# Scientific Memoirs/1/Researches on the Elasticity of Bodies which crystallize regularly

From the Annales de Chimie et de Physique, vol. xl. p. 5, et seq.
Article VII.
Researches on the Elasticity of Bodies which crystallize regularly; by Felix Savart.
(Read to the Academy of Sciences of Paris, January 26th, 1829.)
From the Annales de Chimie et de Physique, vol. xl. p. 5, et seq.
HITHERO precise notions respecting the intimate structure of bodies could be acquired only by two means: first by cleavage, for opake or transparent substances regularly crystallized; secondly, for transparent substances only, by the modifications which they occasion in the propagation of light.
The first of these means has taught us that crystallized bodies are collections of laminæ parallel to certain faces of the crystal; but it has given us no information respecting the force with which these laminæ adhere together nor their elastic state. The second, far more powerful than the first, because it renders evident actions depending on the very form of the particles, has given rise to the discovery of phænomena the existence of which cleavage alone would never have allowed us to suspect. But although these two experimental processes have introduced many new ideas and notions into the science, yet it may be said that the part of physics which treats of the arrangement of the particles of bodies, and the properties resulting from it, as elasticity, hardness, fragility, malleability, &c. is still in its infancy.
The investigations of Chladni respecting the modes of vibration of laminæ of glass or metal, and the researches which I have published on the same subject, especially those which relate to the modes of division of discs of a fibrous substance, such as wood, allow us to suspect that we might acquire by this means new notions respecting the distribution of elasticity in solid bodies; but it was not clearly seen by what process this result might be attained, though the road which it was necessary to follow was one of great simplicity.
But if this mode of experiment, which we are about to describe, is simple in itself, it is not the less surrounded by a multitude of difficulties of detail, which cannot be removed without numerous attempts; and I hope this will serve to excuse the incompleteness of these researches, which I only give as the first rudiments of a more extensive investigation.
§ 1. Statement of the Means of Examination employed in these Researches.
Circular plates which produce normal vibrations are susceptible of several modes of division; sometimes they are divided into a greater or fewer number of equal sectors, always even in number, which perform their vibrations in the same time; at other times they are divided into a greater or fewer number of concentric zones; and these two series of modes of division again may be combined together, so that the acoustic figures which result are circular lines divided into equal parts by diametrical nodal lines.
If the plate which is caused to sound is perfectly homogeneous, circular, and equal in thickness, it is obvious that in the case when the figure consists of diametrical lines only, the system which they form ought to be capable of placing itself in every direction, that is to say, that any point whatever of the circumference of the plate, being taken as the place of excitation, this single condition determines the position of the nodal figure, since the point directly put in motion is always the middle of a vibrating part. In the case of circular lines, under the conditions we have just supposed, these lines would be exactly concentric with the circumference of the plate. These results are a natural consequence of the symmetry which is supposed to exist either in the form or in the structure of the plate; but if this symmetry is deranged, it will easily be conceived that an acoustical figure composed of diametrical nodal lines ought no longer to place itself in a direction depending solely on the position of the point of excitation, and that, with regard to a figure consisting of circular lines, these lines ought to be modified, and will become, for example, elliptical or of some other more complicated form. It is thus that the system of two nodal lines which intersect each other rectangularly, can upon an elliptical plate only place itself in a single position, which is on the axes of the ellipse. There is however a second position in which this mode of division can establish itself; but then it is modified in its form, and it resembles the two branches of a hyperbola, the transverse axis of which corresponds with the greater axis of the ellipse: in this latter case, the number of vibrations is less than in the first, and more so as the axes of the ellipse differ more from each other. A similar phænomenon is observed when the same mode of division is attempted to be produced on a circular plate of brass, of very equal thickness, and in which several parallel saw-cuts have been made, penetrating only to a small distance from the surface: one of the crossed nodal lines always corresponds to a saw-cut which has been made in the direction of a diameter, and the system of the two hyperbolic lines arranges itself in such a manner that the same saw-cut becomes the conjugate axis of the hyperbola. Thus, in both cases, ​the transverse axis of the hyperbola is always in the direction of the least resistance to flexure.
Let us now suppose that, the plate remaining perfectly circular and of equal thickness, it possesses in its plane a degree of elasticity which is not the same in two directions perpendicular to each other; the symmetrical disposition round the centre being then found to be destroyed, although in another manner than in the two examples we have just adduced, an analogous result ought still to be obtained.
Thus, if we take a plate of this description, a plate of wood, for instance, cut parallel to the fibres, and fixing it lightly by its centre, endeavour to make it produce the mode of division consisting of two lines crossed rectangularly, we shall find that when it thus divides itself, the lines of rest always place themselves according to the directions of the greatest and least resistance to flexure, and that putting it afterwards in motion at the extremity of the preceding lines, it may be made to produce a second mode of division, which presents itself under the aspect of a hyperbola the branches of which are much straightened, and which would have for its conjugate axis that line of the cross which corresponds to the direction of the greatest resistance to flexure. In short, when the symmetrical disposition round the centre is destroyed, no matter in what way, the mode of division formed by two nodal lines which intersect each other rectangularly can place itself only in two determinate positions, for one of which it presents frequently the appearance of two hyperbolic branches more or less straightened; and, as we shall soon see, it may even happen that, for certain distributions of elasticity, this mode of division presents itself under the form of two hyperbolic curves in the two positions in which it becomes possible. Lastly, if a similar plate be caused to produce some of the high modes of division, but yet consisting of diametrical lines, experiment shows that they can likewise place themselves in two invariable positions, and pass through certain modifications analogous to those which the system of two lines crossed at right angles undergoes. Thus the immoveability of the nodal figures, and the double position which they can assume, are distinctive characters of circular plates all the diameters of which do not possess a uniform elasticity or cohesion.
It follows therefore from the preceding, that by forming with different substances circular plates of very equal thickness, we may, by the fixed or indeterminate position of an acoustic figure consisting of diametrical nodal lines, ascertain whether the properties of the substance in question are the same in all directions. By applying this mode of examination to a great number of plates formed of different substances regularly or confusedly crystallized, as the metals, glass, sulphur, rock-crystal, carbonate of lime, sulphate of lime, gypsum, &c., it is constantly found that the acoustic figure, formed of two lines crossed rectangularly, can only place ​itself on them in a single position; and that there is a second position in which two hyperbolic curved lines are obtained which are accompanied, according to the different cases, by a sound which differs more or less from that which is produced when the crossed lines occur. Plates are also met with which are incapable of assuming the mode of division formed of two straight lines, and in which only two systems of hyperbolic curves are obtained, sometimes similar, yet giving different sounds. In short, I have yet found no body for which the same nodal figure can place itself in every direction; which seems to indicate that there are very few solid substances which possess the same properties throughout. But what appears still more extraordinary is, that if in the same body, a mass of metal for instance, plates are cut according to different directions, some are susceptible of the mode of division consisting of two lines which cross each other rectangularly, whilst others present only two systems of hyperbolic curves. In both cases, the sounds of the two systems may differ greatly: there may, for example, be an interval between them of more than a fifth.
To arrive at the discovery of the experimental laws of this kind of phænomena, it would be necessary therefore to be able to study them, at first in the most simple cases, for example, upon bodies the elastic state of which, previously known, would differ only according to two directions. This would obtain in a body which might be composed by placing flat plates formed of two heterogeneous substances upon each other in such a manner that all the odd plates might be of one substance, and all the even plates of another, the elasticity in all directions of the plane of each of them being the same. But it has appeared to me difficult to attain this condition, since I have yet found no body the elasticity of which was the same in all directions.
