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-Project Gutenberg's The Molecular Tactics of a Crystal, by Lord Kelvin
-
-This eBook is for the use of anyone anywhere in the United States and most
-other parts of the world at no cost and with almost no restrictions
-whatsoever.  You may copy it, give it away or re-use it under the terms of
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-Title: The Molecular Tactics of a Crystal
-
-Author: Lord Kelvin
-
-Release Date: June 24, 2017 [EBook #54976]
-
-Language: English
-
-Character set encoding: UTF-8
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-*** START OF THIS PROJECT GUTENBERG EBOOK THE MOLECULAR TACTICS OF A CRYSTAL ***
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-
-  THE MOLECULAR TACTICS OF
-  A CRYSTAL
-
-  _LORD KELVIN_
-
-  London
-  HENRY FROWDE
-  OXFORD UNIVERSITY PRESS WAREHOUSE
-  AMEN CORNER, E.C.
-
-[Illustration]
-
-  New York
-  MACMILLAN & CO., 66 FIFTH AVENUE
-
-
-
-
-  THE
-  MOLECULAR TACTICS OF
-  A CRYSTAL
-
-  BY
-
-  LORD KELVIN, P.R.S.
-
-  PROFESSOR OF NATURAL PHILOSOPHY IN THE UNIVERSITY OF GLASGOW
-  AND FELLOW OF PETERHOUSE, CAMBRIDGE
-
-
-  _Being the Second ROBERT BOYLE LECTURE, delivered before
-  the Oxford University Junior Scientific Club
-  on Tuesday, May 16, 1893_
-
-
-  WITH TWENTY ILLUSTRATIONS
-
-
-  Oxford
-  AT THE CLARENDON PRESS
-  1894
-
-  Oxford
-  PRINTED AT THE CLARENDON PRESS
-  BY HORACE HART, PRINTER TO THE UNIVERSITY
-
-
-
-
-ON THE MOLECULAR TACTICS OF A CRYSTAL
-
-By LORD KELVIN, P.R.S.
-
-
-§ 1. My subject this evening is not the physical properties of
-crystals, not even their dynamics; it is merely the geometry of the
-structure--the arrangement of the molecules in the constitution of a
-crystal. Every crystal is a homogeneous assemblage of small bodies
-or molecules. The converse proposition is scarcely true, unless in
-a very extended sense of the term crystal (§ 20 below). I can best
-explain a homogeneous assemblage of molecules by asking you to think
-of a homogeneous assemblage of people. To be homogeneous every person
-of the assemblage must be equal and similar to every other: they must
-be seated in rows or standing in rows in a perfectly similar manner.
-Each person, except those on the borders of the assemblage, must have
-a neighbour on one side and an equi-distant neighbour on the other: a
-neighbour on the left front and an equi-distant neighbour behind on the
-right, a neighbour on the right front and an equi-distant neighbour
-behind on the left. His two neighbours in front and his two neighbours
-behind are members of two rows equal and similar to the rows consisting
-of himself and his right-hand and left-hand neighbours, and their
-neighbours’ neighbours indefinitely to right and left. In particular
-cases the nearest of the front and rear neighbours may be right in
-front and right in rear; but we must not confine our attention to the
-rectangularly grouped assemblages thus constituted. Now let there be
-equal and similar assemblages on floors above and below that which
-we have been considering, and let there be any indefinitely great
-number of floors at equal distances from one another above and below.
-Think of any one person on any intermediate floor and of his nearest
-neighbours on the floors above and below. These three persons must be
-exactly in one line; this, in virtue of the homogeneousness of the
-assemblages on the three floors, will secure that every person on the
-intermediate floor is exactly in line with his nearest neighbours above
-and below. The same condition of alignment must be fulfilled by every
-three consecutive floors, and we thus have a homogeneous assemblage of
-people in three dimensions of space. In particular cases every person’s
-nearest neighbour in the floor above may be vertically over him, but
-we must not confine our attention to assemblages thus rectangularly
-grouped in vertical lines.
-
-§ 2. Consider now any particular person _C_ (Fig. 1) on any
-intermediate floor, _D_ and _D′_ his nearest neighbours, _E_ and _E′_
-his next nearest neighbours all on his own floor. His next next nearest
-neighbours on that floor will be in the positions _F_ and _F′_ in the
-diagram. Thus we see that each person _C_ is surrounded by six persons,
-_DD′_, _EE′_ and _FF′_, being his nearest, his next nearest, and his
-next next nearest neighbours on his own floor. Excluding for simplicity
-the special cases of rectangular grouping, we see that the angles of
-the six equal and similar triangles _CDE_, _CEF_, &c., are all acute:
-and because the six triangles are equal and similar we see that the
-three pairs of mutually remote sides of the hexagon _DEFD′E′F′_ are
-equal and parallel.
-
-[Illustration: FIG. 1]
-
-§ 3. Let now _A_, _A′_, _A″_, &c., denote places of persons of the
-homogeneous assemblage on the floor immediately above, and _B_, _B′_,
-_B″_, &c. on the floor immediately below, the floor of _C_. In the
-diagram let _a_, _a′_, _a″_ be points in which the floor of _CDE_ is
-cut by perpendiculars to it through _A_, _A′_, _A″_ of the floor above,
-and _b_, _b′_, _b″_ by perpendiculars from _B_, _B′_, _B″_ of the floor
-below. Of all the perpendiculars from the floors immediately above
-and below, just two, one from each, cut the area of the parallelogram
-_CDEF_: and they cut it in points similarly situated in respect to
-the oppositely oriented triangles into which it is divided by either
-of its diagonals. Hence if _a_ lies in the triangle _CDE_, the other
-five triangles of the hexagon must be cut in the corresponding points,
-as shown in the diagram. Thus, if we think only of the floor of _C_
-and of the floor immediately above it, we have points _A_, _A′_, _A″_
-vertically above _a_, _a′_, _a″_. Imagine now a triangular pyramid,
-or tetrahedron, standing on the base _CDE_ and having _A_ for vertex:
-we see that each of its sides _ACD_, _ADE_, _AEC_, is an acute angled
-triangle, because, as we have already seen, _CDE_ is an acute angled
-triangle, and because the shortest of the three distances, _CA_, _DA_,
-_EA_, is (§ 2) greater than _CE_ (though it may be either greater than
-or less than _DE_). Hence the tetrahedron _CDEA_ has all its angles
-acute; not only the angles of its triangular faces, but the six angles
-between the planes of its four faces. This important theorem regarding
-homogeneous assemblages was given by Bravais, to whom we owe the whole
-doctrine of homogeneous assemblages in its most perfect simplicity and
-complete generality. Similarly we see that we have equal and similar
-tetrahedrons on the bases _D′CF_, _E′F′C_; and three other tetrahedrons
-below the floor of _C_, having the oppositely oriented triangles
-_CD′E′_, &c. for their bases and _B_, _B′_, _B″_ for their vertices.
