exhausted till the pressure is only x mm. Hydrogen is then admitted till atmospheric pressure is again established, and the mixture is then exhausted till the pressure is again reduced to x mm. Hydrogen is then a second time admitted till atmospheric pressure is established. If the weight of air in the globe at the conclusion of this operation is o' the weight of the hydrogen, what is the value of x? The temperature is supposed constant throughout the operation, and the specific gravity of hydrogen as compared with air is 0*0692. 11. A piece of iron, when plunged in a vessel full of water, makes 10 grammes run over. When placed in a vessel full of mercury, it floats, displacing 78 grammes of mercury. Required the weight, volume, and specific gravity of the iron. 12. A cylinder of steel, 22 cm. long, is to be counterpoised by a cylin¬ der of platinum of the same diameter. What must be the length of the platinum cylinder? (sp. gr. of steel 7*5, of platinum 22*5.) 13. Two liquids are mixed. The total volume is 3 litres, with a sp. gr. of 0*9. The sp. gr. of the first liquid is 1*3, of the second 0*7. Find their volumes. 14. A curved tube has two vertical legs, one having a section of 1 sq. cm., the other of 10 sq. cm. Water is poured in, and stands at the same height in both legs. A piston, weighing 5 kilogrammes, is then allowed to descend, and press with its own weight upon the surface of the liquid in the larger leg. Find the elevation thus produced in the surface of the liquid in the smaller leg. 15. A frustum of a cone of cork, the radii of its ends being 2 and 1 decim. respectively, and its height 1 decim., floats freely in water, with its axis vertical. Find how much of its axis is immersed, the sp. gr. of cork being 0*24. 16. What volume of platinum must be attached to a litre of iron, that the system may float freely at all depths in mercury ? 17. What must be the thickness of a hollow sphere of platinum with an external radius of 1 decim., that it may barely float in water ? 18. A sphere of cork, 3 cm. in radius, is weighted with a sphere of gold. What must be the radius of the latter that the system may barely float in alcohol ? 19. An alloy of gold and silver has density D. The density of gold is PROBLEMS. 1053 d , that of silver d'. Find the proportions by weight of the two metals in the alloy, supposing that neither expansion nor contraction occurs in its formation. 20. Given the weight of a body in air, and in water at maximum den¬ sity ; deduce its weight in vacuo. 21. A vertical cylinder, 1 decim. in diameter and 3 decim. high, com¬ municates at its lower part with a tube 1 centim. in diameter, which is bent up and continued vertically to a sufficient height, its upper end being left open. The cylinder is half full of mercury, its upper half being occupied by air at atmospheric pressure. What additional weight of air must be forced in to produce a fall of 10 cm. in the level of the mercury in the cylinder ? 22. An open manometer, formed of a bent tube of iron whose two branches are parallel and vertical, and of a glass tube of larger size, contains mercury at the same level in both branches, this level being higher than the junction of the iron with the glass tube. What must be the ratio of the sections of the two tubes, that the mercury may ascend half a metre in the glass tube when a pressure of 6 atmospheres is exerted in the opposite branch ? 23. A receiver A, with a capacity of 3 litres, can be put in communi¬ cation either with a forcing pump P, or with the external air. The former communication is established by a valve R, and the latter by a cock R'. The receiver A is initially filled with air at 0° C. and 760 mm. The pump P is supplied from a gasometer containing carbonic acid, at the constant pressure 760 mm. and temperature 0° C., and when R is open the capacity of the pump-barrel is 2 litres. R' is closed. One stroke of the pump is taken, and when time has been allowed for the gases to become thoroughly mixed, R' is opened for an instant, so that equilibrium of pressure is established between A and the external air. R' is then closed, a second stroke is taken, and so on, the cock R' being opened for an instant after each stroke. How many strokes must be taken, that not more than a centigramme of air may be left in the receiver? The external pressure is % supposed to re¬ main constant at 760 mm. 24. There is a glass tube a metre long, with an internal section of 1 square centimetre, the external section being 2 sq. cm., and conse¬ quently the section of the glass itself 1 sq. cm. This tube, being supposed closed at one end by a flat stopper without thickness and without weight, is filled with mercury and inverted in a deep vessel of the same liquid ; 10 cubic centimetres of air at the external pressure and temperature are then introduced, and the tube is 1054 PROBLEMS. left to itself in a vertical position. Required the volume of the air in the tube when equilibrium is attained ; and the difference between the internal and external level of the mercury. Specific gravity of glass, 2*49 ; external pressure, 760 mm. 25. Given that the sp. gr. of the solution, containing 85 parts of water and 15 of salt, which is employed for graduating Baume’s hydrometer for acids, is 1*116, establish the formula D = which gives thesp. gr. in terms of the degree read off at the surface of the liquid. 26. In the graduation of Baume’s hydrometer for spirits, a solution containing 90 parts of water and 10 of salt is employed, its sp. gr. being 1*084. Deduce the formula 129 D =- 119 + N 27. Find, to the nearest millimetre, the edge of a regular tetrahedron of coinage gold, of the value of 1000 francs, the sp. gr. of this gold being 18 [and the value of 1 gramme of it being 3*1 francs]. 28. The pressure indicated by a siphon barometer is 750 mm., and when mercury is poured into the open branch till the barometric cham¬ ber is reduced to half its former volume, the pressure indicated is 740 mm. Deduce the true pressure. 29. A cylindrical test-tube, 1 decim. long, is plunged, mouth down¬ wards, into mercury. How deep must it be plunged that the volume of the inclosed air may be diminished by one-half ? 30. In an air-pump, whose receiver has a capacity of 1 litre, it is found that three strokes reduce the pressure from 0*760 m. to 0*315 m. The experiment is repeated after a body has been introduced into the receiver, and it is found that the same number of strokes reduce the pressure from 0*760 m. to 0*200 m. Deduce the volume of the body. 31. A cylindrical test-tube, 1 decim. in height and 2 centim. in diame¬ ter, floats upright in water, its mouth being downwards, and its top being just level with the surface of the water. To what height does the liquid rise in its interior ? 32. A test-tube floats upright in water, with its mouth downwards. To make its top come down to the surface of the water, it is found necessary to load it with a weight, which, added to its own weight, gives a total weight P. The experiment is repeated with the open end upwards, and the weight which is then necessary to bring its top down to the level of the water, including the weight of the tube itself, is Q. Deduce the atmospheric pressure. 33. A cylinder of wood, 1 decim. long, and with a sp. gr. 0*96, floats PROBLEMS. 1055 upright in water. The vessel is placed in a receiver in which the air can be compressed to 40 atmospheres. Find the change produced by this pressure in the position of the cylinder. II.