The most simple structure after the preceding would be that of a body composed of cylindrical and concentric layers, the nature of which should be alternately different for the layers next each other, as is nearly the case in the branch of a tree free from knots. It is evident that the elasticity ought to be sensibly the same in every direction of the plane of a plate cut perpendicularly to the axis of the cylinder, and it ought to differ greatly from that which is observed in the direction of the axis. Consequently we shall commence by examining this first case; after which we shall pass to that in which the elasticity would be different according to three directions perpendicular to each other, as would take place in a body composed of flat plates alternately of two different substances, and the elastic state of which would not be the same, according to two directions perpendicular to each other. Wood fulfills again these different conditions; for in a tree of very considerable diameter, the ligneous layers may be considered as sensibly plane for a small number of degrees of the circumference; and if we confine ourselves to plates of ​a small diameter, cut at a little distance from the surface, we may suppose without any very notable error, at least for the whole of the phænomena, that the experiments have been made on a body the elasticity of which is not the same, according to three directions rectangular to each other, since, as is well known, this property does not exist in the same degree according to the direction of the fibres, according to that of the radius of the tree, and according to a direction perpendicular to the fibres and tangential to the ligneous layers.
After these two cases — the most simple that we have been able to study — we shall pass to the much more complicated phænomena which regularly crystallized bodies, such as rock crystal and carbonate of lime, present.
§ II. Analysis of Wood by means of Sonorous Vibrations.
Let us suppose that fig. 1 (Plate III.) represents a cylinder of wood the annual layers of which are concentric to the circumference; let $BCDE$, fig. 2, be any plane passing through the axis $AY$ of the cylinder, and let $nn'$ be a line normal to this plane: it is obvious that the plates taken perpendicularly to $BCDE$, and according to the different directions 1, 2, 3, 4, 5, &c. round $nn'$, ought to present different phænomena, since they all will contain the axis of least elasticity $nn'$ in their plane, and the resistance to flexure, according to the lines 1, 2, 3, 4, 5, will go on increasing in proportion as the plates shall more nearly approach being parallel to the axis of greatest elasticity $AY$.
For the plate No. 1, fig. 3, perpendicular to this axis, all being symmetrical around the centre, the mode of division consisting of two lines which intersect each other at right angles, ought to be able to place itself in all kinds of directions, according as the place of excitation shall occupy different points of the circumference: this is really the case; but it is no longer so, for the plate No. 2 inclined 22° 5' to the preceding. In the latter, the elasticity becoming a little greater in the direction $rs$ contained in the plane $BCDE$, than in the direction $nn'$ normal to this plane, this circumstance ought to determine the nodal lines to place themselves according to these two directions. However, as this difference is very slight, the system of these two lines may still be displaced, when the place of excitation is made to vary; but it will change its form a little, and it will assume the appearance of two hyperbolic branches when it has arrived at 45° from its first position. In the plate No. 3, inclined 45° to the axis $AY$, the difference of the two extreme elasticities being greater, the system of crossed lines becomes entirely fixed, or rather it can only move through a few degrees to the right or left of the position which it assumes in preference; but the hyperbolic system, the summits $a$ and $b$ of which recede more from each other than in fig. 2, will present the remarkable peculiarity of
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Savart's Research on Elasticity.
​being capable of transforming itself into the rectangular system, when the position of the point put directly in motion is made to vary.
Examining with care the nodal lines in fig. 2, it is found equally that its two nodal systems can thus change themselves one into the other; and the same phænomenon is reproduced in the plate No. 4, in which the values of the extreme elasticities differ still more, and in which the points $a$ and $b$ recede from each other at the same time as the curves become more straightened. In the plate No. 5, parallel to the axis $AY$, these curves are no longer susceptible of assuming any other position than that indicated in the figure. Thus, in No. 1, the centres $a$ and $b$ coalesce into one, and there is only a single figure consisting of two crossed lines, the system of which can assume any position; these centres afterwards gradually receding, the modes of division can change themselves from one into the other, and at last, when the branches of the curve are nearly straight lines, the two figures become perfectly fixed.
The existence of these nodal points or centres is, without doubt, a very remarkable phænomenon, and which it will be important to study with great care. In order to give an accurate idea of it, I have in fig. 4 indicated by a dotted line the successive modifications which the two hyperbolic lines assume when the plate is fixed at one of the points $a$ or $b$, and the place of excitation moves gradually from $e$ to $e'e''$, passing over a quarter of the circumference of the plate. When the motion is excited in the vicinity of $e''$, the curves are by the union of their summits transformed into two straight lines which intersect each other rectangularly; and it is obvious that if it had been excited near $e'''$, the two branches of the curve would re-appear, but with this peculiarity, that their transverse axis would take the position assumed by the conjugate, when the motion was produced on the other side of $e''$.
As to the numbers of the vibrations which correspond to each mode of division, for the different degrees of inclination of the plates, it will be seen by examining fig. 3, that, at first equal in No. 1, they go on continually increasing and receding from each other up to No. 5, which contains the axis of the cylinder; and it is indeed evident, that the elasticity in the direction perpendicular to the axis remaining the same for all the plates, whilst that which is perpendicular to this direction goes on continually increasing, this ought to be, in general, the progress of the phænomenon.
These experiments were made with plates of oak 8·4 cent. (3·3071 inches) in diameter, and 3ᵐ·7 (·1456 inch) in thickness: they were repeated with plates of beech-wood, and analogous results were obtained; only the ratio between the two elasticities not being the same, the interval between the two sounds of each plate was found to be greater.
The most general consequence that can be deduced from the ​preceding experiments is, that in wood in which the annual layers are nearly cylindrical and concentric, the elasticity is sensibly uniform in all the diameters of any section perpendicular to the axis of the branch. We shall see further on, that plates of carbonate of lime or rock crystal, cut perpendicularly to the axis, very seldom present this uniformity of structure for all their diameters, although the modifications which such plates impress on polarized light appear symmetrical round this same axis.
In the case which we have just examined, two of the three axes of elasticity being equal, the phænomena are, as we have just seen, exempt from any great complication. It is not so when the three axes possess each a different elasticity: it would then be indispensable to cut, first a series of plates round each of the axes, then a fourth series round a line equally inclined with respect to the three axes, and lastly, it would be necessary again to take a series round each of the lines which divide equally into two the angle contained between any two of the axes; and notwithstanding the great number of results which would be obtained by this process, the end would be far from attained, since these different series would want connexion with each other, and consequently this process cannot give a clear idea of the whole of the transformations of the nodal lines. Nevertheless, I shall content myself to follow this route, which appears to me less complicated than any other, and is sufficient to render fully evident all the principal peculiarities of this kind of phænomena.
In order that the relative positions of the lines round which I have cut the different series of plates of which I have spoken, and the relations they have to the planes of the ligneous layers, as well as to the direction of their fibres, may be more easily represented, I shall refer them all to the edges of a cube $AE$ fig. 5, the face of which $AXBZ$ I shall suppose is parallel to the ligneous layers, and the edge $AX$ to the direction of the fibres, which will allow the three edges $AX$, $AY$, $AZ$ to be considered as being themselves the axes of elasticity. Afterwards I shall indicate the different degrees of inclination of the plates of each series, on a plane normal to the line round which they are to be cut; the position and outline of this plane being at the same time referred to the natural faces of the cube.
But before commencing to describe the phænomena which each of these series presents, it is indispensable to endeavour to determine the ratio of the resistance to flexion, in wood, in the direction of each of the three axes of elasticity: this may be easily done by means of vibrations, by cutting three small square prismatic rods, of the same dimensions, according to the three directions just indicated; for, the degree of their elasticity can be ascertained by comparing the numbers of the vibrations which they perform, for the same mode of division, knowing besides ​that, in reference to the transversal motion, the numbers of the vibrations are as the square roots of the resistance to flexion, or, which is the same thing, that the resistance to flexion is as the square of the number of oscillations.