-These three tetrahedrons are equal and heterochirally[1] similar to the
-first three. The consideration of these acute angled tetrahedrons, is
-of fundamental importance in respect to the engineering of an elastic
-solid, or crystal, according to Boscovich. So also is the consideration
-of the cluster of thirteen points _C_ and the six neighbours
-_DEFD′E′F′_ in the plane of the diagram, and the three neighbours
-_AA′A″_ on the floor above, and _BB′B″_ on the floor below.
-
-§ 4. The case in which each of the four faces of each of the
-tetrahedrons of § 3 is an equilateral triangle is particularly
-interesting. An assemblage fulfilling this condition may conveniently
-be called an ‘equilateral homogeneous assemblage,’ or, for brevity, an
-‘equilateral assemblage.’ In an equilateral assemblage _C_’s twelve
-neighbours are all equi-distant from it. I hold in my hand a cluster of
-thirteen little black balls, made up by taking one of them and placing
-the twelve others in contact with it (and therefore packed in the
-closest possible order), and fixing them all together by fish-glue.
-You see it looks, in size, colour, and shape, quite like a mulberry.
-The accompanying diagram shows a stereoscopic view of a similar cluster
-of balls painted white for the photograph.
-
-[Illustration: FIG. 2.]
-
-§ 5. By adding ball after ball to such a cluster of thirteen, and
-always taking care to place each additional ball in some position in
-which it is properly in line with others, so as to make the whole
-assemblage homogeneous, we can exercise ourselves in a very interesting
-manner in the building up of any possible form of crystal of the class
-called ‘cubic’ by some writers and ‘octahedral’ by others. You see
-before you several examples. I advise any of you who wish to study
-crystallography to contract with a wood-turner, or a maker of beads for
-furniture tassels or for rosaries, for a thousand wooden balls of about
-half an inch diameter each. Holes through them will do no harm and may
-even be useful; but make sure that the balls are as nearly equal to one
-another, and each as nearly spherical, as possible.
-
-[Illustration: FIG. 3.]
-
-
-§ 6. You see here before you a large model which I have made to
-illustrate a homogeneous assemblage of points, on a plan first given,
-I believe, by Mr. William Barlow (_Nature_, December 20 and 27, 1883).
-The roof of the model is a lattice-frame (Fig. 3) consisting of two
-sets of eight parallel wooden bars crossing one another, and kept
-together by pins through the middles of the crossings. As you see, I
-can alter it to make parallelograms of all degrees of obliquity till
-the bars touch, and again you see I can make them all squares.
-
-
-§ 7. The joint pivots are (for cheapness of construction) of copper
-wire, each bent to make a hook below the lattice frame. On these
-sixty-four hooks are hung sixty-four fine cords, firmly stretched by
-little lead weights. Each of these cords (Fig. 4) bears eight short
-perforated wooden cylinders, which may be slipped up and down to any
-desired position[2]. They are at present actually placed at distances
-consecutively each equal to the distance from joint to joint of the
-lattice frame.
-
-[Illustration: FIG. 4.]
-
-
-§ 8. The roof of the model is hung by four cords, nearly vertical, of
-independently variable lengths, passing over hooks from fixed points
-above, and kept stretched by weights, each equal to one quarter of
-the weight of roof and pendants. You see now by altering the angles
-of the lattice work and placing it horizontal or in any inclined
-plane, as I am allowed to do readily by the manner in which it is
-hung, I have three independent variables, by varying which I can show
-you all varieties of homogeneous assemblages, in which three of the
-neighbours of every point are at equal distances from it. You see
-here, for example, we have the equilateral assemblage. I have adjusted
-the lattice roof to the proper angle, and its plane to the proper
-inclination to the vertical, to make a wholly equilateral assemblage
-of the little cylinders of wood on the vertical cords, a case, as
-we have seen, of special importance. If I vary also the distances
-between the little pieces of wood on the cords; and the distances
-between the joints of the lattice work (variations easily understood,
-though not conveniently producible in one model without more of
-mechanical construction than would be worth making), I have three
-other independent variables. By properly varying these six independent
-variables, three angles and three lengths, we may give any assigned
-value to each edge of one of the fundamental tetrahedrons of § 3.
-
-
-§ 9. Our assemblage of people would not be homogeneous unless its
-members were all equal and similar and in precisely similar attitudes,
-and were all looking the same way. You understand what a number of
-people seated or standing on a floor or plain and looking the same way
-means. But the expression ‘looking’ is not conveniently applicable to
-things that have no eyes, and we want a more comprehensive mode of
-expression. We have it in the words ‘orientation,’ ‘oriented,’ and
-(verb) ‘to orient,’ suggested by an extension of the idea involved in
-the word ‘orientation,’ first used to signify positions relatively
-to east and west of ancient Greek and Egyptian temples and Christian
-churches. But for the orientation of a house or temple we have only
-one angle, and that angle is called ‘azimuth’ (the name given to an
-angle in a horizontal plane). For orientation in three dimensions of
-space we must extend our ideas and consider position with reference to
-east and west and up and down. A man lying on his side with his head
-to the north and looking east, would not be similarly oriented to a
-man standing upright and looking east. To provide for the complete
-specification of how a body is oriented in space we must have in the
-body a plane of reference, and a line of reference in this plane,
-belonging to the body and moving with it. We must also have a fixed
-plane and a fixed line of reference in it, relatively to which the
-orientation of the moveable body is to be specified; as, for example,
-a horizontal plane and the east and west horizontal line in it. The
-position of a body is completely specified when the angle between the
-plane of reference belonging to it, and the fixed plane is given; and
-when the angles between the line of intersection of the two planes and
-the lines of reference in them are also given. Thus we see that three
-angles are necessary and sufficient to specify the orientation of a
-moveable body, and we see how the specification is conveniently given
-in terms of three angles.
-
-
-§ 10. To illustrate this take a book lying on the table before you with
-its side next the title-page up, and its back to the north. I now lift
-the east edge (the top of the book), keeping the bottom edge north and
-south on the table till the book is inclined, let us say, 20° to the
-table. Next, without altering this angle of 20°, between the side of
-the book and the table, I turn the book round a vertical axis, through
-45° till the bottom edge lies north-east and south-west. Lastly,
-keeping the book in the plane to which it has been thus brought, I turn
-it round in this plane through 35°. These three angles of 20°, 45°,
-and 35°, specify, with reference to the horizontal plane of the table
-and the east and west line in it, the orientation of the book in the
-position to which you have seen me bring it, and in which I hold it
-before you.
