FOOTNOTES:

Fig. 13.

§ 36. Without for a moment imagining the molecules of matter to be hard solids of convex shape, we may derive valuable lessons in the tactics of real crystals by studying the assemblage described in §§24and25and represented in Figs. 12 and 13. I must for the presentforego the very attractive subject of the tactics presented by faces not parallel to one or other of the four faces of the primitive tetrahedrons which we found in§ 24, and ask you only to think of the two sides of a plate of crystal parallel to any one of them, that is to say, an assemblage of such layers as those represented geometrically in Fig. 12 and shown in stereoscopic view in Fig. 13. If, as is the case with the solids10photographed inFig. 13, the under side of each solid is nearly plane but slightly convex, and the top is somewhat sharply curved, we have the kind of difference between the upper and under of the two parallel sides of the crystal which I have already described to you in§ 21above. In this case the assemblage is formed by letting the solids fall down from above and settle in the hollows to which they come most readily, or which give them the stablest position. It would, we may suppose, be the hollowsp′ q′ r′, notp q r, (Fig. 12) that would be chosen; and thus, of the two formations described in§ 25, we should have that in which the hollows abovep′ q′ r′are occupied by the comparatively flat under sides of the molecules of the layer above, and the hollows below the aperturesp q rby the comparatively sharp tops of the molecules of the layers below.

§ 37. For many cases of natural crystals of the wholly asymmetric character, the true forces between the crystalline molecules will determine precisely the same tactics of crystallization as would be determined by the influence of gravity and fluid viscosity in the settlement from water, of sand composed of uniform molecules of the wholly unsymmetrical convex shape represented in Figs.12and13. Thus we can readily believe that a real crystal which is growing by additions to the face seen inFig. 12, would give layer after layer regularly as I have just described. But if by some change of circumstances the plate, already grown to a thickness of many layers in this way, should come to have the side facingfromus in the diagram exposed to the mother-liquor, or mother-gas, and begin to grow from that face, the tactics might probably be that each molecule would find its resting-place with its most nearly plane side in the wider hollows underp′ q′ r′, instead of with its sharpest corner in the narrower and steeper hollows underp q r, as are the molecules in the layer below that shown in the diagram in the first formation. The result would be a compound crystal consisting of two parts, of different crystalline quality, cohering perfectly together on the two sides of an interfacial plane. It seems probable that this double structure may be found in nature, presented by crystalsof the wholly unsymmetric class, though it may not hitherto have been observed or described in crystallographic treatises.

Fig. 14.

§ 38.This asymmetric double crystal becomes simply the well-known symmetrical ‘twin-crystal’11in the particular case in which each of the constituent molecules is symmetrical on the two sides of a plane through it parallel to the plane of our diagrams, and also on the two sides of some plane perpendicular to this plane. We see, in fact, that in this case if we cut in two the double crystal by the plane of Fig. 14, and turn one part ideally through 180° round the intersection of these two planes, we bring it into perfect coincidence with the other part.

This we readily understand by looking at Fig. 14, in which the solid shown in outline may be either an egg-shaped figure of revolution, or may be such a figure flattened by compression perpendicular to the plane of the diagram. The most readily chosen and the most stable resting-places for the constituents of each successive layer might be the wider hollowsp′ q′ r′: and therefore if, from a single layer to begin with, the assemblage were to grow by layer after layer added to it on each side, it might probably grow as a twin-crystal. But it might also be that the presence of a molecule in the wider hollowp′ q′ r′on one side, might render the occupation of the corresponding hollow on the other side by another molecule less probable, or even impossible. Hence, according to the configuration and the molecular forces of the particular crystalline molecule in natural crystallization, there may be necessarily, or almost necessarily, the twin, when growth proceeds simultaneously on the two sides: or the twin growth may be impossible, because the first occupation of the wider hollows on one side, may compel the continuity of the crystalline quality throughout, by leaving only the narrower hollowsp q rfree for occupation by molecules attaching themselves on the other side.

