ELEMENTS OF NATURAL PHILOSOPHY. BY W. H. C. BARTLETT, LL. D., PXOFESSOR OF NATURAL AND EXPERIMENTAL PHILOSOPHY IN THE UNITED STATE; MILITARY ACADEMY AT WEST POINT. I I. -A COUSTICS. I I I.-O P T I C S. FOURTH EDITION, REVISED AND CORRECTED. NEW YOEK: A. S. BARNES & COMPANY, 51, 53 & 55 JOHN STREET. SOLD BY BOOKSELLERS, GENERALLY, THROUGHOUT THE UNITED STATES. 1866. Entered according to Act of Congress, in the year One Thousand Ei^ht Hundred and Fifty-two, BY T fi r . II. C. BARTLETT, , A tie fkrVT JiFcf r' the District Court of the United States for the Southern District of New -York. '. P. JONES & CO., STEREOTYPEKS, 183 William-street. G. W. WOOD, PRINTER, 51 John-st, cor. Dutch. ^ Library. ^California- Baasasss*? PEEF A CE. THOSE wlio are familiar with the subjects of which the present volume professes to treat, will readily recognize the sources whence most of its materials are drawn. In the use of these materials, no distinction of principle is made between SOUND and LIGHT. Both are regarded and treated as the effects of certain disturbances of that particular state of mole- cular equilibrium which determines the ordinary condition of * natural bodies ; the only difference being in the media through which these disturbances are propagated, and in the organs of sense by which their effects are conveyed to the mind. The study of ACOUSTICS is, therefore, deemed to be not only a useful, but almost a necessary preliminary to that of OPTICS. In the preparation of the part relating to Sound, great use was made of the admirable monograph of Sir JOHN HEESCHEL, published in the Encyclopaedia Metropolitana ; and whenever it could be done consistently with the plan of the work, no hesitation was felt in employing the very language of that PREFACE. eminent philosopher. Much valuable matter was also drawn from Mr. AIRY'S Tracts, and from the labors of Mr. ROBISON and M. PESCHEL. In addition to the works of the authors just cited, those of Mr. CODDINGTOIT, Mr. POWELL, Mr. LLOYD, Sir DAVID BREW- STER and M. BABLNET were freely consulted in constructing ,the part relating to Optics. TABLE OF CONTENTS, ACOUSTICS. Page Introductory Remarks and Definitions, .... 9 Waves in general, . . . . . . . 20 Velocity of Sound in Aeriform Bodies, . . . . 24 Velocity of Sound in Liquids, . . , . .38 Velocity of Sound in Solids, . . . . . 44 Pitch, Intensity, and Quality of Sound, . . . .47 Siren, ........ 49 Divergence and Decay of Sound, ..... 54 Molecular Displacement, . . . . . 58 Interference of Sound, . . . . t .61 New Divergence and Inflexion of Sound, .... 70 Reflexion and Refraction of Sound Echos, . . ... 75 Hearing Trumpet, . . . . . 90 Whispering Galleries and Speaking Tubes, . . . .91 Musical Sounds, . . ... . . . 94 Vibrations of Musical Strings, ... . . 97 Monochord, . . . . ,. . . Ill Vibrating Columns of Air, . . . . . .114 Vibrations of Elastic Bars, . . . . . 126 Vibrations of Elastic Plates and Bells, . . . .128 Communication of Vibrations, . . . . . 131 The Ear, ........ 135 Music, Chords, Intervals, Harmony, Scale, and Temperament, . 137 Table of Intervals, ....... 143 Table of Intervals with the Logarithms, . . . . 154 TABLE OF CONTENTS. OPTICS. Fage Introductory Remarks and Definitions, . . . , .167 Reflexion and Refraction of Light, . . , . 171 Table of Refractive Indices and Refractive Powers, . . .176 Deviation of Light at Plane Surfaces, . . . 175 Deviation of Light at Spherical Surfaces, . . .188 Deviation of Light by Spherical Lenses, . . . . 201 Deviation of Light by Spherical Reflectors, . . . .211 Spherical Aberration, Caustics and Astigmatism, . . 216 Oblique Pencil through the Optical Centre, . . . .220 Optical Images, . . . . . . t 222 The Eye and Vision, . ... . . .232 Microscopes and Telescopes, . . . . ' . 238 Common Astronomical Telescope, , . . . .245 Galilean Telescope, . . . . . . 246 Field of View, . . . . 248 Terestrial Telescope, . . . . . . 250 Compound Refracting Microscope, . ... . .252 Astronomical Reflecting Telescope, . . . . 253 Gregorian Telescope, . . . . , .254 Cassegrainian Telescope, . . . . . . 256 Newtonian Telescope, . . . . . . 25*2 Dynameter, . . . . . . . 257 Camera Lucida, . . , , . . .261 Camera Obscura, . . , . . . 263 Magic Lantern, . . . . . . .264 Solar Microscope, . . . . . . 265 Chromatics, ........ 266 Color by Interference, Color of Gratings, . . . . 267 Table of Wave Lengths, . . . . . .276 Colored Fringes of Shadows and Apertures, . , 282 Colors of Thin Plates, ...... 284 Colors of Inclined Glass Plates, ... . . 290 Colors of Thick Plates, . . . 291 TABLE OF CONTENTS. 7 Page Color from Unequal Refrangibility, .... Dispersion of Light, . . . . . . .298 Table of Dispersive Powers, . Chromatic Aberration, . 302 Achromatism, .... Internal Reflexion, .... 309 Absorption of Light, ..... The Rainbow, ... . v 314 Halos, . . . 321 Polarization of Light, . . . .322 Polarization by Reflexion and Refraction, . . 327 Polarization by Absorption, . . . . . .334 Double Refraction, ...... 335 Circular Polarization, . . . .345 Chromatics of Polarized Light, . ELEMENTS OE ACOUSTICS. 1. THE principle which connects us with the external world through the sense of hearing, is called SOUND ; and sound, that branch of Natural Philosophy which treats of sound. __ 'Acoustics. is called ACOUSTICS. To explain the nature of sound, the laws of its propaga- tion through the various media which convey it to our . ears, the mode of its action upon these organs, the modifi- objects of acou cations of which sound is susceptible in speech, in music tics - and in unmeaning noise, as well as the means of pro- ducing and regulating these modifications, are the objects of acoustics. 2. All impressions derived through the senses, imme- jj^onf diately follow and may, therefore, be said to arise from peculiar conditions of relative motion among the elements of which certain parts of our physical organization are constructed. These conditions are mainly determined by Conditions to cause sensation the internal state of the bodies with which we are in sen- determined, sible contact ; and it is entirely from the transfer of work, in the form of molecular living force, from them to our organs of sense, that all impressions from the external world arise. This transfer is unaccompanied by transfer of material, and the agents are the mokcular forces that determine the physical condition, and, therefore, the sensi- ble qualities of all bodies. 3. We have already referred, in the introduction to the first volume, to JBoscovich's views upon this subject, and shall now give some illustration of the mode in which, 10 NATURAL PHILOSOPHY. Exponential curve; according to that distinguished philosopher, all bodies are formed. For this purpose let us resume the exponential curve as exhibited in the annexed figure, and which Boscovich sup- Attractive ordinates; Repulsive ordinates ; Neutral points ; Temporary molecule ; Permanent molecule ; "When permanence exists. Ji" poses to represent the law and intensity of the action of one atom of a body upon another. We have seen that the ordinates of those portions of the curve which lie above the line A (7 5 denote the attractive, while the ordinates of the portions below, represent the repulsive energies of an atom A for another atom situated anywhere upon this ' line. That at the points C", D ', C", D" , in which the curve intersects the line A C, the reciprocal action of the atoms reduces to nothing, and the atoms become neutral. Also that an atom situated at D', D" or D"' and the atom A constitute a temporary molecule, while the molecule formed of the atoms A and (7, A and (7', or A and C'", has a cer- tain degree of permanence, resisting compression and dila- tation, and tending to regain its original bulk when the distending or compressing cause is withdrawn. But this permanence only obtains when the disturbing force is such as to change the interval between the atoms by a distance less than that which separates the consecutive positions of neutrality ; for if the molecule A C", for example, be com- pressed into a less room than A D', the atom originally at (7", will not return to that point, but will be attracted by A, and the molecule will tend to collapse into the bulk A C'. If AC" be stretched beyond the bulk A D", it will tend to take the dimension A C'". The only mole- cule that cannot be permanently changed by compression is A C'. ELEMENTS OF ACOUSTICS. The component atoms of molecules thus constituted are, when in a state of relative equilibrium, in a condition of inactivity upon each other. The approximation or sepa- now the redpro- ration of the atoms by the application of some extraneous cause, gives rise to the exertion of the repulsive or at- cited, tractive forces inherent in the atoms, and thus these forces may be said to be excited or brought into action. The compression or dilatation is the occasion, not the efficient cause of the attractions and repulsions among the atoms. 4. The intensity of the atomical forces determines the Form f tbe * ponential era determined. form of the exponential curve. If a very moderate force produce a sensi- ble displacement of the atoms, the ordinates E' d', and Ed, on each side of the position C\ of inactivity, must be short, and the exponential curve will cross the axis very obliquely, in order that the ordi- nates expressing the attractive and repulsive forces may increase slowly. If, however, it require great force to produce a sensible compression or \~7 C '* dilatation, the curve must cross the ^^' axis almost perpendicularly. But in every case it must be remarked, and the remark is most small important, that when the compression or distension bears 6ion and dlsten " a small proportion to the distance between the neutral positions of the atoms, the degree of compression or dis- tension will be sensibly propor- tional to the intensity of the dis- turbing force. For, when the displacement D' E or D 1 E' is very small in comparison to C' D\ the elementary arc dD'd' will sensibly coincide with a straight line, and the ordinates E d and E' d', be proportional to the compression D' E or distension D' E'. That is to say, because action and 12 NATURAL PHILOSOPHY. Fig. 4. Their reaction are equal, a disturbed consequences. af(m ^ fogged lack tO- wards its position of neutrality ~by a force whose intensity is proportional to the distance of the atom from that point. Moreover, supposing the atom A, Fig. 4, to be kept station- ary, and the points E, and E\ to mark the limits of the disturbance of the other atom, this latter will return to its position of neutrality D', with a living force due to the action of the force of restitution over the path ED', . or E' D '/ it will, therefore, pass the point D', after which the direction of the action will be reversed, the living force will be destroyed, the atom will again return to its Perpetual osciiia- position of neutrality, which it will pass as before, and for tfon; the same reason, and thus be kept in perpetual osqillation. But the action between the two atoms of the molecule be- ing reciprocal, the atom A will not remain stationary, but will move in the same direction as the disturbed atom and tend to preserve its neutral distance, and the oscillation checked. * na ^ would otherwise continue will, therefore, be checked. Action of the sim- 5. Let us next take the case of a molecule of the sim- plest molecule on pl e st constitution, to wit, one composed of two atoms, and examine its action on a third atom situated on the prolon- gation of X Y, joining its elements. Fig. 5. First case; Suppose a molecule X Y, composed of the two atoms -5Tand I 7 ", which are placed, the former at A, and the lat- ELEMENTS OF ACOUSTICS. 13 ter at the last limit of cohesion <7, Fig. 5. The dotted and waving curve beginning at Y and running towards (7, will component represent the exponential curve of the atom JT, in that atoms; direction, while the similar curve beginning at the point E, will represent that of the atom Y, and the full curve C'A'D'R'O'A' &c., of which the ordinate corres- ponding to any point of the line A (7, is equal to the alge- braic suni of the ordinates of the dotted curves correspond- ing to the same point, will be the exponential curve of the That of the molecule X Y, and will give the action of the molecule molecule upon a third atom placed any where on the line A C be- yond Y. The curve has been carefully constructed ac- cording to the conditions of the case, and shows by simple inspection how different the action of even the simplest molecule is from that of a single atom. The neutral posi- Neutral positions tions of an atom with respect to this molecule will be at .J., C, D\ C", D" and so on to G. A curve having a cusp at A, the middle point of the distance X Y, and diverging so as to be asymptotic with the lines c b and c' ~b\ will give the law and intensity of the action on an atom situated between JTand Y. 6. If instead of placing the atoms at a distance apart second case, equal to that of the last limit of cohesion from A, as in the last case, we had supposed them separated by the distance A C", Fig. 1, the resulting exponential curve would have been still more unlike that of a single atom ; for in that case Fig. 6. several of the attractive branches, Fig. 6, of one of the atomi- cal curves would have stood opposed to the repulsive Res o " lt J"f i ^ ctiua braLches of the other, and the molecule thus rendered in- on an atom. NATURAL PHILOSOPHY. active on a third atom till the latter be removed nearly to the furthest limit of the scale of corpuscular action. This third atom will, therefore, admit of considerable latitude of displace- ment without much opposition Exemplification, or any great effort to regain its primitive position ; a fact we often see exemplified in the claas of liquid bodies. Third case; Construction: Construction of the exponential curve giving the action of a molecule on an atom. 7. Let us now take the molecule composed of two atoms placed at the limits A and C"', Fig. 1, and examine its action on a third atom somewhere on the line B B\ which bisects at right angles the distance A. C". Suppose the third atom placed at z. Join & with A and ", and construct the single atomical curves of A and C" in reference to 3, and suppose the atom z in Fig. 7, to have a position with re- spect to A and 6 r ", correspond- ing to any position Between J)" and "", Fig. 1 ; thus situated, it will be repelled both by A and C", Fig. 7. In a pair of dividers take the ordinate z m, Fig. 1, and lay it off from 2, on the prolong- ations of Az and C" z, Fig. 7, and construct the parallelo- gram z m n m'\ the diagonal z n, will represent in direction and intensity the action of the molecule A C" on the third atom. Draw a perpendic- ELEMENTS OF ACOUSTICS. 15 ular to B B' through the point s, and take the distance z R" equal to z n, the point R" will be one point of the exponential curve of the molecule A C" in the direction B B' . Other points being determined in the same way, the waved lines of Fig. 7 will indicate the action sought; the ordinates of the branches A, A\ A", &c., on one side of B B\ denoting attractions, while those of the branches R', R'\ R", &c., on the opposite side, denote repulsions. We see tha this action differs remarkably from that of Action differs a single atom. The curve has, to be sure, like that of a ^7n atom single atom, many alternations of attractions and repul- sions, but these alternations become less marked as they approach the molecule ; and instead of insuperable repul- sion at the greatest vicinity /, we find there a neutral point. Moreover, in approaching the molecule, the repul- sive action ceases at D\ where attraction begins and con- tinues, so far as there is any action, all the way through to D' on the opposite side of A C". This molecule is ever active when approached along the line B B\ except at certain neutral positions where the direction of the action is reversed, and is easily penetrable in this direction, whereas along the line A C" it exerts little or no action within certain limits, and is capable of an infinite repul- sion within its last limit of cohesion. Thus we see that even in this simplest constitution of a molecule, the action on an atom is susceptible of great variety by mere diffe- rence of position and distance between its component atoms ; and it would be easy to show that while the law of the atomic action in all bodies is the same, the reci- T .Law of atomic procal action of the molecules com- action the same pounded of these atoms may be un- Fig . a S^JLto, speakably various according to the ^ of molecules relative position and distance of the jrW4* Qfinitel y variou8 - component atoms. 8. Confining, for the present, the motion of the third atom to the plane of the lines A C" and B B', 16 NATURAL PHILOSOPHY. Fig. 8. Action of a ) We See that w ^ en it is at Z, molecule on an it is repelled by the molecule A C"\ when at 3' it is attracted, and the action is reduced to nothing at the point D". "When the atom is drawn aside from its nentral position D", say to z", Fig. 8, it will be re- pelled by C" and attracted by -4, because the distance from the former will be diminished, while that from A will be increased. Take z" h to represent the intensity of the repulsion and z" o that of the attraction ; complete the parallelogram o z" h q, and we shall find the molecule urged to its neu- tral position D" by a force whose intensity and direction are represented by the diagonal z" q ; so that, so far as the action in the plane AC" D" is concerned, D" is a posi- tion of stable equilibrium, and the three atoms A, C" and jy, ^ const i tllte f or moderate displacements a permanent * * molecule, presenting an elementary surface having length and breadth. The same would be true were the third atom placed at jy or D'", &c., Fig. 7. The disturbed atom when at z" being urged back to its place of neutrality by the molecule A C", will reach that point with a certain amount of living force, due to the ac- tion of the force of restitution over the path from z" to D" ; it will, therefore, pass to the opposite side of J9", where the action being in the opposite direction, its living force will be destroyed, after which it will be brought back and made to oscillate about D" as long as A and C" are sta- tionary. But while the third atom is on the side z", that at C" will be repelled by it, and that at A attracted ; the contrary will be the case when the atom is on the oppo- site side from z", so that the atoms of the molecule A G n will also oscillate, and obviously in such manner as to to cause the neutral position to follow the displaced atom. constitution of an elementary surface. Oscillation of the disturbed atom: That of the atoms of the molecule. Explanation of figure; 9. Now conceive a triangle A 0" #/', each of whose ELEMENTS OF ACOUSTICS. 17 Bides is equal to a distance at which two atoms may form a permanent molecule, and suppose an atom to be placed at each vertex; these atoms /X Explanation. will form a permanent molecule. Place a fourth atom at the vertex D", of a pyramid of which the base is the elementary plane formed by the first three atoms, and each of the edges about the vertex is equal to a distance necessary for two atoms to form a permanent molecule. It will be obvious, from what has already been said, that the fourth atom or that at the vertex cannot be disturbed without being resisted Permanent and urged back to its neutral place by the action of the moleculooffonr atoms; molecules which form the base ; for, if it be moved aside in either of the plane faces of the pyramid, it will, 8, be opposed by the force of restitution due to the action of the molecule of two atoms in the same plane ; and if moved out of these planes, its distance from one at least of the atoms in the triangular base must be altered, thus exciting a force of restitution. "What has been said of the atom at the vertex of the pyramid is equally applicable to each of those in the base when considered in reference , to the three others, and hence the four atoms A, C", C", D", form a permanent molecule ; and from its capa- bility to resist the approach of a fifth atom, another mole- cule or particle, in every direction, we derive the idea of an elementary solid, having length, breadth and thickness. Elementary solid. A disturbance of any one of the four atoms will put the Disturbance win , . . . ... cause the neutral others in motion, and it will appear on the slightest con- points to follow sideration that the directions of these motions will be such thedisturbed atoms. as to cause the neutral positions to shift in the direction of the atoms which have been disturbed from them. 10. What has been said of the action of atoms to samerefls <> nln & form molecules may easily be shown to be true of the reciprocal action of molecules to form particles, and 18 NATURAL PHILOSOPHY. particles to form the bodies which, in all their endless variety of physical characters, come within the reach of our senses. And according to this view, the characteris- tic peculiarities of all bodies are to be understood as aris- ing solely from differences in the action which their atoms, molecules and particles are capable of exerting on each other, and upon those of the bodies with which they may be brought into sensible contact. on ut ft must k e remar k e d that all these differences of confined to small action are confined to small and insensible distances which lie within the limits of physical contact. At all consider- able distances we find nothing but the action of gravita- tion, of which the intensity is proportional to the number of atoms or the mass directly, and to the square of the distance inversely. The most subtile 11. The most subtile and attenuated body of which we fconleivabie- caDr ^ orm anv conception, according to Boscovich, is one composed of atoms arranged at distances from each other equal to that which determines the furthest limit of cohe- sion, or that beyond which gravitation begins. But such a body, when abandoned to itself, would shrink into smaller dimensions in consequence of the gravitating force between the atoms not adjacent to each other, and the contraction would continue till the repulsions which it would develope between the contiguous atoms had in- ittvonid creage( j to an equilibrium with the compressing action, permanent form: when the body would take its permanent form. Such we may suppose to be the constitution of that ethe- real medium which pervades all space, permeates every body, and connects us with the objects of whose existence we are made conscious through the sense of sight. 12. A body similarly constituted, but in which the atoms are replaced by molecules or particles arranged at structure of the distances less than the furthest limit of cohesion may give us an idea of the physical structure of our atmosphere. Here as in the last case the molecules cannot occupy their ELEMENTS OF ACOUSTICS. 19 neutral positions because of the forces of gravitation exist- its molecules ing between those molecules more remote from each other c their neutral than the furthest limit of cohesion, which force will cause positions; the elements to crowd together; but we have seen that when the elements of a body are brought closer than those neutral positions which constitute permanence, the adjacent elements will repel, and can come to rest only when these antagonistic forces of attraction and repulsion balance. Add to these considerations the attraction of the earth for this fluid, and the equilibrium of any molecule will be conditions of u* found to result from the mutual balancing of the weight equilibrium. of the superincumbent column of molecules extending to the top of the atmosphere, and the repulsive action of the molecule in question for that immediately above it ; and since the weight of the pressing column decreases as we ascend, the density must diminish in the same direc- tion, all of which we know to be confirmed by the indica- tions of the barometer, 13. Passing to the denser bodies, whether of the organic or inorganic class, as vegetable or animal tis- sue, water, clay, glass, gold, we find variety of structure without difference in the principles of aggregation. All AH bodies are built up of the same ultimate atomic elements, grouped into molecules, the molecules into particles, and the parti- a tomic element* cles into the various bodies whose places in the scale of gradation, from the hardest to the softest solid, from the most viscous liquid to the most subtile gas, are deter- Their mined solely by the intensity, ran^e and direction o f cbaracteristio J ' properties the atomic actions which mark their internal structure, determined. 14:. All bodies in nature are physically connected AH bodies with each other. Those plunged into the ocean are united ^unlctel by sensible contact with its common element. So of the bodies which exist in the atmosphere. The atmosphere Physical rests upon the ocean, and that ethereal medium which maintained by permeates the atmosphere and the ocean, and extends ocean ' 20 NATURAL PHILOSOPHY. Atmosphere, ether. throughout throughout all space, carries this connection to the hea- venly bodies. Disturbance of a The disturbance of an atom, molecule or particle, will alter its relative distances from the neighboring elements ; the molecular forces on the side of the shortened distances will increase, while those on the opposite side will di- minish. The equilibrium which before existed will be destroyed, and the adjacent elements must also be dis- turbed ; these will disturb others in turn, and thus the agitation of a single element will be transmitted through- out space, and impart motion, in a greater or less degree, to the elements of all bodies. Motion affects 15. Among the bodies thus affected are certain deli- throThor ansof cate an( ^ ne ^i^ e I'^inifications of nervous tissue, which are spread over portions of our organs of sense. These nerves partake of the agitations transmitted to them from without, and by some mysterious process, call up in the mind impressions due to the external commotion. The structure and arrangement of these nerves differ greatly in the different organs, and while they are all subjected to the general laws which control the corpuscular action of bodies, yet each individual class is distinguished by peculiarities which determine them to appeal to the mind on } v w hen addressed in a particular way. We hear, feel an( i see by the operation of a common principle motion ; o f this, there is endless variety in perpetual existence among the elements of the media in which we are im- mersed ; and, according as one or another of the organs of sense becomes involved in the particular motion adapted to excite the mind to action, will our sensation become that of sound, light, heat 3 or electricity. AH our principle OF WAVES. AH sensations 16. All sensations derived from our contact with the physical world depend, according to this view, upon the ELEMENTS OF ACOUSTICS. 21 state of relative motions among the elements of .bodies ; and we now proceed to consider those motions which are T* 1086 P r P er * suited to produce the sensation of sound, and we must be pr careful to distinguish between the properties of solids and fluids in thisrespect. Conceive a perfectly homogeneous solid, that is. one in which the particles occupy the vertices of regular and equal tetrahedrons, and suppose its elements in a state of relative repose. A single particle being disturbed from its place of rest, through a very small distance, compared with the tetrahedral edges, will be urged back by the action of the surrounding elements with an energy which is, 4, proportionate to the disturbance. This particle Orb}tofa will, when abandoned to itself under these circumstances, parl i c i^ ; describe about its position of rest as a centre, an ellipse, or perchance, a circle or right line, the extreme varie- ties of the ellipse whose eccentricities are respectively zero and unity. Moreover, the time of describing a Time or complete revolution will, Mechanics, 180, be constant, descri i )tion ' constant; no matter what the size of the orbit within the limits sup- posed; and the mean velocity of the particle will, there- Mean velocity, fore, be directly proportional to the length of the orbit, or to any linear element of the same, as that of the semi- transverse axis. The disturbed particle being acted upon by its neighbours, these latter will experience from it the action of an equal and contrary force ; tkey must, there- Neighboring fore, move and describe similar orbits ; and the same ^1^" will be tine of the particles next in order, till the disturb- Disturbance ance becomes transmitted indefinitely. The disturbance transmitted ln . T all directions. must take place in all directions from the primitive source, because the displacement of a single particle from its po- sition of rest breaks up the equilibrium on all sides ; and the disturbance must be progressive, since it is to an actual displacement of a particle that the forces are due which give rise to the displacement in others. It follows, therefore, that while the first disturbed particle is describ- ing its elliptical orbit the disturbance itself is being propa- gated from it in all. directions, and that at the : instant this NATURAL PHILOSOPHY. First particle having made one circuit, another j\ist begins to move; A third begins to move. Space including particles in all positions in their orbits; Illustration. Explanation of wave length. particle has completed one entire revolution, and begins a second, the disturbance will have just reached another particle A 2 , in the distance, which particle will then be- gin for the first time to move, so that these two particles will during subsequent revolutions about their respective centres always be at the same angular distance from their starting points ; when the first particle A , has completed its second revolution, and the particle A 2 its first, the dis- tnrbance will have reached a third particle A 3 , still fur- ther in the distance, which begins its first revolution when A 2 begins its second, and A , its third, and so on indefi- nitely. Now, after the disturbance has reached the particle A a it is plain that the particles between A , and A 2 inclu- rig. 10. sive will be in all possible situations in their respective orbits. For example, taking the instant in which A , first returns to its starting point, it will have described three hundred and sixty degrees, -the consecutive particle an arc less than this, the next par- ticle, in order, an arc still less, and so on till we reach A a , which will only just have begun to move. If then, we conceive a series of concentric spheres Fig< 11 - whose radii are re- spectively A l A 2 , it is obvious that within the space in-' eluded between these spherical surfaces, the particles will be in every possible stage of their circuits around their respective centres, and will, as we pass from surface to surface, be found moving in all possible directions in the planes of their several orbits ; and the same would obvi- ously be true, if the radii of any two consecutive surfaces had been increased or diminished by the same length, the only difference being that the particles at the new position ELEMENTS OF ACOUSTICS. 33 of the surfaces, instead of being at the origin or places of rest from which they began their respective circuits, would occupy places more or less remote but equally advanced from these points. Thus, for example, had the radii been taken A^A 2 + \A 2 A^ and A l A 3 + \ A 2 A^ then Wave length not would the particles at the new surfaces have been at an^^ to an7 angular distance from their respective places of primitive position, departure equal to 90, but the surfaces would still have included between them in the direction of the radii, par- ticles in every possible state of progress in their circuits, the particle at the origin of departure being in this case at a distance from the surface of the smaller of the second set of spheres equal to three-fourths of the difference be- tween the radii of any two consecutive spheres of the first set. This particular arrangement of the particles of any body arising from the disturbance of one of its elements, and by which, after a certain lapse of time, all possible positions around their respective places of rest are occu- pied by the particles, in the order of succession, at the same time, is called a Wave. The distance, in the direction Wave - of the radii, between any two of the consecutive spherical surfaces above described, is called the length of the wave. The term phase is used to express both the par-Pbase. ticular displacement and direction of the motion of a par- ticle in any wave. A wave length, therefore, is that interval wave length. of space which comprises particles in every possible phase. Particles which have equal displacements and motions, in the same direction, are said to be in similar phases / Slmilar phases, when the displacements ? and motions are equal and op- posite, the particles are said to be in opposite phases. pp site P ha * The surface which contains those particles of a wave which are in similar phases, is called a wave front. Wave front; In sound this last term will be used to denote the surface containing those particles which are, for the first time, i n sound, beginning to move from their places of rest. In fluids the particles are not, as in solids, invariably connected, but admit of free motion among each other. "When, therefore, a fluid particle is disturbed, it acts on 24 NATURAL PHILOSOPHY. Pulse. the surrounding particles as on detached masses, and having given up its motion after the manner of one body colliding against another, it comes to rest and continues so till disturbed again by some extraneous cause ; in tho meantime, the surrounding particles move to assume with respect to it their positions of relative rest ; other particles, more remote, partake- in turn of this momentary move ment ; one particle after another comes to rest, and thus, but a single wave, denominated a pulse, is transmitted throughout the medium. If, however, instead of aban- dtfning the fluid particle after impressing upon it its primi- tive motion, it were moved to and fro, like air before a vibrating spring, waves would succeed each other in fluids wave recurrence as in solids, the circumstances of wave recurrence being ' determined wholly by the action of the disturbing cause. A wave transmitted through any medium tends to throw the elements of all bodies which it meets in its course into a similar condition of wave motion. "When the elements composing the nervous membranes of the ear become involved in certain of these motions, trans- whence we mitted through the atmosphere or other medium with experience the wm ' cn the ear is in contact, we experience the sensation sound; of sound / when the nerves of the eye partake of a similar class of waving motions conveyed through the ether, we or light; have the sensation of light / and when the waves are of that particular character to agitate the surface or euta- or heat. neous nerves, the sensation becomes that of heat. THE VELOCITY OF SOUND IN AERIFORM BODIES IT. Now, it is important to distinguish between the rate according to which the disturbance is propagated, Velocity of wave and that with which each particle describes its orbit about 3!^ its P lace of rest The first is called the wave veloc ityi tne particle. second the velocity of the wave element. The first deter- determi'ics an n^ 1168 the interval of time from the instant of primitive interval of time; disturbance to that which marks the beginning of motion ELEMENTS OF ACOUSTICS. 25 of any remote particle ; the second, the quantity of action The second a communicated to this particle. The wave is but a form qnantlt 5 r of * * action. or shape, occurring, in the regular lapse of time, at places Wave ig a form more and more remote from the place of first agitation, or sha P e - as from a centre, while the particles whose relative posi- Excursions of tions determine this form 'never depart from their places P articlesver ^ small as of relative rest but by distances which are quite insigni- compared with ficant in comparison with the lengths of the waves. The awavelen s th - wave velocity is called the velocity of sound, of liqliL of Wave velocit J" J J the velocity of hea^ or of electricity, according to the sense to which the sound, of light, waves address themselves. We now proceed to investi- gate the velocity of sound, and shall begin with the aeriform bodies, taking the atmosphere first. From the definition of a wave, 16, it follows that during the time in which the wave element, or single particle a, of air, Fig. 12. describes one entire revolution in its orbit, the front of the wave will have progressed over the distance a a', equal to a wave length. Denoting therefore, the wave velocity by F, the length of the wave a a', by A, and the time required for an element to make one complete cir- cuit by , we shall have, Mechanics Eq. (2), Value for ward velocity. 18. The time t, is, as we have seen in Mechanics, The time *, 180, independent of the distance of the particle from its 2SS^ place of rest, and is determined by the acceleration due the disturbing to the intensity of the central force at the distance unity. force< This intensity, in the case of sound, is the resultant of the antagonistic action of the force of disturbance and that of restitution, and as the latter is always constant for the same medium at the distance unity, or any other given degree of displacement, the value of t must result from the character of the disturbing force. Thus when the par- 26 NATURAL PHILOSOPHY. ticle , is made by any extraneous force to describe a path about its po- Fig. la. sition of rest, the adjacent particles e . .& lustration. ^> c -> &> e i w ^ ^ thrown into motion, and will only return to their places ^ of departure after a has been re- stored by the force of disturbance to d' 'e its position of rest; and since the places occupied at any instant by the particles J, , gives temperature and pressure. F= (t - 32) . 0,00208] . . (5). in which D tl denotes the density of mercury, and D, that of the atmosphere at 32 Fah., the atmosphere being under a pressure of 30 inches of mercury. Barometric 22. The quantity A, does not appear in Equation (5) ; from which we are to infer that the velocity is indepen- ELEMENTS OF ACOUSTICS. 31 dent of the atmospheric pressure, as it should be ; for, an Telocity of sound increase of pressure will increase the elastic force E ; but a " J^e'dc this will increase the density D, in the same ratio, so that, pressure; Equation (3), the velocity should remain unchanged. But an increase of temperature under a constant pressure dilates the air, and therefore reduces D for the same value of K Hence, all other things being equal, the velocity greater ' In warm weatlicr velocity ot sound should be greater in warm than in cold than in cold air ; greater in summer than in winter, and this is what is indicated by the quantity , in Equation (5). 23. If in Equation (5) we make t = 32, we find The density of distilled mercury at 32 Fah. is, Me- chanics, 275, equal to 13,598, and that of air at the same temperature, and under a pressure of 30 inches = 2.5, of Tabnlarvalne9 mercury is 0,001301 ; and the mean value of g is, Media- for the above /. data. nics, 72, Eq. (22), equal to 32,1808, which values in Equa- tion (6) give = 915^69 . . (6V Velocity of sound 0,0013 without increase of temperature. which would be the velocity of sound in our atmosphere under a pressure of 30 inches of mercury and at the tem- perature of freezing water, were it separated from admix- ture with all other media. 24. But it must be remarked that, the value of E, in Equation (3), which is one of the important elements of increase of this estimate, is assumed to be given by the weight due to tem Pf rature ; r the height of the mercurial column. Now, this only mea- vibration, sures the pressure due to the grosser elements of atmo- spheric air, and takes no account whatever of the elasticity eonorous wavee. NATURAL PHILOSOPHY. Elasticity due to due to that vastly more subtile and refined atmosphere of ether which permeates the air, glass, and torricellian vacuum, and which, therefore, presses alike on both ends of the barometric column. A motion among the atmo- spheric strata will give rise to a similar motion in this ether ; the equality in its elasticity on opposite sides of the strata in the direction of the motion will be disturbed ; this inequality will develope a reciprocal action among the strata of ether and those of the atmosphere itself; hence, E, in Eq.' (3), is too small, and consequently F, is also too small. Denote by^ffj a constant co-efficient which, when multi- plied into E, as indicated by the barometer, will give the true elastic force as it actually exists ; then will Equation (5) become Corrected value fcr velocity. v=V / i t g.30 ^f--K- [l + (*-32) . 0,002081 (T). or, replacing the value of the first three factors as given by Equation (6)', Velocity as f / -. affected by y - 915 QQ \/ K . (1 + (tf- 32) . 0,00208 1 . . . (T)'. etherial waves, \ / etherial waves, or increase of temperature. Co-efficient of barometric elasticity, K, To find the constant^", V, e Called the C0 efficient of IdTO- metric elasticity of the air. 25. To find the value of T 7 , corresponding to any tem- p era t ure #, it will be first necessary to know that of K. But I, being constant, if the value of V be found for any particular state of the air, that of K^ will result from e i uation CO'- The velocity T 7 ", is the rate of travel of the front of the wave from a disturbed particle of air taken as an origin. When the wind blows, the whole mass of air, and there- ELEMENTS OF ACOUSTICS. 33 fore this origin, lias a motion of translation ; and to find TO find v V experimentally, the observations should be so con- expen ducted as to eliminate the disturbing effect of the wind. To understand how this may be done, suppose an observer placed at A) midway between two sta- tions B and (7, and the wind to % ^ a blow from B to C. Denote the velocity of the wind by v ; then will the velocity with which sound will travel from B to A, be V 4- v, and from C to A, it will be V v, the mean of which is obviously V. To eliminate therefore the effect of the wind, let four remote stations B, 6 Y , Z>, E, be so chosen that the line connecting C and B, shall be perpendicular, or nearly so, to that joining E and D, and place an observer at the inter- section A. At the stations B, D, (7, E, let signal guns be fired in succession, and the observer at A note, by a stop watch, the intervals of time between his seeing the flash and hearing the report. The distances from A, being carefully measured and each divided by the corres- ponding interval in seconds, will give a value for V. The mean of these values and the reading of the thermome- ter, "which must also be noted, being substituted in Eq. (7)', the value of K will result. The experiments of MOLL, YANBEEK and KUYTEN- Experiments BROUWER, performed in 1823, over a distance of 57839 P /, their respee- D, Solving the equation and introducing these values, Amount of latent heat rendered 0,00208 1(1089^. 1 o L\ 915,697 Difference between computed and observed velocity explained. Effect on the stratum CD resumed. Two cases may arise; First ciwo; This is called the amount of heat given out by an element of air during its condensation in a sound wave. It was to the increased elasticity imparted to air by this sudden change of a portion of its heat from latent to free, that Laplace first attributed the great disparity between *the computed and observed velocity of sound. 27. Before proceeding further we must remark, that nothing has been said of the conduct of the stratum Z>, after it was im- pelled forward from its place of rela- tive rest by the action of the stratum A B, which was brought by the disturbing cause, say the motion of a rigid plane, to the position A' B f . Two cases may occur : either the stratum A B may bo retained in the position A' B 1 , or the disturbing plane may, by an opposite movement, leave this stratum unsup- ported from behind. In the first case, if the medium bo Fig. 15. C ELEMENTS OF ACOUSTICS. 35 homogeneous, the masses of all its particles will be equal, in first case a and the velocity impressed upon those in the stratum toLmltted in CD will, by the principle of the collision of elastic masses, tjn direction of be transferred undiminished to those in the stratum E F, after which the stratum C D will come to rest ; and the same of the succeeding strata in front : Mechanics, 247 ; so that there will simply be a pulse, transmitted along the direction in which the primitive disturbance acted. In the second ease, the stratum A' B ', being left unsupported in from behind, by reason of rarefaction, will be thrust back- ward by the superior elasticity of the medium in front, transmitted ID and this return or backward motion will take place in all direc ** the strata in front, in the same order of time and distance from the original disturbance as in the instance of the forward movement ; so that a second pulse will be trans- mitted in the same direction as before, only differing from the first in the backward motion among the parti- cles. Distances 28. It is easy from the known velocity of sound, to compute the distance between two places which may be sound seen, the one from the other ; and for this purpose let a gun be fired at one place, and the interval of time between seeing the flash and hearing the report at the other be carefully noted. This interval, expressed in seconds, mul- tiplied by 1089,42 %/ 1 + (t - 32) . 0,00208, will give the distance expressed in English feet. The value of t will be given by the Fahr. thermometer. Accuracy slightly The accuracy of this determination will of course be a affected by the wind, should it be blowing at the time. To ascertain the probable amount of this influence, let A be a sta- Rg> 16> tion midway between the places E and (7, and suppose the wind - c to be blowing from B to C, with a velocity denoted by v; denote the distance BA = CA by , then will the actual velocity of sound from B to A, be V+ v, and from C to A, be V r ,' and the intervals 36 NATURAL PHILOSOPHY. of time observed at A, between the flash and report from B and (7, will be, respectively, Intervals of time and V -\- v v v or developing these expressions, Second interval ; Now, the most violent hurricane moves at a rate less than one-tenth that of sound : so that the neglect of the / o terms involving -y 2 , would in the worst case only involve an error less than T V^> an ^ m the ordinary cases likely to be selected for experiment their influence would be quite inappreciable. Neglecting these terms, we see that one of these intervals will be just as much too great as the other is too small, and the true interval, denoted by , will be a mean between them. Hence, True interval; Resulting formula for distance. Jt Example. Distance from Wi-st Point to Uewburgh. or (9). Example. On the occasion of firing a salute of 13 minute guns at Newburgh, the mean of the intervals be- tween noting the flash of each gun and hearing the report at West Point, "N. Y., was 36,2 seconds ; and the temperature of the air, as given by a Fahr. thermometer, was 76 ; required the distance from "West Point to New- burgh. S=t. V= 36,2 . 1089,42 V 1 + (76 - 32) . 0,002D8 ELEMENTS OF ACOUSTICS. 37 S= 36,2.1089,42 V 1,0915 and by logarithms : 36,2 ....... 1,5587086 1089,42 ...... 3,0371954 1,0915, (i), .... 0,0190118 41,202 feet ..... 4,6149158 5280, feet in 1 mile, ac. . , 6,2773661 7,8034 miles, . . . 0,8922819 29. AVe have seen that the velocity of sound through Yelocity the air is independent of the barometric pressure, and independent of experiments show it to be sensibly unaffected by i hygrometrical state of moisture and dryness ; the actual atmosphere, weather characterised by fog, rain, snow, eunshine ;. the round ^ nature of the sound itself, whether produced by a blow, gunshot, the voice or musical instrument ; the original direction of the sound, whether the muzzle of the gun is turned one way or the other ; the nature and position of the ground over which the sound is conveyed, whether smooth or rough, horizontal or sloping, moist or dry. 30. Resuming Eq. (7), and denoting by V and F" velocity of sound the velocities of sound through any two gases whatever, by K' and K" their co-efficients of barometric elasticity, and by D' and D" their densities ; then, supposing the barometric column exposed to the pressures of the gases to be 30 inches, and the temperature of the gases to be the same and equal to t degrees, will, Eq. (7), give V = \/$.30' ^ . K 1 j"l + (t i- 32) . 0,0020Sj| Value In first; and F" = \/g . W**'E. . K" F 1 + (t - 32) . 0,00208J ; Value in 8econd; 38 NATURAL PHILOSOPHY. Dividing the first by the second, we have Velocities compared. n / **L ^ V" - V K" ' D r (10). Conclusion. That is, the velocities of sound in any two gases, at the same temperature, are to each other as the square roots of their coefficients of barometric elasticities directly, and densities inversely. From Equation (10) we readily obtain K' K" V' 2 D r jy (11). Atmospheric air Taking one of the gases atmospheric air, and the other and hydrogen; hydrogen, and assuming the velocity of sound in hydro- gen, as determined by the experiments of YAN REES, FKA- MEYER and MOLL, to wit, 2999,4 English feet, we have, after substituting the known values of the quantities in the second member, liatio of their constant coefficient*; Inference ; Conforms to Boscovich*a theory. K" 1089,42 . 0,0688 = ? 5215. Hence the coefficient of barometric elasticity of air is nearly double that of hydrogen ; a result which appears to indicate that the velocity with which sound is propa- gated through gases is in some way dependent upon their chemical or physical constitution. This would seem but the natural consequence of the views of Boscovich. VELOCITY OF SOUND IN LIQUIDS. Experiments on 31. From the experiments of CANTON, OERSTED, and others, liquids as well as gases are found to be both com- ELEMENTS OF ACOUSTICS. 39 pressible and elastic ; and are therefore fit media for the Experiments on transmission of sound. From the experiments of COLLA- pur ' DON and STURM, on what may be regarded as pure water, SIR JOHN HERSCHEL deduces the compression of this fluid, by one standard atmosphere, to be 0,000049589 = e; that is to say, an increase of pressure equal to that arising from a column of mercury having an altitude of 30 inches and temperature of 32 Fahr., will produce a diminution in the bulk of water equal to 1 4 9 5 8 9 of the entire volume which it had before this increase. 32. All bodies may be stretched or compressed by the application of force, and when unaccompanied by perma- Law { d} nent change of molecular arrangement, the degree of com- tion. pression or extension is directly proportional to the intensity of the force which produces it. 33. Denote by M and B, the intensities of two forces capable of stretching a body, whose cross-section is equal to unity, to double its natural length L, and to L + Z, respectively ; then will j Measure of L:l::M:B-, .-. B = M. j==M . e, elastic force, in which Mia called the coefficient or modulus of elasticity. 34. Let A B, and CD, be two consecutive strata of water, and suppose the stratum AB, to have been suddenly moved by some Fi s- 15 - disturbing cause to the position A B'. Denote the distance BD by a?, and B 1 D by a?,, then, regarding the area of the stratum as unity, will the dif- ference of volume between ABCD and A' B' CD, be represented by a? a?,, and the degree of compression referred to the original volume, by Illustration ; J) X X { Degree of ~ compression; Compressing forc; NxlTURAL PHILOSOPHY. and the force E t , necessary to produce this compression will, 4, be given by the proportion its value. Me : M. - -' : : B : E. x ' in which B = g . D it . A, denotes the pressure due to a stan- dard atmosphere, being the weight of a column of mer- cury whose density is D tl and height A. Whence 7? v v __- JLJ us tC'. x Combined pressure on a stratum below the surface ; But any stratum of water situated below the surface is already subjected to the pressure of the atmosphere, and that arising from the weight of the column of the same fluid above it. Denoting this combined pressure by #>, we shall have the stratum A' B', and therefore CD, since the resistance to compression arises from the reaction of the latter, urged forward toward E F, by E t + p ; but the motion of CD is resisted by the pres- Moving force on sure p, whence the moving force becomes E -f v r> E '. * of*n4i-i-rvt ' -* * ' The mass of the stratum CD will, 20, be stratum; Mass of a stratum ; D.x whence the acceleration due to the moving force, or the velocity generated in a unit of time, becomes, after substi- tution for B, its value, Velocity generated in a nnit of time ; E 1 x x velocity in an anc * tne velocity v, imparted to the stratum CD, in an elementary elementary portion of time t, will be given, Mechanics, portion of time. - but, 20, ELEMENTS OF ACOUSTICS. _ g h D lt t_ X x t Its value; e D x x t 1 , v x x, - = .