—HEAT. 34. A truly conical vessel contains a certain quantity of mercury at 0° C. To what temperature must the vessel and its contents be raised that the depth of the liquid may be increased by T -|-g- of itself ? 35. What temperature is denoted by the same number in the Centi¬ grade as in the Fahrenheit scale ? Is there more than one temperature which fulfils this condition ? 36. A Graham’s compensating pendulum is formed of an iron rod, whose length at 0° C. is l , carrying a cylindrical vessel of glass, which at the same temperature has an internal radius r, and height h. Find the depth x of mercury at 0° C. which is necessary for compensation, sup¬ posing that the compensation consists in keeping the centre of gravity of the mercury at a constant distance from the axis of suspension. 37. A brass tube contains mercury, with a piece of platinum immersed in it ; and the level of the liquid is marked by a scratch on the inside of the tube. On applying heat, it is found that the liquid still stands at this mark. Deduce the ratio of the weight of the platinum to that of the mercury. 38. A glass tube, closed at one end and drawn out at the other, is filled with dry air, and raised to a temperature x at atmospheric pressure. .It is then hermetically sealed. When it has been cooled to the temper¬ ature 100° C, it is inverted over mercury, and its pointed end is broken off beneath the surface of the liquid. The mercury rises to the height of 19 centimetres in the tube, the external pressure remaining at 76 cm. as at the commencement of the experiment. The tube is re-inverted, and weighed with the mercury which it contains. The weight of this mercury is found to be 200 grammes ; when completely full it contains 300 grammes of mercury. Deduce the temperature x. 39. A glass tube, whose interior is a right circular cylinder, 2 milli¬ metres in diameter at 0° C., contains a column of mercury, whose length at this temperature is 2 decim. What will be the length of this column of mercury when the temperature is 80° C., the co-efficient of expan¬ sion of mercury being and the co-efficient of cubical expansion of glass -g-gVo? 40. A closed globe, whose external volume at 0° C. is 10 litres, is im¬ mersed in air at 15° 0. and at a pressure of 0*77 m. Required (1) the 1056 PROBLEMS. loss of weight which it experiences from the action of the air ; (2) the change which this loss would undergo if the pressure became 0*768 m. and the temperature 17° C. 41. Some dry air is inclosed in a horizontal thermometric tube, by means of an index of mercury. At 0° C. and 0*760 m. the air occupies 720 divisions of the tube, the tube being divided into parts of equal capacity. At an unknown temperature and pressure, the same air occu¬ pies 960 divisions. The tube being immersed in melting ice, and the latter pressure being still maintained, the air occupies 750 divisions. Required the temperature and pressure. 42. At what temperature will the density of oxygen at the pressure 0*20 m., be the same as that of hydrogen at 0° C. at the pressure 1*60 m. ? 43. What is the interior volume at 0° C. of a glass bulb which at 25° C. is exactly filled by 53 grammes of mercury ? 44. A barometer at one time reads 770 mm., with a temperature 85° C., and at another time 760 mm., with a temperature 5° C. Find the ratio of the true barometric heights. 45. The specific gravity of copper at 0° C. is 8*8 ; its co-efficient of linear expansion is ^-gxoo- What will be the length at 30° of a roll of copper wire weighing 15 kilogrammes, the section of the wire at 10° C. being 4 square millimetres ? 46. The normal density of air being 0*000154 of that of brass, what change is produced in the apparent weight of a kilogramme of brass when the pressure and temperature of the air change from 713 mm. and —19° C. to 781 mm. and -j-36° C. ? 47. A cylindrical tube of glass is divided into 300 equal parts. It is loaded with mercury, and sinks to the 50th division in water at 10° C. To what division will it sink in water at 50° C. ? The volumes of a given mass of water at 10° and 50° are as 1*000268 and 1*01205. 48. Find the co-efficient of expansion of air, when zero Fahrenheit is the starting point, and the degree Fahrenheit the unit interval. 49. What must be the pressure of air at 15° C., that its density may be the same as that of hydrogen at 0° C. and 760 mm. ? Density of hydrogen 0*0692. 50. At what temperature does a litre of dry air at 760 mm. weigh 1 gramme ? 51. In a cubic metre of air at 20° C., 11*56 grammes of vapour are found. What is the relative humidity of this air ? 52. Calculate the weight of 15 litres of air saturated with aqueous PROBLEMS. 1057 vapour, at 20° C. and 750 mm. The maximum tension of vapour at 20° is 17*39 mm. 53. There is a bent tube, terminating at one end in a large bulb, and simply closed at the other. A column of mercury stands at the same height in the two branches, and thus separates two quantities of air at the same pressure. The air in the bulb is saturated with moisture ; that in the opposite branch is perfectly dry. The length of the column of dry air is known, and also its initial pressure, the temperature of the whole being 0° C. Calculate the displacement of the mercurial column when the temperature of the apparatus is raised to 100° C. The bulb is supposed to have enough water in it to keep the air constantly saturated; and is also supposed to be so large that the volume of the moist air is not sensibly affected by the displacement of the mercurial column. 54. A litre of alcohol, measured at 0° C., is contained in a brass vessel weighing 100 grammes, and [after being raised to 58° C.] is immersed in a kilogramme of water at 10° C., contained in a brass Vessel weighing 200 grammes. The temperature of the water is thereby raised to 27°. What is the specific heat of alcohol ? The specific gravity of alcohol is 0*8 ; the specific heat of brass is 0T. 55. A copper vessel, weighing 1 kilogramme, contains 2 kilogr. of water. A thermometer, composed of 100 grammes of glass and 200 gr. of mercury, is completely immersed in this water. All these bodies are at the same temperature, 0° C. If 100 grammes of steam at 100° C. are passed into the vessel, and condensed in it, what will be the temperature of the whole apparatus when equilibrium has been attained, supposing that there is no loss of heat externally. The specific heat of mercury is 0*033 ; of copper, 0*095 ; of glass, 0*177. III.—ACOUSTICS AND OPTICS. 56. The specific gravity of platinum being taken as 22, and that of iron as 7*8, what must be the ratio of the lengths of two wires, one of platinum and the other of iron, both of the same section, that they may vibrate in unison when stretched with equal forces ? 57. Two strings of the same length and section are formed of materials whose specific gravities are respectively d and d'. Each of these strings is stretched with a weight equal to [1000 times] its own weight. What is the musical interval between the notes which they will yield ? 58. A pipe gives a note of 100 vibrations per second at the tempera¬ ture 10° C. What must be the temperature of the air that the same pipe may yield a note higher by a major fifth ? 1058 PROBLEMS. 59. What is the least height that a plane mirror can have, that the whole of a given vertical object may be seen reflected in it at one view ? 