§ 39. Or the character of the crystalline molecule may be such that when the assemblage grows by the addition of layer after layer on one side only, with a not very strongly decided preference to the wider hollowsp′ q′ r′, some change of circumstances may cause the molecules of one layer to place themselves in a hollowp q r. The molecules in the next layer after this would find the hollowsp′ q′ r′occupied on the far side, and would thus have a bias in favour of the hollowsp q r. Thus layer after layer might be added, constituting a twinned portion of the growth, growing, however, with less strong security for continued homogeneousness than when the crystal was growing, as at first, by occupation of the wider hollowsp′ q′ r′. A slight disturbance might again occur, causing the molecules of a fresh layer to settle, not in the narrow hollowsp q r, but in the wider hollowsp′ q′ r′, notwithstanding the nearness of molecules already occupying the wider hollows on the other side. Disturbances such as these occurring irregularly during the growth of a crystal, might produce a large number of successive twinnings at parallel planes with irregular intervals between them, or a large number of twinnings in planes at equal intervals might be produced by some regular periodic disturbance occurring for a certain number of periods, and then ceasing. Whether regular and periodic, or irregular, the tendency would be that the number of twinnings should be even, and that after the disturbances cease the crystal should go on growing in the first manner, because of the permanent bias in favour of the wider hollowsp′ q′ r′. These changes of molecular tactics, which we have been necessarily led to by the consideration of the fortuitous concourse of molecules, are no doubt exemplified in a large variety of twinnings and counter-twinnings found in natural minerals. In the artificial crystallization of chlorate of potash they are of frequent occurrence, as is proved, not only by the twinnings and counter-twinnings readily seen in the crystalline forms, but also by the brilliant iridescence observed in many ofthe crystals found among a large multitude, which was investigated scientifically by Sir George Stokes ten years ago, and described in a communication to the Royal Society ‘On a remarkable phenomenon of crystalline reflection’ (Proc. R.S., vol. xxxviii, 1885, p. 174).

§ 40. A very interesting phenomenon, presented by what was originally a clear homogeneous crystal of chlorate of potash, and was altered by heating to about 245°-248° Cent., which I am able to show you through the kindness of Lord Rayleigh, and of its discoverer, Mr. Madan, presents another very wonderful case of changing molecular tactics, most instructive in respect of the molecular constitution of elastic solids. When I hold this plate before you with the perpendicular to its plane inclined at 10° or more to your line of vision, you see a tinsel-like appearance, almost as bright as if it were a plate of polished silver, on this little area, which is a thin plate of chlorate of potash cemented for preservation between two pieces of glass; and, when I hold a light behind, you see that the little plate is almost perfectly opaque like metal foil. But now when I hold it nearly perpendicular to your line of vision the tinsel-like appearance is lost. You can see clearly through the plate, and you also see that very little light is reflected from it. As a result both of Mr. Madan’s own investigations, and further observations by himself, Lord Rayleigh came to the conclusion that the almost total reflection of white light which you see is due to the reflection of light at many interfacial planes between successive layers of twinned and counter-twinned crystal of small irregular thicknesses, and not to any splits or cavitiesor any other deviation from homogeneousness than that presented by homogeneous portions of oppositely twinned-crystals in thorough molecular contact at the interfaces.

§ 41. When the primitive clear crystal was first heated very gradually by Madan to near its melting-point (359° according to Carnelly), it remained clear, and only acquired the tinsel appearance after it had cooled to about 245° or 248°12. Rayleigh found that if a crystal thus altered was again and again heated it always lost the tinsel appearance, and became perfectly clear at some temperature considerably below the melting-point, and regained it at about the same temperature in cooling. It seems, therefore, certain that at temperatures above 248°, and below the melting-point, the molecules had so much of thermal motions as to keep them hoveringabout the positions ofp q r,p′ q′ r′, of our diagrams, but not enough to do away with the rigidity of the solid; and that when cooled below 248° the molecules were allowed to settle in one or other of the two configurations, but with little of bias for one in preference to the other. It is certainly a very remarkable fact in Natural History, discovered by these observations, that, when the molecules come together to form a crystal out of the watery solution, there should be so much more decided a bias in favour of continued homogeneousness of the assemblage than when, by cooling, they are allowed to settle from their agitations in a rigid, but nearly melting, solid.