> and = - x V V x which substituted above gives, 33, after clearing the fraction and extracting the square root, "Wave velocity i / L T\ /T? v - \/jL!L~ = \/ - V e.D V D ' and substituting the numerical values of . ed with a number of equal and equidistant holes arranged in the t circumference of a cir- cle concentric with the axis of motion (7, is made to revolve. The tube through which the air passes is so situated that the holes in the disc shall pass in rapid succession over its open end and permit the air to escape, being at the same time so near to the plane of the disc that intervals be- tween the holes serve as a cover to intercept the air. If Construction and the holes be pierced obliquely, the action of the current of air alone will be sufficient to put the disc in motion ; if perpendicular to the surface it must be moved by wheel work, so contrived as to accelerate or retard the rotation at pleasure. The bellows being inflated and the disc put in motion, a series of rapid impulses are communicated to the air in front of the holes ; and, when the rotation is sufficiently rapid, a musical tone is produced whose pitch becomes more acute in proportion as the velocity of rotation increases. To sfrow that the Bellows may be a j r o f the bellows only acts as a mass in motion to im- replacedbya , ., ,. . ,, ,, reservoir of preSS DV its living lOl'Ce SUCCBSSlVe D10WS Upon the ex- water; ternal air, the bellows may be replaced by a reservoir of water, the liquid being under sufficient head to cause it to spout through the holes of the disc as they come successively in front of the duct pipe ; the effect is the same. Connected with the axis of rotation of the disc are a stop-register, which indicates the number of revolutions, and a stop-watch, to mark the time in which these revo- Btop-register and lutions are actually performed. The instrument being put in motion and accelerated to the desired pitch, the i-egister and watch are relieved from the stops, and after ELEMENTS OF ACOUSTICS. 51 the sound has continued for any desired length of time, Reading of the the stops are again interposed, and a simple inspection of the dial plates of the watch and register will give the time and number of revolutions. Now, suppose the disc to be pierced with m holes, the number of revolutions to be n, and the number of seconds to be T. The number of impulses, and therefore the number of waves, will be m . n ; and the number of waves produced in one second w T ill be Number of wave* in one second ; But these waves, generated in one second, occupy the entire distance denoted by V, the velocity of sound ; and hence, denoting by A, the wave length, we have the relation, Equation (8), ^-^ . A - F = 1089,42. V/l-K* -32). 0,00208. Formula; whence, 1089,42 . T. Vl+(t - 32) 0,00208 1ft , Valueforwave - length - Example. Suppose the revolving disc to be pierced with Example; 100 holes, the time of rotation 20 seconds, the number of revolutions in this time 102,4, and the temperature of the air 84 Fahr. Then will m = 100 ; n = 102,4 ; T= 20*- ; t = 84, which in Equation (18), give 1089,42. 20.x/ 1+52. 0,00208 _ * Valne(v , K 100 . 102,4 52 NATURAL PHILOSOPHY Results of experiments; thus making the length of the wave two and a quarter English feet, nearly. The results of the experimental researches of M. BIOT, on this subject, are given in the following table : Number of vibrations in one second. Length of resulting ware in English feet 1 1091,34 2 545,67 4 - . 272,83 64 .... . . . . 17,05 1 128 . . . . ... . . 8,52 1 a 256 .... . 4,26 1 8 512 . . . . . 2,13 1024 .... 1,06 ll 2048 0,53 1 s 4096 .... ...... 0,26 g 8192 . 0,13 Lowest audible pitch. Highest audible pitch. 49. From these experiments it has been inferred that the lowest pitch audible to the human ear, is that pro- duced by a wave whose length is 34,10 English feet, and of which there are generated, in one second of time, 32 in number ; and that the highest audible pitch is given by a wave whose length is 0,13 of an English foot, or about one and a half English inches, and of which 8192 are generated in a second. But in such experiments much must depend upon the ear of the experimenter ; we know that this organ differs greatly in different persons, even among those who are unconscious of any defect in their sense of hearing. Some have contended for a high pro- bability that a body making 24000 vibrations in one second, produces a sound which, to a fine ear, is distinctly audible ; and M. SAVAKT, by means of a rotary cog-wheel, Results vary with so arranged that each tooth should strike a piece of cardj experimenters. f oun( } that 12000 strokes on the card in one second, pro- ELEMENTS OF ACOUSTICS. 53 duced a sound perfectly audible, as a musical tone of high Powers O f th pitch. Although different authorities differ in regard to ** the powers of the ear, they nevertheless all agree in ascribing to them a limit. And thus, of the almost end- less variety of waves which must, from the existence of ceaseless sources of disturbance, pervade the air, our organs of hearing appear to excite the mind to impres- sions of those only whose lengths range within certain prescribed limits. Nor is this limitation peculiar to the ear. "We shall have occasion, wjien speaking of light, to remark the same thing of the eye. We shall find that same tru for the when, from too small or too great lengths, the waves of eye; ether lose the power of stimulating the optic nerve to the sensation of light, they nevertheless do, when addressed to other organs, give rise to the further and obvious sen- sations of heat. And to what extent we are uncon- our senses cannot sciously influenced by those agitations of surrounding "JJ^^. a " media which fall beyond the range of the ordinary senses to appreciate, it would be out of place here to inquire. 50, There is nothing in the constitution of the The sensations of atmosphere to prevent the existence of wave pulses ^a^be^n incomparably shorter and more rapid than those of which wnre ours end; we are conscious ; and we are justified in* the belief that there are animals whose powers in this respect begin where ours end, and which may have the faculty of hear- ing sounds of a much higher pitch than any we actually know from experience to exist. And it is not improba- ble that there are insects endued with a power to excite, and a sense to perceive, vibrations of the same nature as those which constitute our ordinary sounds, yet of wave dimensions so different, that the animal which perceives g ac h animals them may be said to possess a different sense, agreeing may be Bldd to with our own in the medium by which it is excited, yet entirely unaffected by those slower and longer vibrations of which we are sensible. NATURAL PHILOSOPHY. DIVERGENCE AND DECAY OF SOUND. directions; nave already stated, 16, that when the pri- mitive agitation of a medium is confined to a small space, the initial wave front is of a spherical shape, and we have seen, Equation (12), that the sound wave proceeds with equal velocity in all directions in which the density and elastic force are the same/ In all homogeneous media, the wave front will, therefore, retain its sphericity to whatever distance it may be propagated, and a sound produced at a given point, as from the blow of a ham- mer, or the explosion of gunpowder, will be heard equally well in all directions. Not true when density and elastic force vary; ' Illustrated by tuning fork; 52. When, however, sounds proceed from a series, of points situated upon the surface or face of a solid, the body of which interposes to prevent the existence of equal density and elastic force in all directions from the points of disturbance, this equality of transmission in all directions no longer obtains. This is well illustrated by an experiment 'due to Dr. YOUNG. A common tuning fork, a piece of steel, whose shape is repre- sented Jn the figure, being struck sharply and held with its handle A. against some hard substance, is thrown into a state of vibration, its branches B, B, alternately ap- proaching to and receding from each other. Each branch sets the particles of air in motion, and a sound of a certain pitch is produced. But this sound is very unequally audible in different directions. When held with its axis of symmetry vertical and at the distance of about a foot from the ear, and turned gradually about this axis, it is found that at every quarter of a revolution, the ELEMENTS OF ACOUSTICS. 55 sound becomes so faint as scarce- Audible and Iv to be heard, the audible posi- Fig> 20 ' inaudible ' , . . , o positions of the tions of the ear being in planes ni ^ ear. E E and 0, perpendicular and parallel to the broader faces of the fork; the inaudible, in planes E 1 E' and 0' 0', mak- ing with the first, angles of 45. 53. To resume the consideration of sound propagated from a central point. The intensity or loudness of sound sound is, 46, determined by the living force with which the de particles of a medium in sensible contact with the ear act upon the auditory nerves. At the primitive Pi n t Nolos8oflivln of disturbance the living force is impressed by the dis- force in elastic turbing cause, and is transferred from the particles of media; one wave to those of another without loss, provided the molecular arrangements of the medium in the process are not permanently altered, Mechanics, 64, which is the case in all elastic media, such as the air and other gases when not confined. The sum of the living forces Sumofliving of the particles in a wave must, therefore, be constant, forces of particle* to whatever distance the wave be propagated, and equal to double the quantity of work expended by the dis- turbing motor. The living force of any single particle is equal to the product of its mass into the square of its velocity, and from the nature of the wave, 16, the Sa me true for living forces of all the particles on any spherical sur- **Y sp^ricai face whose centre is the point of primitive disturbance must be equal to each other ; for the velocities are equal, and the medium being of homogeneous density, the masses of the particles have the same measure. Denote by R the radius of any spherical surface in- illustration; termediate between the interior and exterior limits of the wave in any assumed position, by n the number of particles on the unit of surface, then will the number of particles on the entire sphere be 56 NATURAL PHILOSOPHY. Number of particles on a spherical surface; Sum of their living forces; n TT JR 2 , and the sum of their living forces n in which Y denotes the velocity common to all the par- ticles, and m the mass of a single particle. For another spherical surface, whose radius is jfr', and the common velocity of whose particles is F"', we will have Same for another spherical surface; n . TT . m V' a Living forces equal; Now, if these spherical surfaces occupy the same rela- tive places in the wave in any two of its positions, be their distances from the centre of disturbance ever so different, these living forces must, from what is said above, be equal ; whence we have, after dividing out the common factors, Consequence ; t m a = or resolving into a proportion Euie first; That is to say, the intensity of sound varies inversely as the square of the distance to which it is transmitted. Again, the particles describe their orbits in equal times ; their greatest velocities will, therefore, 16, be proportional to their greatest displacements, and the in- Euie second. tensity of sound to the squares of these same displace- ments. 54. The greatest distance to which sounds are audi- ble does not admit of precise measurement. It depends ELEMENTS OF ACOUSTICS. 57 principally upon the absolute intensity of the sound itself, 8oundheard J ' i. further in dense the nature of the conducting medium, and the delicacy media; of hearing possessed by individuals. Generally speak- ing, a sound will be heard further, the greater its ori- ginal intensity, and the denser the medium in which it is propagated. The greatest known distance widen sound has been Greatest known carried through the atmosphere is 345 miles, as it is distance ta ***; asserted that the very violent explosions of the volcano at St. Vincent's have been heard at Demerara. Sound travels further and more loudly in the earth's surface earth ' s 8urfac than through the air. Thus, for instance, in 1806, the * cannonading at the battle of Jena was heard in the open fields near Dresden, a distance of 92 miles, though but feebly, while in the casements of the fortifications it was heard with great distinctness. So also it is said that the cannonading of the citadel of Antwerp, in 1832, was Instances J heard in the mines of Saxony, which are about 370 miles distant. When the air is calm and dry, the report of a musket Ee P rt of is audible at 8000 paces ; the marching of a company m may be heard on a still night, at from 580-830 paces off; a squadron of cavalry at foot pace, 750 paces ; trbt- Marchofcavalr y; ting or galloping at 1080 paces distant ; heavy artillery, or artillery. travelling at a foot pace, is audible at a distance of 660 paces, if at a trot or gallop, at 1000 paces. A power- ful human voice in the open air, at an ordinary tem- perature, is audible at a distance of 230 paces, and Captain Parry tells us that in the polar regions a con- versation may be easily carried on between two persons a mile (?) apart. 58 NATURAL PHILOSOPHY. MOLECULAR DISPLACEMENT. TO find the any instant; 55. Let us now seek an expression for the distance of an y mol ecule from its place of rest, at any time, dur- place of rest at ing the transmission of wave motion. This displacement odiously depends upon the intensity of the disturbing cause, the distance of the molecule under consideration from the place of primitive disturbance, the velocity of wave propagation, and the time elapsed since the primi- tive disturbance was made. Disregarding, for the present, the diminution of the amplitude of vibration due to the loss of living force in the successive molecules as we proceed outward from the source of sound, let fa Q p()mt Q f supposed displacement of A an assumed particle; i T> i i disturbance, and B, the place of rest of any assumed mole- cule. Denote by #, the dis- tance of B from A, and by V j? i> \ \ c w j Consequent P the velocity of wave propagation. At the expiration of the time , after the instant of primitive disturbance at A, let the wave front be at TF", and the molecule at B, be disturbed by the distance B b. The distance of W from A, will be V. t. Now, from the nature of the motion transmitted, any otner molecule whose place of rest is C, beyond .#, must experience an equal displacement C c, at the expiration of the time t + ', which is as much in excess over the time required for the wave front to reach (7, as the time , was over that required to reach B. In other words, the displacements must be equal for successive molecules whose places of rest are at equal distances behind the wave front ; and hence the displacement of the must be a function of this distance, that is, of V. t x ; distance v.t-u. an( j denoting the displacement by d, we may write ELEMENTS OF ACOUSTICS. 59 d = F ( V. t X) ; Firstvalueof % displacement. in which F, denotes the form of the function to be em- ployed. Moreover, from the definition of a wave, the nature Function of the function F, must be periodic ; that is to say, it pe must, within a given interval of time, pass through all its possible values, and resume and repeat these values in the same order during the following equal interval of time. This is a property possessed by the circular functions, and hence we may write the above y sin V. t # [ Its form; in which 2 TT, denotes the circumference of a circle whose radius is unity, and y the radius of the small circle of which the sine of some one of its arcs will give the dis- placement sought. The radius y, is equal to the greatest displacement of a molecule in the same wave ; for, a simple inspection will show that the function takes its maxi- mum value when the quotient, "V. t X When a " * J maximum: becomes equal to one^burth, or to any odd multiple of one-fourth ; the value being in that case y . sin 90, or y sin 270, or y . sin 450, &c. = y. Maximum vain* But the intensity of sound diminishes as the square of Law of variation the distance from its source increases ; and the intensity being directly proportional to the square of the greatest displacement, 53, if , denote the radius of the small circle at the distance unity from the source of primitive disturbance, we have 60 NATURAL PHILOSOPHY. Expression of the law; ___ (I) 2 x or Radius of the circle whose sines give the displacements; a which substituted for y above, gives Second value of displacement; The quantity T 7 '. , denotes the linear distance of tl.e front of the wave or pulse from the source ; V. t a?, the dis- tance of the molecule's place of rest from the wave front ; and when this distance contains the length A, either a whole when it is zero; number of times, or a whole number of times plus one- half, d becomes zero. When the remainder, after the divi- sion of V.t w, by A, becomes \ or f, &c., the value of d When a maximum. d becomes 5 its maximum value. If the arc a TT. V.t-x Arbitrary quantity ; Final value of displacement; be increased by an arbitrary quantity A, it is plain that we may assign to A, such a value as to cause any given displacement, and therefore the maximum displacement, to occur at a given place and time. Introducing this arbi- trary quantity, we finally have the general equation a d = sm [ in which * determines the intensity of the sound ; A, its x ELEMENTS OF ACOUSTICS. pitch; and A, the particle whose place of rest is at a dis- Meaning of the tance x from the source^ to have anj ment at the expiration of the time t. tance x from the source., to have any particular displace- q " INTERFERENCE OF SOUND. V ~~ 56. "We have seen, in Mechanics, that a body may be animated by two or more motions at the same time ; that the ultimate result of these motions, as regards the body's position, will be the same as if these motions had taken place successively ; and that one or more of these motions may be destroyed at any instant without affecting in any wise the others. These coexistent motions, estimated in any given direction, become, as it were, superposed upon each Coexistence ^ other, and w r hen very small, give rise to a principle known superposition of as the " coexistence and superposition of small motions" ; a 8DC principle most fruitful of results in sound and light. By it we are taught that when the excursions of the parts of what it teaches; a system from their places of rest are very small, any or all the motions of which, from any cause, they are suscep- tible, may go on simultaneously without disturbing one another. The truth of this important principle will appear from It8 tmth its application to the particular case in question. It has been shown, 4, that when a molecule of any body is very slightly disturbed from its place of rest, as in the case of sound, the forces exerted upon it by the surrounding molecules give rise to a resultant whose in- tensity is proportional to the amount of displacement. This displacement may arise from the action of a sin- gle or from several causes operating at the same time ; but in every case, the expression which gives the value of the resultant action must be a function of those which express the values of the partial actions, and, like each Explanation, of these latter functions, being proportional to the dis- placement it is capable of producing must, as well as NATURAL PHILOSOPHY. Partial, as well as the partial functions, be linear. In any such function, mulu>e unesT- ^ we attribute a slight change to one of the disturbing causes, the corresponding change in the displacement must be proportional thereto; and whether the change in all the partial causes, or in the functions which measure them, be simultaneous or successive, the final result will be the same ; for, the change in the entire function in the first case must be equal to the algebraic sum of the partial changes in the second. To those fa- miliar with the calculus, it will be sufficient to say, that the first power of the total differential of the sum of a number of functions, is always equal to the first power of the sum of the partial differentials. We conclude, therefore, that the function which gives the displacement may be broken up, so to speak, into several partial functions equal in number to that of the disturbing causes ; that these partial functions will be !on; similar to each other and to the entire function; and that this latter will be equal to the algebraic sum of the former. (Analytical Mechanics, 204 and 306.) Blastration; Fig. 22. 57. To illustrate : let the straight line A B, be the locus of a series of molecules in their positions of rest ; the fine waved line 5, that of the same molecules at a particu- lar instant of time, when disturbed and thrown into a wave by the action of some single cause ; and the waved line a' &', that of the same molecules at the same in- stant had they been thrown into a different wave under the operation of some other insulated action. If these disturbing causes had acted simultaneously, the locus of Construction of the disturbed molecules would be represented by the resultant wave. , ,.. -rr -\r- . i ^ A j. heavy waved line A. Jr, constructed in this wise : At the various points of the line A B^ erect perpendiculars Partial waves: ELEMENTS OF ACOUSTICS. 63 and produce them indefinitely; lay off from A. JB, on Resultant curve; these perpendiculars, distances equal to the sum or dif- ference of the corresponding ordinates of the component curves, according as these curves intersect the perpen- diculars on the same or on opposite sides of the line A B, the points thus determined will be points of the resultant curve, which will give the law of displacement at the instant of time in question. Were there three, same for three or four, &c., component curves, the resultant curve would components. be determined by the same rule. 58. Taking it, then, as a fact, that the disturbance Eesultant action of every molecule produced by the coexistence of two of two equal or. more causes will be the algebraic sum of the dis- turbances which they would produce separately, let us consider the nature of the displacement produced by the superposition of the action of two waves of the same length on the same molecule, the waves being supposed to come from any directions whatever. We shall have for the displacement of the molecule by the first wave, Eq. (19), waves on a particle ; and by the second, in which a' and a", determine the intensi- ties of the sound in the two waves at the unit's distance ; and A' and A", the places of the maximum displace- ment at the expiration of the time t. Fig. 22. (20) Displacement by the first wave; = ".sin[8 W .Z^-Hr'];. . . (21) Same by the second; 64 NATURAL PHILOSOPHY. Taking the sum, and developing the circular function bj the usual formula for the sine of the sum of two arcs, we find, after reduction, Bum of , (a'cos;i' + a"cos^") . [~ 2 V^\ , (a 1 sin A' + a" sin A") f K-af| displacements; - -- .sin[_^ ff . - J-J-- -- - -- --cos^ir. - J and making, supposition; a . cos A = a' cos A' + a" cos A!', . . . (a) a . sin A = a' sin A' + a" sin A", . . . (5) the above becomes, after writing d for the total displace- ment, ; '* = ~ [cos A . sin (% TT Zdh?) + sin^l . cos (2 n V ~)] ; Ay- replacing the quantity within the brackets by its equal, viz. : the sine of the sum of the two arcs, we have *=$'**[* *'-^ + A\ ( 33 ) Squaring Equations (a) and (5) and taking the sum, we find, Transformations; ^ = d! 2 + a" 2 + 2 a' V COS (A - A") . . . (23) and dividing Equation (5), by Equation (a), we obtain Seductions; tan A = g'-BinJ/ + g".BJn J" ^ . a . cos ^1 -f a", cos ^1" From Equation (22) we see that the length of tho Conclusions; resulting wave is the same as that of the partial waves ; but the value of A in that equation differing from A', ELEMENTS OF ACOUSTICS. 65 and A". Equation (24), shows that the maximum dis- Timeof maximum placement for a given molecule does not take place displacement in with the same value of t, as for either of the compo- resultantwave; . nent waves. The maximum displacement , which determines the x intensity of the sound, in the resultant wave, is given by Equation (23) to be X V 2 + a" 2 + 2a'a".COs(A'-A") which depends upon the arc A' A". Its greatest va- lue is obtained by making A A" = 0, in which case we have Fig. 23. \ad) General value of this displacement; When this value is greatest; Greatest value; its least value results from making A! - A!' = 180, in which case A Fig. 24. When this value is least; Least value; In the first case Equation (24) gives tan A = (a + a ). = tan A , = First case; whence A, is equal to A', and to A", and the maximum Conclusion- displacement will occur at the same place and at the same time in the resultant wave, and in both compo- nent waves. In the second case, if we substitute in Equation (24) A' = 180 + A", we find 5 NATURAL PHILOSOPHY. Second case ; Conclusion; (a" a') cos A" that is, A is equal to one at least of the arcs A and A") and the greatest displacement in the resultant wave will occur at the same place and time as in one of the component waves. intensity of 59. If the intensity of Bound supposed S()und -^ ^ compone nt equal in component waves be supposed equal at the place of superposition, A then will a' = a", and Eq. Fig. Consequence; (25) becomes a 2a f A - A' = COS xx 2 and Equation (24) reduces to (26) Reduction; Value of arbitrary constant ; Supposition ; cos A' + cos A" or. When A - A' = 0, then will Eq. (26) give J? = , and A = A = A' ; x x (27). Illustration. that is, the intensity of sound in the resultant wave is quad- ruple that in either of the equal component waves ; and the greatest displacement will occur at the same time and place in the component and resultant waves. Fig. 26. ELEMENTS OF ACOUSTICS. 67 If a' and a" continue equal, and we make A' A" '=180, supposition; then will Equation (26), give 126)'. Consequence ; or in words, one of the equal snence sounds will destroy the other. Thus it appears that two equal sounds reaching the same point may be in such relative condition that one will wholly neutralize the other, and the two produce perfect silence. This phenomenon is called the Inter- Interference <* ,. , sound. ference of sound. With any other values for A' and A" than those which give A - A" = 180 or 0, Eq. (26), shows that a Result of partial coincidence of two sound wave* that is, the sound in the resultant wave is less than quad- ruple that in either of the equal component waves. T T . -t .,, will cause two equal waves, which will cause one to destroy the other, eq uai waves to 60. To ascertain the precise relation between two al waves, which will make, in Equation (20), A = A" =t 180 = A' and we have neutralize ach other. d' = sin V.t-x + A" db but 2 r. X TransformationB; NATURAL PHILOSOPHY. and this substituted above, the equation becomes Resultant displacement; d'= - .sin [2*. x V. t-x i x Conditions for interference. When waves Interfere only at point of union. Fig. 27. which becomes identical with Equation (21) by writing x, for x =p X. That is to say, one wave will destroy another of equal length and intensity, if, starting from the same origin, in the same phase, they meet, after trav- elling over routes that differ in distance ly half a wave- length. And since a difference of route equal to any whole num- ber of wave lengths produces no difference of phase in the undulation, it is obvious that a difference of route equal to any odd multiple of half a wave length, produces the same effect as a difference of a single half. Thus, two waves will destroy one another, if they be of the same length, have the same maximum molecular displacement, travel along the same route, and have, at any point, opposite phases. If they travel over different routes and meet, they can only interfere at the point of union. This mutual destruction of two waves, having op- posite phases at their place of union, is illustrated at 52. game 61. The same process of combination may be ap- considerations plied to three, four, &c., waves of equal lengths. Thus let there be the Equations equal waves ; d' = - x Equations to be used; =- sn x ELEMENTS OF ACOUSTICS. 69 adding these, developing the sine of the sum of the two arcs within the brackets, col- lecting the common factors and denoting the resultant displacement by d, we have Operations performed; Fig. 28. Illustration; or The same: in which a cos A = a! cos A' + a" cos A" + a'" cos A" = X a sin A = a'smA! + a" sin J." + a'" sin A" = T Notation. 62. Although it is possible for two waves of sound, whose lengths are the same, to neutralize each other, it is not so when the waves have un- equal lengths; for, Eq. (22) was deduced by making V and X the same in the two component waves, the sum of d f and d" being in that case reducible. If these con- ditions were not fulfilled, this sum would not be reducible, and there would be the two arcs Two unequal waves cannot neutralize each other; Illustration; NATURAL PHILOSOPHY. Explanation; X' in the final value for d, with different coefficients, which could not be made equal to zero at the same time. The values of V, will, J:o be sure, be the same in any two waves of sound, but this need not be so with those of X ; and in waves which produce light, in which subject we shall have most occasion to refer to the doctrine of in- terference, the values of F, as well as those of X, may differ. The discussions of waves of different lengths may, unequal waves; therefore, be kept perfectly separate, as the combined effect of such waves will be the same as the sum of their separate effects, without the possibility of their destroying or modifying one another. NEW DIVERGENCE AND INFLEXION OF SOUND. Any disturbed particle causes subsequent disturbance in another ; Same true for all particles in a wave front; Illustration ; Fig. 30. 63. "We have seen that every disturbance of a mole- cule at one time is truly a cause of disturbance of an- other molecule at some subsequent time. All the mole- cules in a wave 'front become, therefore, simultaneously centres of disturbance, from each one of which a wave proceeds in a spherical front, as from an original dis- turbance of a single molecule. Thus, in the wave front A B, a molecule at x becomes a new centre of dis- turbance as soon as the wave front reaches it ; and if with a radius equal to V.t a circle be described, this circle will represent a section C of the spherical wave front proceed- ing from a?, with the velocity "F, at the end of the interval of time de- noted by t. And the same being true for the molecules a?', a?", &c., of the primitive wave, there will result a series of intersecting circles ELEMENTS OF ACOUSTICS. 71 having equal radii, and the larger circle A' B', tangent to all these smaller circles, will obviously be a front; section of the main wave front at the expiration of the interval , after it was at A B. Any molecule situated at the intersection of the smaller circles will obviously be agitated by the waves transmitted to it from mole- Resultant cules at their respective centres ; and the resultant dis- displacement of placement will, 55, 56, be the algebraic sum of the* 1 displacements due to each when superposed. Hence, to find the disturbing effect of any wave upon a given molecule at a given time, divide the wave 'into a number of small parts, consider each part as a centre of disturbance* and find by summation the aggregate of all the disturbances of the given molecule by the waves coming from all the points of the great wave. The cause which makes the disturbance of a single molecule at one instant the occasion of the simultaneous disturbance of an indefinite number of surrounding mole- Principle of neir cules at a subsequent instant, is called the principle of divergence new divergence, of which frequent use will be made in stated ? the subject of light. 64. Let us trace the consequences of this principle Its application to in its application to the passage of sound through aper- ^^hTugh tures and around the edges apertures and of objects. Take a parti- tion M. N, through which there is an opening A B, and suppose a spherical wave of sound to proceed from a centre C. Only that portion of the wave which comes against the opening can pass through, and the wave front on the opposite side of the partition will be found by taking the diffe- illustration ; rent points of the segment A B, within the opening as centres, and radii equal to V. t, and describing a series of elementary arcs, and drawing a curve tangent to them NATURAL PHILOSOPHY. Explanation and construction ; Fig. 31. Bound that is not reinforced by particles from the primitive wave : Sectors wherein the sound is doe to superposition of waves from the edges; Intensity increased by coincidence; Decreased by interference. Points taken between the all. That portion of this tangent curve included be- tween the lines C A and C B', drawn from (7, and tangent to the limits of the opening, will obviously be the arc of a circle having G for its centre. The elemen- tary circles described about the limits A and B as cen- tres, cannot be intersected at points exterior to the angle A C B by those described with equal radii from points of the wave front lying between A and B ; the wave front within the angles A A M and B' B JV, will have their centres at A and B respectively ; and the sound proceeding from these points will be diffused over the arcs A' M and B' N without reinforcement from molecules of the same primi- tive wave. But other waves from C reaching the opening in suc- cession, a spherical wave diverging from B, and of which the radius is B 0, will be overtaken by a subsequent one from A, having for its radius A / so that, the intensity of sound in the angle A A M will result from the super- position of the disturbances from B and A. The same will be true of the sector B' B N. Now, if B A 0, be equal to X, 2X, 3X, . . .^X, in which n is a whole number, then will the intensity of the sound be increased above that due to either of the com- ponent waves. But if B A O, be equal to \ X, | X . . . . (n + i) X, n being still a whole number, the compo- nent waves will interfere at 6>, and the intensity of the sound will be lessened at that point by the prevention there of the disturbance due to either of these two component waves. Taking another molecule B t , nearer to A, the wave from B^ will interfere with the wave from A, but at a point 0^ nearer to the partition, in order to pre- serve the difference B t O t A O t , the same as before. ELEMENTS OF ACOUSTICS. 73 to wit, (n + J) X. Assuming other points in succession construction; nearer to A, Ve shall find the interference to take place at molecules still " nearer to the partition ; and finally, when we come to a molecule , m the main wave front whose distance A. Q, from A. is equal to (n -f- i) X, the interference will occur at a molecule situ- ated against the partition at P. Now, making n = 0, in which case A Q will equal \ X 5 and applying , . ,. Fig. 82. i X irom a to a l , the waves from a ^ ,..... . * ">* e - ? ,..& Illustratlon 5 (^ jj ji and # y , will inter- fere at P. Applying \ X, from & to T) t , the waves from 5 and 5, will also interfere at the partition ; and in the same way it may be shown that all the partial waves from molecules in the distance A , will interfere with those from the molecules in the distance Q D, Q D being equal to A Q. Commencing the same process at D, we see that the opening may be such that on applying \ X from a' to a' ' t , this latter point a' t may be in the posi- Explanation of tion from which there can be no new divergence to inter- results; fere with that from a' ; and the same for the whole of the arc D B, of the main wave. This latter is, therefore, left, as it were, undisturbed, and sound from it may or may not be audible at P, depending upon the extent of this arc and the intensity with which the sound reaches the opening. The distance A Q is equal to 1 X. But X, we sound heard at have seen, 48, varies with the pitch, whence the sound partition depends ' on pitch, and siz heard at P, will depend upon its pitch and the size of evening. of opening through which it may pass. 65. From what precedes we see that at the line Acoustic shadow A <9, Fig. 31, there begins, as it were, an acoustic shadow, cast which deepens more and more as we approach the partition towards P, where the sound becomes least Inflexion of audible. This bending of sound around the edges of an opening is called the Inflexion of sound. 74: NATURAL PHILOSOPHY. Case of sound bending around corners : Explanation; No perfect neutralization ; Grave sounds more audible .than the acute ; Case of little Inflexion, 66."When the opening is continued indefinitely in one di- rection only, we have the case of sound bending around a cor- ne^. But when the openr ing is continued indefinite- Fig< 38> ly in one direction, there Ji ma smsmm^ B JP ji $,%, JD,. can be no arc of the main wave as D jB, (last figure), without a corresponding arc D t B { , further on, to neutralize it in part at least by interference, and hence, were the component sounds of the same intensity at the point of superposition, they would produce perfect silence, and no sound could be heard at P. The sound from the main wave is of the same intensity throughout on reaching the corner ; the new diverging waves leave their respective centres, which are distri- buted along the front of the main wave, with equal intensities ; they can only interfere after having travel- led over routes which differ by X; the intensity of sound varies inversely as the square of its travelled dis- tance ; and the intensities cannot be equal at the places of interference, and therefore can only partially neutralize each others' effects. This is shown by Equation (26)', in which is zero, only because a?, under the conditions x there imposed, is the same denominator for a' and a". In sound, X varies, as we have seen, 48, from a few inches to many feet, and as the difference of intensities in the interfering waves will be greater as X is greater, the graver sounds would be heard, under the circumstances we have been considering, more audibly than the more acute. If the lengths X, were insensible in comparison with the route travelled, there would be but little in- flexion; since, in that case, the intensities of sound in the interfering waves would be sensibly the same, and it would require but a slight obliquity from the direct course of the main wave to make a difference of route B A 0, Fig. 31, equal to ix, necessary for one wave sensibly to destroy the other. ELEMENTS OF ACOUSTICS. Y5 An auditor placed behind a wall p <*j at P, would hear the bass notes ,,.,. n Person behind a from a band of music playing at a ^^ wan listening to position A on the opposite side, much - p T^mJ a band of music; more distinctly than the acute notes. At P, the notes of the tuba, for instance, might be heard distinctly, while those of the octave flute would be lost to him. In passing from the position P to 0, he would catch in succession the higher notes in order of the ascending scale, and finally, when he attained a position near the p os i t i n whence direct line A 0, drawn from A, tangent to the corner, he a11 the would hear all the instruments with equal distinctness, if ' played with equal intensity and emphasis. The facts arid explanations here given have an important applica- tion in the subject of optics. If we suppose the lengths of sonorous waves propagated through water to be much shorter than those through the air, we have here a full and satisfactory explanation of the phenomenon observed by M. COLLADON, mentioned at the close of 39. Indeed, taking the acoustic shadow Foregoing there referred to as established, it must follow as a conse- deduction3 conformable to quence, from the principle of new divergence, that the experiments lengths of the sonorous waves in liquids are shorter than in aii 1 .* (Analytical Mechanics, 348.) KKFLEXION AND REFRACTION OF SOUND ECHOS. 67. There is no body in nature absolutely hard and inelastic. Whenever, therefore, the molecules of a vi- particles of o brating medium come within the neutral limits of those body agitate , , . those of another, forming the surface of any solid or fluid, they will and transmit a agitate the latter with motions similar to their own, and pulse - a pulse will be transmitted into the solid or fluid with a velocity determined by its density and elastic force. *See Appendix No. 1. 76 NATURAL PHILOSOPHY. Keferences; 68. Referring to the transmission of sound through air, and resuming Equation (2)', we have, after substituting the value of F, as given by Equation (3), Velocity of a particle ; Now, by reference to 34:, it will be seen that Excess of condensation ; expresses the excess of condensation on one side of a molecule over that on the opposite side. Making Expressed by an equation ; the above Equation may be written Velocity of a v = Q . \ _ (28). particle; JJ In the same homogeneous medium E and D are con- v stant, whence we conclude that the actual velocity of a molecule, which is the same as that of the stratum to which it belongs, is directly proportional to the excess of condensation on one side of it, over that on the oppo- site side. when a particle When, therefore, by the forward movement of a mole- /will come to rest ' J ., cule the condensation becomes equal on opposite sides, the molecule comes to rest, and remains s^> till again disturbed by some extraneous force. This explains why it is that a pulse transmitted through a medium of uni- ELEMENTS OF ACOUSTICS. 77 form density sends back no disturbance, but leaves every Livin g foroe molecule behind in a state of rest. The living force im- pressed upon any given stratum is transferred to the next one in front, and this to the next in order, and so on in- definitely. G9. When the stratam . n . Fig. 85. disturbed; stratum A B is mo- ved by some source of disturbance to w A' B', the stratum C" BE' (f J)' AA D JL L' X A pulse CD Will niOVe in transmitted in the same direction, and a pulse will be transmitted on- direction of disturbance; ward towards W, the excess of condensation being on the same side of the moving stratum as the place of the ori- ginal disturbance. But a shifting of the stratum AB to the position A' B ', leaves the excess of condensation And also one in which acts on the stratum C' D on the opposite side*!? * * direction ; from A B / the stratum C' D will therefore close in upon A B', and the same occurring in succession with all the strata on the side towards TF"', a pulse will be trans- mitted in an opposite direction from that which begins with the motion of CD. Thus, every case of an original disturbance of a molecule will give rise to two pulses pro- Eyer 7 ceeding in opposite directions, with the same velocity, the two pulses differing only in this, viz. : in the one the wave velocity will be in the same direction as that of Their difference. the molecules, and in the other in an opposite direc- tion. 70. The elastic force E, of two media in contact and Elastic force of at rest, must be the same ; otherwise motion would ensue. When, therefore, in the progress of a pulse, it reaches a rest stratum X Yj of a density or elasticity different from that of those which precede it, Equation (28), shows that for the same excess CJ of condensation, the velocity of the stratum Effect when the will be altered; that is, the' actual motion of the molecules ^ItTLTof 111 " will either be accelerated or retarded. If the new stra- greater density. 78 NATURAL PHILOSOPHY. turn be of increased density, the next, preceding stratum K Z, will be checked in its progress by the greater mass of X Y, and brought to rest before it reaches its neutral distance from that behind; the excess of elastic force thus retained will react upon the next preceding stratum which has already come to rest, and will thus give rise to a return pulse in which the velocity of propagation and that of the molecules will be in the same direction. Effect when jf on fo Q contrary, the new stratum have a dimin- moving stratum meets one of less ished density, the motion of K L will be accelerated, the density. density in front of the next preceding stratum will be- come less than that between those behind which have come to rest ; these latter strata will therefore move for- ward in succession, and thus a return pulse will be pro- duced as before, but with the difference, that the velocity of propagation and that of the molecules will be in oppo- site directions. wave meeting a 71. It follows, therefore, that when a pulse or wave medium of 60 und in any medium reaches another medium of different density . is resolved into greater or 'less density, it is at once resolved into two, tw 5 one of which proceeds on through the second, while the other is driven back through the first medium. cause of tins This division of an original pulse into two others, arises resolution. entirely from the reciprocal action of the two media on each other. If the media be perfectly elastic, there can be no loss of living force, and the sum of the intensities of sound in the component pulses will be equal to that of the original pulse. If the media be not perfectly elastic, there will be a loss of living force, and the sum of the intensities of the component pulses will be less than that of the original pulse. incident, The original pulse is called the incident that transmit- ted into the second medium, the refracted; and that driven back through the original medium, the reflected pulse. To an ear properly situated, the reflected pulse will be audible, and is, for this reason, called an echo. The sur- ELEMENTS OF ACOUSTICS. 79 face at which the original pulse is resolved into its two Deviating, component pulses, is called the deviating surface. surface ; 72. To find the law which regulates the direction of D5rcctlonoftbe reflected pulse determined; Fig. 86. the reflected pulse ; let A M be a portion of the front of an incident spherical pulse, so small that it may be regarded as a plane. Draw M A", A' N and A 0, normal to the pulse, and suppose the latter, moving in the direction from 2V to A', to meet the face E G of a second me- dium. Each molecule of the pulse as it recoils from the surface E G, becomes the centre of a diverging spherical pulse which will, Eq. (28), be propagated with the velocity of the incident pulse. Accordingly, when the portion M Explanation and reaches the face of the second medium at A", the por- tion A will have diverged into a spherical pulse whose radius is . A B A" M. In like manner, if A' M r be drawn parallel to A M, the portion diverging from A' will, in the same time, have reached the spherical pulse whose centre is A' and radius A' B' = A"M'. The same construction being made for all the points of the incident pulse as they come in succession to the deviating surface, the surface which touches at the same time all these spherical surfaces will obviously be the front of the re- flected pulse. But because A' B' and A B are respec- tively proportional to A! N and A" M, and as this is true for any other similar lines drawn from points of the deviating surface to the corresponding points of the in- cident and reflected pulses, this tangent surface is a plane. i nci(len t an d Moreover, since A B is equal to MA", and the angles reflected P ulses A M A" and A" JB A are right, the angles MA A" and^/^ A" A are equal, and the incident and reflected pulses deviating surface. make equal angles with the deviating surface. Any line which is normal to the front surface of a 80 NATURAL PHILOSOPHY. Ray of sound. Ansle of incidence ; Angle of reflexion ; These angles equal Fig. 87. pulse, is called a ray of sound. The angle N A D, which the normal to the incident pulse makes with the normal to the deviating surface, is called the angle of incidence. The angle BAD, which the normal to the reflected pulse makes with that to the deviating surface, is called the angle of re- flexion and because the angle made by two planes is equal to that made by their normals, we conclude from the foregoing, that in the reflexion of sound, the angles of incidence and of reflexion are equal. Fig. 38. Direction of the g 73. The law which determines the course of the re- dctermined 1 ; 86 fracted pulse is equally simple, and is deduced in a man- ner analogous to the preceding. Let A M be an inclined element- ary plane pulse, incident upon a de- viating surface E G, at any instant. In the interval of time during which the point J/is moving from M to A", the agitation which begins at A will have reached some spherical surface within the second medium of which A B is the radius ; and in like man- ner, the agitation which begins at A\ will have reached some spherical surface of which A' B' is the radius, by the time the portion of the inci- dent pulse at M', will have passed on to A" ; and the same of intermediate points of primitive disturbance on the deviating surface between A and A"^ the first and Ccwrtruction and l as t points of incidence. The surface tangent to all these spherical surfaces will be the front of the transmitted or refracted pulse ; and because A B and A' JS f are respec- tively proportional to A" Jfand A'^ this surface is a plane. G- explanation. ELEMENTS OF ACOUSTICS. 81 The angle N* A D, made by the normal to the refracted pulse and that to the deviating surface, is called the angle of refraction. Denote the angle of incidence N A D, which is equal to the angle M A A"^ Fig. 38, by 9 ; and the angle of refraction W A D, which is equal to the angle A A" B, Fig. 38, by 9' ; then will Refracted sound; Fig. 39. Dlostration; MA" = A" A. sin 9; A B = A" A . sin , this living force will be distributed among the molecules of the surface D D'. After re- flexion, the concavity of the pulse is turned to the front, its extent becomes less and less as it approaches the second focus, and the living force of its molecules will be more and more concentrated, till finally, when the pulse reaches the focus F', the living force of a single molecule will be a maximum, and will be capable of pro- ducing the greatest impression upon the ear. ELEMENTS OF ACOUSTICS. 85 What lias been said of the portion of the pulse within AH the sound - -r-, -r~, . n -i from one focus the sector D F E, is equally true of any other sector concentrated ta and of the whole spherical pulse ; so that all the sound the other. which originated in the focus F, will, after reflexion, be concentrated in the focus F f . 78. When a spherical pulse is incident upon a plane spherical pnin incident on a plane surface deviating surface, it will be easy from the principles now | ) n ] clde explained to' construct both the refracted and reflected pulses. For this purpose, let A E represent the deviating surface ; D, the point of pri- mitive disturbance ; D C, any incident ray ; C G and CE, \\// construction of * ' \j/ the refracted and the corresponding refracted j yfa jg ^ reflected pulses. and reflected rays respec- tively. From the point _Z>, draw ED, perpendicular to the deviating surface. Extend the refracted ray C G, back till it meets this line in the point H. At the point of incidence (7, draw G M parallel to ED. .Denote the angle of incidence D CM= by 9 ; the angle of refraction M C H= C HE\>y 9'; the distance D E by/ ; and the distance HE by/ '. Then will / tan