60. The flame of a candle is placed on the axis of a concave spherical mirror at the distance of 1*54 m., and its image is formed at the distance of 0*45 m. What is the radius of curvature of the mirror ? 61. On the axis of a concave spherical mirror of 1 m. radius, an object 9 cm. high is placed at a distance of 2 m. Find the size and position of the image. 62. What is the size of the circular image of the sun which is formed at the principal focus of a mirror of 20 m. radius ? The apparent diam¬ eter of the sun is 30'. 63. In front of a concave spherical mirror of 2 metres’ radius is placed a concave luminous arrow, 1 decimetre long, perpendicular to the princi¬ pal axis, and at the distance of 5 metres from the mirror. What are the position and size of the image ? A small plane reflector is then placed at the principal focus of the spherical mirror, at an inclination of 45° to the principal axis, its polished side being next the mirror. What will be the new position of the image ? 64. A pencil of parallel rays fall upon a sphere of glass of 1 metre radius. Find the principal focus of rays near the axis, the index of refraction of glass being 1*5. 65. What is the focal length of a double-convex lens of diamond, the radius of curvature of each of its faces being 4 millimetres ? Index of refraction 2*487. 66. An object 8 centimetres high is placed at 1 metre distance on the axis of an equi-convex lens of ordinary crown-glass, the radius of curva¬ ture of its faces being 0*4 m. Find the size and position of the image. 67. What is the ratio of the magnifying power of a diamond lens to that of a glass lens of the same focal length [supposed small] ? Index for glass, 1*5 ; for diamond, 2*481. 68. A Gregorian telescope is constructed in the following manner *.— The rays after reflection from the objective speculum form a real image at the principal focus. Continuing their course, they meet a small con¬ cave mirror, which reflects them so that they form a second image, in¬ verted with respect to the first, and consequently erect with respect to the object. This second image is viewed through an eye-piece, whose tube passes through a hole in the objective. Investigate a formula for the magnifying power of the telescope. 69. Two converging lenses, with a common focal length of 0*05 m., are at a distance of 0*03 m. apart, and their axes coincide. What image PROBLEMS. 1059 will this system give of a circle 0*01 m. in diameter, placed successively at different distances on the prolongation of the common axis ? 70. A convex lens of vocal length /*, is cemented to a concave lens of focal length f. What is the focal length of the system ? 71. The stem of a siren carries a plane mirror, thin, polished on both sides, and parallel to the axis of the stem. The siren gives a note of 345 vibrations per second. The revolving plate has 15 holes. A fixed source of light sends to the mirror a horizontal pencil of parallel rays. What space is traversed in a minute by a point of the reflected pencil at a distance of 4 metres from the axis of the siren ? This axis is supposed vertical. 72. A lamp and a taper are at a distance of 4T5 in. from each other; and it is known that their illuminating powers are as 6 to 1. At what distance from the lamp, in the straight line joining the flames, must a screen be placed that it may be equally illuminated by them both ? 73. A ray of light falls perpendicularly on the surface of an equilateral prism of glass with a refracting angle of 60°. What will be the devia¬ tion produced by the prism? Index of refraction of glass 1*5. 74. What is the length of the cone of the umbra thrown by the earth ? and what is the diameter of a cross section of it made at a distance equal to that of the moon? The radius of the sun is 112 radii of the earth ; the distance of the moon from the earth is 60 radii of the earth ; and the distance of the sun from the earth is 24,000 radii of the earth. Atmospheric refraction is to be neglected. 75. A sphere of glass lying upon a horizontal plane receives the suits rays. What must be the height of the sun above the horizon that the principal focus of the sphere may be in this horizontal plane ? a 2 INDEX A berration, astronomical, 879, — chromatic, 994. — spherical, 894. Absolute temperature and abso¬ lute zero by air thermometer, 298. __by thermo-dynamic scale, 457. — unit of force, 54, 780. -of work, 450, 780. — units, 788. Absorption and emission of radi¬ ant heat, 394. — of gases, 182. Acceleration defined, 53. — uniform, 52. Accidental images, 1009. Accumulation by mutual action, 773. Achromatism, 995. Acoustic pendulum, 812. Actinometer, 462. Adiabatic changes of volume and pressure, 436. Aether, luminiferous, 865,1043. Air, cooling of, by ascent, 498. — density of dry, 140-142, 296. -of moist, 375. — temperature of, 493-498. — vibration of, 788. Air-chamber, 221. Air-engine, 467. Air-film, adherent, 183. Air-pump, 184-202. — Babinet’s, 199. — Bianchi’s, 189. — Deleuil’s, 200. — Geissler’s, 195. — Kravogl’s, 194. — Sprengel's, 197. — condensing, 202. — limits of action of, 192. Alarum, telegraphic, 721. Alcohol at low temperatures, 333. — thermometers, 254. Alphabet, telegraphic, 724. Alternate contact, 530. -discharge by, 573. Alum, its small diathermancy, 406, 410. Amalgam for rubbers, 536. Amalgamated zinc, 651. Ampere’s electro-dynamic for¬ mula, 688. — rule for deflection, 657. Ampere’s stand, 680. — theory of magnetism, 694. Amplitude of vibration, 57,70,786. Analyzer, 1032. Anamorphosis, 906. Andrews’s calorimetric experi¬ ments, 443. — on continuity of liquid and gaseous states, 326. Anemometers, 503. Aneroid barometer, 157. Animal heat and work, 461. Annual variations defined, 165. Anode, 739. Apertures, small, images produced by, 868. Apjohn’s formula, 373. Arago’s rotations, 775. Arc, voltaic, 703. Archimedes’ principle, 104. Aristotle’s experiment on weight of air, 140. Arm of couple, 16. Armstrong’s hydro-electric ma¬ chine, 589. Arrangement of cells in battery, 671. Artificial horizon, 884. Ascent, cooling of air by, 498. — in capillary tubes, 128. Aspirator, 373. Astatic circuits, 693. — galvanometer, 662. — needle, 661. Astronomical telescope, 958. Atlantic cable, 733. -velocity through, 586. Atmosphere, distribution of, over the earth, 501. — pressure of, 142-144. Atmospheric circulation, general, 500. — electricity, 599-611. -modes of observing, 603-606. -results of observation, 607. — refraction, 1018. Atomic weight inversely as spe¬ cific heat, 435. Atoms, 24. Attraction, apparent, due to capil¬ larity, 136. — electrical, laws of, 520-523. — magnetic, laws of, 619. Attwood’s machine, 42. August’s psychometer, 370. I Aurora borealis, 634. Aurum musivum, 536. Austral pole, 616. Autographic telegraph, 730. Automatic system, Wheatstone’s, 735. Axes, optic, in crystals, 1043. Axis, magnetic, 620. — of Iceland spar, 927. Azimuth, 614. B ABBAGE & Herschel’s rota¬ tions, 776. Babinet’s double exhaustion, 199. Bain’s electro-chemical telegraph, 730. Balance, 80-86. — spring, 30. — torsion, 519, 624. Barker’s mill. 102. Barograph, King's. 160. — photographic, 161. — Secchi’s, 160. Barometer, 140-169. — Adie’s, 156. — aneroid, 157. — counterpoised, 159. — Fortin’s, 147. — marine, 156. — siphon, 154. — wheel, 155. Barometric corrections, 150. — measurement of heights. 