-
-
-§ 11. In Figs. 5 and 6 you see two assemblages, each of twelve equal
-and similar molecules in a plane. Fig. 5, in which the molecules are
-all same-ways oriented, is one homogeneous assemblage of twenty-four
-molecules. Fig. 6, in which in one set of rows the molecules are
-alternately oriented two different ways, may either be regarded as
-two homogeneous assemblages, each of twelve single molecules; or one
-homogeneous assemblage of twelve pairs of those single molecules.
-
-[Illustration: FIG. 5.]
-
-
-§ 12. I must now call your attention to a purely geometrical
-question[3] of vital interest with respect to homogeneous assemblages
-in general, and particularly the homogeneous assemblage of molecules
-constituting a crystal:--_what can we take as ‘the’ boundary or ‘a’
-boundary enclosing each molecule with whatever portion of space around
-it we are at liberty to choose for_ _it, and separating it from
-neighbours and their portions of space given to them in homogeneous
-fairness?_
-
-[Illustration: FIG. 6.]
-
-
-§ 13. If we had only mathematical points to consider we should be at
-liberty to choose the simple obvious partitioning by three sets of
-parallel planes. Even this may be done in an infinite number of ways,
-thus:--Beginning with any point _P_ of the assemblage, choose any other
-three points _A_, _B_, _C_, far or near, provided only that they are
-not in one plane with _P_, and that there is no other point of the
-assemblage in the lines _PA_, _PB_, _PC_, or within the volume of the
-parallelepiped of which these lines are conterminous edges, or within
-the areas of any of the faces of this parallelepiped. There will be
-points of the assemblage at each of the corners of this parallelepiped
-and at all the corners of the parallelepipeds equal and similar to
-it which we find by drawing sets of equi-distant planes parallel to
-its three pairs of faces. (A diagram is unnecessary.) Every point of
-the assemblage is thus at the intersection of three planes, which is
-also the point of meeting of eight neighbouring parallelepipeds. Shift
-now any one of the points of the assemblage to a position within the
-volume of any one of the eight parallelepipeds, and give equal parallel
-motions to all the other points of the assemblage. Thus we have every
-point in a parallelepipedal cell of its own, and all the points of the
-assemblage are similarly placed in their cells, which are themselves
-equal and similar.
-
-
-§ 14. But now if, instead of a single point for each member of the
-assemblage, we have a group of points, or a globe or cube or other
-geometrical figure, or an individual of a homogeneous assemblage of
-equal, similar, similarly dressed, and similarly oriented ladies,
-sitting in rows, or a homogeneous assemblage of trees closely planted
-in regular geometrical order on a plane with equal and similar
-distributions of molecules, and parallel planes above and below,
-we may find that the best conditioned plane-faced parallelepipedal
-partitioning which we can choose would cut off portions properly
-belonging to one molecule of the assemblage and give them to the cells
-of neighbours. To find a cell enclosing all that belongs to each
-individual, for example, every part of each lady’s dress, however
-complexly it may be folded among portions of the equal and similar
-dresses of neighbours; or, every twig, leaf, and rootlet of each one
-of the homogeneous assemblage of trees; we must alter the boundary by
-give-and-take across the plane faces of the primitive parallelepipedal
-cells, so that each cell shall enclose all that belongs to one
-molecule, and therefore (because of the homogeneousness of the
-partitioning) nothing belonging to any other molecule. The geometrical
-problem thus presented, wonderfully complex as it may be in cases
-such as some of those which I have suggested, is easily performed for
-any possible case if we begin with any particular parallelepipedal
-partitioning determined for corresponding points of the assemblage
-as explained in § 13, for any homogeneous assemblage of single
-points. We may prescribe to ourselves that the corners are to remain
-unchanged, but if so they must to begin with either in interfaces of
-contact between the individual molecules, or in vacant space among
-the molecules. If this condition is fulfilled for one corner it is
-fulfilled for all, as the corners are essentially corresponding points
-relatively to the assemblage.
-
-
-§ 15. Begin now with any one of the twelve straight lines between
-corners which constitute the twelve edges of the parallelepiped, and
-alter it arbitrarily to any curved or crooked line between the same
-pair of corners, subject only to the conditions (1) that it does not
-penetrate the substance of any member of the assemblage, and (2) that
-it is not cut by equal and similar parallel curves[4] between other
-pairs of corners.
-
-[Illustration: FIG. 7.]
-
-Considering now the three fours of parallel edges of the
-parallelepiped, let the straight lines of one set of four be altered to
-equal and similar parallel curves in the manner which I have described;
-and proceed by the same rule for the other two sets of four edges. We
-thus have three fours of parallel curved edges instead of the three
-fours of parallel straight edges of our primitive parallelepiped with
-corners (each a point of intersection of three edges) unchanged.
-Take now the quadrilateral of four curves substituted for the four
-straight edges of one face of the parallelepiped. We may call this
-quadrilateral a curvilineal parallelogram, because it is a circuit
-composed of two pairs of equal parallel curves. Draw now a curved
-surface (an infinitely thin sheet of perfectly extensible india-rubber
-if you please to think of it so) bordered by the four edges of our
-curvilineal parallelogram, and so shaped as not to cut any of the
-substance of any molecule of the assemblage. Do the same thing with
-an exactly similar and parallel sheet relatively to the opposite face
-of the parallelepiped; and again the same for each of the two other
-pairs of parallel faces. We thus have a curved-faced parallelepiped
-enclosing the whole of one molecule and no part of any other; and by
-similar procedure we find a similar boundary for every other molecule
-of the assemblage. Each wall of each of these cells is common to two
-neighbouring molecules, and there is no vacant space anywhere between
-them or at corners. Fig. 7 illustrates this kind of partitioning by
-showing a plane section parallel to one pair of plane faces of the
-primitive parallelepiped, for an ideal case. The plane diagram is in
-fact a realization of the two-dimensional problem of partitioning the
-pine pattern of a Persian carpet by parallelograms about as nearly
-rectilinear as we can make them. In the diagram faint straight lines
-are drawn to show the primitive parallelogrammatic partitioning.
-It will be seen that of all the crossings (marked with dots in the
-diagram) every one is similarly situated to every other in respect to
-the homogeneously repeated pattern figures: _A_, _B_, _C_, _D_ are four
-of them at the corners of one cell.