that is to say, all the reflected rays will diverge from Reflected puiso a point Z>', as far behind the deviating surface as the spherical; . p ? nt f) O f disturbance is in front of it. The reflected pulse will, therefore, be spherical. From the point D as a centre and radius D K, equal to that of the spherical pulse at any instant, describe the arc KO' ; this will represent a section of the incident pulse by a plane normal to the deviating surface. Make the distance ED' equal to D E, and with D f as a centre, and radius D' K!> equal to D If, Illustration ; ELEMENTS OF ACOUSTICS. 87 describe the arc K' 0' \ this will represent a section, Construction of by the same plane x of the reflected pulse. Draw & n y reflected and incident raj as D C\ through the point H, given by the reflected pulses. value of f in Equation (32), and the point C, draw H C, which being produced will give the refracted ray C X' ; through D' draw the line D' C X, and multiply the inter- cepted portion C X by the ratio of the velocities V and V, and lay off the product from to X', and we have the point X! of the refracted, corresponding to the point X of the reflected pulse. An ear situated at X will hear the direct sound transmitted along the ray D X, and an position whence echo of the same sound reflected at the point 0: the thedirectsound . , . and the echo are interval 01 time, or number ot seconds intervening be- both audible ; tween the two, being equal to DC+ CX- DX Time between 1089,42 V 1+ (ft _ 32) . 0,00208 ' the imprest; on the supposition that the sound is transmitted through Position whenca the atmosphere, and the linear distances are estimated in the transmitted English feet. An ear situated at X will hear the trans- 8ound is bear or, taking the temperature of the air at 32, Bound and echo When the difference of routes exceeds this distance, the interval of time between the two impressions upon the ear becomes distinctly perceptible ; and in proportion as that difference becomes less than a?, will the impression of the echo begin before that of the direct sound ends ; and this overlapping, as it were, of impressions will give rise to confusion, which will continue to a greater or less ex- tent till the difference of routes becomes so small as to afford no sensible interval between the instants that mark *k e beginning of both impressions, in which case the echo will strengthen the effect of the direct sound. distinctly perceptible; Echo causes confusion; lon o echo; 80. Let an observer place him- self at 0, midway between the plane walls A B and C D^ of which the distance apart is some 250 feet or more. The sounds which he utters will be reflected back to him by the ; two walls, and having traversed equal distances will reach him at the same instant; they will, therefore, rein- force each other, and he will hear one distinct echo. JsTow let him move towards one of the walls. At first he will perceive little or no difference of effect, but presently one echo will seem to lag behind the person assuming other, confusion will soon follow, and this will continue different till twice the difference of his distances from the two between two walls becomes equal to or greater than 121 feet, when he walls; w iu h e ar two distinct echos, which will separate more and ELEMENTS OF ACOUSTICS. 9 more from each other as he progresses ; when he gets Effects within sixty feet of the nearest wall, the first echo will begin to confound itself with the sound of his voice heard directly; he will now enter a second space of indistinct- ness, from which he will emerge at a distance from the wall of about fifteen or twenty feet. 81. It thus appears that reflecting surfaces situated at Surfacesat different distances from a speaker may throw back to him ^^a reflect numerous echos of the same sound. Of this many re- man y echos of markable instances are recorded. At Lurley-Fels, on the "" Rhine, 'is a position in which a sound is repeated by echo seventeen times. At the Villa Simonetta, near Milan, is another where it is repeated thirty times. An echo in a building at Pavia used to answer a question by repeat- ing its last syllable thirty times. The rolling of thunder has been attributed to echos from clouds situated at unequal distances from an auditor ; and the propriety of this view has been sustained by the observations of Several ARAGO, MATTHIEU and PRONEY, while experimenting upon the velocity of sound. They found that when the weather was perfectly clear the reports of their guns were always single and sharp; whereas when the sky was overcast or a single cloud of any extent was present, they were fre- Experiments ; quently accompanied with a long continued roll like that of thunder, and occasionally a double sound would arrive from a single shot. Bu-t it is proper to remark that the rolling of thunder admits of another explanation. Thunder is caused by a disturbance of electrical equilibrium in the atmosphere ; experience shows that this takes place over a long and sinuous line, the different points of which are at unequal Roiling of distances from the auditor, and the sounds from these thunder - points can, therefore, only reach him in succession and without sensible intervals. 82. When reflected sound and that proceeding di- rectly from the same source, are made to fall upon the 90 NATURAL PHILOSOPHY. effect of direct sound Fig. Illustrated by the speaking trumpet ; Deflected sound ear simultaneously, or nearly so, they strengthen each may increase the other and become audible in positions where neither could be heard separately. The Speaking Trumpet affords an illustration of this. The Speaking Trumpet is a funnel-shaped tube, of which the object is to throw the voice beyond its ordinary range. In its best form it is parabolic. It is a geometrical property of the parabola that a line FT, drawn from the focus F, to any point T of the curve, and another T K, F drawn from T parallel to the axis FA, make equal angles with the tangent line to the ^\ curve at T. A portion of its construction the diverging rays of sounds proceeding from a mouth at the focus F, will be reflected by the trumpet in directions parallel to the axis A F j and the living forces of the aerial molecules which, without the trumpet, would have been diffused over that portion of the spheri- cal surface on the outside of a cone of which F It and F It are the most diverging elements, become, by its use, concentrated within the limits of a circle whose diameter M N, is equal to that of the trumpet's mouth, By its use sounds and superposed upon the living forces arising from the are rendered action of the direct sound. The axis of the trumpet be- audible that - 1 could not be ing directed upon a person at a distance, sounds of audi- ble intensity may thus be conveyed to him, which he could not hear from the unassisted organs of speech. Leard without it 83. The Hearing Trumpet, which is intended to assist per- sons who are hard of hearing, is similar to the speaking trum- Hearingtrurn P et ; P et ; but the operation is re- versed. The rays of sound en- ter this instrument at the larger Fig. 49. ELEMENTS OF ACOUSTICS, * 91 open in g and are so reflected as to become united at the Construction ATW! TISA smaller end, which is inserted into the ear. and use; Fig. 50. Speaking tubes on same principle. 84. Whispering Galleries, so called from the fact whispering that the faintest whisper uttered at one point may be dis- 8an< tinctly heard at another and distant point, without its be- ing audible at intermediate positions, depend upon the operation of the same principle, to wit, the convergence of the rays of sound by reflexion. The best form for Be8tform - these galleries is that of the ellipsoid of revolution. In such a chamber two persons, one in either focus, could keep up a conversation with each other which would be inaudible at other points. The ear of Dionysius isE celebrated in ancient history ; it was a grotto cut out ^ of the solid rock at Syracuse, in which a person placed at one point could hear every word, however faintly uttered, in the grotto. It was doubtless of a parabolic shape. The same principle is employed in the construction of Spealdng Tubes, used for the purpose of communicating between different apartments of the same building, now coining into very general use. 85. Halls for public speaking, such as lecture rooms, theatres, churches, and the like, should be so constructed as to diffuse the sounds that are uttered throughout the space occupied by the audience, unimpaired by any echo or resound. "Were the speaker to occupy constantly the same position, the parabolic form would,on theoretical grounds, undoubtedly be the best ; but in debating halls, where every speaker occupies a dif- ferent position from another, these conditions are very Principles on difficult to fulfil, especially when the room is large. Every- *^ ^ ey thing should be avoided that would at all interfere with constructed. NATURAL PHILOSOPHY. Experiment; Illustration ; Explanation. the uniform diffusion of sound, and especially all need- less hollows and projections which are likely to gene- rate echos. The following experiment will illustrate, in a very simple manner, the consequences arising from the re- flexion of the rays of sound from the interior of a pa- rabola. Place a watch in the focus A of a parabolic mirror and all the rays of sound that fall on the Fi ~ 51 - concave surface will be reflected in the direc- tion indicated by the arrows. The ticking of the watch will be plainly heard within the space MN P, in which the rays fall, but it will not be audible at a small distance on either side. Now place a second reflector P, opposite to the for- mer, and at some distance from it ; the rays of sound jKrill be received by it and thrown into the focus B. If the ear, or, better still, the mouth of a hearing-trumpet, be applied to this point, the ticking of the watch will be heard as plainly as at A. A /\ N\.i si/\ VI K) V \! 86. "While it is important to diffuse sound uttered Partition walls; or in any way produced, uniformly, so as to render it distinctly and equally audible in all directions, it is also necessary to prevent its passage from one apart- ment to another for which it was not intended. Parti- tions are usually made of solids ; but solids, if elastic, such as wood, metals, and stone, are, as we have seen, better adapted to transmit sound than air itself; an essential condition, however, for this transmission is homoge- neousness of substance and uniformity of structure. Where HOW constructed these are wanting a sonorous pulse transmitted through to prevent the so lid is ever changing its medium, and soon becomes transmission of Bound. broken up by reflexion and refraction, retardation and -ELEMENTS OF ACOUSTICS. 93 acceleration, into a multitude of non-coincident waves, Examples of and these from the laws of interference must, to a greater ^ e , ren or less extent, destroy each other. As an instructive instance of this stifling effect on a sonorous pulse, we may mention the example afforded by a tall glass filled with champagne. As long as the effervescence lasts and the wine is full of bubbles, the glass cannot be made to ring by a stroke on its edge, but will give a dead and puffy sound. As the effer- Glass of vescence subsides tjie tone becomes clearer, and when champagne: the liquid is perfectly tranquil, the glass rings as usual. On re-exciting the bubbles by agitation, the musical tone again disappears So of a solid or union of several solids, in which Heterogeneous there are frequent changes of density and elasticity, Bolid8; and especially where there is a want of adhesion among the different parts; sound penetrates these with great difficulty, and materials so -united as to satisfy to the greatest extent possible the condition of non-homogeneous- ness should, therefore, be employed whenever it is an object to prevent the transmission of sound. The influence of carpets, curtains, and tapestry hangings, in preventing reflexion and echos in large apartments, is due to the causes above mentioned. The mixture of the unelastic carpets, curtains, fibres of the cloth with its numerous layers of entangled &a air, intercepts and deadens the sonorous waves before they reach the more solid and elastic media behind. 94: NATURAL PHILOSOPHY. MUSICAL SOUNDS. Audibi* sounds 87. Every impulse mechanically communicated to produced; foe air or other elastic medium is, as we have seen, propagated onward in a wave or pulse; but in order that it may affect the ear as an audible sound, a cer- tain force and suddenness are necessary. The slow wav- ing of the hand through the air is noiseless, but the sudden displacement and collapse of a portion of that medium by the lash of a whip, produces the effect of impression on an explosion. The impression conveyed to the ear will 8 depend upon the nature and law of the original impulse, which being altogether arbitrary in duration, violence and character, will account for all the variety observed in the continuance, loudness -and quality of sound. The auditory, nerves, by a most refined delicacy of mechan- ism, appear capable of analyzing every pulsation, and of appreciating the laws which regulate the motions of the molecules of air in contact with the ear ; and from this Auditory nerves arise all the qualities grave, acute, harsh, soft, mellow, pulsations- an( ^ nameless other peculiarities which we distinguish whence grave, between the voices of different individuals and different S ' an i ma l s j an( l the tones of different musical instruments musical bells, flutes, cords, &c. instruments. Noise . 88. Every irregular impulse communicated to the air produces what we call noise, in contradistinction to musical sound. If the impulse be short and single, we hear a crack ; and as a proof of the extreme sensibility of the ear, it is to be remarked that the most short and Crack; sudden noise has its peculiar character. The crack of a whip, the blow of a hammer against a stone, the explo- sion of a pistol, are perfectly distinguishable from each other. If the impulse be of sensible duration and irre- gular, we hear a crash ; if long and interrupted, a rattle, ELEMENTS OF ACOUSTICS. 95 or a rumble, according as its parts are less or more con- Rumble, tinuous. 89. The ear retains for a portion of time after the Continuons impulse is communicated to it a perception of excitement, sound produced; If, therefore, a short and sudden impulse be repeated beyond a certain degree of quickness, the ear loses the intervals of silence and the sound appears continuous. The probable frequency of repetition necessary for the production of continuous sound is stated to be not less than sixteen times in a second, though the limit will be Th < ^ uanc r of repetition different for different ears. necessary. 90. If a succession of impulses occur at exactly equal Musical sounds; intervals of time, and if all the impulses be exactly simi- lar in duration, intensity, and law, the sound produced is perfectly uniform and sustained, and takes that pecu- liar and pleasing character called musical. In musical sounds there are three principal points of distinction, viz. : the pitch, the intensity, and the quality. Of these the pitch depends, as we have seen, solely upon the fre- quency of the repetition of the impulses ; the intensity, on their violence ; and the quality, on the peculiar laws which regulate the molecular motions in any particu- lar instance. All sounds, whatever be their intensity or sounds having quality, in which the elementary impulses occur with same pitchi or * the same frequency, have to the ear the same pitch, and are said to be in unison. It is on the pitch alone that the whole doctrine of harmonics is founded. 91. The means by which a series of equidistant im- J Musical sounds pulses can be produced mechanically upon the air are mechanically very various. If a toothed wheel be made to turn with P roduced ; a uniform motion while a steel or other spring is held against its circumference with a constant pressure, each tooth as it passes will receive an equal blow from the spring, and this being communicated to the air, a wave of sound will proceed from the place of collision. The 96 NATURAL PHILOSOPHY. The siren; A series of palisades ; Whistling of a bullet number of such blows in a second will be known when the angular velocity of the wheel and the number of teeth upon its circumference are known, and thus every pitch may be identified with the number of impulses which produce it. The Siren, another instrument by which the same results may be evolved, has been de- scribed in 48. A series of broad palisades, placed edgewise in a line running from the ear, and equidistant from each other, will reflect the sound of a blow struck at the end nearest the auditor, producing a succession of echos which reach the ear at equal intervals of time, thus producing a musical note whose pitch will be determined by the number of reflexions in each second of time. This number will be equal to the quotient arising from dividing the velocity of sound by twice the distance between two adjacent palisades. A similar ac- count may be given of the singing sound produced by a bullet while moving through the air and turning rapidly about its centre of inertia. The angular motion of the bullet being uniform, the actual velocity of its surface on one side will be greater than that on the other, and any inequality in the figure of the bullet will be made to vary its action upon the air periodically, thus producing a musical sound. Most ordinary way of causing musical sounds : Modes considered. 92. The most ordinary way of producing musical sounds is to set in vibration elastic bodies, as stretched strings and membranes, steel springs, bells, glass, co- lumns of air in pipes, &c., &c. All such vibrations con- sist in a regular alternate motion to and fro of the molecules of the vibrating body, and are performed in strictly equal portions of time. They are, therefore, adapted to produce musical sounds by communicating that regularly periodic initial impulse to the aerial mole- cules in contact with them, from which such sounds re- sult. We proceed to consider their modes of production, and especially in the first and last named cases, these being the most simple. ELEMENTS OF ACOUSTICS. 97 VIBRATIONS OF MUSICAL STRINGS. 93. If a string or wire be stretched between two vibrations of fixed points, and then struck or drawn aside from its musical strings ; position of rest and suddenly abandoned, it will vibrate to and fro till its own rigidity and the resistance of the air bring it to rest ; but if a fiddle ~bow be drawn across it, the vibrations will be renewed and may be maintained for any length of time, and a musical sound will be heard whose pitch will depend upon the greater or less ra- pidity of the vibrations. Thus, if MN be any stretched cord, struck at right an- M V- ^ ^ ' V V ^ IIlustratiQn ; -j AJ a ok />// gles to its length at (9, it will be suddenly bent at that point into the curved or waved shape indicated by the dotted line/S, which shape will run along the cord in both directions till it meets with some obstruction to its further progress, when it will be either wholly or partly reflected, and return upon its course in a manner and for the reasons to be explained wave runs along presently, the successive positions in the diverging mo- 11 tion being ', S", &c., on the one side, and S^ S 4i , &c., on the other. 9 i. To find the velocity with which the wave runs TO find the along the cord, it is plain that we may either regard velocity of this ' J wave motion; the cord as continuous, or as being composed of a series of detached points, kept in relative position by their mu- tual attractions for each other, each point being loaded with the mass of so much of the cord as extends half way on either side to the adjacent point, and of which the length is equal to the distance between any two con- secutive points. 7 98 NATURAL PHILOSOPHY. Fig. 53. V -IT Suppose IN to be the cord's position of iiiuBtration; i'est, and the wave to proceed in the direc- 3HL tionfrom to D; let the point A, be just on the eve of motion and the place J5', the position of 'the point B at the same instant. Suppositions; While, therefore, the actual motion of the point B has been from B to B', that of disturbance has been from B to A. The duration of these simultaneous motions is indefi- nitely short ; the motions themselves may, therefore, be regarded as uniform. Hence, denoting the actual velo- city of the point B by v y and the velocity of the dis- turbance by T 7 ", we have Consequences; v : V '. '. B B' I A B. V = V. BB' AB or, denoting the angle BAB' by 9, in which case, 4 BB AB = tan 9, we find, Velocity of a point of the cord; v = V. tan (33). The tension of the cord between A and B, acts to draw the point A, from A towards B, and the tension between A and D acts .to draw the same point from A ' PartS towards # Denote the tension of the cord when at rest by C, that between A and ^?'by C\ then because the ten- sion of the same portion of the cord will be proportional to the length to which it is stretched, will ELEMENTS OF ACOUSTICS. 99 C: O f :: AB : AB E tensions C and whence and this being resolved into two components, one acting from A to E, at right angles to MN, the other in the direction of M N^ will give for the first '. sin

,/~L f* C'. cos ? = O'. ^A = (7; the second will be destroyed by the tension from A to One is destroyed Z>, while the first will alone produce motion in J., and * ^th'e force; is, therefore, the motive force. Denote by t0, the weight of a unit's length of the cord while at rest, then will the mass with which A is loaded be expressed by ^_ ^ Jg Mass on which the motive force acts; in which g denotes the force of gravity ; and the accele- ration due to the motive force will be C. g. tan 9 m Acceleration du W , AB ' to the motive force ; and therefore the velocity of A, in the small time t, which velocity will be equal to that of B' when D begins to move, will be given by the relation v = . tan 9. _ . w A B particle in small time tf But t 1 AB _ ~ ' 100 Value of tension Velocity of particle in small time t. NATURAL PHILOSOPHY. and denoting by Z /5 half the length of the cord whose weight is equal to the tension 6, we have = 2 w. L t which values substituted above give _2gL r tan 9 ~ " and replacing tan 9 by its value found from Equation (33), gives or Velocity of wave along a cord ; Eule. (34). Example ; That is to say, the velocity with which a wave or pulse, will run along a tense cord is constant, and equal to that acquired by a heavy body in falling in vacuo, tin der the action of its own weight, through a height equal to half the length of the cord whose weigJit is equal to the tension. . Example. A cotton thread 73 feet long and weighing 904 grains, is stretched by a weight of 12840 grains; with what velocity will a wave move along this cord ? First whence 904 : 12840 : : 73 : 2 L t Computation- ELEMENTS OF ACOUSTICS. 101 Second V = V 2 f. L. = V 32,18 . 1036,83 = 182,64 Wave veloclt y along the cord; 95. Substituting the above value for V, in Equation (33), the latter becomes v = V 2 g L t . tan N -W W One end of the cord fixed ; 102 NATURAL PHILOSOPHY. Both ends of the the cord and return along its entire length, following after the direct pulse in the same direction. If the second end be fixed, the direct pulse proceeding towards it will conduct itself in the same way ; the reflected pulses will proceed to meet each other, and being on the same side of the cord will conspire at their place of union to Eeflected pulses produce a single resultant pulse, in which the molecules meet and of the cord will depart from their places of rest by the sum of the distances of the same molecules in the com- ponent pulses. These component pulses will, however, separate and are immediately separate, again reflected; an d proceed towards the fixed ends, where they will be reflected -* , f , M\ - ^ - - - T !ar as before, and return -^ to meet again, having ^ _ ^ ^ ^ once more changed ^~ O _ sides. The point of ylfl _ /"""\ _ y Meet a second second meeting will be time at the point a |- ft lQ place of primitive disturbance, from which the of primitive disturbance; waves will depart, as before, to undergo the same round ; and thus, but for the resistance of the air and want of perfect elasticity in the cord, the latter would vibrate for ever. But every pulse communicated to the air, is an elimination from the cord of so much of its living force, and as this must soon become exhausted, the cord cord brought to will come to rest. rest 96. Suppose the whole length of the \Yhoieiengthof cor( j MN, to be de- cord divided into , , , T -, . -,, two parts; noted by L = l t + I , of which l t represents the distance M, from the point of primitive disturb- ance 0, to the fixed obstacle on one side, and T the dis- tance ON to the obstacle on the opposite side. Then denoting the time of describing T by ', and that of de- scribing l t by ^, we have ELEMENTS OF ACOUSTICS I' = Y. t\ Lengths of the whence t v ~rr ' Times required for a pulse to past along them. the first pulse being reflected at JY", will describe the entire length I' + I, in the reverse direction in the time ,, , I' l f V i = "77= 1 y=- > Time in which the first pulse describes the whole length ; and the second pulse being reflected at J/~, will describe the entire length I' + l^ in the same time, or + t= I 1 The same for the second pulse; the first pulse being reflected a second time at describe the length l t in the time will Time in which first pulse passes over second part; and the second pulse being reflected a second time at 2T, will describe the distance Z', in the time tf = l f -=- ') That in which second pulse passes over first part; and at the expiration of all these times the pulses will 104: NATURAL PHILOSOPHY. Pulses return to be at their first starting point, and each haying been ttarting point; twice re fl ec t e d, they will be on the same side of the cord that they were originally ; they will, therefore, pro- duce a resultant pulse precisely the same, abating the qualification due to the air and imperfect elasticity, as that produced by the initial impulse. Hence, if T de- note the time of one complete vibration of the cord, that is to say, the interval between the instant of primitive disturbance and that at which the cord resumes its in- itial condition, .we shall have, by taking the half sum of these several intervals because both pulses are mov- ing during the same time, Conspire and produce a resultant pulse; Time of vibration of the cord ; and replacing V by its value, Equation (34), The same reduced. 2 L (36) Example; Example. Taking the example of 94, in which L = 73, and 2 L t = 1036,83 feet, we find Result. T V 32,18 . 1036,83 . = = 0,799. 97. The truth of the foregoing theory has been fully confirmed by the experiments of WEBEK. He stretched a very uniform and flexible cotton thread fifty-one feet two inches in length, weighing 864 grains, horizontally, by a Experiments; known weight. The thread was struck at six inches from the end, and the time of the wave's running a certain number of times over the length of the string, back- ward and forward, carefully noted by means of a stop- watch that marked thirds (the sixtieth part of a second). ELEMENTS OF ACOUSTICS. 105 The mean of a great many trials, agreeing well with Mean of result* each other, gave the results in the following table : Tension in grains. Length run over by the wave. Time in thirds. Time of running over the length 102 33 in thirds Time by calcu- lation from the formula by observation. 10023 102*4: 46 46 46,012 10023 204,7 92 46 46,012 10023 409,4 184 46 46,012 33292 409,4 99 24,72 25,246 69408 409,4 65 16,25 17,489* Table. A more complete confirmation could not have been desired. The slight discrepancies are doubtless owing to a want of perfect uniformity in so long a thread, which must necessarily have formed a catenary of sen- sible curvature. Denote by N the number of vibrations performed in a given time T^ then will Time of one vibration of the cord; which substituted for T in Equation (36) gives, after taking the reciprocal of both members, N g L, 2Z (37). Eeciprocal of the same. In the foregoing equations 2 Z /5 denotes the length of the cord of which the weight measures the tension. Denote this weight by TF, the diameter of the cord by D, and its density by d / then will W = IT. .2L..d.a Weight of cord whose length measures the tension ; 106 NATURAL PHILOSOPHY. Length of half this cord; whence 4 = 2 W Time of vibration ; which substituted in Equations (36) and (37), give D.L. Vd T= VW (38) Its reciprocal; D.L.Vd' ' ' KuleSrst; Bule second. from which it appears that, the time of vibration of a tense cord varies as its length, diameter and square root of its density, directly / and the square root of the stretch- ing force, inversely. And that, the number of vibrations performed by a tense cord in a given time, varies as the square root of the stretching force directly, and the diame- ter, length and square root of the density inversely. vibrating cord g 93. The tense fixed at both , ends and struck cord MN being fixed in the middle; Fig. 57. -c2T ends and in a M\ ~**-~*r state of vibration, ap- ply the finger, or any other partially obstructing cause, at the middle point F, and then withdraw it. The law of Equation (34), will be suddenly interrupted at this point, the progressing pulse will be resolved into two, one of which will continue to move in the same direc- tion and on the same side of the cord, while the other will be reflected and return along the opposite side. Primitive pulse These component pulses having equal distances to tra- resolved into . t w o; vel before they reach the ends, will be reflected at the fixed points at the same instant, return on opposite sides of the cord, and meet in the centre. They will, ELEMENTS OF ACOUSTICS. 1QT therefore, solicit annul- The two , Fig. 58. reflected pulses taneously the central interfere at the point O, in opposite di- middle of tn * .- , X\ if- \y cord; rections, and it the ^ pulses be equal, they will wholly interfere at that point, which must, therefore, remain stationary. The effect of the reciprocal action of the two waves being to fix the point 0, these waves will both be totally reflected there, will return to the ends, be again reflected, return to the centre, from which they will be thrown back towards on. the ends, and so on till the living force of the cord is totally expended upon the air. Thus the two portions M and N of the cord may vibrate as though the point had been originally fixed. If the finger be ap- ^ Finger applied at plied but for an in- one third the start at 0, at a dis- ^ ^ ^^ ^ ^~ tance from M equal to one-third of the jy, ..-^ . ..^^ {J[r whole length M N, while the wave is pro- jr\ . r~~^ ^ ^ gressing from JLC to- wards 0, the latter will be resolved, as before, into two component waves, Primitive pulse one of which will continue to move towards N on the two; same side of the cord, the other will return to M on the opposite side. The distance MO being equal to one- half of N, the return component will be wholly re- flected and change sides at M, and come back to the point by the time the direct component arrives at N, where the latter will be totally reflected and pass to the opposite side of the cord. The component waves being now on opposite sides of the cord, and moving towards each other, one starting from and the other from N) will meet at ', half way from to JV, component making N 0' equal to one-third of M N. HereP ulse3iuterfere ' they will interfere, be totally reflected, and proceed from 108 NATURAL PHILOSOPHY. Are again reflected ; Again meet at their starting poiut and so on. Finger kept on the cord ; One reflected component resolved into two; Reciprocal action between these two sets of components; Cord broken up into portions, each one vibrating. Fig, 59. Fig. s -or Nodal points. 0' as they did from 0; they will meet again in this latter point and jn\ . ., ? /'"^ w there be totally reflect- ed, and thus each com- M\ ^'"^ ^ ^> \y ponent wave will be _^ made to describe, as Jfl - r ^ & ^^ long as the cord retains any of its living force, alternately one-third on one side and two thirds on the opposite side of the entire length of the cord, as though the point were to become alternately fixed at and 0\ after every reflexion at M and N. Were the finger to be kept at the point 6>, till the first reflected component returned to that point, this compo- nent would be there subdivided, giving rise to a second return as well as a second onward component ; the lat- ter would meet the first onward component at 0', and by its action upon it resolve this also into two com- ponents, the onward one of which would meet the second return component at 0, and being on opposite sides would interfere and hold this point at rest. Thus the whole cord may be broken up into three equal parts, each of which will vibrate as though the points of division, and 0\ had been stationary or fixed. A similar explanation would show that if the finger were applied at any other point of which the distance from one of the fixed ends .were an aliquot part of the whole length of the cord, the cord would in like man- ner be broken up, as it were, into equal aliquot portions, each of which would vibrate as though its extremities were fixed. Molecules or particles of a vibrating body thus ren- dered stationary by the simultaneous action of opposing waves or pulses, are called Nodal points. The interme- ELEMENTS OF ACOUSTICS. 109 diate portions which vibrate, are termed Idlies^ or ven- y e ntrai tml segments. segments. 99. If Z, denote, as before, the entire length of the cord, and w, the number of ventral segments into which it divides itself, then wiU the number of its nodes be n 1, and the length of each segment, ^ Length of a > segment which substituted for L in Equation (36), gives for the time of vibration, 2 L / 4Q x Time of ' Vibr3tion; and in Equation (37), the number of vibrations in the time T N~ n> \f 2 O . L Number in time ' fjj ~7jT ~ o f T ,> -LI * -** and for the number in one second, by making T t equal to one second, r. -j v Number in one * )' second. All of this is confirmed by experience. If the string Above of a violin, or violincello, while maintained in vibration deductions by the action of the bow, be lightly touched by the finger, or a feather, exactly in the middle or at one-third of its length, from either end, it will not cease to vibrate, but its vibrations will be diminished in extent and increased in frequency, and a note will become audible, more faint but more acute than the original, or fundamental note, as it is called, and corresponding, in the former case, to 110 NATURAL PHILOSOPHY. illustrated by flie a double, and in the latter, to a triple rapidity of vibra- vioiin; ^ on> The note heard in the first case being, in the scale of musical intervals, an eighth or octave, and in the second a twelfth, above the fundamental tone. If a small piece of paper cut in the form of an inverted V, be set astride on the string, it will be violently agitated or thrown off if placed on the middle of a ventral seg- ment, but at the node wil* ride quietly as though the Harmonics. string were at rest. The sounds thus produced are termed * Harmonics. Coexistence and superposition of small motions; Its application illustrated ; Explanation; Results confirmed by experience. Fig. 61. 100. But further, according to the principle of the coexistence and superposition of small motions, referred to in 56, any number of the various modes of vibra- tion of which a cord is susceptible, may be going on simultaneously. Thus, if we suppose a mode of vibration repre- sented by figure (a), in which there is no node, and another of the same cord represented by figure (5), with one node, to be going on at the same time, there will be a resultant vibration represented by the curve in figure (c), of which the ordinates are equal to the algebraic sum of the corresponding ordinates of the curves in figures (a) and (5). If a third mode of vibration, represented by figure (d), be superposed upon the other two, there will arise a resultant vibration represented by the curve in figure (e\ of which the ordinates will be equal to the algebraic sum of the corresponding ordinates in figures (a), (b) and (<#), or, which is the same thing, the algebraic sum of the corresponding ordinates of figures (c) and (d). This is also confirmed by experience. It was long known to musicians, that besides the fundamental note ELEMENTS OF ACOUSTICS. HI of a string, an experienced ear could detect in its sound, Harmonic when in motion, . especially when very lightly touched B0 in certain points, other notes related to the fundamental one by fixed laws of harmony, and which, are therefore called harmonic sounds. They are the very sounds that may be heard by the production of distinct nodes as ex- plained in 99, and thus insulated as it were from the fundamental and other coexistin sounds. 101. The Mbnochord is an instrument adapted to e hibit these and other phenomena of vibrating strings. It consists of a single string of catgut or metallic wire stretched over two fixed and well defined edges to- wards its extremities, which effectually terminate its vibrations in the direction of its length ; one end is permanently fixed, and to the other is attached a weight which determines the tension. The interval be- Essential parts; tween the two edges is graduated into aliquot parts, and the instrument is provided with a movable bridge or piece of wood capable of being placed at any point of the graduated scale, and abutting firmly against the string so as to stop' its vibrations, and divide it into two equal or unequal parts, as the case may be. By the aid of this instrument may readily be found its use; the number of vibrations which corresponds to any given note of any particular instrument, as a piano-forte, for in- stance. To this end, it will only be necessary to know, when the note of the monochord is the same as that of the instrument, the distance L between the edges, the stretching weight, and the weight of a unit's length of the string. The quotient obtained by dividing the. former of these weights by the latter will give the va- lue of 2 Z y , in Equation (37), and making T t equal to one second in that Equation, we have for the solution of the question formula. NATURAL PHILOSOPHY. Number of This gives the number of impulses made upon the ear corresponding to ^ n a secon( l, corresponding to the fundamental note. To any higher note, obtain the number which answers to any note sharper, higher, or more acute^ we have but to apply the bridge and slide it to some position such that the portion of the cord between it and one of the edges gives the note in question ; the scale will make known Z, which in Equation (42), will give the number ^Z\T. Harmonic tones 102. The contact of a stretched cord with solid sub- stances is not the only means of producing its fundamen- tal and harmonic tones. The sonorous pulses proceed- * n & ^ rom a vibrating cord are but the consequences of repeated conflicts between the elastic force of the cord and that of the air. The former impresses upon the air a certain amount of living force, and the latter by its reaction transmits this living force through the atmos- phere to a distance. Reverse the process. Impress upon the air the same motion, and subject a stretched cord to its influence. Action and reaction only change names, and the cord must take up the motion of the air. Two TWO cords near cords equally stretched, and in all other respects similar, S^to* but the len g th of one on l v a half > a third > or an y ali - vibrate; quot part of the other, being placed side by side, and the shorter put in motion, the longer will soon assume a mode of vibration by which it will be divided into ventral segments, each equal to the length of the shorter cord. The sonorous pulses diverging from the shorter cord will arrive at the longer; and the mole- cules in the first of these pulses will, in their forward movement, press upon the stationary cord and give it a slight motion in their own direction. On the retreat of these molecules, the excess of aerial condensation will change to the opposite side of the cord ; the latter will yield to the action of this inverted force and that of its its vibrations own elasticity, and pass to some position on the oppo- , site side of its place of rest, where being met by a second transferred to the longer cord; onward pulse, it will be thrown back in the direction of ELEMENTS OF ACOUSTICS. H3 its first motion, and thus made to undergo the same round as before. This process being repeated a number of times, the g nchronal cord will be set in full and audible vibration. But vibrations; these vibrations will obviously be synchronal with the aerial pulsations, and therefore^ with the vibrations of the sJiorter cord, a condition that can oniy be fulfilled ^J Lon(rercor(llsia the longer cord breaking up, as it were, into portions effect broken up; of which the lengths are equal to the length of the shorter cord; for, the tension, diameter and density, of the cords being the same, the times can only be equal, Equation (38), when the vibrating lengths are equal. All motions of the longer cord which are inconsistent with this, though they may be excited for the moment by one pulsation, will be extinguished by the subsequent one. Hence, if two cords can have any mode of vibra- tion in common, that mode may be excited in either of them, and that only, by exciting it in the other. For example, if two cords, in all other respects alike, have lengths which are to each other in the proportion of 2 to 3, and if either be set in motion, the mode of illustration, vibration corresponding to a division of the first into two and of the second into three ventral segments, will, if it exist in the one, be communicated by sympathy to the other. Indeed, if it do not originally exist, it will, after awhile establish itself; for, all the "circumstances which may favor such a division, however minute, will have their effect preserved and continually accumulated, and thus become sensible. And it is important to remark that whether the primi- tive portion disturbed be large or small, whether it occu- py the whole string at once or run along it like a bulge ; w r hether it be a single curve, or composed of several ven- tral segments with intervening nodal points, we must not forget that the motion of a string with fixed ends is H0 tringw j ttl | xed other than an undulation or pulse continually doubled ~back ends & analogous upon itself, and retained within the limits of the cord Stained within instead of running off both ways to infinity. certain limits. 8 114: NATURAL PHILOSOPHY. vibrations plane; Orbits described by particles may be observed; Specimens. 103. It is very seldom that the vibrations of a string can ^ e m ^ e same pl ane - They most commonly consist of rotations more or less complicated, except when pro- duced by the sawing of a bow across the string. The actual orbit described by any one molecule may be made matter of ocular inspection by throwing the solar rays through a narrow slit so as to form a thin sheet of light. A polished wire stretched in such manner as to penetrate ^his sheet at right angles, will appear, when stationary, as a bright spot where it pierces the light, but when in mo- tion, the point of intersection will form a continued lumi- nous orbit, just as a live coal whirled round appears like a circle of fire. The figures exhibit specimens of such orbits observed by Dr. YOUNG. Fig. 62. VIBRATING COLUMN OF AIR OF DEFINITE EXTENT. Vibrating column of air of definite extent; g 104. The circumstances of the molecular vibrations . -i -i i / i * , i * a stretched cord ot indefinite extent, are, as we have seen, similar to those of a sounding column of air ; and the facts which have been stated respecting a vibrating cord are equally true of a vibrating column of air of definite ^extent ELEMENTS OF ACOUSTICS. H5 Thus, if such a cylindrical Tube closed at , , . . Fig. 6a both ends, column be enclosed m a pipe containing air; A B = Z, stopped at both ends by immovable stop- pers, and an impulse be com- municated in the direction - I 11 C A, to one of its sections C, at the distance A C=l, from the end J., and B C=l', from the end B, this impulse will, 69, give rise to two pulses running in opposite impulse directions. In the pulse from C to A the air will be con- densed, and in that from C to B it will be rarefied, direction of the These pulses will be reflected at the stoppers, and the |^ thof the condensed pulse, after passing over the distance I be- fore and I ' after reflexion, will meet the rarefied pulse at the distance I from the end JB, and produce a com- pound agitation in the section C' similar to that of the Two P ulses wil1 original disturbance ; thence the partial pulses will sepa- ^5^^ rate, and after each undergoing another reflexion will opposite unite in their original point of departure, constituting, as it were, a repetition of the first impulse, and so on, Pulses gr adua!] y till the pulses are destroyed by the gradual transmis- destroyed, eion of the whole of their living forces through the sub- stance of the tube to the open air. If the section first set in motion be maintained in a consequence of state of vibration synchronous with the return of the "J^ 1 ^ reflected pulses, it will unite with and reinforce them at section first every return, and the result will be a clear and strong dlsturbei musical sound, resulting from the exact combination of the original periodic impulse with its echos. 105. Let us suppose the section first set in motion and so maintained, to be exactly in the middle of the pipe. Then, when once the pe- riodic pulsation of the contained air is established, the motion will consist of a constant and regular fluctu- Middle section maintained in vibration ; 116 NATURAL PHILOSOPHY. Air condensed in one half and rarefied in the other ; Positions of greatest and least condensations and rarefactions ; Several columns end to end ; Illustration; Nodes; Ventral segments ; Distance between two alternate nodes. ation to and fro of the whole mass, the air being always condensed within one-half of the pipe while it is rare- fied in the other. The greatest excursions from their places of rest will be made by the molecules in the middle, while the molecules at the ends abutting against the solid stoppers will have the least motion, the ex- cursion made by each intermediate molecule being greater in proportion as it is nearer the centre. On the other hand, the rarefactions and condensations are greatest at the .extremities and diminish as we approach the middle, where they are the least. Now, conceive several such columns of vibrating air to be equal and to be placed end to end, so that the condensed portions shall be turned towards each other; it is plain that all the stoppers, except the ex- treme ones, may be removed without in anywise sen- sibly changing the interior motions, and there will re- sult a single column of air broken up into equal por- tions vibrating in a manner similar to that of the ventral segments of a tense cord, -> < < 98, the nodes being at X and IT, where there will be alternately a maximum and minimum of condensation, the bellies lying between in the middle of which the condensation will be the least. It is also obvious that the distance XX, be- tween two alternate nodes, will be the shortest dis- tance from any one section of air to another having the same phase, and that this distance answers to the length of a wave of the same pitch propagated in an indefinite column of air. Fig. 65. X Y 1 An opening in the middle of a segment ; 100. At C, half way be- tween two consecutive nodes, or in the middle of one of the cylinders A. J3, let an opening be made ; and sup- X Fig. 66. c 1 1 ELEMENTS OF ACOUSTICS. pose a vibrating body to be inserted whose vibrations And a vibrating are executed in equal times with those in which the bodyintroduced '' excursions to and fro of the included aerial sections are performed in the stopped pipe. Its vibrations will be communicated to those of the contained air, the latter will be maintained and strengthened, and the sound from the pipe will become full and clear. Such Embouchure. an aperture is called an embouchure. ISText conceive one-half One hair of the ~ -n n L\ v J A T> Fi 67 cylinder replaced * B C, Of the Cylinder A B, by a vibrating to be removed, and in its disc ; place a disc substituted ex- J ^ : E actly closing the aperture, , and maintained by some external cause in a state of constant vibration, such, that the performance of one complete vibration, going and returning, shall occupy as much time as a sonorous pulse would take to tra- verse the whole length of the stopped pipe A B, or double that required for the half pipe A C. Its first impulse on the air will be propagated along the half pipe C A, and reflected at the stopped end J, and will again reach the disc just as the latter is commencing its second impulse. But the absolute velocity of the disc in its vibrations being excessively minute compared with that of sound, the reflected pulse will undergo a second reflexion at the disc as though it were a fixed stopper. It will, therefore, in its return exactly coincide Eeflected puiee and conspire with the second impulse of the disc, and wil1 coincide 1 -with the second the same process being repeated at every impulse, each impulse of the will be combined with all its echos, and a musical tone ^ and so on - will be drawn from the pipe vastly superior to that which the disc vibrating alone in the open air could produce. This is the simplest instance of the resonance of a cavity. Now, it is manifestly of no importance Resonance of * whether the pulses reflected from the closed end A of cav?t7 ' the semi-pipe undergo a second reflexion at the disc and are so turned back, or whether we regard the disc I as penetrable by the pulse, and suppose the latter to 118 NATURAL PHILOSOPHY. disc and be reflected at the other end. Same effects win run on and be reflected at the extremity _Z? uf the other na ^ f * ne en tire tube, and on its return again to pass freely through the disc and be again reflected at the end A. The sound will be the same on the principle of the superposition of vibrations. Thus the fundamental sound of a pipe open at one end is the same as that of a pipe closed at both ends and of double the length, and has the same pitch as that due to waves propa- gated in the open air, and of which the length of each is four times the length of the pipe open at one end. R.vp ftnental k Castration ; Tuning fork and pipe; Flute male to epeak. Fig. 68. 107. The mode here supposed of exciting and sus taining the vibrations of a Column of air in an open tube may easily be put in practice. Take a common tuning- fork and by means of sealing wax fas- ten, a circular disc of card on one of its branches, sufficient- ly large to nearly co- ver the open end of a pipe. The upper joint of a flute with the mouth hole stop- ped will answer well for the purpose ; it may be tuned in unison, that is, made of proper length by the sliding stopper. The fork being set in vibration by a blow 011 the unloaded branch, and held so as to bring the disc just over the mouth of the pipe, a note of great clearness and strength will be heard. Indeed, a flute may be made to "spwde* perfectly well by holding a vibrating tuning-fork close to the embouchure, while the fingering proper to the note of the fork is at the same time performed. 108. But the most usual method of exciting the vibra- tions of a column of air in a pipe is by blowing across the open end, or across an opening made in the side ELEMENTS OF ACOUSTICS. 119 or by introducing a current Resonance of a . . . L . L , Fig.