§ 42. But even in crystallization from watery solution of chlorate of potash the bias in favour of thorough homogeneousness is not in every contingency decisive. In the first place, beginning, as the formation seems to begin, from a single molecular plane layer such as that ideally shown inFig. 14, it goes on, not to make a homogeneous crystal on the two sides of this layer, but probably always so as to form a twin-crystal on its two sides, exactly as described in§ 38, and, if so, certainly for the reason there stated. This is what Madan calls the ‘inveterate tendency to produce twins (such as would assuredly drive a Malthus to despair)13’; and it is to this that he alludes as ‘the inevitable twin-plate’ in the passage from his paper given in the foot-note to § 41 above.

§ 43. In the second place, I must tell you that many of the crystals produced from the watery solution by the ordinary process of slow evaporation and crystallization, show twinnings and counter-twinnings atirregular intervals in the otherwise homogeneous crystal on either one or both sides of the main central twin-plane, which henceforth, for brevity, I shall call (adopting the hypothesis already explained, which seems to me undoubtedly true) the ‘initial plane.’ Each twinning is followed, I believe, by a counter-twinning at a very short distance from it; at all events Lord Rayleigh’s observations14prove that the whole number of twinnings and counter-twinnings in a thin disturbed stratum of the crystal on one side of the main central twin-plane is generally, perhaps always, even; so that, except through some comparatively very small part or parts of the whole thickness, the crystal on either side of the middle or initial plane is homogeneous. This is exactly the generally regular growth which I have described to you (§ 39) as interrupted occasionally or accidentally by some unexplained disturbing cause, but with an essential bias to the homogeneous continuance of the more easy or natural one of the two configurations.

§ 44. I have now great pleasure in showing you a most interesting collection of the iridescent crystals of chlorate of potash, each carefully mounted for preservation between two glass plates, which have been kindly lent to us for this evening by Mr. Madan. In March, 1854, Dr. W. Bird Herapath sent to Prof. Stokes some crystals of chlorate of potash showing the brilliant and beautiful colours you now see, and, thirty years later, Prof. E. J. Mills recalled his attention to the subject by sending him ‘a fine collection of splendidly coloured crystals of chlorate of potash of considerable size, several of the plates having an areaof a square inch or more, and all of them being thick enough to handle without difficulty.’ The consequence was that Stokes made a searching examination into the character of the phenomenon, and gave the short, but splendidly interesting, communication to the Royal Society of which I have already told you. The existence of these beautifully coloured crystals had been well known to chemical manufacturers for a long time, but it does not appear that any mention of them was to be found in any scientific journal or treatise prior to Stokes’ paper of 1885. He found that the colour was due to twinnings and counter-twinnings in a very thin disturbed stratum of the crystal showing itself by a very fine line, dark or glistening, according to the direction of the incident light when a transverse section of the plate of crystal was examined in a microscope. By comparison with a spore of lycopodium he estimated that the breadth of this line, and therefore the thickness of the disturbed stratum of the crystal, ranged somewhere about the one-thousandth of an inch. He found that the stratum was visibly thicker in those crystals which showed red colour than in those which showed blue. He concluded that ‘the seat of the coloration is certainly a thin twinned stratum’ (that is to say, a homogeneous portion of crystal between a twinning and a counter-twinning), and found that ‘a single twin-plane does not show anything of the kind.’