-
-
-§ 16. Confining our attention for a short time to the homogeneous
-division of a plane, remark that the division into parallelograms
-by two sets of crossing parallels is singular in this respect--each
-cell is contiguous with three neighbours at every corner. Any
-shifting, large or small, of the parallelograms by relative sliding
-in one direction or another violates this condition, brings us to
-a configuration like that of the faces of regularly hewn stones in
-ordinary bonded masonry, and gives a partitioning which fulfils the
-condition that at each corner each cell has only two neighbours. Each
-cell is now virtually a hexagon, as will be seen by the letters _A_,
-_B_, _C_, _D_, _E_, _F_ in the diagram Fig. 8. _A_ and _D_ are to be
-reckoned as corners, each with an interior angle of 180°. In this
-diagram the continuous heavy lines and the continuous faint lines
-crossing them show a primitive parallelogrammatic partition by two sets
-of continuous parallel intersecting lines. The interrupted crossing
-lines (heavy) show, for the same homogeneous distribution of single
-points or molecules, the virtually hexagonal partitioning which we get
-by shifting the boundary from each portion of one of the light lines to
-the heavy line next it between the same continuous parallels.
-
-[Illustration: FIG. 8.]
-
-Fig. 8 bis represents a further modification of the boundary by
-which the 180° angles _A_, _D_, become angles of less than 180°. The
-continuous parallel lines (light) and the short light portions of the
-crossing lines show the configuration according to Fig. 8, from which
-this diagram is derived.
-
-
-§ 17. In these diagrams (Figs. 8 and 8 bis) the object enclosed
-is small enough to be enclosable by a primitive parallelogrammatic
-partitioning of two sets of continuous crossing parallel straight
-lines, and by the partitioning of ‘bonded’ parallelograms both
-represented in Fig. 8, and by the derived hexagonal partitioning
-represented in Fig. 8 bis, with faint lines showing the primitive and
-the secondary parallelograms. In Fig. 7 the objects enclosed were
-too large to be enclosable by any rectilinear parallelogrammatic
-or hexagonal partitioning. The two sets of parallel faint lines in
-Fig. 7 show a primitive parallelogrammatic partitioning and the
-corresponding pairs of parallel curves intersecting at the corners of
-these parallelograms, of which _A_,_B_,_C_,_D_ is a specimen, show a
-corresponding partitioning by curvilineal parallelograms. Fig. 9 shows
-for the same homogeneous distribution of objects a better conditioned
-partitioning, by hexagons in each of which one pair of parallel
-edges is curved. The sets of intersecting parallel straight lines in
-Fig. 9 show the same primitive parallelogrammatic partitioning as
-in Fig. 7, and the same slightly shifted to suit points chosen for
-well-conditionedness of hexagonal partitioning.
-
-[Illustration: FIG. 8 bis.]
-
-[Illustration: FIG. 9.]
-
-
-§ 18. For the division of continuous three-dimensional space[5] into
-equal, similar, and similarly oriented cells, quite a corresponding
-transformation from partitioning by three sets of continuous mutually
-intersecting parallel planes to any possible mode of homogeneous
-partitioning, may be investigated by working out the three-dimensional
-analogue of §§ 16-17. Thus we find that the most general possible
-homogeneous partitioning of space with plane interfaces between the
-cells gives us fourteen walls to each cell, of which six are three
-pairs of equal and parallel parallelograms, and the other eight are
-four pairs of equal and parallel hexagons, each hexagon being bounded
-by three pairs of equal and parallel straight lines. This figure, being
-bounded by fourteen plane faces, is called a tetrakaidekahedron. It
-has thirty-six edges of intersection between faces; and twenty-four
-corners, in each of which three faces intersect. A particular case
-of it, which I call an orthic tetrakaidekahedron, being that in
-which the six parallelograms are equal squares, the eight hexagonal
-faces are equal equilateral and equiangular hexagons, and the lines
-joining corresponding points in the seven pairs of parallel faces
-are perpendicular to the planes of the faces, is represented by
-a stereoscopic picture in Fig. 10. The thirty-six edges and the
-twenty-four corners, which are easily counted in this diagram, occur
-in the same relative order in the most general possible partitioning,
-whether by plane-faced tetrakaidekahedrons or by the generalized
-tetrakaidekahedron described in § 19.
-
-
-§ 19. The most general homogeneous division of space is not
-limited to plane-faced cells; but it still consists essentially of
-tetrakaidekahedronal cells, each bounded by three pairs of equal and
-parallel quadrilateral faces, and four pairs of equal and parallel
-hexagonal faces, neither the quadrilaterals nor the hexagons being
-necessarily plane. Each of the thirty-six edges may be straight
-or crooked or curved; the pairs of opposite edges, whether of the
-quadrilaterals or hexagons, need not be equal and parallel; neither
-the four corners of each quadrilateral nor the six corners of each
-hexagon need be in one plane. But every pair of corresponding edges of
-every pair of parallel corresponding faces, whether quadrilateral or
-hexagonal, must be equal and parallel. I have described an interesting
-case of partitioning by tetrakaidekahedrons of curved faces with curved
-edges in a paper[6] published about seven years ago. In this case each
-of the quadrilateral faces is plane. Each hexagonal face is a slightly
-curved surface having three rectilineal diagonals through its centre in
-one plane.
-
-[Illustration: FIG. 10.]
-
-The six sectors of the face between these diagonals lie alternately
-on opposite sides of their plane, and are bordered by six arcs
-of plane curves lying on three pairs of parallel planes. This
-tetrakaidekahedronal partitioning fulfils the condition that the
-angles between three planes meeting in an edge are everywhere each
-120°; a condition that cannot be fulfilled in any plane-faced
-tetrakaidekahedron. Each hexagonal wall is an anticlastic surface of
-equal opposite curvatures at every point, being the surfaces of minimum
-area bordered by six curved edges. It is shown easily and beautifully,
-and with a fair approach to accuracy, by choosing six little circular
-arcs of wire, and soldering them together by their ends in proper
-planes for the six edges of the hexagon; and dipping it in soap
-solution and taking it out.
-
-
-§ 20. Returning now to the tactics of a homogeneous assemblage, remark
-that the qualities of the assemblage as a whole depend both upon the
-character and orientation of each molecule, and on the character of
-the homogeneous assemblage formed by corresponding points of the
-molecules. After learning the simple mathematics of crystallography,
-with its indicial system[7] for defining the faces and edges of a
-crystal according to the Bravais rows and nets and tetrahedrons of
-molecules in which we think only of a homogeneous assemblage of points,
-we are apt to forget that the true crystalline molecule, whatever its
-nature may be, has sides, and that generally two opposite sides of each
-molecule may be expected to be very different in quality, and we are
-almost surprised when mineralogists tell us that two parallel faces on
-two sides of a crystal have very different qualities in many natural
-crystals. We might almost as well be surprised to find that an army in
-battle array, which is a kind of large-grained crystal, presents very
-different appearance to any one looking at it from outside, according
-as every man in the ranks with his rifle and bayonet faces to the front
-or to the rear or to one flank or to the other.