162. — prediction of weather, 167-169. — variations, 165-169. — variation with latitude, 501. Baroscope, 208. Battery, galvanic, 644. -Bunsen’s, 650. -Cruickshank’s, 650. -Daniell’s, 649. -Grove’s, 651. -Hare's, 648. -telegraphic, 715. --Wollaston’s, 647. Battery of Leyden jars, 580. — discharge of, 583. Beats, 813, 860. Beaume’s hydrometers, 118. Becquerel’s phosphoroscope, 979. Bellows of organ, 838. Bells, vibration of, 786, 885. Bernoulli’s laws, 889. Bertsch’s electrical machine, 545. Biaxal crystals, 1042. Bifilar magnetometer, 630. Binocular vision, 948. Part I., p. 1-240. Part II., p. 241-504. Part III., p. 505-784. Part IY., p. 785-1050. 1062 INDEX. Biot’s hypothesis of terrestrial magnetism, 632. Blackburn’s pendulum, 852,1038. Bladder, burst, 191. Block pipe, 837. Boiler of steam-engine, 4S2-486. Boiling, 334. — by bumping, 344. — explosive, 342. — promoted by presence of air, 341. Boiling points, affected by pres¬ sure, 336. -heights determined by, 338. -of solutions, 340. -table of, 335. Boreal pole, 616. Bottle, inexhaustible, 232. — Mariotte's, 239. Bourbouze’s apparatus for falling bodies, 46. — electro-magnetic engine, 711. Bourdon’s gauge, 181. Boutigny’s experiments, 345. Boyle’s law, 170-182. Bramah press, 224. Breezes, land and sea, 499. Breguet’s telegraph, 718. — thermometer, 261. Bridge, Wheatstone’s, 674. Brightness, 965-970. — intrinsic and effective, 966. — of spectra, 992. Bright spot behind eyepiece, 960. British Association unit of resist¬ ance, 760. Brocot’s pendulum, 273. Broken magnet, 618. Brush, electric, 548. Bubbles, filled with hydrogen, 209. — pressure in, 132. Bucket, electric, 558. Bunsen & Kirchhoff’s researches, 441. Bunsen’s cell, 650. Buoyancy, centre of. 105. Burning mirrors, 392. Bursting of boilers, 483. Buys Ballot’s experiment on sound, 827. -law, 169. C AGE electrometer, 597. Cagniard de Latour’s experi¬ ments on vaporization, 325. -siren, 822. Caissons, 206. Calibration, 245. — of thermo-multiplier, 664. Calorescence, 410. Caloric theory. 445. Calorimeter, 430. Calorimetry, 426-444. Camera lucida, 916. — obscura, 941. — photographic, 943. Camphor, movements of, 137. Canton’s phosphorus, 979. Capacity, electric. 565. — of condenser. 568. — specific inductive. 576. Capacity, thermal, 427. Capillarity, 127-138. Carbonic acid, solidification of, 333. Carbon melted, 703. — points, image of, 704. Carnot’s principles, 454. Carre’s two freezing apparatus, 329, 332. Cartesian diver, 108. Cascade, charge by, 582 (2d edi¬ tion). Casselli’s telegraph, 730. Cassegranian telescope, 965. Cathetometer, 146. Cathode, 739. Caustics, 901, 918,1017. Cavendish experiment, 67. Cells arrangement of, for maxi¬ mum current, 671. Centesimal alcoholimeter, 119. Centigrade scale, 250. Centre of buoyancy, 105. — of gravity, 33-39. — of inertia. 72. — of lens, 931. — of mass, 72. — of mirror, 894. — of oscillation, 60. — of parallel forces, 17. — of percussion, 76. Centrifugal force, 62. — pump, 222 (3d edition). — theory of atmospheric circula¬ tion, 501. Character of a musical note, 817, 853. Charge by cascade, 582 (2d edition). — residual, 572. Charts of magnetic lines, 631. — of weather, 168. Chemical action necessary to cur¬ rent, 652. — combination, 442, 462, 435. — harmonica, 789. — hygrometer, 373. Cherra Ponjee, rainfall at, 380. Chimes, electric, 600. Chimneys, draught of; 298. Chromatic aberration, 792. Chromosphere, 980. Circular polarization, 1039,1047. Clarke’s machine, 767. Clearance, see Untraversed Space, 193. Climates, insular and continental, 495. Clink accompanying magnetiza¬ tion, 638. Clocks, electrically controlled, 736. Clothing, warmth of, 423. Clouds, 877-380. Coal, origin of, 462. Coatings, jar with movable. 573. Coefficient of expansion, 2C4. Coercive force, 617. Coil, Ruhmkorff’s induction, 761. Cold of evaporation, 328. Colladon’s experiment at Lake of Geneva, 803, 867. Collimation, line of, 971. Collimator of spectroscope, 983, 985. Colloids, 139. Colour, 1000,1011. — and music. 1010. — blindness. 1010. — by polarized light, 1037-1045. — cone. 1007. — equations, 1004. — mixture of, 1002-1008. — of thin films, 1030. Combination, heat of, 442, 462. Combustion, heat of, 443. -table, 444. Comma, 821. Commutator, 763. Compass, ship’s, 634. Compensated pendulums, 271. Complementary colours, 1008. Compound engines, 477. — magnet, 621, 637. Compressed-air engines, 207. Compressibility, 25. — of water, 26. Concave mirrors, 893-904. Concord, 859. Condensation, 322. Condenser of steam-engine, 474. Condensers, electric, 567. -capacity of, 568. -discharge of, 569. Condensing electroscope, 579. — power, 574. — pump, 202. Conduction of heat, 414-425. -in gases, 423-425. -in liquids, 421-423. Conductivity, comparison of ther¬ mal and electrical, 670. — defined, 415. — determinations of absolute, 420, 421. — electrical, see Resistance. — table of, 420. Conductors, electrical, list of, 507. — lightning, 601-603. Cone of colour, 1007. Congelation, 306. Conjugate foci, 894, 932. — mirrors, 393, 807. Consequent points, 636. Conservation of energy, 79. -motion of centre of mass, 74. Constitution of compound vibra¬ tions, 854. Contact-electricity, note on, 784 (2d edition). Contiguous particles, induction by, 515, 578. Continental climates, 495. Continuity of gaseous and liquid states. 325. Contracted vein, 229. Contractile force of surface-film, 131. Controlled clocks, 736. Convection of heat, 284. — of electricity, 531, 604. Convertibility of centres in pendu¬ lum, 60. Convex mirrors, 905. Cooling, law of, 386. — of air by ascent 498. Copper-cube experiment, 777. Corti’s organ, 861. Coulomb’s torsion-balance.519,624. Counterpoised barometer, 159. Couples, 16. Couronne de tasses, 646. Critical angle, 912. — temperature. Andrews’, 327. Cross-wires of telescope, 971. Cruickshank’s trough, 647. Cryophorus, 330. Crystallization, 307. Crystalloids, 139. Crystals, optical, classification of, 1042. Cup-leathers, 225. Current, deflected by magnetic force, 65S. — direction of, in battery, 643. — induced by motion across lines of force, 752-760. Part I., p. 1-240. Part II., p. 241-504. Part III., p. 505-7S4. Part IV., p. 785-1050. INDEX. 1063 Current, numerical estimate of, 653. Currents, marine, 284. Curvature in connection with ca¬ pillarity, 128, 183. — of rays in air, 1019. Cushions of electrical machine, 585, 536. Cycloidal pendulum, 71 Cyclones, 502, 611. Cylindric mirror, 908. D ALTON’S experiments on vapors, 349. — laws of vapors, 322. Dampers, copper, 777. Daniell’s battery, 649. — hygrometer, 368. Dark ends of spectrum, 978. — lines in spectrum, 978. Davy lamp, 413. — on friction of ice, 447. Dead points, 472. Declination magnet, 626. — magnetic,* 615. -changes of, 633. — theodolite, 626. Deep-water thermometers, 258. Deflagrator, Hare’s. 648. Degree of thermometer, 250. -physical meaning of, 251. Delezenne’s circle, 759. Delicacy of thermometer, 252. Density, 85. — by hydrometers, 114. — by specific gravity bottle, 88. — by weighing in water, 113. — correction of, for temperature, 266. — electric, 528. — of air, 141, 296, 375. — of gases, 294-297. -table of, 297. — of mixtures, 120. — of vapours, 357-363. — table of, 83. — see Air, Vapour, Earth. Depolarization, see Elliptic Polar¬ ization. Depressions, capillary, 128. De Saussure’s hygrometer, 366. Despretz’s experiments on Boyle’s law, 172, -on alcohol at low tempera¬ tures, 333. -on heat of voltaic arc, 703. Deviation, constructions for, 924, 925. — by rotation of mirror, 892. — minimum, 924-926. Dew, 412. — point, 365. -computation of, 371. Dial telegraphs, 718, 722. Dialysis, 139 (3d edition). Diamagnetic bodies, 638; their coefficient of induction nega¬ tive, 781. Diathermancy, 405. — table of, 406. Dielectric, influence of, 575. — polarization of, 578. Difference-tones, 862. Differential galvanometer, 661. — thermometer, 263. Difficulty of commencing change of state, 306, 343. Diffraction, 1013. — by grating, 1025. Diffraction fringes, 1024. — spectrum, 1025. Diffusion, 139. Digester, Papin's, 339. Dimensions of units, 779. Dip, 615. Dip-circle, 628. Direction of vibration in polarized light, 1048. Discharge in rarefied gases, 549- 552, 765. Discharger, jointed, 569. — universal, 584. Discord, 859. Dispersion, chromatic, 973. -in spectroscope, 992. Displacement of spectral lines by motion, 991. Dissipation of charge, 531. — of energy, 466. — of sonorous energy, 799. Distance, adaptation of eye to, 948. — judgment of, 949. Distillation, 347. Distribution of electricity on con¬ ductors. 