&. pipe produced by of air into it through a small aperture of a peculiar con- struction called a "reed" provided with a " tongue" or flexible elastic plate which nearly stops the aperture, and which is alternately forced away by the current of air and brought back by its own elasticity, thus pro- ducing a continued and regularly periodic series of in- terruptions to the uniformity of the stream, and a sound ill the pipe corresponding to their frequency. Except, however, the reed be so constructed as to be in conditions to t> unison with some one of the possible modes of vibration of the column of air in the pipe, the sound of the reed only will be heard, the resonance of the pipe will not be called into play, and the pipe will not speak ; or will speak but feebly and imperfectly and yield a false tone. 109. Let us consider what takes place when the vi- Effect of blowing brations of a column of air are produced by WQWing^^^pJJJ? across the open end of a pipe or an aperture in the side. The current of air being so directed as to graze the opposite edge, a small portion will be caught and turned aside down the pipe, thus giving a first impulse to the contained air and propagating down it a pulse in which the air is slightly condensed. This will be reflected at the end as an echo and return to the aperture where the condensation will go off, the section condensed expanding into the free atmosphere. But in so doing it lifts up and for a moment diverts from its course the impinging cur- rent, and thus suspends its impulse upon the edge of the aperture. The moment the condensation has escaped The production ,, ., f , of sound the current resumes its former course and again explained; touches the opposite edge, creates there a second conden- sation and propagates down the pipe another pulse, and 120 NATURAL PHILOSOPHY. so OIL T ^ ms ^ culTGnt Posing over the end or aperture grazes and misses is kept in a constant state of fluttering agitation, alter- nately grazing and passing free of its edge at regular in- tervals equal to those in which the sonorous pulse can run over twice the length of the pipe ; or more generally, in which the condensation and rarefaction recur in vir- tue of any of the modes of vibration of which the column of air in the pipe is susceptible. excursions of molecules; Point of 110. In general, whenever there is a free communi- cation opened between the column of air in a pipe and * * the free atmosphere, that point becomes a point of maximum excursion of the vibrating molecules, or the middle of a ventral segment. At such a point the rare- faction and condensation assume their smallest possible values by the air reducing itself constantly to an equi- librium of pressure with the external air. Hence, if the pipe speak at all, it will take such a mode of vibration as to satisfy this condition, but, consistently with this, it may divide itself into any number of ventral segments. vibrations of a But nere there is a practical difference between the column of air affections of a vibrating aerial column and those of a tense cord. In the case of the cord both ends in prac- tice must be fixed to secure the requisite elasticity ; this the air possesses in its natural state, and to make the cases analogous we must suppose the cord to be extend- ed in one direction to infinity, so that its pulses like those of the aerial column may run off indefinitely never to return. Cases made Can be no half Not so with pipes; 111. In cords with fixed extremities all the ventral segments must of necessity be complete, no half segment can exist. In pipes < it is otherwise. The air in a pipe closed at one end vibrates as a half, not as a whole of such a segment.' It is owing to this that a pipe open at both ends can, if properly excited, yield a Tig. TO. ELEMENTS OF ACOUSTICS. musical sound. The column of air vibrates in the mode Node in the . . e g middle of the represented in the figure, in which there is a node in entire segment the middle, and each ventral segment is only half a complete one. 112. To find the time of vibration or the number pi ?e open at of vibrations in a given time corresponding to any mode of vibration, denote by m the number of nodes in a pipe open at both ends ; the number of complete ventral segments between them will be both ends ; -, Number of 1 complete ventral segments ; and denoting the length of the pipe in feet by Z /; , the length of each complete ventral segment will be Length of each - j segment; m and denoting the velocity of sound by "F, and the time required for the sonorous pulse to traverse one seg- ment by T, we shall have H TT Time of T . - 'JL ....... (43) describing one m V and this is the time of vibration of the middle section of the segment to which the sound corresponds. The number of vibrations per second being JV^ there will result & = -rf, =i m -f- (44) Nnmberof * -Lt {l vibrations in a second; and the pitches of the series of tones which the pipe can be made to deliver will be expressed by the values 122 NATURAL PHILOSOPHY. pitches of the of ^ determined by making successively m = 1, m 2, tones delivered m ~ q P r ~ .-. 'U,- II v <_> Uuv* ui uy by the pipe. 1- -7 j 3 --F-j3._ , &c. 113. In the case of a pipe stopped at one end, the closed end must be regarded as a node ; and denoting,' as before, Pipe closed at the number of nodes by m, the number of complete ventral seg- ments will be m 1, and one half segment at the open end, or Number of entire 1 J_ i 2i ffl 1 t segments; L + 2 ~ 2 and the length of each complete one, in feet, will be, O T Length of each ; describing one eegment; and the time T, required for a sonorous pulse to tra- verse each segment, will be given by T * = and the pitch by Pitch, or number 2 m 1 V of vibrations per * = - ^r - y .... second; A/ and making m = 1, m = 2, m = 3, &c., the pitches of the tones will become F v r Series of pitches. \ -y- J f "^*- 5 f-y- ELEMENTS OF ACOUSTICS. 123 1 14. Lastly, in the case of a pipe stopped at both pi P e close<1 at ends, the number of nodes, in- cluding the two ends, being m^ the number of ventral segments will be in 1 ; the length in feet of each will be both ends; Fig. T3. m - 1' Length of each segment; the time, T= -T'T' Time of . . (47) describing on* segment . and the pitch, and the series of pitches, , 2 .f;3.f ;& , JL/.. Ju,. Pitch, or number (48) of vibrations per second; (49) Series of pitches* Taking, therefore, the number of vibrations performed in the fundamental note in one second as unity, the series of harmonics will run thus : In a pipe stopped at both ends . 1, 2, 3, 4, 5, &c. series of " " " open at both ends ... 1, 2, 3, 4, 5, &c. harmouics - " " " stopped at one end and ) - open at the other \ * 3 ? 5 ' T ' 9 ' d it being recalled that, Equations (44), (46), and (48), in the last series, the fundamental note is an octave lower These sounds than in the other two. produced by . blowing into a To produce these sounds by blowing into a pipe, it pi pe; NATURAL PHILOSOPHY. Blot's experiments Fundamental will only be necessary to begin with as gentle a blast as tone heard first; w jjj ma k; e the pipe speak, and to augment its force gra- dually. The fundamental tone will first be heard, which will increase in loudness till suddenly it starts up an octave ; that is, passes the interval between notes whose vibrations are as one to two. By adapting an organ- bellows to regulate the blast, M. BIOT succeeded in draw- ing from a pipe all the harmonic notes represented .by the series of natural numbers up to 12, inclusive, except 9 and - 10 ; the reason for failing to produce these two is not stated. The rationale of this continued subdivision of a vibrat- ino- column as the force of the blast increases is obvi- o ous. A quick, sharp current of air is not so easily turned aside from its course as a slow one, and when thrown into a ripple by any obstacle will undulate more rapidly. Consequently, on increasing the force of the blast a pe- riod will arrive in w r hich the current cannot be diverted from its course and return to it as slowly as required for the production of the fundamental note, and the next higher harmonic will be excited. Explanation of these results. sounding body : The air is the 115. That it is the air whicE is the sounding body and not the material of the pipe, appears from the fact that the kind, thickness, or other peculiarities of the latter, make no difference in the tone in regard to pitch. A pipe of paper, lead, glass, or wood, of the same dimensions, gives, under the same circumstances, the same pitch. The qualities of the tone are often different, but this is owing to the feeble vibrations of the molecules of the material of the pipe produced by those of the contained air. 116. Putting the two values of JT, in Equations (41) and (46), equal, we find, Verification. Equation, ELEMENTS OF ACOUSTICS. 125 whence TT _ n L n V 2 g L 4 ,- m Velocity of soond - 1 IT ' ' ' ' ' W" and making m and n each unity, Same reduced; which furnishes a ready means of finding the velocity of sound in any gas or vapor. For this purpose, fill a pipe of known length with the gas in question, and set it to vibrating by any proper means, so as to call forth its fundamental tone. Adjust the bridges of a Mono- Practical use of chord so that the fundamental tone of its string shall fo^mi^ have to the ear the same pitch"; measure the length of the string between the bridges and substitute this length for L in Equation (51), and the velocity sought becomes known. It was by this method that CHLADNI, VAENEES, FKAMEYER and MOLL ascertained the velocity of sound in various media. For a detailed account of the structure and manage- Account of pipes, . reeds, &c. ment of the embouchures 01 pipes, and a vast amount of interesting matter on the subject of reeds, &c., &c., the reader is referred to Sir JOHN HEKSCTIEL'S most va- luable Monograph of Sound, articles 197 to 207, inclu- sive, as published in Yol. IV. of the Encyclopedia Metropolitan*. 126 NATURAL PHILOSOPHY. VIBRATIONS OF ELASTIC BARS. Vibrations of 117. Bars of a cylindrical or prismatic shape are susceptible of sonorous vibrations as well as cords, and columns of air. But as sucli bodies are nearly equally elastic in all directions, transversely as well as longitu- dinally, their vibrations do not obey the same laws as Transversal and those of strings. Transversal vibrations may be excited nal; by striking a bar crosswise, and longitudinal vibrations by striking it in the direction of its length. Laws governing I n bars made of the same substance, the acuteness the pitch in o f the pitch in transversal vibrations is directly as the vibrations; thickness, and inversely as the square of the length of the bar. In bars made of different substances, it is found that the degree of the body's elasticity greatly influences the character of the pitch ; thus steel gives a higher pitch than brass. To produce these vibrations, the bar may be either secured at both ends, or its ends may be made merely to rest on Fi - u some fixed objects ; or one end may be fastened while the other is Meansof J producing these free, or lastly, both ends may w3T* Tlaa vibrations in fa f ree? ftie r0( Jg being SllppOl't- ed at two points. ^^^^^^^ v ^^ SK ^mmm i We have an illustration of these kinds of vibrations in the jews-harp, musical boxes, &c. 118. "When a bar is struck upon one of its ends in the direction of its length, the blow will give rise to a condensed pulse, which will proceed towauAs the other Longitudinal * * vibrations in end like that of a column of air. It will be reflected k" 85 back and forth alternately at the two ends, according to the principles of 106 and 107, and this will con- tinue till its living force is wholly transmitted to the ELEMENTS OF ACOUSTICS. 127 air and wasted in space. If the rod be of glass, the soiia rods give sound emitted will be extremely acute unless its length ""dTthm be very great ; much more so than in the case of a columns of air; column of air of , the same length. The reason of this is, the greater velocity with which sound is propagated in solids than in air. When the bar is short the re- flexions at the ends, which determine the successive impulses upon the air and therefore the pitch, succeed each other with great rapidity. The velocity in cast iron, for example, being 10 J times that in air, a rod Glass ' 8teel> ** of this metal will yield a fundamental sound when lon- gitudinally excited, identical with that of an organ-pipe of T L. of its length, stopped at both ends, or ^ T of its length, open at one end. The laws of longitudinal vibrations have nothing in common with the transversal, except that the acuteness Laws of of the sound emitted varies invers-ly as the length o f longitudinal and the bar, the reason of which is obvious. The sounds vibrations differ; produced by the longitudinal vibrations are, without ex- ception, higher than those yielded by the transverse vibrations of the same body. They are little if at all influenced by the thickness, or, in the case of wires of considerable thickness, by tension. As in the case of transversal vibrations, the sounds emitted from bars of equal dimensions depend upon the nature of the ma- terial. Longitudinal vibrations may be generated in elastic bars, by holding them in the middle between two fin- gers, and rubbing repeatedly one Longitndina. of the ends with the fingers of Fig. T5. produced; the other hand. In experiment- ing on glass tubes the friction- apparatus represented in the figure will be found convenient : #, #, are two pieces of wood hol- lowed out, having their cavity padded with cloth or leather ; 128 NATURAL PHILOSOPHY. Friction apparatus ; Fig. 75. .Nodal points found ; Musical instruments. vibrations c c, is a steel spring connecting them, and d, d, are two rings in- tended to receive the fingers with which the friction is excited. Moisten the padding with spirit of wine, and sprinkle on it a little finely pulverised pumice-stone. If metal or wooden bars are used, the readiest mode will be for the operator to put on a leather glove, on the thumb and index finger of which is some pounded resin, and with these to rub the rods. The existence of nodal points may be verified by sliding small paper rings loosely on the rod. These vibratory movements have been applied to musi- cal purposes in some instruments. KAUFMANN'S Ilarmo- nicJiord and CIILADNI'S Euplion act on this principle. 119. Beside the two species of vibration described al- ready, elastic rods admit of a third, viz., that by rotation. It is most easily generated in cylindrical bodies, by secur- ing one end in a vice, and communicating to the other a rotatory motion by means of a bow or by friction. An alternate expansion and contraction ensue in a direction perpendicular to its axis. Different high and low notes succeed each other, of which, as yet, no use has been made in music. OF THE VIBRATIONS OF ELASTIC PLATES AND BELLS. Vibrations produced in plates; If elastic plates, of glass or metal in particular, be held tightly either 'by the fingers or by means of a clamp, at any one point, and the bow of a violin be drawn across the edge of the plate, sonorous undulations are imme- diately produced. These oscillations resemble those of elastic rods, inas- ELEMENTS OF ACOUSTICS. 129 much as the surface is divided into a greater or less num- ber of perfectly symmetrical parts, and such as are con- terminous, vibrate in opposite directions. The boundary lines of these several parts are all in a state of repose, and form nodal lines' their position de- pends on the places at which the plate is held and ex- cited, as one of these nodal line's invariably runs through the point at which the plate is held, whilst the plate itself receives the vibratory motion at the other point. These lines form certain peculiar figures, called, after their dis- coverer, CHLADNI'S /Sonorous Figures. To make these figures visible, and to render them per- Means of making manent, strew some light sand or dust over the plate; these visible; they may also be seen if a small quantity of water be poured on the plate, nay, even by the rays of light falling on it. WHEATSTONE remarks that, in using the last-named mode, still more delicate divisions in the figures were observable. These sonorous figures are composed sometimes of Their shapes right ' lines, sometimes of curves either parallel to or ^^^ intersecting each other. The shape of the plate greatly p j ate; affects them, as they are differently arranged, accord- ing as it may be a square, a rectangle, a triangle, a circle, an ellipse, or some other figure. A perfectly dis- tinct and welLdefined figure is produced only when the plate gives a very clear sound. By experiments made on such plates the following Laws; laws were detected by CHLADNI : 1. Any particular pitch will always produce the same figure with the same plate; but a small change may often be produced in the figure by slightly changing the place at which the plate is held without causing any difference in the pitch. If the pitch be changed, First law; the existing figure disappears at once, and l a new one arranges itself. 2. The gravest pitch any plate gives is accompanied by the simplest figure, and the higher the pitch the more second law; complex the figure, i. e. the more nodal lines there will be. 130 NATURAL PHILOSOPHY. Third law, Experimental illustration ; 3. If similar plates of various sizes be treated in the same manner, similar figures will be generated in each ; by the same treatment, we mean that they shall be held at the same point, and that the bow shall pass over corresponding points in each. The pitches will,' however, differ, for the larger plate will give out the graver sound ; and if their dimensions be equal, the stronger will give the acuter pitch. 120. If the plates be strewed with fine sand, and held at the point #, whilst the bow be made to pass at J, the figures here depicted will in each case be produced. A striking effect is obtained by making the same figure on several plates of equal size and similar form, and then so arranging them as to make one figure on Union of several a larger scale. The figure thus produced will be both a plates of equal compound and connected one, and such as may not unfrequently be met with on a large plate. If a large square be formed out of four squares, bear- ing the figures I. and II., we shall have the following : Fig. 7T. Fig. T8. The effects. ELEMENTS OF ACOUSTICS. If the large square plates be held at #, touched at a', Particular case; and a bow be drawn across at J, similar compound fig- ures will be generated. Cymbals, the Chinese Tam-tam or Gong, &c., are prac- Examples; tical applications of sonorous plates. 121. The vibrating motions of sonorous bells resem-s on0 rousbeiis; ble those of circular plates. In this case, too, the most acute pitch is accompanied by the most complex figure. To render these vibrations visible, fill a bell-shaped glass rather more than half full of water ; draw a vio- lin bow across the rim, and at the same time touch the glass at two opposite points of the rim with the fingers. The surface of the water will acquire an undulatory mo- Means of making tion, and to make the sonorous figures permanent, strew the surface of the water with any light and exceedingly fine powder, as semen lycopodii. If the point excited by the bow be at a distance of 45 from that touched by the finger, a four rayed star marked III. of the last article will" result ; but if the distance be 30, 60, or 90, the six-rayed star, marked IV., will appear. Such a cup gives musical sounds when rubbed with the r ! They give moistened finger. The vibrations of the glass in this musical sounds. case result from torsion, and this is the principle of the well known finger glass. COMMUNICATION OF VIBRATIONS. 122. The numerous experiments of M. SAVART abun- communication dantly show that the molecular motions of one body are of molecular communicated to another, when there exist between them m< any intervening media, and this the more effectually as the connection is the more perfect But not only this ; they also show that the molecules of the neighboring bodies are agitated by motions both similar in period and parallel in direction to those of the original source of mo- its peculiarity; 132 NATURAL PHILOSOPHY. tion. Of these experiments we have only room for such as have a direct bearing upon the nature and structure of our organs of hearing. 123. Take a thin membrane, moistened tissue paper will answer every purpose, and stretch it over the mouth Experimental O f a C0 mmon bowl or finger glass, place it in a horizontal illustration; . . _ position and strew fine sand over its surface. Hold a glass plate, covered with fine sand or dust, horizontally and di^ rectly over the membrane, and set it in vibration so as to form CHLADNI'S acoustic figures ; these figures will be im- mediately and exactly imitated in the sand on the mem- Effect when the brane, and this will be the case to whatever lateral posi- giass and ^ on w jthin the sphere of sufficient action to move the par- membrane are -i-i-^i -!-! parallel; tides ol sand, the plate may be shitted, provided it retain its parallelism to the membrane. Effect when 1^4. But instead of shifting the plate laterally, let its inclined to one plane be inclined to the horizon. The figures on the membrane will change though the vibrations of the plate remain unaltered, and the change will be greater, the greater the inclination of the plane of the plate. And when it becomes perpendicular to the horizon and there- fore to the surface of the membrane, the figures on the latter will be transformed into a system of straight lines men parallel to the common intersection of the two planes ; perpendicular to an( j f- ne particles of sand, instead of dancing up and down, one another; ' . r will creep in opposite directions to meet on these lines. One of these lines always passes through the centre, and the whole system is analogous to what would be produced by attaching a cord to the centre of the plate, and, having stretched it very obliquely, setting it in vibration when shifted by a bow drawn parallel to the surface. In a word, the laterally vibrations of the membrane are now parallel to its sur- face, and they preserve this character unchanged, how- ever the plate be shifted laterally, provided its plane when the plate is be kept vertical. If the plate be made to revolve about iteTerUcai b01 ^ s ver ^ ca ^ diameter, the nodal lines on the membrane diameter. will rotate, following exactly the motions of the plate. ELEMENTS OF ACOUSTICS. 133 125. Nothing can be more decisive or instructive inference; than this experiment. It shows us that the motions of the aerial molecules in every part of the spherical wave propagated from a vibrating centre, instead of diverging like radii in all directions, so as to be always perpen- Principle of dicular to the wave surface, may be parallel to each ^rltTon^ other and to the wave surface. The same holds good in liquids also. 126. So long as the sound of the plate, its mode of vibration, its inclination to the plane of the membrane, and the tension of the membrane continue unchanged, the nodal figure on the membrane will continue the Conditions to same ; but if either of these be varied, the membrane will not cease to vibrate, but the figure on it will be figure; changed accordingly. Let us consider separately the effects of these changes. 12T. All other things remaining the same, let the pitch of the sounding plate be altered, either by loading Difference it or changing its size. The membrane will still vibrate, ^^rane and differing in this respect from a rigid lamina, which can rigid lamina; only vibrate ~by sympathy with sounds corresponding to its own subdivisions. The membrane will vibrate in sym- pathy with any sound, but every particular sound will be accompanied by its own particular nodal figure, and as the pitch varies, the figure will vary. Thus, if a slow air be played on a flute near the membrane, each note illustration. will call up its particular form, which the next will efface to establish its own. 128. Next suppose the figure of the plate so to vary Change of figure as to change its nodal figures; those on the membrane of thepiate; will also vary ; and if the same note be produced by dif- ferent subdivisions of different sized plates, the nodal figures on the membrane will also be different. 134 Effect of change of tension: Effect of moisture NATURAL PHILOSOPHY. 129. If the tension of the membrane be varied ever so little, material changes will take place in its nodal figures. Hygrometric variations are sufficient to produce these changes. Indeed, the fluctuations arising from this cause were so troublesome in the case of tissue paper, that it became necessary to coat the upper surface with a thin film of varnish. By far the best substance for ex- hibiting the results of these beautiful experiments is var- nished paper. Moisture diminishes the cohesion of the "fibres, and renders them nearly independent of each other, and sensible alike to all impulses. 130. Between the nodal lines formed by the coarser particles of sand, others are occasionally observed, formed only of the finest dust of microscopic dimensions. This secondary nodal is a most important fact, as it goes to show that diffe- rent and higher modes of subdivision coexist with the more elementary divisions which produce the principal figures. The more minute particles are proportionally more re- sisted by the air than the coarser ones, and are thus prevented from making those great leaps which throw the coarser ones into their nodal arrangement. They rise and fall with the greater divisions of the surface, and are only affected by those minute waves which have a smaller amplitude of excursion and occur more frequently, and form their figures as though the others did not exist. These secondary figures often ap- pear as concentric rings between the primary ones, and frequently the centre of the whole system is occupied as a nodal point. lines; Inference; Explanation ; Sensibility of membranes ; Exploring membranes. 131. So sensitive are some varieties of stretched mem- brane to the influence of molecular motion that they have been employed with success in detecting the ex- istence and exploring the extent and limits of the most delicate, continuous and oppositely vibrating portions of air. When so employed they are called exploring mem- ELEMENTS OF ACOUSTICS. 135 "branes. The most highly interesting application of the Application of properties of stretched membrane is in the " membrana ?^^ rila8 i" of the ear. membrane; Fig. 79. Illustrations; Illustrations: THE EAR. 132. The auditory apparatus, called the ear, is a Essential parts collection of canals, chambers, and tense membranes, of the ear; whose office is to collect and convey to the seat of hearing, the vibrations impressed upon the air by sono- rous bodies. Beginning on the exterior and proceeding inwards, 136 NATURAL PHILOSOPHY. Wing; Auditory duct ; Cavity of the drum; Labyrinth; we find a cartilaginous funnel A A, called the wing ; a canal & ft, called the auditory duct, leading to Fi s- 80 - an interior chamber B B, called the cavity of the drum y and behind this a system of canals of con- siderable complexity, call- ed the labyrinth, consist- ing of three semi-circular tubular arches m, m, m, originating and terminat- ing in a common hall n, called the vestibule, which communicates with the cavity of the drum by a small opening I, called the fenestra ovalis, and is prolonged in the opposite direction into a spiral cavity o, called the cochlea. The auditory duct is closed at its junction with the cavity of the drum by a tense ; membrane r, called the drum of the ear, as is also the fenestra ovalis by a similar membrane. The whole ca- vity of the labyrinth is filled with a liquid in which are immersed the branches of the -auditory nerve, wherein is supposed to reside the immediate seat of the first im- pression of sound. "Within the cavity of the drum are four small bones united by articulations so as to form a continuous chain ; the first f, is called the hammer, the second g, the anvil, the third i, the loll, (os orbicularis), and the fourth 7c, the stirrup, from the resemblance which its shape bears to that of the common stirrup. The han- dle of the hammer is attached to the drum, and the stirrup to the membrane which closes the fenestra ova- lis ; and thus the aerial vibrations, first collected by the funnel-shaped wing of the ear, and transmitted through the auditory duct to the drum, are conducted onwards by the articulated bones to the auditory nerve in the labyrinth, which receives them at the window of the vestibule. The cavity of the drum is connected with that EuBtachiantuba;of the mouth by a canal d> called the Eustachian Vestibule; Fenestra ovalis and conchlea ; Drum of the ear Auditory nerve ; Hammer, anvil, ball and stirrup; ELEMENTS OF ACOUSTICS. which serves to keep the cavity of the drum filled with its use. air of uniform density and temperature ; a condition which appears to be necessary in order that the different parts may perform their functions with accuracy. If this be stopped, deafness is said to ensue, but as Dr. WOLLASTON has shown, only to sounds within certain limits of pitch. If the membrane which closes the labyrinth be pierced DeafneS3 and its fluid let out, complete and irremediable deaf- produced, ness ensues. From some experiments of M. FLOUKENS on the ears of birds, it appears that the nerves en- closed in the several arched canals of the labyrinth have other uses besides serving as organs of hearing, and are other uses of th instrumental, in some mysterious way, in giving animals nerves of the the faculty of balancing themselves on their feet and directing their motions. MUSIC, CHORDS, INTERVALS, HARMONY, SCALE AND TEMPERAMENT. 133. Our impression of the pitch of a musical depends, as we have seen, entirely upon the number impression of a of its vibrations in a given time. Two sounds whose musicalsound; vibrations are performed with equal rapidity, whatever be their difference in intensity and quality, affect us with the sentiment of accordance, which we call uni- son, and impress us with the idea that they are simi- lar. This we express by saying that their pitch is the same, or that they are the same note. The impulses Effectoftwo , . , ,, , sounds in unison; which they send to the ear through the medium of the air, occurring with equal frequency, blend and form a compound impulse, differing in quality and intensity from either of its components, but not in the frequency of its recurrence, and we judge of it as of a single note of intermediate quality only. of twonotm But when two notes not in unison are sounded to- 180111 10 138 NATURAL PHILOSOPHY. Concord 01 discord. gether, most persons distinctly perceive both, and can separate them in idea, and attend to one without the other. But besides this, the mind receives an impres- sion from them jointly which it does not receive from either when sounded singly even in close succession ; an impression of concord or of discord, as the case may be, and hence the mind is pleased with some combinations, displeased with others, and it even regards many as harsh and grating. Harmony, chord, melody. Music. Concordant sounds : Discordant sounds ; Limit of simplicity in music. 134. The union of simultaneous and concordant sounds, is called Harmony. Every group of simultaneous and concordant sounds, is called a Chord in harmony. A succession of single sounds makes Melody. To discover and discuss the laws of harmony and melody, are th'e objects of musical science ; to apply these laws to the production of certain effects in musical composition, is the object of musical art. Science and art, thus employed, constitute that department of knowledge properly called Music. ISTow it is invariably found that the concordant sounds are those, and those only, in which the^number of vibra- tions in the same time are in some simple ratio to each other, as 1 to 2, 1 to 3, 1 to 4, 2 to 3, &c., and that the concord is more pleasing the lower the terms of the ratio are and the less they differ from each other. "While, on the other hand, such notes as arise from vibra- tions which bear no simple ratio to each other, as 8. to 15, for instance, produce, when sounded together, a sense of discord, and are unpleasant. By the constitution of the ear, ratios in which 7 and the higher primes occur are not agreeable ; why, cannot be told, but simplicity must end somewhere, and in music this seems to be about the point. This is the natural foundation of all harmony. 135. The .relative effect of any two sounds is found to be always the same as that of any other two in which the ratio of the vibrations is the same. Thus sounds of ELEMENTS OF ACOUSTICS. 139 which the vibrations are respectively 12 and 18, produce compound the same effect as those whose vibrations are 40 and 60, J^J^e f or same effect; 18_ _ 60_* = 3_ . 12 = " 40 r= 2 ' and we say that according as the first and second sounded together, are pleasant or unpleasant, so are the third and fourth ; also, if an air beginning on the first sound re- quire an immediate transition to the second, then, the same air beginning on the third will require an immediate transition to the fourth. 136. The relative pitch of two sounds is called an interval; interval. Its numerical value is expressed in terms of the graver sound, represented by the number of its vibra- tions in a given time, taken as unity. The value of It8numerica] an interval is, therefore, always found by dividing the value found; number of vibrations of the acuter note in a given time by the number of vibrations of the graver note in the same time ; thus, the interval of two sounds, one of which is produced by two and the other by three vibration^ in the same time, has for its measure f. If 18, 23, and 30, be the numbers of vibrations of three sounds Examples; in the same time, and we wish to find a fourth sound which shall be as much above the third as the second is above the first, we say, 18 : 23 : : 30 : x = 3Q ' 23 = 38J . 18 137. Next to unison, wherein the vibrations of the two sounds are in the ratio of 1 to 1, the most satisfactory Yibrations ln t i i ,1 -, ,. ,1 L > ratio of 1 to 2; concord is that in which the vibrations are in the ratio of 1 to 2. The effect of this is not only pleasing, but it always gives rise to the idea of sameness; insomuch that if two instruments were made to play together in such manner that the sounds of the one should always 140 NATURAL PHILOSOPHY. Give the be of twice as many vibrations as the simultaneous sounds dTnCTnTshraL ^ * ne otner ? * ne 7 would be universally admitted to be of the same air; playing the same air, with only that sort of difference which is heard when a niian and a boy sing the same tune together. TWO tense strin s ^" OW 5 take a tense string, and call the sound emitted whose vibrations from it (7, and, for the reasons given above, let the sound are in the ratio of <* , ./? i i i xi i ' r> i , i Ito2 . from a string of double the number of vibrations be called C '. Let us seek for the simplest fractions which lie between 1 and 2, up to the prime Y, and we shall find, * 5 1 J5 =; 3. 6 J 35 35 4) 4 " 2) 55 series of and these, arranged in the order of magnitude, give, vibrations that will produce agreeable pmsical sounds. n, -, _, in after placing 1 and 2 on the extremes, A set of tense strings, or of pipes, so arranged that' the first makes one vibration while the second makes f of a vibration, the third f of a vibration, and so on to the last, which makes -2, will emit sounds every one of which will be agreeable when sounded with the first. Sounds which 138. But it is found that the frequent repetition of are pleasing and soun( i s which are very near to each other is not pleasing to those which are J _ . . . not BO. an uncultivated ear, and that the frequent repetition O A sounds too far from each other is not pleasing to the ear after a little cultivation. Taking the intervals of the above series, we find that from the Examination of the above series; 1st to 2d is | 2d to 3d is f 3d to 4th is f 4th to 5th is | 5th to 6th is 6th to 7th is 2 1 = ! ^ 3" = - \ = I; 2 5 "24 5 II; v; I; ELEMENTS OF ACOUSTICS. The interval between the 1st and 2d, and that between Defects in this the 6th and 7th are too great, while the interval between 8eries ofsounds; the 2d and 3d is too small for frequent repetition. A new sound must, therefore, be substituted for the second one of the scale, and of such value as to increase the interval between the second and third and diminish that Additions between the first and second, while an additional sound re< must be interpolated between the 6th and 7th. Denote the first of these by x and the second by y, then will the series of ratios stand, 1 * a JL A- . l. i f 2 Form of an f I | 45 3> 25 35 2/5 ^> improved series ; making seven in all, for the octave is but the same note with a different pitch. But upon what principle shall the values of these new sounds be determined, seeing we cannot have any Principles by , . , . , , which the series more simple consonances with the fundamental' sound igimproved . whose vibrations are represented by 1 ? The answer is, we must take those sounds which make the simplest con- sonances while they give with the remaining sounds the greatest number of consonances. The consonance indicated by the interval from the to 8th is 2 f = | ; to 8th is 2 4th to 6th is } 5th to 8th is 2 3 -* 4 Consonances; f ) 5. 4 * Now, let the sound a?, have between it and the 5th an First interval equal to that between the 5th and the 8th ; then 8u PP sition ; will whence QJ == 9. . Consequence; and the first three sounds of the series will stand 1, f , f , 142 NATTJR'AL PHILOSOPHY. Intervals. Second supposition ; giving intervals f and y, already found in another part of the series. Again, let the interval between the 5th and 7th be equal to that between the 4th and 6th, and we shall have y - f = and Consequence ; y = V ; Last three sounds ti 1118 ma king the last three sounds f, V an( ^ 2 , and giv- and their ing the consecutive intervals of f and }, both of which are found in another part of the series. Replacing x and y by their values, we have intervals. Its effect universally le: <7, D , E , F , G , A , B , C' Diatonic scale; 1 , f , } , f , f , f , V j 2 5 or multiplying by 24 24,27 ,30,32 ,36 , 40 , 45 ,48, 1st, 2d , 3d , 4th, 5th, 6th, 7th, 8th. This is called the natural, or diatonic scale. "When all its sounds are made to follow each other in order, either upwards or downwards, the effect is universally acknowledged to be pleasing, and all civilized nations have agreed in adopting it as the foundation of their music. Each sound in the scale is called a note^ and takes the name of the letter immediately above it ; and its place in the order of acuteness from the fundamental note is expressed by the ordinal number below it. Thus, count- ing the vibrations of the fundamental note unity, the note whose vibrations are f , is named E, and it is a third above (?, regarded as the fundamental note ; in like man- ner the note whose vibrations are -f, is ^4, and it is a sixth above C. The 8th from the fundamental note, or C', is called an octave above G. Again, we say A, is a fourth above E, and E, is a fourth below J, as would be Note, its name, and place indicated ; Illustrations; ELEMENTS OF ACOUSTICS. 143 manifest by simply sliding the scale to the right or in- verting it, so as to bring the number 1, under the note reference to the of reference regarded as the fundamental note. 139. This diatonic scale, which is obtained from the Essential series of sounds affording the simplest concords with the quakes of the fundamental note, after one alteration on account of the too great proximity of two concordant notes, and one in- terpolation on account of the too great distance of two others, has both of the essential qualities of repetition and variety. Thus, writing CD, for the interval from C to D, and using like notation for the others, and writ- ing the names which have been adopted by musicians for the several intervals, we have the following TABLE. C Z>= F G = A B . . = f ; major tone. DE-GA . . . . = V; minor tone, of a major. EFBG' . . . . =T|J diatonic semitone. C E=FA = GE . = } ; major third. E G = A C' . . . . = | ;- minor third. DF ...... =|f; |f of minor third. C F= D G = EA = GC'=%\ fourth. Table of interval, ; flattened fifth* . . =|; fifth. DA ...... =40. 8 O f a fifth. CA^DB . . . . =| 5 sixth. EG' ...... I; minor sixth. C B ....... = y ; seventh.* D G' ...... = y % ; flattened seventh.f ^ G C' ....... =2; octave. We observe here three different intervals between con- secutive notes, viz. : first, that from G to D F to G * An inharmonious interval when the notes are sour ded tc gether t Decidedly more harmonious than the seventh. 144 NATURAL PHILOSOPHY. Major tone; Minor tone ; Diatonic semitone or limma produce continuous sound; Greatest number; certain individuals ; = A to B = |, and called a major tone / second, that from DtoJ?=Gto A = V, called a minor tone; and third, that from E to F = B to C' = }|, called, though improperly, a diatonic semitone, being in fact much greater than half of either a major or minor tone. This interval is also called by some authors a limma. 140. "When the vibrations are less numerous than 16 a second (M. SAVAKT says 7 or 8), the ear loses the impres- sion of continued sound, and in proportion as the vibra- tions increase in number beyond this, it first perceives a fluttering noise, then a quick rattle, then a succession of distinct sounds capable of being counted. On the other hand, when the frequency of the vibrations exceeds the limit of 24000 a second, all sensation, according to M. SAVAET, is lost ; a shrill squeak or chirp is only heard, and Peculiarities of according to the observation of Dr. WOLL AS'TOX, some in- dividuals, otherwise no way inclined to deafness, are alto- gether insensible to very acute sounds, while others are painfully affected by them. It is probable, however, that it is not alone the frequency of the vibrations which ren- ders shrill sounds inaudible, but also the diminution of intensity which, from the nature of sounding bodies, must ever accompany a rapid vibration among their elements. No doubt if a hundred thousand hard blows per second could be regularly struck by a hammer upon an anvil at precisely equal intervals, they would be heard as a deaf- ening shriek; but in natural sounds the impulses lose in intensity more than they gain in number, and thus the sound grows more and more feeble till it ceases to be heard. If we add to the diatonic scale on both sides the octaves of all its tones, above and below, and again the octaves of these, and so on, we may continue it inde- 'finitely upwards and downwards. But the considera- tions above show that we shall soon reach practical limits in both directions, growing out of the limited powers of the ear. Causes which render sounds inaudible. Diatonic scale continued in both directions ; Practical limits. ELEMENTS OF ACOUSTICS. 145 141. By the aid of the ascending and descending series of sounds thus obtained, pieces of music which are perfectly pleasing may be played, and they are said to be in the key of that note which is taken as the Key; fundamental, sometimes called the tonic note of the Tonic note; scale, and of which the vibrations are represented by 1. And if such pieces be analyzed they will be found to consist chiefly if not entirely of triple or quadruple combinations of several simultaneous sounds called chords, chords; such as the following : <7, D, E,F, G,A,B ,C f ,D',E',F',G',A',B',C" 1 9 5 4 3 5 1_5 9 1_8 1 JL 6 1_0 3_0 A. X 5 854)3?25358?' J 58 5 453 52535 8)^' 1st, 2d , 3d , 4th, 5th, 6th , 7th , 8th , 9th , 10th, llth, 12th , 13th, 14th , 15tb. 1st. The common or fundamental c7tord, called also chord of the the chord of the tonic, which consists of the 1st, 3d and tonic; 5th ; or the 3d, 5th and octave. This is the most harmo- nious and satisfactory chord in music, and when sounded the ear is satisfied and requires nothing further. It is, therefore, more frequently heard than any other, and its continued recurrence in a piece of music determines the key in which the piece is played. 2d. The chord of the dominant. The fifth of the key chord of the note is called the dominant, by reason of its oft re cur- dominant; ring importance in harmonic combinations of a given key. The chord of the dominant is constructed like that of the tonic, but on the dominant -as a fundamental note, and consists of the 5th, 7th and ftth, being the 5th and 7th of one scale, and the 2d on the next following scale of octavt s ; or, replacing the latter note by its octave be- low, the notes of this chord will be 2d, 5th and 7th. 3d. The chord of the sub-dominant; that is, the chord Chord of the constructed upon the 4th note next below the dominant. 8U It consists of the 4th, 6th and 8th ; or, replacing the lat- ter note by its octave below, the notes of this chord be- come the 1st, 4th and 6'th. 10 146 NATURAL PHILOSOPHY. False close. Dissonance of the 7th. 4th. The false close, which is the chord of the 6th, its notes being 6th, 8th and 10th, or replacing the last two notes by their octaves below, 1st, 3d and 6th. The term false close arises from this, viz. : A piece of mu- sic frequently before its termination (which is always on the fundamental chord) comes to a momentary close on this chord, which pleases only for a short time, and requires the strain to be taken up again and closed as usual, to give full satisfaction. 5th. The dissonance of the *lth, or the combination of the 2d, 4th, 5th and 7th. It consists of four notes, and is the common chord of the dominant with the note immediately below it, or the 7th in order above it. Bhort piecs of 142. With these chords and a few others, music may be arranged in short pieces so as to possess considera- ble variety, but long pieces would appear monotonous. In the latter the fundamental note would occur so often as to appear to pervade the whole composition, and the change of key ear would require a change of key to avoid the feel- n Q f tedium which would naturally arise from such a t cause. This change of key is called modulation. But the change is not possible without introducing other notes than those already enumerated. Suppose, for example, it were desirable to change from the key of C to that of G. The chord of the tonic in the key of C is composed of the notes C EG ; in the key of G, of the notes G B D r , giving the intervals, or modulation, avoids monotony; Example for illustration; = 'and In the chord of the dominant, in the key of (7, the notes are G B D\ giving the intervals, =l, and the same as before. But the chord of the dominant in the key of #, if it could be formed at all from existing ELEMENTS OF ACOUSTICS. 147 notes, would consist of D\F',A', giving the intervals Necessity for other notes, shown ; D'F' = and Z>' J/ = which are very different from the intervals of the com- mon chord to which they ought to be equal ; and in order that we may be able to make them equal, we must have other notes for the purpose. Now D\ being the dominant of 6*, must be the com- what the new mencement of the interval, and cannot be altered ; new notes must, therefore, be substituted for F* and A'. Denote the vibrations of the new notes by x and y ; then, passing to the octave below to avoid the com- mon factor 2, we must have, To find the new notes; J= i,and| = f; whence, substituting the value f iwD, x = f . f and y = f . f . That is to say, a change from the key of C to that of 6r, requires for the formation of the chord of the dominant in the latter key, two new notes, whose vibra- tions would be represented respectively by the ratios f and f multiplied by the vibrations in the dominant of 6r, Now, as any note may be taken as the key note, and Multiplicityof as the dominant changes with the latter, the number new notes must . . ITT j be avoided; of requisite notes would be so numerous as to render the generality of musical instruments excessively com- plicated and unmanageable. It becomes necessary, there- fore, to inquire how the number may be reduced, and what are the fewest notes that will answer. For this purpose we remark, that if we multiply the values of x and y by 24, to reduce them to the same unit as that of the scale of whole numbers in 138, we find 148 NATURAL PHILOSOPHY. How this is accomplished ; Place of first ne\ note determined : Sharp, flat; Place of second new note ; 24 y = -^ -^ ' 24 = 40 J. In the scale just referred to we find the numbers 32 and 36, so that the note whose vibrations are x, is almost half way between these two notes, and may be in- terpolated at that place. It will, therefore, stand between F and G, and is designated in music either by the sign #. sharp, or b, flat, according as it takes the name of the first or second of these letters. Thus it is written either %F, or ^>G. With regard to the note whose vibrations are y, and of which the value is 40 J, it comes so near to the note A, whose value in the same scale is 40, that the ear can- hardly distinguish the difference between them, so that the latter may be used for it ; and though a small error of one vibration in 80 is introduced in using A as the dominant of D, yet it is not fatal to harmony, and it is far better to encounter it than to multiply pipes or strings to our instruments for its sake. Besides, these errors are modified and in a great measure subdued, by Temperament what is called temperament, of which the foregoing is the origin. 143. The highest note of the perfect chord of the dominant of G, is three perfect fifths abo*e C, and the note A', which we have adopted in its place, is the oc- tave of the 6th above C. The vibrations of the first are denoted, by (J) 3 = y, and of the second, by 2 . f = y 5 and the interval between will be v + v = it- comma in music. Tliis interval of two notes, one of which rises three per- fect fifths, and the other an octave of the 6th above the same origin is called, in music, a comma. ELEMENTS OF ACOUSTICS. 149 144. Were any other note selected for the fundamen- NO two keys tal one, similar changes w^ould be required ; and no two ^ . e ' keys can agree in giving identically the same scale. All, however, may be satisfied by the interpolation of a new note within each of the intervals of the major and minor tones in the scale of article (138), thus, Interpolation required; and the scale thus obtained is called the Chromatic chromatic scale; scale. 145. But what shall be the numerical values of the Numerical values interpolated notes? If it were desirable to make the scale of article (138), which is in the key of (7, (the vi- notes; brations of this note being represented by unity,) as per- fect as possible, at the expense of the others, there would be but little difficulty, as the mere bisection of the lar- ger intervals would possibly answer every practical pur- pose, and %C = b7>, might be represented by v/l-f; = ^E, by V f f , and so on ; but as in practice no such preference is given to this particular key, and as variety is purposely studied, we are obliged to depart from the pure and perfect diatonic scale ; and to do so Necessity for a with the least possible offence to the ear, is the object of a system of temperament. If the ear required perfect concords, there could be no music but a very limited and monotonous one. But this is not the case ; per- fect harmony is never heard, and if it were, would be appreciated only by the most refined ears; and it is this fortunate circumstance which renders musical com- Perfect harmony position, in the exquisite and complicated state in which never hearB = T= 51; 156 NATURAL PHILOSOPHY. giving by addition 8n PP o6it!on; - A = T + = 51 + 2S = 79 = major third. The quantity by which B must be flattened for this purpose is obviously Consequence ; Interpolation necessary to render the two scales nearly perfect in one particular case. Another case supposed ; T- 0= 51 - 28 =23; and this is the amount by which, in this case, a note differs from its flat. As to the remaining three inter- vals, the difference between jTand t being small, amount- ing only to 5, (which answers to the logarithm of a comma }J,) the sequence T t 6 is Jhardly distinguishable from t TO, and if the note D be tempered flat by an T-t interval = H~-? or half a comma, this sequence will in both cases be the same, and our two scales of G and F will be rendered as perfect as the nature of the case will permit by the interpolation of only one new note. But, on the other hand, suppose we would modulate from G to B. In this case the scale of G will stand Scale as it stands in this case ; G D E F G whereas it should be A B' Scale as it should stand; Conclusions. t E t The intervals from j? to E, and from E to B, are the only ones that are equal, and to make the others equal would require (7, D, F, G and A to be sharpened, and consequently the introduction of no less than five new notes. But to confine ourselves to the change from A to %A we have ELEMENTS OF ACOUSTICS. 