-
-
-§ 21. Consider, for example, the ideal case of a crystal consisting
-of hard equal and similar tetrahedronal solids all same-ways oriented.
-A thin plate of crystal cut parallel to any one set of the faces of
-the constituent tetrahedrons would have very different properties on
-its two sides; as the constituent molecules would all present points
-outwards on one side and flat surfaces on the other. We might expect
-that the two sides of such a plate of crystal would become oppositely
-electrified when rubbed by one and the same rubber; and, remembering
-that a piece of glass with part of its surface finely ground but not
-polished and other parts polished becomes, when rubbed with white
-silk, positively electrified over the polished parts and negatively
-electrified over the non-polished parts, we might almost expect that
-the side of our supposed crystalline plate towards which flat faces
-of the constituent molecules are turned would become positively
-electrified, and the opposite side, showing free molecular corners,
-would become negatively electrified, when both are rubbed by a rubber
-of intermediate electric quality. We might also from elementary
-knowledge of the fact of piezo-electricity, that is to say, the
-development of opposite electricities on the two sides of a crystal
-by pressure, expect that our supposed crystalline plate, if pressed
-perpendicularly on its two sides, would become positively electrified
-on one of them and negatively on the other.
-
-
-§ 22. Intimately connected with the subject of enclosing cells for
-molecules of given shape, assembled homogeneously, is the homogeneous
-packing together of equal and similar molecules of any given shape. In
-every possible case of any infinitely great number of similar bodies
-the solution is a homogeneous assemblage. But it may be a homogeneous
-assemblage of single solids all oriented the same way, or it may be
-a homogeneous assemblage of clusters of two or more of them placed
-together in different orientations. For example, let the given bodies
-be halves (oblique or not oblique) of any parallelepiped on the two
-sides of a dividing plane through a pair of parallel edges. The two
-halves are homochirally[8] similar; and, being equal, we may make a
-homogeneous assemblage of them by orienting them all the same way
-and placing them properly in rows. But the closest packing of this
-assemblage would necessarily leave vacant spaces between the bodies:
-and we get in reality the closest possible packing of the given bodies
-by taking them in pairs oppositely oriented and placed together to form
-parallelepipeds. These clusters may be packed together so as to leave
-no unoccupied space.
-
-Whatever the number of pieces in a cluster in the closest possible
-packing of solids may be for any particular shape, we may consider each
-cluster as itself a given single body, and thus reduce the problem to
-the packing closely together of assemblages of individuals all sameways
-oriented; and to this problem therefore it is convenient that we should
-now confine our attention.
-
-
-§ 23. To avoid complexities such as those which we find in the familiar
-problem of homogeneous packing of forks or spoons or tea-cups or bowls,
-of any ordinary shape, we shall suppose the given body to be of such
-shape that no two of them similarly oriented can touch one another
-in more than one point. Wholly convex bodies essentially fulfil this
-condition; but it may also be fulfilled by bodies not wholly convex, as
-is illustrated in Fig. 11.
-
-[Illustration: Fig. 11.]
-
-
-§ 24. To find close and closest packing of any number of our solids
-_S_{1}_, _S_{2}_, _S_{3}_ ... of shape fulfilling the condition of § 23
-proceed thus:--
-
-(1) Bring _S_{2}_ to touch _S_{1}_ at any chosen point _p_ of its
-surface (Fig. 12).
-
-(2) Bring _S_{3}_ to touch _S_{1}_ and _S_{2}_, at _r_ and _q_
-respectively.
-
-(3) Bring _S_{4}_ (not shown in the diagram) to touch _S_{1}_, _S_{2}_,
-and _S_{3}_.
-
-(4) Place, any number of the bodies together in three rows continuing
-the lines of _S_{1}S_{2}_, _S_{1}S_{3}_, _S_{1}S_{4}_, and in three
-sets of equi-distant rows parallel to these. This makes a homogeneous
-assemblage. In the assemblage so formed the molecules are necessarily
-found to be in three sets of rows parallel respectively to the three
-pairs _S_{2}S_{3}_, _S_{3}S_{4}_, _S_{4}S_{2}_. The whole space
-occupied by an assemblage of _n_ of our solids thus arranged has
-clearly _6n_ times the volume of a tetrahedron of corresponding points
-of _S_{1}_, _S_{2}_, _S_{3}_, _S_{4}_. Hence the closest of the
-close packings obtained by the operations (1) ... (4) is found if we
-perform the operations (1), (2), and (3) as to make the volume of this
-tetrahedron least possible.
-
-[Illustration: FIG. 12]
-
-
-§ 25. It is to be remarked that operations (1) and (2) leave for (3)
-no liberty of choice for the place of _S_{4}_, except between two
-determinate positions on opposite sides of the group _S_{1}_, _S_{2}_,
-_S_{3}_. The volume of the tetrahedron will generally be different for
-these two positions of _S_{4}_, and, even if the volume chance to be
-equal in any case, we have differently shaped assemblages according as
-we choose one or other of the two places for _S_{4}_.
-
-This will be understood by looking at Fig. 12, showing _S_{1}_
-and neighbours on each side of it in the rows of _S_{1}S_{2}_,
-_S_{1}S_{3}_, and in a row parallel to that of _S_{2}S_{3}_. The plane
-of the diagram is parallel to the planes of corresponding points of
-these seven bodies, and the diagram is a projection of these bodies
-by lines parallel to the intersections of the tangent planes through
-_p_ and _r_. If the three tangent planes through _p_, _q_, and _r_,
-intersected in parallel lines, _q_ would be seen like _p_ and _r_ as a
-point of contact between the outlines of two of the bodies; but this is
-only a particular case, and in general _q_ must, as indicated in the
-diagram, be concealed by one or other of the two bodies of which it is
-the point of contact. Now imagining, to fix our ideas and facilitate
-brevity of expression, that the planes of corresponding points of the
-seven bodies are horizontal, we see clearly that _S_{4}_ may be brought
-into proper position to touch _S_{1}_, _S_{2}_, and _S_{3}_ either from
-above or from below; and that there is one determinate place for it if
-we bring it into position from above, and another determinate place for
-it if we bring it from below.
-
-
-§ 26. If we look from above at the solids of which Fig. 12 shows the
-outline, we see essentially a hollow leading down to a perforation
-between _S_{1}_, _S_{2}_, _S_{3}_, and if we look from below we see a
-hollow leading upwards to the same perforation: this for brevity we
-shall call the perforation _pqr_. The diagram shows around _S_{1}_
-six hollows leading down to perforations, of which two are similar
-to _pqr_, and the other three, of which _p′q′r′_ indicates one, are
-similar one to another but are dissimilar to _pqr_. If we bring _S_{4}_
-from above into position to touch _S_{1}_, _S_{2}_, and _S_{3}_, its
-place thus found is in the hollow _pqr_, and the places of all the
-solids in the layer above that of the diagram are necessarily in the
-hollows similar to _pqr_. In this case the solids in the layer below
-that of the diagram must lie in the hollows below the perforations
-dissimilar to _pqr_, in order to make a single homogeneous assemblage.