528. Diurnal variations defined, 165. Diver, Cartesian, 108. Divided circuits, 673. Divisibility, 23. Donny’s experiment, 841. Doppler’s principle, 991. Double-action air-pump, 189. — water-pump, 222. Double refraction, 926, 1035. Doubly-exhausting air-pump, 199. Draught of chimneys, 298. Drion’s experiments, 326. Dry pile, 651. Duality of electricity, 593. Duboscq’s regulator, 706. Dufour’s experiment, 342. Duhamel’s vibroscope, 824. Dulong & Petit's law, 435. -law of cooling, 338. Dumas’ method for vapour den¬ sities, 359. Dynamics of rigid bodies, 72-77. Dynamometer, 30. Dynamo-electric machines, see Accumulation by Mutual Ac¬ tion. E AR, how affected by discord, 861. Earth, action of, on currents, 689. — as a magnet, 632. — mean density of, 67. Earth-currents, 634. Ebullition, 334. Eccentric of slide-valve, 473. Echo, 808. Efficiency of engines, 710. — of pumps, 218. — of thermic engine, 453; rever¬ sible, 454. Efflux of liquids, 226. Elasticity. 27. — Young’s modulus of, 29. Electrical force at a point defined, 559. — machines. 533, see Machine. Electric chimes, 600. — egg, 550. — light, 702, 769. — pendulum, 509. — spark, 546, see Spark. — telegraph, 713-736. Eclectric whirl, 558. Electricity, 505. — atmospheric, 599. — voltaic, 642. Electrodes of battery, 644, 739. Electro-dynamics, 680. -gilding, 746. —magnetic engines, 710. -magnets, 697. -medical machines, 778. -motors, 710. Electrolysis, 738-744. Electrolytes, conduction in, 746. Electrometer, absolute, 592. — attracted disc, 591. — cage, 597. — portable, 593. — quadrant, 595. Electrometers, 591-598. Electro-motive force, 665, 677. -its value for different bat¬ teries, 679. Electrophorus, 544. Electro-plating, 746. Electroscope, 517. — Bohnenberger’s, 652. — condensing, 579. Electrotype, 747. Elementary tones, 863. Elements of currents, mutual ac¬ tion of, 6S8. Ellicott’s pendulum, 272. Ellipsoid, 1044. — distribution of electricity on, 529. Elliptic polarization, 1037. Elmo's fire, St., 602 Emissive power, 394.' Endosmose. 138. Energy, available sources of, 465. — conservation of. 79. — dissipation of, 466. — of motion, 76. — of position, 78. — of rotation, 75. — of sonorous -vibrations, 799. — transformation of. 79. Engines, thermic, 453; see Steam- engine. Equipotential surfaces. 561. Equivalent simple pendulum, 60. Equivalents of heat and work, 449. Errors and corrections, signs of, 153. Evaporation. 317. — cold of. 328. — latent heat of, 441. Exchanges, theory of, 396. Exhaustion, calculation of, 185. — limit of, 193. Expansion, apparent and real, of liquids, 275. — by heat, 242. 264. — coefficient of. 264. — cubic and linear, 265. — force of, 273. — formulae relating to, 264. — heat lost in, 451. — in freezing. 311. — linear, modes of observing, 269. -table of. 270. — of gases. 287. -table of, 292. — of liquids, table of, 277, 2S0. — of mercury, 280. Expansion-factor, 264. Expansive working in steam-en¬ gine, 476. Explosion of boilers, 484. Part I., p. 1-240. Part II., p. 241-504. Part III., p. 505-7S4. Part IV., p. 785-1050. 1064 : INDEX. Extra current, 7(51. Extraordinary index, 928. — rays, 928,1035. Eye, 946. Eye-pieces, 996. F AHRENHEIT’S barometer, 160. — hydrometer, 116. — scale of temperature, 250. Falling bodies, laws of, 49. Fall in vacuo, 41. Faraday’s experiments within electrified box, 527. -on liquefaction of gases, 323. -on solidification of gases, 333. — views regarding electro-static induction, 578, 515. Favre and Silbermann’s calori¬ meter, 442. Field, magnetic, 620. -intensity of, 620. -uniform, 757. Filings, lines formed by, 612. Film of air, adhei'ent, 183. Films, colours of, 1030. — tension in, 131-13S. Fire-engine, 221. Fizeau’s measurement of velocity of light, 873. Flames, manometric, 846, 857. — singing, 789. Flexure, resistance to, 29. Floating, conditions of, 107. Floating bodies, attraction be¬ tween, 136. -stability of, 109. Floating needles, 110. Flowers of ice, 308. Flue-pipe, 837. Fluids, 21. — electric, theories of, 510. — imaginary magnetic, 618. Fluorescence, 980, 410. Flute mouthpiece, 837. Fly-wheel, 75, 474. Focal lines, 903. Foci, conjugate, 894, 932. — explained by wave theory, 1017. — primary and secondary, 901. — principal. 893, 930. —■ virtual, 897. Focometer, 940. Forbes’ experiments on conduc¬ tivity, 420. — observations on glacier motion, 314. Force, 11. — lines and tubes of, 560-563. -their movement, 757. -their relation to induced currents, 754-760. — unit of, 54. 780. Force-pump, 220. Fortin's barometer, 147. Foucault's experiment on velocity of light, 875, 1016. — magneto-thermic experiment, 448. — prism, 1037. — regulator. 707. Fountains, 230. — intermittent. 233. — in vacuo, 192. Fourier’s theorem, 853. Franklin’s experiment on ebulli¬ tion, 337. -on lightning. 599. Fraunhofer’s lines, 978. Free-piston air-pump, 200. Free-reed, 845. Freezing at abnormally low tem¬ peratures, 306, 460. — by evaporation, 328-333. — by the spheroidal state, 345. — expansion in, 311. — mercury in red-hot crucible, 345. — mixtures, 305. Freezing-peint lowered by pres¬ sure, 312. -; computation, 459. -by stresses, 313. Frequency, 817. Fresnel’s rhomb, 1047. — wave-surface, 1043. Friction, heat of, 446. Friction in connection with con¬ servation of energy, 79. Fringes, diffraction, 1024. Frog, experiment with, 645. Froment’s engine, 712. Frost, hoar, 413. Fuse, Statham’s, 764. Fusion, 302. — latent heats of, 439. — temperatures of, 302. G alilean telescope. 962. Galileo’s experiments on falling bodies, 40. — explanation of suction-pump, 144. Galvani, 644. Galvanic battery, 644. — electricity, 642. Galvanometers, 659-664. — choice of, 677. Gamut, 819. Gas-battery, 745. — engine. 490. Gases distinguished from liquids, 21 . — table of densities of, 297. — their expansion by heat. 287. — their tendency to expand, 22. — two specific heats of, 435, 451. Gauss’ unit of force, 54. Gay-Lussac’s experiments on ex¬ pansion of gases, 287. -method for vapour densities, 362. Geissler's air-pump, 195. — tubes, 765. Giffard’s injector, 485. Gimbals, 634. 