157 B A = T = 51 : Particular case taken. and B - *A = 6 =28; consequently, by subtraction, A - *A = 23 = B - *B, Eesult; as before determined. But since the whole interval from B to A = T = 51, is more than double this interval, the flattened note b.Z?, will lie nearer to B, and the Explanation ; sharpened note #A nearer to the lower one A than a note arbitrarily interpolated half way between A and B, (to answer both purposes approximately,) would be, Diesis left in and thus a gap or diesis, as it is called, would be l e ft thiscase; between %A and ^>B. The diesis in this case only amounts to T-2 (T- 0) ^ bat ifc amounts = 51 46 = 5, equal to a comma, or the tenth part of a major tone T\ in other cases it would be greater. But in all cases the interval between any note and its sharp is considered to be equal to that between the same note and its fiat. 151. Taking each note of the diatonic scale as the Eachnoteof & 6 fundamental or key note in succession, we shall find, take^astheLy by the same mode of comparison, the following sets of not ; ! notes in the several scales the accent at the top of the letter denoting one octave above the key note. Names of the Keys. <7, D, E, F, G, A, B, C'. (natural, 0) D, E, *F, G, A, B, *C, D'. (two sharps, 2) sets r notes E, *F, *G, A, B, *C, *D, E'. (four sharps, E) " F, G, A, \>B, G, D, E, F r . (one flat, F) G, A, B, C, D, E, *F, G'. (one sharp, G) A, B,*C, D, E, *F, *G, A. (three sharps, A) B, *C, *D, E, *F, *G, *A, B'. (five sharps, B) 158 NATURAL PHILOSOPHY. These scales j n these scales which have the natural notes of the defective by two sharps; diatonic scale for the key, there are but five sharps, whereas there should be seven. Where are the other two? If we take #F and #C as the key notes, we shall find Names of the Keys. *J? , #, A, , #O, *D, * E, #1". (six sharps, iff) *^> *&, *% *F, *G, *^, *B, *0'. (seven sharps, *C ) In like manner, constructing a diatonic scale on ^>B, and on each new flat as it is successively introduced, we find the following, in which the accent at the bottom of a letter denotes one octave below the key. Names of the Keys. B t , C, D^E, F, G, A, \>B. (two flats, bJB) Same for another \> ? f ^ & , \> A ,*> B , C', D\ \> E' . (three flats, \> E) \>A t , *B t , C,*>D, *E, F, G^A. (four flats, bA) \>D , ^E, F, \>G, *>A, \>B, C'^D'. (five flats, b) \>G , M, ^B, bC', b2) r , \>E\ F', \>G F . (six flats, *&) *>C , *D, *E, *>F, b&, bA, bJS 9 bC r . (seven flats, \>C) several systems 152. Assuming, the principle that the interval tewbeBB* 11 * * Between any note and its sharp is to be equal to that devised; between the same note and its flat, a variety of systems of temperament have been devised for producing the best harmony by a system of twenty-one fixed notes, viz : the seven notes of the diatonic scale with their seven some of the sharps and seven flats. Among the most remarkable sys- most remarkable terns may be mentioned those of HUYGENS, SMITH, YOUNG and LAGIEK, for an account of which the reader is referred to the Encyclopoedia Metropolitan^ article, Sound. Vol. IV., page 797. Peculiarity of 153. But the piano-forte, an instrument in almost >rte ' universal use, and of the highest interest to all lovers of music, admits of only twelve keys from any one note to its octave, and a temperament must be devised which will accommodate itself to this condition. ELEMENTS OF ACOUSTICS. 159 "We have already spoken of the division of the octave Argumentj in , favor of eqtal into twelve equal parts, and have seen that this makes temperament; the fifths all too flat, the thirds all too sharp, and gives a harmony equally imperfect in all the keys. It is urged in favor of equal temperament that all the keys are made equally good, and that in no one does the temperament amount to a striking defect; also, that in the orchestra there is little chance of any uniform temperament if it be not this. Against equal temperament it is urged, how- ever, as before stated, that it takes away all distinctive Against equal character from the different keys, and after all, leaves no tem P erament - one of them perfect. A piano-forte perfectly tuned by the system of equal temperament has to some persons a certain insipidity which only wears off as the effect of this tuning disappears ; insomuch that the best phase of illustration by the piano-forte the instrument is exhibited during the period which pre- cedes its becoming disagreeably out of tune, or, more properly, while it is assuming a state of maltonatwn j for, the transition is only a change from equal to unequal temperament, in which the several keys begirt to exhibit variety of character, until maltonation arrives and makes the instrument offensive. The best practicable way of obtaining a given tempera- use of the ment, equal or unequal, is by means of the monochord. monochord *' The proper lengths of the strings of this instrument, to form the required notes, are first calculated, and after- wards those of the instrument to be tuned are brought into unison with them. No tuner can get an equal tempera- ment by trial ; so that the question in practice generally Generalaimln lies between all sorts of approximations to equal tempera- practice, ment, and as many approximations to some other tem- perament. 154. The mode of proceeding by approximation to The most nsual equal temperament is simply to tune all the fifths a little proceeding; flat ; and the following order is the most usual. The first letters represent the note already tuned, the second the one which is to be tuned from it ; a chord in parenthesis 160 NATURAL PHILOSOPHY. First step, V tuning fork; represents a trial that should be made on notes already tuned, to test the success of the operations as far as it has gone. The first step is to put C ' in tune by the tuning fork; # ' / O'O; CO; GG t ; G,V; DA; AA t ; A t E; (GEG); EB; (G E G ; D G B) ; BB t ; B*F; (D*FA); *F*F,\ *F,*C: (A t *C E)\ *C*G; (E#GB); C'F; (FAC'); F*A t ; (*A,DF); *A t *A ; (#A^ *D ; (*D G *A) ; *D * G t ; (# G t C Explanation of method, and of results that should be obtained : Bearings. All the semitones are written as sharps whether tuned from above or below. Since the fifths are all to be a little too small in their intervals, the upper notes must be flattened when tuned from below, and the lower notes sharpened when tuned from above. In the preced- ing, the octave C C' is completely tuned, and also the adjacent interval *F t C. The rest of the instrument is tuned by oqtaves. The thirds should come out a little sharper than perfect, as the several trials are made, and when this does not happen, some of the preceding fifths are not equal. The parts which are first tuned by fifths, and from which all the others are tuned by octaves, are called bearings. on | 1 55. I n unequal temperament, some of the keys are temperament; kept more f ree from error than others, both for the sake of variety and because keys with five or six sharps or flats are comparatively but little used; these latter keys are left less perfect, and this is called throwing the wolf into these keys. From equal intervals to those which produce what has been called maltonation, there is abundant room for the advocates of unequal tempera- ment to select that particular system most congenial to the views of each, and, accordingly, many systems have been proposed. Of these we shall only mention two, Smith's system; viz. : that denominated by Dr. SMITH the system of mean ELEMENTS OF ACOUSTICS. tones, and that which bears the name of its author, Dr. YOUNG. The system of mean tones supposes the octave divid- B 7 stcm of meatt ed into five equal tones, of which we shall denote the value of each by a, and two equal limrnas, each hav- ing the value /3, succeeding each other in the order a a {3 a a a j3 instead of Tt & Tt T&, as in the diatonic scale, and such that the thirds shall be perfect, and the fifths tempered a little flat. These conditions are sufficient to .determine the values of a and j3, for, 5 a -h 2 (3 = 1 octave = 2 a = 1 third = T + t whence nr< j / rp Use of this a . - - Q Q _j_ - - 5 " system explained 2 . 4: and illustrated; and substituting the values from the table 51 +46 51 - 46 - = 48,5 ; j3 = 28 + - -- - 28,125 and since the interval from the 1st to the 5th of the scale is 3 a + j3. = the fifth by this scale is flatter than the perfect fifth by the quantity 1 (T t\ that is, by a quarter of a com- Results. ma. In this system the sharps and flats are inserted by bisecting the larger intervals. Dr. YOUNG'S first system is as follows, viz. : Tune Young 3 flm downwards from the key note six perfect fifths, then up- s y st m ; wards from the key note six imperfect fifths, dividing the excess of twelve perfect fifths, above seven octaves, 11 162 NATURAL PHILOSOPHY. Explanation. equally among the imperfect fifths, and observing to as* cend in the first case, and descend in the second, by octaves, when necessary, to keep between the key note and its octave. Scale of the Chinese, Hindoos, &c. Effect of small intervals. Principles of nwsic a_ conversation. Misor scales. 156. If we take from the diatonic scale the notes F, and j5, which rise from those immediately preceding them by semitones, there will remain C f , D, E, G, A and G ' for all the sounds of the octave. This is the original scale of the Chinese, Hindoos, the Eastern Islands and the nations of Northern Europe. It is the scale of the Scotch and Irish music, and the Chinese have preserved it to the present time. The character of this scale is exhibited by playing on the black keys alone of the piano-forte. 157. The effect of making an interval smaller is to give the consonance a more plaintive character. It may easily be observed, for example, that the intervals of the minor third, E 6r, and minor sixth, E C ' on any instru- ment, have a sad or plaintive effect as compared with the major third, C E, and major sixth, C A. Almost all per- sons in ordinary conversation are constantly varying the tone in which they speak, and making intervals which approach to musical correctness, and the effect of sorrow, regret, and the like, is to make these intervals minor. Any one with a musical ear, noticing the method of say- ing " I cannot" pronounced as a determination of the will, and comparing the same uttered as an expression of regret for want of ability, will understand what is here meant. Why this is so, no one can. tell. But the asso- ciation exists, and resort is had to those modifications of the diatonic scale which are known from experience to produce the emotions here referred to. The results of these modifications, of which tbere are several, are called Minor Scales, in contradistinction to the diatonic, which is called the Major Scale. The change from a minor to the major scale is one of the most effective of musical resources. ELEMENTS OF ACOUSTICS. 163 If we return to the fundamental note G and its conso- nances, viz. : C *>E E F G A C' Fundamental 654350. noteandits * j JjTJs'^aJsJ^J consonances ; and instead of rejecting ^>E as too near to E, we discard this latter note, and finish by inserting D and B of the diatonic scale, we shall have what is called the common ascending minor scale, as follows : G , D , *E , F , G , A , J? , G' Amending minor 1 9 6 4 3 5 1 5 O ''''' But it is not easy to recognize this as a minor scale in Not easily descent, because, in going from C ' to C, there is no dis- tinction between it and the major scale till we come to descent; *E, or until the scale has produced its principal effect upon the ear. To remedy this, A and B are both lowered a semitone ; that is, A is made ^A, and B is made b_Z?, thus making ^>A a fourth to ^>E^ and b a fifth to and giving t , * , t, *V * , f , 8; which being reversed, is called the common mode of Descending the descending the minor scale. minor scale. Again, if we retain B of the major scale and lower ., we have C , D , ^E , F , G , M , B , C' 1 , I , ! , t , } , ! , V , 2, which is a mild and pleasing scale both in ascent and _ * Schneider's descent, notwithstanding the wide interval between ^A principal minor and B. Its harmonics are more easy and natural than scale ' the other, and SCHNEEDEE makes it, in his Elements of 164 NATURAL PHILOSOPHY. Harmony, a principal minor scale, and treats all others as incidental deviations. Any system of 158. "We shall now show how we may, from the temperament . ' . may be examined theory of the scale, examine any system of tempera- .ment ; and as the method will be rendered the more obvious by applying it to a particular example, we shall take the system of Dr. YOUNG just described. Let all the intervals be expressed in mean semitones, as the unit. There being twelve semitones in the oc- tave, we have one semitone equal to the logarithm of 2 divided by 12, or 0,30103 12 0,0250858 ; Method explained ; f , that System of Di Young taken ; Example for illustration ; and dividing the logarithm of the major tone of the minor tone = y, that of the diatonic semitone = }f, and the excess of twelve perfect fifths over seven octaves = 0,00588 by this value of the mean semitone, we shall find 1 major tone = 2,039100 1 minor tone = 1,824037 1 diatonic semitone = 1,117313 " Excess of 12 fifths over 7 octaves = 0,234600 mean semitones, u u u In tuning upwards, each fifth is to be flattened by one-sixth of 0,234600, or by 0,039100. In the equal tem- perament 'the wolf is replaced by twelve equal whelps ; here by six, but of double the size. JSTow, a perfect fifth is composed of 2 major tones = 4,078200 1 minor tone = 1,824037 1 diatonic semitone = 1,117313 Perfect fifth = 7,019550 Deduct . . . 0,039100 Flattened fifth = 6,980450 ELEMENTS OF ACOUSTICS. 165 -5< A F F' #A -5* *D' -5* #C +s tA *C king C for . 12,00000 . 7,01955 the key note, C . +5< A . . (1) G . +5 D' . -8< A . . (2) D . +5' A . . (3) A . E' . -8< A . . (4) E . . (5) B . +5< A . -8< A . . (6) *F . . 0,00000 . 6,98045 Example continued; (i) (2) .(3) ^ ' The saras .(5) (6) . 4,98045 12 . 6,98045 . 6,98045 . 16,98045 . 7,01955 . 13,96090 . 12 . 9,96090 . 7,01955 . 1,96090 . 6,98045 . 2,94135 12 . 8,94135 . 6,98045 . 14,94135 . 7,01955 . 15,92180 . 12 . 7,92180 . 7,01855 . 3,92180 . 6,98045 . 0,90225 12 . 10,90225 . 6,98045 . 12,90225 . 7,01955 . 17,88270 . 12 . 5,88270 . 5,88270 Collecting these intervals for all the notes from C to ', we have C . . . 0,00000 *F . . . 5,88270 *c . . . 0,90225 a . . . 6,98045 D . . . 1,96090 *G. . . 7,92180 *D . . .. 2,94135 A . . . 8,94135 E . . . 3,92180 *A. . . 9,96090 F . . . 4,98045 B . . 10,90225 Kesults collected As the most important chord is that of the tonic, we form our idea of the effect of each key, from the effect of the temperament upon this chord, judging of the character of the key by the amount and direction of 166 NATURAL PHILOSOPHY. Explanation; the temperament upon the third and fifth, which with the key make, as w^e have seen, the chord in question. Now, a major third is composed of 1 major tone 1 minor tone = 2,03910 mean semitones, = 1,82404 " " Value of a major third; Major third . . 3,86314 " A minor third is composed of 1 major tone = 2,03910 mean semitones, '1 diatonic semitone =1,11731 " " Value of a minor third; Minor third . 3,15641 and hence the intervals for the chord of the tonic are For a major key minor . 3,86314 and 7,01955 . 3,15641 and 7,01955. Conclusions. Method of To examine any particular key, take the numbers from the preceding table opposite the notes of the tonic chord, adding twelve to make the octave when necessary ; sub- tract the number of the key note from each of the other two, and the remainders will give the tempered examining any , particular key. intervals ; from these remainders subtract the correct in- tervals above, and these second remainders will give the amount and direction of the temperament. For exam- ple, let us examine the key of A; we find A . 8,94135; #C' . 12,90225; E' . 15,92180 8,94135 8,94135 Tempered intervals 3,96090 . . . 6,98045 Perfect intervals . .3,86314. . . 7,01955 Temperaments . . + 0,09776 . . 0,03910 Example for illustration. whence we see that the first interval is sharper and the second flatter than perfect, the sign +, indicating sharper, and the sign , flatter. END OF ACOUSTICS. ELEMENTS OF OPTICS, 1. THE principle by whose agency we derive our Light sensations of external objects through the sense of sight, is called LIGHT ; and that branch of Natural Philosophy which treats of the nature and properties of light, is called OPTICS. 2. There exists throughout space an extremely at- n t dple of tenuated and highly elastic medium called ether. This ether permeates all bodies, and the pulsations or waves propagated through it, constitute the principle of light. The eye admitting the free passage of the ethereal Sensation of J sight produced ; waves into it, the sensation of sight arises from the motions which these waves communicate to cer- tain nerves which are spread over a portion of the internal surface of that organ; we therefore see by a Anal gy between > . J t the sensations of principle in every respect analogous to that by which g i g ht and sound. we hear the only difference being in the nature of the medium employed to impress upon us the motions proper to excite these different kinds of sensations. In the former case it is the ether agitating the nerves of the eye, in the latter, the air communicating its vibra- tions to the nerves of the ear. 3. Some bodies, as the sun, stars, &c., possess, in their ordinary condition, the power of exciting light, while many others do not. The first are called self- luminous^ and the second non-luminous bodies. All substances, however, become self-luminous when their f ' bodies; temperature is sufficiently elevated, or when in a state 168 NATURAL PHILOSOPHY. insects that of chemical transition ; and some organisms, as the glow- worm, fire-fly, and the like, are provided with an appa- ratus capable of exciting ethereal undulations and of becoming self-luminous when thrown into a state of vibration by these insects. Self-luminous bodies are seen in consequence of the light proceeding directly from them; whereas, non-lu- minous bodies only become visible because of the light which they receive from bodies of the self-luminous Non-luminous " rendered so. class, and reflect from their surfaces. power of exciting light. Self-luminons bodies visible ; Medium. "Waves of light spherical in homogeneous media ; Geometrical illustration. Wave front not spherical in heterogeneous media. Fig. 1. 4. Whatever affords a passage to light is called a me- dium. Glass, .water, air, Torricellian vacuum, &c., are media. 5. Waves of light, like those of sound, proceed from any disturbed molecule as a centre, with a constant velocity in all directions, through media of homogeneous density. The front of the luminous wave in such media is, there- fore, always on the surface of a sphere whose centre is at the place of primitive disturbance, and whose radius is equal to the velocity of propagation multiplied into the time since the wave began. Thus, if a molecule of ether be disturbed at (7, and the velocity of propagation be denoted by V y and the time elapsed since the disturbance by , then will the front of the wave at the expiration of this time be upon the surface of a sphere whose centre is at C and radius C A = V. t. If the medium through which the wave moves be not homogeneous, the shape of the wave front will not be spherical, but will vary from that figure in proportion as the medium de- parts from perfect homogeneousness. 6. The circumstances attending the propagation of luminous and sonorous waves are similar. The intensity ELEMENTS OF OPTICS. of light, like that of sound, depends upon, and is directly intensity of proportional to the amount of molecular displacement. hght It is, therefore, Acoustics, 53, inversely proportional to the square of the distance from the original luminous source. 7. We have seen, Acoustics, 16, that in wave pro- TO demonstrate pagation through a homogeneous me- the rectilineal propagation of dium, the displacement of a mole- Fi g . 2. Hght in cule (9, from its place of rest at one time, becomes a source of displace- ment at a subsequent time for an in- definite number of molecules situat- ed on the surface of a sphere .#/"ffi whose centre is at 0, and of which the radius is equal to V. t ; that these numerous disturbances become in their turn so many sources of disturb- ance for any single molecule as 0', in front of the wave, and that the amount of 0' 's displacement from its place of rest will be found by compounding the displacements due to all these sources, after estimating the amount due to each separately. To ascertain the effect of this process of composition, Geometrical denote by X, the length of a luminous wave ; join and 0' by a right line, and take the distances A B = B C CD =DE=\\ and with 0' as a centre and the distances 0' B, 0' C, 0' D, 0' E, &c., successively as radii, describe the arcs Bbi, Cc, D d, Ee, &c., cutting the section of the wave MN, in the points 5, S" by refraction ; also drawing the normal P P' to the deviating surface, the angle P D 8, which the incident ray makes with this normal, is called the angle of incidence ; the angle P D S', which the reflected ray makes with the normal, is called the angle of reflexion, and the angle P D S'" = P' D S", which the refracted ray makes with the normal, is called the angle of refraction. 12. These angles are always estimated from that part of the normal draw r n through the point of incidence of the ray, which lies in the medium of the incident wave. ELEMENTS OF OPTICS. 173 negative; Fig. 6. Illustration. They are accounted positive when on the same side of when positive the normal as the incident ray, and negative when O n andwhel the opposite side. Thus, the angle of incidence P D S, is always positive, as also the angle of re- fraction PDS'", while the angle of reflexion PDS', will always be negative, as it should be, since the velocity of the reflected light must be counted negative, the reflected wave being dri- ven back from the de- viating surface. 13. When the deviating, surface is curved, we con- ceive a tangent plane drawn to it at the point of incidence, and treat this plane as the deviating surface for that portion of the wave which is incident immediately about the tangential point. 14. The angle which any ray after deviation, makes with the prolonga- tion of the same ray be- fore incidence, is called the deviation. Thus, S lv - D ,' is the devia- tion by reflexion ; and S" D IV -, the deviation by refraction. The deviation ; By reflexion and by refraction. 15. If we make V T' = m, (2) NATURAL PHILOSOPHY. Equation applicable to refraction ; Equation (1) becomes sin 9 = m sin 9 (3) which answers to any refracted ray. For the reflected ray, V becomes equal to F', and 1 = this in Equation (3) gives Equation applicable to reflexion ; sm 9 = sin 9 which applies to all cases of reflexion. we may consider the Equation ... .(4) And generally General equation for all deviations. sin 9 = m sm

sin " = 48 15'; Atmosphere and water; that is to say, the greatest angle of refraction which can exist when light passes from air into crown glass, is 41 5' 30"; and from air into water, 48 15'. If the ray pass from a medium to another less dense, 180 NATURAL PHILOSOPHY. Light pacing m" will be less than unity, from denser to an( j e q ua i to the reciprocal rarer medium; t of its former value ; Equa- tion (7) will then give sin 9" > sin 9' ; Angle of taking the maximum value refraction taken 90; for sin 9 = 1, we shall ob- tain from the same Equation, Fig. 10. h Consequence ; sn = (9) Analogy; Examples ; Conclusion; Angle of total reflexion. this value for the sine of the angle of incidence, which corresponds to the greatest angle of refraction when light passes from any medium to one less dense, is the same as that found before for the greatest angle of refraction, when the incidence was taken a maximum, in the pas- sage of light from one medium to another of greater den- sity, In the case of air and glass, it is 0,657 ; correspond- ing to an angle of 41 5' 30" ; for air and water, the angle is 48 15'. If the angle 9' be taken greater than that whose sine is _ 7/ , the angle of refraction, or emergence from the denser medium, will be imaginary, and the light will be wholly reflected at the deviating surface. This maximum value for 9' is called the angle of total reflexion. Light cannot, therefore, pass out of crown glass into air under a greater 'angle of incidence than 41 5 ' 30 " , nor out of water into air under a greater angle than 48 15'. 22. The maximum limit of refraction, and the cases of total reflexion, are attended with many interesting ELEMENTS OF OPTICS. 181 results. If an eye be placed in a more refracting medium Appearances due than the atmosphere, as that of a fish under water, it will tothelimitof r . . . ' . refraction and perceive, by the limit of refraction, all objects in the total reflexion ; horizon elevated in the air, and brought within 48 15 ' of the zenith, while some objects in the water would ap- pear to occupy the belt included between this limit and the horizon by total reflexion. Those remarkable cases of mirage, where objects are seen suspended in the air, and oftentimes inverted, are Those due to explained by ordinary refraction and total reflexion. ordinar y * * refraction and The phenomena of mirage most frequently occur when total reflexion, there intervenes between the suspended object and spec- tator a large expanse of water or wet prairie, and towards the close of a hot and sultry day, when the air is calm, so that the different strata may arrange themselves ac- cording to their different densities. "When the wind rises the phenomena cease. Fig. 1L Illustration It is well known that in the ordinary state of the at- Apparent mosphere, its density decreases as we ascend; a ray of effectofthe 1 atmosphere on light, therefore, entering the atmosphere at /S, would un- the positions of dergo a series of refractions, and reach the eye at B, with celestial bodtoi J an increased inclination to the surface of the earth ; and would appear to come from a point, S', in the heavens above that at /#, 'occupied by a body from which it pro- 182 NATURAL PHILOSOPHY. ceeded. Hence, the effect of the atmosphere is to in- crease apparently the altitudes of all the heavenly bodies. Relative index 23. Dr. WOLL ASTON suggested a method, founded on determined by the limit of total reflexion, to determine the relative in- dices and refractive powers of different substances. If the angle of incidence, 9', be measured by any device, Equation (9) will give, <-.4r, sm

a ' r . an( j fa Q re f rac tive power may then be deduced from Equation (6). optical prism; 24. The deviating surfaces have, thus far, been supposed parallel. .If they be inclined to each other, as M N, M N r , we shall have what is called an optical prism, which consists of any re- fracting substance bounded by plane surfaces intersecting each other. Deviating planes M N and M ' N' ', are called the deviating planes, and and refracting the j under which they are inclined, is called the angle. J refracting angle of the prism. Deviation of a ray of light in passing through a prism; 25. To find the deviation of a ray of light in passing through a prism, let SD be the incident, D D r the first, and I)' S' the second re- fracted ray. The total deviation will be ELEMENTS OF OPTICS. 183 which denote by ; then, calling the refracting angle of the prism a, and adopting the notation of the figure, we shall have 180 or* =a + MD D r + MD' D = a +~ 9' + - Equations ; a""'" 1 " iy or a = -4/ 4- 9' (10) Kefracting angle, hence =

= 77i sn by a simple process of the calculus, or by trial, it may be shown, that when the angles of incidence and emer- condition for gence are equal, the deviation will be a minimum, or the least possible.* Making 9 equal to ^, in Equations (11) and (10), we find, *See Appendix No. 2. 184: NATURAL PHILOSOPHY Its use;

, and con- tained in a plane perpendicular to the intersection of the reflectors ; this ray will be devia- ted at the point Z), of the first reflec- tor, again at the point D f , of the second, and so on. Required the circumstances attending these deviations. Call the first angle of incidence 9 , second, ..... 9 3 third ...... 9 3 Deviations of a ray of light by two plane reflectors, the plane of incidence being normal to their intersection ; Notation; 9, In the triangle P D D\ the angle at P is equal to the inclination of the reflectors, which denote by $', and we shall' have 9 n - 2 9 - i = 9 n _ , -9 ra = .... (14) and by addition, Equations from the figure; Sum of theso equations; (15) 186 NATURAL PHILOSOPHY. if 9 1 be a If 9 1 be any multiple of a, as n 1 . i, multiple of *; 9 t - iT=TL . i = 0, (16) The ray will that is to say, the nth incidence will be perpendicular to the reflector, and the ray will, consequently, return upon itself. Example 1st. ' Suppose the angle made by the reflec- "Fvimrilp first* required the number of reflexions before the ray retraces its course. These values in Equation (16), give, Data; 60 - fT^l . 6 = Eesult n = 11. Example %d. The angle of the reflectors being 15, Example second; and the first angle of incidence 80, required the fourth angle of incidence. These values in Equation (15), give

the ray will be reflected by one of the reflectors into a direction parallel to the other, and in the second, this last reflexion will give the ray such a direction that it will meet the other reflector only on being produced back. 28. Adding the first two Equations in group (14), we have 6 2 i Angle made b7 the incident ray and the same ray after two reflexions; SS'D' = 2i. That is, the angle made by the incident ray and the 188 NATURAL PHILOSOPHY. Equal to double same ray after two reflexions, is equal to double the an- the angle made gj e O f the reflectors. It follows, therefore, that if the C.3 ft angle of the reflectors be increased or diminished by giv- ing motion to one of the reflectors, the angular velocity of the reflected ray will be double that of the reflector. This is the principle upon which reflecting instruments for the measurement of angles are constructed. by the reflectors. Application of this principle. DEVIATION OF LIGHT AT SPHERICAL SURFACES. Deviation of light at spherical surfaces ; Illustration and notation ; Vertex. Kule first; Eule second. 29. Let MD N, be a section of a spherical surface separating two me- dia of different den- Fi s- 15 - sities, as air and glass, having its cen- tre at C, on the line C, which will be called the axis of the deviating sur- face ; FD a ray of light, incident at D, and D S, the direction of this ray after deviation, which being produced back will intersect the axis at F '. The point 0, where the axis meets the surface, is called the vertex, which will, for the present, be taken as the origin. Call FD, u; F' D, u' ; CD,r; OF',f>; OF,f; and the angle CD, 0. Now, distances estimated in the direction of wave pro- pagation, from any origin whatever ', are always negative / those estimated in the contrary direction, positive. And, when light is incident on a concave surface, the radius of curvature is always positive j when incident on a convex surface, negative. In the triangle CD F, we have the relation, sin

A pencil with the axis of the deviating surface, is called a direct pen- cil / and if such a pencil be taken very small, the quantysj, in Equation (18), will be so small that General equation the products of joying itg centra | ray coincident 16 - M may, without mate- rial error, be omitted. This will reduce Equation (18) to Equation for a email direct pencil; or r-&= and talking the reciprocal, mrf (19) Reciprocal of the f m r 1 m .f . . (20) If f be con- stant, or the rays all proceed Fig, IT. from the same Conclusion for a point F On the small direct . , - , pencil axw Before de- viation, f will also be constant for the same medium and curvature, and all the rays after deviation will meet in ELEMENTS OF OPTICS. 191 some other point F' on the axis. The first of these points is called a radiant, and the eecond a focus ; and because of the mutual dependence of these points upon each other with respect to their positions, they are called conjugate foci, and the distances / and /', are conjugate focal distances. The radiant is a point common focal distances; to the undeviated, and a focus to the deviated rajs. Then, a radiant is the centre of curvature of the undeviated wave ; and a focus of the deviated wave. When a wave turns its convexity to the front, its molecular living force becomes more and more diffusive as the wave progresses ; Eeai and virtual when it turns its concavity to the front, more and more 0< concentrative. A radiant is real, when the undeviated wave turns its convexity to the front; and virtual, when it turns its concavity to the front. A focus is real, when the deviated wave turns its concavity to the front; and Eeal and *irtni - radiants. virtual, when it turns its convexity to the front. 31. Luminous waves, like waves of sound, Acoustics Living force of f>Kr,T T T re i A molecules, or 53, become more and more dinused in proportion as intens it y O f light they recede further and further from the place of primi- decreases for tive disturbance, provided their convexities continue to be inlrTSeTfoT turned to the front, and more and more concentrated converging ray*, after they have been so deviated as to turn their con- cavities to the front. ' In other words, the living force of the wave molecules, which determines the intensity of light, will become less and less for divergent, and greater and greater for convergent rays. That portion of the living force imparted to the ethereal Living force of molecules at any one place, as a radiant, and which proceeds J^^ 011 upon a spherical segment embraced by the bounding rays segment of a small direct pencil, can, therefore, Equations (19) and (20), be concentrated upon the ethereal molecules at another place, as a focus, by the action of a spherical devi- ating surface ; and the focus, whether real or virtual, be- And the focus comes a source of light as well as the radiant, and is as becomes a sourc distinctly visible. "When the focus is real, the deviated of u * ht wave first becomes concentrated in, and subsequently 192 NATURAL PHILOSOPHY. emanates from it ; when virtual, the deviated wave pro- Whence the deviated wave proceeds for real ceeds only from the deviating surface, but with dimen- and for virtual g - ons fa Q game ag though a had departed from the virtual focus. 32. If the raj which is deviated at the first, be incident First deviated . upon a second sur- rayincidentupon f Jf' JP 7 having a second surface ; a radius /, and situated at a dis- tance t, from the first, measured on the axis, we may suppose this ray to have proceeded originally from F' / and denoting the distance from the new vertex ', to the point F", in which this ray, after deviation at the second surface, meets the axis, by f", and the index of refraction of the second medium by m', we shall have from Equation (20), Equation applicable to the second deviation; Second deviated ray incident upon a third surface ; . (21) /AT" Equation applicable to the third deviation ; And by the same process for a third deviating surface, "I f t ~\ ~i m"r" . . . (22) ELEMENTS OF OPTICS 193 Fig. 20. M':. Third deviated ray incident upon a fourth surface ; And for a fourth, 1 _ ( m "' - f"" , (23) Equation m'" r'" m'" (f" + t") V applicable to the fourth deviation, and so on. And so on for any number of surfaces, the law being ma- nifest. 33. The value of /' 4- , deduced from Equation (20) Direct relation and substituted in Equation (21), will give a direct re la- found between ' the first radiant tion between f" and/, in terms of r, r, ra, m and t; distance and final and the value of f" + t' found from this derived equa- focal distance - tion and substituted in Equation (22) will give a direct relation between f" and /, in terms of f, /, /', m, m', m", and t' / and by the same process of elimination a direct relation may be found between the radiant distance / and the final focal distance f'"-* . 34. But in practice the distance , is so small that it Practical relation may, without sensible error, be neglected. Omitting , ^ we shall find that the first member of each of the preced- omitting '<; ing equations becomes a factor in the last term of the second member of that which immediately follows it, and proceeding to eliminate these factors by their values, we obtain from Equations (20) and (21) fi - . m' T' 13 t \ m' \ - mr mf Resulting / 9 . v equation for two 8urfaces 194: NATURAL PHILOSOPHY. K elation -^ between this value of _-, substituted in Equation (22), gives, conjugate focal J distances for three surfaces, omitting*; _^ -1 , 1 j m'-l . 1 (m~l , 1 \) /"' m" r" + m"\ m f r' + m' V^T" + ^f) \ and this value of -7^-, in Equation (23), gives, J Same for four 1 m'" \ surfaces, and so ~/>//// = /// /// r J^rm^-1 J_ ,mr-l 1 , m -l ix.-i m^Lm'V" * m" 1 m'/ + m'l"^7" ^m/ N and so on for additional surfaces. Medium between 35. If we now suppose the medium between the second and third, second and third, fourth and fifth, sixth and seventh, supposed^ & G "> deviating surfaces, the same as that in which the same as that of light moved before the first deviation, we shall have the case of a number of refracting media bounded by spherical surfaces, situated in a homogeneous medium, such as the atmosphere, for example, and nearly in contact. Hence, Corresponding , 1 < m 1 , 1 values of ^ = > m = - 77 j *** = > <* refractive m Indices ; and the foregoing Equations reduce to N Resulting 1 fo*, 1| 3* ^l_\^ l^\ f* = lj ' (^ -yf + T two, three, four, equations for \ r r' > f * J m"-l 1 - /I 1 =^--T.-+^i.-+- (29) &C., &C. ELEMENTS OF OPTICS. 195 36. Any medium bounded by curved surfaces and Lens defined; used for the purpose of deviating light by refraction, is called a lens. Equation (27) relates, therefore, to the deviation of a small pencil of light by a single spheri- cal lens ; /, denoting the distance of the radiant, and Equations /", that of the focus from the lens. Equation (28), lates to the refraction or deviation by a single lens and lensea - a second medium of indefinite extent bounded on one side by a spherical surface nearly in contact with the lens. Equation (29), relates to deviation by two spheri- cal lenses close together, / and /"" denoting, as before, the radiant and focal distances. 37. If the rays be parallel before the first deviation, incident raj* supposed / will be infinite, or = 0, and Equations (20), (27), J (28), and (29), will reduce to 1 Resulting form of the preceding mr equations; 1 _ m"-l 1 f - /I 1\1 777 == ^V 7 "" ~ri' [ m ~ L ' 17 "' 7' ) \ The values of/', /", /"/, /"", &c., deduced from these Principal focal Equations, are called the principal focal distances^ being distance ' the focal distances for parallel rays. Denoting these distances by F tA F tli , F UI &c, and (J- - 1), (^ - i) &c., by _, __ r , J_, &c., we shaU have the Mowing table, P ?" P viz. : 196 NATURAL PHILOSOPHY. Table of reciprocals of principal focal distances; 1 m-1 1 m-1 1 n P i" 1 1 /-: ^ F nt ~' m"r" ' m"\ P / 1 / *m ~~~ 1 f nfi/~~~\- -r~1 n T /;// P p 1 m""-l 1 /m"-l , m-l\ F tun m""r"" l m"" V // i / P p / 1 \ w"-l m 1 F ////// P"" P 7 P &C., &C. , &c. (30) Bole. An examination of the alternate formulas of the above table, beginning with the second, leads to this result, viz., that the reciprocal of the principal focal distance of any combination of lenses, is equal to the sum of the re- ciprocals of the principal focal distances of the lenses taken separately / which may be expressed in a general way by the Equation, Value for the -9 ? =2(-^J (31) reciprocal of the principal focal distancfesof any of wherein ( ), denotes the reciprocal of the principal \. \ F i tecaj distance of any one lens in the combination, the Greek letter 2, that the algebraic sum of these is to be taken, and _^>, the reciprocal for the combination. First members of Substituting the first member of the first Equation, substituted in in group (30), and the first members of the alternate preceding Equations, beginning with the second, for their corres- ponding values in Equations (20), (27), (29), &c., we finally obtain, ELEMENTS OF OPTICS. 197 1 1 , 1 7 ^ i i mf 1 /" " " -^ i i / f 1 i i 1 /unit TJI nun *"/ (34) (35). Equations (33), (34), and (35), are of a convenient form for discussing the circumstances attending the deviation of light by refraction through a single lens, or a com- bination of lenses placed close together; and Equation (32), the deviation at a single surface. 38. The several terms of these Equations are the re- ciprocals of elements involved in the discussions which are to follow. The pencil of light being small, the versed sine of half the arc !>/>', has been- disregard- ed, and the arc itself may be regarded as coinciding with the tangent line at the vertex 0, and as having been described about either of the points <7, F', or F) as a centre, indifferently ; and denoting the length of the arc D by , and the number of degrees in this arc when referred to the centre F, corresponding to the radius /, by n, we shall have the proportion, (32) Resulting equations for the discussion of the /oo\ deviation of light ^ 'by one or more lenses or by a single surface. To find relative measures for the vergency of incident and deviated rays ; Eays supposed to diverge both before and after deviation, and arc taken; whence, n = a . 360 1 Number of degrees in this arc referred to the centre F; 198 NATURAL PHILOSOPHY. Number of arc referred to the centre F' \ in which if denotes the ratio of the diameter of a circle to its circumference. When this arc a is referred to the centre F' , corres- ponding to a radius f ', its number of degrees, denoted by ri, becomes, a . 360 1 and dividing the first of these Equations by the second, we find, Ratio of the above values ; n / 1 /' Conclusion for . . diverging rays. w ^ ence we conclude that and f measure the relative J J divergence of the incident and deviated rays. When the devi- Fig. 22. ated rays meet the axis at F', on the opposite side of the deviating surface from the radiant, the value /', being laid off in a contrary di- rection from the Conclusion for \ converging rays ; vertex 0, becomes negative, and the relative measure -^y, for the convergence of these rays will be negative. Again, if the incident rays converge to a point F, before deviation, f for the same reason, would be ne- gative, and the measure for the corresponding conver- gence would be negative. And, generally, we shall find that, referring the radiant and focal distances to the ELEMENTS OF OPTICS. 199 vertex as an origin, di- n -u Fig. 23. vergence will be mea- sured by a positive and convergence by a nega- tive quantity ; and for , ., ^ - ., is General rule for Convenience We shaU, ^g^S_ ^ vergency of raya. therefore, hereafter em- ploy the general term vergency to express either of these conditions of the rays, indifferently. 39. The power of a lens t's its greater or less capacity Power of a Ien8 . to deviate the rays that pass through it. In Equations (33), (34), (35,) Ac, * _L, J_, &c., // //// llllil will measure the vergency of parallel rays after devia- tion ; and as these measures are expressed in functions of the indices of refraction, and . or ( ) &c., p \r r I they will be constant for the same media and curvature, and may be employed as terms of comparison for the other two terms which enter into the Equations to which they respectively belong. From what has been said, it is apparent that JL,in F Equation (31), will measure the vergency of parallel r ays Meag eonho after deviation by any combination of spherical lenses power of a lens whatever, and will consequently be the measure of the or comt)ination of lenses; power of the combination / and as ( ), is the measure \F ' of the power of any one lens of the combination, we hate this rule for finding the power of any system of lenses, viz. : Find the power of each lens separately, and take the algebraic sum of the whole. 40. It will be convenient to express the relation in Equations (32), (33), (34), &c., by referring to the centre 200 NATURAL PHILOSOPHY. TO find a relation of curvature of the deviating surfaces as an origin. For between the ^ purpose l et Q J) conjugate focal distances when be a section of the the centre of deviating Slir f ace curvature is taken as the and denote the dis- origin; tances of the radiant and focal points from the centre C, by c and c', respectively ; we have by inspection, Substitutions and reductions ; Fig. 24. f Eelation for one surface ; which in Equation (19), give, after reduction, m 1 m l ~d (36) For a second surface; and for a second deviating surface whose centre of curva- ture is at a distance , from that of the first, we ob- tain from Equation (36), m' - 1 y c'-t . . . . (37) and for a third, whose centre is at a distance ', from that of the second. For a third surface ; m"-l m c"-t' . . . (38) ^f deviating light by refraction ; the surfaces are generally various spherical spherical. lenses. A, called a double convex lens, is bounded by two spherical sur- faces, having their cen- tres and the surfaces to which they corres- pond, on opposite sides of the lens. When the Fig. 25. ir Geometrical representations of the spherical lenses. 202 NATURAL PHILOSOPHY. Double convex lens; Plano-convex : Double concave: Plano-concave ; Meniscus; curvature of the two surfaces is the same, the lens is said to be equally convex. B, is a lens with one of its faces plane, the other spherical, this latter face and its cen- tre being on opposite sides of the lens, and is called a plano-convex lens. C, is a double concave lens / each curved face and its centre lying on the same side of the lens. D, is a plano-concave lens, having one face plane and the other concave. jEJ has one face concave and the other convex, the con- vex face having the greater curvature ; this lens is called a meniscus. -concavo-convex. F, like the meniscus, has one face concave and the other convex, but the concave face has the greater curvature ; this is called a concavo-convex lens. The line containing the centres of the spherical surfaces, is called the axis. Different cases arise from the 43. A moment's consideration will show that all the circumstances of vergency attending the deviation of light Bigns of the radii; * by any one of these lenses, will be made known by Equation (33), it being only necessary to note the dif- ferent cases arising out of the various combinations of surfaces by which the lenses are formed ; these cases de- pend upon the signs of the radii. Equations (33), (34), (35), &c., were deduced on the Ru.e for signs of supposition that r is positive, the concave side of the surface being turned towards incident light; it will, of course, 29, be negative when the convex side is turn- ed in the same direction. Besides, / was taken positive for a Teal radiant, or when the rays are supposed to di- verge from any point upon the axis of the lens, before deviation ; on the contrary, it will become negative when radii; ELEMENTS OF OPTICS. 203 the rajs are received by the deviating surface in a state ^^ of conjugate of convergence to a point behind the lens. The signs 0< of /', /", &c., will be positive when the deviated rays meet the axis 011 being produced back. The foci are then virtual. When the rays meet the axis on the opposite side of the lens or lenses, /*', /", &c., become negative, and will correspond to real foci. The several lenses may be characterized as follows : 1 Double Convex, ..... 1 Piano-Convex, convex side to in- cident light, Do. plane side to inci- dent light, . . ( Meniscus, convex side turned to 3 j incident light, . . (. Same, concave side do. do. 4 Double Concave, ..... (Piano- Concave, concave side to 5 J incident light, . r and + / r and + /= = oo and + r' T r _{_ T _|- tjf + /,+/ = '^Same, plane side to do. do. _|- 7^ = oo, and T' f Co7icavo- Convex, concave side to incident light, ? <^ f' ^ -\- T^~\- T' Same, reversed, . . r ^> T', ^, T r [ Characteristics of the yariou8 lenses. 8 44. To discuss the properties of any one of these Discussion of the properties of any lenses, resume lens ^ Equation (33), de- termine the sign of -L, by refe- rence to its gen- eral value in Equations (30), and the table above, and then proceed to make various suppositions in regard to the position of the radiant and deduce the corresponding places of the focus. 204: NATURAL PHILOSOPHY. Double convex lens taken as an example; 45. As an example, let us take the double convex lens. Equation (33), is General equation ; I f " ~ ~F~ ~? j a j and, Equation (30), and Table (A\ Value for reciprocal of principal focal distance ; Equation for discussion ; m Real radiants between principal focus and infinity; Give real foci. Real radiants within the principal focus ; ~ P and as long as m > 1, we shall have, f" F. t f (40) For L > , or / > F lt , f" will be negative, and F ' f it J Fig. 27. the vergency of the deviated rays will be ne- gative. That is to say, if a wave proceed from a point upon the axis in front of the lens between the limits F tl , the principal focus, and infinity, it will be concentrated after deviation, into a point upon the same line behind, and the focus will be real. 1- < 4-> r Fig. 28. For positive, and the ver- gency of the deviated rays will be positive. That is, if the wave pro- ceed from a point in front and situated be- ELEMENTS OF OPTICS. 205 tween the lens and the principal focus, it will, after Give vlrtual focL deviation, proceed from some other point in front, and the focus will be virtual. _ = 0, or/"=oo Fig. 29. i t-fi Eeal radiant at principal focus. For JL = JL, or / That is, the vergency of the deviated rays will be zero, and a spherical wave pro- ceeding from the prin- cipal focus will be con- verted, by deviation, into a plane wave which can only be concentrated into a point at an infinite distance. If the rays be received by the lens in a state of con- For virtual vergence to a point behind, that is, if the concavity radiants; of the wave be turned to the front before deviation, then - or f will be negative, and Equation (40), becomes J J_ /J_ /" ~\F U / The equation becomes ; ^/And the foci TT$! always be real and the vergency of the deviated rays will always be ne- gative. In other words, to whatever point behind the lens the wave may be tending to con- centration before de- viation, the deviation will cause it to concentrate in some other point behind. If the rays proceed from a point in front and at the Eeal radiant at ~ r distance equal to distance ot twice the principal focal distance, / becomes 2 F,, , equal to 2 F t # and Equation (40) reduces to 206 NATURAL PHILOSOPHY. The focus will be real and at a distance behind the lens equal to Divergence of rays decreased ; I /" 1 1 or 2 *' Fig. 27. fft __ and the wave will be concen- trated at the same distance behind the lens. For all cases of positive vergency, both before and after deviation, we find which shows us that a positive vergency will be dimin- ished by the deviation. cases ^ ne g at i ye vergency, we find numerically but algebraically, r < / Hence the effect is to say, when the rays diverge before deviation, of a convex lens they will diverge less after; and when they converge be- f re deviation, they will converge more after. Hence we conclude, that the tendency of a convex lens is to collect the rays, or concentrate the waves of light deviated by it. The focal distance of the double convex lens is given by Equation (27), ELEMENTS OF OPTICS* 207 f . f . f> f Focal distance; If the lens be supposed of glass, m = |, nearly, and _ For lens of glasc . If the lens be equally convex, r = ?', and /"=-^; For lens equallj convex; and if the rays be supposed parallel before deviation, f = oo , and f = i For parallel rays, 46. Each of the other lenses described may be sub- jected to a similar discussion. This being done, the re- sults will conform to those exhibited in the following TABLE 1 Lens. Incident pencil. 7" Sign of/" Kefrac-pencil. x / _^_ s r Table for convex "^ + 7 Convex j Converging ) \ I r _ i j Converges Ll -/ f I f,, S) \f"7/ } 208 NATURAL PHILOSOPHY. light. covex lenses ^ similar table may also be constructed by formula collect, and J J concave lenses (34), for a combination of any of the spherical lenses disperse the taken two and two, and. by formula (35), for any com- bination taken three and three, and so on. In general, it may be inferred from the preceding table, that convey lenses tend to collect the incident rays, while concave lenses, on the contrary, tend to scatter them. TO construct the 47. Transposing, in Equation (33), -7 to the first member, we get 1 /" __ / inustration Interpretation. which shows that the vergency after, diminished by that before deviation, gives a constant vergency measured by the power of the lens. Hence, to construct the focus, draw the extreme ray FD, and from the point Fig> 81< Z>, the line D H, mak- m g w ^h the incident ray7^J9, produced, the angle HDK, equal to ---- the power of the lens ; D II will be the de- viated ray, and the point -F' ', where it meets the axis, will be the focus. For, in the triangle F D F", the angle D F" 0, 1 measured by _, diminished by D F F", measured by , is equal to H D K, measured by - ; which is / ' *n the geometric interpretation of the above equation. conjugate foci g 43. Suppose the conjugate foci to be in motion, and denote any two consecutive values of / by x and a/. ELEMENTS OF OPTICS. 