Fig. 6 shows the results of an experiment of this kind which was made upon the same piece of beech-wood from which I cut all the plates which I shall mention hereafter. In this figure I have, to impress the mind more strongly, given to these rods directions parallel to the edges $AX$, $AY$, $AZ$ of the cube fig. 5, and I have supposed that the faces of the rods are parallel to those of the cube. It is to be remarked that two sounds may be heard for the same mode of division of each rod, according as it vibrates in $ab$ or $cd$; but when they are very thin the difference which exists between them is so slight that it may be neglected. The inspection of fig. 6 shows, therefore, that the resistance to flexion is the least in the direction $AZ$, and is such, that being represented by unity, the resistance in the direction $AY$ becomes 2.25, and 16 in the direction of $AX$. It is evident that the elasticity in any other direction must be always intermediate to that of the directions we have just considered.
This being well established, we shall proceed to the examination in detail of the different series of plates we have mentioned above.
First Series Plates taken round the axis $AY$ and perpendicular to the face $AXBZ$ of the cube.
In the plates of this series, one of the modes of division remains constantly the same. (See figs. 5, 7 and 8.) It consists of two lines crossed rectangularly, one of which, $ay$, places itself constantly on the axis $AY$ of mean elasticity; but although this system always presents the same appearance, it is not accompanied, for the different inclinations of the plates, by the same number of vibrations; this ought to be the case, since the influence of the axis of greatest elasticity ought to be more sensible as the plates more nearly approach containing it in their plane: the sound of this system ought therefore to ascend in proportion as the plates become more nearly parallel to the plane $CYAX$. As to the hyperbolic system, it undergoes remarkable transformations, which depend on this circumstance, that the line $ay$ remaining the axis of mean elasticity in all the plates, the line $cd$, which is the axis of least elasticity in No. 1, transforms itself gradually into that of the greatest elasticity, which is contained in the plane of the plate No. 6. It hence follows that there ought to be a certain degree of inclination for which the elasticities, according to the two directions $ay$, $cd$, ought to be equal: now, this actually happens with respect to the plate No. 3; and this equality may be proved by cutting in this plate, in the direction of $ay$ and its perpendicular, two small rods of the same dimensions: it ​will be seen, on causing them to vibrate in the same mode of transversal motion, that they produce the same sound. It also follows, because the elasticity in the direction $ay$ is sometimes smaller and sometimes Greater than that which exists in the direction of $cd$, that the first axis of the nodal hyperbola ought to change its position to be able to remain always perpendicular to that of the lines $ay$, $cd$, which possess the greatest elasticity; thus, in Nos. 1 and 2, $cd$ possessing the least elasticity, it becomes the transverse axis of the hyperbola, whilst in Nos. 4, 5 and 6, the elasticity being greater in the direction c d than in that of $ay$, the transverse axis of the hyperbola places itself on the latter line. As the ratio of the two elasticities varies only gradually, it is obvious that the modifications impressed on the hyperbolic system ought in the same manner to be gradual: thus the summits of these curves, at first separated in No. 1 by a certain distance (which will depend on the nature of the wood), will approach nearer and nearer, for the following plates, until they coalesce as in No. 3, at a certain degree of inclination, which was 45° in the experiment to which I now refer, but which might be a different number of degrees for another kind of wood. At the point where we have seen that the elasticities are equal in the direction of the axis, the two curves transform themselves into two straight lines which intersect each other rectangularly, after which they again separate; but their separation is effected in a direction perpendicular to that of their coalescence. The sounds of the hyperbolic system follow nearly the same course as those of the system of crossed lines, that is to say, they become higher in proportion as the plates more nearly approach being parallel to the axis of greatest elasticity; but it deserves to be remarked, that the plate No. 3, for which the elasticity is the same in the two directions $ay$, $cd$, is that between the two sounds [of which there is the greatest interval: this evidently depends on the elasticity in the two directions $ay$, $cd$ being very different from that which exists in the other directions of the plate.
Lastly, it is to be remarked that, in the four first plates, the sound of the hyperbolic system is sharper than that of the system of crossed lines, and that it is the contrary for the plate No. 6, which renders it necessary that there should be between No. 4 and No. 6 a plate, the sounds of which are equal, which in the present case is exemplified in No. 5, although its two modes of division differ greatly from each other. There is another thing remarkable in this plate; its two modes of division can transform themselves gradually into each other by changing the position of the place of excitation, so that the two points $c$ and $c'$ becoming two nodal centres, are in every respect in the conditions indicated by fig. 4.
The interval included between the gravest and the sharpest sounds of this series was an augmented sixth.
​It is almost useless to observe that the plates taken in the directions I, II, III, inclined on the other side of the axis $AX$ the same number of degrees as the plates 1, 2, 3, would present exactly the same phænomena as these latter. This observation being equally applicable to the following series, we shall not mention it again.
Second Series. — Plates taken round the axis $AZ$ of least elasticity and perpendicular to the plane $CYAX$; figs. 9 and 10.
As in the preceding case, one of the nodal systems of the plates of this series consists of two lines crossed rectangularly, one of which, $az$, corresponds with the axis $AZ$; whence it follows that the second may be considered as the projection of the two other axes on the plane of the plate, which, whatever its inclination may be, ought consequently to possess a greater elasticity in the direction $fg$ than in the direction $az$: thus the hyperbolic system of this series cannot present the transformations which we saw in the preceding series, where $cd$, fig. 8, possesses sometimes a less, at other times a greater elasticity than that of $ay$. In the present case, $az$ remaining constantly the axis of least elasticity, the resistance to flexion in the direction $fg$ goes on gradually increasing from the plate No. 1 to the plate No. 6 parallel to the plane $AXBZ$, and the branches of the hyperbola straighten themselves in proportion as the plates more nearly approach this last position. As to the sounds which correspond to each of these nodal systems, it is observed that they ascend gradually from No. 1 to No. 6, and that the sound of the hyperbolic system is sharper in a part of the series than that of the system of crossed lines, whilst they become graver in the other part. There is therefore a certain inclination for which the sounds of the two systems ought to be equal; and this evidently would have taken place in the present experiment for a plate intermediate to No. 4 and No. 5.
The interval between the gravest and the sharpest sound of each series was an augmented fifth.
Third Series. — Plates taken round the axis $AX$ of greatest elasticity, and perpendicular to the plane $AYDZ$; figs. 11 and 12.
The elastic state of these plates cannot present such remarkable differences as those we have observed in the preceding series; for, being all cut round the axis of greatest elasticity, they can only contain in their plane that of least or that of mean elasticity, or lastly, those intermediate between these limits, which do not vary greatly from each other. Thus it is seen that their modes of division differ very little from each other, and that the sounds which correspond to them present rather slight differences, although they go on ascending in proportion as the plates more nearly approach containing the axis of mean elasticity in their plane. Here, as in the other series, one of the nodal ​systems consists of two lines crossed rectangularly, one of which, $ax$, places itself always on the axis of greatest elasticity, and this line serves as the second axis to the hyperbolic curves which compose the nodal system. Doubtlessly these curves are not entirely similar in the different plates; but I have not been able to perceive any very remarkable difference between them, unless that it appears that their summits gradually approach by a very small quantity, in proportion as the plates more nearly approach containing the intermediate axis in their plane.
Fourth Series Plates cut round the diagonal $AD$, and perpendicular to the plane $BCYZ$; figs. 13 and 14.
These plates present much more complicated phænomena than those we have hitherto observed. Except for the first and the last, neither of the two nodal systems consists of lines crossed rectangularly, which shows that this kind of acoustic figure can only occur on plates which contain at least one of the axes of elasticity in their plane, since Nos. 2, 3, 4, 5, which are inclined to the three axes, present only hyperbolic lines, whilst No. 1, which contains two of the axes of elasticity, and No. 6, which contains only one, are susceptible of assuming this kind of division.