§ 45. A year or two later Lord Rayleigh entered on the subject with an exhaustive mathematical investigation of the reflection of light at a twin-plane of a crystal (Philosophical Magazine, September, 1888), by the application of which, in a second paper ‘On the remarkablephenomenon of Crystalline Reflection described by Prof. Stokes,’ published in the same number of thePhilosophical Magazine, he gave what seems certainly the true explanation of the results of Sir George Stokes’ experimental analysis of these beautiful phenomena. He came very decidedly to the conclusion that the selective quality of the iridescent portion of the crystal, in virtue of which it reflects almost totally light nearly of one particular wave-length for one particular direction of incidence (on which the brilliance of the coloration depends), cannot be due to merely a single twin-stratum, but that it essentially is due to a considerable number of parallel twin-strata at nearly equal distances. The light reflected by this complex stratum is, for any particular direction of incident and reflected ray, chiefly that of which the wave-length is equal to twice the length of the period of the twinning and counter-twinning, on a line drawn through the stratum in the direction of either the incident or the reflected ray.

§ 46. It seems to me probable that each twinning is essentially followed closely by a counter-twinning. Probably three or four of these twin-strata might suffice to give colour; but in any of the brilliant specimens as many as twenty or thirty, or more, might probably be necessary to give so nearly monochromatic light as was proved by Stokes’ prismatic analysis of the colours observed in many of his specimens. The disturbed stratum of about a one-thousandth of an inch thickness, seen by him in the microscope, amply suffices for the 5, 10, or 100 half wave-lengths required by Rayleigh’s theory to account for perceptible or brilliant coloration. But whatcanbe the cause of any approach to regularperiodicity in the structure sufficiently good to give the colours actually observed? Periodical motion of the mother-liquor relatively to the growing crystal might possibly account for it. But Lord Rayleigh tells us that he tried rocking the pan containing the solution without result. Influence of light has been suggested, and I believe tried, also without result, by several enquirers. We know, by the beautiful discovery of Edmond Becquerel, of the prismatic colours photographed on a prepared silver plate by the solar spectrum, that ‘standing waves’ (that is to say, vibrations with stationary nodes and stationary places of maximum vibration), due to co-existence of incident and reflected waves, do produce such a periodic structure as that which Rayleigh’s theory shows capable of giving a corresponding tint when illuminated by white light. It is difficult, therefore, not to think that light may be effective in producing the periodic structure in the crystallization of chlorate of potash, to which the iridescence is due. Still, experimental evidence seems against this tempting theory, and we must perforce be content with the question unanswered:—What can be the cause of 5, or 10, or 100 pairs of twinning and counter-twinning following one another in the crystallization with sufficient regularity to give the colour: and why, if there are twinnings and counter-twinnings, are they not at irregular intervals, as those produced by Madan’s process, and giving the observed white tinsel-like appearance with no coloration?

§ 47. And now I have sadly taxed your patience: and I fear I have exhausted it and not exhausted my subject! I feel I have not got halfway through what I hoped I might be able to put before you this eveningregarding the molecular structure of crystals. I particularly desired to speak to you of quartz crystal with its ternary symmetry and its chirality15; and to have told you of the etching16by hydrofluoric acid which, as it were, commences to unbuild the crystal by taking away molecule after molecule, but not in the reverse order of the primary up-building; and which thus reveals differences of tactics in the alternate faces of the six-sided pyramid which terminates at either end, sometimes at both ends, the six-sided prism constituting generally the main bulk of the crystal. I must confine myself to giving you a geometrical symbol for the ternary symmetry of the prism and its terminal pyramid.

Fig. 15.

