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

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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 , fig. 2, be any plane passing through the axis of the cylinder, and let be a line normal to this plane: it is obvious that the plates taken perpendicularly to , and according to the different directions 1, 2, 3, 4, 5, &c. round , ought to present different phænomena, since they all will contain the axis of least elasticity 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 .

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 contained in the plane , than in the direction 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 , 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 and of which recede more from each other than in fig. 2, will present the remarkable peculiarity of 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 and recede from each other at the same time as the curves become more straightened. In the plate No. 5, parallel to the axis , these curves are no longer susceptible of assuming any other position than that indicated in the figure. Thus, in No. 1, the centres and 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 or , and the place of excitation moves gradually from to , passing over a quarter of the circumference of the plate. When the motion is excited in the vicinity of , 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 , 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 .

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 fig. 5, the face of which I shall suppose is parallel to the ligneous layers, and the edge to the direction of the fibres, which will allow the three edges , , 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 , , 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 or ; 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 , and is such, that being represented by unity, the resistance in the direction becomes 2.25, and 16 in the direction of . 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 and perpendicular to the face 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, , places itself constantly on the axis 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 . As to the hyperbolic system, it undergoes remarkable transformations, which depend on this circumstance, that the line remaining the axis of mean elasticity in all the plates, the line , 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 , , 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 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 is sometimes smaller and sometimes Greater than that which exists in the direction of , 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 , , which possess the greatest elasticity; thus, in Nos. 1 and 2, 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 , 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 , , is that between the two sounds [of which there is the greatest interval: this evidently depends on the elasticity in the two directions , 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 and 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 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 of least elasticity and perpendicular to the plane ; 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, , corresponds with the axis ; 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 than in the direction : thus the hyperbolic system of this series cannot present the transformations which we saw in the preceding series, where , fig. 8, possesses sometimes a less, at other times a greater elasticity than that of . In the present case, remaining constantly the axis of least elasticity, the resistance to flexion in the direction goes on gradually increasing from the plate No. 1 to the plate No. 6 parallel to the plane , 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 of greatest elasticity, and perpendicular to the plane ; 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, , 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 , and perpendicular to the plane ; 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 , and perpendicular to the plane ; figs. 5.

Among all the plates which may be cut round the diagonal 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 , and through the edge , contains the diagonal in its plane; also, the plate No. 4, fig. 10, which passes through one of the diagonals or , and which is perpendicular to the plane , contains also in its plane; and lastly, the plate No. 3 of fig. 12, parallel to the plane , is circumstanced in the same manner. Thus, if , fig. 15, is a plane perpendicular to the diagonal , 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 , the projection of which is in , 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 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, , places itself upon the axis , 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 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 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 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 of greatest elasticity; after which they redescend by degrees in Nos. 2 and 1, (the latter contains the axis 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 , and the other round the diagonal ; 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.