149. Glaciers, motion of, 314. Glaisher’s balloon-ascents, 497. -- tables, 371. Glass, expansion of. 276, — strained, exhibits colours by polarized light, 1044. Gold-leaf electroscope, 517. Governor-balls, 474. Gradient, barometric, 168. Gramme-degree, 427. Graphical method of interpolation, 120 . Gratings for diffraction, 1026. Gravesande’s apparatus, 13. Gravitation, universal, 66. Gravity, centre of. 33. — formula for variation, with lati¬ tude, 61. — proportional to mass, 55. — terrestrial, 31. Gregorian telescope, 964. Gridiron-pendulum, 271. Grotthus’ hypothesis, 739. Grove’s battery, 651. Gulf stream, 285. H ADLEY’S sextant, 892. Hail, 383. — Yolta’s theory of, 610. Hare’s deflagrator, 648. Harmonics, 832, 854, see Over¬ tones. Harrison’s gridiron-pendulum.271. Head, 226, 298. Heat, effects of, on magnets, 638. — mechanical equivalent of, 449. — of combustion, table of, 444. — polarization of, 1049. — produced by discharge of Ley¬ den jars, 584, 590. -by electric currents, 699. — quantity of, 426. — required for a cyclic change,459. -for change of volume and temperature, 457. — units, 427. Heating by hot water, 284. Heights measured by barometer, 162. -by boiling point, 338. Ileliostat, 977. Helmholtz’s colour-observations, 1004. — resonators, 856. — theory of dissonance, 860. Hemispheres, Magdeburg, 191. Herschelian telescope, 963. High-pressure engines, 478. Hirn on animal heat, 461. Hoar-frost, 413. Holtz’s electrical machine, 541. Homogeneous atmosphere, height of, 162. Hope’s experiment, 279. Horse-power, 19. Houdin’s regulator, 708. Howard’s cloud nomenclature, 378. Hughes’ printing telegraph, 726. Humidity of air, 364. Huygens’ construction for wave- front, 1014. — principle, 1012. Hydraulic press, 93, 224. — tourniquet, 101. Hydro-electric machine, 539. Hydrogen, conductivity of, 425. — heat of combustion of, 444. — soap-bubbles filled with, 209. Hydrometers, 113-121. Hygroscopes and hygrometers, 365-374. Hypsometer, 338. Hypsometry, 163. TCE-CALORIMETER, 429. JL -flowers, 308. -pail experiment, 526, 564. — regelation of. 314. Iceland-spar, 926. Images, accidental, 1009. — brightness of, 967. — electric, 5C6. — formation of. 898. — in mid air, 901. — on screen, 900. — produced by small apertures, 868 . — size of, 898. 937. Imaginary magnetic matter, 619. Inclination, magnetic, 615. Inclined plane, 41. Part I., p. 1-240. Part II., p. 241-504. Part III., p. 505-784. Part IV., p. 785-1050. INDEX. 1065 Index errors, 158. Index of refraction, 911. -table of, 912. -of air, 1019. Induced currents, 750-760. Induction coil, 761. — electro-static, 513-527. -its relation to force-tubes, 563. . . — magnetic, 617, coefficient of, 781. Inductive capacity, specific, 576. Inertia, 9. Inexhaustible bottle, 232. Ingenhousz’s experiment, 416. Injector, Giffard’s, 485. Insects walking on water, 110. Insular climates, 495. Insulators, list of, 507. Intensity, horizontal, vertical, and total, 623. — of field, 620, 781. — of magnetization, 621, 781. Interference, 810, see Diffraction. Intervals, musical, 818. Iodine, solution of, in bisulphide of carbon, 410. Isobaric lines and charts, 168. Isochronism, condition of, 70. — of pendulum, 58. Isoclinic and other magnetic lines, 632. Isothermal lines, 494. J ET PUMP, 223 (3d edition). Jets, liquid, 227. Jones’ controlled clocks, 736. Joule’s equivalent, 450, 452. — experiment in stirring water, 449. — law for energy of current, 699- 702. Jupiter's satellites, eclipses of, 878. K aleidoscope, 890. Kater’s pendulum, 60. Key, Morse’s telegraph, 724. Kienmayer’s amalgam, 536. King’s barograph, 160. Kinnersley’s thermometer, 555. Konig’s manometric flames, 846, 857. Kravogl’s air-pump, 194. L ADD’S machine, 774. Land and sea breezes, 495. Lantern, magic, 945. Laplace and Lavoisier’s experi¬ ments, 269. Laplace’s correction of sound-velo¬ city, 803. Laryngoscope, 907. Latent heat of fusion, 303. -of steam, 441. -of vaporization, 328. -of water, 304. -below freezing-point, 460. Latitude, 33. — its influence on gravity, 61. Least time, principle of, 1016. Leidenfrost’s phenomenon, 345. Lenses, 929. — centre of lens, 931. Lenz’s law, 753. Le Roy’s hygrometer, 367. Leslie’s differential thermometer, 263. — experiment (freezing by evapo¬ ration), 328. Levelling, 124. Levelling, corrections in, 1018. Lever, 15. Leyden battery, 5S0. — jar, 571. -capacity of, 568. -with movable coatings, 573. Lichtenberg’s figures, 5S1. Light, 865-1050. — electric, 702. -for lighthouses, 769. Lightning, 599. -conductors, 601. — duration of, 600. Limma, 819. Linear dimensions, in sound, 836, 839. Line of collimation, 971. Lines, isoclinic, isodynamic, iso- gonic, 632. Lines of force, 560. -caution regarding, 778. -due to current, 657, 689. -magnetic, 619. -— shown by filings, 613. Link-motion, 489. Liquefaction of gases, 322-323. — of solids, see Fusion. Liquefiable and non - liquefiable gases, 173. Liquid and gaseous states continu¬ ous, 325-328. Liquids, 21. Lissajou’s curves, 819. -equations to, 850. — experiments, S47. Local action, 651. Locomotive, 4S6. Lodestone, 612. Longitudinal vibrations of rods and strings, 843. Looking-glasses, 886. Loudness, 816. Luminiferous aether, 865,1043. Lycopodium on vibrating plate, 788. M achine, electrical. 531 . -Bertsch’s, 545. -Guericke’s, 533. -Holtz’s, 541. -Nairne’s, £37. -Ramsden’s, 535. -Winter’s, 538. — hydro - electric, Armstrong’s, Machines, magneto-electric, 766- 774. Magdeburg hemispheres, 191. Magic funnel, 232. — lantern, 945. Magnet, ideal simple, 620. — moment of, 621, 622. — natural, 612. Magnetic attraction and repulsion, 619. — charts, 631. — curves formed by filings, 613. — fluids, imaginary, 618. — meridian, 615, 631. — potential, 619. — storms, 634. — variations, 633. Magnetism, remnant or residual, 698. Magnetization, methods of, 635, 696. — specification of, 619. Magneto-crystallic action, 64ft. -electric machines, 766-775. Magneto-optic rotation, 1045. Magnetometers, 630. Magnification, 951. — by lens, 954. — by microscope, 959. — by telescope, 959, 961, 964. Malus’ polariscope, 1033. Manometers, 177. Manometric flames, 846, 857. Marine barometer, 156. Mariotte’s bottle, 239. — law, 170. — tube, 171. Mason’s hygrometer, 370. Mass, 54. — centre of, 72. Matches for collecting electricity, 605. Maximum thermometers, 254. Maxwell’s colour-box, 1005. — rule for action between circuits, 689. Mean temperature, 493. Mechanical equivalent of heat, 449. Mechanics, 11. Melloni’s experiments, 399-406. — method of evaluating deflec¬ tions, 664. Melting-points, table o£ 302. Meniscus, 135, 929. Mercury, density of, 88. — expansion of, 280, 253. Meridian, 615. Meridians, chart of magnetic, 631. Metacentre, 110. Metallic barometers. 157. — thermometers, 261. Meteoric theory of sun’s heat, 464. Mica plates for circular polariza¬ tion, 1047. Micrometers, 972. Microscope, compound, 956. — electric, 945. — simple, 955. — solar, 944. Mines, firing by electricity, 590. Minimum deviation by prism, 924- 926. Mirage, 913,1022. Mirror electrometer. 594. — galvanometer, 663. Mirrors, 886. — concave. 893. — conjugate, 393, 807. — convex, 905. — cylindric, 906. — parabolic. 894. — plane, 886. Mist, 377. Mixture of colours, 1002-1008. — of gases and vapours. 181, 321. Mixtures, density of, 120. — method of, 429. Modulus of elasticity. 29. Moist air, density of, 375. Moment of couple, 16. — of force about axis, 75. — of inertia, 74. — of magnet, 621, 622. Momentum, 76. — angular. 75. Monochord, see Sonometer. Monochromatic light, 992. Monsoons, 499. Morin’s apparatus, 47. Morse’s telegraph, 722. — telegraphic alphabet, 724. Mortar, electric, 555. Part I., p. 1-240. Part II., p. 241-504. Part III., p. 505-784. Part IV., p. 785-1050. 1066 INDEX. Motions, composition of, 52. Mountain-barometer, theory of, 162. Mousson’s experiment, 312. Mouth-pieces of organ-pipes, 837, 844. Multiple images. 