§ 48.Make an equilateral equiangular hexagonal prism, with its diagonal from edge to edge ninety-five hundredths17of its length. Place a number of these close together, so as to make up a hexagonal plane layer with its sides perpendicular to the sides of the constituent hexagonal prisms: see Fig. 15 and imagine the semicircles replaced by their diameters. You see in each side of the hexagonal assemblage, edges of the constituent prisms, and you see at each corner of the assemblage a face (not an edge) ofoneof the constituent prisms. Build up a hexagonal prismatic assemblage by placing layer after layer over it with the constituent prisms of each layer vertically over those in the layer below; and finish the assemblagewith a six-sided pyramid by building upon the upper end of the prism, layer after layer of diminishing hexagonal groups, each less by one circumferential row than the layer below it. You thus have a crystal of precisely the shape of a symmetrical specimen of rock crystal, with the faces of its terminal pyramid inclined at 38° 13′ to the faces of the prism from which they spring. But the assemblage thus constituted has ‘senary’ (or six-rayed symmetry). To reduce this to ternary symmetry, cut a groove through the middle of each alternate face of the prismatic molecule, making this groove in the first place parallel to the edges: and add a corresponding projection, or fillet, to the middles of the other three faces, so that two of the cylinders similarly oriented would fit together, with the projecting fillet on one side of one of them entering the groove in the anti-corresponding side of the other. The prismatic portion of the assemblage thus formed shows (see Fig. 15), on its alternate edges, faces ofmolecules with projections and faces of molecules with grooves; and shows only orientational differences between alternate faces, whether of the pyramid or of the prism. Having gone only so far from ‘senary’ symmetry, we have exactly the triple, or three-pair, anti-symmetry required for the piezo-electricity of quartz investigated so admirably by the brothers Curie18, who found that a thin plate of quartz crystalFig. 16. Fig. 17.cut from any position perpendicular to a pair of faces of a symmetrical crystal, becomes positively electrified on one side and negatively on the other when pulled in a direction perpendicular to those faces. But this assemblage has not the chiral piezo-electric quality discovered theoretically by Voigt19, and experimentally in quartz and in tourmaline by himself and Riecke20, nor the well-known optic chirality of quartz.

§ 49. Change now the directions of the grooves and fillets to either of the oblique configurations shown in Fig. 16, which I call right-handed, because the directions of the projections are tangential to the threads of a three-thread right-handed screw, and Fig. 17 (left-handed). The prisms with their grooves and fillets will still all fit together if they are all right-handed, or all left-handed.

Fig. 18.

Fig. 18 shows the upper side of a hexagonal layer of an assemblage thus composed of the right-handed molecule ofFig. 16.Fig. 15unchanged, still represents a horizontal section through the centres of the molecules. A prism built up of such layers, and finished at each end with a pyramid according to the rule of§ 48, has all the qualities of ternary chiral symmetry required for the piezo-electricity of quartz; for the orientational differences of the alternate pairs of prismatic faces; for the absolute difference between the alternate pairs of faces of each pyramid which are shown in the etching by hydrofluoric acid; for the merely orientational difference between the parallel faces of the two pyramids; and for the well-known chiro-optic21property of quartz. Look at two contiguous facesA,Bof our geometrical model quartz crystal now before you, with its axis vertical. You will see a difference between them: turn it upsidedown;Bwill be undistinguishable from whatAwas, andAwill be undistinguishable from whatBwas. Look at the two terminal pyramids, and you will find that the face aboveAand the face belowBare identical in quality, and that they differ from the face aboveBand belowA. This model is composed of the right-handed constituent molecules shown inFig. 16. It is so placed before you that the edge of the prismatic part of the assemblage nearest to you shows you filleted faces of the prismatic molecules. You see two pyramidal faces; the one to your right hand, overB, presents complicated projections and hollows at the corners of the constituent molecules; and the pyramidal face next your left hand, overA, presents their unmodified corners. But it will be the face next your left hand which will present the complex bristling corners, and the face next your right hand that will present the simple corners, if, for the model before you, you substitute a model composed of left-handed molecules such as those shown inFig. 17.

§ 50. To give all the qualities of symmetry and anti-symmetry of the pyro-electric and piezo-electric properties of tourmaline investigated theoretically by Voigt22, and experimentally by himself and Friecke23, make a hollow in one terminal face of each of our constituent prisms, and a corresponding projection in its other terminal face.