-In the other case, _S_{4}_ brought up from below finds its place on the
-under side of the hollow _pqr_, and all solids of the lower layer find
-similar places: while solids in the layer above that of the diagram
-find their places in the hollows similar to _p´q´r´_. In the first
-case there are no bodies of the upper layer in the hollows above the
-perforations _similar_ to _p´q´r´_, and no bodies of the lower layer in
-the hollows below the perforations _similar_ to _pqr_. In the second
-case there are no bodies of the upper layer in the hollows above the
-perforations _similar_ to _pqr_, and none of the under layer in the
-hollows below the perforations _similar_ to _p´q´r´_.
-
-
-§ 27. Going back now to operation (1) of § 23, remark that when the
-point of contact _p_ is arbitrarily chosen on one of the two bodies
-_S_{1}_, the point of contact on the other will be the point on it
-corresponding to the point or one of the points of _S_{1}_, where its
-tangent plane is parallel to the tangent plane at _p_. If _S_{1}_
-is wholly convex it has only two points at which the tangent planes
-are parallel to a given plane, and therefore the operation (1) is
-determinate and unambiguous. But if there is any concavity there will
-be four or some greater even number of tangent planes parallel to any
-one of some planes, while there will be other planes to each of which
-only one pair of tangent planes is parallel. Hence, operation (1),
-though still determinate, will have a multiplicity of solutions, or
-only a single solution, according to the choice made of the position of
-_p_.
-
-Henceforth however, to avoid needless complications of ideas, we shall
-suppose our solids to be wholly convex; and of some such unsymmetrical
-shape as those indicated in Fig. 12 of § 25, and shown by stereoscopic
-photograph in Fig. 13 of § 36. With or without this convenient
-limitation, operation (1) has two freedoms, as _p_ may be chosen
-freely on the surface of _S_{1}_; and operation (2) has clearly just
-one freedom after operation (1) has been performed. Thus, for a solid
-of any given shape, we have three disposables, or, as commonly called
-in mathematics, three ‘independent variables,’ all free for making a
-homogeneous assemblage according to the rule of § 22.
-
-
-§ 28. In the homogeneous assemblage defined in § 24, each solid,
-_S_{1}_, is touched at twelve points, being the three points of
-contact with _S_{2}_, _S_{3}_, _S_{4}_, and the three 3’s of points
-on _S_{1}_ corresponding to the points on _S_{2}_, _S_{3}_, _S_{4}_,
-at which these bodies are touched by the others of the quartet. This
-statement is somewhat difficult to follow, and we see more clearly
-the twelve points of contact by not confining our attention to the
-quartet _S_{1}_, _S_{2}_, _S_{3}_, _S_{4}_ (convenient as this is
-for some purposes), but completing the assemblage and considering
-six neighbours around _S_{1}_ in one plane layer of the solids as
-shown in Fig. 12, with their six points _prq″p′r′q″′_ of contact with
-_S_{1}_; and the three neighbours of the two adjacent parallel layers
-which touch it above and below. This cluster of thirteen, _S_{1}_
-and twelve neighbours, is shown for the case of spherical bodies in
-the stereoscopic photograph of § 4 above. We might of course, if we
-pleased, have begun with the plane layer of which _S_{1}_, _S_{2}_,
-_S_{4}_ are members, or with that of which _S_{1}_, _S_{3}_, _S_{4}_
-are members, or with the plane layer parallel to the fourth side
-_S_{2}_ _S_{3}_ _S_{4}_ of the tetrahedron: and thus we have four
-different ways of grouping the twelve points of contact on _S_{1}_ into
-one set of six and two sets of three.
-
-
-§ 29. In this assemblage we have what I call ‘close order’ or ‘close
-packing.’ For closest of close packings the volume of the tetrahedron
-(§ 24) of corresponding points of _S_{1}_, _S_{2}_, _S_{3}_, and
-_S_{4}_ must be a minimum, and the least of minimums if, as generally
-will be the case, there are two more different configurations for
-each of which the volume is a minimum. There will in general also be
-configurations of minimax volume and of maximum volume, subject to
-the condition that each body is touched by twelve similarly oriented
-neighbours.
-
-
-§ 30. Pause for a moment to consider the interesting kinematical
-and dynamical problems presented by a close homogeneous assemblage
-of smooth solid bodies of given convex shape, whether perfectly
-frictionless or exerting resistance against mutual sliding according to
-the ordinarily stated law of friction between dry hard solid bodies.
-First imagine that they are all similarly oriented and each in contact
-with twelve neighbours, except outlying individuals (which there must
-be at the boundary if the assemblage is finite, and each of which is
-touched by some number of neighbours less than twelve). The coherent
-assemblage thus defined constitutes a kinematic frame or skeleton
-for an elastic solid of very peculiar properties. Instead of the six
-freedoms, or disposables, of strain presented by a natural solid it has
-only three. Change of shape of the whole can only take place in virtue
-of rotation of the constituent parts relatively to any one chosen row
-of them, and the plane through it and another chosen row.
-
-
-§ 31. Suppose first the solids to be not only perfectly smooth but
-perfectly frictionless. Let the assemblage be subjected to equal
-positive or negative pressure inwards all around its boundary. Every
-position of minimum, minimax, or maximum volume will be a position of
-equilibrium. If the pressure is positive the equilibrium will be stable
-if, and unstable unless, the volume is a minimum. If the pressure
-is negative the equilibrium will be stable if, and unstable unless,
-the volume is a maximum. Configurations of minimax volume will be
-essentially unstable.
-
-
-§ 32. Consider now the assemblage of § 31 in a position of stable
-equilibrium under the influence of a given constant uniform pressure
-inwards all round its boundary. It will have rigidity in simple
-proportion to the amount of this pressure. If now by the superposition
-of non-uniform pressure at the boundary, for example equal and opposite
-pressures on two sides of the assemblage, a finite change of shape is
-produced: the whole assemblage essentially swells in bulk. This is the
-‘dilatancy’ which Osborne Reynolds has described[9] in an exceedingly
-interesting manner with reference to a sack of wheat or sand, or an
-india-rubber bag tightly filled with sand or even small shot. Consider,
-for example, a sack of wheat filled quite full and standing up open. It
-is limp and flexible. Now shake it down well, fill it quite full, shake
-again, so as to get as much into it as possible, and tie the mouth very
-tightly close. The sack becomes almost as stiff as a log of wood of
-the same shape. Open the mouth partially, and it becomes again limp,
-especially in the upper parts of the bag. In Reynolds’ observations on
-india-rubber bags of small shot his ‘dilatancy’ depends, essentially
-and wholly, on breaches of some of the contacts which exist between the
-molecules in their configuration of minimum volume: and it is possible
-that in all his cases the dilatations which he observed are _chiefly_,
-if not wholly, due to such breaches of contact.