888, 919. Multiple-tube barometer, 160. -manometer, 170. Musical sound, 790. AIRNE’S electrical machine, 537. Needle, magnetized, 614. Negretti’s maximum thermome¬ ter, 257. Newtonian telescope, 964. Newton’s law of cooling, 386. — rings, 1030. — spectrum experiment, 975. — theory of refraction, 1015. Nicholson’s hydrometer, 115. Nicol’s prism, 1036. Nobili’s thermo-pile, 397. Nodal lines on plate, 788. Nodes and antinodes in air, 811. -in pipes, 840. Noise and musical sound, 790. Non-liquefiable gases, 173. Notes m music, 820. O BSCURE radiation, 409, 978. (Ersted’s experiment, 656. — piezometer, 26. Ohm as unit of resistance, 758- 760. Ohm’s law, 665. Opera-glass, 963. Ophthalmoscope, 907. Optical centre of lens, 931. — examination of vibrations, 847- 851. Optic axes in biaxal crystals, 1043. — axis in uniaxal crystals, 927, 1042. Order, thermo-electric, 654. Ordinaryand extraordinary image, 928. -index, 928. -rays, 1035. Organ-pipes, 837-842. -effect of temperature on, 845. -overtones of, 840-843, 855. Oscillating engines, 480. Overtones, 882-S46. Oxyhydrogen blow-pipe, 444. P APIN’S digester, 339. Parabolic mirrors, 894, 393, 807. Parachute, 211. Paradox, hydrostatic, 100. Parallel currents, 682. — forces, 14. — mirrors, 889. Parallelogram of forces. 12. Paramagnetic bodies, 638, 781. Pascal’s experiment at Puy-de- Dome, 145. — principle, 91. — vases. 97. Peltier effect, 70S. Pendulum, 56. — compensated. 271. — compound. 60. — convertibility of centres in, 60. — cycloidal, 71. — electric, 509. Pendulum electrically controlled, 737. — isochronism of, 58, 70. — time of vibration of, 58. Penumbra, 872. Pepper’s ghost, 892. Percussion, centre of, 76. Perforation by electric discharge, 588. Period of vibration, 57, 70, 786. Permanent gases, 173. Perpetual motion schemes, 20. Person on specific heat of ice, 460. Phantom bouquet, 899. Phial of four elements, 112. Phillips’ electrophorus, 544. — maximum thermometer, 257. Phonautograph, 825. I Phosphorescence, 979. j Phosphoroscope, 979. ! Photographic registration, 160. I Photography, 943. Photometers, 881. Piezometer, 26. Pile, dry, 651. — Volta's, 645. Pipes, vibration of air in, 788; see Organ-pipes. Pipette. 231. Pistol, Volta’s, 556. Pitch, 816. — modified by motion, 826. — standards of, 820. Pixii’s machine, 767. Plane mirrors, 886. Plane of polarization, 1034,1048. Plasticity of ice, 314. Plateau’s experiments, 134. Plates, refraction through, 918. — vibration of, 787, 835. Plumb-line, 31. Plunger, 220. | Pluviometer, see Rain-gauge. Pneumatic despatch, 206. — tinder-box. 445. Points discharge electricity, 530. — wind from, 557. Polarization by absorption, 1032. — by double refraction, 1035. — by reflection and transmission, 1033. — circular, 1047. — elliptic, 1037. — in batteries, 649, 675. — of dark rays, 1050. — of dielectric, 578. — of light. 1032. — plane of, 1034. Polarizer. 1032. Poles of battery, 644. — of magnet, 612. — — their names, 616. Porosity, 25. Portable electrometer, 592. Portative force, 637. Portrait, electric, 585. Potential, 559. — analogous to level, 561. — curve of, in battery, 676. — energy, 78. — equal to sum of quotients, 564. ! — its relation to force and work, ! 559-561. I — strong and feeble, 579. j Pouillet’s apparatus for compress- I ing gases, 173. i Pound, a standard of mass, 54. I Pressure, centre of. 102. 1 — hydrostatic, 90-101. Pressure, intensity of, f5. — reduction of, to absolute mea¬ sure, 154. — total amount of, 103. Pressure and volume when no heat enters or escapes, 436. Pressure-gauges, 177-182. Prevost’s theory of radiation, 396. Primary colour-sensations, 1008. Principal focus, 893, 930. Principle of Archimedes, 104. — of Huyghens, 1012. — of Pascal, 91. Prism in optics, 920-926. — Nicol’s and Foucault’s, 1036. Problem’s in Acoustics and Optics, 1057. -Dvnamics and Hydrostatics 1051. -Heat, 1055. Projectiles, motion of, 50. Projection by lenses, 944. Proof-plane, 524. Propagation of light, 1012. — of sound, 792. Psychrometer, 370. Pumps, centrifugal, 222 (3d-edit.) — for air, 184; see Air-pump. — forcing, 220. — for liquids, 215. — Galileo on, 144. — jet, 223 (3d edition). — suction, 216. Puncture by electric discharge, 588. Pure spectrum, 976. Purity numerically measured, 993. Pyrheliometer, 463. Pyrometer, 262, 293. Pythagorean scale, 821. Q UADRANT electrometer, 594. — electroscope, 536. Quantity of heat, 426. Quarter-wave plates, 1047. Quartz rotates plane of polariza¬ tion, 1045. — transparent to ultra-violet rays, 410, 981. R ADIANT heat and light, 408. Radiation. 385. — coefficient of, 394. — selective, 410. Rain, 381. Rainbow, 997. Rainfall, British, 382. Rain-gauge, 382. Ramsden and Roy’s experiments, 270. Ramsden’s electrical machine, 535. Rarefaction by Alvergniat’s meth¬ od, 551. — in air-pump, 185. — in Sprengel’s air-pump, 198. Rarefied gases, discharge in, 549, 765. Reaction of issuing jet, 101. Real and apparent expansion, 275. Reaumur’s scale, 250. Recomposition of white light, 9S2. Rectilinear propagation of light, 866,1013. Reed-pipes, 844. Reflecting power, 393. -403 (table). Reflection of heat, 390. — of light, 8S3. Part I., p. 1-240. Part II., p. 241-504. Part III.,p. 505-784. Part IV., p. 785-1050. INDEX. 1067 Reflection of light, irregular, 885. -total, 913. — of sound, 806. Refraction, 908. — atmospheric, 1018. — double. 926,1035. — N ewtonian explanation of, 1015. — of sound, 808. — table of indices of, 912. — undulatory explanation of, 1014. Refrangibility, change of, 410, 981. Regelation, 314. Regnault’s hygrometer, 369. — hypsometer, 338. — experiments onBoyle’slaw,173. -on expansion of gases, 288. -on sound, 798. -on specific heat, 432. -on vapour-tensions, 350. Regulators for electric light, 705- 708. Relay, 725. Remanent magnetism, 698. Replenisher, 597. Repulsion, see Attraction. Repulsion a more reliable test than attraction, 516. Residual charge, 609. — magnetism, 698. Resistance, electrical, 666. -and thermal, compared, 670. — in battery, 677. — of wires, 667. — specific, 667. — table of, 670. — unit of, 758, 782. Resonance, 833. Resonators, 856. Resultant, 12. — tones, 862. Reversal of bright lines, 412, 98S. Reversible engine, perfect, 454. Reversing of locomotive, 489. Rheostat, 668. Rhomb, Fresnel’s, 1047. Rings by polarized light, 1040. — Newton’s, 1031. Rock-salt, its diathermancy, 407, 411. Rods, vibrations of, 843. Rotating vessel of liquid, 96. Rotation of earth as affecting wind, 500. — plane of polarization, 1045. Rotations, electro-dynamic, 683. — electro-magnetic, 695. Rotatory engines, 480. Roy and Ramsden’s measures of expansion, 270. Rubbers of electrical machine, 535, 536. Ruhmkorff’s coil, 761. Rumford on heat of friction, 446. — on radiation in vacuo, 3S5. Rumford’s thermoscope, 263. Rupture of magnet, 618. Rutherford’s self-registering ther¬ mometers, 255. S ACCHARINE solutions, by polarized light, 1045. Safety-valve, 483. Saturated vapour, 318. Saturation, magnetic, 686. Sawdust battery, 651. Scales measure mass, 30. Scales, musical, 818. — thermometric, 250. Scattered light, 393. Schiehallien experiment, 67. Schweiger’s multiplier, 660. Sea-breeze and land-breeze, 499. Secondary axis, 894, 932. — coil, 762. — pile, 746. Segmental vibration, 832. Selective emission and absorption, 410. Selenite by polarized light, 1037. Semitone, 820. Sensibility of balance, 86. — of thermometer, 252. Series, arrangement of cells in, 672. Sextant, 892. Shadows, 