§ 51. Coming back to quartz, we can now understand perfectly the two kinds of macling which are well known to mineralogists as being found in many natural specimens of the crystal, and which I call respectivelythe orientational macling, and the chiral macling. In the orientational macling all the crystalline molecules are right-handed, or all left-handed; but through all of some part of the crystal, each of our component hexagonal prisms is turned round its axis through 60° from the position it would have if the structure were homogeneous throughout. In each of the two parts the structure is homogeneous, and possesses all the electric and optic properties which any homogeneous portion of quartz crystal presents, and the facial properties of natural uncut crystal, shown in the etching by hydrofluoric acid; but there is a discontinuity at the interface, not generally plane, between the two parts, which in our geometrical model would be shown by non-fittings between the molecules on the two sides of the interface, while all the contiguous molecules in one part, and all the contiguous molecules in the other part, fit into one another perfectly. In chiral macling, which is continually found in amethystine quartz, and sometimes in ordinary clear quartz crystals, some parts are composed of right-handed molecules, and others of left-handed molecules. It is not known whether, in this chiral macling, there is or there is not also the orientational macling on the two sides of each interface; but we may say probablynot; because we know that the orientational macling occurs in nature without any chiral macling, and because there does not seem reason to expect that chiral macling would imply orientational macling on the two sides of the same interface. I would like to have spoken to you more of this most interesting subject; and to have pointed out to you that some of the simplest and most natural suppositions we can make as to the chemical forces (or electrical forces,which probably means the same thing) concerned in a single chemical molecule of quartz,SiO2, and actingFig. 19.between it and similar neighbouring molecules, would lead essentially to these molecules coming together in triplets, each necessarily either right-handed or left-handed, but with as much probability of one configuration as of the other: and to have shown you that these triplets of silica 3(SiO2) can form a crystalline molecule with all the properties of ternary chiral symmetry, typified by our grooved hexagonal prisms, and can build up a quartz crystal by the fortuitous concourse of atoms. I should like also to have suggested and explained the possibility that a right-handed crystalline molecule thus formed may, in natural circumstances of high temperature, or even of great pressure, become changed into a left-handed crystal, orvice-versa. My watch, however, warns me that I must not enter on this subject.

§ 52. Coming back to mere molecular tactics of crystals, remark that our assemblage of rounded, thoroughly scalene, tetrahedrons, shown in the stereoscopic picture (§ 36,Fig. 13above), essentially has chirality because each constituent tetrahedron, if wholly scalene, has chirality24. I should like to have explained to you how a single or double homogeneous assemblage of points has essentially no chirality, and how three assemblages of single points, or a single assemblage of triplets of points, can have chirality, though a single triplet of points cannot have chirality. I should like indeed to have brought somewhat thoroughly before you the geometrical theory of chirality; and inillustration to have explained the conditions under which four points, or two lines, or a line and two points, or a combination of point, line and plane, can have chirality: and how a homogeneous assemblage of non-chiral objects can have chirality; but in pity I forbear, and I thank you for the extreme patience with which you have listened to me.

1Seefoot-noteon § 22 below.

2The holes in the cylinders are bored obliquely, as shown inFig. 4, which causes them to remain at any desired position on the cord and allows them to be freed to move up and down by slackening the cord for a moment.

3‘On the Homogeneous Division of Space,’ by Lord Kelvin,Royal Society Proceedings, vol. lv, Jan. 18, 1894.

4Similar curves are said to be parallel when the tangents to them at corresponding points are parallel.

5Seefoot-noteto § 12 above.

6‘On the Division of Space with Minimum Partitional Area,’Philosophical Magazine, vol. xxiv, 1887, p. 502, andActa Mathematicaof the same year.

7A. Levy,Edinburgh Philosophical Journal, April, 1822; Whewell,Phil. Trans. Royal Society, 1825; Miller,Treatise on Crystallography.