209 and the corresponding values of f" by y and ^', then Notation and Equation (33), 1 1 1 T ~^7 + ^' subtracting the second from the first we find, y y' W &' Transformations and reductions; reducing to a common denominator, and writing for the products y y' and a? a?', the quantities f" 3 and/ 3 , to which they will be sensibly equal, the Equation becomes y' y _ x' x . ~~~ and dividing by the interval of time , during which Time t, the change from x to x' takes place, which is the same introduccd ' as that from y to y', we have y y 1 77 x or Relation between conjugate focal distances and velocities of F" F - - f"* ' fa ' in which V denotes the velocity of the radiant, and V" C that of its conjugate focus; and since the denomina- tors must always be positive, being squares, the signs of the two velocities must .always be alike. Whence we conclude, that in lenses a change in the place of the radiant will always be accompanied by a change of its conjugate in the same direction, and that the rate of change in the one will be to that of the other , i / , - . . -, . . . , Conclude, that in as the squares of their respective distances from the lensesconjugate lens directly. This has an important application in the focialwa y smove action of lenses when employed to form images. 14 210 NATURAL PHILOSOPHY. If the lens be a 11 sphere. *" " the lens be a sphere, m = > = 0, and r> y/t C from Equation (36), being substituted in Equation (37), we obtain G 50. If in Equation (20), we make T infinite, we get Deviation at a plane surface by refraction : or, /=/', which answers to the case of a small pencil deviated at a plane surface separating two media of different densities, as air and water. On the supposition that the Radiant in 1 denser medimn- radiant is in the denser medium, m becomes , and m this in the preceding Equation gives /=/'; Elustration ; Appearances accounted for. that is, to an eye situated without this medium, the dis- tance of the radiant from the deviating sur- face will appear dimin- ished in the ratio of unity to the relative in- dex of refraction of the ray in passing from the denser to the rarer me- dium. This accounts for the apparent eleva- Fig. 32. 1 tion above their true positions of all bodies beneath the surface of fluids, as the bottom of a vessel partly filled with water, and the apparent bending of a straight stick at the surface when partly immersed in the same fluid. ELEMENTS OF OPTICS. 211 APPLICATION TO THE DEVIATION OF LIGHT BY SPHERICAL REFLECTORS. 51. In reflexion, we have only to consider one de- E< i ation applicable to a viating surface. Equation (20) applies here by making sphericalconcaye reflector; = - 1, which reduces it to, (43) But two cases can arise, and these are distinguished by the sign of the radius. The reflector may be concave towards incident light, in which case r will be positive, or it may be convex towards the same direction, when r will be negative. Equation (43) relates to the first case, which will now be discussed. If the incident rays be parallel, = 0, and Incident rays parallel ; _ f or, Fig. 83. Principal focal distance ; Hence the principal focal distance is equal to half radius, and Equation (43), reduces to 1 "F. / I 1 \ Equation for V / discussion; !N"ow, this Equation is only concerned with the re- flected wave, and if this wave be concentrated at all after deviation, it must be upon that part of the axis on the side of the incident light, and hence /', for a 212 NATURAL PHILOSOPHY. Real radiants rea | f ocus mus t be positive, and for a virtual focus ne- beyond the principal focus gatlV6. As long as _ - > _ , or f > F^ f will be positive, F, J and the vergency of the deviated rays will be positive ; that is, a wave proceeding from a point in front of the reflector between the principal focus and infinity will, after deviation, be concentrated into some other point in front. 1 1 ^T < 7 the vergency will be negative ; in other words, a wave proceeding from a point on the axis between the vertex and principal focus, will never be concentrated after de- viation, but will appear to pr-oceed from a virtual focus behind. If the radiant be at the centre of curvature, / = 2 F { , and Eeai radiants within the principal focus; "When . . < , or /< F^f will be negative, and Radiant at the centre of curvature. Real radiants beyond the centre . that is, a wave proceeding from the centre of curvature will, after deviation, return to that point. For we have r f ' 2 F] focus will be between the reflector and centre, and / "^' Give real foci between the .111 /? 1 1 centre and since < , W6 find < _ , Or /' > F, \ SO principal focus; J? f f " t f & , that the focus will be found between the centre and prin- cipal focus. ELEMENTS OF OPTICS. 213 If Real radiants between the centre and r<'r principal focus ; then will -| 1 Give real foci < , OP /' > r; beyond the / * -F j centre; that is, the focus will be at a greater distance from the reflector than the centre. "When / = F t we shall have - = ; that is, the ver- Real radiant at J the principal gency will be zero, which shows that a spherical wave focus, proceeding from the principal focus will be transformed by deviation into a plane wave, which can only be con- centrated at a distance /' co . If the vergency before incidence be negative, / will be negative, and Equation (44), becomes r, = -j=T H -JT (45) Virtual radiants Hence, /' will always be positive, and the vergency Alwfty8givereal positive ; that is, when a wave is proceeding to con- foci, centration in a point behind a concave reflector, it will, after deviation, be concentrated into some other point in front. Equations (44) and (45), show that -L, which mea- sures the vergency of deviated rays, is always algebrai- rejectors 1 , . , analogous to cally greater than , which measures the vergency of convex lenses. the incident rays. Hence, concave reflectors, like con- vex lenses, tend to collect the rays of light which are deviated by them. 214: NATURAL PHILOSOPHY Different cases 52. By discussing the several cases that will arise in m reflector^ attr ik u ti n g different signs to r and/, and various values to the latter, we shall find the results in the following TABLE. Reflector Incident pencil. Table for convex and concave reflectors ; Sign of -j; Eeflectpencil Concave Diverging I < _1 _ jj \{^ F ' \{ + H J Diverges more. - l\ Diverges. ( 1 1 ) 1 < --- r- > ^-.BrM + M Converges I -P, /> I ff l\ less. conclusions. from which we perceive that convex reflectors tend to scatter the rays and concave reflectors to collect them. 53. If , in Equation (44), be transferred to the first J member, we find \_ F. SMTI of the constant; which shows that the vergency after, increased by that before deviation, is a constant vergency, which is mea- sured by the power of the reflector; and to construct ELEMENTS OF OPTICS. 215 J) the focus, draw the extreme ray FD, and the line DF\ mak- ing with the normal DC, the angle C DF' equal to the angle of incidence, the V point F', where this line meets the axis, will be the focus, obvious. Construction of foci for reflectors. The reason is 54. By a process entirely similar to that of 48, we For reflectors may find from Equation (44), which appertains equally to ^T te c J* 5 ^ a concave or convex reflector by assigning to -=- its pro- directions. per sgn, (46) and because V and Fhave contrary signs, we conclude that the conjugate foci in the case of spherical reflectors proceed, when in motion, in opposite directions. 55. Equation (43), by making r infinite, reduces to l^ J_ /' = "/ Deviation by reflexion at plane surfaces ; or, Which shows, that in all cases of deviation of a pencil by a plane reflector, the divergence or convergence will not be altered ; and if the rays diverge before deviation, they will appear after deviation to proceed from a point conclusion. as far behind the reflector as the real radiant is in front ; but if they converge before deviation, they will be brought to a focus as far in front as the virtual radi- ant is behind the reflector. 216 NATURAL PHILOSOPHY. Spherical aberration : Incident pencil not small ; Illustration ; Longitudinal aberration ; Lateral aberration ; SPHERICAL ABERRATION, CAUSTICS, AND ASTIGMATISM. 56. Tims far the discussion has been conducted upon the supposition that the pencil is very small, and that z, the versed-sine of the angle 0, included between the axis and the radius drawn to the point of incidence of the extreme rays of the pencil, is so small, that all the products of which it is a factor may be neglected. If, however, z be retained, and Equation (18) be solved with reference to f, the value of this latter quantity will be expressed in terms of m, /*, r and z, and may be written (47) Fig. 85. and if the semi-arc of the deviating surface, denoted by 0, and of which z is the versed- sine, be made to vary from zero to any magni- tude sufficient to embrace the ex- terior rays of any definite pencil, it is obvious that //, must have an infinite number of values, and that each value will give the focus for those rays only which make up the surface of a cone and are incident at equal distances from the vertex. This wan- dering of the deviated rays from a single focus is called aberration, and when caused by a spherical deviating surface, as it is in the case under consideration and in practice generally, it is called sp?ierical aberration. "When estimated in the direction of the axis, it is called longitu- dinal, and at right angles to the axis, lateral aberration. If we represent the second member of Equation (19) by Jf, that Equation may be written (19)' ELEMENTS OF OPTICS. 217 and subtracting this from Equation (47), we find Measure of longitudinal and lateral f ' f> _ flf jyr (48) aberration > and V ' their laws of variation; in which the first member denotes the length of the por- tion F' FV of the axis along which the different foci will be distributed, and will measure the longitudinal aber- ration. The lateral aberration is measured by the length of the line F' Z, drawn through the focus of the rajs near the axis of the pencil and perpendicular to the axis of the deviating surface. The linear length of the arc, D = r . d, is called the radius of aperture, and. it is found Radius of that in all cases of ordinary practice, the longitudinal aper< aberration varies as the square, and the lateral aber- ration as the cube of the radius of aperture. If in Equation (48), we make m = - 1, we shall have Aberration for a reflector. the longitudinal aberration for a spherical reflector. If the value of // in Equation (47), be substituted for / in Equation (18), and we write /" for /', then solve the equation with reference to / ", still retaining z, and take the difference between this value of f" and thatiens. given by Equation (27), we shall find the longitudinal aberration for a single lens; and that for any number of lenses placed close together might be found by the same process. 57. We perceive that a spherical wave of any con- siderable extent deviated at a spherical surface, will not, . -i -, , T .,, ., General effects of in general, be concentrated at, nor will it appear to pro- 6p hericai ceed from, the same point ; but if we conceive the wave aberration; to be divided into an indefinite number of elementary zones by planes perpendicular to the axis of the devi- ating surface, each zone will have its particular point of concentration or of diffusion, according as the foci are real or virtual. Moreover, longitudinal aberration di- minishes the focal distance, that is, in general, // is less than /', and the deviated rays which are in the same 3^' nal plane and on the same side of the axis, will intersect ablation; 218 NATURAL 'PHILOSOPHY. '"eometrical f "ustration ; Explanation of tne figure; each other before they do this latter line. Thus, if FD Fig. 36. Caustic curve; Caustic surface : Section of the deviated wave by a plane through the axis of the surface ; "When the caustic will be real be the exterior, and F D' its consecutive incident ray, D F z and D' F", the corresponding deviated rays, these latter will intersect each other at some point as c', on the same side of the axis F j in like manner, if D" F' be the next consecutive deviated ray to D' F", it will intersect this latter in same point as c", and so for other deviated rays up to' that one which coincides with the axis. The locus of these intersections 87 - being formed by the doubling over, as it were, of the deviated wave up- on itself, thus pro- ducing at the cusp c' double the ethe- real agitation due ELEMENTS OF OPTICS. 219 to either segment F z c' or c' c t separately. If, on the con- Virtual caustic, trary, the wave recede from the caustic on being devia- ted, the caustic will be virtual. Caustics are finely illus- trated on the surface of milk when the light is reflected upon it from the interior edge of the vessel in which Illustration - it is contained. 58. We have only spoken of a pencil of light whose Oblique pencil; radiant is on the axis, which is usually called a direct pencil. When the radiant is off the axis, the axial ray of the pencil becomes oblique to the deviating surface, and the pencil is said to be oblique. In the case of an oblique pencil, however small, the deviated rays will not, in general, meet the axis as in the case of the direct pencil, but will all intersect two lines at right angles to each other and not situated in the same plane. These lines are called focal lines, and the property of the de-Focai lines; viated rays by which all of them intersect both of these lines, is called astigmatism. Astigmatism. 59. It is, generally, not possible to deviate a spherical Aberration ware of sensible magnitude by a single lens or surface of spherical form without aberration, and yet the practi- cal difficulties in grinding lenses and reflectors to any other figure render it necessary to adhere to this shape. Fortunately, however, two or more lenses may be so united that the aberration of one shall counteract that of another, and light may thus be deviated without aberration. When such combinations are used, a wave proceeding from one point .may be made by deviation to proceed from, or concentrate in, some other point. Such points are called aplanatic foci, and the combi- Apianatic fod, nations which produce them are said to be aplanatic. 220 NATURAL PHILOSOPHY. OBLIQUE PENCIL THROUGH THE OPTICAL CENTRE. Explanation. Oblique pencil a QQ t Tff Q ^ ave seen ar ti c le (19), that a ray undergoes through the optical centre; no ultimate deviation when it passes through a medium bounded by two parallel planes. If, then, in the case of an oblique pencil the rays diverge sufficiently to cover the entire face of a lens, there may always be found one at least which will enter and leave the lens at points where tangent planes to its surfaces are pa- rallel. This ray being taken as the axis of a very small pencil proceeding from the assumed radiant, will con- tain the focus of the others, the distance of which from the lens, in very moderate obliquities, will be measured by /"", given in Equation (27). This is obvious from the fact that in the immediate vicinity of the tangen- tial points the surfaces, which are spherical, will be symmetrical in respect to the line which joins them. To find where the ray referred to intersects the axis of the lens after deviation at the first surface, let M N N' M' repre- TO find the sen ^ a section of a concavo- opticai centre of convex lens, of which the ra- dius G of the first surface is r, and C' 0' of the second is r' ; /SPand S' P' the traces of two parallel tangent planes. Denote by t the distance 0' , between the surfaces measur- ed on the axis, and by e the distance OK, from the first surface to the intersection of the line joining the tangen- tial points P, P', with the axis. Then, since the radii C P and C' P', drawn to the tangential points, must be parallel, the similar triangles OP jfiTand C' P 1 K, will give the relation, Relation from figure; CO C' 0' CK~ C' K ELEMENTS OF OPTICS. 221 and replacing these quantities by their values, T T' Same in other torm3; from which we find Kesult. But this value of e is constant, whence we infer that all rays which emerge from a lens parallel to their di- optical centra rections before entering it, proceed after deviation at the de first surface in directions having a common point on the axis. This point is called the optical centre, and may lie between the surfaces or not, depending upon the figure of the lens. If we suppose but one deviating surface, then the medium behind must be of indefinite extent, in which case r' and t will become infinite and sensibly equal, and Equation (49) reduces to e = r. That is to say, the optical centre of a single deviating optical centre of surface is at the centre of curvature. a single surface; If the lens be double concave, the radius r r becomes negative, and the value of r', whence Of a meniscus ; rt e = r r Of a plano-convex Ian. In a plano-convex lens having its plane face turned towards incident light, r will be infinite, and r' finite and positive, and e = t. which brings the optical centre to the vertex of the curved face. The student may determine in the same way the optical centre of the other lenses. OPTICAL IMAGES. Optical images ; 61. The surface of every luminous body is made up of a vast number of radiants, from each of which waves of light proceed in all directions. These waves cross each other; and if any deviating surface be presented, it be- comes the common base of a multitude of pencils, whose vertices are the radiants which make up the surface of the body. Some one ray of each of these pencils will pass * through the optical centre of the surface, and those rays in the immediate vicinity of this one constituting a small pen- cil will be brought to a focus upon it as an axis, and hence for each radiant in the surface of the body there will be a corresponding conjugate. These conjugate foci make up a second luminous surface, from which waves will pro- ceed as from the original body ; and this surface is called Explanatory remarks; ELEMENTS OF OPTICS. 223 an image of the body, because to an eye so situated as to Image of a body receive these new waves, the object, though often modi- fied in shape and size, will seem to occupy the position of the new surface.. An optical image is, therefore, an assemblage of foci optical image conjugate to a series of contiguous radiants on the de ied; surface of some object; and its formation consists, in so deviating portions of the waves of light which proceed its formation from the object, .as either to concentrate them in some consis ' ts in ; new positions from which they may proceed as from the object itself, or to cause them to move from these new positions without having at any time occupied them. In the first case the image will, be real and in the second Eeai image; virtual. In general, but a part of each wave can be de- viated by the use of spherical deviating surfaces to sat- isfy these conditions, for t those portions remote from the Yirtual image> undeviated ray of each pencil cannot, in consequence of aberration and astigmatism, be brought to accurate ver- gency. 62. To ascertain the relation between an object and TO find the its image, let us suppose the deviation to be produced ^^ct'anT*^ by a lens, so thin that its thickness may be neglected, image formed *> which is the usual case in practice. The optical centre alcns; (r, may be taken ,1 /> Fig. 89. as the origin 01 co- ordinates. Denot- ing by Z, the dis- tance from this point to any as- sumed point P in the object, and writing this quantity for /, in Equation (33), which we may do without sensible error, we get Conjugate corresponding to an assumed radiant point, 224: Section of the NATURAL PHILOSOPHY. Let the object be a plane, perpendicular to the axis of the lens ; its section will be a right line P Q. Call 0, the angle included between the axis of any oblique pen- cil and the axis of the lens. "When the pencil becomes direct, & will be zero, and I will equal f. But, generally, we have General relation ; 1 = cos 4 ' Equation of the image of a right line; this in Equation (50), reduces it to F /// -* // TT (51) which is the polar equation of the image referred to the optical centre as a pole. It is the same in form as the polar equation of a conic section, which is Is the same in form as that of a conic section ; T A(l-e 2 ) 1 + e cos v conclusion; Whence we conclude that the image of a straight line perpendicular to the axis of the lens which forms it, is a conic section, and comparing the two Equations, we find, % Equations compared ; (52) 6 = f' (53) = V. ELEMENTS OF OPTICS. 225 For the same lens, F tl is constant ; its value in 3 * ination (58 > " ^ shows that the Equation (52), which is the radius of curvature at the uneViii be one vertex, is also constant. of the nio From Equation (53), it is easily seen that the curve will be the arc of a circle, ellipse, parabola, hyperbola, or a right line, one of the varieties of the hyperbola, ac- cording as Conditions for the different conic sections T or according as the distance of the object is infinite ; greater than the principal focal distance of the lens; equal to this distance ; less than this distance ; or zero. If the section P Q be supposed to revolve about the axis of the lens, it will generate a plane, and the image a curved surface whose nature will depend upon the dis- tance of the object. We have seen that a positive value for /", answers sign of the focal to a virtual, and a negative value to a real focus ; ^"^^ BO, if the points of the image -be indicated "" -y ^sitive whether the values for /", the image will be virtual ; if by nega- '~ tive values, real. For a concave lens, F n is positive, and Equation (51), answers to this case. For a convex lens, F tl is negative, and Equation (51), becomes 226 NATURAL PHILOSOPHY. linage will be real for a convex lens as long as the object is beyond the principal focus; /" =- (54) 1 TT+ COS and the image will always be real as long as F. 11 COS d < 1, or Fig. 40. That is, if from the optical centre, with a radius equal to the principal focal distance, we describe the arc of a circle, and this arc cut the object, the image of all that part of the object in- cluded between the points of intersection A and A' will be vir- tual, while that of the parts without these lim- its will be real ; if the distance of the object exceed that of the prin- cipal focus, the whole image will be real. 63. Multiplying both members of Equation (51), by sin 6 it becomes Equation (51) transformed ; *, . /. tan A + F ' *- a . (55) cos 6 and giving to 0, its -greatest value for any assumed object, /tan 6 will be the length of that portion of the object on ELEMENTS OF OPTICS. 227 the positive side of the axis as long as & is positive and less Explanation of than 90 ; /" sin 0, is the distance of the extreme limit O f terms; the image of this portion of the object from the axis ; and writing f tan d = #, Substitutions; Equation (55) becomes, after dividing both members by /tani, -p - Equation (55) J , -TJ transformed; COSd If the linear dimensions of the object be small as com- pared with its distance from the optical centre, we write unity for cos d, the image will, 48, and Eq. (52), lts distance fron ., -, . . -I .,-. ^ -. ., optical centre ; sensibly coincide with $, and the above equation reduces to 5 // (56). In which the essential signs of all the quantities correspond to a concave lens. For a convex lens, F lt is negative, and Equation (56) becomes (K*7\ Equation for a convex lens; Equations (51) and (56), show that the image of every real object formed by a concave lens is virtual, erect, and less than the object, while Equations (54) and (57), show that the image of every real object formed by a convex lens is real as long as the object is beyond the principal focus, is inverted, and less or greater than the object, depending upon the distance of the latter from convex lenses. 228 NATURAL PHILOSOPHY. when the image the optical centre. When the distance of the object is the object twice that of the principal focus, Equation (57) becomes other cases. and the object and image are equal in size. "When the object is within twice the principal focal distance, it is less, and when beyond this same distance it is greater than the image. Eolation between 64. If we make 6 equal to nothing in Equation (51), linear dimensions fn iu co ' mQ ^ Q ^fa fa axig Q f fa leng itg l QJ1 gfa of the object and * image; will measure the distance of the image from the opti- cal centre, while / will measure that of the object on the same line. Denoting these distances by D tl and D, respectively, substituting them in Equation (51), clear- ing the fraction in the second member, and dividing both members by D, we find which, in Equation (56), gives 1> D same in words. That is to say, the corresponding linear dimensions of an object and of its image are to each other directly as their respective distances from the optical centre. image formed by 65. If an image be formed by deviation at a sin- j surface. its points will be readily found by means of Jngle surface; Equation (36) ; the optical centre, in this case, being at the centre of curvature 60. Writing f for c, and f for c', that Equation becomes ELEMENTS OF OPTICS 229 1 m 1 m 7" ~ + 7 Equation applicable; making f = oo, 1 _m-l 7" distance ; hence, or, / = Equation for discussion ; Same in another form: For an oblique pencil passing through the optical cen- tre, we have, on the supposition that the object is a right line perpendicular to the axis of the surface, /'= Same for an oblique pencil through the optical centre. wherein 521* = 1, as in article (62). / ^ 66. If the image be formed by reflexion, m and Equation (60) becomes -1, /'=- 1+ -5-cosd sf*-t\ Image formed by ^ U ) reflexion; 230 NATURAL PHILOSOPHY. since for a concave re- flector, F0 Eq. (59), becomes negative. This is a polar equation of a conic section, the nature of which will result from the relation of F t to /. It will, g 62, be an ellipse, parabola, or hyperbola, according as Fig. 41. Imageofaright line will be a conic section. Eel ation between dimensions of object and image, f>F i \f = F,\ Or / < F'. 67. By a process entirely similar to that of 63 and V * J m , we shall find that the linear dimension of the 6b- ^ s ^ ^ & COTreS p 0n ^i n g dimension of the image, as the distance of the object from the centre is to that of the image from the same point. And a moment's reflec- tion will show us that all real images must be in front, while all virtual images must be behind the reflector. 68. "We get the point in which the image cuts the axis by making Equation for discussing a concave reflector; or + 1 (62) Interpretation of results ; This value of /' being negative, the image will be found between the reflector and the centre, the distance/ being positive on the opposite side. As long as/is posi- tive, the image will lie between the centre and reflec- tor, /' will be less than / and the image, consequently, less than the object. When / is zero, /' will also equal zero, and the object and image will be equal and occupy ELEMENTS OF OPTICS. 331 the centre. "When / becomes negative, or the obiect positionsand ^ ' J relative size of passes between the centre and reflector, / will be posi- the image when tive as long as / < F.* and the image will pass without, theob J ecti8 & ^ '' ' between the f will be greater than/, or the image will be greater than centre and the the object. "When/ being still negative, is equal to F t vertex - or the object is in the principal focus, the image will be infinitely distant. The object still approaching the reflec- tor, / will be greater than F t ; f becomes negative again and the image will approach the reflector from behind it, and will be greater than the object till/ = 2 F t or the object be in contact with the reflector, when/' will equal /, and the image and object be of the same size. 69. When the reflector is convex, r is negative, the convex reflector; principal focal distance F t Equation (59), is positive, and Equation (60) becomes F t , ft ox Equation / ~~F~ V 00 ; applicable; 1 TT COS & and making 4 = 0, /' = J . , . . (64). Equation for f ' discussion; This value of f is always positive, greater than F# and Eelations less than 2 F fl for all values of/, between 2 F t and in- between the finity, or for any position of the object from the surface J of the reflector to a point infinitely distant in front. In the latter position, f is equal to F# or the image is in the principal focus. It follows also, that the image, which will always be virtual for real objects, will be elliptical, erect, and smaller than the object. 70. If we make f positive, greater than F t and less same for virtual than 2 F t , the object will be virtual ; the image real, objed * erect, and greater than the object. 232 NATURAL PHILOSOPHY. The eye; OF THE EYE AND OF VISION. 71. The eye is a collection of refractive media which, concentrate the waves of light proceeding from every point of an external object, on a tissue of delicate nerves, called the retina, there forming an image, from which, by some process unknown, our perception of the object arises. These media are contained in a globular en- Four coatings velope composed of four coatings, two of which, very refractive media; unequal in extent, make up the external enclosure of the eye, the others serving as -lining to the larger of these two. Fig. 42. Graphic representation of the eye ; The cornea ; Shape of the eye; The shape of the eye is spherical except immediately in front, where it projects beyond the spherical form, as indicated at d e d", which represents a section of the human eye through the axis by a horizontal plane. This- part is called the cornea, and is in shape a segment of an ellipsoid of revolution about its transverse axis which coincides with the axis of the eye, and which has to the conjugate axis, the ratio of 1,3. It is a strong, horny, and delicately transparent coat. Immediately behind the cornea, and in contact with ELEMENTS OF OPTICS. 233 it, is the first refractive medium, called the aqueous Aqueoos humour, which is found to consist of nearly pure wa- ter, holding a little muriate of soda and gelatine in solution, with a very slight quantity of albumen. Its refractive index is found to be very nearly the same as that of water, viz. : 1,336, and parallel rays having the direction of the axis of the eye will, in consequence of the figure of the cornea, after deviation at the surface of this humour, converge accurately to a single point. At the posterior surface of the chamber A, in con- tact with the aqueous humour, is the iris, g g, Iris ; which is a circular opaque diaphragm, consisting of muscular fibres by whose contraction or expansion an aperture in the centre, called the pupil, is diminished pn P n ' or increased according to the supply of light. The ob- ject of the pupil seems to be, to moderate the illumi- nation of the image on the retina. The iris is seen through the cornea, and gives the eye its color. In a small transparent bag or capsule, immediately behind the iris and in contact with it, closing up the pupil, and thereby completing the chamber of the aque- ous, lies the crystalline humour, B / it is a double con- Crystalline vex lens of unequal curvature, that of the anterior sur- bmnour: face being least ; its density towards the axis is found to be greater than at the edge, which corrects the spherical aberration that would otherwise exist ; its mean refractive index is 1,384, and it contains a much greater portion of albumen and gelatine than the other humours. The posterior chamber C, of the eye, is filled with the vitreous humour, whose composition and specific vitreous gravity differ but little from those of the aqueous. Its re- humour; fractive index is 1,339. At the final focus for parallel rays deviated by these hu- mours, and constituting the posterior surface of the cham- ber C, is the retina, hhh, which is a net-work of nerves of exceeding delicacy, all proceeding from one great Q tionerw . branch 0, called the optic nerve, that enters the eye 234 NATURAL PHILOSOPHY. Fig. 42. Graphic representation of the eye; Choroid coat : Sclerotic coat ; Inverted images formed on the retina : obliquely on the side of the axis towards the nose. The retina lines the whole of the chamber C, as far as ii^ where the capsule of the crystalline commences. Just behind the retina is the cTioroid coat, Jc &, cov- ered with a very black velvety pigment, upon which the nerves of the retina rest. The office of this pig- ment appears to be to absorb the light which enters the eye as soon as it has excited the retina, thus prevent- ing internal reflexion and consequent confusion of vision. The next and last in order is the sclerotic coat, which is a thick, tough envelope d d f d", uniting with the cor- nea at d d", and constituting what is called the white of the eye. It is to this coating that the muscles are attached which give motion to the whole body of the eye. From the description of the eye, and what is said in arti- cle (62), it is obvious that inverted images of external objects are formed on the retina. This may easily be seen by removing the posterior coating of the eye of any re- cently killed animal and exposing the retina and cho- roid coating from behind. The distinctness of these im- ages, and consequently of our perceptions of the objects from which they arise, must depend upon the distance ELEMENTS OF OPTICS. 235 of the retina from the crystalline lens. The habitual Habitual position position of the retina, in a perfect eye, is nearly at the focus for parallel rays deviated by all the humours, be- cause the diameter of the pupil is so small compared with the distance of objects at which we ordinarily look, that the rays constituting each of the pencils employed in the formation of the internal images may be regarded as parallel. But we see objects distinctly at the distance of a few inches, and as the focal length of a system of lenses, such as those of the eye, Equation (25), increases with the diminution of the distance of the radiant or object, it is certain that the eye must possess the power E e ossessesthe of self-adjustment, by which either the retina may be power of made to recede from the crystalline humour and the 6elf ' ad J ust eye lengthen in the direction of the axis, or the curva- ture of the lenses themselves altered so as to give, greater convergency to the rays. The precise mode of this ad- justment does not seem to be understood. There is a limit, however, with regard to distance, within which vision becomes indistinct ; this limit is usually set down at six inches, though it varies with different eyes. The Limlt of distinct limit here referred to is an immediate consequence of vision; the relation between the focal distances expressed in Equation (25), for when the radiant or object is brought within a few inches, the corresponding conjugate or im- agb is thrown behind the point to which the retina may be brought by the adjusting power of the eye. With age the cornea loses a portion of its convexity, the power of the eye is, in consequence, diminished, and distinct images are no longer formed on the retina, the Long sightedneM ' IT -,-> and its remedy ; rays tending to a focus behind it. Persons possessing such eyes are said to be long sighted, because they see objects better at a distance ; and this defect is remedied by convex glasses, which restore the lost power, and with it, distinct vision. The opposite defect arising from too great convexity in the cornea is also very common, particularly in young persons. The power of the eye being too great, the 236 NATURAL PHILOSOPHY. Images on the retina are iBverted: But objects appear erect shortsightedness image is formed in the vitreous humour in front of the ae 7 ' retina, and the remedy is in the use of concave glasses. Cases are said to have occurred, however, in which the prominence of the cornea was so great as to render the convenient application of this remedy impossible, and relief was found in the removal of the crystalline lens, a process common in cases of cataract, where the crys- talline loses its transparency and obstructs the free pas- sage of light to the retina. The fact that inverted images are formed upon the retina, and we, nevertheless, see objects erect, has given rise to a good deal of discussion. "Without attempting to go behind the retina to ascertain what passes there, it is believed that the solution of the difficulty is found in this simple statement, viz. : that ive look at the object, not at the image. This supposes that every point in an image on the retina, produces, without reference to its neighboring points, the sensation of the existence of the corresponding point in the object, the position of which the mind locates somewhere in the axis of the pencil of rays of which this point is the vertex ; all the axes cross at the optical centre of the eye, which is just within the pupil, and although the lowest point of an object will, in consequence, agitate by its waves the highest Explanation of point of the retina affected, and the highest point of the object the lowest of the retina, yet the sensations be- ing referred back along the axes, the points will appear in their true positions and the object to which they belong erect. In short, instead of the mind contemplat- ing the relative positions of the points in the image, the image is the exciting cause that brings the mind to the contemplation of the points in the object. Base of the optic It may be proper to remark here, that the base of to ^ i ve h 1 t nsensible the optic nerve, where it enters the eye, is totally insen- sible to the stimulus of light, and the reason assigned for this is, that at this point the nerve is not yet divided into those very minute fibres which are capable of being affected by this delicate agent. ELEMENTS OF OPTICS. 237 72. All other things being equal, the apparent magnitude Apparent magni an object ii by its image. of an object is determined by the extent of retina covered determined If, therefore, Ii R f be a section of the retina, by a plane through the optical centre (7, of the eye, and A B = I, al> = X, the linear dimensions of an object and its image in the same plane, we shall have, from the similar triangles CA B and C a 5, \=-Ca. L denoting by s, the distance of the object. And for any other object whose linear dimension is V and distance s y , calling the corresponding dimension of the image \, Dimension of image of an object on the retina; X =- Ca.L, and since C a is constant, or very nearly so, Same for a second Proportion that is, the apparent linear dimensions of objects are as their real dimensions directly, and distances from the eye inversely. But _, may be taken as the measure of the angle B C A = I G a, which is called the visual an- Rule flret ; 238 Euie second.- Example for illustration , NATURAL PHILOSOPHY. gle, and hence the apparent linear magnitudes of objects are said to ~be directly proportional to their visual angles. Small and large objects may, therefore, be made to ap- pear of equal dimensions by a proper adjustment of their distances from the eye. For example, if X = X y? we have I' or, *,'. Numerical data; and if I = 1000 feet, s = 20000, and I' = 0,1 of a foot, or little more than an inch, Result the distance of the small object at which its apparent magnitude will be as great as that of an object ten thou- sand times larger, at the distance of 20000 feet. Microscopes ; Explanatory remarks ; MICROSCOPES AND TELESCOPES. 73. From what has just been said, it would appear that there is no limit beyond which an object may not be magnified by diminishing its distance from the optical centre of the eye. But when an object passes within the limit of distinct vision, what is gained in its apparent in- crease of size, is lost in the confusion with which it is seen. If, however, while the object is too near to be distinctly visible, some refractive medium be interposed to assist the eye in bending the 'rays to foci upon its retina, distinct vision will be restored, and the magnifying process may ELEMENTS OF OPTICS. 239 be continued. Such a medium is called a single micros- cope, and usually consists of a lens, whose principal focal mi distance is negative and numerically less than the limit of distinct vision. To illustrate Fi & 44 - the operation of this instrument, let MN be a section of a dou- ble convex lens whose optical centre is G / Q P an object in front and at a distance from C equal to the principal focal distance of the lens; E the optical centre of the eye, at any distance behind the lens. The rays Q and P C, containing the optical centre, will undergo no deviation, and all the rays proceeding from the points Q and P, will be respectively parallel to these rays after passing the lens ; some rays, as JV E Ex lanation of from Q, and J/^from P, will pass through the optical the figure; centre of the eye, and will belong to two beams of light whose boundaries will be determined by the pupil, and whose foci will be at q and p on the retina, giving the visual angle, MEN = P C Q; Eelation from same; or the apparent magnitude of the object P Q, the same as if the optical centre of the eye were at that of the lens. And this will always be the case when an object occupies the principal focus of a lens whatever the dis- tance of the eye, provided the latter be within the field of the rays. Without the lens, the visual angle is QEP < P C Q; Effect of the hence, the apparent magnitude of the object will be in- 8il creased by the lens. Calling X and X y , the apparent magnitudes of the ob- ject as seen with, and without the lens, we shall have, 240 NATURAL PHILOSOPHY. Magnitudes of PQ PQ 1 1 ill"! ^'''OQ'-TQ-C-Q'-WQ- lens compared ; or, by using the notation employed in Equation (33), and calling E Q, the limit of distinct vision, unity, // _ (66) when the lens As long as F tl < 1, or the principal focal length of l sa the lens is less than the limit of distinct vision, the ap- parent size of the object will be increased, and the lens may be used as a single microscope. We can now understand what is meant by the power of a lens or combination of lenses, referred to at the close What is meant -t by the of article (39). , which was there said to measure magnifying F fl -^ tbe P ower of a lens > we see f rom Equation (66), expresses the apparent magnitude of an object compared to that at the limit of distinct vision, taken as unity ; and what- ever has been demonstrated of the powers of lenses gen- erally, is true of magnifying powers. Thus, in Equation (31), we have the magnifying power of any combination of lenses equal to the algebraic sum of the magnifying powers taken separately. Should any of the individuals of the combination be concave, they will enter with signs contrary to those of the opposite curvature. The power of a single microscope is, Equation (66), e< l ua ^ to the limit of distinct vision divided ~by its princi- pal focal distance, and the numerical value of the power will be greater as the refractive index and curvature are grwter. 74. To obtain a general expression for the visual an- gle under which the image of an object formed by a lens, and having any position in reference to the eve. ELEMENTS OF OPTICS 241 Fig. 45. is seen, let Q P, be an object in front of a concave lens. From P, draw through the optical centre E, the line PE; from P, draw the extreme ray P M, and from M draw M S) making with P 2f produced the angle SMT equal to the power of the lens ; then will, 47, MS be the corresponding deviated ray, and its intersection ^, with the ray P E, through the optical centre, will be a point in the image; from p, draw p q, parallel to P Q, till it is cut by the ray Q E, through the other extreme of the object and optical centre; p q will be the image. Let (9, be the optical centre of the eye ; then denoting the visual angle p q by JL, we have, To find the visual angle under which an image formed by : Oq~ Eq-OE' and representing the distances Q E by /*, Eq\yj /", and EO by d, we find, Value of visual angle ; =QP.-; Eq-EO=f"-d; and hence A f Same in other terms; and denoting the visual angle P EQ by -4/ 5 A /" j_ A'-f'-d-, d 16 Ratio of visual (67) angles with and without the lens, 242 NATURAL PHILOSOPHY Sign of this ratio depends upon; Eye placed so as toseetheimage formed by a concave lens: The angles A and A' will have contrary signs when on opposite sides of the axis of the deviating surface. The relation expressed by this equation answers to a concave lens in which f" will, Equation (27), be posi- tive for a real object. Moreover, d is positive, the eye being on the same side of the lens as the object ; but that seen the eye must - ,T ., be on the opposite side, in which case d will be negative, and the Equation becomes Equation corresponding to this case; A_ A 1 (68) Real image formed by a convex lens ; whence we conclude that objects will always appear dimin- ished when seen through concave lenses. If the lens be convex and the object be situa- ted beyond its principal focus /" will be nega- tive, and Equa- tion (68) becomes Equation , corresponding; A_ A' (69) Distinct vision supposed possible an d supposing distinct vision possible for all positions for all positions _ , of the eye. * the eye, it appears, ELEMENTS OF OPTICS. 243 1st. That when the object is at a distance from the conclusion em lens greater than that of the principal focus, in which case there will be a real image, the lens 'will make no difference in the apparent magnitude of the object, provided the eye is situated at a distance from the lens equal to twice that of the image. 2d. At all positions for the eye between this limit second, and the image, the apparent magnitude of the object is increased by the lens. 3d. At a position half way between this limit and Third, the lens, the apparent magnitude of the object would be infinite. 4th. The eye being placed at a distance greater than Fonrth . twice that of the image, the apparent magnitude of the object will be diminished by the lens. 5th. When the distance of the object from the lens is equal to that of the principal focus, in which case f" becomes infinite, the apparent magnitude will be the same as though the eye were situated at the optical centre of the lens, no matter what its actual distance behind the lens. 75. The visual angle under which the image formed TO find the visual by a reflector is seen, is found in the same way. Thus, let angle under J '. J which an image P Q be an object in formed by a front Of a Convex re- *V & reflector i8 seen; fleeter M N. From the extreme point P, of the object, and through the optical centre O, draw the ray P C\ from the same point jP, draw to the extreme of the reflector the ray P M, and from M draw MS, making with P M, the angle P MS equal to the power of the Explanation; reflector; Jf /Swill, 53, be the deviated ray, and its intersection with P <7, will give the image of the point 244: NATURAL PHILOSOPHY. construction of P. Draw p ^, paral- the Image formed ii f p ft .jii . by a reflector; ^ V> l intersected by Q (7, drawn through the op- posite extreme of the object and optical centre, and we have the image. Let the optical centre of the eye be at ; then, de- denoting the visual angle p q by A, will Value of visual angle with the reflector; _ ' Oq C Q Eatio of visual angles with and without the reflector. and representing, as before, C Q, Cfr and C 0, by /, f\ and d, respectively, and the visual angle __*iby^i', Q we have A_ A' f - d-f We shall not stop to discuss this Equation. (70) In practice distinct vision is not possible for all positions of the eye; 76. We have supposed, in the preceding discussion, distinct vision to be possible for all positions of the eye ; but this we know depends upon the state of convergence or divergence of the rays. If, however, the image, when one is formed, instead of being seen by the naked eye, be viewed by the aid of another lens, so placed that the rays composing each pencil proceeding from the object shall, after the second deviation, be parallel, or within such limits of vergency that the eye can command them, the object will always be seen distinctly, And the image is and either larger or smaller than it would appear to ^^ e a ^ e e ^ ed the unassisted eye, depending upon the magnitude of lens. the image, and the power of the lens used to view it. ELEMENTS OF OPTICS. 245 As most eyes see distinctly with plane waves or parallel Position of the rays, this second lens is usually so placed that the image eye lena> shall occupy its principal focus ; and where this is the case, we have seen that the apparent magnitude of the image will be the same as though the eye were at its optical centre. Fig. 49. Refracting telescope ; N The image p q, being in the principal focus of the lens m n, draw from the point p, the line construction for p 0, to the optical centre of this lens; the rays fi^m^^tfaLe^to will, 73, be deviated parallel to this line, and the line the retioa - 0' K, through the optical centre 0' of the eye, paral- rel to p 0, will determine by its intersection K, with the retina, the place upon that membrane of the image of the point P. Calling the principal focal distance of this lens, (F^ ; #, in Equation (67), will equal /" + (j^), and that equa- tion will become, by first making f " and d negative and then replacing d by this value, A f" .General equation 7-f "= ' ( ^ 1) made applicable " ( & u ) to this telescope ; and if the object P Q, be so distant that the rays com- posing each of the small pencils whose common base is M N, may be regarded as parallel, f" becomes F n , and we have, Tj* Eatio of visual ...... (72) angles forparallel rays; 246 NATURAL PHILOSOPHY. Refracting telescope; Fig. N Compound microscope; Fiuld and eye lenses; Bale for magnifying power. Objects appear Inverted. Galilean telescope ; Construction of image on the retina ; Equation (71) exhibits the principles of the com- pound refracting microscope, and refracting telescope; and Equation (72), which is a particular case of (71), those of the astronomical refracting telescope. The lens M N, next the object, is called the object or .field lens, and m n, the eye lens. The magnifying power in the first case, is equal to the distance of the image from the field lens divided l>y the principal focal length of the eye lens / and in the second, to the principal focal length of the field lens, divided by that of the eye lens. The ratio of A to A!, being negative, shows that ob- jects appear inverted through these instruments, the vis- ual angles of corresponding parts of the object and im- age being on opposite sides of the axis. 77. If instead of a convex, a concave lens be used for the eye lens, the combination will be of the form used by GALILEO, who invented this instrument in 1609. In this construction, the eye lens is placed in front of th# image at a distance equal to that of its principal focus, E(O that the rays composing each pencil shall emerge from it parallel. Draw through the point p, where the image of P would be formed, the line p 0, to the optical centre of the eye lens, and through the optical centre 0' of the eye, the line 0' TTparallel to p 0, its intersection K, with the retina will give the image of the point JP on the back part of the eye. ELEMENTS OF OPTICS. 24/f Fig. 50. Galilean telescope ; The rule for finding tlie magnifying power of this Magnifying instrument is the same as in the former case; for we powerfound analytically; have, d=f"-(F H ); which in Equation (67), after makingy", and d, negative, gives A -f" _ A ; (7 3} Katioof visual A' f TJ* \* \ lv y A (J? ) angles; / and for parallel rays, ^ A! The second member being positive, shows that objects objects appear seen through the Galilean telescope appear erect. 