In this series, neither of the modes of division remains constantly the same for the different degrees of inclination of the plates: setting out from the plate No. 1, one of the systems gradually passes from two crossed lines to two hyperbolic branches, which are nearly transformed into parallel straight lines in No. 6; on the contrary, the other system appears in No. 1 under the form of two hyperbolic curves, the summits of which approach nearer and nearer until they coalesce in No. 6, where they assume the form of two straight lines which cut each other at right angles and this contrary course in the modifications of the two systems is such, that there is a certain inclination (No. 3) for which the two modes of division are the same, although the sounds which correspond to them are very different.
As in the preceding series, and for the same reasons, the sound of each nodal system goes on always ascending in proportion as the plate more nearly approaches containing the axis of greatest elasticity in its plane.
Fifth Series. — Plates cut round the diagonal $AE$, and perpendicular to the plane $rst$; figs. 5.
Among all the plates which may be cut round the diagonal $AE$ of the cube fig. 5, there are three each of which contains one of the axes of elasticity, and which consequently we have already had occasion to observe; thus the plate No. 3, fig. 8, which passes through the diagonal $AB$, and through the edge $AY$, contains the diagonal $AE$ in its ​plane; also, the plate No. 4, fig. 10, which passes through one of the diagonals $XY$ or $AC$, and which is perpendicular to the plane $CYAX$, contains also $AE$ in its plane; and lastly, the plate No. 3 of fig. 12, parallel to the plane $ADEX$, is circumstanced in the same manner. Thus, if $rst$, fig. 15, is a plane perpendicular to the diagonal $AE$, and if the lines 1, 3, 5 indicate the directions of the three plates we have just spoken of, in order to become acquainted with the progress of the transformations which connect the modes of division of these plates together, it will be sufficient to take round $AE$, the projection of which is in $c$, a few other plates such as 2, 4, 6. The Nos. 1, 2, 3 of fig. 16 represent this series thus completed, and the dotted line $ae$ indicates in all the direction of the diagonal of the cube.
The nodal syytem represented by the unbroken lines consists, for No. 1, of two crossed nodal lines, one of which, $ay$, places itself upon the axis $AY$, and the other in a perpendicular direction it transforms itself in No. 2 into hyperbolic curves, which by the approximation of their summits again become straight lines in No. 3, which contains the axis $AY$ of greatest elasticity: these curves afterwards recede again, No. 4, and in the same direction as No. 2; they then change a third time into straight lines in No. 5, which contains the axis $AZ$ of least elasticity; and lastly, they reassume the appearance of two hyperbolic branches in No. 6.
The transformations of the dotted system are much less complicated, since it appears as two straight lines crossed rectangularly in No. 1, and afterwards only changes into two hyperbolic branches, which continue to become straighter until a certain limit, which appears to be at No. 3, and the summits of which afterwards approach each other, Nos. 5 and 6, in order to coalesce again in No. 1.
As to the general course observed by the sounds of the two nodal systems, it is very simple, and it was easy to determine it previously. Thus, the plate No. 5, containing in its plane the axis $AZ$ of least elasticity, the two gravest sounds of the entire series is heard; these sounds afterwards gradually rise until No. 3, which contains the axis $AX$ of greatest elasticity; after which they redescend by degrees in Nos. 2 and 1, (the latter contains the axis $AY$ of intermediate elasticity in its plane,) and they return at last to their point of departure in the plates Nos. 6 and 5.
The transformations of the nodal lines of this series, by establishing a link between the three series of plates cut round the axes, makes us conceive the possibility of arriving at the determination of nodal surfaces, which we might suppose to exist within bodies having three rectangular axes of elasticity, and the knowledge of which might enable us to determine, a priori, the modes of division of a circular plate inclined in any manner with respect to these axes. But it is obvious, that to attempt such an investigation it would be necessary to base it on ​experiments made with a substance the three axes of which shall be accurately perpendicular to each other, which is not entirely the case in wood.
It would now remain for us to examine two other series of plates, one taken round the diagonal $AB$, and the other round the diagonal $AC$; but as it is evident that the arrangements of nodal lines which they would present would differ very little from those of the fourth series, we may dispense with their examination.
Such are, in general, the phænomena which are observed in bodies which, like that we have just examined, possess three axes of elasticity: collected into a few propositions, the results we have obtained are reducible to the following general data.
1st. When one of the axes of elasticity occurs in the plane of the plate, one of the nodal figures always consists of two straight lines, which intersect each other at right angles, and one of which invariably places itself in the exact direction of this axis; the other figure is then formed by two curves which resemble the branches of a hyperbola.
2nd. When the plate contains neither of the axes in its plane, the two nodal figures are constantly hyperbolic curves; straight lines never enter into their composition.
3rd. The numbers of vibrations which accompany each mode of division are, in general, higher as the inclination of the plane to the axis of greatest elasticity becomes less.
4th. The plate which gives the sharpest sound, or which is susceptible of producing the greatest number of vibrations, is that which contains in its plane the axis of greatest elasticity and that of mean elasticity.
5th. The plate which is perpendicular to the axis of greatest elasticity is that from which the gravest sound is obtained, or which is susceptible of producing the least number of vibrations.
6th. When one of the axes is in the plane of the plate, and the elasticity in the direction perpendicular to this axis is equal to that which itself possesses, the two nodal systems are similar; they each consist of two straight lines which intersect each other rectangularly, and they occupy positions 45° from each other. In a body which possesses three unequal axes of elasticity there are only two planes which possess this property.
7th. The transverse axis of the nodal curves always occurs in the direction of the least resistance to flexion; it hence follows, that when in a series of plates this axis places itself in the direction at first occupied by the conjugate axis, it is because the elasticity in this direction has become relatively less than in the other.
8th. In a body which possesses three unequal axes of elasticity, there are four planes in which the elasticity is so distributed that the two ​sounds of the plates parallel to these planes become equal, and the two modes of division gradually transform themselves into each other, by turning round two fixed points, which, for this reason, I have called nodal centres.
9th. The numbers of vibrations are only indirectly connected with the modes of division, since two similar nodal figures, as in No. 3, fig. 8, and in No. 3, fig. 14, are accompanied by very different sounds; whilst, on the other side, the same sounds are produced on the occurrence of very different figures, as is the case for No. 5 of fig. 8.
10th. Lastly, a more general consequence which may be deduced from the different facts we have just examined is, that when a circular plate does not possess the same properties in every direction, or, in other words, when the parts of which it consists are not symmetrically arranged round its centre, the modes of division of which it is susceptible assume positions determined by the peculiar structure of the body; and that each mode of division, considered separately, may always, subject however to alternations more or less considerable, establish themselves in two positions equally determined, so that it may be said that, in heterogeneous circular plates, all the modes of division are double.
By the aid of these data, which are no doubt still very few and imperfect, a notion may be formed, to a certain point, of the elastic state of crystallized bodies, by submitting them to the same mode of investigation: this is what we have attempted for rock crystal, in a series of experiments which will be the subject of § iii. of this Memoir.
§ III. Analysis of Rock Crystal by means of Sonorous Vibrations.