-
-But it is possible, it almost seems probable, that in bags or boxes
-of sand or powder, of some kinds of smooth rounded bodies of any
-shape, not spherical or ellipsoidal, subjected persistently to unequal
-pressures in different directions, and well shaken, stable positions
-of equilibrium are found with almost all the particles each touched by
-twelve others.
-
-Here is a curious subject of Natural History through all ages till
-1885, when Reynolds brought it into the province of Natural Philosophy
-by the following highly interesting statement:--‘A well-marked
-phenomenon receives its explanation at once from the existence of
-dilatancy in sand. When the falling tide leaves the sand firm, as the
-foot falls on it the sand whitens and appears momentarily to dry round
-the foot. When this happens the sand is full of water, the surface of
-which is kept up to that of the sand by capillary attractions; the
-pressure of the foot causing dilatation of the sand more water is
-required, which has to be obtained either by depressing the level of
-the surface against the capillary attractions, or by drawing water
-through the interstices of the surrounding sand. This latter requires
-time to accomplish, so that for the moment the capillary forces are
-overcome; the surface of the water is lowered below that of the sand,
-leaving the latter white or drier until a sufficient supply has been
-obtained from below, when the surface rises and wets the sand again.
-On raising the foot it is generally seen that the sand under the foot
-and around becomes momentarily wet; this is because, on the distorting
-forces being removed, the sand again contracts, and the excess of water
-finds momentary relief at the surface.’
-
-This proves that the sand under the foot, as well as the surface around
-it, must be dry for a short time after the foot is pressed upon it,
-though we cannot see it whitened, as the foot is not transparent. That
-it is so has been verified by Mr. Alex. Galt, Experimental Instructor
-in the Physical Laboratory of Glasgow University, by laying a small
-square of plate-glass on wet sand on the sea-shore of Helensburgh, and
-suddenly pressing on it by a stout stick with nearly all his weight.
-He found the sand, both under the glass and around it in contact with
-the air, all became white at the same moment. Of all the two hundred
-thousand million men, women, and children who, from the beginning
-of the world, have ever walked on wet sand, how many, prior to the
-British Association Meeting at Aberdeen in 1885, if asked, ‘Is the sand
-compressed under your foot?’ would have answered otherwise than ‘Yes!’?
-
-(Contrast with this the case of walking over a bed of wet sea-weed!)
-
-
-§ 33. In the case of globes packed together in closest order (and
-therefore also in the case of ellipsoids, if all similarly oriented),
-our condition of coherent contact between each molecule and twelve
-neighbours implies absolute rigidity of form and constancy of bulk.
-Hence our convex solid must be neither ellipsoidal nor spherical
-in order that there may be the changes of form and changes of bulk
-which we have been considering as dependent on three independent
-variables specifying the orientation of each solid relatively to rows
-of the assemblage. An interesting dynamical problem is presented by
-supposing any mutual forces, such as might be produced by springs, to
-act between the solid molecules, and investigating configurations of
-equilibrium on the supposition of frictionless contacts. The solution
-of it of course is that the potential energy of the springs must be a
-minimum or a minimax or a maximum for equilibrium, and a minimum for
-stable equilibrium. The solution will be a configuration of minimum or
-minimax, or maximum, volume, only in the case of pressure equal in all
-directions.
-
-
-§ 34. A purely geometrical question, of no importance in respect to
-the molecular tactics of a crystal but of considerable interest in
-pure mathematics, is forced on our attention by our having seen (§ 27)
-that a homogeneous assemblage of solids of given shape, each touched
-by twelve neighbours, has three freedoms which may be conveniently
-taken as the three angles specifying the orientation of each molecule
-relatively to rows of the assemblage as explained in § 30.
-
-Consider a solid _S_{1}_ and the twelve neighbours which touch it,
-and try if it is possible to cause it to touch more than twelve of
-the bodies. Attach ends of three thick flexible wires to any places
-on the surface of _S_{1}_; carry the wires through interstices of the
-assemblage, and attach their other ends at any three places of _A_,
-_B_, _C_, respectively, these being any three of the bodies outside
-the cluster of _S_{1}_ and its twelve neighbours. Cut the wires across
-at any chosen positions in them; and round off the cut ends, just
-leaving contact between the rounded ends, which we shall call _f′f_,
-_g′g_, _h′h_. Do homogeneously for every other solid of the assemblage
-what we have done for _S_{1}_. Now bend the wires slightly so as to
-separate the pairs of points of contact, taking care to keep them
-from touching any other bodies which they pass near on their courses
-between _S_{1}_ and _A_, _B_, _C_ respectively. After having done this,
-thoroughly rigidify all the wires thus altered. We may now, having
-three independent variables at our disposal, so change the orientation
-of the molecules, relatively to rows of the assemblage, as to bring
-_f′f_, _g′g_, and _h′h_ again into contact. We have thus six fresh
-points of _S_{1}_; of which three are _f′_, _g′_, _h′_; and the other
-three are on the three extensions of _S_{1}_ corresponding to the
-single extensions of _A_, _B_, _C_ respectively, which we have been
-making. Thus we have a _real_ solution of the interesting geometrical
-problem:--It is required so to form a homogeneous assemblage of solids
-of any arbitrarily given shape that each solid shall be touched by
-eighteen others. This problem is determinate, because the making of
-the three contacts _f′f_, _g′g_, _h′h_, uses up the three independent
-variables left at our disposal after we have first formed a homogeneous
-assemblage with twelve points of contact on each solid. But our manner
-of finding a shape for each solid which can allow the solution of the
-problem to be real, proves that the solution is essentially imaginary
-for every wholly convex shape.
-
-
-§ 35. Pausing for a moment longer to consider afresh the geometrical
-problem of putting arbitrarily given equal and similar solids together
-to make a homogeneous assemblage of which each member is touched by
-eighteen others, we see immediately that it is determinate (whether it
-has any real solution or not), because when the shape of each body is
-given we have nine disposables for fixing the assemblage: six for the
-character of the assemblage of the corresponding points, and three for
-the orientation of each molecule relatively to rows of the assemblage
-of corresponding points. These nine disposables are determined by the
-condition that each body has nine pairs of contacts with others.