8I call any geometrical figure, or group of points,chiral, and say that it has chirality, if its image in a plane mirror, ideally realized, cannot be brought to coincide with itself. Two equal and similar right hands are homochirally similar. Equal and similar right and left hands are heterochirally similar or ‘allochirally’ similar (but heterochirally is better). These are also called ‘enantiomorphs,’ after a usage introduced, I believe, by German writers. Any chiral object and its image in a plane mirror are heterochirally similar.

9Philosophical Magazine, vol. xx, 1885, second half year, p. 469, andBritish Association Report, 1885, Aberdeen, p. 896.

10The solids of the photograph are castings in fine plaster of Paris from a scalene tetrahedron of paraffin wax, with its corners and edges rounded, used as a pattern.

11‘A twin-crystal is composed of two crystals joined together in such a manner that one would come into the position of the other by revolving through two right angles round an axis which is perpendicular to a plane which either is, or may be, a face of either crystal. The axis will be called the twin-axis, and the plane to which it is perpendicular the twin-plane.’ Miller’sTreatise on Crystallography, p. 103. In the text the word ‘twin-plane,’ quoted from the writings of Stokes and Rayleigh, is used to signify the plane common to the two crystals in each of the cases referred to: and not the plane perpendicular to this plane, in which one part of the crystal must be rotated to bring it into coincidence with the other, and which is the twin-plane as defined by Miller.

12‘A clear transparent crystal of potassium chlorate, from which the inevitable twin-plate had been ground away so as to reduce it to a single crystal film about 1 mm. in thickness, was placed between pieces of mica and laid on a thick iron plate. About 3 cm. from it was laid a small bit of potassium chlorate, and the heat of a Bunsen burner was applied below this latter, so as to obtain an indication when the temperature of the plate was approaching the fusing-point of the substance (359°Caccording to Prof. Carnelly). The crystal plate was carefully watched during the heating, but no depreciation took place, and no visible alteration was observed, up to the point at which the small sentinel crystal immediately over the burner began to fuse. The lamp was now withdrawn, and when the temperature had sunk a few degrees a remarkable change spread quickly and quietly over the crystal plate, causing it to reflect light almost as brilliantly as if a film of silver had been deposited upon it. No further alteration occurred during the cooling; and the plate, after being ground and polished on both sides, was mounted with Canada balsam between glass plates for examination. Many crystals have been similarly treated with precisely similar results; and the temperature at which the change takes place, has been determined to lie between 245° and 248°, by heating the plates upon a bath of melted tin in which a thermometer was immersed. With single crystal plates no decrepitation has ever been observed, while with the ordinary twinned-plates it always occurs more or less violently, each fragment showing the brilliant reflective power above noticed.’—Nature, May 20, 1886.

13Nature, May 20, 1886.

14Philosophical Magazine, 1888, second half year, p. 260.

15Seefoot-noteto § 22 above.

16Widmanstätten, 1807. Leydolt (1855, Wien. Akad. Ber. 15, 59, T. 9, 10. Baumhauer, Pogg. Ann. 138, 563 (1869); 140, 271; 142, 324; 145, 460; 150, 619.) For an account of these investigations, see Mallard,Traité de Crystallographie(Paris, 1884), Tome II, chapitre xvi.

17More exactly .9525, being 3/4 × cot 38° 13′; see p. 53.

18J. and P. Curie and C. Friedel,Comptes Rendus, 1882, 1883, 1886, 1892.

19Allgemeine Theorie der piëzo- und pyroelectrischen Erscheinungen an Krystallen. W. Voigt, Königl. Gesellschaft der Wissenschaften zu Göttingen, August 2, 1890.

20Wiedemann,Annalen, 1892, xlv, p. 923.

21Generally miscalled ‘rotational.’

22Seefoot-note (2)to p. 54 above.

23Seefoot-note (3)to p. 54 above.

24Seefoot-noteto § 22 above.

THE END

OxfordPRINTED AT THE CLARENDON PRESSBY HORACE HART, PRINTER TO THE UNIVERSITY

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