ROCK Crystal most ordinarily occurs under the form of a hexahedral prism, terminated by pyramids with six faces (fig. 1. pl. IV.). Although this substance does not admit of cleavage by the ordinary means, it is assumed, from analogy, that its primitive form is a rhombohedron, like that which would be obtained if the crystal were susceptible of cleavage parallel to the three non-adjacent faces of the pyramid, such, for example, as $aXb$, $eXf$, $cXd$, and their parallels $a^{\prime }Yb^{\prime }$, $e^{\prime }Yf^{\prime }$, $c^{\prime }Yd^{\prime }$. The accuracy of this induction is besides confirmed by a very simple experiment, which consists in making a prism of rock crystal red hot, and suddenly cooling it; an operation which determines its fracture, and which most frequently, gives as the result pieces of crystal which have the form of rhombohedrons.
Setting out with these notions with which mineralogy furnishes us, it is obvious that circular plates taken parallel or perpendicular to the axis, parallel to a face of cleavage or of non-cleavage of the pyramid, &c. ought to present different phænomena with respect to sonorous vibrations, since the cohesion and elasticity are not the same in these different directions. Consequently, to simplify as much as possible the examination of these phænomena, we have had cut, from different pieces of rock crystal, a considerable number of circular plates, at first taken in different azimuths of a plane perpendicular to the axis, fig. 2. and fig. 2, bis; then, according to the azimuths of a plane perpendicular to two parallel faces of the hexahedron, and passing through its axis, fig. 3. and fig. 3, bis; lastly, according to the different azimuths of a plane passing through the axis and two opposite edges of the crystal fig. 4. and 4 bis.
As it was necessary to support this general disposition of the experiments by facts, it was indispensable to ascertain first, that the elastic state of the crystal is the same for all the planes parallel to the natural faces of the hexahedron, and next, that it is also the same for all the ​planes perpendicular to the preceding and passing through the axis, although it be different in the latter from what it is in the former; lastly, it was necessary to verify whether the plates cut parallel to the faces $aXb$, $eXf$, $cXd$ of the pyramid were really susceptible of assuming the same modes of division, and whether these modes were different from those of the three plates cut parallel to the faces $bXc$, $dXe$, $aXf$, these latter being besides similar to each other. Experiment having shown the affirmative of these positions, it is evident that all the series of plates perpendicular to a plane normal to any two parallel faces of the prism, and passing through its axis, ought to present identical phænomena for the same degrees of inclination, and that the same ought to be the case for the series of plates perpendicular to any plane passing through two opposite edges of the hexahedron. All the plates we have employed are 23 or 27 lines in diameter and 1 line in thickness; they have been cut with great care and are polished, in order that the phænomena they exhibit with respect to light might be compared with those they present relative to sonorous vibrations. Lastly, although they have been taken from five or six different crystals and from different countries, it may be supposed that they belong to the same piece of quartz, because, whenever it was necessary to pass from one crystal to another, the precaution was taken of causing to be cut in the new specimen a certain number of plates, for the sole purpose of repeating the experiments already made; and by this process we may assure ourselves that crystals of very different appearance, such as those of Madagascar and of Dauphiny, do not however present remarkable differences in their structure.
Before proceeding to the description of the phænomena which are related to each series of plates, we shall observe, that in all the figures the line $xy$ represents the axis itself of the crystal when it is contained in the plane of the plate, or its projection in the contrary case, and that the position of this axis has been determined with great care, for each plate individually, by means of polarized light; so that with this datum and the details into which we shall enter, the position occupied by any plate within the mass of the crystal may easily be represented to the mind.
First Series. Plates parallel to the Axis of the Hexahedron.
If we consider first the plates i., v., ix., fig. 2. and 2, bis, which are parallel to the faces of the hexahedron, we see that they assume exactly the same modes of division: one of these modes, that which is represented by dotted lines, consists of two nodal lines, which cross each other rectangularly, whilst the other resembles the two branches of a hyperbola, to which the two preceding lines serve as axes. The sound of the first system being F, that of the second is the D# of the ​same octave. Thus, in any plate taken parallel to the faces of the hexahedron, one of the nodal lines of the rectangular system always corresponds with the axis of the crystal. In this case everything occurs the same as in plates composed of parallel fibres and which contain in their plane at least one of the axes of elasticity; but this is no longer the case for the plates iii., vii., xi., perpendicular to two parallel faces of the hexahedron, although they are also parallel to the axis like the preceding: instead of a system of lines crossed rectangularly and a hyperbolic system, they exhibit only two hyperbolic systems, which appear exactly similar, and which however are accompanied by very different sounds, since one of them gives D and the other F# of the same octave. The principal axes $lm$, $l^{\prime }m^{\prime }$ of each of the two hyperbolic curves appear to intersect each other at the centre of the plate; their mutual inclination is from 51° to 52°, so that the branches of these curves intersect each other; and if a line $op$ be drawn through the centre of the plate equally inclined to each of the axes $lm$, $l^{\prime }m^{\prime }$ and this line be supposed to be the section of a plane perpendicular to the plate, this plane will, for the plate iii., be parallel to the face $eXf$ of the pyramid fig. 1.; for the plate vii., to the face $aXb$; and lastly, for the plate xi., to the face $cXd$; so that it must hence be concluded that the six faces of the pyramid do not possess the same properties, and that the three we have just indicated perform an important part in the phænomena in question. It must be remarked that the modes of division of these plates are exactly the same as those of the plate No. 3 of fig. 14, Pl. III., which contains neither of the axes of elasticity in its plane. Now, if we consider the plates ii., iv., vi., viii., x., xii. intermediate to the preceding and to those which are parallel to the faces of the hexahedron, we find also in them properties which seem to de-pend on both jointly, as well with respect to the nodal lines of the two systems as to the sounds they produce. Thus with reference to the process of investigation which we have employed, all the plates parallel to the axis do not possess the same properties, whilst with regard to light it is well known that they exhibit exactly the same appearances.
Although this result has been verified many times, it was still important to verify it again, which I did in the following manner: I took, first, two plates like Nos. i. and v., and then two plates like iii. and vii., and after having crossed their optic axes, I placed successively each of these pairs in the path of a large pencil of light polarized by a black glass, the plane of the plates being placed perpendicularly to the luminous rays, and their axes making an angle of 45° with the plane of polarization. It is known that if we look through a similar pair by means of a tourmaline, the axis of which is in the plane of polarization, we perceive two systems of coloured hyperbolas, the tints of which ​appear to follow sensibly in their succession the order of those of Newton's rings: it was required therefore only to compare the phænomena observed in the two cases, and to see whether they presented any hitherto unobserved differences; but it was impossible to recognise any. Thinking that perhaps a considerable augmentation of thickness in the plates might bring to view some appreciable differences, I repeated the experiment with pieces of rock crystal which were eight centimetres (3·149 inches) in thickness, and I saw nothing that could indicate that all the plates parallel to the axis did not comport themselves in the same manner with regard to light: whence it must be concluded that what we can learn respecting the structure of crystals by means of light, is not of the same order as that which sonorous vibrations may enable us to discover. It would appear from what precedes, that this latter process indicates more specially the elastic state and the force of cohesion of the integrant particles in the different directions of every plane, whilst the phænomena of light, depending more specially on the form of the particles and on the position they assume round their centre of gravity, are, to a certain point, independent of the mode of junction of the different plates of which the crystal is formed.
$ab$
$mnXopY$
One of the modes of division of all the plates of this series remains constantly the same, fig. 3, bis; it is formed of two straight lines crossing each other rectangularly, and $xy$, one of these lines, is always the projection of the axis of the crystal on the plane of the plate. The other mode of division consists of two hyperbolic curves, which undergo various modifications depending on the inclination of the plates to the axis of the hexahedron, and which are in general analogous to those we have observed in the two first series of plates belonging to bodies possessing three rectangular axes of elasticity.