-
-Suppose now a homogeneous assemblage of the given bodies, in open
-order with no contacts, to be arbitrarily made according to any nine
-arbitrarily chosen values for the six distances between a point of
-_S_{1}_ and the corresponding points of its six pairs of nearest and
-next nearest neighbours (§ 1 above), and the three angles (§ 9 above)
-specifying the orientation of each body relatively to rows of the
-assemblage. We may choose in any nine rows through _S_{1}_ any nine
-pairs of bodies at equal distances on the two sides of _S_{1}_ far or
-near, for the eighteen bodies which are to be in contact with _S_{1}_.
-Hence there is an infinite number of solutions of the problem of which
-only a finite number can be real. Every solution of the problem of
-eighteen contacts is imaginary when the shape is wholly convex.
-
-[Illustration: 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 §§
-24 and 25 and represented in Figs. 12 and 13. I must for the present
-forego 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 solids[10] photographed in Fig. 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 § 21 above. 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 hollows _p′ q′ r′_, not _p 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 above _p′ q′ r′_ are
-occupied by the comparatively flat under sides of the molecules of
-the layer above, and the hollows below the apertures _p q r_ by 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. 12 and 13.
-Thus we can readily believe that a real crystal which is growing
-by additions to the face seen in Fig. 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 facing _from_ us 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
-under _p′ q′ r′_, instead of with its sharpest corner in the narrower
-and steeper hollows under _p 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 crystals of the wholly unsymmetric
-class, though it may not hitherto have been observed or described in
-crystallographic treatises.
-
-[Illustration: FIG. 14.]
-
-
-§ 38. This asymmetric double crystal becomes simply the well-known
-symmetrical ‘twin-crystal’[11] in 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 hollows _p′ 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 hollow _p′ 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 hollows _p q r_ free 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 hollows _p′ q′ r′_, some change of circumstances may cause
-the molecules of one layer to place themselves in a hollow _p q r_.
-The molecules in the next layer after this would find the hollows
-_p′ q′ r′_ occupied on the far side, and would thus have a bias in
-favour of the hollows _p 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
-hollows _p′ q′ r′_. A slight disturbance might again occur, causing
-the molecules of a fresh layer to settle, not in the narrow hollows
-_p q r_, but in the wider hollows _p′ 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 hollows _p′ 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 of the 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 cavities or 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 hovering about the positions of _p 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 in Fig. 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
-at irregular 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 observations[14] prove 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 area of 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 remarkable
-phenomenon of Crystalline Reflection described by Prof. Stokes,’
-published in the same number of the _Philosophical 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 what _can_ be the cause
-of any approach to regular periodicity 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
-evening regarding the molecular structure of crystals. I particularly
-desired to speak to you of quartz crystal with its ternary symmetry
-and its chirality[15]; and to have told you of the etching[16] by
-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.
-
-[Illustration: FIG. 15.]
-
-
-§ 48. Make an equilateral equiangular hexagonal prism, with
-its diagonal from edge to edge ninety-five hundredths[17] of 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) of _one_ of
-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
-assemblage with 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 of molecules 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 Curie[18], who found that a thin plate of
-quartz crystal 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 Voigt[19], and experimentally in
-quartz and in tourmaline by himself and Riecke[20], nor the well-known
-optic chirality of quartz.
-
-[Illustration: FIG. 16.]
-
-[Illustration: FIG. 17.]
-
-
-§ 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.
-
-[Illustration: FIG. 18.]
-
-Fig. 18 shows the upper side of a hexagonal layer of an assemblage
-thus composed of the right-handed molecule of Fig. 16. Fig. 15
-unchanged, 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-optic[21] property of quartz. Look at two contiguous faces _A_,
-_B_ of our geometrical model quartz crystal now before you, with its
-axis vertical. You will see a difference between them: turn it upside
-down; _B_ will be undistinguishable from what _A_ was, and _A_ will be
-undistinguishable from what _B_ was. Look at the two terminal pyramids,
-and you will find that the face above _A_ and the face below _B_ are
-identical in quality, and that they differ from the face above _B_
-and below _A_. This model is composed of the right-handed constituent
-molecules shown in Fig. 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, over _B_, presents complicated projections and
-hollows at the corners of the constituent molecules; and the pyramidal
-face next your left hand, over _A_, 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 in Fig. 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 Voigt[22], and experimentally by himself
-and Friecke[23], 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
-respectively the 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 probably _not_; 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, _SiO_{2}_, and acting 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(_SiO_{2}_) 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, or _vice-versa_.
-My watch, however, warns me that I must not enter on this subject.
-
-[Illustration: FIG. 19.]
-
-
-§ 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. 13 above), essentially has
-chirality because each constituent tetrahedron, if wholly scalene, has
-chirality[24]. 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 in illustration 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.
-
-
-
-
-FOOTNOTES:
-
-[1] See foot-note on § 22 below.
-
-[2] The holes in the cylinders are bored obliquely, as shown in Fig.
-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.
-
-[4] Similar curves are said to be parallel when the tangents to them at
-corresponding points are parallel.
-
-[5] See foot-note to § 12 above.
-
-[6] ‘On the Division of Space with Minimum Partitional Area,’
-_Philosophical Magazine_, vol. xxiv, 1887, p. 502, and _Acta
-Mathematica_ of the same year.
-
-[7] A. Levy, _Edinburgh Philosophical Journal_, April, 1822;
-Whewell, _Phil. Trans. Royal Society_, 1825; Miller, _Treatise on
-Crystallography_.
-
-[8] I 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.
-
-[9] _Philosophical Magazine_, vol. xx, 1885, second half year, p. 469,
-and _British Association Report_, 1885, Aberdeen, p. 896.
-
-[10] The 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’s _Treatise 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° _C_ according 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.
-
-[13] _Nature_, May 20, 1886.
-
-[14] _Philosophical Magazine_, 1888, second half year, p. 260.
-
-[15] See foot-note to § 22 above.
-
-[16] Widmanstätten, 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.
-
-[17] More exactly .9525, being 3/4 × cot 38° 13′; see p. 53.
-
-[18] J. and P. Curie and C. Friedel, _Comptes Rendus_, 1882, 1883,
-1886, 1892.
-
-[19] Allgemeine Theorie der piëzo- und pyroelectrischen Erscheinungen
-an Krystallen. W. Voigt, Königl. Gesellschaft der Wissenschaften zu
-Göttingen, August 2, 1890.
-
-[20] Wiedemann, _Annalen_, 1892, xlv, p. 923.
-
-[21] Generally miscalled ‘rotational.’
-
-[22] See foot-note (2) to p. 54 above.
-
-[23] See foot-note (3) to p. 54 above.
-
-[24] See foot-note to § 22 above.
-
-
-  THE END
-
-  Oxford
-  PRINTED AT THE CLARENDON PRESS
-  BY HORACE HART, PRINTER TO THE UNIVERSITY
-
-
-
-
-
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