78. If we divide both numerator and denominator of Equation (72), by F lt . (F,,), it becomes, A (F t/ ) Magnifying 77 ~~~ -< > power in terms " of the powers of F lt the lenses; and denoting by Z, the power of the field, and by Z, that of the eye lens, we have _ = __ JL. (75) Eatfo of visual A' L angles; 248 NATURAL PHILOSOPHY. that is, the magnifying power of the astronomical tele- scope is equal to the quotient arising from dividing the power of the eye lens ly that of the field lens. Fig. 51. Geometrical illustration of the field of view ; Q Q' deneral explanation ; Field of view ; Determined by construction ; 79. If E, be the optical centre of the field, and O that of the eye lens of an astronomical telescope, the line E 0) passing through the points E and 0, is called the axis of the instrument. Let Q' P' be any object whose centre is in this axis, and < p' its image. !N"ow, in order that all points in the object may appear equally bright, it is obvious from the figure, that the lens must be large enough to embrace as many rays from the points P' and (>', as from the intermediate points. It is not so in the figure ; a portion, if not all the rays from those points will be excluded from the eye, and the object, in consequence, appear less luminous about the exterior than towards the centre, the brightness increasing to a certain boundary, within which all points will appear equally bright. The angle subtended at the centre of the field lens, by the greatest line that can be drawn within this boundary, is called the field of mew. To find this angle, draw m N and 31 n to the opposite extremes of the lenses, intersecting the image in^> and an( i w ^h their planes at right an- moved tangent to gles to its axis ; an image of the common centre of the disks will be formed on the retina of an eye viewing them through the lens, at m". If one of the disks be moved to the position m', so that its circumference be tangent to that of the other, the image of its centre will be at m" f , determined by drawing from 0, the optical centre of the eye, a line parallel to that joining ELEMENTS O,F OPTICS 259 the lens : the optical centre of the lens and the centre of the Takeone, Fig. ca Graphic representation; behind which is a concave reflector E. The rays pro- ELEMENTS OF OPTICS. 265 eeeding from any point in a figure, painted with some P tical transparent pigment upon the glass slide and strongly illuminated by the lens A^ upon which the direct light explained; from the lamp, as^well as that from the reflector E, is concentrated, will be brought to a focus by the lens j??, on a screen MJV, placed at a distance in front of the instrument ; here the light being reflected will pro- ceed as from a new radiant, and a magnified image of the figure will thus appear upon the screen. Should, the screen be partially transparent, a portion of the light will be transmitted, and the image will be visible to an observer behind it. The linear dimensions of the object or figure, will Eelationbetween , ,, . , . the dimensions of be to those ol the image, as their respective distances the object and from the lens B ; if, therefore, the lens B be mounted image ; in a tube which admits of a free motion in that con- taining the lens A, its distance from the figure may be varied at pleasure, and the image on the screen made larger or smaller, the instrument, at the same time, be- ing so moved as to keep the screen in the conjugate corresponding to the focus occupied by the glass sli&e. The instrument with an arrangement by which this can be accomplished, is called the phantasmagoria. In or- Phantasmagoria, der, however, that the deception may be complete, there must be some device to regulate the light, so that the illumination of the image may be increased with its increase of size, not diminished, as it would be without such contrivance. SOLAR MICROSCOPE. 90. This is the same as the magic lantern, except Solar that the light of the sun is used instead of that from microsc P e ' 266 NATURAL PHILOSOPHY. Solar microscope ; Essential parts and manner of using. a lamp. D E, is a long reflector on the outside of a window shutter, in which there is a hole occupied by the tube containing the lenses. Fig. 64. The object to be exhibited is placed near the focus of the illuminating lens A, so as to be perfectly en- lightened and not burnt, which would be the case were it at the focus. CHROMATICS pitch and harmony in sound ; Explanatory remarks ; 91. Chromatics is a name given to that branch of optics which treats of Color. Color is to light what pitch and harmony are to sound. We have seen, in Acoustics, that by the principle of the coexistence and superposition of small motions, any number of sonorous waves may exist at the same time and place, and pro- duce, through the organs of hearing, an impression dif- ferent from that produced by either of the waves when acting singly. The united tones proceeding from the various voices of a full choir of music, for example, im- press the ear very differently from the insulated note of the acute treble, the medium tenor, or the full, deep-toned bass ; and as each voice is partially or wholly suppressed in succession from a full strain of concordant sounds, while others are reinforced, the mind ELEMENTS OF OPTICS. 267 marks the change, and attributes to it a distinct and Analogy between specific character. So it is with the luminous waves which act upon the luminous waves; organs of sight. These come to us from the sun, and other self-luminous bodies, of every variety of length ca- pable of affecting the eye; they coexist and are super- posed upon the retina, and by their united influence give us the impression of white light ; and when one white light after another of these waves is enfeebled, while others produced; are strengthened, each new combination gives us a dif- ferent impression, and each impression we call a color. The longest waves capable of affecting the eye corres- pond to red, and the shortest to violet or lavender grey. But how are individual waves either suppressed or separated from the group which produce the sensation Principles of white light? The answer is, by the principles of in- terference and of unequal ref Tangibility. , produce colors. COLOR BY INTERFERENCE. Colors of Gratings. Fig. 65, Colors of gratings; 92. Recalling the expla- nation of 7, let J/^be a wave front proceeding from a source 0. Assume any point O r , in front of the wave, and draw the straight line O r 0. Take the dis- tance A B equal to half the length X r , of the longest, and A B' equal to half the length X r , of the shortest wave capable of affecting the organs of sight ; and make B C = CD A B = \. construction of With O r as a centre, and the radii O r B', O r B, O r 268 NATURAL PHILOSOPHY. Colors of gratings; Fig. 65. Construction of figure; Even numbered portions of main wave opposed to ^ Consequence of stopping the even portions; Effect of longest waves most increased at O r ; increased at O O r D, &c., describe arcs cutting the wave front in J', 5, c, d, &c. ; then will the portions A 5, 5 c, c d, &c., in the immediate vicinity of A) partially, and those remote from the same point, wholly, interfere, 7, and neutralize each other's ef- fects at O r : for, at this point the secondary waves from the successive points of the portion A 1), beginning at A, will be opposed to those from the corresponding points in the portion 5 c, beginning at 5, being in opposite phases ; and it is plain that if the several portions be numbered in order from A, that those distinguished by the even will be opposed to those designated by the odd numbers, the odd portions tending to displace the molecule at O r in one direction, and the even ones in the opposite direction. Now, conceive the even portions 5 r, d b r , on one end of A 5, will interfere and neutralize those from an equal portion A a, at the other end, so that the effect of the longest waves at O r , will be increased in a much greater proportion than that of the shortest. From the construction O r c O r A = \. Take a point O v , such that O v c O v A = X r , then will the point O v , for the same reasons, receive the greatest possible dis- turbance from the action of the shortest waves which are here in the same phase, while the effect of the longest waves, at this point, will be less than at O r , being no longer in the same phase. The effect of the waves whose ELEMENTS OF OPTICS. 269 lengths are intermediate between \ and \, will have their preponderance upon jnolecules between these two points, and the space O r O v , should exhibit correspond- ing effects. And this is found by experiment to be the case. For when a grating is formed by fine parallel Effectsofftirrowl wires, or by a series of fine furrows cut in the face of a piece of well polished glass, and held iii front of any luminous source, there will be formed upon a screen placed behind, a series of richly colored fringes, sepa- rated by dark intervals, and arranged along a line per- pendicular to .the furrows. The furrows intercept the light, while the intermediate spaces between permit it to pass ; the former correspond to the even and the lat- ter to the odd portions of the luminous wave referred to above, and form, as it were, a series of parallel linear Illustration , radiants. The molecules O n 2r , &c., where the longest waves prevail, exhibit red, and those at O v , 2V , &c., Explanation of where the shortest preponderate, violet or lavender grey, the color8 ' the molecules between exhibiting orange, yellow, green, blue and indigo, in the order named, beginning at the red. The line IK Y, being drawn from the luminous source to the middle of an opaque portion of the grating, the first fringe on either side of this line is formed by secondary waves whose radii differ by X, the second by 2 X, the third by 3 X, and so on ; that is, every fringe is 270 NATURAL PHILOSOPHY. Collateral fringe Effects of light transmitted through furrowed glass ; Effects of light reflected from the same ; Fraunhofer and Barton's experiments. formed by the conspiring of secondary waves whose radii differ by some even multiple of JX, while the dark spaces between are produced by the opposition of waves of which the radii differ by some odd multiple of |X. Fig. 67. Fig. 68. 93. If the furrowed glass be interposed between the eye at 6>, and any lu- minous source L, say a small hole in a window shutter, the latter will ap- pear flanked on either side by similar fringes with in- termediate dark spaces between also arranged on a right line perpendicular to the direction of the fur- rows, the red appearing on the outside, the violet on the inside, with the other colours in the order just named between. And to an eye at 0', so placed as to receive the light reflect- ed from the ridges of plane glass between the furrows, the whole furrowed space will appear covered by the most beautiful irised hues which change with every change of position of the eye. By means of a fine diamond point, FEAUKIIOFEE suc- ceeded in forming a ruled surface of glass in which the striae were actually invisible under the most powerful microscope, the interval of the furrows being only ^ Jo-o- of an inch. In some furrowed surfaces produced by Mr. BAKTON, the lines are so close that 10000 of them would occupy only the space of an inch in breadth. The light reflected from surfaces so minutely divided, exhibits the purest colors of which we have any knowledge. Simi- lar appearances are exhibited when light is reflected ELEMENTS OF OPTICS. 271 from metallic surfaces which have been polished by a coarse powder, and from surfaces of glass over which the finger is passed after being moistened by the breath. The beautiful colors of mother of pearl are natural Effects of the instances of the same phenomena. This substance is ^ f n ^ar" "* composed of a vast number of thin layers, which are gradually and successively deposited within the shell of the oyster, each layer taking the form of the preceding. When it is wrought, therefore, the natural 'joints are cut through in a great number of sinuous lines, and the result- ing surface, however highly polished, is covered by an immense number of undulating ridges formed by the out- cropping edges of the layers. These striae may be ob- served by the aid of a powerful microscope, although they are so close that 5000 of them occupy but a single inch. That they are the cause of the brilliant colors displayed Experiment vf by this substance has been placed beyond doubt by Sir DAVID BREWSTER, who received from an impression of the surface of pearl on soft wax the same display of colors as from the pearl itself. 9i. Knowing the space AC^ occupied by a single TO find the furrow and one of its adjacent transparent intervals, it will be easy to find the lengths of the waves which cor- respond to the different colors. For this purpose we remark, 1. That the first colored fringe F^ , seen through the glass on either side of the luminous source L is formed by secondary waves whose radii differ by X, the second F 2 by 2 X, the third by 3 X and so on, and are called respec- Eemarklst. Fig. 69. 272 NATURAL PHILOSOPHY. Eeinark 2 i i ~\r ~\T~ nn_ particular fringe fro m the central line X Y. They from the central depend upon the sum AG=s, of the width of one opaque and one adjacent transparent interval, and upon the distance X Y, of the screen from the grating. The place 0, of the fringe of any order, say' the ?ith, is determined by the condition that the difference of its distances A and C 0, from A and (7, is an integer multiple of the length X, of the wave of the particular color con- sidered. Now, drawing the lines A A' and C C ', paral- lel to X I" and denoting the distance X Y by d, and Y ' 0, by a?, we find JCJ! Y C' O JV Equations from A O = = d _ ^ = _ Difference between these equations ; the distance d, being very great in comparison with x and 5. Hence, 4 O But this difference is equal to n X ; that is, Equal to n X ; S . X Value for distance of any fringe from central line. whence x = n . X . d (88) ELEMENTS OF OPTICS. 277 From which it appears that that the fringes will be crowd- conditions that ed together more and more in proportion as d decreases ^^^*^ relatively to 5, or 5 increases relatively to d a result towards the confirmed by experience, for when the screen is either ce made to approach the grating, or the furrows are in- creased in size, the fringes will be observed to contract and crowd in upon the centre Y, till they become so narrow as not to be perceptible. 97. Again, let X r and \, denote the lengths of the waves which give red and violet colors respectively ; then will Equation (88) give n . d . X r Distances of the ^r ) red and violet colors of the nth fringe from the centre In which, because X r is greater than X c , a? r , which de- notes the distance from Y to the red color of the nth fringe, will be greater than a? B , which represents the dis- tance of the corresponding violet color from the same point. Subtracting the second from the first, we get The colors separate from each other ; (89) From which we see that the different colors will sep- arate more and more as the fringe to which they belong An d the dark recedes from the centre Y. The black intervals will, interval8 fina11 ^ disappear. therefore, be encroached upon, and at no great distance from Y will disappear. To find the order of the last insulated fringe, denote by a? M+1 and x n the distances of the (n+l)th and ^th fringes of the]ant from Y\ and by \ the length of the wave for red, then insulated fringe; will Equation (88) give 278 NATURAL PHILOSOPHY. Notation and equations ; n . \ . d s whence, taking the difference, we obtain for the inter- val between the reds of two consecutive fringes, Interval between the reds of two consecutive fringes ; \.d and placing the second member of this Equation and that of Equation (9) equal, ^ we find - (X ,_X^ - f ' s - ' ' ~ s whence Order of the last insulated fringe. n = (90) Experimental illustration. Experiment performed in vacuum ; That is to say, the order of the last insulated fringe is denoted by the number of units in the quotient arising from dividing the length of the red wave by the dif- ference of the lengths of the red and violet waves. This result is beautifully illustrated by interposing between the screen and grating some medium which will arrest all the waves but those which correspond to a particular color. When this is done, the fringes will be greatly multiplied in number beyond that of the nth order determined by Equation (90). 98. Thus far the waves have been supposed to pro- ceed, after passing the grating, in the atmosphere. But when the experiment is performed in vacuum, with the same grating and same position of the screen, the fringes are found to dilate and separate from each other ; when per formed in a medium of greater density than the air, as ELEMENTS OF OPTICS. 279 in water or glass, the fringes are reduced in width and Experiments crowded towards the centre ; and what is remarkable, adenseTmedinm and important to observe, this latter effect is found, by thanair ; careful experiments, to be exactly proportional to the in- dex of refraction of the medium as referred to that of atmospheric air. JSTow, referring to Equation (88), it is easy to see that this change in the position and width of the fringes Causeofthe can only arise from a change in X, which denotes the position and length of the waves, since s and d are, by the conditions width of the of the experiment, constant ; and from the relations of " x and X, in that Equation, it follows that the length of luminous waves of the same color are shorter in propor- tion as the indexes of refraction of the media in which they exist are greater. But, Equation (2), these indexes Lengths of wave* vary inversely as the velocities of wave propagation, and in different hence the lengths of the waves are .directly proportional to the velocities with which they are transmitted through different media. The cause by which the lengths of the waves are thus altered in the direct proportion to their Princ1 p ]e of wav velocities, is called the principle of wave acceleration retardation. and retardation. 99. Returning to the experiment in air ; if a very thin plate of glass be interposed in front of one of the interposing a grate openings, and parallel to the plane of the grating, P lateof g ]ass *e - .-,11 *f under different the whole system 01 fringes will be shifted towards conditions, thu side of the interposed glass. If an exactly similar plate be placed in front of the other opening, and parallel to the first plate, the fringes will be re- stored to their original position. If one of the plates be slightly inclined, so as to cause the waves passing through it to traverse a greater thickness, the fringes will all move towards that side, and by gradually in- creasing the inclination, they will pass entirely out of eight. Taking plates of any other medium, possessing a greater refractive index than glass, and of the same 280 NATURAL PHILOSOPHY. Effect of interposing a plate of any medium. Effect on the lengths of the rays; This last effect investigated ; Illustration; Explanation ; Fig. 72. C thickness as before, it is found that the effects just no- ticed will be increased, and in the direct ratio of the refractive indexes of the media. In the shifting of the fringes, it is evident that the lengths of the rays which correspond to the central one are made unequal, and that the differences as to lengths existing among the rays which appertain to the other fringes, are not the same as before the interposition of the medium. We will now investigate this change. For this purpose, let the waves from both openings pass through a prism of any medium, as glass, hav- ing a very small refracting angle, i, the first face being held pa- rallel to the plane of the grating. The thickness of the prism tra- versed by two interfering waves will be different ; call this diffe- rence, which is r n in the figure, d. Draw nn', parallel to KL\ with 0, as a centre and Or as a radius, describe the arc rr'. It i& obvious that the number of waves in the length A n + r will be equal to the number in the length Cn'+0r f , sinco the circumstances are the same in both routes ; the only difference, if there be any, must lie in the paths n r and n f r'. Since the angle made by the rays A and C 0, is very small, these rays will enter the first sur- face under very small angles of incidence, and both be- ing refracted towards the perpendicular, their direction through the prism will be nearly normal to that surface ; hence, denoting by &, the distance r n\ we have Equation from figure; d = 5 . sin i ; but under the above supposition, the angle of incidence at the second surface will be equal to ; and denoting ELEMENTS OF OPTICS. 281 the corresponding angle of refraction by 9, we also get Sin

Uglit^ and were supposed to arise from some peculiar action exerted by the edges of bodies on the rays as they passed near them. Effect of If the refractive index of the medium in which the increasing the experiments are performed be increased, the phenomena of the medium, indicate a diminution in the lengths of the waves in the same ratio. COLORS OF THIN PLATES. AH media 104. Transparent, and indeed all media, when re- exhibit colors duced to very thin films, are found to exhibit colors when reduced to _ thin films which vary with the thickness of the film. These are called the colors of thin plates, and the easiest way to exhibit them is by means of a soap bubble blown from the end of a quill or the bowl of a common smoking *See Appendix No. 1. ELEMENTS OF OPTICS. 285 exhibition of the pipe. As the bubble increases in diameter, and the Familiar , . , T exhibitio VL u fluid envelope is reduced m thickness at the top by colorg of tbin gradual subsidence toward the bottom, many colored plates; arid concentric rings will be seen around the point of least thickness. At this, point, the color will be found to change, first appearing white, then passing through blue to perfect blackness, the rings the while dilating till the bubble is destroyed. ' The same is true of any other medium, whether gase- ous, liquid, or solid. These different colors being exhibited upon the same when the plate plate of variable thickness, no single color can be iden- tified with its chemical composition. When of uniform thickness, a single color only will be seen, and this will change as the thickness of the plate changes. A thin plate is very con- veniently formed of air ; and Fig. 73. for this purpose, let A. B, be Thin p i a tecfair a plano-convex, and CD a plano-concave lens, placed one upon the other, as represented in the figure. When this arrangement is viewed on either of the plane faces by reflected light, colors will be seen in the form of con- Appearances by centric circles about the point of contact, which, should reflectedlight; the pressure be sufficient, will be totally black. If view- ed by transmitted light, rings whose colors when united with those of the first, form white light, and which By transmitted colors are, therefore, said to be complementary, will ap- light; pear about the central spot, which will now be per- fectly white. With waves of a single length, as yellow, these rings are alternately bright and dark, begin- ning with the central spot; and by reflected light, dark and bright. They are broadest and have the great- est diameter in the red, and narrowest with least diam- eter in the violet ; the breadths and diameters in the . . Effects due to other colors being intermediate and varying in magni- wa tude in the order of the spectrum from red to violet. It leu & th ; is by the superposition of these rings, or the waves 286 NATURAL PHILOSOPHY. Newton's scale; which produce them, that the different colors appear in common light. Seventh. These colors, which are of different orders as regards tint, 'constitute what is called Newton's scale ; and by reflected light, occur as follows, beginning with the cen- tral spot. Black, very faint blue, brilliant white, yel- low, orange and red. Dark violet, blue, yellow-green, bright yel- low, crimson and red. Purple, blue, rich green, fine yellow, pink and crimson. Dull blue-green, pale yellow-pink, and red. Pale blue-green, white and pink. Pale blue-green, pale pink. - The same as 6th, very faint. The other or- ders being too faint to be distinguishable. First order ; 19^ order. Second ; 3d order. Third; 3d order. Fourth, &c. ; 4-f7), nrfJar. 5th order. th order. 1th order. Waves which interfere to produce the colors by reflexion ; These colors arise from the in- terference of waves reflected from the first, with those reflected from the second surface of the air plate. Suppose a small beam incident perpendicularly or nearly so, on the first surface MN of the plate, where the thickness is t. A part A will be reflected back, the rest A B, being transmitted, will illustration and ^ raverse tne thickness t. At the second surface, again explanation; a part B C, is reflected, and the reflected portion return- ing through the thickness , will emerge at the first sur- face in the direction G 0, and be superposed on that first reflected at this surface, and these will either conspire and reinforce each other or will interfere and partially or wholly neutralize each other, according to any of the conditions explained in 7, depending upon the differ- ELEMENTS OF OPTICS. 287 ence of route 2 t. Whenever 2 t is equal to any even if 2 1 be an even multiple of \ X, for any color, this color will be increased, mi and when equal to any odd multiple of | X, it will be suppressed. Now, 2 , will vary from a value sensibly nothing to one equal to many times X, for even the long- est waves, in passing outward from the point in which the spherical surfaces are tangent to each other, and if an odd hence the colored fringes and the intermediate dark multlple * rings. But the portion reflected at the second surface will, Transmitted in part, be again reflected at the first, and will traverse ^^ re fl g e h c t ed the thickness t, a third time, and emerge below super- ""s 3 are dark; posed upon the portion first transmitted at the second surface. The difference of route of these portions will also be 2 , so that the effects should be the same on either side of the lenses. Experiment shows, however, that this is not the case, for wherever there is total darkness by reflexion, there is a maximum of bright- ness by transmission. Hence, there must be half a wave lennth subtracted from the route at each internal re- This difference . accounted for ; flexion / the cause of the loss being a change m density and elasticity at the surfaces of contact of the glass and air. This will give for the interfering rays, in case of reflected rings, a difference of route expressed by and for the transmitted, 2 t + X. Fig. 75. To ascertain the value of , at the different rings, call d, the diameter 2 PII, of one of them, as determined by actual measurement; r and / the radii of the surfaces, v and -z/, the cor- responding versed sines of the arcs whose sines P H and P' Il\ are equal to the semi-diameter of the ring in question. Difference of route for reflected rings ; Same for transmitted ringat To find t, at the 'different rings; 288 NATURAL PHILOSOPHY Equation from the figure ; Then, for very small arcs, we have _(4): and Another equation ; Id); Yalne of t; whence d 2 (I IV = v v = ) & \r r 1 same for first In this way NEWTON found the thickness at the brightest part of the first ring nearest the central black spot, to be 0,00000561 of an inch. Pie also found the diameters of Law of variation ,-,-,. v of diameters of the darkest rings to be as the square roots of the even dark and bright num b e rs 0,2,4,6, &c., and those of the brightest as the square roots of the odd numbers 1,3,5,7, &c. The radii of the surfaces being great compared with the diameters of the rings, the value of t at the alternate points of greatest obscurity and illumination are as the natural numbers Law of variation of t, at the dark and bright rings ; Eule. Above results compared with X for yellow ; 0,1 ,2,3 &c. hence, the value of , just found, multiplied by these num- bers, will give the thickness at the different rings. On comparing the value for the thickness at the first bright ring, with the numbers in the table of article (95), it will be found just equal to one-fourth of the interval denoted by X, for the yellow ray, which is the most illuminating of the elements of white light. Taking this value for , we shall have for the difference ELEMENTS OF OPTICS. 289 of route for the interfering rays producing the dark Difference of rings by reflexion^ including the central black spot, rings; ^ "* V 5 * !i AC r\ ) r ' n ' r ' ^^"J these being the even multiples of X, increased by | X for the retardation caused by one internal reflexion. The odd multiples, increased by X, give X, 2 X, 3 X > &C. Same for the bright reflected rings. The transmitted rings will be complementary to those seen by reflexion. The phenomena we have just considered are equally Same produced, whatever may be the medium interposed phenomena between s the glasses, the only difference being in the different media, contraction or expansion of the rings, depending upon the refractive index of the medium. It is found that as the refractive index of the medium increases, the diame- ter .of the rings will decrease, which might have been inferred from article (99). 105. If any one of the rings at a particular color be conceived to be expanded in all directions in the plane of the ring and to retain the same thickness, it is obvious that the plate thus produced would present the same color over its entire surface. If a second plate of the colors of natural same thickness and material be placed behind this one, it bo would act upon the waves transmitted through the first just as the latter did upon the incident waves, and the same would be true of any number of plates, so that a body made up of a series of such plates would present a uniform, distinct, and characteristic color. These con- siderations, in connection with those relating to the color of minute gratings or striae, furnish an explana- tion of the colors of natural bodies. 19 290 NATURAL PHILOSOPHY. Colors of inclined glass plates ; Circumstances attending the deviation of light by such plates ; Emergent waves will generally have travelled routes differing in length ; Two will emerge after having travelled different routes of equal length ; Illustration ; COLORS OF INCLINED GLASS PLATES 106. If a luminous object be viewed through two plates of glass of precisely equal thickness, slightly in- clined to each other, it will be evident that besides the transmitted image, there will be a number of images formed* by the successive reflexions between the glasses. The first or brightest of these is formed by the waves which have all undergone two reflexions and at different pairs of the four surfaces. On entering the first plate they undergo a partial 1'eflexion at every surface they successively encounter, each of the reflected waves un- dergoing a similar series of partial reflections at each surface. Thus it will appear evident that the different portions into which the waves have been separated must go through a length of route differing by the length of the interval between the glasses and the thickness of the glasses, or the different multiples of those which they have respectively traversed. They will, therefore, in general, emerge after traversing routes which differ by considerable quantities. Among these portions, however, there are two which, (if we abstract the very small difference in the in- terval between the glasses at the points where they re- spectively pass,) will have gone through different routes of precisely equal length. These two waves will be, 1st. One which passes di- rectly through the first plate A B, equal to t, and through the interval B C, equal to i, between the plates, is then reflected at (7, in the first surface of the second plate, re- turns along CD, equal to i, and a thickness D E, equal to the first, or t\ Fig. 76. ELEMENTS OF OPTICS. 291 at the first surface it is reflected again and passes the whole system EF + F G + G H, equal to 2 t + i\ or Explanation; upon the whole, it has travelled over 4 + 3 i. 2d. Another portion proceeds directly through the whole A B -f B C+ Cd, equal to 2 t + i, is reflected at d, in the last surface, retraces the distance de -\- ef, equal to t + i, is reflected at the second surface of the first glass and pursuing the course f g + g A, equal to i + , emerges after having, on the whole, passed through 4 t 4- 3 *, or a route exactly equal in length to that of the former, neglecting, as before, the difference in i. It will be seen that out of all the possible combina- NO other waves tions of different successive reflexions, these two are the only ones which will give routes precisely equal ; all the others will differ by quantities amounting to some mul- tiple of t or i. If we now recur to the small difference in the interval i, for the points at which the rays respec- tively pass, it is obvious that by slightly altering the in- clination of the plates we may diminish the difference of routes to any amount, and may consequently make them colored fringes, differ by half a wave length, or any multiple of the same; and we shall thus produce colored fringes sepa- rated by dark bands, parallel to the intersection of the planes of the glasses. COLORS OF THICK PLATES. 107. Another phenomenon, which depends upon the same principle, and called the colors of thick plates, will be readily understood from pre- ceding considerations, The effect is observed to take place under these circumstances, viz. : Light being transmitted through a small hole A, in a screen, and al- lowed to fall upon a spherical con- Colors of thick plates; How thej may be exhibited ; 292 NATURAL PHILOSOPHY. Facts with regard to these colors ; How produced. ninstration and cave glass reflector M. ' JV^ with con- on ' centric surfaces, the back being sil- vered, and its centre of curvature situated at the aperture, there will be formed upon the screen about the aperture a series of colored rings, or in luminous waves of a single length, alternate bright and dark cir- cles. These become faint and disap- pear if the distance of the screen be increased or diminished beyond a small difference from its original position. They diminish in diameter as the glass is thicker. They arise from the interference of waves which emerge from different points of the first, after being reflected from the second surface. Denote by y, the radius A D, of one of the rings, either dark or bright; by t, the thickness C ' E, of the reflector; and by r, the radius A C. The equivalent interval to t, in air, will be mt, in which m denotes the relative index of refraction for air and glass. The question is to find the difference of the routes Notation; Equation'; and or to find, Kow, AC+CE+ED; EC+CD-ED; 0+ CD = 2mt+ yV 2 + y 2 = ?/ 2 2? by neglecting the fourth and higher powers of y ; and Another; whence, Difference of routes equal to <2/2 some multiple of E C -\~ C D ED ~ u. 2/> ELEMENTS OF OPTICS. 293 But this difference of route must be equal to some mul- tiple of J A ; whence, Value for radio* = i n ^ of ring . and neglecting 2mtin comparison with r, in value of y, we find, , 91) Same reduced. This accords precisely with the most exact measure- ments of Sir ISAAC NEWTON. COLOR FROM UNEQUAL REFR AN GIB ILIT Y. 108. It is demonstrated in the " Analytical Mechanics," 316, that the velocity of wave propagation through an elastic medium, is given by the equation ,A 7* Velocity of wave in which V denotes the velocity of wave propagation, H a function of the elasticity and density of the medium, r the distance between the adjacent molecules, A r the projection of this distance on the direction of the wave motion, and X the wave length. Now, when r is very small in comparison to X, the arc of jwhich the sine enters the last factor above will be small ; the ratio of the sine to its arc will be equal to unity, and the velocity will be simply equal to V H. In other words, ve i city wm when the distances between the consecutive molecules of the8amefor , waves of all the medium are small compared to the wave lengths, i eng th 8; the velocity becomes the same for waves of all lengths. 294: NATURAL PHILOSOPHY. This is the case This is the case with sonorous waves, Equation (3), with sonorous Acoustics, whose lengths vary from several inches to seve- ral feet; compared to which distances those between the consecutive molecules of air may be regarded as insignificant. Butisnottruo 109. Iii the ethereal medium, whose vibrations, pro- duce light, however, as it exists in the various forms wav of natural bodies, the above conditions do not obtain. In the ether of the atmosphere, for example, the lumin- ous waves, we have seen, vary in length from 0,0000167 to 0,0000266 of an inch, compared to which distances those between the adjacent molecules have a sensible value ; the last factor in Equation (92), cannot, therefore, be unity, and the velocity of wave propagation must depend upon the \vave length. A consideration of the velocity greatest equation shows that the velocity will be greatest for the for red and least j.^j and least for the violet waves. for violet waves. . Ken-active index The index of ret raction of any substance is, Equation varies with the ^>), the ratio of the velocity of the luminous wave through the ether of a Torricellian vacuum to that through the ether of the body. And the relative index of two bodies is the ratio of the velocities through their respective ethers. Hence, both the absolute and rela- tive indices vary with the wave lengths, being greatest in lavender grey and least in red, those of violet, indigo, is greatest for blue, green, yellow, and orange, lying intermediate be- lavendergrey fc } ' * and least for red. Effect of an When, therefore, the waves which constitute white light fall obliquely and simultaneously upon the face of a new medium, they will all be deviated on account of the change of density and elasticity of the ether which they then encounter, and the intromitted waves will be unequally deviated, because of their difference of wave length ; these waves will, hence, separate from each other and proceed in different directions ; and, if intercepted by any reflecting surface, as a screen, will exhibit thereon their respective colors. ELEMENTS OF OPTICS. 295 110. This is well illustrated by the action of an bpti- Experimental illustration; Fig. 78. T , cai prism. .Let a beam S S\ of solar light, be admitted in- to a dark room through a small hole in a window shutter, and received upon a screen X Y\ it will exhibit a round lu- minous spot at T, in the direction of 'SS' produced ; but if the face of a re- fracting prism A. C, be interposed, the spot .T'will dis- appear, and there will be formed upon the screen an elongated image of the sun, variously and beautifully caused to colored, beginning with red on the side of the refract- pass through a ing angle A, of the prism, and passing in succession pnsm; through orange, yellow, green, Hue, indigo, and terminat- ing in violet and lavender grey, making eight in all. These colors are not separated by well-defined bounda- Coloreproducod; ries, but run imperceptibly into each other ; nor are the colored spaces of the same length. The following table exhibits the relative lengths of these spaces as obtained by Sir ISAAC NEWTON with the glass prism used by him, and by FKAUNIIOFER, with a prism made of flint glass. Eed Beam of white Orange - t - Yellow - - - - Green ... Blue Indigo Violet and lavender grey, Total length, 360 Newton. Fraunhofer. 45 56 27 27 48 27 60 46 60 48 40 47 80 109 Relative lengths of the colored 360 This property of luminous waves by which they pos- Unequal sess different indices of refraction and are deviated re 296 NATURAL PHILOSOPHY. solar spectram. through different angles for the same angle of incidence, is called the unequal refrangibility of light ; and the colored image thence arising is called the solar spectrum. Effect of Effect varies with the light Use of these 111. "When the light is admitted through a very narrow slit parallel to the refracting edge of the prism, and the prism is of pure homogeneous glass and held in the position of minimum deviation, 25, the whole spectrum appears mark- ed by dark and bright lines, all parallel to the slit, some being broader Fig. 79. and better defined and more conspicuous than others. "With an ordinary prism of flint glass, the eye distinguishes about twelve ; FRATJN- HOFER, with a fine prism of his own glass, distinguished, by the aid of a telescope, six hundred. Certain of these lines are at unequal intervals, which also differ for dif- ferent media, though they are of the same order and in the same colored spaces. They differ essentially with the light employed : the light of the clouds, of the Moon, and of Yenus, show them exactly as in the direct light of the sun. The bright fixed stars give lines peculiar to themselves, as also do electric lights. The light of flames shows none, or at least only certain dark intervals under peculiar circumstances. These lines furnish the means of measuring the refractive indices of different media for different colors. considered ; Number of 112. A question often proposed, as to the number of primary colors primary colors, can only be answered with reference to nnsirW*rt 4 . ... the sense in which it is asked. If it be meant to apply to the number of tints distinguishable in the spectrum, this will be a matter of individual judgment to different eyes. NEWTON distinguished seven. Sir JOHN HERSCHEL eight, Sir DAVID BREWSTER three / but perhaps most ob- servers would admit that it is impossible to fix on any definite number, since the light appears to go through ELEMENTS OF OPTICS. 297 every possible shade of color, from the deep red to faint colors of the violet or lavender grey. If we understand the question r ^ as applying to the number of definite points at each of eight classes. which a wave of different length occurs, their number must be considered as infinite. These waves resolve themselves into eight classes, distinguished by the color they excite in the mind, the same color of different shades being produced by waves whose lengths vary between certain limits. 113. To find the index of refraction for any one of TO find the these different CO- refractive index Fig. 80. for any color; lors, let A be a re- fracting prism,made of any transparent medium ; m n, a gra- duated circle, to the centre of which a small telescope is attached in such a manner that its line of colli- ination shall move in a plane parallel to that -of the gra- duated circle, which is held in a position at right angles Exp]anation . to the edges of the prism. The telescope, being pro- vided at its solar focus with a fine wire perpendicular to the plane of the circle, is directed to some distant source of light, and the reading of the vernier noted. It is then directed so as to receive the colored rays from the prism, and the reading again noted when the prism is turned to the position giving the deviation a minimum. We shall then have or neglecting the very small angle subtended by D C at the distance of the object, S = DCS, Same reduced, 298 NATURAL PHILOSOPHY. of refractive index. Any color deviated a second time. Result of reuniting the colors of the spectrum. which is the difference of the readings ; and this in Equation (12), will give the value of m. If the color occupying the middle of the spectrum be taken, we shall find the value of ???-, which answers to what is called the mean deviation, and which is the same as that given in the table of article 18. If a hole be made in the screen, Fig. (78,) at any one of the colors, as green, for example, and this color, after passing through, be deviated by a second prism jp, no further separation of the waves will be found to take place, but a green image, of the shape and size of the hole in the first screen, will be formed upon a second screen held behind at G '. The colors of the spec- trum being received, each upon a separate mirror, may, by vary- ing the relative position of the mirrors, be re- united, by reflexion, on a screen at TF, where a white spot will be formed as though it were illuminated with common light. Fig. 81. DISPERSION OF LIGHT. Dispersion of light ; Dispersive power. 114:. From what has just been explained, it appears that the waves which constitute white light may be sepa- rated from each other by refraction. The act of such separation is called the dispersion of light, and that pro- perty of any medium by which this is performed, is called its dispersive power. 115. Supposing the light to be incident under a very ELEMENTS OF OPTICS. 299 small angle on any prism, we may replace the sine of the angle of incidence and that of refraction by the arcs, and we shall have from Equations (10), (3) and (3)', by accenting the refractive index and refracting angle of the prism, = m and this in Equation (11), by accenting , gives $' (in' 1) . a', ...... (93) Equation for on* prism ; from which it appears that the deviation will increase with the refractive index and refracting angle of the prism. For a second prism, we have in like manner, aild denotin g tne devia - tion T n F, of the violet, Tn R, of the red, and Tn 6*, of the green waves by w , S r , and S e , re- spectively, the green being the mean of the spectrum, we have Fig. Equations for the extreme and mean colors ; *. = - 1) a', S r =(m' r -!)',, *,= -!)'; m' r , and in' g , denote respectively the in- in which dices of refraction of the violet, red, and green waves. Subtracting the second from the first, and dividing by the third, there will result Reductions; 8. - 8. . . (95) Notation ; whence, the quotient arising from dividing the angle RnV, subtended by the spectrum, by the angle Tn G, of mean deviation, is constant for the same medium, and is therefore taken as the measure of the dispersive power of the medium. And denoting this quotient by D, the foregoing Equation gives, omitting the accents. ELEMENTS OF OPTICS. 301 Value for dispersive power. By this formula, after finding the values of m v , m r , and m f , in the manner indicated, the dispersive powers of the substances named in the following table, as well as those of many others, were obtained. TABLE OF DISPERSIVE POWERS. Substances. V-jr m v -m r -! Realgar melted, 0,267 0,394 Table of Chromate of Lead, 0,262 0,388 dispersive Oil of Cassia, 0,139 0,089 powers. Flint Glass, 0,050 0,032 Crown Glass, 0,033 0,018 Olive Oil, 0,038 0,018 Water, 0,035 0,012 Muriatic Acid, 0,043 0,016 There is a circumstance connected with this subject which has been already alluded to, and which should be carefully noticed, owing to its importance i the con- struction of lenses. If the lengths of spectra formed by two prisms of different media be the same, the colored spaces in the one will not, in general, be equal in length irrationaiitj of to the corresponding spaces of the other. This circum- dis P ersion - stance has been called the irrationality of dispersion. 117. It is one of the4 popular, and at first view objection to plausible, objections to the theory, just explained, of the ^J re g f ^ constitution of white light, and especially of the unequal constitution of velocities of waves of different lengths, that a star when whlte llght ; shut out from view, by the interposition between it and the earth of any opaque and non-luminous body, should exhibit at its disappearance tints of color due to the suc- cessive elimination from its light of the red, orange, yel- low, and so on, in the order of the spectrum, while at 302 NATURAL PHILOSOPHY. Objection answered. its reappearance it should present the complementary hues of these tints in the reverse order as to time ; whereas no such phenomena are found to take place. The objec- tion^ however, assumes what we have no right to grant, viz. : that the relations of the wave lengths to the distances between the adjacent molecules in the great atmosphere of ether which connects us with the plane- tary and stellar regions, are the same as in the ether which pervades the bodies that make up the materi- als of the earth. But we have just seen that the wave lengths, as a general rule, diminish as the densities of the bodies in whose ether the waves exist, increase, while, on the contrary, the distances between the ethereal mole- cules may increase. It would be more consonant to the principles of in- duction, to adopt the law expressed by Equation (92), which is but the simple consequence of known physi- cal principles, and conclude from the non-appearance ol color at the occultation of a star, that the distances be- tween the ethereal molecules which occupy the celestial regions are insignificant in comparison to the wave lengths. This would bring the final waves at disappear- ance, of whatever length, all to the spectator at the And appearances game j ngtanfc . an( j t he same being true of the first waves thus accounted Conclusion drawn from known principles ; for. at reappearance, there should be no color. CHROMATIC ABERRATION Chromatic aberration ; Illustration : 118. It fol- lows from the un- equal refrangibili- ty of the elements 'of white light, that the action of a lens will be, to separate these el- Fig. ELEMENTS OF OPTICS. 393 ements and direct them to different foci, since the value Elements of of/", in Equation (27), depends upon that of m. Sub- ^ted to ,.,,..,, ,. 1 /. / 1 1 \ different foci; stituting in that equation for ( . ), in the case p \r r' I of a spherical lens ; and writing f v and / r , for the focal distances of the violet and red rays, we obtain 1 1 1 Relation between - = (in v 1) . -{ - the conjugate J v ? J focal distances for red and 1 = K- i).JL + -1 fr P / in which m^, being greater than m r , f v , will be less than / r , and the violet rays will be brought to a focus soonest. This departure from accurate convergence, caused by the unequal refrangibility of the elements of white light, when deviated by a lens, is called chromatic aberration, and aberration depends upon the nature of the lens and not on its de figure. It is measured, along the axis of the lens, by the its measure, value of f r -f v . The intersection of the cone of violet rays, with that of the red rays, will give what is called the circle of circle of least least chromatic aberration. The diameter and position c * iromatic * t aberration ; of this circle can readily be found. From the point s, demit the perpendicular s ' = y, to the axis ; this will divide f r /, into two parts v x, and r = w ; and calling the radius of aperture of the lens a , we shall obtain from the similar .triangles of the figure, y W X - Relations from = /, ~f> the figure; a Jr Jv whence we deduce =s L-l-P^.-f\ Same reduced a 304 NATURAL PHILOSOPHY. Radius of circle Or of least chromatic aberration ; Diameter of the same ; (97) The denominator of this expression is equal to twice the mean value of /", and therefore, = ,-, -* and from Equation (27), we have 7 m r or Measure of chromatic aberration. Same in a different form ; Final value for diameter of circle of least chromatic aberration. /, /. Jr Jv = ff / by substituting f" 2 ^ forf r .f v , to which it is nearly equal. Substituting the value of p, from second equation of group (30), the above becomes m 1 hence, m 1 In the case of parallel rays, the last factor is unity, 4-om which we conclude, that the diameter of the circle of least chromatic aberration is equal to the radius of aperture of the lens, multiplied by the dispersive power. The distance of this circle from the lens is, ELEMENTS OF OPTICS. 3Q5 Distance of this f | Q, -P i J v ' y circle from the a lens ? replacing ^ by its value in Equation (97), we have a ua? *r + # = - /./. ..... V jy j reduced. Jr The effect of chromatic aberration is to give color to Effect of the image of an object, and to produce confusion of chromatic ... r ii T/r- i r aberration; vision in consequence of the different degrees of conver- gence in the differently colored waves proceeding from the same point of an object. The vertices of the cones composed of the rays of these waves, lying in the axis, every section perpendicular to this line will have its brightest point in the' centre, and the yellow waves con- in part verging nearly to the mean focus, and having by far the de greatest illuminating property, the bad effects which would otherwise arise from this aberration are in part destroyed. Besides, these effects may be lessened by reducing the aperture of the lens, though not in the May . be same degree as those arising from spherical aberration, diminished, ACHROMATISM. 