No. 1. represents the two modes of division of the plate perpendicular to the axis $XY$; they are both composed of straight lines; or, if either is formed of two curves, their summits are so near each other that they appear to coalesce. Rock crystal being a crystal with one axis, in respect to light, it w£is natural to presume that the elasticity would be equal in every direction of the plane of the plate in question, and that, in consequence, this plate might assume only a single mode of division, having the property of placing itself in any direction; but this is not the case, even in plates cut with extreme care, and which by their optical properties appear sensibly perpendicular to the axis. ​Nevertheless, the interval which is observed between the sounds of the two systems being always very small, and not being constant in different crystals, it appears more natural to attribute this difference of elasticity to an irregularity of structure than to suppose that it depends on a determinate and regular arrangement, the more so as in very large crystals, like those I have employed, it is very rare not to meet with irregularities of structure sufficiently obvious even to be recognised by the naked eye.
The plate No. 2, inclined 78° to the axis, begins to present a difference in the disposition of these two systems of nodal lines; one of the two transforms itself into two hyperbolic branches, which become more straightened in the plate No. 3, inclined 75° to the axis, and which afterwards approach each other again, and become two straight lines, which intersect each other at right angles in the plate No. 4, inclined about 51° to the axis, and which consequently is nearly perpendicular to the face $aXb$ of the pyramid fig. 1; the inclination of the faces of the pyramid to those of the hexahedron being 140° 40′.
The numbers of vibrations which were nearly the same for No. 1, from which only the sounds D and D+ were obtained, differ more as the plate approaches No. 4, when the gravest sound being C, the second is the G of the same octave, although the two modes of division are the same as those of No. I . It is this sound C, given by one of the modes of division of the plate perpendicular to the face of the pyramid, which I have taken as the term of comparison, and to which the sounds of all the other plates are referred. Recommencing with the plate No. 4, the variable system separates once more, but in the contrary way; the curves which form it continue to straighten, whilst their summits recede from each other, and at the same time the two sounds approximate until they are sensibly the same in No. 8, inclined about 12° to the axis. The hyperbolic system ceases to assume a determined position, and it can, without the sound undergoing any change, transform itself gradually into the rectangular system which form the axes, so that this plate appears to be exactly in the same conditions as No. 5 of fig. 8, PL III. In a crystal of quartz there are three planes analogous to the preceding, since the phænomena which are presented by the plates cut round the edge $ab$ of the base of the prism, are, as I have satisfied myself, precisely the same as those which are presented, for the same degrees of inclination, by plates cut round the two other edges $cd$, $ef$.
Beyond No. 8. the sounds begin to differ from each other, and the branches of the hyperbola continue to straighten until No. 11, parallel to the second face of the pyramid. There the distance between their summits is greater than for any other degree of inclination of the plates, and the sound of the rectangular system is the same as that of ​the same mode of division in No. 4, perpendicular to the face $aXb$ of the pyramid. Lastly, from No. 11 until the plate perpendicular to the axis, the sounds approximate again, as well as the summits of the hyperbolic curves, and at the same time the two systems of nodal lines again become rectangular; the sounds thus become almost the same.
Among the plates which we have just examined there are two which merit particular attention; these are Nos. 5 and 11, parallel to the the faces $eXd$ and $aXb$ of the pyramid, and the elastic state of which undoubtedly differs very much, since in one it is the hyperbolic system which gives the gravest sound, whilst in the other it is the rectangular system, and that, besides, there is a great difference between the sounds which correspond to each of their nodal systems. The faces $aXb$ and and $eXd$ of the pyramid being opposite, one of the two ought to be susceptible of cleavage, whilst the other ought not to be capable of this mechanical division; consequently if we knew which of the two plates Nos. 5 and 11 possesses this property, we might, by examining its acoustic figures, determine which are the faces of the pyramid parallel to the faces of the primitive rhombohedron. Rock crystal not yielding in the least to any attempt at dividing it into regular layers in any direction, it was impossible for me to ascertain directly which of the two faces $aXb$ or $eXd$ were those of cleavage; but this question can be resolved with ferriferous carbonate of lime, a substance which is cleaved with almost the same facility as pure carbonate of lime, and which appears to possess, in reference to sonorous vibrations, properties in general analogous to those of rock crystal. Now, if we cut in such a crystal two plates,—one taken parallel to a natural face of the rhombohedron, the other corresponding with a plane inclined to the axis by the same number of degrees as these faces, and which are besides equally inclined to the two faces which form one of the obtuse solid angles,—we find that the first possesses the same properties as No. 11, whilst the second has a structure analogous to that of No. 5; whence it ought to be concluded, from analogy, that the face $aXb$ of the pyramid fig. 1. is that which is susceptible of cleavage. This once established, it is not even requisite, in order to ascertain which of the faces is susceptible of cleavage, to cut a plate parallel to one of these faces; it is obvious that a plate parallel to the axis and normal to two parallel faces of the hexahedron should be sufficient to attain this end. Thus, let fig. 5, $abcdef$, be the horizontal projection of the prism represented fig. 1; according to what has been said, $rstv$ will be the projection of the primitive rhombohedron; again, let $ll^{\prime }$ be the projection of a plate parallel to the axis and equally inclined to the two faces of $a$ and $f$ of the hexahedron; according to what we have above said, this plate will assume the mode of division of No. 3, fig.2, bis, and the line $op$ will be parallel to the plane $rstu$ normal to the plate, that is to say, ​to one of the cleavage planes; thus the direction of this line, in a plate parallel to the axis and normal to two faces of the hexahedron, is sufficient to enable us to ascertain which of the faces of the pyramid are susceptible of cleavage.
In order to complete all that relates to the transformations of the nodal lines of this series of plates, it would have been important to determine with accuracy the degree of inclination to the axis, of the plane situated between No. 3 and No. 4, for which the summits of the nodal hyperbola are at the greatest distance from each other: but, having been stopped in these investigations by the difficulty of procuring a sufficient quantity of rock crystal very pure and regularly crystallized, I have been reduced to determine this maximum of recession on another substance, and I have chosen for this purpose the ferriferous carbonate of lime, a substance whose primitive form is a rhombohedron, which differs from that of rock crystal only in the angles formed by its terminating planes. As we have already observed, there is a sufficiently great analogy between the phænomena presented by these two substances, with respect to sonorous vibrations, to enable us to admit that what occurs in one occurs also in the other: thus, let $AE$, fig. 6, be a rhombohedron of carbonate of lime, of which $A$ is one of the obtuse solid angles; $ABCD$ corresponding to the face of cleavage of the pyramid of rock crystal, the diagonal $BD$ will be the line round which all the plates must be supposed to be cut; and they are consequently normal to $ACEG$, represented separately in fig. 7, in which the lines 1, 2, 3, &c., are their projections, and indicate at the same time the angles which they make with the axis $AE$. We will first remark that the modes of division of the plate No. 1, fig. 7, bis, perpendicular to the axis, are the same as those of the corresponding plate of rock crystal, and that the plate No. 5, perpendicular to $AC$, assumes also the same modes of division as the plate perpendicular to the cleavable face of the pyramid of rock crystal, which establishes a sufficient analogy between the two orders of phænomena. The inspection of fig. 7, bis, shows then that the branches of the nodal hyperbola of No. 3, parallel to $AG$, consequently to the plane $BDFH$, are straighter than those of the plates which precede or follow it; and admitting that this maximum of recession occurs equally in quartz for the corresponding diagonal plane of its rhombohedron, as this plane forms with the cleavable face of the pyramid an angle of 96° 0′ 13″, the plate in question will be inclined 57° 40′ 13″ to the axis of the crystal, the face of the pyramid forming with this axis an angle of 38° 20′; thus the projection of this plate on the plane $mnXopY$ of fig. 3. will be the line $AB$.
Now since the maximum of recession of the summits of the nodal hyperbola is in this manner determined, it is easy to recognise a great analogy between the phænomena of fig. 8, Pl. III., and those of fig. 3, bis,
(Improve this image)
Savart on the Elasticity of Crystallized Bodies.