870. Siemens’ armature, 771. — and Wheatstone’s machine, 773. Simple harmonic motion, 68, 70. — magnet, ideal, 620. — tones arise from simple vibra¬ tions, 863. — vibrations, 68, 70. Sine-galvanometer, 659. Sines, law of, 910. Singing flames, 786. Sinuous currents, 688. Siphon, 234. -barometer, 154. -temperature correction of, 153. Siren, 822. Sirius, motion of, 991. Six’s thermometer, 254. Slide-valve, 473. Snow, 3S3. Soap-bubbles, pressure -within, 134. -with hydrogen, 209. — films, 133. Sodium line, 987. Solar heat, 462. -sources of, 464. — microscope, 944. — spectrum, 975, 978. Solenoids, 690. Solidification, change of volume in, 310. — of gases, 333. — of liquids, 306. Solution, 304. Solutions, boiling points of, 340. Sondhaus’ experiment, 808. Sonometer, 831. Sound, 785-864. — in exhausted receiver, 791. — propagation of, 792. — reflection of, 807. — refraction of, 808. — shadows in water, 867. Sources of energy, 465. Spangled tube, 553. Spark, electric, 546. -colour of, 552. -duration of, 549. -heating effects of, 556. -in rarefied air, 550. Speaking-trumpet, 808. Specific gravity, 86. -correction of, for tempera¬ ture, 266. -for weight of air, 213. -determination of, by hydro¬ meters, 114. -by weighing in water, 113. -flask, 88. — — of mixtures, 120. -table of, 88. Specific heat, 427-436. Specific heat, at constant pres¬ sure and constant volume, 435, 451. -tables of, 434, 439. Specific inductive capacity, 576. Spectacles, 952. Spectra, 986-994. — brightness and purity of, 791. — by diffraction, 1026-1030. Spectroscope, 983. Spectrum analysis, 986. Specula, silvered, 965. Speculum-metal, 886. Sphere, electric capacity of, 565. Spherical mirrors, 893-905. — aberration, 894. Spheroidal state, 344. Spirit-level, 124. -thermometer, 254. Sprengel’s air-pump, 197. Springs and spring-balances, 30. — vibration of, 785. Squares, inverse, 389. -in electricity, 520-528. Stable equilibrium, 36. Stars, brightness of, 969. — motion of, 991. — spectra of, 986. Statham’s fuse, 764. Stationary undulations, 841. Steam, volume of, 363. Steam-engine, 469-490. — locomotive, 486. Steel, its magnetic properties, 617, Step-by-step telegraphs, 718-722. Stereoscope, 949. Still, 347. Stirling’s air-engine. 468. Storms, magnetic, 634. Storm-warnings, 169. Stoves, 299. — Norwegian, 424. Strained glass, by polarized light, 1044. Stratification in electric discharge, 765. Strength of pole, 620. — of current, 653. Striking reed, 845. Stringed instruments, 835. Strings, overtones of, in longitudi¬ nal vibration. 843. — vibration of, 788, 829-835, 854. Strutt’s (Hon. J. W.) diffraction gratings, 1026. Submarine telegraphs, 733. Successive reflections, 888. Suction, 215. -pump, 216. Sulphate of soda, 310. Summation-tones, 862. Sun, atmosphere of, 987. — distance of, 879. — see Solar. Superheating of steam, 480. Supersaturated solutions, 310. Surface, electricity resides on, 523. Surface-condensers, 478. — tensions, table of, 138. Sympiesometer, 156. — Synthesis of sounds, 858. Syringe, pneumatic, 445. Swan on the sodium line, 987. T ANGENT galvanometer, 660. Tantalus’ vase, 237. Tartini’s tones, 863. Telegraph, autographic, 730. Part I., p. 1-240. Part II., p. 241-504. Part III., p. 505-784. Part IT., p. 785-1050. 1068 INDEX. Telegraph, automatic, 735. — dial, 718, 722. — electric, 713-736. — electro-chemical, 730. — Morse’s, 722. — printing, 726. — single-needle, 716. — submarine, 733. Telegraphic alarum, 721. — alphabet, 724. Telescopes, 958-964. Telespectroscope, 989. Temperament, 819. Temperature, 241. — absolute, 293, 456. — mean, 493. — of a place, 493. — of the air, 493. -decrease upwards, 497. — of the soil, 421, 495. -*-increase downwards, 496. — scales of, 250. Tempering of metals, 29. Tension, electric, 579. Terrestrial refraction, 1018. — temperatures, 493. Thermochrose, 408. Thermo-dynamics, 445. — first law of, 450. — second law of, 455. Thermo-electricity, 652-655. Thermographs, 260. Thermometer, 244-252. — alcohol, 254. — differential, 262. — metallic, 260. — self-registering, 254. Thermopile, 397, 654. Thilorier’s apparatus, 324. Thin films, colours of, 1030. Thomson, J., on glacier motion, 315. Thomson’s galvanometer, 663. Thunder, 601. Tickling by electricity, 554. Timbre, 817. Tones, major and minor, 819. — resultant, 862. Tonometer, 825. Tornadoes, 611, 503. Torricellian experiment, 143. Torricelli’s theorem on efflux, 226. Torsional rigidity, 29. Torsion-balance, 519, 624. Total reflection, 913. Tourmalines, 1032. Tourniquet, hydraulic, 101. Transmission of sound, 793. Transport of elements, 739. Transverse and longitudinal vibra¬ tions, 795, 828. Trevelyan experiment, 789. Trumpet, speaking and hearing, 809. Tubes of force, 562. -movement of, 757. -relation of, to induced cur¬ rents, 754-760. Tuning-fork, 836. Twaddell’s hydrometer, 119. Tyndall on magneto-crystallic ac¬ tion, 640. — on moulding of ice, 315. Volume and pressure, changes of. when no heat enters or es¬ capes, 437. Vowel-sounds, 857. U MBRA and penumbra, 872. Unannealed glass, by polar¬ ized light, 1044. Underground temperature, 495, 421. Undulation, definition of, 798. — nature of, 795, 1012. — stationary, 841. Uniaxal crystals, 1042, 1044, 927. Uniform acceleration, 52. — field, 757. Unit-jar, 587. Unit of resistance, B. A., 760. Units and their dimensions, 779- 783. — of heat, 427. Unstable equilibrium, 36. Untraversed space, 193. V APOUR, 317. — apparatus to illustrate, 319. — at maximum tension, 318. Vapour-density, 357-363. -related to chemical combina¬ tion, 357. Vapour-tension, measurement of, 349-356. Variation of magnetic elements, 633. Vegetable growth, 462. Velocity of electricity, 585. — of light, 873-880. — of sound in air, 800. -in gases, 803, 844. -in liquids, 803. -in solids, 805, 844. -mathematically investi¬ gated, 814. Vena contracta, 229. Vernier, 148. Vertical, 31. Vesicular state, 377. Vessels in communication, 122. -with two liquids, 127. Vibrations of ordinary light, 1049. — of plane polarized light, 1048. Vibrations, simple, 68. — single and double, 785. — tranverse and longitudinal, 795, 828. Vibroscope, 824. Virtual images, 904, 939, 940. Vision, 948. Visual angle, 951. Vitreous and resinous electricity, 510. Volta, 645. Voltaic arc, 703. — electricity, 642. — element, 643. Voltameter, 738, 742. Volume, change of, in congelation, 310. -in vaporization, 363. W ALFERDIN’S maximum thermometer, 259. Water, compressibility of, 26. -dropping collector, 604. — equivalent of calorimeter, 431. — level, 123. — maximum density of, 278. — specific heat of, 434. -spouts, 611. Watering-pot, electric, 558. Watt’s improvements in steam- engine, 470. Wave-front, 1012. -lengths of light, 1030. -of sound, 794-817. -relation of, to velocity and frequency, 794, 866. — -surface, 1013,1014.1043. — theory of light, 1012. Weighing, double, 81. — in air, 213. — in water, 113. — with constant load, 84. Weight-thermometer, 253. Well-thermometers, 258. Wertheim’s experiments on velo¬ city of sound, 844. Wet and dry bulb, 370. Wheatstone's automatic system, 735. — bridge, 674, — rotating mirror, 549, 586. — universal telegraph, 721, 775. — and Cooke’s telegraphs, 716. Wheel-barometer, 155. Whirl, electric, 558. Wiedemann and Franz's experi¬ ments, 419. Wild’s machine, 772. Williams’, Major, experiment with ice, 311. Wind, causes of, 499. — from points, 557. — measurement of, 503. — trade, 500. 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