119. It is, then, impossible, by the use of a single Achromatism; homogeneous lens, to deviate the different waves of white light accurately to a single focus, and^ consequently, im- possible, by the use of such a. lens, to form a colorless image of any object ; both, however, may be done by the union of two or more lenses of different dispersive powers. The principle according to which this maybe accomplished, Achromatic is termed Achromatism, and the combination is said to co be achromatic. Let us suppose two lenses of different dispersive powers placed close together. The focus of the combination will, 20 NATURAL PHILOSOPHY. TWO lenses Equation (34), and the fourth Equation of group (30), for any one of the elementary colors as red, be given by Focus for red; 1 ^ ^ __ 1 m j _ 1 1 n it T r T 7T * Jr P . P / and for violet, = _ -- If f r " and //', were equal, the chromatic aberration, as regards these colors, would be destroyed; equating them we have, Equating these (m r 1) p' + (m r ' 1) p = (m v 1) p' + (m v ' - 1) p focal distances ; whence, _p_ _ (m v 1) (m r 1) m v m r ' = ' = : the second member being negative, because m' r , i& greater than wi' r . Multiplying both members of this equation by - m 1' it may be put under the form, w 1 m v fn r Same in a 1 ^ different form; ?> = m * . . . (100) m 1 Relation m' -I Explanation of Th^ second member expresses the ratio of the disper- the result; g i ve powers of the media, and the first, the inverse ratio of the powers of the lenses for the mean waves; this being, negative, one of the lenses must be concave, the other convex ;, and the powers of the lenses being inversely ELEMENTS OF OPTICS. 307 as the focal distances, we conclude, that chromatic dber- Ell] e for ration, as regards red and violet, may ~be destroyed l)y uniting a concave with a convex lens, the principal focal for red and lengths heing taken in the ratio of their dispersive powers. V1< The usual practice is to unite a convex lens of crown glass with a concave lens of flint glass, the focal distance usual of the first being to that of the second as 33 to 50, combination ' these numbers expressing the relative dispersive pow- ers as determined by experiment; (see Table 116). The convex lens should have the greater' power, and, there- fore, be constructed of the crown-glass ; otherwise, the effect of the combination would be the same as that of convex lens a concave lens with which it is impossible to form a 8hould have the greater power; real image of a real object. Fig. 84 Illustration ; To illustrate : let parallel rays be received by the lens A ; its action alone would be, to spread the different colors over the space- VR, whose central point m, is dis- tant from A, 33 units of measure, (say inches), the violet being at V, and the red at R\ the action of the lens B, alone would be, to disperse the -rays as though they pro- ceeded from different points of the line V R', whose Explanation of central point ra', is distant from B, 50 inches, the violet theaction of thc x ' ' compound lens ; appearing to proceed from V, and the red from R' '; and the effect of their united action would be, to concentrate the red and violet at F, whose distance from the lens is equal to the value of F, deduced from the formula JL F ~ L 33 50 97,06 inches. 308 NATURAL PHILOSOPHY. Point in which QJ red and violet wauld be united ; Geometrical illustration ; 97 , 06 inches. Fig. 84. generally be concentrated in the same point ; why the other N~ow, any one of the colors, orange for example, at colors would not Q - ^ ft y whi h ig throwil by the * convex lens in advance of the centre m, and the same ^^ ai Q > {R tlie gpace y* R\ which is thrown by the concave lens behind the centre m', will, it is obvious, be united with the violet and red at F, by the joint action of both lenses ; and the same would be true of any other color, but for the irrationality of dispersion of the me- dia of which these lenses are composed, which prevents it, < and hence an image formed by such a combination of lenses will be fringed with color ; the colors of the fringe constituting what is called a secondary spectrum. An additional lens is sometimes introduced to complete the achromaticity of this arrangement. Secondary spectrum. substances perfect Clt7 120. If two lenses, constructed of media between . wn ^ cn there is no irrationality of dispersion, be united according to the conditions of Equation (100), the com- bination will be perfectly achromatic. It is found that between a certain mixture of muriate of antimony with muriatic acid, and crown-glass, and between crown-glass and mercury in a solution of sal ammoniac, there is lit- tle or no irrationality of dispersion. These substances have therefore been used in the construction of com- pound lenses which are perfectly achromatic. The figure ELEMENTS OF OPTICS. 3Q9 represents a section of one of these, Fi s- &- Representation consisting of two double convex ^^f^^^^ < * an achromatio \-^MOMU*S?^I lens; lenses of crown-glass, holding be- tween them, by means of a glass cylinder, a solution of the muriate in the shape of a double concave lens, the whole combined agreeably to the relations expressed by Equation (100). The focal distance of the convex lenses is determined from Equation (31). 121. From Equation (98), we infer, that the circle circle of least of least chromatic aberration is independent of the focal c j iromatic * aberration length of the lens, and will be constant, provided the independent of aperture be not changed. Now, by increasing the focal focal length : length of the object glass of any telescope, the eye lens remaining the same, the image is magnified ; it follows, therefore, that by increasing the focal length of the field lens, we may obtain an image so much enlarged that the color will almost disappear in comparison. Besides, Telescopes an increase of focal length is attended with a diminution former] 7 Ver 7 long' of the spherical aberration. This explains why, when single lenses only were used as field lenses, they were of such enormous focal length, some of them being as much as a hundred to a hundred and fifty feet. The use of achromatic combinations has rendered such lengths unnecessary, and reduced to convenient limits, instru- Modem ones ments of much greater power than any formerly made 8horter - with single lenses. INTERNAL REFLEXION. 122. "Whenever the waves of light in their motion Internal through any medium meet with a change of density and reflexion; elasticity, they will be both reflected and refracted. In consequence, objects seen by reflexion from a plate of 310 NATURAL PHILOSOPHY. When objects seen by reflexion from glass appear doable ; Relative brightness of the images when the second surface is in contact with various substances ; glass, in the atmosphere, appear double when llu ibices of 'the glass are not parallel, there being an image formed by reflexion from each face. The image from the second surface will be brighter in proportion as the obliquity or angle of incidence of the incident waves becomes greater. If the second surface of the glass be placed in contact with water, the brightness of the image from that surface will be diminished ; if olive oil be substituted for the water, the diminution will be greater, and i the oil be replaced by pitch, softened by heat to produce accurate contact, the image will disappear. If, now, the contact be made with oil of cassia, the image will be restored ; if with sulphur, the image will be brighter than with oil of cassia, and if with mercury or an amalgam, as in the common looking-glass, still brighter, much more so indeed than the image from the first surface. The mean refractive indices of these substances are as follows : Refractive indices of these substances ; Air, ----- 1,0002 "Water, - - - - 1,336 Olive Oil, 1,470 Pitch, - - 1,531 to 1,586 Plate Glass, 1,514 to 1,583 Oil of Cassia, - - 1,641 Sulphur, - - - 2,148 indices plate glass; Taking the differences between the index of refraction for compared with p] ate g } ass an( j fo^g f Of fa e o ft ier su | )S tances of the the index for table, and comparing these differences with the forego- ing statement, we are made acquainted with the fact, which is found to be general, viz. : that when two media are in perfect contact, the intensity of the light reflect- ed at their common surface will be less, the nearer their refractive indices approach to equality ; and when these are exactly equal, reflexion will cease altogether. This I 7 G is an obvious consequence of the rationale of reflexion, given in Acoustics, ^ 1 Conclusion. ELEMENTS OF OPTICS. 123. Different substances, we have seen, have, in Owing to a general, different dispersive powers. Two media may, d ^^J^ f therefore, be placed in contact, for each of which the power the light same color, as red, for example, may have the same in- ^ ns ^i t t ed b a e t dex of refraction, while for the other elements of white the second light, the indices may be different ; when this is the case, according to what has just been said, the red would be wholly transmitted, while portions of the other colors would be reflected and impart to the image from the second surface the hue of the reflected beam; and this would always occur, unless the media in contact pos- sessed the same refractive and dispersive powers. ABSORPTION OF LIGHT. 124. The waves of light which enter any body are Absorption of not transmitted without diminution ; but in consequence of Iight; a want of perfect elasticity due to the reciprocal action of the molecules of the ether and the particles of the body, and owing to the absence of perfect contact of the elements of bodies, these waves undergo a series of internal reflexions which give rise, as in the case of HOW produced, sound, to interferences and consequent loss of intensity. This action of bodies upon light is called absorption. The quantity absorbed is found to vary not only from Qnantity one medium to another, but also in the same medium absorbed varies; for different colors ; this will appear by viewing the pris- matic spectrum through a plate of almost any transpa- rent, colored medium, such as a piece of smalt blue glass, when the relative intensity of the colors will appear al- tered, some colors being almost wholly transmitted, while others will disappear or become very faint. Each color may, therefore, be said to have, with respect to every [ndex of medium, its peculiar index of transparency as well S of 11 transparency ; refraction. 312 NATURAL PHILOSOPHY. Quantity absorbed depends upon ; Extreme colors transmitted longest. Herschel's hypothesis to account for the extinction of a homogeneous wave ; The quantity of each color transmitted, is found to depend, in a remarkable degree, upon the thickness of the medium; for, if the glass just referred to be extremely thin, all the colors are seen ; but if the thickness be about 2-V of an inch, the spectrum will appear in detached portions, separated by broad and perfectly black inter- vals, the rays corresponding to these intervals being to- tally absorbed. If the thickness be diminished, the dark spaces will be partially illuminated ; but if the thickness be increased, all the colors between the extreme red and violet will disappear. Sir JOHN HERSCHEL conceived that the simplest hypothesis with regard to the extinction of a wave of homogeneous light, passing through a homogeneous me- dium i. So that, calling c the intensity of the extreme red waves in white light, c' that of the next degree of re- frangibility, c" that of the next, and so on, the incident light will, according to Sir J. H., be represented in in- tensity by c + c' + c" + G'" + &c. Intensit y of incident light; and the intensity of the transmitted light, after travers- ing a thickness , by c y l + c f y rt + c" y nt + &c. . . . (101) at of transmitted light ; Wherein y, represents the fraction -, which will m depend upon the waves and the medium, and will, of course, vary from one term to another. From this it is obvious, that total extinction will be Total extinctiol impossible for any medium of finite thickness ; but if for fini;e the fraction ?/, be small, then a moderate thickness, which thickness ; enters as an exponent, will reduce the fraction to a value perfectly insensible. Numerical values of the fractions y, y r , y'\ &c., may indices of be called the indices of transparency of the different transparcnc y- waves for the medium in question. There is no body in nature perfectly transparent, though No bod y i all are more or less so. Gold, one of the densest of me- tals, may be beaten out so thin as to admit the passage of light through it : the most opaque of bodies, charcoal, becomes one of the most beautifully transparent under a different state of aggregation, as in the diamond, " and all colored bodies, however deep their hues and however seemingly opaque, must necessarily be rendered visible by waves which have entered their surface ; for if re- Hected at their surfaces they would all, appear white T14 NATURAL PHILOSOPHY. colors of todies alike. Were the colors of bodies strictly superficial, no not superficial ; variation in their thickness could affect their hues ; but so far is this from being the case, that all colored bodies, however intense their tint, become paler by diminution of thickness. Thus, the powders of all colored bodies, or the streak they leave when rubbed on substances har- Powdersand ^ er tnan themselves, have much paler colors than the streaks. same bodies in mass." THE RAINBOW. Rainbow defined: Fig. 86. Illustration by prisms of water Order in which the colors wilt appear in the primary bow ; 125. The rainbow is a circular arch, frequently seen in the heavens during a shower of rain, in a direction from the observer opposite to that of the sun. If AB C, be a section of a prism of water at right angles to its length by a ver- tical plane, and Sr a beam of light pro- ceeding from the sun ; a part of the latter will be refracted at T, reflected at D, and again refracted at /, where the constituent elements of white light, which had been separated at r, will be made further divergent, the red taking the direction of / 7?, and the violet the direction r r V, making, because of its greater refractive index, a greater angle than the red with the normal to the refracting surface at /. To an observer whose eye is situated at E, the point / will appear red, the other colors passing above the eye ; and if the prism be de- pressed so as to occupy the position Ji 1 B' ' C ', making r" V, parallel to r' V, the point r" would appear of a violet hue, the remaining colors from this position of the ELEMENTS OF OPTICS. 315 prism foiling below the eye. In passing from the first to the second position, the prism would, therefore, pre- water ; sent, in succession, all the colors of the solar spectrum. If, now, the faces of the prism be regarded as tangent planes to a spherical drop of water at the points where the two refractions and intermediate reflexion take place, the prism may be abandoned and a drop of water sub- stituted without altering the effect ; and a number of these drops existing at the same time in the successive po- sitions occupied by the prism in its descent, would exhi- bit a series of colors in the order of the spectrum with the red at the top. Aline ES, passing through the eye and the sun, is Axis of always parallel to the incident rays ; and if the vertical plane revolve about this line, the drops will describe con- centric circles, in crossing which, the rain in its descent will exhibit all the colors in the form of concentric arches having a common centre on the line joining the eye and the sun, produced in front of the observer. "When this . line passes below the horizon, which will always be the case when the sun is above it, the bow will be less than ig sem i- C ir C uiar, a semi-circle ; when it is in the horizon, the bow will be &c - semi-circular. Fig. 87. Illustration for primary bow, m r i i 111 i To find the angle subtended at the eye by the radii subtended at of these colored arches, let A BD, be a section of a drop thee 7 eb y th * of rain through its centre; SA the incident, AD the colored arches; 316 NATURAL PHILOSOPHY. Fig. ST. Illustration for primary bow ; Notation ; Equation for one internal reflexion ; refracted, D E the reflected, and B R the emergent raj. Call the angle OA.m = the angle of incidence, m' ', is equal to to angle of the angle of incidence C Am. incidence; The angle A B, in Fig. (87), is the supplement of the total deviation of the emergent from the incident ray, and is equal to the angle B E F, subtended by the ra- inferences to dius of the bow; in Fig. (88), it is the excess of total de- flgures; viation above 180. Calling this angle $. we shall have Notation and equation ; Illustration for secondary bow ; the upper sign referring to Fig. (87), and the lower to Fig. (88) ; replacing d, by its value in Equation (103,) the above reduces to d rs General value (104:) forradiusofa colored arch ; this, with equation sin 9 m . sin 9 (105) From which the radius of any particular color will enable us to determine the value of , when 9 and m can be found ; are given for any particular color. For any value of 9, assumed arbitrarily, will, in gen- eral, correspond to rays of the same color so much dif- fused as to produce little or no impression upon the eye ; but if 9 be taken such as to give $ a maximum or mini- 318 NATURAL PHILOSOPHY. what waves mu m. then will the rays corresponding to m. emerge pa- appertain to the ^ . . * * rainbow. rallel, or nearly so, for a small variation in the angle 9 on either side of that from which this maximum or minimum value of results ; hence, the waves which en- ter the eye in this case will be sufficiently copious to produce the impression of color, and these are the waves that appertain to the rainbow. 126. By an easy process of the calculus it is found that the relation which will satisfy these conditions, is Eelauonttat will fulfil the conditions for color ; 1 cos 9 * n + 1 in cos 9' Clearing the fraction, squaring both members, adding ra 2 sin 2 9' sin 2 9 and reducing, we get Corresponding angle of incidence : bame for one internal reflexion ; Radii of the colors of primary bow deduced ; cos 9 /m 2 -1 . . . . (106) n 2 + 2 n For one internal reflexion, which answers to Fig. (87), cos 9 =v< m 2 1 and substituting in succession the values of m,- answering to the different colors for water, we shall have values for 9, and consequently for 9', Equation (105), which substi- tuted in Equation (104), will give the angles subtended by the radii of the colored arches which make up what is called the primary ~bow. Example, red of the primary; For red, m 1,3333, hence cos 9 = 0,5092 = cos 59 21', sin 9 = \/l cos 2 9 = 0,8603 ; See Appendix No. 3. ELEMENTS OF OPTICS. 319 tliis last, in Equation (105), gives substitution and reduction; 9' = 40 11', and these values of 9 and 9', in Equation (104), give S = - 118 42' + 160 44' = 42 02'. value of Si For the violet, m = 1,3456, Example, violet COS 9 = 0,5199 = COS 58 41i', of the primary; sin 9 = 0,8543, 9' = 39 25', tfaa- 117 23' + 157 40' = 40 17'; Valueof S'l hence, the width of the primary bow is S - etween these J bows accounted reflected can reach the eye till the drops arrive at the prim- for. ary, and none which is twice refracted and twice reflected, can arrive at the eye after the drops pass the secondary ; hence, while the drops are descending in the space be- tween the bows, the light twice refracted with one or two intermediate reflexions, will pass, the first above, and the second below or in front of the observer. The same discussion will, of course, apply to the lunar rainbow which is sometimes seen. 128. Luminous and colored rings, called halos, are Halos; occasionally seen about the sun and moon ; the most re- markable of these are generally at distances of about twenty-two and forty-five degrees from these luminaries, and may be accounted for upon the principle of unequal refrangibility of light. They most commonly occur in cold climates. It is known that ice crystallizes in minute prisms, having angles of 60 and sometimes 90 ; floating in the atmosphere constitute a kind of mist, and having their axes in all possible directions, a number will always' be found perpendicular to each plane, pass- ing through the sun or moon, and the eye of the obser- ver. One of these planes is indicated in the Figure. /Sm. being a beam of light pa- Fig. 90. rallel to S E, drawn through the v^ * Illustration and sun and the eye, j^ Jk* j explanation; and incident upon the face of a prism whose refracting angle is 90 or 60, we shall have the value of , corresponding to a minimum from Equation (12), by substituting the proper values of m for ice. The mean value being 1,31, we have 21 322 Example first, NATURAL PHILOSOPHY. sin i(+ 60) = 1,31. sin 30 i S = 40 55' 10" 30 = 10 55' 10" = 21 50' 20" and Example second ; sin i(+ 90) = 1,31. sin 45 ' j = 67 52' 45 = 22 52' S = 45 44'. ' Retrospective view of the phenomena of unpolarized light Remarks on the disturbance of molecular equilibrium ; Other phenomena of a similar nature will be noticed hereafter. PO IGHT. 129. We have thus far been concerned with the pro- pagation of luminous waves through homogeneous media, with the deviation which these waves .undergo on meet- ing with a change of density, and with the superposi- tion of two or more waves, by which their effects are increased, diminished, or totally destroyed. "VYe now come to a class of optical phenomena whose explanation depends upon considerations affecting the particular mode of molecular vibrations in these waves. When an ethereal molecule is displaced from its posi- tion of equilibrium, the forces of the neighboring mole- cules are no longer balanced, and their resultant tends to drive the displaced molecule back to its position of rest. The displacement being supposed very small in comparison with the distance between the molecules, the forces thus excited will, we have seen in Acoustics, be ELEMENTS OF OPTICS. proportional to the displacement ; and according to prin-. ciples explained in Mechanics, the trajectory described by the molecule will be an e'lipse whose centre coincides with the position of equilibrium. Hence, the vibration A disturbed 7 molecule in of the ethereal molecules is, in general, elliptic, and the general, describes nature of the light thence arising depends upon the re- an ellipse; lative directions and magnitudes of the axes. These el- liptic vibrations are in planes parallel to the wave front, and consequently transverse to the direction of wave pro- pagation. The axes of the ellipses may either preserve constantly the same direction in their respective planes, or may be continually shifting. In the former case the light Distinction . . . . . -, between is said to be polarized / in the latter, it is unpolanzed po i ar i ze d and or common light. common ]i ^ 130. The relative magnitude of the axes of the e llip- Natureof tLe -, . , -ivn polarization ses determines the nature, ox the polarization. VVnen determined . the axes are equal the ellipses become circles, and the light is said be circularly polarized, when the lesser axis vanishes, the ellipse becomes a right line, and the light is said to be plane polarized \k\z vibrations being in this case confined to a single plane passing normally through the wave front. In intermediate cases the polarization is called elliptical, and its character may vary indefi- Cirfukr -r |lane * J ami elliptical nitely between the two extremes of plane and circular polarization, polarization. The term polarization in optics has come to be a misno- Usc of the term mer. It was introduced before the theory of luminous e undulations had gained much favor with the scientific world ; and was intended at the time of its adoption to express certain fancied affections, analogous to the polari- ties of a magnet, conceived to exist in the material ema- nations which, according to NEWTON, constituted the es- sence of light. It would be better were it replaced by some other term more expressive of the actual condition of the light; but at present this seems to be impossible, owing to its very general acceptation, and it is accord- ingly retained. 324: NATURAL PHILOSOPHY. illustration by a 131. To conceive the manner in which an undulation stretched cord ; mav ^ Q propagated by transversal vibrations, imagine a cord stretched horizontally, one end being attached to a fixed point and the other held in the hand. If the lat- ter extremity be made to vibrate by moving the hand up and down, each particle of the cord will, in succes- sion, be thrown into a similar state of vibration, and a series of waves will be propagated along the cord with a constant velocity. The vibrations of each succeeding .particle of the cord being similar to that of the first, will all be performed in the same plane, and the whole ethereal "articles w ^ represent the state of the ethereal particles along a in a plane plane polarized wave. The plane of vibration is called polarized wave; ^^ pUm O f polarization. If, after a certain number of vibrations in the vertical plane, the extremity of the cord be made to vibrate in some other plane, and then in another, and so on in rapid succession each particle of the cord will, after a certain time, proportional to its -distance from the hand, assume in succession all these varied vibrations ; and the whole cord instead of taking the form of a curve lying in one plane, will be thrown into a species of helical curve, depending on the nature of the original disturbance. condition of the g^^ | s ^ Q con dition of the ethereal molecules in waves ethereal particles . . in common light of common or unpoldwzed light. Undulation When, therefore, we admit a connection among the taT^J b7 molecllles of etlier > similar to that which exists among vibrations. the particles of the cord, there is no difficulty in con- ceiving how a vibration may be propagated in a diiec- tion perpendicular to that in which it is executed. The particles of ether, it is true, are not held together by -cohesive forces like those of a cord, but the molecular forces which subsist among them, are of the same kind, ' and produce similar effects. Neither the particles of the cord nor the ethereal molecules are in contact. 132. These illustrations being understood, conceive a transversal vibration to proceed from a disturbed mole- ELEMENTS OF OPTICS. 325 Cllle at A, towards C, and Transversal suppose the vibration to take ** 91 * vibration supposed to place in the plane of the pa- *" -^ proceed from a per, and let Jf N be the front \ \ disturbed particle of the wave at the expiration ^ of any time , after the be- ginning of motion. The dis- placement #, of the molecule at (7, will, 55, Acoustics, be giren by the equation 0SB .' Sin' * ^ * Consequent ,, I X / displacement of another particle at C-, in which a denotes the amplitude of the disturbance at unit's distance from A ; any angle less at rest, the angle of incidence on the glass M f N', will remain unchanged, refraction Fig. 94. Thus, M N and 3 r N f , representing two plates of glass, mounted upon swing frames, attached to two tubes A and B, which move freely one within the other about a common axis, let the beam SD, of homogeneous light, be received upon the first under an angle of incidence equal to 56 ; reflexion and refraction will take place ac- cording to the ordinary law, and if the reflected beam DD', which is sup- posed to coincide with the common axis of the tubes, be incident upon the second re- flector under the same of incidence,. the reflector being per- pendicular to the plane of first reflex- ion, it will be totally reflected, there being none refracted. But if the tube B, be turned about its A Fig. 95, ELEMENTS OF OPTICS. 329 will begin, and the re- Fi s- 96> The same for a fracted portion will in- ^ , and b > c, these sections, by the planes be, a c, and a &, are respectively, x = ; (y 9 + z> - a 2 ) (6V + iraxe8 * Borax ..... 28 42 Feldspar 6300 Sulphate of magnesia . 37 24 Carbonate of potassa . . 80 30 Sulphate of barytes . 37 42 Cyanite 81 48 Spermaceti .... 37 40 Sulphate of iron . . 90 00 ELEMENTS OF OPTICS. 313 148. If a plane Wave Planewaveof W TF', of common light be A common light . . -i , . T |U incident upon a incident on the upper sur- | . crystal of Land facet>f a crystal of Iceland w -^ -, 8 P ar ; spar to which it is parallel, this wave will be resolved into two components, one of which will take the direction of and be normal to an ob- lique line P e, and will be refracted according to the extraordinary law ; the other will preserve its original course and pass through without deviation. These waves will both leave the crystal normal to that plane of prin- cipal section which is perpendicular to its upper face, the waves themselves becoming parallel; each will be plane polarized, the plane of polarization of the ordinary wave coinciding with the plane of principal section just Effect of this named, and that of the extraordinary wave being at rys right angles -to it. If these component waves be received upon the upper Emergent waves surface of a second crystal of the same kind, and whose eecon^crystai of optical axis is parallel to that of the .first, they will take Iceland spar ; the directions e' e" and o' o" , parallel, respectively, to the directions P e, and P o, and will not be again divided, the first undergoing extraordinary, and the latter ordi- nary refraction ; and if the crystals be of equal thick- ness, the distance e" 0", will be double e o. If either or both of the component waves whose directions are e e', and o o', had been polarized by reflexion, refraction or absorption, the action of the second prism would have been the same ; this is, therefore, another characteristic property of plane polarized light, viz. : that it will not un- dergo double refraction when its plane of polarization is either parallel or perpendicular to the plane of principal sec- Another tion ; being in the former case wholly refracted according to characteristic of . s plane polarized the ordinary, and the latter according to the extraord^ u g ht 344 NATURAL PHILOSOPHY. Reverse tme for nary law. The reverse would have been the case if the positive crystals, crystal, like quartz, had possessed a positive axis. The second crystal supposed to turn on its Effect on the ordinary wave; Effect on the ordinary wave; 149. When the crystal M r JV', is turned around on its base so that the prin- cipal sections of the crys- tals, which are normal to the upper surfaces, make an angle with each other, each of the component waves of which the direc- tions are o o' and e e\ will be again divided into an ordinary and extraordi- nary wave, whose relative intensities will depend up- on the inclination of the principal sections to each other. To avoid complica- tion, let us suppose the wave moving along P 0, to be arrested by sticking a piece of wafer to the lower surface of the first crystal at e ; then will the intensi- ties of the portions into which the wave moving along o o', is divided by the second crystal, be ex- pressed by the formulas = A . cos 2 a, "Wherein A represents the intensity of the wave o o' ; Fig. 104. o ELEMENTS OF OPTICS. 345 a, the angle made by the principal sections of the crys- Notation; tals ; , the intensity of the ordinarily refracted wave ; and O e , that of the wave refracted according to the ex- traordinary law. Removing the wafer from When laminae of different thicknesses are inter- variable thicknesses; posed between the polarizer and analyzer, so as to re- ceive the polarized wave parallel to their surfaces, the tints are found to vary with the thickness. The colors pro- duced by plates of the same crystal, of different thick- nesses, follow, in fact, the same law as the colors reflect- ed from thin plates of air ; the tints rising in the scale Law followed by as the thickness is diminished, until finally, when this thick-, . -. , 5 .. . T.-I ness is reduced below a certain limit, the colors disap- pear altogether, and the central space appears ~black, as the colors ; ELEMENTS QF OPTICS. 351 when the crystal is removed. The thickness producing Results of corresponding tints is, however, much greater in crystal- experiments; line plates exposed to polarized light, than in thin plates of air, or any other medium of homogeneous structure. . The Hack of the first order appears in a plate of sul- phate of lime, when the thickness is the ^V 7 of an inch ; between ^oV o- anc ^ jo f an inch, we have the whole succession of colors of Newton's scale; and when the thickness exceeds the latter limit, the transmitted light ' P Effects of is always white. The tint produced by a plate of mica, different in polarized light, is the same as that reflected from a substancc3 compared. plate of air of only the j^^h part of the thickness. The same subject has been investigated for oblique in- Oblique cidc'iices, and the laws which connect the tint developed incidences. w T ith the number of wave lengths and parts of a length within the crystal, for a wave of given refrangibility, have been determined, both for uniaxal and biaxal crys- tals. 159. Let us now apply the principles already estab- App]ication of lishecl, to explain the appearances. preceding It has been shown, that a wave of common light, on pr entering a crystalline plate, is resolved into two waves, which traverse the crystal with different velocities, and in different directions. One of these waves, therefore, will lag behind the other, and they will be in different phases of vibration at emergence. "When the plate is thin, this Preliminary retardation of one wave upon the other will amount only re Iarks; to a few wave lengths and parts of a length ; and it would, therefore, appear that we have here all the condi- tions necessary for their interference, and the consequent production of color. But here we are met by a difficulty. So far as this An apparent explanation goes, the phenomena of interference and of difficulty aris * 8 ' color should be produced by the crystalline plate alone, and in common light, without either polarizing or ana- lyzing plate. Such, however, is not the fact ; and the real difficulty in this case is, not so much to explain 352 NATURAL PHILOSOPHY. Its solution inquiry suggested. Experimental how the phenomena are produced, as to show why they are not always produced. In seeking for a solution of this difficulty, it may be remarked, that the two waves, whose interference is sup- posed to produce the observed results, are not precisely in the condition of those whose interference we have hitherto examined ; they are polarized^ and in planes at right angles to each other. We are led, then, to inquire whether there is anything peculiar to the interference of polarized waves which may influence these results ; and the answer to .this inquiry will be found to remove the difficulty. 160. The subject of the interference of polarized light researches on the wag exam j ne( j w ith reference to. this question, by FRES- interference of * polarized light ; NEL and AsAGO, and its laws experimentally developed. It was found that two waves of light, polarized in the same plane, interfere and produce fringes, under the same circumstances as two waves of common light ; that when the planes of polarization of the two waves are inclined to each other, the interference is diminished, and the fringes decrease in intensity ; and that, finally, when the angle between these planes is a right angle, no fringes whatever are produced, and the waves no lon- ger interfere at all. These facts may be established by taking a plate of tourmaline which has been carefully worked to a uniform thickness, cutting it in two, and placing one-half in the path of each of the interfering Bni< deduced, waves. It will be thus found that the intensity of the fringes depends on the relative position of the axes of the tourmalines. When these axes are parallel, and con- sequently the two waves polarized in the same plane, the fringes are best defined ; they decrease in. intensity when the axes of the tourmalines are inclined to one another ; and, finally, they vanish altogether when these Experimental ' J ' J illustration. ^axes form a right angle. 161. The non-interference of waves, polarized in ELEMENTS OF OPTICS. 353 planes at right angles to one another, is a necessary result Experiments of the mechanical theory of transversal vibrations. In ^l,^ fact, it is a mathematical consequence of that theory, that theory of . the intensity of the resultant light in that case is constant, ^^3! and equal to the sum of the intensities of the two compo- nent waves, whatever be the phases of vibration in which they meet. But although the intensity of the light does not vary with the phase of the component vibrations, the character of the resulting vibration will. It appears from Equation (107), that two rectilinear and rectangular vibrations com- pose a single vibration, which will be also rectilinear when the phases of the component vibrations differ by an exact number of semi-wave lengths; that, in all other cases, the resulting vibration will be elliptic; . and that the ellipse will become a circle, w r hen the component vibrations have equal amplitudes, and the difference of Results of thl9 theory and their their phases is an odd multiple of a quarter of a wave experimental length. These results have been completely confirmed by confirmation - experiment. In the above mentioned law we find the explanation of Apparent the fact, that no phenomena of interference or color are^ved. f produced, under ordinary circumstances, by the two waves which emerge from a crystalline plate, for these waves are polarized in planes at right angles to one an- other ; and we see that, to produce the phenomena of color in perfection, the planes of polarization of the two waves must be brought to coincide by the analyzer. 162. FKESNEL and ARAGO discovered, further, that Law deduccd . i - -I i from experiment; two waves polarized in planes at right angles to each other, will not interfere, even when their planes of po- larization are made to coincide, unless they "belong to a wave, the whole of which was originally polarised in one vlane ; and that, in the interference of waves which had undergone double refraction, half a wave length must be supposed to be lost or gained, in passing from the ordi- nary to the extraordinary system, just as in the transi- 23 354- NATURAL PHILOSOPHY. Another ^ on f rom the reflected to the transmitted system, in the confirmation oT _ , _ : ' ' the theory of colors formed by thin plates. transversal The principle of the allowance of half a wave length vibrations; . \ . is a beautiiul and simple consequence of the theory of transversal vibrations. In fact, the vibration of the wave incident on the crystal is resolved into two within it, at right angles to one another, one in the plane of prin- cipal section, and the other in a plane perpendicular to it. Each of these must be again resolved, in two fixed directions which are also perpendicular ; and it will easily appear from the process of resolution, that, of the four Experiment components into which the original vibration is thus re- solved, the pair in one of the final directions must con- spire, while in the other, at right angles to it, they are opposed. Accordingly, if the vibrations of the one pair be regarded as coincident, those of the other must differ ly half a wave length. Hence, when the plane of reflex- ion of the analyzer coincides successively with these two positions, the colors, which result from the interference of the portions in the plane of reflexion, those in the per- com icmenta pendicular plane being not reflected, will be complemen- coiors. tary. office of the 163< The f ormer O f the two laws explains the office polarizer ; of the polarizer in the phenomena. To account mechani- cally for the non-interference of the two waves, when the light incident upon the crystal is unpolarized, we may, 133, regard a wave of common light as composed of two waves of equal intensity, polarized in planes at right angles to one another, and whose vibrations are therefore perpendicular. Each of these vibrations, when resolved into two within the crystal, and these two r again resolved in the plane of -reflexion of the ana- lyzer, will exhibit the phenomena of interference. But the amount of retardation will differ by half a wave length in the two cases ; the tints produced will therefore be complementary, and the light resulting from their union will be white. ELEMENTS OF OPTICS. 355 16 i. The preceding laws of interference being kept Keason of the in mind, the reason of all the phenomena is apparent. P henomena ; The" wave is originally polarized in a single plane, by means of the polarizer ; it is then resolved into two waves within the crystal, which are polarized in planes at right angles to each other ; and these are finally reduced to the same plane by means of the analyzer. The two waves will, therefore, interfere, and the resulting tint will depend on the retardation of one of the waves behind the other, produced by the . difference of the velocities Resultant tint with which they traverse the crystal. dependent upon ; 165. It is plain, Equation (107), that the light issu- Emergent light, ing from the crystal is, in general, elliptically polarized, ^f^ 113 inasmuch as it is the resultant of two waves, in which polarized; the vibrations are at right angles, and differ in phase. Hence, when homogeneous light is used, and the emer- gent wave is analyzed with a double-refracting prism, the two waves into which it is divided vary in intensity as the prism is turned, neither, in general, ever vanish- ing. "When, however, the thickness of the crystal is such thickness give8 that the difference of phase of the two waves is an exact the differ ence of number of semi-wave lengths, they will constitute a plane LmLTof XaC polarized wave at emergence, the plane of polarization semi-wave either coinciding with the plane of primitive polarization, ei or making an equal angle with the principal section of the crystal on the other side, according as the difference of phase is an even or odd multiple of half a wave length. Accordingly, one of the waves into which the light is divided by the analyzing prism, will vanish in two posi- tions of its principal section ; and it is manifest that the successive thicknesses of the crystalline plate, at which this takes place, form a series in arithmetical progres- sion. On the other hand, when the difference of phase is a quarter of a wave length, or an odd multiple of that quantity, and when, at the same time, the principal difference of section of the crystal is inclined at an angle of 45 to the pbasei8a * quarter of a plane of primitive polarization the emergent light will be wave length , 356 NATURAL PHILOSOPHY. Circular polarization perfect for one , with the axis R ay coinciding of the crystal. The ray which traverses the crystal in with the axis undergoes no the direction of the axis, P E, will undergo no change change; whatever ; and will consequently be reflected or not from the analyzing plate, according as the plane of reflexion there coincides with, or is perpendicular to, the plane of first Veflexion. But the other rays composing the cone will be modified in their passage through the crystal, and th'e changes which they will undergo will depend on their inclination to the optical axis, and on the position of the principal section with respect to the plane of pri- other rays wiii mitive polarization. Let the circle represent the sec- tion of the emergent cone of rays made by the surface A B of the crystal ; and let M M ' and NN', be two lines drawn through its centre at right angles, being the intersection of the same surface by the plane of primitive polariza- tion, and by the perpendicular plane, respectively. Now, the vi- brations which emerge at any point of these lines will not be resolved into two within the * f Vibrations that crystal, nor will their places of polarization, that is, of win not be vibration, be altered ; because the principal section O f resolved; Fig. 109. be modiSed. Section of ttie emergent pencil by the face of the crystal ; Illustration! ; 358 NATURAL PHILOSOPHY Fig. 109. Illustrations; Vibrations that the crystal, for these vibrations, in the one case coincides resolved; with the plane of primitive polarization, and in the other is perpendicular to it. These waves, therefore, will be reflect- ed, or not, from the analyzer, according as the plane of reflexion there coincides with, or is perpendicular to, the plane of first reflexion. In the latter case, a Hack cross wiute or black will be displayed on the screen, and in the former a white one. But the case is different with the vibrations which ' emerge at any other point, such as L. The principal section of the crystal for these vibrations, neither coincides with, nor is perpendicular to, the plane of primitive po- larization ; and consequent- ly the incident polarized wave will be resolved into two, within the crystal, whose planes of polariza- Vibratiens that will be resolved Fig. 107. tion are respectively paral- lel and perpendicular to the principal section L. The vibrations in these two waves are reduced to the same plane by means of the Reduced to the analyzer; they will, therefore, interfere, and the extent the analyzer, and of that interference will depend upon their difference of interfere, ^ phase. ELEMENTS OF OPTICS. 359 Now, the difference of phase of the two waves varies Extent of with the interval of retardation. When this interval is lnterference dependent on an odd multiple of half a wave length, the two waves difference of will be in complete discordance ; and, on the other hand, phase< they will be in complete accordance, and will unite their strength, when the retardation is an even multiple of the same quantity. The successive dark and bright lines will, therefore, be arranged in circles. 168. "We have been speaking here of homogeneous Phenomena light. When white or compound light is used, the rings pr of different colors will be partially superposed, and the result will be a series of iris-colored rings separated by dark intervals. All the phenomena, in fact, with the ex- ception of the cross, are similar to those of NEWTON'S f rings ; and we now see that they are both cases of the same fertile principle, the principle of interference. These rings are exhibited even in thick crystals, because the difference of the velocities of the two waves is very Analogous to small for rays slightly inclined to the optic axis. Newton's m,^ Fig. 110. Illustrations ; 169. We will now consider briefly the case of liaxal crystals. Let a plate of such a crystal br-r cut perpon- 360 NATURAL PHILOSOPHY. their f and amen tal property ; Effects of biaxai dicularly to the line bisecting the optic axes, and let it crystals. ]) e interposed, as before, between the polarizer and ana- lyzer. In this case, the bright and dark bands will no longer be disposed in circles, as in the former, but will form curves which are symmetrical with respect to the lines drawn from the eye in the direction of the twc axes. The points of the same band are those for which the interval of retardation of the two waves, is constant. The curve formed by each band is the Lemniscata of JAMES BERNOUILLI, the fundamental property of which is, that the product of the radii vectores, drawn from any point to two fixed poles, is a constant quantity. The exactness of this law has been verified, in the most com- plete manner, by the measurements of Sir JOHN HEK- SCHEL. The constant varies from one curve to another, being proportional to the interval of retardation, and in- creasing, therefore, as the numbers of the natural series for the successive dark bands ; for different plates of the same substance, the constant varies inversely as the thick- ness. The form of the dark brushes, which cross the entire system of rings, is determined by the law which governs the planes of polarization of the emergent waves. It may be shown that two such dark curves, in general, pass through each pole; and that they are rectangular hyperbolas t whose common centre is the middle- point of the line which connects the projections of the two axes. Form of the dark brushes determined. END OF OPTICS. APPENDIX, No. I. Suppose a general wave front, sensibly plane, to have reached an open- ing A B, in a partition MN; it is proposed to find the displacement which it will pro- duce in a molecule situated behind and anywhere, as at 0, on the arc of a semi- circle MO JV, of which the plane is normal to the partition, and the centre at the mid- die point of the opening. Take any molecule as Q ; draw Q and (7; make CO = r ; Q = y; C Q =z; C A b ; the angle C Q = 6 ; and denote the whole dis- placement at by Z>, then y = and by Maclaurin's formula, 2 r cos 6 z y r cos 6 - z -f &c. ( a ) The displacement at 0, produced by the wave from Q, will, Eq. (19), be and from the molecules in the distance d z. adz and from those in the entire distance A B, Vt To facilitate the integration, suppose the greatest value of z to be very small 362 APPENDIX. as compared to r, and also the greatest displacements at 0, by the par- tial waves from the molecules on A B, to be equal to one another, then will, Equation (a), y = r cos & z, and writing r for y in the coefficient of the circular function, Equation (b) becomes, a /*+* 2* D = - / sin rJ -b X and performing the integration without regard to limits, aX 2*,, D = -cos ( Vt r -f- cosQz)' 2*rcosd X v and between the limits b and -+- 6, a X 2 * , , 2 * . __ D = cos ( Vt r coseo) cos- (Vt r 4- cosd 2 5X 7X - > J 5 &C. '5 2 2 2 So that, when the radius r is very great, in comparison with 6, there wijl be upon the semicircular arc alternate places of sound or silence, light or APPENDIX. ._ 363 darkness, symmetrically disposed upon either side of the point E, correspond- ing to which 6 is 90. Sound decays rapidly as the distance it has travelled increases, and within the range of ordinary experience r cannot be very great. The relation assumed between r and 6, to integrate Equation (b), can only be obtained, therefore, for audible sounds, by making 6 very small. And since X may be many feet, let us take the case in which the fraction - is so small as to A. justify the substitution of the arc 2*6 cos & in Equation (c), for its sine ; in which case the intensity will be deter- mined by a X 2 tf 6 cos & 2 a b ~~ tr cos & in other words, the sound passing through a small opening will be diffused with equal intensity in every direction behind the partition. Light follows the same law of decay as sound, but the value of X for the waves of ether being extremely small, the greatest not exceeding the 0,0000260 of an inch, the limitations supposed with regard to the fraction -, in the case of sound, will not apply in that of light, and there must exist X the alternations of light and shade abort? referred to. When 6 approaches nearly to 90, cos 6 will be exceedingly small, and the arc 2< 6 cos 6 ~ X may again "be substituted for its sine, in which case, Equation (c), which determines the intensity directly opposite the opening. The maxi- mum value for D / in Equation (c), will arise when , 2 if . b . cos & .in - - - =1, which gives, Equation (c), *," = s n r cos 4 and as the intensity of light varies as the square of the greatest displace- ment, 53, we have APPENDIX. whence Substituting the value of X for the longest wave of light, it is obvious that for any appreciable value for the cos $, th'e intensity of light becomes insignifi- cant, and the only sensible illumination will be immediately opposite the opening. This explains the rectilinear propagation of light ; and why it is, " we may not see, and yet may hear around a corner" No. II. Differentiating Equation (11), we have or differentiating Equations (3) and (3)', we obtain from them c?4/ cos 9 cos 4/ rf-v}/ d 9 cos 9' cos 4> d 9' * * and from Equation (10), and this, combined with Equations (a) and (b), gives cos 9 cos 4/ cos 9' cos -^ which will be satisfied by making 9 = 4> ; 9' =4/. That is, the deviation becomes a minimum when the angles of incidence and of emergence are equal. APPENDIX. No. III. Differentiating Equation (104), we find '-. ...... (a) and from Equation (105), d 9' cos 9 which substituted above, gives, whence 1 cos 9 n 4 1 ~ m cos 9' which is the first equation of 126. Differentiating Equation (a) again, we find 'n 4 1) sin 9', cos 9 < cos 9' , therefore the last factor must be positive ; whence d is a maximum in the primary, and a minimum in the secondary bow. UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY Return to desk from which borrowed. This book is DUE on the last date stamped below. MAY 14 1948 8Apr'59GM MAR 2 5 1959 LD 21-100m-9,'47(A5702sl6)476 THE UNIVERSITY OF CALIFORNIA LIBRARY