​Pl. IV.; for, supposing several intermediate plates between Nos. 3 and 4, that which would be inclined 57° to the axis would correspond to No. 1 of fig. 8, Pl. III.; No. 4 in the crystal would correspond to No. 3 in the wood; and lastly, No. 11 of the crystal plates, in which a second maximum of recession of the summits of the hyperbola occurs, would correspond to No. 6 of the plates of wood; so that the same phænomena, which are included, in a body having three rectangular axes of elasticity, in an arc only of 90°, to be afterwards reproduced in a contrary direction in the following quadrant, are included in rock crystal in an arc of 96° 0′ 13″, and cannot be entirely reproduced, because similar phænomena to those we have just observed for a series of plates cut round $ab$, fig. 1 , Pl. IV., occurring, for the same degrees of inclination, in the two series of plates which might be cut round $cd$ and $ef$, both are confounded together in the vicinity of the plate perpendicular to $XY$.
$ac$
$be^{\prime }Yb^{\prime }eX$
These plates present phænomena much more complicated than those of the two preceding series. It may be easily conceived that this ought to be the case, since the plates parallel to the two adjacent faces of the pyramid assume very different modes of division, which supposes that their elastic state also greatly differs: consequently the plates perpendicular to the plane which passes through the two opposite edges of the hexahedron ought to participate in the properties of both. It is thus that the plates perpendicular to two parallel faces of the prism, and passing through its axis, assume a disposition of nodal lines in which the direction of the planes of cleavage, parallel to one of the faces of the pyramid, exercises a considerable influence.
In the plates of this series (fig. 4, bis,) neither mode of division is constant; nevertheless, in order that they may be easily distinguished from eacli other, I have continued to indicate them, one by uninterrupted lines, the other by dotted lines. And for the purpose of preserving, in all the plates, the projection $xy$ of the axis parallel to the axis $XY$ of fig. 1, I have here supposed that the crystal had been turned round until its edge $be^{\prime }$ was in front. This is besides sufficiently indicated by figure A, which represents the modes of division of the plate perpendicular to the axis, as it also does the section of the hexahedron by a plane parallel to this plate.
The inspection of figures A, B, C, D, E. … shows that the nodal system indicated by the perfect lines is formed of two hyperbolic branches which at first straighten themselves, and the summits of which recede more from each other, so far as the plate E inclined 51° to the axis, ​beyond which they approach each other until they coalesce in $K$, after which they again diverge until the plate $N$, which is parallel to the axis.
The nodal system indicated by the dotted lines follows another course; the summits of the two curves which compose it at first recede from each other, but they soon reapproach each other, and these curves transform themselves into two straight lines in the plate $E$, where the curves of the other mode of division attain their maximum of recession: beyond this limit they separate, but in a perpendicular direction to that in which they approached, and they attain their maximum of recession towards the plate $H$, for which the two systems of curves are nearly similar: they afterwards approach each other, and like those of the other system, they transform themselves, in $K$, into two straight lines, which intersect each other at right angles. Lastly, starting from this point, they diverge again, until the plate $N$, for which the two systems again become equal, assuming, with respect to the axis of the crystal, a direction different from that which they had taken at $I$ and at $H$. I must observe that my supply of rock crystal having failed at the end of my experiments, I have not been able to cut the plate $K$; but the transformations of the nodal lines so clearly indicate that there ought to be a plate which presents these modes of division, that I have not hesitated to admit its existence.
The course which the two sounds follow, in this series of plates, is much more simple than that of the nodal figures: at first those of the dotted system become lower, commencing with the plate $A$, and proceeding as far as the plate $E$, inclined 51° to the axis, and which gives the sound $C$ like the plate No. 4, inclined the same number of degrees to the axis; aftem^ards the sound of this system gradually ascends until the plate $N$ parallel to the axis, where it attains its maximum of elevation. As to the sounds of the other series of modes of division, it is observed that they gradually ascend from the plate perpendicular to the axis unto $K$, in which the nodal systems both consist of lines crossed rectangularly, and that they afterwards descend again until the plate $N$ parallel to the axis. It is obvious that it is not necessary to examine such plates as $A^{\prime }$, $B^{\prime }$, $C^{\prime }$, $D^{\prime }$, fig. 4, since they ought to present the same phænomena as the corresponding plates $A$, $B$, $C$, $D$: only, that which was inclined to the right of the axis in the plates $B$, $C$, $D$ is found inclined to the left in the plates $B^{\prime }$, $C^{\prime }$, $D^{\prime }$.
There is none of the modes of division of this series which is not analogous to some one of those which have been presented to us by bodies in which there are evidently three rectangular axes of elasticity; nevertheless, considered all together, the transformations we have just described present peculiarities which do not exist in the fourth series of plates of wood, fig. 14, Pl. III. The most striking consists in this, that in the transformations of this last series, none of the systems, except ​the first and the last, was rectangular, whilst in rock crystal this mode of division may establish itself.
Summary.
1st. The elasticity of all the diameters of any plane perpendicular to the axis of a prism of rock crystal, may be considered as being sensibly the same.
2nd. All the planes parallel to the axis are far from possessing the same elastic state; but if any three of these planes be taken, restricting ourselves only to this condition, that the angles which they form with each other are equal, then their elastic state is the same.
3rd. The transformations of the nodal lines of a series of plates cut round one of the edges of the base of the prism are perfectly analogous to those which are observed in a series of plates cut round the intermediate axis in bodies which possess three unequal and rectangular axes of elasticity.
4th. The transformations of a series of plates perpendicular to any one of the three planes which pass through two opposite edges of the hexahedron are, in general, analogous to those of a series of plates cut round a line which divides into two equal parts the plane angle included between two of the three axes of elasticity in bodies where these axes are unequal and rectangular.
5th. By means of the acoustic figures of a plate cut in a prism of rock crystal, nearly parallel to the axis, and not parallel to the two faces of the hexahedron, we can always distinguish which are the faces of the pyramid susceptible of cleavage. The same result may be obtained by the disposition of the modes of division of a plate taken nearly parallel to one of the faces of the pyramids.
6th. Whatever be the direction of the plates, the optical axis, or its projection on their plane, always occupies a position on them which is intimately connected with the arrangement of the acoustic lines: thus, for example, in all the plates cut round one of the edges of the base of the prism, the optical axis, or its projection, invariably corresponds with one of the two straight lines which compose the nodal system formed of two lines which intersect each other rectangularly.
Although there is doubtless a great analogy between the phænomena which rock crystal has just presented to us, and those we have observed in bodies in which the elasticity is different according to three directions perpendicular to each other, nevertheless we are forced to acknowledge that, with respect to the mode of experiment we have employed in these researches, rock crystal cannot be placed in the number of substances with three rectangular and unequal axes of elasticity, and still less in the number of those all the parts of which are symmetrically arranged round a single straight line. For the same phænomena ​are constantly reproduced in it in three different positions; and it seems that everything in it has reference to the different directions of cleavage, to the faces, and to the edges of the primitive rhombohedron. Thus all the plates cut parallel to the natural faces of the hexahedron possess exactly the same properties, and these properties are very different from those of the plates equally parallel to the axis, but which are normal to two faces of the hexahedron. Likewise, the plates parallel to the cleavable faces of the pyramid produce the same sounds, and exhibit the same acoustic figures; whilst the plates parallel to the three other faces present figures different from those of the preceding plates. It appears therefore to result from this identity of phænomena for three distinct positions, that there are in rock crystal three systems of axes or principal lines of elasticity.
But in this point of view, what would be the precise directions of these axes for each system? This can, to a certain point, be determined by comparing the phenomena we have observed in rock crystal with those presented to us by wood. For, all the plates cut round one of the edges which result from the junction of a face of the pyramid with the adjacent face of the hexahedron, producing a nodal system composed of two lines cutting each other rectangularly, one of which always corresponds with the edge in question; and the transformations of the acoustic lines in it being entirely analogous to those of a series of plates cut round the intermediate axis in wood, it follows that this edge, which is nothing else than the great diagonal of the primitive rhombohedron, ought to be regarded as the intermediate axis of elasticity. In the next place, as the maximum of straightening and of deviation of the branches of the nodal hyperbola takes place in the plate No. 11 , fig. 3, bis, parallel to the cleavable face of the pyramid, and as at the same time this plate is a limit for the sounds which it produces, it is equally natural to suppose that it ought also to contain in its plane another axis of elasticity, which can correspond only to the second of the crossed nodal lines, that is to say, to that which serves as the second axis of the nodal hyperbola, and which is, at the same time, the smaller diagonal of the lozenge face of the primitive rhombohedron. This line may therefore be considered as the axis of greatest elasticity of each system. Lastly, following the same analogy, as the plate which is cut parallel to the diagonal plane, the intersection of which with the lozenge face of the rhombohedron forms its great diagonal, is besides a maximum of deviation for the summits of the nodal hyperbola, it must thence be concluded that this plane contains the axis of least elasticity, and, at the same time, that this axis is perpendicular to the intermediate axis, and forms with that of greatest elasticity an angle of 57° 40′ 13″, since such is the inclination of the face of the rhombohedron to the diagonal plane. Thus, first, the axis of ​greatest elasticity and the intermediate axis are contained in the plane which forms the face of the rhombohedron, and they are perpendicular to each other; secondly, the intermediate axis and the axis of least elasticity are contained in the diagonal plane, and they are in like manner perpendicular to each other.
Such are the consequences to which the analogy observed between the successive transformations of the nodal lines in plates of wood and of rock crystal seems to lead. The co-existence of three systems of axes of elasticity in the latter body, introduces however so great a complication in the particulars of the phænomenon, especially in the progression of the sounds, that the elastic state of this substance can only be definitively determined by a method analogous to that which I have above employed for wood, that is to say, by comparing together the numbers of vibrations of a series of small rods of the same dimensions, and cut according to the different directions in which the preceding experiments appear to indicate that the elasticity differs the most. Without in the least prejudging the results to which these new researches might lead us, we may even now foresee that there ought to be a great difference between the greatest and the least degree of elasticity in rock crystal, since, among the various plates of beech-wood, a substance in which these two extremes are as one to sixteen, there is none the sounds of which have a greater interval than that of a major third between them, whilst, among the plates of crystal, there are some, the two sounds of which are a fifth from each other.
As we have already remarked above, the transparent carbonate of lime and the ferriferous carbonate of lime appear to possess elastic properties which are, for the most part, analogous to those of rock crystal; three systems of principal lines of elasticity, which appear exactly similar to each other, are likewise recognised in them; but the extreme facility with which carbonate of lime may be cleaved, enables us to discover in it a peculiarity which cannot be perceived in rock crystal, and which may explain why it is that the plates cut round one of the edges of the base of the hexahedron, all present a nodal system composed of two lines crossed rectangularly.
It is well known that the rhombohedron of carbonate of lime is frequently susceptible of a mechanical division according to the directions parallel to its diagonal planes; now, these planes cutting each other perpendicularly two and two, the intersection of each of these pairs with the lozenge faces of the crystal, forms the great and small diagonal of each of them, so that, if a plane be imagined which turns round the great diagonal, it ought always to remain normal to the supernumerary joint which passes through the small diagonal. It hence results that, if a series of plates be cut round the same line, their structure, considered in the different directions of their plane, will differ according to two ​directions perpendicular to each other; this explains the production of the nodal lines crossed at right angles, as in the series of plates cut round one of the axes of elasticity, in bodies in which these axes are rectangular. It appears therefore that we may conclude from this observation that rock crystal possesses, like carbonate of lime, supernumerary planes of cleavage parallel to the diagonal planes of its primitive rhombohedron, and that it is to the existence of these supernumerary joints that the principal peculiarities of the elastic state of this substance must be attributed.
The only striking difference there appears to be between the structure of carbonate of lime and that of quartz consists in this, that, in the first of these substances, the small diagonal of the rhombohedron is the axis of least elasticity, whilst it is that of greatest elasticity in the second. To be convinced of the accuracy of this assertion, it is sufficient to cut, in a rhombohedron of carbonate of lime, a plate taken parallel to one of its natural faces, and to examine the arrangement of its two nodal systems, one of which consists of two lines crossed rectangularly, which are always placed on the diagonals of the lozenge, the primitive outline of the plate, and the other is formed of two hyperbolic branches, to which the preceding lines serve as axes (see fig. 7, bis, No. 6); but with this peculiarity, that it is the small diagonal which becomes the first axis of the hyperbola, whilst it is its second axis in the corresponding plate of rock crystal (see fig. 3, bis, No. 11). It may be here asked how far this difference of structure may influence the phænomena of light which are peculiar to each of these two substances, one of which is a crystal with attractive (positive) double refraction, and the other with repulsive (negative) double refraction.
It appears, therefore, to result from this approximation between the phænomena presented by carbonate of lime and rock crystal, with respect to sonorous vibrations, that the arrangement of the acoustic figures, and the numbers of vibrations by which they are accompanied, are always found intimately connected with the directions of cleavage in each plate; and it may be said in general, that if these directions intersect each other at right angles, in the plane of the plate, one of the two modes of division will always consist of two lines crossed rectangularly; whilst if they are inclined to each other the two nodal systems will be hyperbolic curves.
The disposition of the nodal lines upon circular plates of sulphate of lime gives additional support to tliis conclusion. For thin plates of this substance break according to two directions inclined to each other at 113° 8′; and experiment shows that the two modes of division of which they are susceptible are two nearly similar hyperbolic curves, one of which appears to have for its asymptotes the directions of cleavage, and the other for its principal axis that one of these two directions in ​which the plates do not break clean off; for there is, it is well known, an obvious difference in the manner in which sulphate of lime breaks according to one direction or the other. We will, on concluding, remark, that these modes of division are precisely the same as those of a disc of rock crystal parallel to the axis and perpendicular to two faces of the hexahedron, and that the mean of the optic axes in sulphate of lime occupies in it the same position relatively to the nodal curves, as the projection of the single axis of rock crystal assumes in that of the plates of this substance of which we have just spoken. (See fig. 2, bis, No. 3.)
The preceding researches are, doubtless, far from deserving to be considered as a complete examination of the elastic state of rock crystal and of carbonate of lime; nevertheless we hope they will be sufficient to show that the mode of experiment we have employed may hereafter become a powerful means of studying the structure of solid bodies, regularly or even confusedly crystallized. Thus, for instance, the relations which exist between the modes of division and the primitive form of crystals allow us to presume that the primitive form of certain substances which do not at all yield to a mere mechanical division may be determined by sonorous vibrations. It is equally natural to think that less imperfect notions respecting the elastic state and cohesion of crystals than those we now possess, may throw light upon many peculiarities of crystallization: for example, it is not impossible that the degrees of elasticity of a determinate substance may not be exactly the same, for the same direction referred to the primitive form, when the secondary form is different; and, if it be so, as some facts induce me to suspect, the determination of the elastic state of crystals will lead to the explanation of the most complicated phænomena of the structure of bodies. Lastly, it appears that the comparison of the results furnished concerning the constitution of bodies, on the one hand by means of light, and on the other hand by means of sonorous vibrations, ought necessarily to contribute to the progress of light itself, as well as to that of acoustics.